Master Thesis

Maastricht University

School of Business and Economics

Master’s Thesis

Jump-Sensitive RV Models:

Forecasting and Transmissionof

Realized Volatility

AuthorAli Nabbi

SupervisorDr. Alain Hecq

August 28, 2016

Abstract

In this thesis, univariate and multivariate simple component models are developed withtime-varying and jump-sensitive parameters to forecast realized volatility in equity markets.Moreover, the dynamics of volatility transmission across financial centers are analyzed. In par-ticular, following indexes have been considered: S&P500, FTSE100 and HSI (Hang Seng Index).

Keywords: High-frequency data, long-memory models, simple component models, time-varyingparameters, jump-sensitive parameters, realized volatility, volatility forecast, volatility transmis-sion.

Declaration

I, Ali Nabbi, hereby do declare that the thesis entitled Jump-Sensitive RV Models: Fore-

casting and Transmission of Realized Volatility is my original work and it has been

written by me in its entirety. I have acknowledged all the sources of information which have

been used in the thesis.

Date:

August 28, 2016Signature:

i

Acknowledgments

I would like to express my deepest gratitude to my supervisor, Dr. Alain Hecq who has supported

me throughout my thesis with his vast knowledge, valuable comments and constructive criticism.

I would also like to thank my parents, my older sister, my younger brother, my uncle Mohsen

and my aunt Mechthild. They were always supporting me and encouraging me with their best

wishes.

ii

Contents

List of Tables v

List of Figures viii

List of Abbreviations x

1 Introduction 1

2 Volatility Models for Forecasting 3

2.1 Volatility Estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Jump Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.3 Volatility Models as Benchmark . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Jump-sensitive Models 9

3.1 Baseline Quarticity Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.2 Proposed Models in Univariate cases . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.3 Proposed Models in Multivariate cases . . . . . . . . . . . . . . . . . . . . . . . . 12

4 Empirical Analysis of Univariate Models 15

4.1 Data Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.2 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.3 Forecast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.3.1 In-Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4.3.2 Out-of-the-Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5 Empirical Analysis of Multivariate Models 31

5.1 Granger-Causality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.2 Volatility Transmission and In-Sample Forecast . . . . . . . . . . . . . . . . . . . 33

5.2.1 The Whole Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5.2.2 Pre-Crisis Sub-Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.2.3 Post-Crisis Sub-Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

iii

CONTENTS iv

5.3 Out-of-the-Sample Forecast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

6 Conclusion 52

Appendix 54

References 77

List of Tables

4.1 Descriptive statistics of whole sample. . . . . . . . . . . . . . . . . . . . . . . . . 16

4.2 Descriptive statistics of pre-crisis sub-sample. . . . . . . . . . . . . . . . . . . . . 17

4.3 Descriptive statistics of post-crisis sub-sample. . . . . . . . . . . . . . . . . . . . 17

4.4 Estimation of S&P500 on the whole sample . . . . . . . . . . . . . . . . . . . . . 19

4.5 Estimation of S&P500 on the pre-crisis sub-sample . . . . . . . . . . . . . . . . . 20

4.6 Estimation of S&P500 on the post-crisis sub-sample . . . . . . . . . . . . . . . . 20

4.7 One-day-ahead static in-sample forecast performance and accuracy of S&P500

on the whole sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21


on the pre-crisis sub-sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22


on the post-crisis sub-sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.10 Out-of-the-sample forecast performance S&P500 on the whole sample . . . . . . 25

4.11 Out-of-the-sample forecast performance S&P500 on the pre-crisis sub-sample . . 25

4.12 Out-of-the-sample forecast performance S&P500 on the post-crisis sub-sample . . 25

4.13 One-day-ahead out-of-sample forecast accuracy of S&P500 on the whole sample . 26

4.14 One-week-ahead out-of-sample forecast accuracy of S&P500 on the whole sample 26

4.15 One-month-ahead out-of-sample forecast accuracy of S&P500 on the whole sample 27

4.16 One-day-ahead out-of-sample forecast accuracy of S&P500 on pre-crisis sub-sample 27

4.17 One-week-ahead out-of-sample forecast accuracy of S&P500 on pre-crisis sub-

sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.18 One-month-ahead out-of-sample forecast accuracy of S&P500 on pre-crisis sub-

sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.19 One-day-ahead out-of-sample forecast accuracy of S&P500 on post-crisis sub-

sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.20 One-week-ahead out-of-sample forecast accuracy of S&P500 on post-crisis sub-

sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

v

LIST OF TABLES vi

4.21 One-month-ahead out-of-sample forecast accuracy of S&P500 on post-crisis sub-

sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

5.1 Granger-causality test of S&P500 with BpV as jump component . . . . . . . . . 32

5.2 Granger-causality test of S&P500 with MedRV as jump component . . . . . . . . 32

5.3 In-Sample Estimation of S&P500 with Residual-Modification on the whole sample

and the BpV as jump component . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

5.4 In-Sample Estimation of S&P500 with RV-Modification on the whole sample and

the BpV as jump component . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5.5 One-day-ahead in-sample forecast accuracy of S&P500 on the whole sample with

FTSE100 as the secondary index. . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5.6 One-day-ahead in-sample forecast accuracy of S&P500 on the whole sample with

HSI as the secondary index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5.7 In-Sample Estimation of S&P500 with Residual-Modification on pre-crisis sub-

sample and the BpV as jump component . . . . . . . . . . . . . . . . . . . . . . . 38

5.8 In-Sample Estimation of S&P500 with RV-Modification on pre-crisis sub-sample


5.9 One-day-ahead in-sample forecast accuracy of S&P500 on pre-crisis sub-sample

with FTSE100 as the secondary index. . . . . . . . . . . . . . . . . . . . . . . . . 41

5.10 One-day-ahead in-sample forecast accuracy of S&P500 on pre-crisis sub-sample

with HSI as the secondary index. . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.11 In-Sample Estimation of S&P500 with residuals-Modification on post-crisis sub-

sample and the BpV as jump component . . . . . . . . . . . . . . . . . . . . . . . 44

5.12 In-Sample Estimation of S&P500 with RV-Modification on post-crisis sub-sample


5.13 One-day-ahead in-sample forecast accuracy of S&P500 on post-crisis sub-sample

with FTSE100 as the secondary index. . . . . . . . . . . . . . . . . . . . . . . . . 46

5.14 One-day-ahead in-sample forecast accuracy of S&P500 on post-crisis sub-sample

with HSI as the secondary index. . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.15 Out-of-the-sample forecast performance of S&P500 . . . . . . . . . . . . . . . . . 48

5.16 Out-of-the-sample forecast performance of S&P500 . . . . . . . . . . . . . . . . . 49

5.17 One-day-ahead out-of-the-sample forecast accuracy of S&P500 on the whole sam-

ple with FTSE100 as the secondary index. . . . . . . . . . . . . . . . . . . . . . . 50

5.18 One-day-ahead out-of-the-sample forecast accuracy of S&P500 on the whole sam-

ple with HSI as the secondary index. . . . . . . . . . . . . . . . . . . . . . . . . . 51

LIST OF TABLES vii

A.1 Estimation of FTSE100 on the whole sample . . . . . . . . . . . . . . . . . . . . 62

A.2 Estimation of HSI on the whole sample . . . . . . . . . . . . . . . . . . . . . . . 62

A.3 Out-of-the-sample forecast performance FTSE100 on the whole sample . . . . . . 63

A.4 Out-of-the-sample forecast performance FTSE100 on the pre-crisis sub-sample . 63

A.5 Out-of-the-sample forecast performance FTSE100 on the post-crisis sub-sample . 63

A.6 Out-of-the-sample forecast performance HSI on the whole sample . . . . . . . . . 64

A.7 Out-of-the-sample forecast performance HSI on the pre-crisis sub-sample . . . . . 64

A.8 Out-of-the-sample forecast performance HSI on the post-crisis sub-sample . . . . 64

A.9 Granger-causality test of FTSE100 with BpV as jump component . . . . . . . . . 65

A.10 Granger-causality test of FTSE100 with MedRV as jump component . . . . . . . 65

A.11 Granger-causality test of HSI with BpV as jump component . . . . . . . . . . . . 66

A.12 Granger-causality test of HSI with MedRV as jump component . . . . . . . . . . 66

A.13 In-Sample Estimation of S&P500 with Residual-Modification on the whole sample

and the MedRV as jump component . . . . . . . . . . . . . . . . . . . . . . . . . 67

A.14 In-Sample Estimation of S&P500 with RV-Modification on the whole sample and

the MedRV as jump component . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

A.15 In-Sample Estimation of S&P500 with Residual-Modification on pre-crisis sub-

sample and the MedRV as jump component . . . . . . . . . . . . . . . . . . . . . 69

A.16 In-Sample Estimation of S&P500 with RV-Modification on pre-crisis sub-sample


A.17 In-Sample Estimation of S&P500 with residuals-Modification on post-crisis sub-

sample and the MedRV as jump component . . . . . . . . . . . . . . . . . . . . . 71

A.18 In-Sample Estimation of S&P500 with RV-Modification on post-crisis sub-sample


A.19 One-day-ahead out-of-the-sample forecast accuracy of S&P500 on pre-crisis sub-

sample with FTSE100 as the secondary index. . . . . . . . . . . . . . . . . . . . . 73

A.20 One-day-ahead out-of-the-sample forecast accuracy of S&P500 on post-crisis sub-

sample with FTSE100 as the secondary index. . . . . . . . . . . . . . . . . . . . . 74

A.21 One-day-ahead out-of-the-sample forecast accuracy of S&P500 on pre-crisis sub-

sample with HSI as the secondary index. . . . . . . . . . . . . . . . . . . . . . . . 75

A.22 One-day-ahead out-of-the-sample forecast accuracy of S&P500 on post-crisis sub-

sample with HSI as the secondary index. . . . . . . . . . . . . . . . . . . . . . . . 76

List of Figures

2.1 Observed returns of S&P500 and estimated returns by RV of the benchmark models 7

2.2 PDF of observed RV, estimated RV using benchmarks in the whole sample of

S&P500. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 ACF of observed RV, estimated RV using benchmarks in the whole sample of

S&P500. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.1 PDF of observed RV, RV estimated by benchmark and proposed models in the

whole sample of S&P500. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.2 ACF of observed RV, RV estimated by benchmark and proposed models in the

whole sample of S&P500. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.1 RV, BpV, MedRV and their respective jumps of S&P500. . . . . . . . . . . . . . 18

A.1 Observed returns of FTSE100 and estimated returns by RV of the benchmark

models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

A.2 Observed returns of HSI and estimated returns by RV of the benchmark models . 55

A.3 PDF of observed RV, estimated RV using benchmarks in the whole sample of

FTSE100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

A.4 ACF of observed RV, estimated RV using benchmarks in the whole sample of

FTSE100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

A.5 PDF of observed RV, estimated RV using benchmarks in the whole sample of HSI. 57

A.6 ACF of observed RV, estimated RV using benchmarks in the whole sample of HSI. 57

A.7 PDF of observed RV, RV estimated by benchmark and proposed models in the

whole sample of FTSE100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

A.8 ACF of observed RV, RV estimated by benchmark and proposed models in the

whole sample of FTSE100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

A.9 PDF of observed RV, RV estimated by benchmark and proposed models in the

whole sample of HSI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

A.10 ACF of observed RV, RV estimated by benchmark and proposed models in the

whole sample of HSI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

viii

LIST OF FIGURES ix

A.11 RV, BpV, MedRV and their respective jumps of FTSE100. . . . . . . . . . . . . . 60

A.12 RV, BpV, MedRV and their respective jumps of HSI. . . . . . . . . . . . . . . . . 61

List of Abbreviations

ACF Autocorrelation Function

ARJ Autoregressive Model with Jump-sensitive parameter

ARQ Autoregressive Quarticity Model

BpV Bi-power Variation estimator

BpV Square root of bi-power variation estimator

DM Diebold-Mariano forecast accuracy test

FTSE100 Financial Times Stock Exchange 100 index (UK - London)

HAR(3)1 Heterogeneous Autoregressive Model

HAR(3)-J1 Heterogeneous Autoregressive Model with Jump component

HARJ Heterogeneous Autoregressive Model with Jump-sensitive parameter

HARJ-semiF Heterogeneous Autoregressive semi-Full model with Jump-sensitive parameters

HARJ-F Heterogeneous Autoregressive Full-model with Jump-sensitive parameters

HARQ Heterogeneous Autoregressive Quarticity Model

HARQ-F Heterogeneous Autoregressive Quarticity Full-model

HSI Hang Seng Index (China - Hong Kong)

IV Integrated Variance

JB Jarque-Bera Normality test

LB22 Ljung-Box serial autocorrelation test up to 22th lag order

MAE Mean Absolute Errors

MedRV Median-truncated Realized Variance

MedRV Median-truncated Realized Volatility

PDF Probability Density Function

RMSE Root Mean Square Errors

RQ Realized Quarticity

RV Realized Variance

1Number in the parenthesis indicates that model contains three volatility components, namely short-term(daily), mid-term(weekly) and long-term (monthly).

x

LIST OF ABBREVIATIONS xi

RV Realized Volatility

S&P500 Standard & Poor’s 500 index (US - New York)

VHAR(3) Vector Heterogeneous Autoregressive RV-modified model

VHAR(3)-J Vector Heterogeneous Autoregressive RV-modified model with Jump component

VHAR(3)-η Vector Heterogeneous Autoregressive residual-modified model

VHAR(3)-J-η Vector Heterogeneous Autoregressive residual-modified model with Jump

component

VHARJ Vector Heterogeneous Autoregressive RV-modified model with Jump-sensitive

parameter

VHARJ-semiF Vector Heterogeneous Autoregressive RV-modified semi-Full model with

Jump-sensitive parameters

VHARJ-F Vector Heterogeneous Autoregressive RV-modified Full model with Jump-sensitive

parameters

VHARJ-η Vector Heterogeneous Autoregressive residual-modified model with Jump-sensitive

parameter

VHARJ-semiF-η Vector Heterogeneous Autoregressive residual-modified semi-Full model with


VHARJ-F-η Vector Heterogeneous Autoregressive residual-modified Full model with


Chapter 1

Introduction

Volatility plays a major role in pricing financial securities and risk management. Specifically,

accurate forecasting volatility is crucial in option pricing, portfolio management, market predic-

tions and regulations. Various settings of GARCH models and stochastic volatility models have

been proposed to estimate and predict the market’s volatility. However, GARCH models are

not able to produce financial data characteristics such as fat-tailed and leptokurtic distribution

of returns, persistent autocorrelation and slow convergence to Normal distribution, particularly

in high-frequency data. To simulate long-memory models, fractionally integrated ARMA and

GARCH models have been introduced —also denoted as ARFIMA and FIGARCH— which are

not fully able to produce financial data features for long-memory models. Besides, the above-

mentioned class of volatility models are inefficient and biased in large order of autoregressive

and moving average (Comte and Renault, 1998) and the performance of these parametric mod-

els are significantly low in high-frequency data (Bollerslev et al., 2016). Furthermore, fractional

integration concept is a mathematical tool to implement the long-memory behavior of intraday

returns which results in lack of economic interpretation and loss of many observations.

Realized variance is defined by the summation of the square intraday returns and Realized

Volatility (hereinafter referred to as RV) is defined by square root of realized variance. RV is

easy-to-implement and highly persistent for most of the financial assets. The performance of

GARCH and stochastic volatility models increases by implementing RV estimators (Andersen

et al., 2003).

Due to the complications to distinguish between simple component models and true long-

memory processes in terms of performance, forecastability, simplicity in implementation and in-

terpretation, simple component models were used in the present thesis to estimate and forecast

RV which are arguably well-performed in forecasting using consistent RV estimators e.g. Corsi

(2009) introduced a simple and well-performing model using first order lagged of RV in differ-

1

CHAPTER 1. INTRODUCTION 2

ent time horizons, namely short-term (daily), medium-term (weekly) and long-term (monthly)

to forecast RV and Bollerslev et al. (2016) formulated the mixed-data sampling structure in

the Heterogeneous Autoregressive Quarticity model which applies time-varying parameter as a

function of realized quarticity to consider measurement errors.

Models converge significantly faster to the true RV when discrete jump component is ex-

cluded in the return process. Additionally, RV is consistent in absence of jumps (Andersen

et al., 2007). However, constructing RV process to continuous sample path and a jump process

despite of slower convergence can improve the predictive power of volatility models.

The rest of this thesis is organized as follows. In the next chapter, volatility estimators, jump

processes and simple component volatility models will be briefly reviewed. In chapter 3, the

new setting of simple component models will be introduced where the constructed parameters

allow the model to consider variations in time and effect of jumps. Descriptive statistics, data

cleaning process, estimation, In-Sample and Out-of-the-Sample forecasts are included in chapter

4 for univariate models and further analysis for multivariate models is included in the chapter

5 which contains Granger-causality tests, volatility transmission and forecast performance and

accuracy. This author concludes the outcome of the empirical analysis in chapter 6 with several

suggestions on improving the volatility models with jump component for future research.

Chapter 2

Volatility Models for Forecasting

In this chapter, volatility estimators will be reviewed, namely realized variance, bi-power vari-

ation and median-truncated realized variance. Jump processes and their asymptotics will be

discussed briefly, and finally, univariate and multivariate volatility models will be introduced to

use as benchmarks.

2.1 Volatility Estimators

Daily realized variance is defined as the summation of intraday squared returns

RV(d)t = RV t ≡

M∑i=1

r2t,i (2.1)

where ∆ = 1d/M and ∆-frequency return is defined by rt,i = log(Pt−1+i.∆)− log(Pt−1+(i−1).∆).

One working day is 390 minutes (the pit hours 8:30AM–4:00PM excluding lunch break) and the

value of M will be 390, 78 and 39 for the frequencies ∆ = 1, 5 and 10 minutes, respectively. It

is evident that RV can be computed using square root of realized variance. Furthermore, RV

over time horizon, h is defined as

RV(h)t−i = RVt−i|t−h ≡ 1

h

h∑k=i

RVt−k (2.2)

let i = 1 and if h = 1, 5, 22 is assigned to (2.2), it results in first order lagged of daily, weekly and

monthly RV respectively. Realized variance is a consistent estimator for returns variance as ∆

converges to zero (Andersen and Bollerslev, 1998). It can be concluded that RV is a consistent

estimator for return process volatility which is highly persistent by the construction of realized

variance. As a consequence, RV is the best candidate to produce daily returns characteristics.

More jump robust estimators of volatility have been developed e.g. Bauwens et al. (2012)

3

CHAPTER 2. VOLATILITY MODELS FOR FORECASTING 4

suggest to compute bi-power variation estimator introduced by Barndorff-Nielsen and Shephard

(2004) as follows

BpV t ≡√

2

π.

M

M − 1.

M∑i=2

|rt,i|.|rt,i−1| (2.3)

and median-truncated realized variance is introduced by Andersen et al. (2012) and is defined

as

MedRV t ≡π

π − 2.( M

M − 1

).M−1∑i=2

med(|rt,i−1|.|rt,i|.|rt,i+1|

)2(2.4)

Note that to find corresponding volatility estimator, square root of above-mentioned esti-

mators needs to be considered which will be referred as BpV and MedRV. Similar to (2.2), BpV

and MedRV estimators over time period, h can be computed

BpV(h)t−i = BpVt−i|t−h ≡ 1

h

h∑k=i

BpVt−k (2.5)

MedRV(h)t−i = MedRVt−i|t−h ≡ 1

h

h∑k=i

MedRVt−k (2.6)

2.2 Jump Processes

Let r(t) denoted as the logarithmic transformation of an asset price at time t (also known

as return process). Traditionally, return process is constructed as follows to include the non-

continuous jump processes (Andersen et al., 2007).

dr(t) = µ(t)dt+ σ(t)dW (t) + κ(t)dq(t), 0 ≤ t ≤ T (2.7)

where µ(t) is a continuous process, σ(t) is a positive stochastic process, W (t) is a standard

Brownian motion and q(t) is a counting process with P[dq(t) = 1] = λ(t)dt and κ(t) refers to

size of the jump. Realized variance converges in probability to increment of quadratic variation

process of the underlying return process as return increases.

RV t+1(∆) →∫ t+1

tσ2(s)ds+

∑t≤s≤t+1

κ2(s) (2.8)

It is evident that in the absence of jumps, realized variance is consistent as ∆ → 0 (and so

does RV). This does not limit modeling RV t+1(∆) using a simple procedure that can not

distinguish between continuous and discrete components of the return process. Through this


intuition, Andersen et al. (2001) showed the asymptotic result of bi-power variation

BpV t+1(∆) →∫ t+1

tσ2(s)ds (2.9)

same result has been found by Andersen et al. (2012) for median-truncated realized variance

MedRV t+1(∆) →∫ t+1

tσ2(s)ds (2.10)

Combining (2.8), (2.9) and (2.10) leads to following consistent asymptotics of the discrete

component of the return process as ∆ → 0

RV t+1(∆)−BpV t+1(∆) →∑

t≤s≤t+1

κ2(s) (2.11)

similar result can be obtained for median-truncated realized variance

RV t+1(∆)−MedRV t+1(∆) →∑

t≤s≤t+1

κ2(s) (2.12)

Since the results of (2.11) and (2.12) can lead to negative values and the positive jumps are

the major point of focus, truncation at zero is suggested by Barndorff-Nielsen and Shephard

(2004). Therefore, the jump components on volatility are defined as follows

Jt+1(∆) ≡√[RV t+1(∆)−BpV t+1(∆)

]+(2.13)

Jt+1(∆) ≡√[RV t+1(∆)−MedRV t+1(∆)

]+(2.14)

2.3 Volatility Models as Benchmark

Several sets of simple component models are developed to model financial intraday data char-

acteristics including but not limited to heterogeneous autoregressive model of RV (HAR(3))

of Corsi (2009), autoregressive realized quarticity and heterogeneous autoregressive realized

quarticity of Bollerslev et al. (2016), non-linear heterogeneous autoregressive model with jump

(HAR(3)-J) and continuous heterogeneous autoregressive model of Andersen et al. (2007), semi-

variance heterogeneous autoregressive model of Patton and Sheppard (2015). All models —with

the exception of HAR(3) and HAR(3)-J— are formulated using realized variance rather than

RV. HAR(3) and HAR(3)-J have been formulated as follows


RVt = β0 + β1.RVt−1 + β5.RVt−1|t−5 + β22.RVt−1|t−22 + ut. (2.15)

RVt = β0 + β1.RVt−1 + β5.RVt−1|t−5 + β22.RVt−1|t−22 + βJ .Jt−1 + ut. (2.16)

where Jt−1 ≡√[RV t−1(∆)−BpV t−1(∆)

]+.

where ut are independent and identically distributed with mean of zero and finite second and

fourth moment. Among the above mentioned models, HAR(3) and HAR(3)-J are chosen as

benchmark.

HAR(3) can be transferred to multivariate models, also known as Vector Heterogeneous

Autoregressive models (VHAR). The following VHAR specification is used by Bubak et al.

(2011)

RVt = β0 + β1RVt−1 + β5RVt−1|t−5 + β22RVt−1|t−22 + ut (2.17)

where β are square matrices of daily, weekly and monthly stacked RVs and ut are independent

and identically distributed with 0 mean, positive definite covariance matrix,Σ and finite fourth

moment. Similar specification can be applied to HAR(3)-J. The multivariate model VHAR

can be simplified into bivariate model to analyze Granger-causality of index j on index i,

volatility transmission and forecasting RV of index i. Following setting will be referred to as

RV-modification.

RV it = βi

0+βi1RV i

t−1 + βi5RV i

t−1|t−5 + βi22RV i

t−1|t−22

+βj1RV j

t−1 + βj5RV j

t−1|t−5 + βj22RV j

t−1|t−22 + uit

(2.18)

In order to account for new information and the jumps from other markets, the following

modification is applied which is introduced by Soucek and Todorova (2013) and will be referred

to as residuals-modification. The bivariate VHAR specification of index i with index j as the

secondary index is formulated as follows

RV it = βi

0+βi1RV i

t−1 + βi5RV i

t−1|t−5 + βi22RV i

t−1|t−22

+βj1η

jt−1 + βj

5ηjt−1|t−5 + βj

22ηjt−1|t−22 + uit

(2.19)

where ηjt−1|t−h expresses the fluctuation of RV jt−1|t−h which can not be explained by RV i

t−1|t−h.

It can be computed using the following regression


RV jt−1|t−k = αj

0 + αi→j1 RV i

t−1|t−k + ηjt−k, k = 1, 5, 22. (2.20)

Note that in (2.19), residuals on different time periods (daily, weekly and monthly) are

needed to estimate the bivariate model setting. Similar to (2.2), (2.5), (2.6) and (3.11) residuals

over time horizon, h is defined as

ηj(h)t−i = ηjt−i|t−h ≡ 1

h

h∑k=i

ηjt−k. (2.21)

Observed returns and estimated returns of S&P500 index by RV of the benchmark models

in univariate case of S&P500 are illustrated in the figure (2.1). The figures related to FTSE100

and HSI can be found in the appendix, figures (A.1) and (A.2).

FIGURE 2.1: Observed returns of S&P500 and estimated returns by RV of thebenchmark models


The probability density function (PDF) of observed RV and estimated RV using benchmark

models HAR(3) and HAR(3)-J in the whole sample of index S&P500 are shown in figure (2.2)

and figure (2.3) illustrates the Autocorrelation Function (ACF) up to 100th lag order which

shows the long-memory of the empirical RV and estimated RV using benchmark models. Figures

(A.4) and (A.6) shows the comparison of ACF of observed and RV estimated by benchmarks

in the whole sample of FTSE100 and HSI. The PDF of FTSE100 and HSI are shown in (A.3)

and (A.5).

FIGURE 2.2: PDF of observed RV, estimated RV using benchmarks in the whole sampleof S&P500.

FIGURE 2.3: ACF of observed RV, estimated RV using benchmarks in the whole sampleof S&P500.

Chapter 3

Jump-sensitive Models

In this chapter, quarticity models will be reviewed which are the conceptual baseline models to

develop new models. The new setting of volatility models (with jump-sensitive and time-varying

parameters) will also be introduced and formulated in both univariate and multivariate cases.

3.1 Baseline Quarticity Models

Bollerslev et al. (2016) introduced the quarticity models to forecast realized variance by consid-

ering measurement error in the parameters which resulted in improvement of forecast errors.

RV t = β0 + (β1 + β1QRQ1/2t−1)RV t−1 + ut (3.1)

RV t = β0 + (β1 + β1QRQ1/2t−1)RV t−1 + β5RV t−1|t−5 + β22RV t−1|t−22 + ut (3.2)

RV t = β0+(β1 + β1QRQ1/2t−1)RV t−1

+(β5 + β5QRQ1/2t−1|t−5)RV t−1|t−5

+(β22 + β22QRQ1/2t−1|t−22)RV t−1|t−22 + ut

(3.3)

where RQ denotes realized quarticity and ut is independent and identically distributed inno-

vations with mean of zero and finite second and fourth moment. For short, the equations (3.1),

(3.2) and (3.3) will be referred as ARQ, HARQ and HARQ-F (Full model). Note that jump

components have not been considered in above-mentioned models. As a result, Quadratic Vari-

ation (QV ) will be a summation of the latent Integrated Variance (IV ) and an error term (ηt)

9

CHAPTER 3. JUMP-SENSITIVE MODELS 10

where IV is defined as

IV t =

∫ t

t−1σ2(s)ds (3.4)

which in the absence of jumps, realized variance is its consistent estimator.

To consider estimation error, the asymptotics of Barndorff-Nielsen and Shephard (2002) is

concluded as follows

RV = IV + ηt ηt ∼ N(0, 2∆IQt) (3.5)

where IQt ≡∫ tt−1 σ

4(s)ds denotes the integrated quarticity with RQ as its consistent estimator.

RQt ≡M

3

M∑i=1

r4t,i (3.6)

Bollerslev et al. (2016) implemented this methodology on realized variance which can be

estimated using standard OLS. In order to describe long-run memory, HARQ has been suggested

which can be further evolved to a full-model of HARQ.

3.2 Proposed Models in Univariate cases

Since this thesis focuses on the jumps rather than measurement errors, the specification of

linearly dependent parameter on jump component is employed.

RVt = β0 + (β1 + β1JJt−1)RVt−1 + ut (3.7)

RVt = β0 + (β1 + β1JJt−1)RVt−1 + β5RVt−1|t−5 + β22RVt−1|t−22 + ut (3.8)

RVt = β0+(β1 + β1JJt−1)RVt−1

+(β5 + β5JJt−1|t−5)RVt−1|t−5

+β22RVt−1|t−22 + ut

(3.9)

RVt = β0+(β1 + β1JJt−1)RVt−1

+(β5 + β5JJt−1|t−5)RVt−1|t−5

+(β22 + β22JJt−1|t−22)RVt−1|t−22 + ut.

(3.10)


Error terms specifications are set to the settings of baseline models in the section (3.1). For

simplicity, the models (3.7), (3.8), (3.9) and (3.10) will be referred as ARJ, HARJ, HARJ-semiF

(Semi-full model) and HARJ-F (Full model). Note that in the semi-full and full models, the

jump-sensitive parameter is extended to the higher lag orders, namely to weekly and monthly

RV. All models will be estimated using jump component with BpV from (2.13) and with MedRV

from (2.14). Furthermore, the jump components over time horizon, h can be computed similar

to (2.2), (2.5) and (2.6)

J(h)t−i = Jt−i|t−h ≡ 1

h

h∑k=i

Jt−k (3.11)

These specifications are estimated by standard OLS which is straightforward both for esti-

mation and forecasting. To consider possible presence of heteroskedasticity and serial aurocor-

relations, the Newey-West covariance correction is employed for all the proposed models.

The PDF of observed RV and estimated RV using benchmark models and the proposed

models in the whole sample of index S&P500 are shown in figure (3.1). Corresponding figures

for FTSE100 and HSI are illustrated in (A.7) and (A.9).

FIGURE 3.1: PDF of observed RV, RV estimated by benchmark and proposed models inthe whole sample of S&P500.


Figure (3.2) illustrates the ACF of the observed RV and estimated RV by the benchmarks

and the proposed models with both BpV and MedRV as jump component up to 50th lag order

in S&P500 index. Based on the value of ACF, the proposed models have succeeded to replicate

the long-memory feature of the financial data. ACF of observed and estimated RV for FTSE100

and HSI are shown in figures (A.8) and (A.10) in the appendix.

FIGURE 3.2: ACF of observed RV, RV estimated by benchmark and proposed models inthe whole sample of S&P500.

It is worth mentioning that homogeneous model ARJ can be compared to a homogeneous

benchmark for forecastibility and performance e.g. AR(1) process. Note that in absence of

jumps ARJ behaves similar to AR(1). Similar results can be drawn for constant realized quar-

ticity in the model ARQ formulated in section (3.1) (Bollerslev et al., 2016). It is expected

to observe significantly higher capability of producing returns characteristics in heterogeneous

model versus homogeneous models (Corsi, 2009).

3.3 Proposed Models in Multivariate cases

Corsi (2009) and Bollerslev et al. (2016) showed the accuracy of the returns simulations in

heterogeneous models and thus their improved predictive power and unbiasedness. Therefore,

homogeneous models are excluded from the multivariate analysis.

The jump-sensitive parameters introduced in the section (3.2), can be extended to the

bivariate structure similar to the heterogeneous models explained in section (2.3). HARJ,


HARJ-semiF and HARJ-F (models (3.8)–(3.10)) can be adopted to the bivariate structure of

model (2.18) —will be referred to as RV-modified models.

RV it = βi

0+(βi1 + βi

1JJit−1)RV i

t−1 + βi5RV i

t−1|t−5 + βi22RV i

t−1|t−22

+(βj1 + βj

1JJjt−1)RV j

t−1 + βj5RV j

t−1|t−5 + βj22RV j

t−1|t−22 + uit

(3.12)

RV it = βi

0+(βi1 + βi

1JJit−1)RV i

t−1 + (βi5 + βi

5JJit−1|t−5)RV i

t−1|t−5 + βi22RV i

t−1|t−22

+(βj1 + βj

1JJjt−1)RV j

t−1 + (βj5 + βj

5JJjt−1|t−5)RV j

t−1|t−5 + βj22RV j

t−1|t−22 + uit

(3.13)

RV it = βi

0+(βi1 + βi

1JJit−1)RV i

t−1 + (βi5 + βi


t−1|t−5 + (βi22 + βi


t−1|t−22

+(βj1 + βj

1JJjt−1)RV j

t−1 + (βj5 + βj


t−1|t−5 + (βj22 + βj


t−1|t−22 + uit

(3.14)

The models (3.12), (3.13) and (3.14) will be referred to as VHARJ, VHARJ-semiF (Semi-

full model) and VHARJ-F (Full model). Similarly, proposed models adopted to (2.19) that are

formulated below —will be referred to as residual-modified models.

RV it = βi

0+(βi1 + βi

1JJit−1)RV i

t−1 + βi5RV i

t−1|t−5 + βi22RV i

t−1|t−22

+(βj1 + βj

1JJjt−1)η

jt−1 + βj

5ηjt−1|t−5 + βj

22ηjt−1|t−22 + uit

(3.15)

RV it = βi

0+(βi1 + βi

1JJit−1)RV i

t−1 + (βi5 + βi


t−1|t−5 + βi22RV i

t−1|t−22

+(βj1 + βj

1JJjt−1)η

jt−1 + (βj

5 + βj5JJ

jt−1|t−5)η

jt−1|t−5 + βj

22ηjt−1|t−22 + uit

(3.16)

RV it = βi

0+(βi1 + βi

1JJit−1)RV i

t−1 + (βi5 + βi


t−1|t−5 + (βi22 + βi


t−1|t−22

+(βj1 + βj

1JJjt−1)η

jt−1 + (βj

5 + βj5JJ

jt−1|t−5)η

jt−1|t−5 + (βj

22 + βj22JJ

jt−1|t−22)η

jt−1|t−22 + uit

(3.17)

where ηjt−1|t−k expresses the fluctuation of RV jt−1|t−k which can not be explained by RV i

t−1|t−k.


It can be computed using the following regression.

RV jt−1|t−k = αj

0 + αi→j1 RV i

t−1|t−k + ηjt−1|t−k, k = 1, 5, 22 (3.18)

where

ηj(h)t−i = ηjt−i|t−h ≡ 1

h

h∑k=i

ηjt−k. (3.19)

The models (3.15), (3.16) and (3.17) will be referred to as VHARJ-η, VHARJ-semiF-η (Semi-

full model) and VHARJ-F-η (Full model). Since capturing the second moment of volatility and

investigation the correlation of volatilities are not the major interests of this thesis, suggested

GARCH-type innovations in Corsi (2009) and Bollerslev (1990) are excluded.

Chapter 4

Empirical Analysis of Univariate

Models

This chapter contains data description including indexes, sampling, missing value technique and

descriptive statistics followed by models in-sample estimations, in-sample and out-of-the-sample

forecast performance using mean absolute error (MAE) and root mean square error (RMSE)

and forecast accuracy using Diebold-Mariano (DM) pairwise test (Diebold and Mariano, 1995).

4.1 Data Description

The data contains realized variance, bi-power variation and median-truncated realized variance

estimators of equity markets from Jan. 3, 2000 to Jul. 1, 2016. All estimators can be trans-

formed to RV, BpV and MedRV as discussed in the section (2.1). In this thesis, the 5-minutes

realized variance, bi-power variation and median-truncated realized variance estimators are cho-

sen which can not be beaten significantly by the alternatives (Liu et al., 2015). Samples are

divided into low volatile and high volatile sub-samples, namely pre-crisis and post-crisis in order

to control the performance and accuracy. Division is considered on Jan. 1, 2008. To conclude

generality of the proposed models in empirical analysis, three indexes have been chosen based

on the region and their impact on global economy, namely New York S&P500, London FTSE100

and Hong Kong Hang Seng Index(HSI).

It is common to use logarithm of RV estimators which has two beneficial aspects. First,

no requirements to model non-negative parameters and second, the distribution of logarithmic

RV is much closer to Normal distribution which from statistical standpoint is favorable. In this

thesis, non-negativity restriction is satisfied by construction of RV as square root of realized

variance and similarly for BpV, MedRV and jump components. Logarithmic transformation

15

CHAPTER 4. EMPIRICAL ANALYSIS OF UNIVARIATE MODELS 16

smoothens the process significantly that can interrupt analysis of jumps. Hence, no logarithmic

transformation of RV estimators are employed in this study.

Missing data can be due to the holidays or days without any data stored (for any reason)

which can have a significant impact on the conclusions drawn from the analysis. Several methods

are suggested to overcome the pitfall of missing values e.g. deleting missing values from the data,

replacing them with the value from the day before or take an average of last week etc. Since

proposed models will be used to analyze the volatility transmission and volatility forecasting in

multivariate case, deleting the missing values can have an outstanding and detrimental impact

on analysis due to the loss of a big proportion of the data. To address this issue, the following

methodology is used in the present study. Every missing data is replaced by taking average of

past and next week observations. For the missing values in the first 5 observations of Jan. 3.

2000, average of next week is considered and the average of past week is used for the last 5

observation to Jul. 1, 2016.

Descriptive statistics of RV, BpV and MedRV of S&P500, FTSE100 and HSI are summarized

in tables (4.1), (4.2) and (4.3) containing Jarque-Bera Normality test and Ljung-Box serial

autocorrelation up to 22 lagged order in addition to the common summary measures, namely

mean, standard deviation, skewness, kurtosis, maximum, minimum.

TABLE 4.1: Descriptive statistics of whole sample.

S&P500 FTSE100 HSI

RV BpV MedRV RV BpV MedRV RV BpV MedRV

Mean 0.9197 0.8121 0.7242 0.8040 0.7569 0.6725 0.9134 0.8494 0.8112

Std.Dev. 0.6121 0.5481 0.4990 0.4923 0.4622 0.3909 0.5061 0.4715 0.3859

Skewness 3.1313 3.3591 4.0566 2.8480 3.1245 3.1878 3.5857 3.5442 2.6176

Kurtosis 20.8270 23.1360 34.5312 18.7410 23.2143 22.3502 25.2563 24.7871 17.0023

Max 8.8021 7.7577 8.6607 6.8019 7.2114 5.0879 6.6129 6.6057 4.9950

Min 0.1273 0.1030 0.1364 0.1947 0.1131 0.1463 0.2252 0.2040 0.2502

JB 0.0010 0.0010 0.0010 0.0010 0.0010 0.0010 0.0010 0.0010 0.0010

LB22 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

Note: The table summarizes descriptive statistics of daily RV, BpV and MedRV of each equity market in the

whole sample from Jan. 3, 2000 to Jul. 1, 2016 including p-value of Jarque-Bera Normality test and p-value

of Ljung-Box serial autocorrelation test up to 22nd order.

The financial data characteristics are evident in the descriptive statistics i.e. leptukortic

distribution and highly persistent autocorrelation based on Ljung-Box test which rejects the

null hypothesis of no autocorrelation regardless of the index and sampling. Also, the null

hypothesis of the distribution of the volatility estimators being Normal is significantly rejected.


TABLE 4.2: Descriptive statistics of pre-crisis sub-sample.

S&P500 FTSE100 HSI


Mean 0.8874 0.7980 0.6740 0.7947 0.7259 0.6230 0.9096 0.8402 0.8916

Std.Dev. 0.4615 0.4224 0.3123 0.4962 0.4389 0.3309 0.4055 0.3779 0.3366

Skewness 1.7735 1.9390 1.4615 2.7962 3.4643 3.7351 1.9294 1.9842 0.6003

Kurtosis 7.9705 9.5381 6.0672 19.8513 33.1509 35.7318 10.2552 10.2735 3.3581

Max 3.8494 3.7476 2.3931 6.8019 7.2114 5.0879 4.3417 3.5166 2.8472

Min 0.2379 0.1706 0.1625 0.2205 0.2037 0.1879 0.3000 0.2694 0.2963

JB 0.0010 0.0010 0.0010 0.0010 0.0010 0.0010 0.0010 0.0010 0.0010

LB22 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

Note: The table summarizes descriptive statistics of daily RV, BpV and MedRV of each equity market in

pre-crisis sub-sample from Jan. 3, 2000 to Dec. 31, 2007 including p-value of Jarque-Bera Normality test

and p-value of Ljung-Box serial autocorrelation test up to 22nd order.

TABLE 4.3: Descriptive statistics of post-crisis sub-sample.

S&P500 FTSE100 HSI


Mean 0.9504 0.8256 0.7716 0.8129 0.7861 0.7191 0.9172 0.8582 0.7359

Std.Dev. 0.7244 0.6441 0.6219 0.4885 0.4813 0.4349 0.5851 0.5450 0.4133

Skewness 3.1580 3.4665 3.7204 2.9023 2.8685 2.8110 3.8878 3.8063 4.0116

Kurtosis 18.8893 21.6381 26.2851 17.6430 16.6630 16.4545 25.1194 24.5103 26.0355

Max 8.8021 7.7577 8.6607 5.5838 5.0899 4.7003 6.6129 6.6057 4.9950

Min 0.1273 0.1030 0.1364 0.1947 0.1131 0.1463 0.2252 0.2040 0.2502

JB 0.0010 0.0010 0.0010 0.0010 0.0010 0.0010 0.0010 0.0010 0.0010

LB22 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

Note: The table summarizes descriptive statistics of daily RV, BpV and MedRV of each equity market in

post-crisis sub-sample from Jan. 1, 2008 to Jul. 1, 2016 including p-value of Jarque-Bera Normality test and

p-value of Ljung-Box serial autocorrelation test up to 22nd order.

Figure (4.1) illustrates RV, BpV, MedRV and their corresponding jump components of

S&P500 index in the whole sample from Jan. 3, 2000 to Jul. 1,2016. The figures of FTSE100

and HSI can be found in the appendixes (A.11)-(A.12).


FIGURE 4.1: RV, BpV, MedRV and their respective jumps of S&P500.


4.2 Estimation

Estimation of ARJ, HARJ, HARJ-semiF and HARJ-F models are summarized in (4.4), (4.5)

and (4.6) which contain the in-sample results of the S&P500 on the whole sample, pre-crisis and

post-crisis sub-sample together with benchmarks HAR(3) and HAR(3)-J. All parameters are

estimated by applying simple OLS regression. As mentioned in the section (3.2), Newey-West

HAC estimators are applied. Coefficient estimation of FTSE100 and HSI are summarized in

tables (A.1)-(A.2) in the appendix.

TABLE 4.4: Estimation of S&P500 on the whole sample

+BpV +MedRV

HAR(3) HAR(3)-J ARJ HARJ HARJ-semiF HARJ-F ARJ HARJ HARJ-semiF HARJ-F

β0

0.0511* 0.0618* 0.0744* -0.0066 -0.0466** 0.0324 0.0616* 0.0019 -0.0303 0.0271

(2.9034) (3.9871) (3.4605) (-0.3000) (-1.7992) (1.4087) (2.4252) (0.0815) (-1.1882) (0.9377)

β1

0.3437* 0.5432* 0.9873* 0.5275* 0.4819* 0.4711* 1.0207* 0.4894* 0.4456* 0.4388*

(8.0304) (12.044) (31.644) (11.129) (10.781) (10.610) (22.189) (6.7690) (5.5149) (5.5827)

β1J

-0.1283* -0.0883* -0.0723* -0.0727* -0.1161* -0.0580* -0.0426 -0.0418

(-6.5459) (-6.0897) (-4.4907) (-4.6013) (-4.1025) (-2.7103) (-1.5214) (-1.5311)

β5

0.4323* 0.3632* 0.3951* 0.5305* 0.6570* 0.4109* 0.5333* 0.6402*

(6.5500) (5.9250) (6.0229) (7.2473) (6.8509) (5.7308) (5.1928) (5.4208)

β5J

-0.0864* -0.1691* -0.0567 -0.1117*

(-2.6103) (-3.9602) (-1.4651) (-2.4442)

β22

0.1681* 0.1556* 0.1314* 0.1179* -0.1473 0.1412* 0.1252* -0.0854

(4.0702) (4.0521) (2.8943) (2.4665) (-1.3128) (3.2914) (2.8782) (-0.9499)

β22J

0.2259* 0.1292*

(2.8487) (2.3837)

βJ

-0.3277*

(-9.8408)

AIC 2823.69 2525.01 3248.86 2626.41 2600.63 2552.34 3407.46 2765.51 2755.12 2735.35

BIC 2849.15 2556.83 3267.96 2658.23 2638.81 2596.88 3426.56 2797.33 2793.31 2779.89

R2 0.6995 0.7199 0.6683 0.7132 0.7150 0.7184 0.6559 0.7037 0.7046 0.7061

Note: In-sample estimation of OLS regression of the models in the whole sample from Jan. 3, 2000 to Jul. 1, 2016. t-Statistics are reported

in the parentheses computed with Newey-West standard errors correction for heteroskedasticity and serial correlation and respective Akaike

information criteria (AIC), Bayesian information criteria (BIC) and regression R2. * and ** denote the significance at 5% and 10% level.

It was expected to find negative and highly significant coefficients corresponding to jump

components. Intuitively, the negative sign of jump coefficient implies that the model loses its

instructiveness by a certain level to predict the future RV in presence of jumps.

Comparing HARJ with BpV as its jump component with HAR(3), HARJ assigns larger

weight to the daily lag and smaller weights to weekly and monthly components which have less

sensitivity to jumps. Further investigation on the impact of jumps in higher lags are applied

particularly in the semi and full models. HARJ-semiF and HARJ-F models coefficients are

slightly different from HARJ and they improved the goodness-of-fit. However, statistically

significant estimators have not been observed by applying proposed models to the different sub-

samples and various indexes. The identical conclusion can be drawn from HARJ with MedRV


as its jump process and estimation of the pre-crisis and post-crisis sub-samples.

TABLE 4.5: Estimation of S&P500 on the pre-crisis sub-sample

+BpV +MedRV


β0

0.0543* 0.0589* 0.1168* -0.0088 -0.0438 -0.0199 0.0346 -0.0011 -0.0187 -0.0228

(3.2130) (3.8497) (5.1454) (-0.4706) (-1.7590) (-0.7007) (1.3008) (-0.0474) (-0.6339) (-0.7451)

β1

0.2953* 0.4507* 0.9536* 0.4934* 0.4437* 0.4393* 1.0864* 0.4556* 0.4137* 0.4147*

(7.2734) (11.829) (27.246) (11.113) (8.2319) (8.0311) (20.498) (7.0092) (5.0024) (5.0450)

β1J

-0.2034* -0.1592* -0.1273* -0.1263* -0.1708* -0.0661* -0.0492 -0.0496

(-7.6755) (-7.2364) (-3.8930) (-3.8273) (-4.9080) (-2.4481) (-1.2846) (-1.2984)

β5

0.4618* 0.3854* 0.3972* 0.5111* 0.5643* 0.4383* 0.5244* 0.5142*

(6.0052) (5.5533) (5.6261) (4.7901) (4.6869) (5.5481) (4.6448) (4.1434)

β5J

-0.1402** -0.1898* -0.0418 -0.0365

(-1.8570) (-2.3490) (-0.6491) (0.4710)

β22

0.1806* 0.2049* 0.1856* 0.2017* 0.0944 0.1549* 0.1468* 0.1644

(3.5342) (4.1394) (3.6392) (4.0830) (0.9094) (3.1104) (3.1638) (1.7179)

β22J

0.1334 -0.0106

(1.2852) (-0.1646)

βJ

-0.2921*

(-8.5409)

AIC 469.74 335.92 716.27 342.13 329.83 327.91 780.60 453.64 454.07 456.02

BIC 492.28 364.09 733.20 370.30 363.30 367.34 797.53 481.81 487.87 456.02

R2 0.6568 0.6787 0.6135 0.6777 0.6799 0.6805 0.6014 0.6598 0.6601 0.6601

Note: In-sample estimation of OLS regression of the models in the pre-crisis sub-sample from Jan. 3, 2001 to Dec.31, 2007. t-Statistics

are reported in the parentheses computed with Newey-West standard errors correction for heteroskedasticity and serial correlation and

respective Akaike information criteria (AIC), Bayesian information criteria (BIC) and regression R2. * and ** denote the significance at

5% and 10% level.

TABLE 4.6: Estimation of S&P500 on the post-crisis sub-sample

+BpV +MedRV


β0

0.0495* 0.0697* 0.0490** -0.0132 -0.0696 0.0432 0.0635* 0.0012 -0.0292 0.0484

(2.0287) (3.3835) (1.7314) (-0.4415) (-1.5640) (1.0460) (1.9659) (0.0388) (-0.9329) (1.2222)

β1

0.3492* 0.5821* 1.0314* 0.5703* 0.5126* 0.5018* 1.0147* 0.4917* 0.4546* 0.4451*

(5.7910) (8.4240) (23.765) (7.8771) (7.8140) (7.5459) (18.896) (5.2928) (4.4265) (4.4734)

β1J

-0.1318* -0.0900* -0.0743* -0.0727* -0.1056* -0.0563* -0.0438 -0.0435

(-5.8319) (-4.7120) (-3.8842) (-3.9946) (-3.5436) (-2.3755) (-1.3994) (-1.4404)

β5

0.4360* 0.3624* 0.3904* 0.5880* 0.7057* 0.4154* 0.5254* 0.6532*

(4.8444) (4.2686) (4.4213) (4.6002) (4.9078) (4.2388) (3.8491) (4.2945)

β5J

-0.1069* -0.1791* -0.0498 -0.1160*

(-2.4575) (-3.1094) (-1.0596) (-2.1481)

β22

0.1610* 0.1312* 0.1087** 0.0781 -0.2405 0.1337* 0.1181** -0.1424

(2.8246) (2.5086) (1.7227) (1.1401) (-1.5822) (2.2546) (1.9456) (-1.1916)

β22J

0.2523* 0.1612*

(2.4448) (2.1044)

βJ

-0.3613*

(-6.7407)

AIC 2028.87 1859.67 2231.24 1907.04 1886.34 1859.66 2366.28 1994.48 1990.47 1972.38

BIC 2051.66 1888.16 2248.36 1935.52 1920.52 1899.54 2383.40 2022.97 2024.65 2012.26

R2 0.7177 0.7389 0.6963 0.7332 0.7359 0.7393 0.6772 0.7224 0.7231 0.7256

Note: In-sample estimation of OLS regression of the models in the post-crisis sub-sample from Jan. 1, 2008 to Jul. 1, 2016. t-Statistics

are reported in the parentheses computed with Newey-West standard errors correction for heteroskedasticity and serial correlation and

respective Akaike information criteria (AIC), Bayesian information criteria (BIC) and regression R2. * and ** denote the significance at

5% and 10% level.


HAR(3)-J model performs slightly better than the proposed heterogeneous models with

respect to goodness-of-fit. Note the contribution of jump component in HAR(3)-J in comparison

with heterogeneous proposed models i.e. additive structure versus multiplicative structure.

4.3 Forecast

4.3.1 In-Sample

S&P500 forecasts obtained from estimations on respective samples are summarized in the tables

(4.4), (4.5) and (4.6) which contain performance of the models using MAE and RMSE in the

whole sample as the forecast window size and accuracy of models forecastability using DM test.

e21 − e22 = α+ ut (4.1)

where e2i denotes the square of forecast errors of model i for i = 1, 2 and error terms being white

noise. The null hypothesis of equal predictive power accuracy will be tested using t-Statistics

of Newey-West HAC estimators.

TABLE 4.7: One-day-ahead static in-sample forecast performance and accuracy ofS&P500 on the whole sample

+BpVs +MedRVs

RMSE MAE ARJ HARJ HARJ-semiF HARJ-F ARJ HARJ HARJ-semiF HARJ-F

HAR(3) 0.3361 0.2098-0.011t 0.005s 0.006s 0.007s -0.016t 0.002 0.002s 0.002s

(0.000) (0.008) (0.005) (0.009) (0.002) (0.127) (0.044) (0.016)

HAR(3)-J 0.3246 0.2022-0.019t -0.003t -0.002 -0.001 -0.024t -0.006t -0.006t -0.005t

(0.000) (0.010) (0.060) (0.688) (0.000) (0.002) (0.002) (0.003)

+BpVt

ARJ 0.3526 0.22650.017s 0.017s 0.018s -0.005 0.013s 0.013s 0.014s

(0.000) (0.000) (0.000) (0.302) (0.000) (0.000) (0.000)

HARJ 0.3285 0.20680.001 0.002 -0.021t -0.004 -0.003 -0.003

(0.206) (0.081) (0.001) (0.077) (0.100) (0.152)

HARJ-semiF 0.3274 0.20620.001 -0.022t -0.004t -0.004 -0.003

(0.151) (0.000) (0.049) (0.053) (0.081)

HARJ-F 0.3256 0.2054-0.023t -0.005 -0.005 -0.005

(0.001) (0.053) (0.058) (0.070)

+M

edRVt

ARJ 0.3592 0.22890.018s 0.018s 0.019s

(0.000) (0.000) (0.000)

HARJ 0.3338 0.20920.000 0.001

(0.502) (0.202)

HARJ-semiF 0.3334 0.20910.001

(0.228)

HARJ-F 0.3326 0.2088

Note: One-day-ahead in-sample static forecast of the models in the whole sample from Jan. 3, 2000 to Jul. 1, 2016. Performance can be

checked with RMSE and MAE. Diebold-Mariano test estimated intercept and corresponding p-values are reported is performed to check

accuracy of the models. p-Values are reported in the parentheses computed with respective Newey-West HAC standard error corrections.

s and t indicate which model is more accurate. Otherwise both models have equal predictive accuracy (Based on 5% significance level).


TABLE 4.8: One-day-ahead static in-sample forecast performance and accuracy ofS&P500 on the pre-crisis sub-sample

+BpVs +MedRVs


HAR(3) 0.2708 0.1828-0.009t 0.004s 0.005s 0.005s -0.011t 0.001 0.001 0.001

(0.000) (0.000) (0.001) (0.000) (0.000) (0.177) (0.167) (0.161)

HAR(3)-J 0.2621 0.1772-0.013t 0.000 0.000 0.000 -0.016t -0.004t -0.004t -0.004t

(0.000) (0.642) (0.635) (0.514) (0.000) (0.002) (0.004) (0.004)

+BpVt

ARJ 0.2871 0.19990.013s 0.014s 0.014s -0.003 0.009s 0.009s 0.009s

(0.000) (0.000) (0.000) (0.265) (0.000) (0.000) (0.000)

HARJ 0.2625 0.17780.000 0.001 -0.016t -0.004t -0.004t -0.004t

(0.370) (0.283) (0.000) (0.007) (0.111) (0.011)

HARJ-semiF 0.2617 0.17720.000 -0.016t -0.004t -0.004t -0.004t

(0.532) (0.000) (0.003) (0.005) (0.004)

HARJ-F 0.2615 0.1768-0.017t -0.004t -0.004t -0.004t

(0.000) (0.002) (0.004) (0.004)

+M

edRVt

ARJ 0.2915 0.20320.012s 0.012s 0.012s

(0.000) (0.000) (0.000)

HARJ 0.2697 0.18310.000 0.000

(0.750) (0.738)

HARJ-semiF 0.2697 0.18290.000

(0.934)

HARJ-F 0.2697 0.1828

Note: One-day-ahead in-sample static forecast of the models in pre-crisis sub-sample from Jan. 3, 2000 to Dec. 31, 2007. Performance can

be checked with RMSE and MAE. Diebold-Mariano test estimated intercept and corresponding p-values are reported is performed to check



In one-day-ahead static in-sample forecast, it is noticeable that heterogeneous models are

clearly preferred to homogeneous models. As we discussed before, homogeneous models are not

able to produce financial data characteristics e.g. long-memory and fat-tailed and leptokurtic

distribution. As a result, they have not been captured interest in forecasting in empirical

analysis.

All the proposed heterogeneous models outperform HAR(3) in terms of forecast errors re-

gardless of sampling and considering DM, all models are either more accurate or have the same

accuracy as HAR(3). There is barely any improvement in forecast errors in comparison to

HAR(3)-J but the proposed models with BpV as their jump component have the equal predic-

tive accuracy no matter which sample is chosen.

It was expected that the models with MedRV (more jump robust RV estimator) as their

jump component do not improve forecast errors mostly because MedRV estimator smoothens the

process more than BpV such that at the cost of adversely affecting the impact of jumps in these

models. Hence, they predict future RV with less sensitivity to the observed jumps. However,

they have similar predictive power to the models with BpV in the high-volatile periods. Same


result can be concluded from the whole sample mainly because of the financial crisis in 2008.

TABLE 4.9: One-day-ahead static in-sample forecast performance and accuracy ofS&P500 on the post-crisis sub-sample

+BpVs +MedRVs


HAR(3) 0.3833 0.2317-0.011 0.008 0.009 0.011 -0.021t 0.002 0.003 0.004s

(0.080) (0.053) (0.051) (0.051) (0.018) (0.191) (0.091) (0.033)

HAR(3)-J 0.3688 0.2219-0.022t -0.003 -0.002 0.000 -0.032t -0.009t -0.008t -0.007t

(0.000) (0.153) (0.521) (0.936) (0.002) (0.021) (0.023) (0.041)

+BpVt

ARJ 0.3995 0.24850.019s 0.020s 0.022s -0.010 0.013s 0.014s 0.015s

(0.000) (0.000) (0.000) (0.259) (0.026) (0.018) (0.011)

HARJ 0.3727 0.22830.001 0.003 -0.029t -0.006 -0.005 -0.004

(0.255) (0.148) (0.010) (0.175) (0.195) (0.304)

HARJ-semiF 0.3709 0.22800.002 -0.030t -0.007 -0.007 -0.005

(0.209) (0.010) (0.154) (0.158) (0.229)

HARJ-F 0.3686 0.2272-0.032t -0.009 -0.008 -0.007

(0.012) (0.136) (0.143) (0.178)

+M

edRVt

ARJ 0.4118 0.25230.023s 0.024s 0.025s

(0.004) (0.004) (0.003)

HARJ 0.3802 0.23040.000 0.002

(0.623) (0.218)

HARJ-semiF 0.3798 0.23050.001

(0.272)

HARJ-F 0.3782 0.2298

Note: One-day-ahead in-sample static forecast of the models in post-crisis sub-sample from Jan. 1, 2008 to Jul. 1, 2016. Performance can

be checked with RMSE and MAE. Diebold-Mariano test estimated intercept and corresponding p-values are reported is performed to check



4.3.2 Out-of-the-Sample

In the out-of-the-sample forecast, three different forecast horizons are chosen, namely, one-day-

ahead, one-week-ahead and one-month-ahead. Models reestimated daily on the rolling window

and the estimation is used to predict a-day-ahead RV. For longer forecast horizons, similar

approach is applied simultaneously with dynamic forcasting i.e. h-step-ahead predicted RV is

the average of h number of predicted RVs computed from dynamic forecasting. Performance

of the forecast in a rolling window technique is highly sensitive to the size of the window

(Ferraro et al., 2015). Inoue et al. (2014) showed that the window size picked by minimizing the

conditional mean square forecast errors is indeed optimal and forecasting improves in comparison

with alternatives. However, the forecastability, performance and accuracy of the proposed

models in various indexes with different sample sizes in univariate and multivariate cases along

with finding the optimal window size needs tremendous amount of time to investigate. As a

consequence, this author decided to choose the window size arbitrarily however, further analysis

on the impact of the rolling window size on our finding needs to be undoubtedly undertaken.


The chosen size of the rolling window for whole sample is 2000 and 1000 for pre- and post-crisis

sub-sample.

Tables (4.10), (4.11) and (4.12) summarize MAE and RMSE of volatility models in out-of-

the-sample in forecast horizon of a day, a week and a month into the future. In comparison

with HAR(3), all the proposed heterogeneous models have lower forecast errors in a-day-ahead

forecast in the whole sample and pre-crisis sub-sample with lower RMSE which implies less

sensitivity to outliers. In low-volatile samples, HARJ with BpV as its jump component performs

better even in longer forecast horizons. However, these models carry higher MAE in high-volatile

period regardless of the forecast horizon and the RMSE are relatively lower in one-day-ahead

and one-week-ahead forecast.

In comparison with HAR(3)-J, HARJ with BpV as its jump component results in improve-

ment of forecast error in dynamic forecasting in longer forecast horizons regardless of the sam-

pling. It was expected to see better performance in HARJ models rather than HARJ-semiF

and HARJ-F while HARJ models has a clear economic interpretation and each parameter and

its setting has intuitive an economic sense. On the other hand, weekly and monthly jump

components are designed artificially to extend HARJ model to semi-full and full models which

suffer from lack of economic interpretation. As mentioned in section (4.2), semi-full and full

models provide a better goodness-of-fit. As a result, these models will be considered for further

analysis in other indexes and in multivariate case in the next chapter.

Regardless of forecast horizon and sampling, all the proposed heterogeneous models have

equal accuracy compared with HAR(3) excluding one-week-ahead dynamic forecast on pre-

crisis sub-sample in which HAR(3) outperforms. The majority of the proposed heterogeneous

models have equal predictive precision as HAR(3)-J in particular in dynamic forecasts. Most

likely, models with MedRV as their jump component lose their forecast accuracy make them

empirically difficult to generalize the outcome of the analysis and to conclude their performance

and forecastability.

Note that the analysis above is solely for S&P500 and for further generalization, estimation

results, forecast performance and forecast accuracy of FTSE100 and HSI need to be analyzed.

Related outcomes are summarized in appendixes (A.3)-(A.8).


TABLE 4.10: Out-of-the-sample forecast performance S&P500 on the whole sample

One-day-ahead One-week-ahead One-month-ahead

MAE RMSE MAE RMSE MAE RMSE

HAR(3) 0.2354 0.3866 0.0713 0.1080 0.1339 0.2188

HAR(3)-J 0.2247 0.3727 0.0721 0.1073 0.1479 0.2209

+BpV

ARJ 0.2499 0.4051 0.2382 0.3884 0.2954 0.4377

HARJ 0.2318 0.3804 0.0623 0.1068 0.1414 0.2145

HARJ-semiF 0.2312 0.3798 0.0664 0.1348 0.1520 0.2208

HARJ-F 0.2314 0.3781 0.0706 0.1364 0.1657 0.2307

+M

edRV ARJ 0.2541 0.4129 0.2493 0.3926 0.3222 0.4588

HARJ 0.2340 0.3847 0.0630 0.0983 0.1373 0.2139

HARJ-semiF 0.2347 0.3851 0.0631 0.1089 0.1432 0.2167

HARJ-F 0.2349 0.3844 0.652 0.1147 0.1563 0.2249

Note: Out-of-the-sample dynamic forecast performance of the models in the

whole sample from Jan. 3, 2000 to Jul. 1, 2016 using MAE and RMSE.

TABLE 4.11: Out-of-the-sample forecast performance S&P500 on the pre-crisissub-sample



HAR(3) 0.1392 0.1958 0.0494 0.0684 0.0718 0.1018

HAR(3)-J 0.1376 0.1910 0.0447 0.0619 0.0681 0.0999

+BpV

ARJ 0.1613 0.2208 0.1785 0.2362 0.2149 0.2727

HARJ 0.1377 0.1937 0.0400 0.0586 0.0681 0.0977

HARJ-semiF 0.1363 0.1935 0.0386 0.0606 0.0687 0.0991

HARJ-F 0.1366 0.1937 0.0376 0.0588 0.0720 0.1005

+M

edRV ARJ 0.1609 0.2221 0.1807 0.2331 0.2623 0.3097

HARJ 0.1390 0.1964 0.0426 0.0585 0.0763 0.1057

HARJ-semiF 0.1382 0.1953 0.0402 0.0599 0.0808 0.1110

HARJ-F 0.1373 0.1947 0.0408 0.0625 0.0784 0.1117

Note: Out-of-the-sample dynamic forecast performance of the models in pre-crisis

sub-sample from Jan. 3, 2000 to Dec. 31, 2007 using MAE and RMSE.

TABLE 4.12: Out-of-the-sample forecast performance S&P500 on the post-crisissub-sample



HAR(3) 0.1884 0.3038 0.0745 0.1089 0.0905 0.1327

HAR(3)-J 0.1814 0.2940 0.0767 0.1130 0.0972 0.1374

+BpV

ARJ 0.2040 0.3274 0.2114 0.3230 0.2689 0.3644

HARJ 0.1875 0.3144 0.0683 0.1131 0.0907 0.1271

HARJ-semiF 0.1859 0.3146 0.0664 0.1186 0.0940 0.1285

HARJ-F 0.1862 0.3164 0.0681 0.1276 0.1032 0.1337

+M

edRV ARJ 0.2063 0.3168 0.2198 0.3425 0.2912 0.3887

HARJ 0.1887 0.3277 0.0660 0.1203 0.0965 0.1345

HARJ-semiF 0.1887 0.3334 0.0639 0.1273 0.1005 0.1366

HARJ-F 0.1892 0.3370 0.0650 0.1388 0.1089 0.1413

Note: Out-of-the-sample dynamic forecast performance of the models in post-

crisis sub-sample from Jan. 1, 2008 to Jul. 1, 2016 using MAE and RMSE.


TABLE 4.13: One-day-ahead out-of-sample forecast accuracy of S&P500 on the wholesample

+BpVs +MedRVs

ARJ HARJ HARJ-semiF HARJ-F ARJ HARJ HARJ-semi HARJ-F

HAR(3)

-0.015t 0.005 0.005 0.006 -0.021t 0.001 0.001 0.002

(0.013) (0.145) (0.105) (0.293) (0.039) (0.407) (0.582) (0.453)

HAR(3)-J

-0.025t -0.006t -0.005t -0.004 -0.032t -0.009t -0.009t -0.009t

(0.000) (0.006) (0.012) (0.337) (0.008) (0.007) (0.008) (0.010)

+BpVt

ARJ0.019s 0.020s 0.021s -0.006 0.016s 0.016s 0.016s

(0.002) (0.001) (0.013) (0.394) (0.003) (0.002) (0.001)

HARJ0.000 0.002 -0.026t -0.003 -0.004 -0.003

(0.840) (0.655) (0.033) (0.341) (0.372) (0.441)

HARJ-semiF0.001 -0.026t -0.004 -0.004 -0.003

(0.754) (0.023) (0.283) (0.222) (0.282)

HARJ-F-0.028 -0.005 -0.005 -0.005

(0.063) (0.445) (0.444) (0.471)

+M

edRVt

ARJ0.023s 0.022s 0.023s

(0.015) (0.015) (0.014)

HARJ0.000 0.000

(0.905) (0.907)

HARJ-semiF0.001

(0.541)

Note: One-day-ahead out-of-the-sample static forecast accuracy check of the models in the whole sample from Jan. 3,

2000 to Jul. 1, 2016 with Diebold-Mariano test. Estimated intercept and corresponding p-values are reported. p-Values

are stated in the parentheses computed with respective Newey-West HAC standard error corrections. s and t indicate

which model is more accurate (Based on 5% significance level).

TABLE 4.14: One-week-ahead out-of-sample forecast accuracy of S&P500 on the wholesample

+BpVs +MedRVs


HAR(3)

-0.047t 0.000 -0.007 -0.007 -0.048t 0.002s 0.000 -0.001

(0.000) (0.829) (0.126) (0.057) (0.000) (0.000) (0.881) (0.365)

HAR(3)-J

-0.047t 0.000 -0.007 -0.007 -0.048t 0.002s 0.000 -0.002

(0.000) (0.940) (0.138) (0.058) (0.000) (0.002) (0.819) (0.365)

+BpVt

ARJ0.047s 0.040s 0.040s -0.001 0.049s 0.047s 0.045s

(0.000) (0.000) (0.000) (0.818) (0.000) (0.000) (0.000)

HARJ-0.007t -0.007t -0.048t 0.002 0.000 -0.002

(0.034) (0.008) (0.000) (0.136) (0.676) (0.252)

HARJ-semiF0.000 -0.041t 0.009s 0.006 0.005

(0.773) (0.000) (0.046) (0.088) (0.185)

HARJ-F-0.041 0.009 0.007 0.005

(0.000) (0.013) (0.027) (0.066)

+M

edRVt

ARJ0.050s 0.048s 0.046s

(0.000) (0.000) (0.000)

HARJ-0.002 -0.003t

(0.070) (0.033)

HARJ-semiF-0.001

(0.102)

Note: One-week-ahead out-of-the-sample dynamic forecast accuracy check of the models in the whole sample from Jan.

3, 2000 to Jul. 1, 2016 with Diebold-Mariano test. Estimated intercept and corresponding p-values are reported. p-

Values are stated in the parentheses computed with respective Newey-West HAC standard error corrections. s and t

indicate which model is more accurate (Based on 5% significance level).


TABLE 4.15: One-month-ahead out-of-sample forecast accuracy of S&P500 on thewhole sample

+BpVs +MedRVs


HAR(3)

-0.054t 0.002 -0.001 -0.005 -0.080t 0.002 0.001 -0.003

(0.000) (0.692) (0.870) (0.414) (0.000) (0.503) (0.784) (0.560)

HAR(3)-J

-0.053t 0.003 0.000 -0.004 -0.079t 0.003 0.002 -0.002

(0.000) (0.305) (0.996) (0.258) (0.000) (0.421) (0.684) (0.651)

+BpVt

ARJ0.055s 0.053s 0.048s -0.027t 0.056s 0.055s 0.051s

(0.000) (0.000) (0.000) (0.026) (0.000) (0.000) (0.000)

HARJ-0.003 -0.007 -0.082t 0.000 -0.001 -0.005

(0.321) (0.079) (0.000) (0.908) (0.728) (0.143)

HARJ-semiF-0.004 -0.079t 0.003 0.002 -0.002

(0.290) (0.000) (0.423) (0.552) (0.639)

HARJ-F-0.075t 0.007 0.006 0.003

(0.000) (0.155) (0.227) (0.412)

+M

edRVt

ARJ0.082s 0.081s 0.078s

(0.000) (0.000) (0.000)

HARJ-0.001 -0.005

(0.493) (0.138)

HARJ-semiF-0.004

(0.214)

Note: One-month-ahead out-of-the-sample dynamic forecast accuracy check of the models in the whole sample from

Jan. 3, 2000 to Jul. 1, 2016 with Diebold-Mariano test. Estimated intercept and corresponding p-values are reported.

p-Values are stated in the parentheses computed with respective Newey-West HAC standard error corrections. s and t


TABLE 4.16: One-day-ahead out-of-sample forecast accuracy of S&P500 on pre-crisissub-sample

+BpVs +MedRVs


HAR(3)

-0.010t 0.001 0.001 0.001 -0.011t 0.000 0.000 0.000

(0.000) (0.333) (0.392) (0.408) (0.000) (0.735) (0.745) (0.482)

HAR(3)-J

-0.012t -0.001 -0.001 -0.001 -0.013t -0.002t -0.002t -0.001

(0.000) (0.071) (0.239) (0.188) (0.000) (0.011) (0.027) (0.062)

+BpVt

ARJ0.011s 0.011s 0.011s -0.001 0.010s 0.011s 0.011s

(0.000) (0.000) (0.000) (0.496) (0.000) (0.000) (0.000)

HARJ0.000 0.000 -0.012t -0.001 -0.001 0.000

(0.799) (0.985) (0.000) (0.108) (0.312) (0.557)

HARJ-semiF0.000 -0.012t -0.001 -0.001 0.000

(0.762) (0.000) (0.168) (0.364) (0.568)

HARJ-F-0.012t -0.001 -0.001 0.000

(0.000) (0.207) (0.427) (0.644)

+M

edRVt

ARJ0.011s 0.011s 0.011s

(0.000) (0.000) (0.000)

HARJ0.000 0.001

(0.159) (0.044)

HARJ-semiF0.000

(0.279)

Note: One-day-ahead out-of-the-sample static forecast accuracy check of the models in pre-crisis sub-sample from Jan.

3, 2000 to Dec. 31, 2007 with Diebold-Mariano test. Estimated intercept and corresponding p-values are reported.




TABLE 4.17: One-week-ahead out-of-sample forecast accuracy of S&P500 on pre-crisissub-sample

+BpVs +MedRVs


HAR(3)

-0.026t 0.001s 0.001s 0.001s -0.024t 0.001s 0.001s 0.001

(0.000) (0.000) (0.007) (0.000) (0.000) (0.001) (0.020) (0.177)

HAR(3)-J

-0.027t 0.000s 0.000 0.000 -0.025t 0.000 0.000 0.000

(0.000) (0.017) (0.691) (0.231) (0.000) (0.081) (0.569) (0.874)

+BpVt

ARJ0.028s 0.027s 0.028s 0.002 0.028s 0.028s 0.027s

(0.000) (0.000) (0.000) (0.313) (0.000) (0.000) (0.000)

HARJ0.000 0.000 -0.026t 0.000 0.000 0.000

(0.358) (0.934) (0.000) (0.950) (0.636) (0.290)

HARJ-semiF0.000 -0.025t 0.000 0.000 0.000

(0.134) (0.000) (0.550) (0.845) (0.656)

HARJ-F-0.026t 0.000 0.000 0.000

(0.000) (0.937) (0.749) (0.398)

+M

edRVt

ARJ0.026s 0.025s 0.025s

(0.000) (0.000) (0.000)

HARJ0.000 0.000

(0.510) (0.207)

HARJ-semiF0.000

(0.159)

Note: One-week-ahead out-of-the-sample dynamic forecast accuracy check of the models in pre-crisis sub-sample from

Jan. 3, 2000 to Dec. 31, 2007 with Diebold-Mariano test. Estimated intercept and corresponding p-values are reported.



TABLE 4.18: One-month-ahead out-of-sample forecast accuracy of S&P500 onpre-crisis sub-sample

+BpVs +MedRVs


HAR(3)

-0.033t 0.001 0.001 0.000 -0.060t 0.001 -0.002t -0.002

(0.000) (0.133) (0.463) (0.730) (0.000) (0.261) (0.022) (0.053)

HAR(3)-J

-0.034t 0.000 0.000 0.000 -0.060t -0.001 -0.002t -0.002t

(0.000) (0.131) (0.684) (0.811) (0.000) (0.055) (0.000) (0.001)

+BpVt

ARJ0.0034s 0.034s 0.033s -0.027t 0.032s 0.031s 0.031s

(0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000)

HARJ0.000 -0.001 -0.061t -0.002t -0.003t -0.003t

(0.0345) (0.088) (0.000) (0.000) (0.000) (0.000)

HARJ-semiF0.000 -0.060t -0.001t -0.002t -0.003t

(0.312) (0.000) (0.028) (0.000) (0.000)

HARJ-F-0.060t -0.001 -0.002t -0.002t

(0.000) (0.057) (0.001) (0.005)

+M

edRVt

ARJ0.059s 0.0585s 0.058s

(0.000) (0.000) (0.000)

HARJ-0.001t -0.001

(0.000) (0.050)

HARJ-semiF0.000

(0.708)

Note: One-month-ahead out-of-the-sample dynamic forecast accuracy check of the models in pre-crisis sub-sample from

Jan. 3, 2000 to Dec. 31, 2007 with Diebold-Mariano test. Estimated intercept and corresponding p-values are reported.




TABLE 4.19: One-day-ahead out-of-sample forecast accuracy of S&P500 on post-crisissub-sample

+BpVs +MedRVs


HAR(3)

-0.015t -0.007 -0.007 -0.008 -0.028 -0.015 -0.019 -0.021

(0.005) (0.464) (0.499) (0.483) (0.077) (0.372) (0.363) (0.358)

HAR(3)-J

-0.021t -0.012 -0.013 -0.014 -0.034t -0.021 -0.025 -0.027

(0.002) (0.206) (0.241) (0.252) (0.046) (0.239) (0.253) (0.259)

+BpVt

ARJ0.008 0.008 0.007 -0.013 0.000 -0.004 -0.006

(0.112) (0.171) (0.327) (0.237) (0.986) (0.811) (0.736)

HARJ0.000 -0.001 -0.021t -0.009 -0.012 -0.015

(0.938) (0.581) (0.004) (0.286) (0.299) (0.302)

HARJ-semiF-0.001 -0.021t -0.008 -0.012 -0.015

(0.382) (0.002) (0.239) (0.267) (0.275)

HARJ-F-0.020t -0.007 -0.011 -0.013

(0.001) (0.220) (0.256) (0.267)

+M

edRVt

ARJ0.013s 0.009 0.007

(0.000) (0.139) (0.423)

HARJ-0.004 -0.006

(0.330) (0.326)

HARJ-semiF-0.002

(0.323)

Note: One-day-ahead out-of-the-sample static forecast accuracy check of the models in post-crisis sub-sample from Jan.

1, 2008 to Jul. 1, 2016 with Diebold-Mariano test. Estimated intercept and corresponding p-values are reported. p-

Values are stated in the parentheses computed with respective Newey-West HAC standard error corrections. s and t


TABLE 4.20: One-week-ahead out-of-sample forecast accuracy of S&P500 on post-crisissub-sample

+BpVs +MedRVs


HAR(3)

-0.032t -0.001 -0.002 -0.004 -0.044t -0.003 -0.004 -0.007

(0.001) (0.716) (0.591) (0.465) (0.011) (0.571) (0.518) (0.441)

HAR(3)-J

-0.031t 0.000 -0.001 -0.004 -0.043t -0.002 -0.003 -0.006

(0.001) (0.994) (0.744) (0.553) (0.012) (0.712) (0.608) (0.497)

+BpVt

ARJ0.031s 0.030s 0.027s -0.012 0.029s 0.028s 0.024s

(0.000) (0.000) (0.000) (0.144) (0.000) (0.000) (0.000)

HARJ-0.001 -0.004 -0.043t -0.002 -0.003 -0.006

(0.450) (0.334) (0.003) (0.427) (0.418) (0.362)

HARJ-semiF-0.002 -0.042t 0.000 -0.002 -0.005

(0.258) (0.002) (0.767) (0.447) (0.353)

HARJ-F-0.039t 0.002 0.000 -0.003

(0.000) (0.399) (0.960) (0.428)

+M

edRVt

ARJ0.041s 0.039s 0.036s

(0.001) (0.000) (0.000)

HARJ-0.002 -0.005

(0.433) (0.348)

HARJ-semiF-0.003

(0.293)

Note: One-week-ahead out-of-the-sample dynamic forecast accuracy check of the models in post-crisis sub-sample from





TABLE 4.21: One-month-ahead out-of-sample forecast accuracy of S&P500 onpost-crisis sub-sample

+BpVs +MedRVs


HAR(3)

-0.041t 0.001 0.001 0.000 -0.056t 0.000 -0.001 -0.002

(0.000) (0.322) (0.657) (0.926) (0.000) (0.651) (0.578) (0.314)

HAR(3)-J

-0.040t 0.003s 0.002 0.001 -0.055t 0.001 0.000 -0.001

(0.000) (0.018) (0.075) (0.565) (0.000) (0.604) (0.872) (0.501)

+BpVt

ARJ0.042s 0.042s 0.041s -0.015t 0.041s 0.040s 0.039s

(0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000)

HARJ0.000 -0.002 -0.058t -0.002t -0.003t -0.004t

(0.751) (0.268) (0.000) (0.001) (0.000) (0.000)

HARJ-semiF-0.001 -0.057t -0.002 -0.002t -0.003t

(0.107) (0.000) (0.308) (0.009) (0.000)

HARJ-F-0.056t 0.000 -0.001 -0.002t

(0.000) (0.915) (0.553) (0.014)

+M

edRVt

ARJ0.056s 0.055s 0.054s

(0.000) (0.000) (0.000)

HARJ-0.001 -0.002

(0.527) (0.173)

HARJ-semiF-0.001

(0.066)

Note: One-month-ahead out-of-the-sample dynamic forecast accuracy check of the models in post-crisis sub-sample from




It is worth mentioning the dynamic forecast of benchmark model HAR(3)-J in details. It is

evident that for longer forecast horizons, the estimate for the jump component is needed. This

author decided to recalculate jump component using following model

BpVt = β0 + β1BpVt−1 + β5BpVt−1|t−5 + β22BpVt−1|t−22 + ut (4.2)

where ut has the same specification as the error terms of the models introduced so far. By

applying (2.15) and (4.2), the jump component of HAR(3)-J can be computed using (2.13) in

order to estimate HAR(3)-J for dynamic forecasting.

Note that the outcome of the empirical analysis in univariate case strongly shows that

homogeneous models have been beaten by all the heterogeneous models in estimation, in-sample

and out-of-the-sample forecast and forecast accuracy. Hence, these models are excluded for

further analysis in multivariate case in the next chapter.

Chapter 5

Empirical Analysis of Multivariate

Models

In this chapter, the results of the multivariate extension to HARJ, HARJ-semiF and HARJ-F

models will be discussed. RV and orthogonalized RV-modifications are employed to analyze

Granger-causality, volatility transmission and performance of one-day-ahead forecast. Further-

more, accuracy of the forecasts will be considered by applying DM test. Analysis is done for

the whole sample as well as pre- and post-crisis sub-samples.

5.1 Granger-Causality

The Granger-causality tests are based on the bivariate models introduced in section (3.3). F-

statistics are used to verify whether the predictive power of volatility of the other index is

significant e.g. in VHAR(3) and VHAR(3)-η models, restriction βj1 = βj

5 = βj22 = 0 is applied.

Table (5.1) summarizes the statistics of the bivariate heterogeneous models to test whether

RV of FTSE100 or HSI Granger-causes RV in S&P500, both in RV- and residual-modifications

in the whole sample, pre- and post-crisis sub-samples. All the models have BpV as their jump

component. In most of the models, lagged RV components of FTSE100 seems to contain

persistent information for determining volatility in S&P500 regardless of sampling and type

of modification. However, predictive information provided by HSI is more plausible subject

to high-volatile periods. In the appendix, tables (A.9)-(A.12) summarize Granger-causality

F-statistics of FTSE100 and HSI to consider bidirectional causality and existence of spillover

effect across equity markets.

Table (5.2) shows the F-value of the Granger-causality test on S&P500 with the residual

and RV-modified models where all have MedRV as jump component.

31

CHAPTER 5. EMPIRICAL ANALYSIS OF MULTIVARIATE MODELS 32

TABLE 5.1: Granger-causality test of S&P500 with BpV as jump component

VHAR(3)-η VHAR(3)-J-η VHARJ-η VHARJ-semiF-η VHARJ-F-η

FTSE100 HSI FTSE100 HSI FTSE100 HSI FTSE100 HSI FTSE100 HSI

u The whole sample:

F-value9.1123* 5.0967* 4.1616* 1.7685 5.3849* 2.7540* 5.1377* 2.1288 5.4355* 1.8869

(0.0000) (0.0016) (0.0023) (0.1320) (0.0002) (0.0264) (0.0001) (0.0589) (0.0000) (0.0789)

u Pre-crisis sub-sample:

F-value6.1205* 0.6326 2.6069* 0.8857 3.9674* 0.9494 2.7552* 1.6783 3.5555* 1.9906

(0.0004) (0.5939) (0.0338) (0.4714) (0.0033) (0.4341) (0.0171) (0.1359) (0.0016) (0.0632)

u Post-crisis sub-sample:

F-value6.4243* 6.8094* 2.7688* 3.1062* 4.9722* 4.1407* 4.3342* 3.0686* 4.6883* 2.4133*

(0.0002) (0.0001) (0.0257) (0.0145) (0.0005) (0.0023) (0.0006) (0.0090) (0.0001) (0.0247)

VHAR(3) VHAR(3)-J VHARJ VHARJ-semiF VHARJ-F


u The whole sample:

F-value10.039* 4.7524* 4.5255* 1.8804 5.9242* 2.6833* 4.4588* 2.3620* 3.8670* 1.4927

(0.0000) (0.0026) (0.0012) (0.1108) (0.0001) (0.0295) (0.0005) (0.0375) (0.0007) (0.1761)


F-value6.1597* 0.4410 2.6993* 0.3236 4.5906* 0.4148 3.9380* 1.1807 3.5555* 1.9906

(0.0004) (0.7237) (0.0289) (0.8623) (0.0010) (0.7981) (0.0014) (0.3157) (0.0016) (0.0632)


F-value7.2987* 5.5254* 3.4983* 2.8780* 5.1012* 3.1369* 4.4183* 2.6720* 4.3536* 1.6568

(0.0001) (0.0009) (0.0073) (0.0214) (0.0004) (0.0137) (0.0005) (0.0202) (0.0002) (0.1272)

Note: Table provides Granger-causality F-statistic of S&P500 with BpV as jump component in the proposed models. p-Values are

reported in the parentheses computed with Newey-West standard errors. * indicates whether corresponding index Granger-causes

S&P500 (Based on 5% significance level).

TABLE 5.2: Granger-causality test of S&P500 with MedRV as jump component



u The whole sample:

F-value9.1123* 5.0967* 4.1616* 1.7685 6.6433* 3.7684* 5.2868* 3.1044* 6.1288* 3.4868*

(0.0000) (0.0016) (0.0023) (0.1320) (0.0000) (0.0046) (0.0001) (0.0084) (0.0000) (0.0019)


F-value6.1205* 0.6326 2.6069* 0.8857 4.1453* 0.6610 4.2713* 1.1678 3.9056* 1.5674

(0.0004) (0.5939) (0.0338) (0.4714) (0.0023) (0.6191) (0.0007) (0.3222) (0.0007) (0.1521)


F-value6.4243* 6.8094* 2.7688* 3.1062* 5.2680* 5.2975* 3.9676* 4.5290* 4.5532* 3.6901*

(0.0002) (0.0001) (0.0257) (0.0145) (0.0003) (0.0003) (0.0013) (0.0004) (0.0001) (0.0011)



u The whole sample:

F-value10.0386* 4.7524* 4.5255* 1.8804 7.5629* 3.3900* 5.7162* 3.0441* 5.7006* 2.8139*

(0.0000) (0.0026) (0.0012) (0.1108) (0.0000) (0.0088) (0.0000) (0.0095) (0.0000) (0.0097)


F-value6.1597* 0.4410 2.6993* 0.3236 4.4182* 0.3226 3.4977* 0.2465 3.0950* 0.2306

(0.0004) (0.7237) (0.0289) (0.8623) (0.0014) (0.8630) (0.0037) (0.9417) (0.0050) (0.9668)


F-value7.2987* 5.5254* 3.4983* 2.8780* 3.9991* 4.4081* 2.8980* 5.6414* 3.2750* 4.1834*

(0.0001) (0.0009) (0.0073) (0.0214) (0.0030) (0.0015) (0.0128) (0.0000) (0.0032) (0.0003)

Note: Table provides Granger-causality F-statistic of S&P500 with MedRV as jump component in the proposed models. p-Values

are reported in the parentheses computed with Newey-West standard errors. * indicates whether corresponding index Granger-

causes S&P500 (Based on 5% significance level).


The outcome of Granger-causality test is fixed in comparison with the benchmarks in all

models with FTSE100 as their secondary index regardless of the sampling and the jump type.

In comparison with the benchmarks, fixed outcome is observed in all the models with HSI as

their secondary index regardless of jump type. However, the result is inconclusive in the whole

sample.

Note that the outcome of Granger-causality test in the benchmark does not coincide in the

models with HSI as their secondary index. Furthermore, proposed models confirm the result of

VHAR(3) model rather than VHAR(3)-J which has jump component as an additive term.

5.2 Volatility Transmission and In-Sample Forecast

It can be investigated whether additional information from another equity market can increase

the forecasting power of RV system by testing significance of coefficients e.g. Granger-causality

test in section (5.1). However, the excessive size of the lags in the explanatory variables could

result in multiple testing issues (Soucek and Todorova, 2013). Therefore, the coefficient analysis

of the models is considered to reveal the additional information by controling significance of

causalities. Similar to section (4.3.1), the size of the forecast window is the same as the sample

size.

5.2.1 The Whole Sample

Table (5.3) summarizes the estimation, goodness-of-fit and in-sample forecast errors of residual-

modified models on S&P500. As expected, S&P500 is affected by its own volatility components

and its jump component. Nevertheless, long-term volatility components are significant at large

levels in semi and full-models. In VHAR(3), VHAR(3)-J and VHARJ models, FTSE100 does

not provide any explanatory power to predict RV for S&P500. However, mid-term and long-term

jump components of FTSE100 seems to have predictive information for S&P500 in VHARJ-

semiF and VHARJ-F models.

In the model with HSI as its secondary index, short-term components of the models VHAR(3),

VHAR(3)-J and VHARJ do provide new information for RV in S&P500 and unlike FTSE100,

mid-term and long-term jump components of HSI have no impact on S&P500 in semi and full

models. Note that all the proposed models have higher fit to the data than VHAR(3). In terms

of forecast errors, proposed models have lower forecast errors compared to benchmark VHAR(3)

and forecast errors are only slightly higher than the errors in VHAR(3)-J.


TABLE 5.3: In-Sample Estimation of S&P500 with Residual-Modification on the wholesample and the BpV as jump component



u Intercept:

β0

0.0637* 0.0618* 0.0730* 0.0702* 0.0066 0.0049 -0.0337 -0.0372 0.0614* 0.0523*

(3.4809) (3.5930) (4.5625) (4.2805) (0.2809) (0.2178) (-1.3182) (-1.4198) (2.6812) (2.4670)

u Primary index parameters:

β1

0.3856* 0.3826* 0.5667* 0.5718* 0.5690* 0.5641* 0.5270* 0.5171* 0.514* 0.5031*

(8.3826) (8.2814) (12.240) (11.505) (11.732) (11.039) (11.482) (11.073) (11.126) (10.866)

β1J

-0.0861* -0.0875* -0.0722* -0.0716* -0.0728* -0.0748*

(-6.1064) (-6.2950) (-4.5409) (-4.7733) (-4.5786) (-4.8571)

β5

0.3947 0.3881* 0.3449* 0.3296* 0.3573* 0.3602* 0.4992* 0.5055* 0.6342* 0.6432*

(0.5385) (5.3828) (4.9697) (4.8258) (4.7704) (4.8610) (5.8660) (6.6865) (5.9271) (5.8261)

β5J

-0.0853* -0.0907* -0.1723* -0.1806*

(-2.7695) (-2.6994) (-4.2438) (-3.7084)

β22

0.1502* 0.1619* 0.1426* 0.1483* 0.1147* 0.1175* 0.0964** 0.0998* -0.2088** -0.2006

(3.3829) (3.8825) (3.5123) (3.7355) (2.3601) (2.4789) (1.8791) (2.0409) (-1.8316) (-1.5360)

β22J

0.2600* 0.2771*

(3.3037) (2.7167)

βJ

-0.3088* -0.3241*

(-9.0557) (-9.7620)

u Secondary index parameters:

β1

0.1384 0.1523* 0.0765 0.0929** 0.1535 0.1115** 0.1274 0.0814 0.1158 0.0416

(1.4327) (2.7691) (0.8191) (1.7268) (1.5049) (1.8460) (1.2441) (1.3986) (1.1840) (0.6910)

β1J

-0.0639 -0.0076 -0.0218 0.0101 -0.0056 0.0316

(-1.5856) (-0.2375) (-0.4668) (0.2399) (-0.1202) (0.7198)

β5

79.641 -120.51 93.268 -21.733 90.517 -18.251 196.37 35.755 224.40 142.72

(0.4524) (-1.1213) (0.6081) (-0.2317) (0.5260) (-0.1770) (0.9650) (0.2962) (1.1996) (1.2547)

β5J

-257.77* -16.840 -445.83* -156.69

(-1.9715) (-0.2389) (-2.6508) (-1.4008)

β22

-22.984 7.1163 -13.4187 -12.288 -30.605 -9.8464 -34.003 -16.472 -47.274 -27.051

(-0.8007) (0.3692) (-0.5580) (-0.7897) (-1.0691) (-0.5445) (-1.1721) (-0.8628) (-1.3235) (-1.3011)

β22J

90.722* 10.864

(2.0078) (0.3095)

βJ

-0.0497 0.0063

(-1.4661) (0.2158)

AIC 2763.44 2802.97 2499.95 2518.60 2581.55 2612.32 2553.05 2585.77 2494.33 2530.77

BIC 2807.98 2847.51 2557.22 2575.87 2638.82 2669.59 2623.05 2655.78 2577.06 2613.50

R2 0.7041 0.7014 0.7220 0.7208 0.7167 0.7147 0.7188 0.7167 0.7119 0.7206

MAE 0.2076 0.2087 0.2013 0.2017 0.2053 0.2061 0.2051 0.2056 0.2038 0.2047

RMSE 0.3334 0.3349 0.3232 0.3239 0.3263 0.3274 0.3250 0.3263 0.3226 0.3240

Note: Coefficient estimation, Goodness-of-fit, in-sample forecast errors of models with residuals-modification on the whole sample from

Jan. 3, 2000 to Jul. 1, 2016 are reported. t-Statistics are stated in the parentheses computed with Newey-West standard errors. * and

** denote the significance at 5% and 10% level.

Table (5.4) summarizes the estimation, goodness-of-fit and in-sample forecast errors of RV-

modified models on S&P500. In all the models, the short-term volatility component of the

secondary index exhibits significant causality to S&P500. Additionally, long-term volatility

component of FTSE100 appears to be significant at 10% level in VHAR(3), VHARJ and

VHARJ-semiF. Similar to the residual-modification, proposed models outperform VHAR(3)


and same conclusion can be drawn for the goodness-of-fit and information criteria of the pro-

posed RV-modified models.

In terms of magnitude of the coefficients, residual-modification assigns more weight to recent

lags than RV-modified models in the primary index parameters.

TABLE 5.4: In-Sample Estimation of S&P500 with RV-Modification on the whole sampleand the BpV as jump component



u Intercept:

β0

0.0383* 0.0352 0.0506* 0.0598* -0.0188 -0.0099 -0.0508* -0.0380 0.0224* 0.0135

(2.5533) (1.6000) (3.9225) (3.1979) (-0.9843) (-0.4041) (-2.1803) (-1.2709) (0.7203) (0.5510)


β1

0.2849* 0.3259* 0.4995* 0.5282* 0.4683* 0.5128* 0.4320* 0.4649* 0.4246* 0.4581*

(6.5797) (8.0074) (10.835) (12.487) (9.4990) (11.472) (9.0947) (10.862) (8.9015) (10.907)

β1J

-0.0830* -0.0875* -0.0698* -0.0709* -0.0710* -0.719*

(-6.4844) (-6.4412) (-5.0216) (-4.6645) (-4.8966) (-4.8581)

β5

0.3380* 0.4427* 0.2772* 0.3625* 0.2961* 0.3901* 0.4249* 0.5369* 0.5677* 0.6515*

(4.6492) (6.6974) (4.0825) (5.9917) (4.3608) (6.0763) (5.0644) (7.0459) (5.0961) (6.4314)

β5J

-0.0740* -0.0977* -0.1595* -0.1721*

(-2.3567) (-2.9428) (-3.6086) (-4.0117)

β22

0.2519* 0.1403* 0.2103* 0.1650* 0.2391* 0.1245* 0.2341* 0.1141* -0.1282 -0.1500

(3.5231) (3.0237) (3.2605) (3.7079) (3.4552) (2.5884) (3.3443) (2.1169) (-1.0039) (-1.2366)

β22J

0.2433* 0.2425*

(2.8827) (2.8132)

βJ

-0.3081* -0.3234*

(-9.1696) (-9.9508)


β1

0.1473* 0.1202* 0.0955** 0.0877* 0.1514* 0.0867* 0.1485* 0.0918* 0.1431* 0.0760*

(3.0246) (2.9752) (1.8652) (2.3962) (2.7279) (2.0400) (3.0873) (2.3906) (3.0254) (1.9740)

β1J

-0.0261 0.0094 -0.0268 0.0047 -1.1891 0.0082

(-1.4420) (0.5251) (-1.1806) (0.1780) (-1.1891) (0.2993)

β5

0.0954 -0.1154 0.1084 -0.0638 0.1075 -0.0552 0.0894 -0.0637 0.0366 -0.0075

(1.0010) (-1.4987) (1.2290) (-0.8704) (1.0519) (-0.7886) (1.0079) (-0.9165) (0.3845) (-0.0833)

β5J

-0.0025 0.0279 0.0458 -0.0330

(-0.0367) (0.4903) (0.4526) (-0.4225)

β22

-0.1475** 0.0483 -0.0996 -0.0191 -0.1809** -0.0049 -0.1869** -0.0302 -0.0347 -0.0209

(-1.6800) (0.8104) (-1.3088) (-0.3517) (-1.8328) (-0.0926) (-1.8975) (-0.5841) (-0.3480) (-0.2652)

β22J

-0.0672 -0.0084

(-0.6049) (-0.1241)

βJ

-0.0489 0.0064

(-1.4257) (0.2215)

AIC 2758.46 2800.76 2496.40 2517.83 2575.45 2611.64 2559.84 2582.01 2511.17 2545.27

BIC 2803.01 2845.31 2553.67 2575.10 2632.73 2668.91 2629.84 2652.02 2593.90 2628.00

R2 0.7045 0.7016 0.7223 0.7209 0.7171 0.7147 0.7184 0.7169 0.7218 0.7196

MAE 0.2075 0.2088 0.2013 0.2017 0.2053 0.2061 0.2049 0.2055 0.2042 0.2050

RMSE 0.3332 0.3349 0.3230 0.3238 0.3260 0.3274 0.3253 0.3261 0.3233 0.3246

Note: Coefficient estimation, Goodness-of-fit, in-sample forecast errors of models with RV-modification on the whole sample from Jan.

3, 2000 to Jul. 1, 2016 are reported. t-Statistics are stated in the parentheses computed with Newey-West standard errors. * and **

denote the significance at 5% and 10% level.


Tables (A.13) and (A.14) summarize estimation, the goodness-of-fit and in-sample forecast

errors on S&P500 with MedRV as jump type. The outcome of these models are identical to the

models with BpV as jump component. However, impact of recent jump components of primary

index decreases and at the same time, strong significance of the recent jumps of the secondary

index is observable.

Table (5.5) shows the forecast accuracy of the models with FTSE100 as the secondary

index using DM test. The models have equal accuracy no matter which estimator is picked

as jump component. However, models with BpV as their jump component are more accurate

compared to VHAR(3). Besides, these models are quite capable of producing same accuracy as

the benchmark VHAR(3)-J particularly in semi and full models.

TABLE 5.5: One-day-ahead in-sample forecast accuracy of S&P500 on the whole samplewith FTSE100 as the secondary index.

+BpVs +MedRVs

VHARJ-η VHARJ-semiF-η VHARJ-F-η VHARJ-η VHARJ-semiF-η VHARJ-F-η

VHAR(3)-η

0.005s 0.006s 0.007s 0.001 0.002 0.002s

(0.010) (0.008) (0.012) (0.133) (0.055) (0.015)

VHAR(3)-J-η

-0.002t -0.001 0.000 -0.005t -0.005t -0.004t

(0.024) (0.248) (0.933) (0.008) (0.011) (0.017)

+BpVt

VHARJ-η0.001 0.002 -0.003 -0.003 -0.002

(0.159) (0.066) (0.114) (0.145) (0.226)

VHARJ-semiF-η0.002 -0.004 -0.004 -0.003

(0.089) (0.086) (0.101) (0.148)

VHARJ-F-η-0.006 -0.005 -0.005

(0.072) (0.083) (0.100)

+M

edRVt

VHARJ-η0.000 0.001

(0.398) (0.135)

VHARJ-semiF-η0.001

(0.217)

+BpVs +MedRVs

HARJ HARJ-semiF HARJ-F HARJ HARJ-semi HARJ-F

VHAR(3)

0.005s 0.005s 0.007s 0.002 0.002s 0.003s

(0.012) (0.008) (0.017) (0.055) (0.041) (0.021)

VHAR(3)-J

-0.002t -0.001 0.000 -0.005t -0.004t -0.004t

(0.032) (0.116) (0.914) (0.003) (0.003) (0.011)

+BpVt

VHARJ0.000 0.002 -0.003 -0.002 -0.002

(0.254) (0.109) (0.070) (0.077) (0.197)

VHARJ-semiF0.001 -0.003 -0.003 -0.002

(0.166) (0.052) (0.050) (0.125)

VHARJ-F-0.004 -0.004 -0.003

(0.061) (0.062) (0.090)

+M

edRVt

VHARJ0.000 0.001

(0.597) (0.126)

VHARJ-semiF0.001

(0.169)

Note: One-day-ahead in-sample forecast accuracy check of the models in the whole sample from Jan. 3, 2000 to Jul. 1,

2016 with Diebold-Mariano test. Estimated intercept and corresponding p-values are reported. p-Values are stated in

the parentheses computed with respective Newey-West HAC standard error corrections. s and t indicate which model

is more accurate (Based on 5% significance level).


Table (5.6) shows the forecast accuracy of the models with HSI as the secondary index using

DM test. Similar to the models with FTSE100 as the secondary index, accuracy is not affected

by the choice of the jump component. Including the jump components in the models increases

accuracy of the forecast in comparison with the models without jump component i.e. VHAR(3).

Note that semi and full models with BpV and VHAR(3)-J have identical precision in forecast.

TABLE 5.6: One-day-ahead in-sample forecast accuracy of S&P500 on the whole samplewith HSI as the secondary index.

+BpVs +MedRVs


VHAR(3)-η

0.005s 0.006s 0.007s 0.002 0.002s 0.003s

(0.008) (0.005) (0.013) (0.081) (0.019) (0.013)

VHAR(3)-J-η

-0.002t -0.002 0.000 -0.006t -0.005t -0.005t

(0.010) (0.095) (0.950) (0.002) (0.002) (0.012)

+BpVt

VHARJ-η0.001 0.002 -0.003 -0.003 -0.002

(0.225) (0.112) (0.084) (0.114) (0.247)

VHARJ-semiF-η0.001 -0.004 -0.004 -0.003

(0.159) (0.055) (0.060) (0.134)

VHARJ-F-η-0.006 -0.005 -0.004

(0.066) (0.069) (0.111)

+M

edRVt

VHARJ-η0.000 0.001

(0.486) (0.153)

VHARJ-semiF-η0.001

(0.152)

+BpVs +MedRVs

VHARJ VHARJ-semiF VHARJ-F VHARJ VHARJ-semiF VHARJ-F

VHAR(3)

0.005s 0.006s 0.007s 0.002 0.002s 0.003s

(0.006) (0.004) (0.007) (0.091) (0.008) (0.004)

VHAR(3)-J

-0.002t -0.001 0.000 -0.006t -0.005t -0.004t

(0.010) (0.124) (0.724) (0.002) (0.003) (0.005)

+BpVt

VHARJ0.001 0.002 -0.003 -0.003 -0.002

(0.132) (0.089) (0.081) (0.145) (0.225)

VHARJ-semiF0.001 -0.004t -0.004 -0.003

(0.165) (0.048) (0.068) (0.101)

VHARJ-F-0.005 -0.005 -0.004

(0.053) (0.064) (0.081)

+M

edRVt

VHARJ0.001 0.001

(0.448) (0.202)

VHARJ-semiF0.001

(0.154)

Note: One-day-ahead in-sample forecast accuracy check of the models in the whole sample from Jan. 3, 2000 to Jul. 1,

2016 with Diebold-Mariano test. Estimated intercept and corresponding p-values are reported. p-Values are stated in

the parentheses computed with respective Newey-West HAC standard error corrections. s and t indicate which model

is more accurate (Based on 5% significance level).

5.2.2 Pre-Crisis Sub-Sample


modified models on S&P500. Each index depends on their own lagged volatility components.


However, statistically significant long-term components and their respective jumps have not

been observed in the full models regardless of the index.

TABLE 5.7: In-Sample Estimation of S&P500 with Residual-Modification on pre-crisissub-sample and the BpV as jump component



u Intercept:

β0

0.0681* 0.0574* 0.0651* 0.0613* 0.0017 -0.0081 -0.0324 -0.0441** -0.0142 -0.0313

(4.1779) (3.2989) (4.2829) (4.2805) (0.0939) (-0.4091) (-1.3729) (-1.6897) (-0.5483) (-0.9399)


β1

0.3299* 0.3010* 0.4598* 0.4546* 0.5074* 0.4951* 0.4612* 0.4460* 0.4529* 0.4412*

(8.1658) (7.3236) (11.000) (11.777) (10.773) (10.7165) (8.2357) (7.8383) (8.0731) (7.6864)

β1J

-0.1538* -0.1605* -0.1244* -0.1285* -0.1249* -0.1268*

(-7.0228) (-6.4980) (-3.7583) (-3.8244) (-3.7173) (-3.7185)

β5

0.4372* 0.4688* 0.3878* 0.3985* 0.3935* 0.4078* 0.4992* 0.5115* 0.5169* 0.5421*

(5.9161) (6.0257) (5.5718) (5.6127) (5.5974) (5.6482) (5.0680) (4.5792) (4.6906) (4.3299)

β5J

-0.0853* -0.1410* -0.1626* -0.1696*

(-1.7420) (-1.9859) (-2.0049) (-2.1633)

β22

0.1550* 0.1651* 0.1842* 0.1889* 0.1619* 0.1723* 0.1801* 0.1957* 0.1311 0.1394

(3.1828) (3.2183) (3.8137) (3.7406) (3.5349) (3.3917) (3.9933) (3.9218) (1.4204) (1.1874)

β22J

0.0680 0.0780

(0.6827) (0.6538)

βJ

-0.2805* -0.2966*

(-8.1541) (-8.2389)


β1

0.0741 -0.040 0.0081 -0.0829 0.0323 -0.0870 0.0429 -0.0471 0.0213 -0.0398

(1.0693) (-0.6650) (0.1233) (-1.4595) (0.4702) (-1.2464) (0.6155) (-0.6787) (0.3151) (-0.5735)

β1J

-0.0138 0.0251 -0.0171 -0.0435 0.0101 -0.0590

(-0.3476) (0.4474) (-0.4052) (-0.6905) (0.2265) (-0.9190)

β5

7.8717 74.184 24.765 106.50** 17.916 97.047** 8.480 52.022 86.998 13.860

(0.1336) (1.1135) (0.4193) (1.8042) (0.2946) (1.6970) (0.1297) (0.8170) (1.2896) (0.1979)

β5J

-9.8368 108.22 -173.94* 199.75**

(-0.1676) (1.4504) (-2.4906) (1.8922)

β22

4.1034 -10.930 4.2026 -17.003** 5.2481 -16.277** 7.1293 -18.1423* -34.421* -4.1326

(0.3644) (-1.0165) (0.3925) (-1.7595) (0.4530) (-1.7661) (0.5858) (-2.0237) (-1.9977) (-0.3501)

β22J

103.4786* -33.9457

(3.6011) (-1.4767)

βJ

-0.0068 -0.0046

(-0.1988) (-0.1518)

AIC 447.619 470.665 330.479 337.192 332.057 343.992 323.669 327.653 307.207 322.831

BIC 487.052 510.098 381.179 387.892 382.758 394.692 385.636 389.620 380.440 396.065

R2 0.6615 0.6577 0.6807 0.6797 0.6805 0.6787 0.6824 0.6818 0.6855 0.6832

MAE 0.1809 0.1826 0.1761 0.1771 0.1765 0.1776 0.1760 0.1768 0.1757 0.1761

RMSE 0.2687 0.2702 0.2610 0.2614 0.2611 0.2618 0.2603 0.2605 0.2590 0.2600

Note: Coefficient estimation, Goodness-of-fit, in-sample forecast errors of models with residuals-modification on pre-crisis sub-sample

from Jan. 3, 2000 to Dec. 31, 2007 are reported. t-Statistics are stated in the parentheses computed with Newey-West standard errors.

* and ** denote the significance at 5% and 10% level.

Mid-term and long-term volatility components are significant at 10% level in the VHAR(3)-J,

VHARJ models with HSI as their secondary index and there is stronger significance on long-term


component in VHAR-semiF compared to other models. Furthermore, mid-term jump compo-

nents seem to provide new information for S&P500 future at certain level in the full models.

Long-term jump component has an impact on S&P500 at 10% level in the full model with ex-

planatory FTSE100 components. Considering the goodness-of-fit and in-sample performance,

proposed models outperform the benchmark models particularly in semi and full-models.

Table (5.8) summarizes the estimation, goodness-of-fit and in-sample forecast errors of RV-

modified models on S&P500. Starting S&P500 volatility lags and jumps, all the short- and

mid-terms affect S&P500 in addition to long-term components with the exception of the full

models. However, long-term jumps seem to provide information to predict RV in these models.

The impact of daily volatility of the secondary index is barely noticeable in low-volatile sub-

sample regardless of index and modification. However, RV-modified VHAR(3) and VHARJ

increase the predictive power of RV in S&P500. In the semi and full models, mid and long-

term jumps provide additional explanatory information for future RV in S&P500 especially with

FTSE100 as the secondary index. Same significant causality can be observed in the full model

with HSI as the side index.

Similar to the residual-modified models, all the proposed models have the higher fit to the

data and forecast errors have been improved significantly compared to VHAR(3) and slight im-

provement is observed against VHAR(3)-J. Furthermore, residual-modified models have greater

effects from the recent lags in terms of magnitude versus RV-modified models in the primary in-

dex. Additionally, residual-modified models assigns more weight to the jumps than RV-modified

models. Effect of the jumps in both modifications is in the same direction with the exception of

RV-modified VHARJ with HSI as its secondary index where it has a negative effect on S&P500

rather than a positive effect in the same model with residual-modification on future RV of

S&P500.

In the high volatile periods, the effect of short-term volatility components is not as large(maybe

significant) as the models in the low volatile periods. However, the effect of long-terms com-

ponents is lower than the RV-modified models. Note that both modifications have identical

direction in the effect of jumps.


TABLE 5.8: In-Sample Estimation of S&P500 with RV-Modification on pre-crisissub-sample and the BpV as jump component



u Intercept:

β0

0.0650* 0.0420* 0.0671* 0.0617* -0.0016 -0.0061 -0.0570* -0.0120 -0.0015 -0.0471

(4.0123) (2.3729) (4.4437) (3.6726) (-0.0920) (-0.3297) (-2.1724) (-0.3683) (-0.0497) (-1.2764)


β1

0.2629* 0.2943* 0.4324* 0.4548* 0.4680* 0.4990* 0.4176* 0.4474* 0.4075* 0.4412*

(5.7907) (7.1956) (9.8273) (11.692) (9.5510) (10.872) (6.8684) (8.2852) (6.8030) (8.0954)

β1J

-0.1527* -0.1621* -0.1220* -0.1300* -0.1195* -0.1293*

(-6.9409) (-6.8686) (-3.7000) (-4.2000) (-3.6212) (-4.2117)

β5

0.3905* 0.4471* 0.3277* 0.3636* 0.3390* 0.3759* 0.4627* 0.4946* 0.5085* 0.5161*

(4.3197) (5.5957) (3.7323) (5.2092) (3.7542) (5.2795) (3.4974) (4.7375) (3.3945) (4.3737)

β5J

-0.1380** -0.1530* -0.1899* -0.1738*

(-1.8260) (-2.1908) (-2.2852) (-2.3392)

β22

0.1619** 0.1845* 0.1777* 0.2278* 0.1493** 0.2117* 0.1560** 0.2434* 0.0928 0.2087**

(1.8695) (3.3183) (2.1461) (4.2107) (1.6663) (3.8491) (1.6649) (4.6450) (0.6610) (1.6899)

β22J

0.0680 0.0534

(0.6827) (0.4484)

βJ

-0.2790* -0.2950*

(-8.0899) (-8.2921)


β1

0.0796* 0.0136 0.0316 -0.0080 0.0616** 0.0009 0.0456 0.0241 0.0410 0.0263

(2.1170) (0.4198) (0.9968) (-0.2206) (1.6694) (0.0222) (1.3029) (0.6148) (1.1326) (0.6726)

β1J

-0.0140 -0.0088 -0.0020 -0.0350 0.0001 -0.0378

(-0.7407) (-0.3682) (-0.1089) (-1.2509) (0.0049) (-1.3452)

β5

0.0592 0.0431 0.0681 0.0819 0.0566 0.0768 0.1091 0.0159 0.1833** -0.0715

(0.9122) (0.4898) (1.0350) (1.0161) (0.8179) (0.9309) (1.1937) (0.1781) (1.9254) (-0.7157)

β5J

-0.0710** 0.0930 -0.1463* 0.2064**

(-1.6850) (1.0345) (-2.7920) (1.9236)

β22

-0.0154 -0.0314 0.0025 -0.0780 0.0091 -0.0866 0.0188 -0.1170** -0.1543 0.0572

(-0.2110) (-0.3985) (0.0360) (-1.0969) (0.1204) (-1.2443) (0.2214) (-1.7315) (-1.3619) (0.6384)

β22J

0.1948* -0.2399*

(2.1863) (-2.9328)

βJ

-0.0080 -0.0100

(-0.2339) (-0.3158)

AIC 447.404 473.114 331.187 341.346 331.517 347.163 317.653 329.810 307.207 322.831

BIC 486.838 512.548 381.888 392.046 382.217 397.864 379.621 391.777 380.440 396.065

R2 0.6615 0.6573 0.6806 0.6791 0.6806 0.6782 0.6833 0.6815 0.6855 0.6832

MAE 0.1809 0.1825 0.1762 0.1771 0.1768 0.1776 0.1764 0.1767 0.1757 0.1761

RMSE 0.2687 0.2704 0.2610 0.2617 0.2610 0.2620 0.2699 0.2907 0.2590 0.2600

Note: Coefficient estimation, Goodness-of-fit, in-sample forecast errors of models with RV-modification on pre-crisis sub-sample from

Jan. 3, 2000 to Dec. 31, 2007 are reported. t-Statistics are stated in the parentheses computed with Newey-West standard errors. *

and ** denote the significance at 5% and 10% level.

Table (5.9) summarizes the conclusion of the forecast accuracy of the models with FTSE100

as the secondary index. In contrast to the whole sample forecast analysis, models with BpV as

their jump component are more accurate in forecasting than the models with MedRV. However,

including the jump components in the models results in more precise forecast regardless of the

choice of jump type. The models with BpV as their jump component have identical prediction

power compared to VHAR(3)-J. Besides, forecast errors decline by employing jump component


over higher time horizons.

The outcome of the DM accuracy test of the models with HSI as their secondary index is

reported in table (5.10). Proposed models with BpV as jump component have greater predictive

power in comparison with the benchmark VHAR(3) where models with MedRV as jump type

have equal accuracy. Regardless of the model modification, accuracy of the models with MedRV

jump type is outperformed by the models with BpV as their jump component. Note that the

models with BpV jump type and benchmark VHAR(3)-J have equal accuracy.

TABLE 5.9: One-day-ahead in-sample forecast accuracy of S&P500 on pre-crisissub-sample with FTSE100 as the secondary index.

+BpVs +MedRVs


VHAR(3)-η

0.004s 0.004s 0.005s 0.001 0.001 0.001s

(0.002) (0.000) (0.000) (0.120) (0.079) (0.039)

VHAR(3)-J-η

0.000 0.000 0.001 -0.003t -0.003t -0.003t

(0.903) (0.506) (0.112) (0.003) (0.009) (0.010)

+BpVt

VHARJ-η0.000 0.001s -0.003t -0.003t -0.003t

(0.402) (0.049) (0.008) (0.019) (0.020)

VHARJ-semiF-η0.001 -0.003t -0.003t -0.003t

(0.066) (0.003) (0.007) (0.006)

VHARJ-F-η-0.005t -0.004t -0.004t

(0.002) (0.005) (0.003)

+M

edRVt

VHARJ-η0.000 0.000

(0.339) (0.129)

VHARJ-semiF-η0.000

(0.329)

+BpVs +MedRVs


VHAR(3)

0.004s 0.005s 0.005s 0.001 0.001 0.001

(0.001) (0.000) (0.001) (0.137) (0.137) (0.095)

VHAR(3)-J

0.000 0.001 0.001 -0.003t -0.003t -0.003t

(0.979) (0.346) (0.151) (0.003) (0.006) (0.004)

+BpVt

VHARJ0.001 0.001 -0.003t -0.003t -0.003t

(0.269) (0.080) (0.007) (0.011) (0.009)

VHARJ-semiF0.000 -0.004t -0.004t -0.004t

(0.197) (0.002) (0.003) (0.002)

VHARJ-F-0.004t -0.004t -0.004t

(0.003) (0.004) (0.003)

+M

edRVt

VHARJ0.000 0.000

(0.634) (0.503)

VHARJ-semiF0.000

(0.738)

Note: One-day-ahead in-sample forecast accuracy check of the models in pre-crisis sub-sample sample from Jan. 3, 2000

to Dec. 31, 2007 with Diebold-Mariano test. Estimated intercept and corresponding p-values are reported. p-Values




The forecast accuracy of the models with HSI as their secondary index is shown in table

(5.10). The conclusion is mostly similar to the accuracy of the models with FTSE100 as the

secondary index. However, VHARJ-F-η with MedRV jump type has greater predictive power

than VHAR(3) from the in the table (5.9) but it has equal accuracy by applying this model

with HSI as the secondary index.

TABLE 5.10: One-day-ahead in-sample forecast accuracy of S&P500 on pre-crisissub-sample with HSI as the secondary index.

+BpVs +MedRVs


VHAR(3)-η

0.004s 0.005s 0.005s 0.001 0.001 0.001

(0.002) (0.001) (0.001) (0.196) (0.146) (0.052)

VHAR(3)-J-η

0.000 0.000 0.001 -0.004t -0.004t -0.004t

(0.631) (0.451) (0.322) (0.002) (0.004) (0.004)

+BpVt

VHARJ-η0.001 0.001 -0.004t -0.004t -0.004t

(0.206) (0.116) (0.008) (0.011) (0.013)

VHARJ-semiF-η0.000 -0.005t -0.004t -0.004t

(0.332) (0.003) (0.005) (0.005)

VHARJ-F-η-0.005t -0.005t -0.004t

(0.002) (0.003) (0.004)

+M

edRVt

VHARJ-η0.000 0.000

(0.547) (0.268)

VHARJ-semiF-η0.000

(0.376)

+BpVs +MedRVs


VHAR(3)

0.004s 0.005s 0.006s 0.001 0.001 0.001

(0.002) (0.001) (0.000) (0.189) (0.175) (0.167)

VHAR(3)-J

0.000 0.001 0.001 -0.004t -0.004t -0.004t

(0.679) (0.402) (0.194) (0.003) (0.004) (0.004)

+BpVt

VHARJ0.001 0.001 -0.004t -0.004t -0.004t

(0.293) (0.154) (0.009) (0.012) (0.012)

VHARJ-semiF0.000 -0.005t -0.005t -0.005t

(0.118) (0.003) (0.004) (0.004)

VHARJ-F-0.005t -0.005t -0.005t

(0.002) (0.002) (0.002)

+M

edRVt

VHARJ0.000 0.000

(0.763) (0.740)

VHARJ-semiF0.000

(0.892)

Note: One-day-ahead in-sample forecast accuracy check of the models in pre-crisis sub-sample sample from Jan. 3, 2000




A summary of coefficient estimation, goodness-of-fit and in-sample forecast errors on S&P500

with MedRV as jump type are shown in tables (A.15) and (A.16). Similar to the section (5.2.1),

models with MedRV as jump type have weak or no significance for the recent jump components


in primary index. Simultaneously, recent jumps in the secondary index provide information to

explain RV in S&P500.

5.2.3 Post-Crisis Sub-Sample


modified models on S&P500. by including the jump components, long-term volatility compo-

nents are no longer capable of producing predictive power for RV in S&P500. None of the

FTSE100 volatility components increase the performance of the models both in the proposed

models and the benchmarks which is controversial conclusion compared to Granger-causality

test in the section (5.1). However, strong significance can be observed in VHARJ model by

short-term jump component. Considering HSI as the secondary explanatory index, all the mod-

els with exception of VHARJ-F provide new information to predict RV in S&P500 with strong

significance in the lagged daily volatility measure.

Table (5.12) lists the estimation, goodness-of-fit and in-sample forecast errors of RV-modified

models on S&P500. In RV-modified models, long-term volatility component of S&P500 provides

predictive power which was not instructive under residual-modification. On the other hand,

mid-term component loses its predictive power in VHAR(3)-J and VHARJ when the secondary

index is FTSE100. In the RV-modified models, lagged daily volatility components are highly

informative to produce additional prediction power for future RV in S&P500 regardless of the the

choice of secondary index. In the high-volatile periods, higher lags seem to have less informative

data for RV forecast e.g. weekly and monthly RV. This effect is regardless of the secondary

index choice excluding VHAR(3) with HSI and VHAR(3)-J with FTSE100. Short-term jump

components are significant at higher levels in VHARJ-semiF and VHARJ-F with FTSE100 as

their secondary index. Besides, mid-term jumps highly increase the predictive power of the

models.

The goodness-of-fit is improved compared to the benchmark VHAR(3). Furthermore, semi

and full models have higher fit than benchmark models. In terms of forecast errors, the proposed

models outperform VHAR(3). Additionally, semi and full models produce higher MAE with

lower RMSE which can be interpreted in less sensitivity to outliers in high volatile periods.

Note that the conclusion of Granger-causality test in the table (5.1) and the analysis of

volatility transmission in sections (5.2.1), (5.2.2) and current section do not coincide. This au-

thor applied backward elimination in order to reach the models with highly significant parameter

which results in returning back to the univariate cases. Since, the setting of the bivariate mod-

els improve performance and forecast errors in addition to clear economic interpretation, the


modifications and model structures are fixed for further analysis. The outcome of the Granger-

causality test and the volatility transmission analysis needs to be carefully considered for other

indexes.

TABLE 5.11: In-Sample Estimation of S&P500 with residuals-Modification on post-crisissub-sample and the BpV as jump component



u Intercept:

β0

0.0632* 0.0670* 0.0819* 0.0783* -0.0005 0.0064 -0.0438 -0.0559 0.0829* 0.0725*

(2.5079) (2.9911) (3.9375) (3.6934) (-0.0160) (0.2085) (-1.0355) (-1.2205) (2.0725) (2.0597)


β1

0.4008* 0.4251* 0.6018* 0.6248* 0.6274* 0.6355* 0.5759* 0.5759* 0.5664* 0.5575*

(5.7257) (6.3638) (8.6840) (8.2755) (8.8867) (8.2212) (8.6602) (8.4567) (8.5688) (8.1150)

β1J

-0.0902* -0.0869* -0.0783* -0.00730* -0.0801* -0.0766*

(-4.6495) (-4.8011) (-3.7644) (-4.3976) (-3.9458) (-4.4277)

β5

0.4045* 0.3438* 0.3583* 0.2872* 0.3529* 0.3167* 0.4967* 0.5334* 0.6333* 0.6470*

(3.7247) (3.4107) (3.4854) (2.8952) (3.3292) (2.9962) (3.9296) (3.9836) (4.3111) (3.7356)

β5J

-0.0764** -0.1114* -0.1576* -0.1901*

(-1.8454) (-2.5147) (-2.9792) (-2.8247)

β22

0.1278* 0.1592* 0.1042** 0.1310* 0.0792 0.0658 0.0648 0.0582 -0.3037* 0.2910

(1.9662) (2.8479) (1.7691) (2.4486) (1.0879) (1.4648) (0.8963) (0.8039) (-1.9954) (-1.6422)

β22J

0.2900* 0.3179*

(2.9472) (2.5012)

βJ

-0.3365* -0.3508*

(-6.6766) (-6.6064)


β1

0.1161 0.3001* 0.0206 0.2065* 0.1512 0.2368* 0.1827 0.1618** 0.1909 0.1279

(2.1170) (3.6553) (0.1034) (2.4790) (0.7135) (2.6137) (0.9677) (1.8814) (1.0564) (1.3948)

β1J

-0.1542* -0.0117 -0.2209 0.0396 -0.2251 0.0547

(-2.6495) (-0.2740) (-1.2466) (0.8049) (-1.1694) (0.9699)

β5

173.53 -157.92 155.67 -100.45 168.29 -82.748 47.276 -17.793 34.810 10.453

(0.7703) (-2.0002) (0.8268) (-1.3953) (0.7616) (-1.0493) (0.2536) (-0.1960) (0.1737) (0.1379)

β5J

314.64 -34.79 215.17 -89.78

(0.5397) (-0.7732) (0.3175) (-1.3095)

β22

-24.545 13.886 -9.7797 2.6841 -26.523 2.4113 -17.472 -2.3167 -14.648 -3.1323

(-0.7745) (0.9944) (-0.3755) (0.2344) (-0.8409) (0.1885) (-0.6497) (-0.1684) (-0.3337) (-0.2081)

β22J

73.811 -0.4225

(0.4249) (-0.0177)

βJ

-0.0584 0.065

(-0.7654) (1.3374)

AIC 1983.61 2002.36 1838.72 1845.57 1869.34 1891.22 1856.23 1869.16 1824.84 1842.21

BIC 2023.48 2042.24 1889.99 1896.84 1920.61 1942.49 1918.89 1931.82 1898.90 1916.26

R2 0.7242 0.7219 0.7423 0.7415 0.7387 0.7361 0.7407 0.7392 0.7448 0.7428

MAE 0.2291 0.2295 0.2206 0.2206 0.2258 0.2266 0.2254 0.2262 0.2248 0.2251

RMSE 0.3785 0.3801 0.3659 0.3665 0.3685 0.3703 0.3670 0.3681 0.3641 0.3656

Note: Coefficient estimation, Goodness-of-fit, in-sample forecast errors of models with residuals-modification on post-crisis sub-sample

from Jan. 1, 2008 to Jul. 1, 2016 are reported. t-Statistics are stated in the parentheses computed with Newey-West standard errors.



TABLE 5.12: In-Sample Estimation of S&P500 with RV-Modification on post-crisissub-sample and the BpV as jump component



u Intercept:

β0

0.0044 0.0295 0.0242 0.0648* -0.0551** -0.0083 -0.0777* -0.0664 0.0171 0.0166

(0.2037) (0.9641) (1.2222) (2.5714) (-1.8552) (-0.2508) (-2.0887) (-1.4098) (0.4385) (0.3962)


β1

0.2741* 0.3178* 0.5341* 0.5502* 0.5017* 0.5407* 0.4462* 0.4836* 0.4363* 0.4794*

(4.4642) (5.6548) (7.3873) (8.4692) (6.4156) (8.0342) (6.4294) (7.7749) (6.2063) (7.8206)

β1J

-0.0848* -0.0881* -0.0726* -0.0718* -0.0738* -0.0732*

(-5.1394) (-5.1520) (-3.9457) (-4.4596) (-4.0109) (-4.6038)

β5

0.1989 0.4622* 0.1367 0.3795* 0.1250 0.4027* 0.4175* 0.6031* 0.5701* 0.7119*

(1.6397) (5.0847) (1.2518) (4.4752) (1.0073) (4.7432) (3.0123) (4.8403) (3.7830) (4.4972)

β5J

-0.1365* -0.1178* -0.2211* -0.1824

(-2.8556) (-2.6895) (-3.7285) (-3.0968)

β22

0.3293* 0.0997 0.2387* 0.1191** 0.3160* 0.0733 0.2601* 0.0541 -0.2681** -0.2688

(2.7580) (1.4705) (2.1389) (1.8323) (2.5525) (1.0843) (2.8522) (0.7017) (-1.7197) (-1.5915)

β22J

0.3400* 0.2811*

(3.4448) (2.4810)

βJ

-0.3361* -0.3514*

(-6.6819) (-6.8902)


β1

0.1951* 0.2088* 0.0968 0.1426* 0.1780** 0.1377* 0.2219* 0.1258* 0.2171* 0.1045**

(2.1510) (3.2473) (1.0342) (2.5239) (1.8717) (2.1185) (2.8522) (2.3340) (2.8566) (1.8966)

β1J

-0.0340 0.0216 -0.0880 0.02485 -0.0939** 0.0265

(-1.0658) (1.1192) (-1.6418) (0.7923) (-1.6888) (0.7637)

β5

0.3555 -0.2163* 0.3807** -0.1574 0.4173 -0.1371 0.1345 -0.0861 0.0829 -0.0510

(1.4950) (-1.8100) (1.7991) (-1.3795) (1.5269) (-1.2069) (0.6841) (-0.8138) (0.4284) (-0.3775)

β5J

0.2815* 0.0014 0.3271* -0.0551

(1.9672) (0.0196) (2.2312) (-0.5488)

β22

-0.3253** 0.0986 -0.2238 0.0235 -0.3902** 0.0387 -0.3034 -0.0093 -0.0225 0.0261

(-1.7285) (1.1519) (-1.3019) (0.2990) (-1.7212) (0.5013) (-1.5567) (-0.1157) (-0.1201) (0.2399)

β22J

-0.2541 -0.0210

(-0.9736) (-0.2253)

βJ

-0.0661 0.0645

(-0.8335) (1.3136)

AIC 1976.39 2003.55 1830.88 1846.33 1859.44 1890.57 1824.17 1868.35 1789.08 1851.62

BIC 2016.27 2043.43 1882.15 1897.60 1910.71 1941.84 1886.84 1931.01 1863.13 1925.67

R2 0.7251 0.7217 0.7432 0.7414 0.7398 0.7361 0.7444 0.7393 0.7489 0.7417

MAE 0.2284 0.2298 0.2204 0.2208 0.2255 0.2265 0.2248 0.2263 0.2242 0.2261

RMSE 0.3779 0.3802 0.3653 0.3666 0.3677 0.3703 0.3644 0.3681 0.3612 0.3663

Note: Coefficient estimation, Goodness-of-fit, in-sample forecast errors of models with RV-modification on post-crisis sub-sample from

Jan. 1, 2008 to Jul. 1, 2016 are reported. t-Statistics are stated in the parentheses computed with Newey-West standard errors. *


Similar to the previous sections, the forecast accuracy of the in-sample forecast is verified

by DM test. Table (5.13) contains the outcome of DM test for the models with FTSE100 as

their secondary index. Note that with exception of VHARJ-η, all the models have the same

predictive power compared to benchmark models regardless of modification and choice of jump


component.

TABLE 5.13: One-day-ahead in-sample forecast accuracy of S&P500 on post-crisissub-sample with FTSE100 as the secondary index.

+BpVs +MedRVs


VHAR(3)-η

0.007 0.009 0.0011 0.002 0.005 0.007

(0.056) (0.060) (0.052) (0.259) (0.157) (0.087)

VHAR(3)-J-η

-0.002 -0.001 0.001 -0.008t -0.005 -0.003

(0.352) (0.724) (0.665) (0.029) (0.151) (0.398)

+BpVt

VHARJ-η0.001 0.003 -0.006 -0.003 -0.001

(0.307) (0.118) (0.172) (0.394) (0.784)

VHARJ-semiF-η0.002 -0.007 -0.004 -0.002

(0.146) (0.157) (0.266) (0.568)

VHARJ-F-η-0.009 -0.006 -0.004

(0.124) (0.172) (0.297)

+M

edRVt

VHARJ-η0.003 0.005

(0.358) (0.188)

VHARJ-semiF-η0.002

(0.209)

+BpVs +MedRVs

HARJ VHARJ-semiF VHARJ-F VHARJ VHARJ-semiF VHARJ-F

VHAR(3)

0.008 0.010 0.012 0.004 0.004 0.006

(0.077) (0.109) (0.088) (0.170) (0.186) (0.101)

VHAR(3)-J

-0.002 0.001 0.003 -0.006t -0.005 -0.004

(0.443) (0.0873) (0.539) (0.048) (0.059) (0.196)

+BpVt

VHARJ0.002 0.005 -0.004 -0.004 -0.002

(0.278) (0.137) (0.148) (0.140) (0.431)

VHARJ-semiF0.002 -0.006 -0.006 -0.004

(0.104) (0.151) (0.134) (0.269)

VHARJ-F-0.009 -0.008 -0.007

(0.112) (0.100) (0.161)

+M

edRVt

VHARJ0.000 0.002

(0.684) (0.157)

VHARJ-semiF0.002

(0.193)

Note: One-day-ahead in-sample forecast accuracy check of the models in post-crisis sub-sample sample from Jan. 1,




Table (5.14) summarizes the conclusion of the accuracy test for the models with HSI as their

secondary index. All the proposed models have either same accuracy or greater predictive power

compared to VHAR(3) regardless of the modification and choice of the jump type. However,

the models perform as well as VHAR(3)-J by choosing BpV as jump type in contrast to MedRV

which causes the loss of predictive power.


TABLE 5.14: One-day-ahead in-sample forecast accuracy of S&P500 on post-crisissub-sample with HSI as the secondary index.

+BpVs +MedRVs


VHAR(3)-η

0.007 0.009 0.0011 0.003 0.003s 0.004s

(0.054) (0.056) (0.067) (0.116) (0.031) (0.022)

VHAR(3)-J-η

-0.003 -0.001 0.001 -0.008t -0.007t -0.006

(0.116) (0.583) (0.833) (0.025) (0.025) (0.056)

+BpVt

VHARJ-η0.002 0.004 -0.005 -0.004 -0.003

(0.273) (0.185) (0.205) (0.237) (0.401)

VHARJ-semiF-η0.002 -0.006 -0.006 -0.005

(0.202) (0.177) (0.182) (0.275)

VHARJ-F-η-0.008 -0.008 -0.006

(0.168) (0.173) (0.227)

+M

edRVt

VHARJ-η0.001 0.002

(0.560) (0.213)

VHARJ-semiF-η0.001

(0.192)

+BpVs +MedRVs


VHAR(3)

0.007s 0.009s 0.010s 0.003 0.004s 0.005s

(0.036) (0.034) (0.041) (0.142) (0.008) (0.006)

VHAR(3)-J

-0.003 -0.001 0.000 -0.008t -0.007t -0.005t

(0.123) (0.594) (0.951) (0.024) (0.031) (0.044)

+BpVt

VHARJ0.002 0.003 -0.005 -0.004 -0.002

(0.209) (0.166) (0.183) (0.279) (0.417)

VHARJ-semiF0.001 -0.006 -0.005 -0.004

(0.216) (0.146) (0.187) (0.247)

VHARJ-F-0.008 -0.007 -0.005

(0.141) (0.170) (0.200)

+M

edRVt

VHARJ0.001 0.002

(0.435) (0.194)

VHARJ-semiF0.001

(0.218)

Note: One-day-ahead in-sample forecast accuracy check of the models in post-crisis sub-sample sample from Jan. 1,




Tables (A.17) and (A.18) show estimation, goodness-of-fit and in-sample forecast errors

on S&P500 with MedRV as jump component in the proposed models. In the post-crisis sub-

sample, all the models are not capable of producing predictive information for RV in S&P500

using jumps components of primary index. On the other hand, recent jump components of

the secondary index result in greater impact on RV of S&P500 in comparison with the models

having BpV as their jump type.

Note that performance of the models with BpV and MedRV as their jump component need

to be meticulously analyzed together with the outcome of the DM test. While the models with

MedRV decline the forecast error, models with BpV increase accuracy.


5.3 Out-of-the-Sample Forecast

All the setting are identical to the setting discussed in the section (4.3.2). However, the choice

of forecast horizon is limited to one day in the multivariate models since the estimation of

the secondary index can not be obtained in the multivariate case. Tables (5.15) and (5.16)

summarize forecast performance of MAE and RMSE of residual and RV-modified models of

S&P500 on the whole sample, pre-crisis and post-crisis sub-sample.

TABLE 5.15: Out-of-the-sample forecast performance of S&P500

The Whole Sample Pre-Crisis Post-Crisis


VHAR(3)-η 0.2329 0.3848 0.1390 0.1959 0.1874 0.3020

VHAR(3)-J-η 0.2239 0.3730 0.1372 0.1917 0.1809 0.2941

+BpV

VHARJ-η 0.2303 0.3797 0.1380 0.1955 0.1856 0.3089

VHARJ-semiF-η 0.2303 0.3779 0.1368 0.1960 0.1847 0.3090

VHARJ-F-η 0.2304 0.3757 0.1377 0.1972 0.1854 0.3109

+M

edRV VHARJ-η 0.2323 0.3839 0.1386 0.1973 0.1865 0.3194

VHARJ-semiF-η 0.2329 0.3854 0.1381 0.1951 0.1878 0.3225

VHARJ-F-η 0.2330 0.3853 0.1375 0.1951 0.1887 0.3271

VHAR(3) 0.2326 0.3839 0.1389 0.1956 0.1869 0.3018

VHAR(3)-J 0.2238 0.3723 0.1372 0.1915 0.1810 0.2941

+BpV

VHARJ 0.2300 0.3787 0.1378 0.1944 0.1861 0.3081

VHARJ-semiF 0.2295 0.3787 0.1372 0.1954 0.1848 0.3079

VHARJ-F 0.2296 0.3771 0.1375 0.1956 0.1852 0.3098

+M

edRV VHARJ 0.2335 0.3820 0.1388 0.1962 0.1880 0.3230

VHARJ-semiF 0.2340 0.3837 0.1380 0.1949 0.1881 0.3276

VHARJ-F 0.2356 0.3840 0.1392 0.1957 0.1885 0.3313

Note: Out-of-the-sample forecast performance of the models of S&P500 using MAE

and RMSE with FTSE100 as the secondary index.

In the residual-modified models with FTSE100 as the secondary index, all the proposed

models improve forecast errors compared to benchmark VHAR(3)-η. However, higher RMSE

is reported in the pre- and post-crisis sub-samples especially in the semi and full-models, which

can be interpreted as higher sensitivity of the models to outliers. Models fail to improve fore-

cast errors compared to benchmark model VHAR(3)-η. However, in the pre-crisis sub-sample,

proposed models with BpV as jump type slightly improve errors. Meanwhile, extreme values

can affect forecast errors with greater impact than that of the benchmarks.

In the RV-modified models, models with MedRV as jump component are outperformed by

the benchmarks. However, models with BpV outperform VHAR(3) with exception of fore-

castability in the post-crisis sub-sample. Similar to residual-modified models, there is a slight

improvement in semi and full models with BpV as their jump component in comparison with


benchmark VHAR(3)-J.

Considering the analysis with HSI as the secondary index, most of the models show lower

forecast errors in comparison with benchmark VHAR(3) and VHAR(3)-η regardless of the

choice of jump type particularly in the pre-crisis sub-sample. However, higher RMSE has been

observed in high-volatile periods. Forecast errors of the models are relatively lower than forecast

errors of benchmark VHAR(3)-J in pre-crisis sub-sample regardless of the modification.

TABLE 5.16: Out-of-the-sample forecast performance of S&P500

The Whole Sample Pre-Crisis Post-Crisis


VHAR(3)-η 0.2339 0.3856 0.1410 0.1969 0.1883 0.3040

VHAR(3)-J-η 0.2241 0.3728 0.1398 0.1928 0.1817 0.2941

+BpV

VHARJ-η 0.2310 0.3807 0.1395 0.1952 0.1877 0.3146

VHARJ-semiF-η 0.2314 0.3818 0.1396 0.1958 0.1860 0.3139

VHARJ-F-η 0.2328 0.3819 0.1395 0.1961 0.1868 0.3151

+M

edRV VHARJ-η 0.2324 0.3839 0.1402 0.1973 0.1865 0.3194

VHARJ-semiF-η 0.2333 0.3857 0.1413 0.1982 0.1878 0.3345

VHARJ-F-η 0.2346 0.3862 0.1400 0.1972 0.1878 0.3271

VHAR(3) 0.2343 0.3857 0.1412 0.1968 0.1883 0.3043

VHAR(3)-J 0.2245 0.3729 0.1401 0.1929 0.1819 0.2942

+BpV

VHARJ 0.2309 0.3804 0.1394 0.1951 0.1883 0.3154

VHARJ-semiF 0.2306 0.3810 0.1379 0.1948 0.1864 0.3139

VHARJ-F 0.2336 0.3823 0.1377 0.1945 0.1877 0.3157

+M

edRV VHARJ 0.2325 0.3842 0.1408 0.1976 0.1901 0.3287

VHARJ-semiF 0.2325 0.3842 0.1398 0.1966 0.1897 0.3331

VHARJ-F 0.2352 0.3861 0.1385 0.1951 0.1918 0.3372

Note: Out-of-the-sample forecast performance of the models of S&P500 using MAE

and RMSE with HSI as the secondary index.

One-day-ahead forecast accuracy is controlled by DM test on the whole sample with S&P500

as the primary and FTSE100 as the secondary index. The outcome is summarized in the table

(5.17). Note that proposed models have equal predictive power regardless of modification and

the jump type. However, with the exception of residual-modified semi and full-models and

RV-modified full model —both with BpV jump type— the rest of the models fail to improve

accuracy against benchmarks VHAR(3)-J-η and VHAR(3)-J. Furthermore, forecast accuracy is

identical to the benchmark models without jump components.

Tables (A.19) and (A.20) contain the result of DM test for models with S&P500 as the

primary index and FTSE100 as the secondary index in pre-crisis and post-crisis sub-samples.

All the models have equal precision of forecasting in the post-crisis sub-sample. Same conclusion

can be drawn for pre-crisis sub-sample with the exception of VHARJ with MedRV as its jump

component.


TABLE 5.17: One-day-ahead out-of-the-sample forecast accuracy of S&P500 on thewhole sample with FTSE100 as the secondary index.

+BpVs +MedRVs


VHAR(3)-η

0.004 0.005 0.007 0.001 0.000 0.000

(0.186) (0.220) (0.293) (0.699) (0.864) (0.896)

VHAR(3)-J-η

-0.005t -0.004 -0.002 -0.008t -0.009t -0.009t

(0.006) (0.178) (0.662) (0.020) (0.027) (0.039)

+BpVt

VHARJ-η0.001 0.003 -0.003 -0.004 -0.004

(0.554) (0.483) (0.368) (0.342) (0.374)

VHARJ-semiF-η0.002 -0.005 -0.006 -0.006

(0.584) (0.372) (0.307) (0.335)

VHARJ-F-η-0.006 -0.007 -0.007

(0.401) (0.366) (0.376)

+M

edRVt

VHARJ-η-0.001 -0.001

(0.583) (0.636)

VHARJ-semiF-η0.000

(0.962)

+BpVs +MedRVs


VHAR(3)

0.004 0.004 0.005 0.001 0.000 0.000

(0.199) (0.210) (0.365) (0.315) (0.919) (0.975)

VHAR(3)-J

-0.005t -0.005t -0.004 -0.007t -0.009t -0.009t

(0.009) (0.028) (0.364) (0.006) (0.009) (0.012)

+BpVt

VHARJ0.000 0.001 -0.003 -0.004 -0.004

(0.984) (0.716) (0.375) (0.305) (0.314)

VHARJ-semiF0.001 -0.003 -0.004 -0.004

(0.739) (0.423) (0.224) (0.247)

VHARJ-F-0.004 -0.005 -0.005

(0.504) (0.420) (0.402)

+M

edRVt

VHARJ-0.001 -0.001

(0.463) (0.458)

VHARJ-semiF0.000

(0.848)

Note: One-day-ahead out-of-the-sample forecast accuracy check of the models in the whole sample from Jan. 3, 2000

to Jul. 1, 2016 with Diebold-Mariano test. Estimated intercept and corresponding p-values are reported. p-Values are

stated in the parentheses computed with respective Newey-West HAC standard error corrections. s and t indicate


The conclusion of the forecast accuracy of models with S&P500 as the primary and HSI

as the secondary index is quite similar to the models with FTSE100 as the secondary index.

However, semi-models lose their accuracy compared to VHAR(3)-J-η and VHAR(3)-J when the

jump type is BpV. The outcome is summarized in the table (5.18).

Tables (A.21) and (A.22) contain the result of DM test on the models with S&P500 as

primary and HSI as the secondary index in pre- and post-crisis sampling. Models VHARJ-η

and VHARJ-semiF-η show less precision in comparison with VHAR(3)-J-η in the pre-crisis sub-

sample. The remaining models behave identical to the models with FTSE100 as the secondary


index in accuracy. Furthermore, DM test outcome in post-crisis sub-sample is identical to the

models with FTSE100 as their secondary index.

TABLE 5.18: One-day-ahead out-of-the-sample forecast accuracy of S&P500 on thewhole sample with HSI as the secondary index.

+BpVs +MedRVs


VHAR(3)-η

0.004 0.003 0.003 0.001 0.000 0.000

(0.235) (0.336) (0.670) (0.626) (0.983) (0.864)

VHAR(3)-J-η

-0.006t -0.007t -0.007 -0.009t -0.010t -0.010t

(0.005) (0.002) (0.154) (0.006) (0.008) (0.007)

+BpVt

VHARJ-η-0.001 -0.001 -0.003 -0.004 -0.004

(0.695) (0.848) (0.392) (0.359) (0.316)

VHARJ-semiF-η0.000 -0.002 -0.003 -0.003

(0.986) (0.530) (0.379) (0.335)

VHARJ-F-η-0.002 -0.003 -0.003

(0.780) (0.683) (0.638)

+M

edRVt

VHARJ-η-0.001 -0.001

(0.655) (0.547)

VHARJ-semiF-η0.000

(0.716)

+BpVs +MedRVs


VHAR(3)

0.004 0.004 0.003 0.001 0.001 0.000

(0.172) (0.233) (0.682) (0.480) (0.608) (0.917)

VHAR(3)-J

-0.006t -0.006t -0.007 -0.009t -0.009t -0.010t

(0.006) (0.007) (0.143) (0.006) (0.014) (0.006)

+BpVt

VHARJ0.000 -0.001 -0.003 -0.003 -0.004

(0.806) (0.729) (0.359) (0.479) (0.294)

VHARJ-semiF-0.001 -0.002 -0.003 -0.004

(0.739) (0.455) (0.521) (0.305)

VHARJ-F-0.001 -0.001 -0.003

(0.827) (0.838) (0.670)

+M

edRVt

VHARJ0.000 -0.001

(0.982) (0.610)

VHARJ-semiF-0.001

(0.376)

Note: One-day-ahead out-of-the-sample forecast accuracy check of the models in the whole sample from Jan. 3, 2000

to Jul. 1, 2016 with Diebold-Mariano test. Estimated intercept and corresponding p-values are reported. p-Values are

stated in the parentheses computed with respective Newey-West HAC standard error corrections. s and t indicate


Chapter 6

Conclusion

This thesis introduces various setting of volatility models in which jump components are con-

sidered together with the heterogeneous market hypothesis in a multiplicative structure both

in univariate and bivariate cases. The model is termed HARJ where jump component can be

structured by BpV or MedRV. The simple and additive cascade structure of the model can

be extended to semi and full models where time-varying and jump-sensitive parameters are

included in the lags over longer time periods.

In the univariate case, improvement in the goodness-of-fit and information criteria has been

achieved specially in pre-crisis sub-sample. In-sample forecast errors have been decreased in

models with BpV as their jump component in low-volatile periods. Furthermore, equal accuracy

in predictive power is observed in these models in comparison with models with jump compo-

nent with additive structure. Not surprisingly, all the models with BpV jump type outperform

heterogeneous models, which do not distinguish between continuous process and discrete jump

component in goodness-of-fit, information criteria, forecast errors and forecast accuracy regard-

less of sub-sampling. However, both have equal precision to predict RV in high-volatile periods.

Furthermore, models succeed in reproducing the financial intraday data characteristics.

In the out-of-the-sample forecast, all the proposed models improve performance in one-day-

ahead static forecast versus models without jump component. Additionally, models with BpV

as their jump type decrease one-day-ahead static forecast in low-volatile periods in comparison

with benchmark model which has an additive structured jump component. In the one-week-

ahead out-of-the-sample dynamic forecast, all heterogeneous models outperform benchmark

models. High performance of these models can be seen in longer forecast horizons. In terms of

accuracy, these models do not lose power to predict RV in comparison with benchmark models.

In the multivariate case, two modifications have been applied, namely residual- and RV-

modifications. The models are termed by VHARJ-η for residual-modified and VHARJ for

52

CHAPTER 6. CONCLUSION 53

RV modified models. Similar to univariate case, these models are extended to semi and full

models. All models have identical outcome for Granger-causality test to the results of the

test for the benchmarks in the sub-samples no matter which modification is chosen. Similar to

univariate case, models outperform the benchmark model without jump component in goodness-

of-fit, information criteria, in-sample forecast errors and accuracy. However, improvement can

be barely observed when comparing to benchmark model with additive jump component.The

models with BpV have identical predictive power specially in semi and full models.

In the out-of-the-sample forecast, conclusion is similar to that of the one-day-ahead static

forecast in univariate case. Models outperform the benchmark without jump component in all

aspects but fail to show improvement when compared to model with additive jump component.

However, the level of forecast accuracy is identical specifically in sub-samples.

In conclusion, the new set of volatility models proposed in this thesis successfully achieve to

provide the long-memory behavior of high-frequency financial data in addition to improvements

in dynamic forecasting in univariate case without loss of accuracy. In multivariate case, accu-

rate outcome of Granger-causality is achieved by considering discrete jump component in the

return process and constructing time-varying and jump-sensitive parameters with multiplicative

structure.

For future research on this topic, the optimal size of the rolling window for out-of-the-sample

forecast needs to be considered in both univariate and multivariate cases. The dynamic forecast

in longer horizons needs to be implemented for further investigation in the multivariate case. For

future attempts, this author suggests using recursive approach to predict RV of the secondary

index for longer forecast horizons by switching primary and secondary indexes in the model. To

estimate jump components, VHAR(3) or VHAR(3)-η on BpV or MedRV can be used instead

of RV estimators.

Appendix

FIGURE A.1: Observed returns of FTSE100 and estimated returns by RV of thebenchmark models

54

APPENDIX . 55

FIGURE A.2: Observed returns of HSI and estimated returns by RV of the benchmarkmodels

APPENDIX . 56

FIGURE A.3: PDF of observed RV, estimated RV using benchmarks in the whole sampleof FTSE100.

FIGURE A.4: ACF of observed RV, estimated RV using benchmarks in the whole sampleof FTSE100.

APPENDIX . 57

FIGURE A.5: PDF of observed RV, estimated RV using benchmarks in the whole sampleof HSI.

FIGURE A.6: ACF of observed RV, estimated RV using benchmarks in the whole sampleof HSI.

APPENDIX . 58

FIGURE A.7: PDF of observed RV, RV estimated by benchmark and proposed models inthe whole sample of FTSE100.

FIGURE A.8: ACF of observed RV, RV estimated by benchmark and proposed models inthe whole sample of FTSE100.

APPENDIX . 59

FIGURE A.9: PDF of observed RV, RV estimated by benchmark and proposed models inthe whole sample of HSI.

FIGURE A.10: ACF of observed RV, RV estimated by benchmark and proposed models inthe whole sample of HSI.

APPENDIX . 60

FIGURE A.11: RV, BpV, MedRV and their respective jumps of FTSE100.

APPENDIX . 61

FIGURE A.12: RV, BpV, MedRV and their respective jumps of HSI.

APPENDIX . 62

TABLE A.1: Estimation of FTSE100 on the whole sample

+BpV +MedRV


β0

0.0440* 0.0432* 0.1221* 0297* 0.0225 0.0300 0.0044 -0.0185 -0.0158 0.0081

(4.1905) (4.0374) (4.5730) (2.1838) (1.5306) (1.5228) (0.2126) (-1.0335) (-0.8977) (0.3990)

β1

0.3360* 0.3688* 8791* 0.3920* 0.3817* 0.3811* 1.0937* 0.5805* 0.5869* 0.5810*

(6.7879) (6.4930) (21.337) (6.0775) (6.2166) (6.1967) (29.801) (8.5619) (7.1139) (7.0595)

β1J

-0.0990* -0.0440* -0.0369** -0.0364** -0.1679* -0900* -0.0923* -0.0902*

(-3.6803) (-2.2111) (-1.7005) (-1.6774) (-10.627) (-5.2326) (-3.7368) (-3.6667)

β5

0.4307* 0.4191* 0.4180* 0.4504* 0.4674* 0.3596* 0.3459* 0.4519*

(7.6365) (7.1764) (7.0968) (7.2064) (7.4190) (7.1491) (4.0887) (4.7619)

β5J

-0.0348 -0.0543 0.0070 -0.0538

(-0.8700) (-1.2121) (0.1667) (-1.0508)

β22

0.1783* 0.1802* 0.1670* 0.1613* 0.1278* 0.1360* 0.1375* -0.0142

(5.2908) (5.3952) (4.5879) (4.2447) (2.0882) (3.9650) (14.129) (-0.2000)

β22J

0.0479 0.1023*

(0.6221) (2.1953)

βJ

-0.0806*

(-2.2898)

AIC 996.65 981.14 1700.73 970.43 969.89 970.31 1331.58 868.84 870.70 860.36

BIC 1022.11 1012.96 1719.84 1002.25 1008.07 1014.85 1350.69 900.66 908.88 904.91

R2 0.6964 0.6976 0.6419 0.6984 0.6985 0.6986 0.6713 0.7054 0.7054 0.7063



information criteria (AIC), Bayesian information criteria (BIC) and regression R2. * and ** denote the significance at 5% and 10% level.

TABLE A.2: Estimation of HSI on the whole sample

+BpV +MedRV


β0

0.0467* 0.0457* 0.1427* 0.0297 0.0374 0.0120 0.1522* 0.0592* 0.0621* 0.0154

(2.0216) (1.9698) (5.3849) (1.2273) (1.4960) (0.5607) (4.9577) (2.5852) (3.3031) (0.8750)

β1

0.2174* 0.2442* 0.8586* 0.2476* 0.2575* 0.2617* 0.8412* 0.1968* 0.2082* 0.2248*

(5.8284) (6.000) (24.254) (6.0686) (5.9060) (6.0299) (16.690) (4.1000) (2.7946) (3.1617)

β1J

-0.0384 -0.0239 -0.0308* -0.0347* -0.0125 0.0080 0.0037 -0.0057

(-1.2387) (-1.8244) (-2.2482) (-2.4964) (-0.4112) (0.4124) (0.1156) (-0.1906)

β5

0.6357* 0.6334* 0.6448* 0.6215* 0.5514* 0.6315* 0.6164* 0.3868*

(9.5450) (9.4537) (9.1331) (7.9884) (6.7906) (10.120) (6.4008) (4.6885)

β5J

0.0179 0.0660 0.0063 0.0874*

(0.5234) (1.3866) (0.2540) (3.0453)

β22

0.0950 0.0930 0.0836 0.0847 0.1947* 0.1010 0.1006 0.3835*

(1.5200) (1.4715) (1.2981) (1.3173) (2.7372) (1.7119) (1.7109) (6.7996)

β22J

-0.0852* -0.1069*

(-1.8089) (-5.4821)

βJ

-0.0674*

(-2.3241)

AIC 508.57 498.19 1741.35 500.68 501.42 490.90 1758.18 508.65 510.38 452.92

BIC 534.02 530.01 1760.46 532.50 539.61 535.44 1777.29 540.47 548.56 497.47

R2 0.7373 0.7380 0.6579 0.7379 0.7380 0.7387 0.6566 0.7374 0.7374 0.7410



information criteria (AIC), Bayesian information criteria (BIC) and regression R2. * denotes the significance at 5% level.

APPENDIX . 63

TABLE A.3: Out-of-the-sample forecast performance FTSE100 on the whole sample



HAR(3) 0.1536 0.2547 0.0527 0.0831 0.0835 0.1329

HAR(3)-J 0.1541 0.2548 0.0500 0.0792 0.0827 0.1275

+BpV

ARJ 0.1541 0.2548 0.0500 0.0792 0.0827 0.1275

HARJ 0.1542 0.2548 0.0497 0.0797 0.0837 0.1279

HARJ-semiF 0.1545 0.2550 0.0474 0.0786 0.0865 0.1300

HARJ-F 0.1543 0.2555 0.0473 0.0804 0.0849 0.1290

+M

edRV ARJ 0.1703 0.2633 0.1583 0.2396 0.2000 0.2810

HARJ 0.1562 0.2524 0.0500 0.0741 0.0889 0.1375

HARJ-semiF 0.1566 0.2529 0.0478 0.0727 0.0947 0.1424

HARJ-F 0.1558 0.2522 0.0433 0.0691 0.1053 0.1527



TABLE A.4: Out-of-the-sample forecast performance FTSE100 on the pre-crisissub-sample



HAR(3) 0.1037 0.1664 0.0484 0.0644 0.0530 0.0718

HAR(3)-J 0.1046 0.1662 0.0460 0.0621 0.0506 0.0673

+BpV

ARJ 0.1395 0.1972 0.2038 0.2539 0.2823 0.3339

HARJ 0.1039 0.1656 0.0462 0.0625 0.0501 0.0671

HARJ-semiF 0.1034 0.1656 0.0422 0.0600 0.0507 0.0693

HARJ-F 0.1049 0.1666 0.0474 0.0644 0.0570 0.0753

+M

edRV ARJ 0.1195 0.1902 0.1304 0.1941 0.3282 0.4738

HARJ 0.1044 0.1690 0.0374 0.0547 0.0631 0.0874

HARJ-semiF 0.1054 0.1702 0.0440 0.0620 0.0808 0.1102

HARJ-F 0.1061 0.1709 0.0413 0.0596 0.0802 0.1106



TABLE A.5: Out-of-the-sample forecast performance FTSE100 on the post-crisissub-sample



HAR(3) 0.1138 0.1762 0.0439 0.0583 0.0568 0.0766

HAR(3)-J 0.1140 0.1763 0.0435 0.0578 0.0573 0.0783

+BpV

ARJ 0.1307 0.1940 0.1468 0.2038 0.1860 0.2402

HARJ 0.1139 0.1760 0.0437 0.0580 0.0588 0.0797

HARJ-semiF 0.1139 0.1761 0.0445 0.0588 0.0585 0.0794

HARJ-F 0.1142 0.1763 0.0451 0.0599 0.0563 0.0782

+M

edRV ARJ 0.1271 0.1905 0.1239 0.1811 0.1494 0.2002

HARJ 0.1157 0.1783 0.0395 0.0545 0.0584 0.0781

HARJ-semiF 0.1157 0.1783 0.0366 0.0524 0.0621 0.0819

HARJ-F 0.1159 0.1786 0.0375 0.0532 0.0626 0.0825



APPENDIX . 64

TABLE A.6: Out-of-the-sample forecast performance HSI on the whole sample



HAR(3) 0.1664 0.2816 0.0483 0.0848 0.1104 0.1932

HAR(3)-J 0.1671 0.2814 0.0451 0.0806 0.1078 0.1912+BpV

ARJ 0.1914 0.3187 0.2028 0.3458 0.2606 0.4442

HARJ 0.1667 0.2821 0.0442 0.0809 0.1120 0.1942

HARJ-semiF 0.1674 0.2830 0.0456 0.0893 0.1105 0.1907

HARJ-F 0.1674 0.2831 0.0456 0.0878 0.1056 0.1890

+M

edRV ARJ 0.1912 0.3213 0.2135 0.3571 0.2862 0.4582

HARJ 0.1664 0.2824 0.0503 0.0879 0.1127 0.2105

HARJ-semiF 0.1670 0.2840 0.0490 0.0903 0.1154 0.2282

HARJ-F 0.1652 0.2823 0.0539 0.1012 0.0989 0.2373



TABLE A.7: Out-of-the-sample forecast performance HSI on the pre-crisis sub-sample



HAR(3) 0.1441 0.2208 0.0571 0.0808 0.0345 0.0508

HAR(3)-J 0.1436 0.2201 0.0552 0.0779 0.0338 0.0482

+BpV

ARJ 0.1714 0.2528 0.1936 0.2777 0.2139 0.3040

HARJ 0.1435 0.2203 0.0555 0.0784 0.0336 0.0476

HARJ-semiF 0.1438 0.2207 0.0545 0.0769 0.0356 0.0513

HARJ-F 0.1438 0.2209 0.0557 0.0776 0.0346 0.0510

+M

edRV ARJ 0.1697 0.2540 0.2041 0.2904 0.2328 0.3232

HARJ 0.1442 0.2218 0.0564 0.0801 0.0359 0.0537

HARJ-semiF 0.1436 0.2225 0.0590 0.0834 0.0354 0.0570

HARJ-F 0.1442 0.2231 0.0597 0.0846 0.0356 0.0554



TABLE A.8: Out-of-the-sample forecast performance HSI on the post-crisis sub-sample



HAR(3) 0.1403 0.2205 0.0442 0.0698 0.0663 0.0985

HAR(3)-J 0.1406 0.2208 0.0438 0.0688 0.0663 0.0973

+BpV

ARJ 0.1573 0.2392 0.1588 0.2437 0.1805 0.2621

HARJ 0.1408 0.2212 0.0434 0.0689 0.0681 0.1002

HARJ-semiF 0.1406 0.2221 0.0451 0.0709 0.0684 0.1020

HARJ-F 0.1401 0.2219 0.0441 0.0702 0.0678 0.1010

+M

edRV ARJ 0.1555 0.2359 0.1576 0.2344 0.1912 0.2616

HARJ 0.1404 0.2203 0.0427 0.0667 0.0664 0.0976

HARJ-semiF 0.1404 0.2205 0.0433 0.0670 0.0672 0.1013

HARJ-F 0.1399 0.2207 0.0459 0.0692 0.0661 0.1017



APPENDIX . 65

TABLE A.9: Granger-causality test of FTSE100 with BpV as jump component


S&P500 HSI S&P500 HSI S&P500 HSI S&P500 HSI S&P500 HSI

u The whole sample:

F-value14.6669* 0.4398 12.1258* 0.2990 13.0058* 1.6061 11.1455* 1.4630 9.2339* 1.8877

(0.0000) (0.7245) (0.0000) (0.8788) (0.0000) (0.1696) (0.0000) (0.1982) (0.0000) (0.0788)


F-value10.6342* 0.2255 7.5575* 0.1547 7.4398* 0.1943 5.6880* 2.1902 5.2958* 0.6733

(0.0000) (0.8787) (0.0000) (0.9610) (0.0000) (0.9415) (0.0000) (0.0524) (0.0000) (0.6713)


F-value7.2284* 1.1533 6.7238* 0.8373 7.9112* 3.1612* 8.0661* 2.4631* 5.1520* 2.5961

(0.0001) (0.3260) (0.0000) (0.5012) (0.0000) (0.0131) (0.0000) (0.0307) (0.0000) (0.0162)



u The whole sample:

F-value16.5458* 0.6664 13.9354* 0.4120 15.3294* 2.0833 13.4893* 1.9710 11.1355* 2.1444*

(0.0000) (0.5725) (0.0000) (0.8001) (0.0000) (0.0801) (0.0000) (0.0795) (0.0000) (0.0452)


F-value12.4350* 0.2996 8.6617* 0.1924 8.3185* 0.2862 6.5913* 0.9940 5.2958* 0.6733

(0.0000) (0.8257) (0.0000) (0.9425) (0.0000) (0.8871) (0.0000) (0.4196) (0.0000) (0.6713)


F-value7.6927* 3.0048* 7.2117* 2.3662 9.1107* 3.5824* 7.6402* 2.9037* 5.7479* 2.5358*

(0.0000) (0.0291) (0.0000) (0.0505) (0.0000) (0.0063) (0.0000) (0.0126) (0.0000) (0.0186)

Note: Table provides Granger-causality F-statistic of FTSE100 with BpV as jump component. p-Values are reported in the

parentheses computed with Newey-West standard errors. * indicates whether corresponding index Granger-causes S&P500

(Based on 5% significance level).

TABLE A.10: Granger-causality test of FTSE100 with MedRV as jump component



u The whole sample:

F-value14.6669* 0.4398 12.1258* 0.2990 10.9283* 1.0386 8.8320* 1.0025 8.2427* 2.9033

(0.0000) (0.7245) (0.0000) (0.8788) (0.0000) (0.3855) (0.0000) (0.4143) (0.0000) (0.0079)


F-value10.6342* 0.2255 7.5575* 0.1547 6.7825* 0.6955 7.1310* 1.4460 5.9498* 1.6766

(0.0000) (0.8787) (0.0000) (0.9610) (0.0000) (0.5950) (0.0000) (0.2041) (0.0000) (0.1222)


F-value7.2284* 1.1533 6.7238* 0.8373 7.7064* 3.6656* 8.1273* 3.4074* 9.1700* 3.5302*

(0.0001) (0.3260) (0.0000) (0.5012) (0.0000) (0.0055) (0.0000) (0.0044) (0.0000) (0.0017)



u The whole sample:

F-value16.5458* 0.6664 13.9354* 0.4120 11.8093* 0.7949 9.8340* 0.8363 7.3297* 2.0051

(0.0000) (0.5725) (0.0000) (0.8001) (0.0000) (0.5283) (0.0000) (0.5236) (0.0000) (0.0613)


F-value12.4350* 0.2996 8.6617* 0.1924 7.5240* 0.6587 6.1896* 0.7020 5.2492* 0.5965

(0.0000) (0.8257) (0.0000) (0.9425) (0.0000) (0.6207) (0.0000) (0.6219) (0.0000) (0.7335)


F-value7.6927* 3.0048* 7.2117* 2.3662 10.3034* 3.2909* 8.3571* 5.6196* 7.1864* 6.3072*

(0.0000) (0.0291) (0.0000) (0.0505) (0.0000) (0.0105) (0.0000) (0.0000) (0.0000) (0.0000)

Note: Table provides Granger-causality F-statistic of FTSE100 with MedRV as jump component. p-Values are reported in

the parentheses computed with Newey-West standard errors. * indicates whether corresponding index Granger-causes S&P500

(Based on 5% significance level).

APPENDIX . 66

TABLE A.11: Granger-causality test of HSI with BpV as jump component


S&P500 FTSE100 S&P500 FTSE100 S&P500 FTSE100 S&P500 FTSE100 S&P500 FTSE100

u The whole sample:

F-value15.2027* 7.6197* 14.9924* 5.9652* 12.0710* 5.1495* 9.9756* 5.2995* 9.5673* 4.5333*

(0.0000) (0.0000) (0.0000) (0.0001) (0.0000) (0.0004) (0.0000) (0.0001) (0.0000) (0.0001)


F-value6.8180* 2.7489* 6.3895* 2.1963 5.5637* 2.1358 5.0002* 2.4877* 4.9875* 2.0663

(0.0001) (0.0412) (0.0000) (0.0667) (0.0002) (0.0736) (0.0001) (0.0292) (0.0000) (0.0537)


F-value11.2443* 8.3169* 10.1549* 6.7974* 9.3706* 6.3731* 8.6809* 6.7972* 8.4605* 6.1316*

(0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000)



u The whole sample:

F-value15.1483* 7.8019* 15.0350* 6.1387* 12.6619* 5.1830* 10.1311* 4.2668* 11.0086* 3.6999*

(0.0000) (0.0000) (0.0000) (0.0001) (0.0000) (0.0004) (0.0000) (0.0007) (0.0000) (0.0011)


F-value7.1087* 2.9501* 6.5663* 2.2894 6.3765* 2.2740 5.6139* 2.2788* 4.9875* 2.0663

(0.0001) (0.0313) (0.0000) (0.0573) (0.0000) (0.0587) (0.0000) (0.0441) (0.0000) (0.0537)


F-value12.0880* 8.8046* 11.7451* 7.1714* 10.0704* 7.2875* 9.0243* 7.7405* 9.1697* 5.9137*

(0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000)

Note: Table provides Granger-causality F-statistic of HSI with BpV as jump component. p-Values are reported in the parentheses com-

puted with Newey-West standard errors. * indicates whether corresponding index Granger-causes S&P500 (Based on 5% significance

level).

TABLE A.12: Granger-causality test of HSI with MedRV as jump component



u The whole sample:

F-value15.2027 7.6197 14.9924 5.9652 13.3303 8.2823 10.8482 6.7271 10.8396 5.9852

(0.0000) (0.0000) (0.0000) (0.0001) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000)


F-value6.8180 2.7489 6.3895 2.1963 5.3943 3.4470 4.7790 2.8331 4.0918 2.4190

(0.0001) (0.0412) (0.0000) (0.0667) (0.0002) (0.0080) (0.0002) (0.0146) (0.0004) (0.0244)


F-value11.2443 8.3169 10.1549 6.7974 13.8163 8.2809 10.4468 6.6157 8.2767 5.8082

(0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000)



u The whole sample:

F-value15.1483* 7.8019* 15.0350* 6.1387* 14.1735* 8.4678* 11.5775* 6.7306* 12.5588* 6.5384*

(0.0000) (0.0000) (0.0000) (0.0001) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000)


F-value7.1087* 2.9501* 6.5663* 2.2894 6.0943* 3.1963* 5.0121* 2.5525* 4.2415* 2.2622*

(0.0001) (0.0313) (0.0000) (0.0573) (0.0001) (0.0124) (0.0001) (0.0257) (0.0003) (0.0348)


F-value12.0880* 8.8046* 11.7451* 7.1714* 14.0257* 7.9443* 10.3342* 6.3270* 9.1186* 5.1248*

(0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000) (0.0000)

Note: Table provides Granger-causality F-statistic of HSI with MedRV as jump component. p-Values are reported in the paren-

theses computed with Newey-West standard errors. * indicates whether corresponding index Granger-causes S&P500 (Based on 5%

significance level).

APPENDIX . 67

TABLE A.13: In-Sample Estimation of S&P500 with Residual-Modification on the wholesample and the MedRV as jump component



u Intercept:

β0

0.0637* 0.0618* 0.0730* 0.0702* 0.0211 0.0125 –0.0070 -0.0220 0.0581* 0.0429**

(3.4809) (3.5930) (4.5625) (4.2805) (0.8941) (0.5208) (-0.2629) (-0.8884) (2.0104) (1.6824)


β1

0.3856* 0.3826* 0.5667* 0.5718* 0.5277* 0.5380* 0.4963* 0.4911* 0.4876* 0.4867*

(8.3826) (8.2814) (12.240) (11.505) (6.4119) (7.9001) (5.3251) (6.4029) (5.3406) (6.2638)

β1J

-0.0523* -0.0594* -0.0410 -0.0440** -0.0399 -0.0438**

(-2.4326) (-3.0777) (-1.4555) (-1.6591) (1.4669) (-1.6718)

β5

0.3947 0.3881* 0.3449* 0.3296* 0.3697* 0.3665* 0.4544* 0.5050* 0.5701* 0.5871*

(0.5385) (5.3828) (4.9697) (4.8258) (4.4976) (4.8479) (3.8738) (4.9220) (4.29622) (4.6818)

β5J

-0.0440* -0.0630 -0.1027* -0.1007*

(-1.2137) (-1.5547) (-2.3235) (-1.9553)

β22

0.1502* 0.1619* 0.1426* 0.1483* 0.1218* 0.1276* 0.1180** 0.1032* -0.1114 -0.0977

(3.3829) (3.8825) (3.5123) (3.7355) (2.7309) (3.0673) (2.5268) (2.4340) (-1.1978) (-0.9607)

β22J

0.1406* 0.1211**

(2.4973) (1.8863)

βJ

-0.3088* -0.3241*

(-9.0557) (-9.7620)


β1

0.1384 0.1523* 0.0765 0.0929** 0.1993** 0.1492* 0.2427* 0.1258** 0.2453* 0.0754

(1.4327) (2.7691) (0.8191) (1.7268) (1.8557) (2.2640) (2.0208) (1.7046) (2.0966) (1.0928)

β1J

-0.0390* -0.0067 -0.0600* 0.0017 -0.0626* 0.0206

(-2.1910) (-0.3032) (-2.3372) (0.0531) (-2.2847) (0.7384)

β5

79.641 -120.51 93.268 -21.733 66.935 -62.213 -53.300 -37.190 -89.113 149.55

(0.4524) (-1.1213) (0.6081) (-0.2317) (0.3946) (-0.6245) (-0.2556) (-0.2666) (-0.3904) (1.1757)

β5J

-96.727 1.8551 119.92 -95.467

(-1.4521) (0.0403) (1.0310) (-1.4847)

β22

-22.984 7.1163 -13.4187 -12.288 -24.812 -3.8100 -22.470 -6.9250 -0.6936 -43.055**

(-0.8007) (0.3692) (-0.5580) (-0.7897) (-0.8906) (-0.1907) (-0.7911) (-0.3248) (-0.0162) (-1.9367)

β22J

-17.694 22.230

(-0.4471) (1.2160)

βJ

-0.0497 0.0063

(-1.4661) (0.2158)

AIC 2763.44 2802.97 2499.95 2518.60 2709.57 2743.15 2702.13 2733.19 2680.55 2707.92

BIC 2807.98 2847.51 2557.22 2575.87 2766.84 2800.42 2772.13 2803.20 2763.28 2790.65

R2 0.7041 0.7014 0.7220 0.7208 0.7081 0.7058 0.7089 0.7068 0.7106 0.7088

MAE 0.2076 0.2087 0.2013 0.2017 0.2070 0.2082 0.2069 0.2082 0.2066 0.2076

RMSE 0.3334 0.3349 0.3232 0.3239 0.3312 0.3325 0.3307 0.3319 0.3297 0.3308




APPENDIX . 68

TABLE A.14: In-Sample Estimation of S&P500 with RV-Modification on the wholesample and the MedRV as jump component



u Intercept:

β0

0.0383* 0.0352 0.0506* 0.0598* -0.0365 -0.0010 -0.0414** -0.0421 0.0153 0.0095

(2.5533) (1.6000) (3.9225) (3.1979) (-1.4202) (-0.0389) (-1.7542) (-1.4223) (0.6000) (0.3322)


β1

0.2849* 0.3259* 0.4995* 0.5282* 0.3759* 0.4805* 0.3506* 0.4370* 0.3448* 0.4314*

(6.5797) (8.0074) (10.835) (12.487) (5.0866) (6.8545) (4.3716) (5.6242) (4.4896) (5.6319)

β1J

-0.0405** -0.0610* -0.0324 -0.0453** -0.0316 -0.0448**

(-1.7920) (-3.0653) (-1.1956) (-1.7626) (-1.2154) (-1.7849)

β5

0.3380* 0.4427* 0.2772* 0.3625* 0.3090* 0.4106* 0.4083* 0.5354* 0.5044* 0.6049*

(4.6492) (6.6974) (4.0825) (5.9917) (4.3768) (5.6324) (3.6423) (5.2853) (3.8357) (5.0789)

β5J

-0.0391* -0.0634 -0.0961* -0.1053*

(-0.9726) (-1.5890) (-2.1356) (-2.3245)

β22

0.2519* 0.1403* 0.2103* 0.1650* 0.2344* 0.1201* 0.2245* 0.1109* -0.0571 -0.0851

(3.5231) (3.0237) (3.2605) (3.7079) (3.0681) (2.6166) (2.8893) (2.3298) (-0.4313) (-0.8939)

β22J

0.1624* 0.1385*

(2.2873) (2.5840)

βJ

-0.3081* -0.3234*

(-9.1696) (-9.9508)


β1

0.1473* 0.1202* 0.0955** 0.0877* 0.3080* 0.0971** 0.3244* 0.0918* 0.3194* 0.0060

(3.0246) (2.9752) (1.8652) (2.3962) (3.7699) (1.7186) (3.2801) (2.3906) (3.2625) (0.1087)

β1J

-0.0595* 0.0092 -0.0654* 0.0047 -0.0645* 0.0434*

(-2.1636) (0.6479) (-1.9758) (0.1780) (-1.9846) (2.000)

β5

0.0954 -0.1154 0.1084 -0.0638 0.0711 -0.0848 -0.0152 -0.0637 0.0430 0.1180

(1.0010) (-1.4987) (1.2290) (-0.8704) (0.8089) (-1.1601) (-0.1232) (-0.9165) (0.2728) (1.0333)

β5J

0.0376 0.0279 0.0143 -0.0719

(0.7705) (0.4903) (0.2058) (-1.5837)

β22

-0.1475** 0.0483 -0.0996 -0.0191 -0.1714** 0.0180 -0.1617** -0.0302 -0.1253 -0.0652

(-1.6800) (0.8104) (-1.3088) (-0.3517) (-1.7526) (0.3279) (-1.6705) (-0.5841) (-0.8762) (-0.7460)

β22J

-0.0147 0.0182

(-0.1624) (0.4789)

βJ

-0.0489 0.0064

(-1.4257) (0.2215)

AIC 2758.46 2800.76 2496.40 2517.83 2679.41 2741.22 2678.28 2723.05 2651.80 2704.28

BIC 2803.01 2845.31 2553.67 2575.10 2736.68 2798.49 2748.28 2793.05 2734.53 2787.01

R2 0.7045 0.7016 0.7223 0.7209 0.7102 0.7060 0.7105 0.7075 0.7126 0.7090

MAE 0.2075 0.2088 0.2013 0.2017 0.2077 0.2081 0.2075 0.2079 0.2073 0.2078

RMSE 0.3332 0.3349 0.3230 0.3238 0.3300 0.3324 0.3298 0.3315 0.3286 0.3306




APPENDIX . 69

TABLE A.15: In-Sample Estimation of S&P500 with Residual-Modification on pre-crisissub-sample and the MedRV as jump component



u Intercept:

β0

0.0681* 0.0574* 0.0651* 0.0613* 0.0146 0.0040 0.0001 -0.0090 0.0098 -0.0051

(4.1779) (3.2989) (4.2829) (4.2805) (0.6266) (0.1619) (0.0034) (-0.3072) (0.3025) (-0.1541)


β1

0.3299* 0.3010* 0.4598* 0.4546* 0.4989* 0.4600* 0.4519* 0.4258* 0.4628* 0.4271*

(8.1658) (7.3236) (11.000) (11.777) (8.1254) (6.8862) (5.6558) (5.0933) (0.3025) (5.1896)

β1J

-0.0659* -0.0642* -0.0446 -0.0486 -0.0483 -0.0485

(-2.6789) (-2.3431) (-1.2355) (-1.2623) (-1.3233) (-1.2630)

β5

0.4372* 0.4688* 0.3878* 0.3985* 0.4079* 0.4446* 0.4703* 0.5085* 0.4650* 0.5043*

(5.9161) (6.0257) (5.5718) (5.6127) (5.4242) (5.5093) (4.0648) (4.4179) (3.6102) (4.0539)

β5J

-0.0462 -0.0355 -0.0432 -0.0348

(-0.7380) (-0.5662) (-0.5791) (-034615)

β22

0.1550* 0.1651* 0.1842* 0.1889* 0.1256* 0.1398* 0.1381* 0.1355* 0.1159 0.1313

(3.1828) (3.2183) (3.8137) (3.7406) (2.6667) (2.8825) (3.0419) (2.9140) (1.2252) (1.3170)

β22J

0.0028 -0.0033

(0.0447) (-0.0505)

βJ

-0.2805* -0.2966*

(-8.1541) (-8.2389)


β1

0.0741 -0.040 0.0081 -0.0829 0.1084 -0.0175 0.1751* 0.0280 0.1434** 0.0049

(1.0693) (-0.6650) (0.1233) (-1.4595) (1.3987) (-0.2212) (2.1020) (0.2969) (1.7445) (0.0521)

β1J

-0.0132 -0.0193 -0.0419* -0.0513 -0.0286 -0.0406

(-1.1000) (-0.6349) (-2.3807) (-1.2451) (-1.5264) (-0.9783)

β5

7.8717 74.184 24.765 106.50** 2.4643 70.738 -82.355 36.341 7.1121 101.02

(0.1336) (1.1135) (0.4193) (1.8042) (0.0411) (1.1211) (-1.1341) (0.4974) (0.0883) (1.2650)

β5J

-63.239* 34.417 -10.150 -28.828

(-2.1256) (1.3352) (-0.2015) (-0.6576)

β22

4.1034 -10.930 4.2026 -17.003** 3.6303 -11.767 7.3635 -11.152 -25.383 -28.733*

(0.3644) (-1.0165) (0.3925) (-1.7595) (0.3047) (-1.1954) (0.6019) (-1.1151) (-1.4045) (-2.1701)

β22J

39.600* 23.1468**

(3.0625) (1.7491)

βJ

-0.0068 -0.0046

(-0.1988) (-0.1518)

AIC 447.619 470.665 330.479 337.192 432.390 457.241 429.251 458.258 425.977 457.043

BIC 487.052 510.098 381.179 387.892 483.090 507.941 491.219 520.226 499.211 530.277

R 0.6615 0.6577 0.6807 0.6797 0.6646 0.6605 0.6658 0.6610 0.6669 0.6619

MAE 0.1809 0.1826 0.1761 0.1771 0.1810 0.1828 0.1807 0.1823 0.1806 0.1825

RMSE 0.2687 0.2702 0.2610 0.2614 0.2675 0.2691 0.2670 0.2689 0.2666 0.2686

Note: Coefficient estimation, Goodness-of-fit, in-sample forecast errors of models with residuals-modification on pre-crisis sub-sample

from Jan. 3, 2000 to Dec. 31, 2007 are reported. t-Statistics are stated in the parentheses computed with Newey-West standard errors.


APPENDIX . 70

TABLE A.16: In-Sample Estimation of S&P500 with RV-Modification on pre-crisissub-sample and the MedRV as jump component



u Intercept:

β0

0.0650* 0.0420* 0.0671* 0.0617* 0.0087 -0.0139 -0.0051 -0.0302 -0.0073 -0.0363

(4.0123) (2.3729) (4.4437) (3.6726) (0.4028) (-0.5148) (-0.1723) (-0.8162) (-0.2288) (-1.0254)


β1

0.2629* 0.2943* 0.4324* 0.4548* 0.4156* 0.4506* 0.3669* 0.4131* 0.3694* 0.4138*

(5.7907) (7.1956) (9.8273) (11.692) (5.7483) (6.8584) (4.2963) (4.9711) (4.3104) (5.0036)

β1J

-0.0640* -0.0644* -0.0448 -0.0493 -0.0456 -0.0496

(-2.2939) (-2.3676) (-1.1728) (-1.2839) (-1.1844) (-1.2950)

β5

0.3905* 0.4471* 0.3277* 0.3636* 0.3714* 0.4256* 0.5121* 0.5047* 0.4644* 0.4982*

(4.3197) (5.5957) (3.7323) (5.2092) (4.1130) (5.2221) (3.7961) (4.8669) (2.6751) (4.2042)

β5J

-0.0666 -0.0380 -0.0484 -0.0340

(-0.9610) (-0.6061) (-0.5284) (-0.4427)

β22

0.1619** 0.1845* 0.1777* 0.2278* 0.1338 0.1617* 0.1192 0.1550* 0.2336 0.1684

(1.8695) (3.3183) (2.1461) (4.2107) (1.4801) (2.9889) (1.3592) (3.0754) (1.1082) (1.5578)

β22J

-0.0590 -0.0088

(0.5008) (-0.1310)

βJ

-0.2790* -0.2950*

(-8.0899) (-8.2921)


β1

0.0796* 0.0136 0.0316 -0.0080 0.0939 0.0256 0.1064** 0.0235 0.1036** 0.0234

(2.1170) (0.4198) (0.9968) (-0.2206) (1.5676) (0.5235) (1.7189) (0.3760) (1.6764) (0.3744)

β1J

-0.0038 -0.0075 -0.0076 -0.0064 -0.0068 -0.0064

(-0.2275) (-0.4573) (-0.4720) (-0.2723) (0.4172) (-0.2712)

β5

0.0592 0.0431 0.0681 0.0819 0.0511 0.0373 -0.0312 0.0397 0.0505 0.0212

(0.9122) (0.4898) (1.0350) (1.0161) (0.7813) (0.4182) (-0.2301) (0.3567) (0.2785) (0.1473)

β5J

0.0351 -0.0036 0.0014 0.0071

(0.7389) (-0.1184) (0.0197) (0.1284)

β22

-0.0154 -0.0314 0.0025 -0.0780 -0.0134 -0.0358 0.0023 -0.0367 -0.1467 -0.0143

(-0.2110) (-0.3985) (0.0360) (-1.0969) (-0.1703) (-0.4767) (0.0280) (-0.4823) (-0.6578) (-0.1252)

β22J

0.0758* -0.0145

(0.6750) (-0.2562)

βJ

-0.0080 -0.0100

(-0.2339) (-0.3158)

AIC 447.404 473.114 331.187 341.346 433.114 459.827 434.588 462.540 437.283 466.426

BIC 486.838 512.548 381.888 392.046 483.814 510.527 496.555 524.507 510.517 539.660

R2 0.6615 0.6573 0.6806 0.6791 0.6645 0.6601 0.6649 0.6603 0.6651 0.6603

MAE 0.1809 0.1825 0.1762 0.1771 0.1811 0.1828 0.1809 0.1826 0.1809 0.1826

RMSE 0.2687 0.2704 0.2610 0.2617 0.2675 0.2693 0.2674 0.2692 0.2673 0.2692

Note: Coefficient estimation, Goodness-of-fit, in-sample forecast errors of models with RV-modification on pre-crisis sub-sample from

Jan. 3, 2000 to Dec. 31, 2007 are reported. t-Statistics are stated in the parentheses computed with Newey-West standard errors. *


APPENDIX . 71

TABLE A.17: In-Sample Estimation of S&P500 with residuals-Modification on post-crisissub-sample and the MedRV as jump component



u Intercept:

β0

0.0632* 0.0670* 0.0819* 0.0783* 0.0207 0.0190 0.0177 -0.0170 0.1131* 0.0752*

(2.5079) (2.9911) (3.9375) (3.6934) (0.6530) (0.6271) (0.5364) (-0.5556) (2.7188) (2.1860)


β1

0.4008* 0.4251* 0.6018* 0.6248* 0.5281* 0.5872* 0.5304* 0.5435* 0.5172* 0.5384*

(5.7257) (6.3638) (8.6840) (8.2755) (4.6652) (6.4670) (4.0030) (5.6147) (3.9663) (5.4329)

β1J

-0.0484* -0.0586* -0.0486 -0.0457 -0.0483 -0.0465**

(-1.9755) (-2.7642) (-1.4639) (-1.5979) (-1.5094) (-1.6489)

β5

0.4045* 0.3438* 0.3583* 0.2872* 0.3844* 0.3251* 0.3376* 0.4687* 0.4714* 0.5615*

(3.7247) (3.4107) (3.4854) (2.8952) (3.2521) (3.0612) (2.3842) (3.5806) (3.1787) (3.5493)

β5J

0.0217 -0.0599 -0.0462 -0.1129**

(0.3848) (-1.2909) (-0.6958) (-1.8151)

β22

0.1278* 0.1592* 0.1042** 0.1310* 0.1059** 0.1142* 0.1321* 0.0832 -0.1576 -0.1778

(1.9662) (2.8479) (1.7691) (2.4486) (1.6495) (2.0000) (2.2017) (1.3867) (-1.4840) (-1.2619)

β22J

0.1775* 0.1762**

(2.2669) (1.9131)

βJ

-0.3365* -0.3508*

(-6.6766) (-6.6064)


β1

0.1161 0.3001* 0.0206 0.2065* 0.1873 0.3065* 0.4586* 0.2549* 0.4905* 0.2057*

(2.1170) (3.6553) (0.1034) (2.4790) (0.7721) (3.0197) (2.6072) (2.6197) (2.8206) (2.1953)

β1J

-0.1082 -0.0153 -0.4253* 0.0039 -0.4627* 0.0184

(-1.3836) (-0.5083) (-2.8952) (0.1124) (-2.8094) (0.6093)

β5

173.53 -157.92 155.67 -100.45 165.83 -110.64 -321.63 -76.387 -438.58** 28.461

(0.7703) (-2.0002) (0.8268) (-1.3953) (0.7611) (-1.4517) (-1.6318) (-0.7837) (-1.7533) (0.3133)

β5J

715.13** -6.2471 846.76** -54.213

(1.9322) (-0.2331) (1.7929) (-1.4035)

β22

-24.545 13.886 -9.7797 2.6841 -29.407 6.3430 0.9564 4.3068 53.742 -16.803

(-0.7745) (0.9944) (-0.3755) (0.2344) (-0.9898) (0.4216) (0.0385) (0.2711) (1.0693) (-1.0120)

β22J

-68.010 8.1170

(-0.7365) (0.6965)

βJ

-0.0584 0.065

(-0.7654) (1.3374)

AIC 1983.61 2002.36 1838.72 1845.57 1960.61 1965.78 1917.17 1961.84 1891.48 1945.13

BIC 2023.48 2042.24 1889.99 1896.84 2011.88 2017.05 1979.84 2024.51 1965.54 2019.19

R2 0.7242 0.7219 0.7423 0.7415 0.7276 0.7270 0.7334 0.7279 0.7370 0.7305

MAE 0.3785 0.3801 0.3659 0.3665 0.3762 0.3766 0.3722 0.3760 0.3697 0.3742

RMSE 0.2291 0.2295 0.2206 0.2206 0.2283 0.2284 0.2276 0.2285 0.2268 0.2275

Note: Coefficient estimation, Goodness-of-fit, in-sample forecast errors of models with residuals-modification on post-crisis sub-sample

from Jan. 1, 2008 to Jul. 1, 2016 are reported. t-Statistics are stated in the parentheses computed with Newey-West standard errors.


APPENDIX . 72

TABLE A.18: In-Sample Estimation of S&P500 with RV-Modification on post-crisissub-sample and the MedRV as jump component



u Intercept:

β0

0.0044 0.0295 0.0242 0.0648* -0.0722* -0.0072 -0.0529 -0.0577 0.0021 0.0021

(0.2037) (0.9641) (1.2222) (2.5714) (-2.0867) (-0.1983) (-1.4860) (-1.2937) (0.0504) (0.0494)


β1

0.2741* 0.3178* 0.5341* 0.5502* 0.3693* 0.4745* 0.3582* 0.4354* 0.3522* 0.4273*

(4.4642) (5.6548) (7.3873) (8.4692) (4.3601) (5.1464) (3.9536) (4.4383) (4.1387) (4.3736)

β1J

-0.0327 -0.0612* -0.0299 -0.0474** -0.0297 -0.0474**

(-1.3625) (-2.6725) (-1.0418) (-1.6809) (-1.0761) (-1.7299)

β5

0.1989 0.4622* 0.1367 0.3795* 0.1761 0.4255* 0.2425** 0.5514* 0.3681* 0.6166*

(1.6397) (5.0847) (1.2518) (4.4752) (1.5488) (4.2764) (1.6576) (4.1057) (2.3491) (4.0039)

β5J

-0.0231* -0.0660 -0.0908** -0.1145*

(-0.4724) (-1.3608) (-1.8455) (-2.1048)

β22

0.3293* 0.0997 0.2387* 0.1191** 0.3873* 0.0777 0.3422* 0.0594 -0.0471 -0.2285

(2.7580) (1.4705) (2.1389) (1.8323) (2.9186) (1.1580) (2.7289) (0.8204) (-0.2861) (-1.4317)

β22J

0.2031* 0.2343*

(2.3317) (2.2041)

βJ

-0.3361* -0.3514*

(-6.6819) (-6.8902)


β1

0.1951* 0.2088* 0.0968 0.1426* 0.4229* 0.1854** 0.4688* 0.0627 0.4629* 0.0590

(2.1510) (3.2473) (1.0342) (2.5239) (3.3013) (1.9272) (3.0032) (0.7741) (3.0137) (0.7347)

β1J

-0.1204* 0.0095 -0.1423* 0.0512* -0.1434* 0.0515*

(-12.338) (0.4524) (-2.1892) (2.0562) (-2.3317) (2.2103)

β5

0.3555 -0.2163* 0.3807** -0.1574 0.2847 -0.1722 0.1068 -0.0054 0.1573 0.1451

(1.4950) (-1.8100) (1.7991) (-1.3795) (1.3486) (-1.5584) (0.5042) (-0.0404) (0.6426) (0.9242)

β5J

0.1075 -0.0453 0.0746 -0.1091**

(0.8187) (-0.9042) (0.4574) (-1.8554)

β22

-0.3253** 0.0986 -0.2238 0.0235 -0.4532** 0.0584 -0.3778* 0.0367 -0.2443 -0.0148

(-1.7285) (1.1519) (-1.3019) (0.2990) (-2.0507) (0.7327) (-2.0724) (0.4576) (-1.147) (-0.1132)

β22J

-0.0394 0.0011

(-0.2037) (0.0209)

βJ

-0.0661 0.0645

(-0.8335) (1.3136)

AIC 1976.39 2003.55 1830.88 1846.33 1922.76 1967.22 1917.17 1961.84 1899.70 1937.56

BIC 2016.27 2043.43 1882.15 1897.60 1974.03 2018.49 1979.84 2024.51 1973.76 2011.62

R2 0.7251 0.7217 0.7432 0.7414 0.7322 0.7268 0.7334 0.7279 0.7360 0.7314

MAE 0.3779 0.3802 0.3653 0.3666 0.3730 0.3768 0.3722 0.3760 0.3704 0.3736

RMSE 0.2284 0.2298 0.2204 0.2208 0.2292 0.2282 0.2276 0.2285 0.2286 0.2280

Note: Coefficient estimation, Goodness-of-fit, in-sample forecast errors of models with RV-modification on post-crisis sub-sample from

Jan. 1, 2008 to Jul. 1, 2016 are reported. t-Statistics are stated in the parentheses computed with Newey-West standard errors. *


APPENDIX . 73

TABLE A.19: One-day-ahead out-of-the-sample forecast accuracy of S&P500 onpre-crisis sub-sample with FTSE100 as the secondary index.

+BpVs +MedRVs


VHAR(3)-η

0.000 0.000 -0.001 0.000 0.000 0.000

(0.881) (0.975) (0.704) (0.984) (0.672) (0.709)

VHAR(3)-J-η

-0.001 -0.002 -0.002 -0.002 -0.001 -0.001

(0.090) (0.128) (0.077) (0.092) (0.139) (0.188)

+BpVt

VHARJ-η0.000 -0.001 0.000 0.000 0.000

(0.557) (0.183) (0.901) (0.888) (0.894)

VHARJ-semiF-η0.000 0.000 0.000 0.000

(0.207) (0.966) (0.778) (0.793)

VHARJ-F-η0.001 0.001 0.001

(0.724) (0.561) (0.585)

+M

edRVt

VHARJ-η0.000 0.000

(0.373) (0.464)

VHARJ-semiF-η0.000

(0.976)

+BpVs +MedRVs


VHAR(3)

0.000 0.000 0.000 0.000 0.000 0.000

(0.615) (0.947) (0.986) (0.731) (0.671) (0.959)

VHAR(3)-J

-0.001 -0.002 -0.002 -0.002t -0.001 -0.002

(0.076) (0.086) (0.083) (0.031) (0.115) (0.080)

+BpVt

VHARJ0.000 0.000 -0.001 0.000 0.000

(0.278) (0.408) (0.347) (0.822) (0.611)

VHARJ-semiF0.000 0.000 0.000 0.000

(0.897) (0.735) (0.843) (0.920)

VHARJ-F0.000 0.000 0.000

(0.807) (0.813) (0.962)

+M

edRVt

VHARJ0.001 0.000

(0.153) (0.706)

VHARJ-semiF0.000

(0.397)

Note: One-day-ahead out-of-the-sample forecast accuracy check of the models in pre-crisis sub-sample from Jan. 3, 2000




APPENDIX . 74

TABLE A.20: One-day-ahead out-of-the-sample forecast accuracy of S&P500 onpost-crisis sub-sample with FTSE100 as the secondary index.

+BpVs +MedRVs


VHAR(3)-η

-0.004 -0.004 -0.005 -0.011 -0.013 -0.016

(0.512) (0.546) (0.501) (0.395) (0.383) (0.369)

VHAR(3)-J-η

-0.009 -0.009 -0.010 -0.015 -0.018 -0.021

(0.205) (0.240) (0.242) (0.247) (0.254) (0.261)

+BpVt

VHARJ-η0.000 -0.001 -0.007 -0.009 -0.012

(0.956) (0.483) (0.304) (0.305) (0.305)

VHARJ-semiF-η-0.001 -0.007 -0.009 -0.012

(0.290) (0.262) (0.271) (0.280)

VHARJ-F-η-0.005 -0.007 -0.010

(0.268) (0.274) (0.282)

+M

edRVt

VHARJ-η-0.002 -0.005

(0.338) (0.316)

VHARJ-semiF-η-0.003

(0.313)

+BpVs +MedRVs


VHAR(3)

-0.004 -0.004 -0.005 -0.013 -0.016 -0.019

(0.506) (0.562) (0.528) (0.347) (0.344) (0.338)

VHAR(3)-J

-0.008 -0.008 -0.009 -0.018 -0.021 -0.023

(0.194) (0.239) (0.256) (0.229) (0.244) (0.249)

+BpVt

VHARJ0.000 -0.001 -0.009 -0.012 -0.015

(0.824) (0.617) (0.264) (0.281) (0.283)

VHARJ-semiF-0.001 -0.010 -0.013 -0.015

(0.401) (0.225) (0.251) (0.258)

VHARJ-F-0.008 -0.011 -0.014

(0.204) (0.238) (247)

+M

edRVt

VHARJ-0.003 -0.005

(0.335) (0.317)

VHARJ-semiF-0.002

(0.299)

Note: One-day-ahead out-of-the-sample forecast accuracy check of the models in post-crisis sub-sample from Jan. 1,




APPENDIX . 75

TABLE A.21: One-day-ahead out-of-the-sample forecast accuracy of S&P500 onpre-crisis sub-sample with HSI as the secondary index.

+BpVs +MedRVs


VHAR(3)-η

0.001 0.000 0.000 0.000 -0.001 0.000

(0.447) (0.682) (0.753) (0.819) (0.530) (0.898)

VHAR(3)-J-η

-0.001 -0.001 -0.001 -0.002t -0.002t -0.002

(0.110) (0.152) (0.108) (0.046) (0.021) (0.057)

+BpVt

VHARJ-η0.000 0.000 -0.001 -0.001 -0.001

(0.531) (0.398) (0.210) (0.152) (0.379)

VHARJ-semiF-η0.000 -0.001 -0.001 0.000

(0.625) (0.474) (0.328) (0.608)

VHARJ-F-η0.000 -0.001 0.000

(0.560) (0.385) (0.683)

+M

edRVt

VHARJ-η0.000 0.000

(0.566) (0.916)

VHARJ-semiF-η0.000

(0.310)

+BpVs +MedRVs


VHAR(3)

0.001 0.001 0.001 0.000 0.000 0.001

(0.433) (0.480) (0.316) (0.637) (0.891) (0.431)

VHAR(3)-J

-0.001 -0.001 -0.001 -0.002t -0.001 -0.001

(0.140) (0.364) (0.425) (0.033) (0.091) (0.348)

+BpVt

VHARJ0.000 0.000 -0.001 -0.001 0.000

(0.793) (0.613) (0.125) (0.383) (0.994)

VHARJ-semiF0.000 -0.001 -0.001 0.000

(0.758) (0.173) (0.388) (0.916)

VHARJ-F-0.001 -0.001 0.000

(0.080) (0.276) (0.765)

+M

edRVt

VHARJ0.000 0.001

(0.212) (0.081)

VHARJ-semiF0.001

(0.243)

Note: One-day-ahead out-of-the-sample forecast accuracy check of the models in pre-crisis sub-sample from Jan. 3, 2000




APPENDIX . 76

TABLE A.22: One-day-ahead out-of-the-sample forecast accuracy of S&P500 onpost-crisis sub-sample with HSI as the secondary index.

+BpVs +MedRVs


VHAR(3)-η

-0.007 -0.006 -0.007 -0.016 -0.019 -0.022

(0.461) (0.507) (0.698) (0.341) (0.332) (0.332)

VHAR(3)-J-η

-0.012 -0.012 -0.013 -0.022 -0.025 -0.028

(0.203) (0.235) (0.245) (0.217) (0.227) (0.235)

+BpVt

VHARJ-η0.000 0.000 -0.020 -0.013 -0.015

(0.535) (0.819) (0.239) (0.251) (0.263)

VHARJ-semiF-η-0.001 -0.010 -0.013 -0.016

(0.482) (0.201) (0.222) (0.238)

VHARJ-F-η-0.009 -0.013 -0.015

(0.185) (0.211) (0.229)

+M

edRVt

VHARJ-η-0.003 -0.006

(0.291) (0.308)

VHARJ-semiF-η-0.002

(0.341)

+BpVs +MedRVs


VHAR(3)

-0.007 -0.006 0.007 -0.015 -0.018 -0.021

(0.438) (0.507) (0.467) (0.344) (0.350) (0.335)

VHAR(3)-J

-0.013 -0.012 -0.013 -0.021 -0.024 -0.027

(0.184) (0.220) (0.213) (0.212) (0.235) (0.233)

+BpVt

VHARJ0.001 0.000 -0.009 -0.012 -0.014

(0.121) (0.847) (0.257) (0.291) (0.278)

VHARJ-semiF-0.001 -0.010 -0.012 -0.015

(0.212) (0.209) (0.252) (0.246)

VHARJ-F-0.008 -0.011 -0.014

(0.219) (0.263) (0.254)

+M

edRVt

VHARJ-0.003 -0.006

(0.386) (0.312)

VHARJ-semiF-0.003

(0.228)

Note: One-day-ahead out-of-the-sample forecast accuracy check of the models in post-crisis sub-sample from Jan. 1,




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