The generalized Additive Nonparametric GARCH Model --With application to the Chinese stock market

Ai Jun Hou

Department of Economics

School of Economics and Management

Lund University

Lund Fudan Economic Forum , Nov 2007

1. Introduction1. Introduction

2. The basic Model2. The basic Model

3. Simulation experiments 3. Simulation experiments

4. Real Examples4. Real Examples

5. Conclusion 5. Conclusion

1. Introduction Stationary Times series showing Volatility

asymmetries and clustering Provides the impetus grounded under GARCH

family models ARCH (Engle,1982), standard GARCH (Bollerslev, 1986),

EGARCH and Threshold GARCH models Volatility parametrically depends on Lagged

volatility and innovations Nonparametric GARCH model (BÜlman and McNeil, 2002) (NP Model)

Local polynomial smoothing iteration (procedures )algorithm

Introduction (con.)

Attractive Iterative method: no requirement of specification of the functional form of volatility and the innovation distributions

Problem of -the curse of dimensionality (Härdle, 2004)

To apply an generalized additive model (Hastie and Tibshirani, 1986) lower dimensional smoothing the backfitting algorithm

Introduction (Con.) We apply the iterative algorithm of NP model to the

Generalized Additive GARCH Model (NP GAM model), use it to adjust the volatility estimated from the GARCH model

Results from the simulation and real data: NP and NP GAM model outperform parametric GARCH models if

market got asymmetric information NP GAM model dominates NP model in most cases there is a moderate improvement in In-sample forecast, and a more clear

improvement in Out-of-sample forecast

Why Chinese stock market? The Chinese market is still young and yet develops

quickly. SHSE and SZSE were established in 1990 and 1991 Mid October, 856 companies (SHSE), floated market value RMB 6 trillion,

644 companies (SZSE), RMB 2.7 trillion

Introduction (con.) However, Studies on the Chinese stock market are still

limited (Tang and Chen, 2002), to our knowledge, no one has applied the NP GAM model to the Chinese stock market, besides:

-- Zou and Wang (2007) examine currency market, , Lu ( 2004) examines the Chinese stock market with NP model , but not NP GAM model

A distinct asymmetric effect exists in the Chinese stock market (Wang, et al.,2005)

--it is attractive to fit the Chinese data with the new iterative method

Our contributions: --apply a newly proposed method to examine a new market --fit the models with residuals under both t and normal distributions --show that the nonparametric model could be an effective auxiliary

tool to test if the parametric model is an appropriate one which fits the volatility well

2. The basic model

(2.1)

The Basic Model (con.)

2.1 Estimation algorithm

and lagged

Estimation algorithm (con.) and

For computational convenience, we perform the NP GAM (1,1)and compare our results with GARCH(1,1) , EGARCH(1,1), and TGARCH (1,1) models, and also with the NP (1,1) model

The parametric models are simulated and estimated in Matlab with a maximum likelihood method, while the nonparametric procedures and backfitting algorithm are performed in S-PLUS student version

3. Simulation experiments We consider three designed process: Process A:

Process B:

Process C:

For both processes, we work with n=1000 observations, generate 50 realizations, and the maximum iterating is M=8, and a final smooth is performed by averaging over the last four (K=5) iterations. We fit the processed with both T and normal distributed innovations.

The performance of each models are evaluated by MSE and MAE, which are the average of the volatility estimation errors of each realization, the first 20 points are omitted from the calculation

Simulation experiments (con.)

The above figure shows t´the volatility surface of process A and B, under the asymmetric information effects, there is a significant brokensegment on the volatility surface, the results from our simulations show that the nonparametric models smooth surface quite well and outperformthe parametric GARCH models

Simulation results : process AModels

NP Regression GAM NP Regression NP Regression GAM NP Regression

Normal Student-t

MSE Std. MAE Std. MSE Std. MAE Std. MSE Std. MAE Std. MSE Std. MAE Std.

GARCH 0.231 0.103 0.361 0.093 0.231 0.103 0.361 0.093 0.227 0.103 0.358 0.094 0.227 0.103 0.358 0.094

Iteration 1 0.313 0.102 0.426 0.079 0.321 0.112 0.429 0.089 0.313 0.102 0.426 0.079 0.321 0.112 0.429 0.089

Iteration 2 0.346 0.115 0.449 0.082 0.345 0.138 0.443 0.096 0.346 0.115 0.449 0.082 0.345 0.138 0.443 0.096

Iteration 3 0.372 0.130 0.468 0.087 0.375 0.150 0.466 0.101 0.372 0.130 0.468 0.087 0.375 0.150 0.466 0.101

Iteration 4 0.371 0.136 0.470 0.093 0.393 0.169 0.476 0.106 0.371 0.136 0.470 0.093 0.393 0.168 0.476 0.106

Iteration 5 0.368 0.131 0.468 0.089 0.396 0.157 0.482 0.101 0.368 0.130 0.468 0.089 0.396 0.157 0.482 0.101

Iteration 6 0.377 0.140 0.473 0.093 0.400 0.166 0.483 0.105 0.377 0.140 0.473 0.093 0.399 0.165 0.482 0.105

Iteration 7 0.386 0.153 0.479 0.100 0.401 0.154 0.484 0.100 0.385 0.152 0.479 0.100 0.401 0.155 0.484 0.101

Iteration 8 0.373 0.140 0.471 0.094 0.404 0.159 0.487 0.102 0.373 0.140 0.471 0.094 0.406 0.158 0.488 0.100

Final 0.363 0.138 0.465 0.093 0.385 0.160 0.475 0.104 0.363 0.138 0.465 0.093 0.385 0.161 0.475 0.104

EGARCH 0.254 0.115 0.375 0.093 0.254 0.115 0.375 0.093 0.281 0.143 0.394 0.108 0.281 0.143 0.394 0.108

GJR 0.253 0.105 0.376 0.091 0.253 0.105 0.376 0.091 0.250 0.105 0.374 0.091 0.250 0.105 0.374 0.091

When there is no asymmetric effect, the standard GARCH model dominates EGARCH, TGARCH, and nonparametric models. The estimations with t distributed errors perform slightly better than the normal fitting. However, nonparametric models provide the nearly identical results, which disregarding the innovation distribution.

Simulation results: process BModels

NP Re g r e s s io n GA M NP Re g r e s s io n NP Re g r e s s io n GA M NP Re g r e s s io n

No r m al Stu d e n t- t

M SE Std. M A E Std. M SE Std. M A E Std. M SE Std. M A E Std. M SE Std. M A E Std.

GA RC H 0.555 0.047 0.615 0.020 0.555 0.047 0.615 0.020 0.555 0.046 0.616 0.020 0.555 0.046 0.616 0.020

Iteration 1 0.347 0.043 0.460 0.027 0.286 0.048 0.393 0.031 0.347 0.043 0.460 0.027 0.286 0.048 0.393 0.031

Iteration 2 0.281 0.045 0.414 0.032 0.237 0.053 0.349 0.041 0.281 0.045 0.414 0.032 0.237 0.053 0.349 0.041

Iteration 3 0.267 0.045 0.406 0.034 0.225 0.056 0.339 0.044 0.267 0.045 0.406 0.034 0.225 0.056 0.339 0.043

Iteration 4 0.262 0.044 0.405 0.033 0.221 0.055 0.338 0.044 0.262 0.044 0.405 0.033 0.221 0.055 0.338 0.044

Iteration 5 0.264 0.046 0.406 0.035 0.221 0.058 0.337 0.045 0.264 0.046 0.406 0.035 0.221 0.057 0.337 0.045

Iteration 6 0.263 0.047 0.406 0.037 0.222 0.058 0.339 0.045 0.263 0.047 0.406 0.037 0.222 0.058 0.339 0.045

Iteration 7 0.262 0.048 0.405 0.037 0.224 0.061 0.341 0.049 0.262 0.048 0.405 0.037 0.224 0.061 0.341 0.049

Iteration 8 0.263 0.048 0.406 0.037 0.224 0.060 0.340 0.047 0.263 0.048 0.406 0.037 0.224 0.060 0.340 0.047

Fin al 0.261 0.048 0.405 0.037 0.221 0.058 0.339 0.046 0.261 0.048 0.405 0.037 0.221 0.058 0.339 0.046

EGA RC H 0.300 0.045 0.430 0.030 0.300 0.045 0.430 0.030 0.298 0.038 0.427 0.023 0.298 0.038 0.427 0.023

GJR 0.390 0.042 0.507 0.027 0.390 0.042 0.507 0.027 0.390 0.042 0.507 0.027 0.390 0.042 0.507 0.027

0 2 0 4 0 6 0 8 0 1 0 0

34

56

true Volatility and estim ated volatility f rom nonparam etric G arch

ra ins uns po ts

Simulation results : process B iteration1 iteration 2 iteration 3

iteration 4 iteration 5 iteration 6

iteration 7 iteration 8 final smooth

Simulation results: process CModels

NP Regression GAM NP Regression NP Regression GAM NP Regression

Normal Student-t

MSE Std. MAE Std. MSE Std. MAE Std. MSE Std. MAE Std. MSE Std. MAE Std.

GARCH 0.290 0.067 0.399 0.037 0.290 0.067 0.399 0.037 0.322 0.069 0.415 0.037 0.322 0.069 0.415 0.037

Iteration 1 0.261 0.032 0.399 0.031 0.069 0.012 0.188 0.023 0.262 0.032 0.402 0.031 0.068 0.012 0.188 0.022

Iteration 2 0.204 0.018 0.329 0.024 0.056 0.011 0.170 0.025 0.203 0.018 0.329 0.024 0.056 0.011 0.171 0.025

Iteration 3 0.202 0.027 0.329 0.028 0.052 0.011 0.164 0.024 0.201 0.022 0.328 0.025 0.052 0.011 0.164 0.024

Iteration 4 0.201 0.024 0.326 0.028 0.051 0.010 0.161 0.024 0.203 0.028 0.328 0.031 0.050 0.011 0.160 0.024

Iteration 5 0.202 0.022 0.325 0.026 0.050 0.011 0.160 0.024 0.202 0.022 0.326 0.025 0.050 0.011 0.160 0.024

Iteration 6 0.203 0.026 0.327 0.029 0.050 0.010 0.160 0.024 0.203 0.025 0.327 0.028 0.050 0.011 0.160 0.024

Iteration 7 0.204 0.025 0.326 0.028 0.050 0.011 0.159 0.024 0.203 0.025 0.326 0.028 0.050 0.011 0.159 0.024

Iteration 8 0.204 0.025 0.327 0.028 0.050 0.011 0.159 0.024 0.204 0.025 0.327 0.028 0.050 0.011 0.159 0.024

Final 0.201 0.021 0.325 0.025 0.050 0.011 0.160 0.024 0.201 0.021 0.325 0.025 0.050 0.011 0.160 0.024

EGARCH 0.250 0.104 0.345 0.047 0.250 0.104 0.345 0.047 0.300 0.161 0.364 0.065 0.300 0.161 0.364 0.065

GJR 0.293 0.065 0.399 0.036 0.293 0.065 0.399 0.036 0.319 0.065 0.413 0.036 0.319 0.065 0.413 0.036

4. Application to China Stock market

• Widely accepted SHCI and SZCI • the daily price of SHCI and SZCI

from 2nd January, 1997 to 31st August,  2007.

• Are Converted to daily log return and multiply 100

• In-sample group (from 2nd January, 1997 to 31 August, 2006) and an out-of-sample group (from 1st September 2006 to 31 August 2007)

• 2379 observations for in-sample and 243 observations for out-of -sample forecast

• Realized volatility, extracted from high frequency data (5 minutes) for true volatility proxy for out-of-sample forecast

0

500

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97-01-0

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CI

SHCI SZCI

Performance evaluation criteria

For in-sample forecast we use three indicators: Mean Squared Error between

squared innovation and squared volatility (LL2)

MSE MAE

we use :

For out-of-sample forecast, we use two criteria:

MSE MAE

We use both

As the true volatility proxy

and realized volatility

as true volatility

Data description

8,0728,747Kurtosis

0,0610,008Skewness

0,0010,001JB Test

-46,644-48,491DF Test

1,6521,514Std.

0,0110,025Mean

SZCISHCI

In-Sample Result

Fit with AR(0)-GARCH (1,1)

Table 4.2 GARCH, EGARCH, GJR Result during In-Sample Period

Shanghai Composite Index

GARCH EGA RCH GJR

Normal T Normal T Normal T

μ 0,000 0,014 -0,011 0,009 -0,018 0,004

ω 0,095 0,093 0,038 0,027 0,082 0,090

α 0,139 0,117 0,242 0,239 0,098 0,077

β 0,829 0,848 0,964 0,957 0,845 0,844

DoF 4,638 4.87 4,725

Leverage -0,036 -0,063 0,060 0,095

Q(20) 24,32(0,23) 24,43(0,22) 24,91(0,20) 25,48(0,18) 25,12(0,20) 25,53(0,18)

Shenzhen Component Index

GARCH EGA RCH GJR

Normal T Normal T Normal T

μ -0,021 -0,029 0,040 -0,034 -0,034 -0,037

ω 0,067 0,097 0,080 0,027 0,062 0,093

α 0,100 0,102 0,276 0,216 0,082 0,080

β 0,879 0,865 0,932 0,967 0,882 0,863

DoF 4,848 4,952 4,882

Leverage 0,028 -0,035 0,036 0,055

Q(20) 27,29 (0.13) 27,17(0.13) 26,10(0.16) 27,55(0.12) 28,28(0.10) 28,97(0.09)

all models appear to be adequate in describing the linear dependence in the return and volatility series. The leverage parameters from EGARCH and GJR indicates moderate asymmetric characteristics.

In Sample forecast

Iteration 1 Iteration 2 Iteration 3

Iteration 4 Iteration 5 Iteration 6

Iteration 7 Iteration 8 Final Sm ooth

In Sample forecast (con.)

In-sample forecast resultsModel Distribution

SHCI SZCI

LL2 MSE MA E LL2 MSE MA E

Para

metr

ic

GARCHNormal 39.123 1.312 0.880 48.509 1.472 0.934

T 38.908 1.299 0.879 48.380 1.461 0.935

EGARCHNormal 38.037 1.248 0.860 48.232 1.462 0.944

T 37.883 1.242 0.858 47.956 1.431 0.926

GJRNormal 38.884 1.301 0.875 48.564 1.471 0.933

T 38.787 1.295 0.874 48.457 1.462 0.934

NonP

are

metr

ic

NPNormal 37.846 1.263 0.871 47.631 1.439 0.930

T 37.851 1.263 0.871 47.817 1.438 0.929

GAM

Normal 37.700 1.238 0.859 47.858 1.429 0.924

T 37.708 1.238 0.859 47.874 1.430 0.924

GARCH model with student t distributed errors performs better than the one fitted with normal distributed innovations

Nonparametric models outperform parametric ones (5% for SHCI, 3% for SZCI)

Nonparmetric models fit data better than EGARCH and TGARCH although EGARCH with T distribution got very good result

Nonparametric models disregarding the innovation distributions

No need to see other models for the dynamic changes in the market

Out-of-sample forecast results

The improvement of the nonparametric models are more significant in the out-of-sample forecast. (e.x. For MSE, 10% for SHCI and 5% for SZCI)

0,6190,6921,2072,4720,5330,5281,0551,928T

0,6200,6901,2052,4680,5330,5261,0561,930NormalNP

0,6040,6281,1922,4030,5300,5171,0451,903T

0,6040,6261,1952,4110,5300,5151,0471,905NormalGAM

0,6550,7441,2382,5630,5900,6251,1092,120T

0,6430,7241,2562,6070,5930,6391,1232,138NormalJGR

0,6200,6831,2182,4900,5520,5771,0641,983T

0,6440,6961,2282,5310,5540,5731,0872,026NormalEGARCH

0,6420,7091,2412,5590,5760,5871,1142,088T

0,6530,7311,2572,6020,5810,5961,1292,130NormalGARCH

MAEMSEMAEMSEMAEMSEMAEMSE

Realized Volatility Realized

Volatility DistributionModel 

Shenzhen Component IndexShanghai Composite Index 

Out-the-Sample Predictability

Param

etricN

onparametri

c

Out-of-sample forecast results

50 200

4

9

GA

RC

H

50 200

4

9

EG

AR

CH

50 200

4

9

GA

M

50 200

3

7

GA

RC

H

50 200

3

7

EG

AR

CH

50 200

3

7

GA

M

Shanghai Com posite Index Out-the-Sam ple Period Volatility

Shenzhen Com ponent Index Out-the-Sam ple Period Volatility

Conclusion apply the iterative algorithm of the nonparametric GARCH model (NP

model), which is first proposed by the BÜhlman and McNeil (2002), to the Generalized Additive Model (NP GAM model), and use it to adjust the volatility estimated by the parametric GARCH model.

nonparametric iterative technique can provide an improvement for the estimation of the hidden volatility process when the market is complicated e.g. exists asymmetric effects, and this improvement is more clear for an out-of-sample forecast.

The NP GAM model appears to be a more stable method with the computational convenience, and in the most cases outperforms the NP model.

An attractive method: no specification of the functional form of the volatility process nor that of the innovation distributions is required for such an additive algorithm.

It could be also used to test if the parametric model is an appropriate one which fits the volatility process well

Conclusion (con.)

limitations of this method: e.g. Several assumptions of the models have not been able to be

proved (BÜhlman and McNeil, 2002). This additive iteration can not fit the stochastic volatility model,

where the volatility process is fully hidden. Furthermore, it is well known that the volatility jumps.