Financial Time Series Analysis (FTSA) · 2009. 10. 27. · Financial Time Series Analysis (FTSA) Lecture 5: Conditional Heteroscedastic Models The goal of volatility analysis must

Financial Time Series Analysis (FTSA)

Lecture 5: Conditional Heteroscedastic Models

The goal of volatility analysis must ultimately be to explain the causes of volatility. While time series structure is valuable for forecasting, it does not satisfy our need to explain volatility. .... Thus far, attempts to find the ultimate cause of volatility are not

very satisfactory. ~ Robert Engle (2001)

Stock volatility is defined as conditional standard deviation of stock returns.

Unfortunately, it is NOT directly observable! Its importance is seen in many

applications such as:

Option (derivative) pricing – Black-Scholes option pricing formula

Risk management – VaR

Asset allocation – Minimum variance portfolios

Interval forecasts

Methods to Calculate Volatility

1. Use high-frequency data a. Realized volatility of daily returns b. Use daily high, low, and closing price

2. Implied volatility of options data, eg: VIX

3. Econometric modeling --- our focus for now

4. Others -- http://www.sitmo.com/eqcat/4

Characteristics of Volatility

1. Not directly observable 2. Volatility clusters 3. Volatility evolves over time in a continuous manner (that is jumps are rare) 4. Volatility DOES NOT diverge to infinity – implying volatility is stationary 5. Volatility reacts differently to upward v/s downward price ranges – leverage

effect.

http://www.sitmo.com/eqcat/4

Time Series Analysis – Lecture 4 Edited: Oct 18, 2009 Page 2 Professor: Nina Kajiji

Structure of a Model

Volatility plays an important role in studying asset returns because the latter are NOT

serially correlated, but dependent. That is, dependence is nonlinear. This is

evidenced by the ACF of returns and absolute returns.

Ln Returns(IBM) – No Serial Correlation Abs Ln Returns(IBM) -- No Serially Independent

Sqr Ln Returns(IBM) -- Not Serially Independent

Sqr Ln Returns(IBM) – Order ??

Some spikes indicating some ARCH effects


Basic Form

From the above charts we can say that volatility models follow a simple time series

model such as an ARMA(p,q) models with possible explanatory variable effects as

studied in Chapter 2. Thus we have,

Where:

The above equation is often referred to as the mean equation of

As mentioned above volatility evolves with time, therefore the volatility models should

be concerned with time-evolution. Recall, a simple autoregressive model. The

conditional mean and variance of given can be written as:

And,

Or what we call the conditional variance of a return or the volatility equation of .

Note: is the shock or innovation of an asset at time t.


Example: Returns of the Daily closing index of the S&P500 index from 1950 to 2008

Suggesting a MA(2) model

If one looks at the ACF of squared residuals you will observe that the volatility is not

constant over time.


Modeling the Evolving Volatility

Models that study the evolution of over time are characterized into two general

categories:

Fixed function – such as the family of GARCH models where volatility is modeled

by an exact function.

Stochastic function – that is, volatility is modeled by a stochastic function such

as the Heston Model.

Univariate Volatility Models

1. Autoregressive conditional heteroscedastic (ARCH) model of Engle (1982).

2. Generalized ARCH (GARCH) model of Bollerslev (1986).

3. GARCH-M models

4. IGARCH models

5. Exponential GARCH (EGARCH) model of Nelson (1991).

6. Treshold GARCH model of Zakoian (1994) or GJR Model of Glosten, Jagannathan,

and Runkle (1993).

7. Conditional heteroscedastic ARMA (CHARMA) model of Tsay (1987).

8. Random coefficient autoregressive (RCA) model of Nicholls and Quinn (1982).

9. Stochastic volatility (SV) models of Melino and Turnbull (1990), Harvey, Ruiz, and

Shephard (1994), and Jacquier, Polson, and Rossi (1994).


ARCH Model

A series is said to have an ARCH effect if:

a) The shock of an asset return is serially uncorrelated, but dependent. Large shocks tend to be followed by another large shock, and

b) The dependence of can be described by a simple quadratic function of its lagged values.

The ARCH(m) model therefore assumes that:

Where: is a sequence of iid r.v. with mean 0 and variance 1, and for

i > 0. Distribution of can be standard normal, standardized Student-t, generalized

error dist (GED), or skewed Student-t.

Properties of an ARCH Model

Consider and ARCH(1) model: where and

. We then have:

1. E( = 0

2. Var( = if 0 < < 1

3. Under normality we have the fourth moment written as:

Provided 0 < . Implying heavy tails.


Advantages of an ARCH Model

Simplicity

Generates volatility clustering

Heavy tails (high kurtosis)

Weaknesses of an ARCH Model

Symmetric to both positive and negative prior returns

Restrictive because 0 < -- otherwise we will have an infinite fourth

moment.

Provides no explanation as to what causes the variation in volatility

Not sufficiently adaptive in prediction – because they react slowly to large

isolated shocks.

Building an ARCH Model

1. Modeling the mean effect and testing it using the Q-statistics of squared

residuals; presented by McLeod and Li (1983), or Engle (1982).

2. Order determination using PACF of the squared residuals

3. Estimation: Conditional MLE

4. Model Checking: Look at: Q-stat of standardized residuals, residuals and

squared standardized residuals. Skewness and kurtosis of standardized

residuals.


Example IBM

The ACFs and PACFs of the squared returns showed the existence of conditional

heteroscedasticity. The sample PACF indicates an ARCH(3) model might be

appropriate.

Sqr(Ln(IBM Returns))

Executing the model in SAS

* set library;

filename retin disk 'c:\timeseries\ibmarchexamp.csv';

* create dataset;

data ibm;

infile retin;

input lnibm;

time = _n_;

run;

* Solve an ARCH(3);

proc autoreg data=ibm;

model lnibm = / garch=(q=3); * ARCH(3);

run;


The SAS System 15:23 Sunday, October 18, 2009 29

The AUTOREG Procedure

Dependent Variable lnibm

Ordinary Least Squares Estimates

SSE 0.04443893 DFE 355

MSE 0.0001252 Root MSE 0.01119

SBC -2183.7718 AIC -2187.6467

MAE 0.00811462 AICC -2187.6354

MAPE 99.9730532 Regress R-Square 0.0000

Durbin-Watson 2.2892 Total R-Square 0.0000

Standard Approx

Variable DF Estimate Error t Value Pr > |t|

Intercept 1 0.0000140 0.000593 0.02 0.9811

Algorithm converged.

GARCH Estimates

SSE 0.04448295 Observations 356

MSE 0.0001250 Uncond Var 0.00012795

Log Likelihood 1102.06252 Total R-Square .

SBC -2180.6253 AIC -2196.125

MAE 0.00810079 AICC -2196.0111

MAPE 99.2983682 Normality Test 36.0218

Pr > ChiSq <.0001

Standard Approx


Intercept 1 0.000366 0.000557 0.66 0.5111

ARCH0 1 0.0000858 8.1212E-6 10.56 <.0001

ARCH1 1 -2.25E-18 1.4797E-7 -0.00 1.0000

ARCH2 1 0.1675 0.0615 2.72 0.0065

ARCH3 1 0.1623 0.0634 2.56 0.0104

Model :

Example INTEL


* set library;

filename retin disk 'c:\timeseries\inteltosas.csv';

* create dataset;

data intel;

infile retin;

input lnintel;

time = _n_;

run;

* Solve an ARCH(3);

proc autoreg data=intel;

model lnintel = / garch=(q=3); * ARCH(3);

run;




Dependent Variable lnintel


SSE 6.6350799 DFE 371

MSE 0.01788 Root MSE 0.13373

SBC -436.25745 AIC -440.17634

MAE 0.09839854 AICC -440.16553



Standard Approx


Intercept 1 0.0180 0.006934 2.59 0.0099

Algorithm converged.

GARCH Estimates


MSE 0.01784 Uncond Var 0.0178877


SBC -437.05696 AIC -456.65143

MAE 0.09846411 AICC -456.48749


Pr > ChiSq <.0001

Standard Approx


Intercept 1 0.0165 0.006663 2.47 0.0135

ARCH0 1 0.0121 0.001151 10.55 <.0001

ARCH1 1 0.1960 0.0840 2.33 0.0197

ARCH2 1 0.0748 0.0503 1.49 0.1372

ARCH3 1 0.0502 0.0753 0.67 0.5049

Model :


* Solve an ARCH(1);


model lnintel = / garch=(q=1); * ARCH(1);

run;





SSE 6.6350799 DFE 371

MSE 0.01788 Root MSE 0.13373

SBC -436.25745 AIC -440.17634

MAE 0.09839854 AICC -440.16553



<<< output snipped >>

GARCH Estimates


MSE 0.01784 Uncond Var 0.01960984


SBC -442.734 AIC -454.49068

MAE 0.0984596 AICC -454.42546


Pr > ChiSq <.0001

Standard Approx


Intercept 1 0.0166 0.006173 2.68 0.0073

ARCH0 1 0.0125 0.001247 10.02 <.0001

ARCH1 1 0.3627 0.0899 4.03 <.0001

Model :

Implications:

Expected monthly log return is 1.66%

so the 4th moment exists.


* Solve an ARCH(1) with t innovations;


model lnintel = / garch=(q=1) dist=T; * ARCH(1);

run;





SSE 6.6350799 DFE 371

MSE 0.01788 Root MSE 0.13373

SBC -436.25745 AIC -440.17634

MAE 0.09839854 AICC -440.16553



GARCH Estimates


MSE 0.01785 Uncond Var 0.01813331


SBC -888.17087 AIC -903.84645

MAE 0.09837097 AICC -903.73746


Pr > ChiSq <.0001

Standard Approx

Variable DF Estimate Error t Value Pr > |t| Variable Label

Intercept 1 0.0216 0.005997 3.60 0.0003

ARCH0 1 0.0134 0.002021 6.64 <.0001

ARCH1 1 0.2597 0.1194 2.17 0.0296

TDFI 1 0.1672 0.0447 3.74 0.0002 Inverse of t DF

Model :

The t-distribution has 1/0.1672 = 5.98 degrees of freedom.

Implications:

Using a heavy-tailed distribution for reduces the ARCH effect.

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Financial Time Series Analysis (FTSA) · 2009. 10. 27. · Financial Time Series Analysis (FTSA) Lecture 5: Conditional Heteroscedastic Models The goal of volatility analysis must