Constant Mean and Conditional Variance A

Constant Mean and Conditional Variance Autoregressive

Heteroscedasticity Models Selection Analysis for Indian

Market Returns

Joydip Dhar,Manisha Pattanaik,Utkarsh Shrivastava,Harendra Sharma,Vishal Pradhan,

Mithlesh Kumar,Tarun Motwani

jdhar@iiitm.ac.in, manishapattanaik@iiitm.ac.in,utkarsh@students.iiitm.ac.in

ABV-Indian Institute of Information Technology and Management,Gwalior-474010

Abstract

An overview of market returns after few years can assist greatly in making important financial decisions like pricing options, financial derivatives and hedge funds. Nested Constant Mean and Conditional Variance GARCH and GJR-GARCH models can play an important role in predicting stock market returns over long run. This paper tries to indentify the model among GARCH(p,q) and GJR(p,q) where p,q = 1,2,3, which best fits historical Indian market returns series. All possible generalized autoregressive conditional heteroscedasticity models for different combinations of p and q are simulated and compared with last 17 years market .

Likelihood Ratio Tests over .05 significance levels are applied to models with mean comparable to observed data, as a models selection analysis tests for best fit models for Indian Markets.

Parameter analysis for GARCH and GJR models has confirmed that GJR(1,1) is the

model which best fits the return series

.Comparision is done using time-series data of S&PCNX Nifty, a value-weighted index of 50 stocks traded on the National Stock Exchange (NSE), Mumbai from 3-july-1990 to 31-Dec-2007 for greater accuracy.

Keywords:- Heteroscedasticity,GARCH, Likelihood Ratio Tests.

1.Introduction

Following the pioneering work of Engle and Bollerslev in eighties on developing models (ARCH/GARCH type models) to capture time-varying characteristics of volatility and other stock return properties, extensive research has been done world over in modeling volatility for estimation and forecasting. The autoregressive conditional heteroskedasticity (ARCH) model, introduced by Engle (1982) and later generalized by [1], spawned numerous empirical studies modeling volatility in developed markets. Later in [2], there have been quite a few studies focusing on emerging stock markets as well (see, [3]). Researchers have increasingly used conditional volatility models such as ARCH, generalized autoregressive conditional heteroskedasticity (GARCH), and their extensions as these models have helped them to model some of the empirical regularities.

a. Indian Market Behaviour from 3-jul-1990 to 31-dec-2007 [10]

Figure 1 Figure 2

Closing stock prices Returns over the period

Trading Days (4168) (Y axis)

min: 1 max: 4168

mean: 2.0845e+003 median: 2.0845e+003 mode: 1

std: 1.2033e+003 range: 4167

b.Conditional Variance Models

Conditional variance models, unlike the traditional or extreme value estimators,

incorporate time varying characteristics of second moment/volatility explicitly.

Following models fall into the category of conditional volatility models:

Closing Stock Price (X axis)

min: 279.0200 max: 6.1593e+003 mean: 1.4947e+003 median: 1.1198e+003 mode: 954.7500 std: 1.0433e+003 range: 5.8803e+003

Returns (Y axis)

a) ARCH (m) Model (Auto Regressive Conditional Heteroscedasticity)[4]

b) EWMA Model (Exponentially Weighted Moving Average Model).[5]

c) GARCH (P, Q) Model (Generalized Autoregressive Conditional

Heteroscedasticity). d) EGARCH Model.

e) GJR-GARCH Model.

The distinctive features of the above listed models is that they recognize that volatilities

and correlations are not constant .During some periods, a particular volatility or

correlation may be relatively low, whereas during other periods it be relatively high, in a

nutshell the above models attempt to keep track of the variations of the volatility or

correlation through time.

c.GARCH (P, Q) (GENERALIZED AUTOREGRESSIVE CONDITIONAL

HETEROSCEDASTICITY

This model was proposed by T.Bollerslev in 1986[6]it was a remarkable improvement

over other conditional volatility models, it combined ARCH(m) and EWMA models and

thus addressed many issues which were not addressed earlier , though we can use many

combinations of (P,Q) here ,but the most popular and the most oftenly used model is the

GARCH(1,1) model. In GARCH (1,1),

is calculated from a long run average variance

rate ,V, as well as from ( n-1) and (Un-1).The equation for GARCH(1,1)is :

Here,

is the weight assigned to V,

is the weight assigned to

, is the weight

assigned to

,because the weights must sum to one ,we have :

+ + =1 ---

(2)

d .GJR-GARCH Model (Glosten-Jagannathan-Runkle Model)

--- (3)

Where, = V, =Weights assigned to lagged residuals (

), =Weights assigned to

conditional variances, =Weights assigned to modified lagged residuals, I (.) is an

indicator function which is 1 if,

and 0 if,

.

The above model is also sometimes referred as a SIGN-GARCH model. The GJR

formulation is closely related to the TGARCH model

1

by Zakoian (1994), and the

AGARCH model of Engle (1990).When estimating the GJR model with index equity

returns, is typically found to be positive, so that volatility increases proportionally more

following negative than positive shocks. We can easily notice that the condition for

non-negativity will be >0, >0, >=0 & + >=0,the model is still admissible even if

<0,provided that + >=0.

e.Likelihood Ratio Tests

:- Of the two models, GJR(1,1) and GJR(2,1), that are

associated with the same return series: The default GJR(1,1) model is a restricted model.

That is, you can interpret a GJR(1,1) model as a GJR(2,1) model with the restriction that

G2 = 0. The more elaborate GJR(2,1) model is an unrestricted model. In Likelihood ratio

tests[9] context, the unrestricted GJR(2,1) model serves as the alternative hypothesis; that

is, the hypothesis the example gathers evidence to support. The restricted GJR(1,1) model

serves as the null hypothesis, that is, the hypothesis the example assumes is true, lacking

evidence to support the alternative. The LRT statistic is asymptotically chi-square

distributed with degrees of freedom equal to the number of restrictions imposed.

2.Observations

:- This section does the analysis of simulated GARCH(p,q) and

GJR(p,q) for p,q = 1,2,3 and observed returns series of Indian Markets over last

seventeen years. Observation table displays minimum, maximum, mean, mode, median,

standard deviation, range of the residuals(simulated return – observed return) of

simulated curve and observed curve. The models which have lesses value of residual’s

mean will be considered more suitable to represent Indian Market returns observed curve

and likelihood ratio test analysis would also be done to confirm the observations.

Observation Tables

1.GARCH(1,1)

min: -0.1316 max: 0.1198 mean: -2.7267e-004 median: 6.2456e-005 mode: -0.0010 std: 0.0175 range: 0.2514

2.GARCH(1,2)

min: -0.1316 max: 0.1198 mean: -2.7236e-004 median: 6.2771e-005 mode: -0.0010 std: 0.0175 range: 0.2514

3.GARCH(2,2)

From the above observations it is clear that GJR models outperform GARCH models ver

fitting the returns series of Indian Markets as the mean of the residuals is lesser in the

case of GJR models.Now we will apply Likelihood ratio test for best three models i.e

GJR(1,1),GJR(3,1),GJR(2,1).In this test the input parameters are LLF (maximized

log-lkelihood function value), degree of freedoms which is one in case of GJR(1,1) and

significance level which is .05 in all cases and output values are H value, p-value.

Test observations:

a. GJR(1,1) and GJR(2,1)

[H,pValue,Stat,CriticalValue] = lratiotest(LLF21,LLF11,...1,0.05); [H,pValue,Stat,CriticalValue]

ans =

0 0.0609 3.5117 3.8415

as value of H=0 hence its evident that GJR(1,1) fits better than GJR(2,1)

b. GJR(1,1) and GJR(3,1)

[H,pValue,Stat,CriticalValue] = lratiotest(LLF11,LLF31,1,0.05); [H,pValue,Stat,CriticalValue]

ans =

0 1.0000 -3.5117 3.8415

as value of H=0 hence its evident that GJR(3,1) is better selection than GJR(3,1)

c. GJR(2,1) AND GJR(3,1)

[H,pValue,Stat,CriticalValue] = lratiotest(LLF21,LLF31,1,0.05); H,pValue,Stat,CriticalValue

ans =

0 1.0000 -2.3285 3.8415

as value of H=0 hence its evident that GJR(3,1) is better selection than GJR(2,1)

3.Results:- Parameter analysis for GARCH and GJR models have confirmed that GJR(1,1) is the models which best fits the return series with mean error((mean of residuals/Observed mean)*100) = 19.69% ,while GJR(3,1) is next best fit with mean error = 22.068% ,GJR(2,1) has mean error = 22.066%While for GARCH models the mean error percentage ranges from (36.74-38.55). The mean error is maximum for GARCH(3,3) while its minimum for GARCH(1,2). Likelihood ratio test analysis shows that GJR(3,1) is better selection over GJR(1,1) and GJR(2,1) over .05 significance level

REFERENCES

[1] Bollerslev,T.,"GeneralizedAutoregressiveConditional Heteroskedasticity," Journal of GARCH, Vol. 31, 1986, pp. 307–327.

[2] Box, G.E.P., G.M. Jenkins, and G.C.Reinsel, Time Series Analysis:Forecasting and Control, Third edition,Prentice Hall, Upper Saddle River, NJ,1994.

[4] R.Engle (1982): “Autoregressive Conditional Heteroscadesticity with Estimates of the Variance of UK Inflation,”Econometrica, 50: 987-1008.

[5] Hull, John (2006).Options, Futures and Other Derivatives, New Delhi: Pearson Education. [6] T.Bollerslev,”Generalized Autoregressive Conditional Heteroscedasticity,”Journal of Econometrics, 31(1986):307-27.

[7] . Chen, Y.T &Kuan, C.M (2002),”The Pseudo-Score Encompassing Test For Non- Nested Hypothesis”, Journal of Econometrics 106,271-295.

[8] . Lee, J.H & Brorsen, B.W (1997),”A Non-Nested Test of GARCH vs. EGARCH Models”, Applied Economics Letters 4,627-636.