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
A time series is defined as a set of quantitative observations arranged in chronological order. We
generally assume that time is a discrete variable. Time series have always been used in the field
of econometrics. Already at the outset, JAN T INBERGEN (1939) constructed the first econometric
model for the United States and thus started the scientific research programme of empirical
econometrics. At that time, however, it was hardly taken into account that chronologically
ordered observations might depend on each other. The prevailing assumption was that, according
to the classical linear regression model, the residuals of the estimated equations are stochastically
independent from each other. For this reason, procedures were applied which are also suited for
cross section or experimental data without any time dependence.Economic time series
characterstics probability distributions for asset returns often exhibit fatter tails than the standard
normal, or Gaussian, distribution. The fat tail phenomenon is called excess kurtosis. Time series
that exhibit a fat tail distribution are often called leptokurtic. Volatility clustering (a type of
heteroscedasticity) accounts for some but not all of the fat tail effect (excess kurtosis) typically
observed in financial data.Finally, certain classes of asymmetric GARCH models can also capture
the leverage effect. This effect often results in observed asset returns being negatively correlated
with changes in volatility.
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
Closing stock prices
Figure 2
Returns over the period
Trading Days (4168) (Y axis)
Closing Stock Price (X axis)
min: 1
max: 4168
mean: 2.0845e+003
median: 2.0845e+003
mode: 1
std: 1.2033e+003
range: 4167
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)
min: -0.1305
max: 0.1209
mean: 7.4180e-004
median: 0.0011
mode: 0
std: 0.0175
range: 0.2514
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:
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
,
assigned to
,because the weights must sum to one ,we have :
is the weight
+ + =1 ------- (2)
d .GJR-GARCH Model (Glosten-Jagannathan-Runkle Model)
One of the primary restrictions of GARCH Models is that they enforce a symmetrical
response of volatility to positive and negative shocks .This phenomenon arises because
the conditional variance in the GARCH equation is a function of lagged residuals(Un1)and not their signs (by squaring them, the sign is lost ). However, it has been observed
that a negative shock to financial time series is likely to cause volatility to rise by more
than a positive shock of the same magnitude, however, this kind of behavior is generally
observed in equities returns where the conventional GARCH(1,1) model fails. The above
stated phenomenon is popularly known as “Leverage Effect “ in stock markets. So, in
order to handle the asymmetric news GJR-GARCH model was proposed by the three
eminent scientists Glosten, Jagannathan and Runkle in the year 1993[7], this model
incorporated asymmetricity which was earlier absent in GARCH model. The model can
be simply regarded as an extension of GARCH with an additional term added to account
for possible asymmetries. The conditional variance equation [8] of GJR Model is given
below:-
----- (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 model1 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 nonnegativity will be
>0, >0, >=0 & + >=0,the model is still admissible even if
=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)
2.GARCH(1,2)
3.GARCH(2,2)
min: -0.1316
max: 0.1198
mean: -2.7267e-004
median: 6.2456e-005
mode: -0.0010
std: 0.0175
range: 0.2514
min: -0.1316
max: 0.1198
mean: -2.7236e-004
median: 6.2771e-005
mode: -0.0010
std: 0.0175
range: 0.2514
min: -0.1316
max: 0.1198
mean: -2.8598e-004
median: 4.9150e-005
mode: -0.0010
std: 0.0175
range: 0.2514
4.GARCH(2,1)
9.GARCH(2,3)
14.GJR(3,1)
min: -0.1316
max: 0.1198
mean: -2.8010e-004
median:5.5028e-005
mode: -0.0010
std: 0.0175
range: 0.2514
min: -0.1316
max: 0.1198
mean: -2.8602e-004
median: 4.9106e-005
mode: -0.0010
std: 0.0175
range: 0.2514
min: -0.1314
max: 0.1200
mean: -1.6378e-004
median: 1.7135e-004
mode: -9.0557e-004
std: 0.0175
range: 0.2514
5.GARCH(3,1)
10.GJR(1,1)
15.GJR(3,2)
min: -0.1316
max: 0.1198
mean: -2.8047e-004
median: 5.4656e-005
mode: -0.0010
std: 0.0175
range: 0.2514
min: -0.1314
max: 0.1200
mean: -1.4618e-004
median: 1.8895e-004
mode: -8.8797e-004
std: 0.0175
range: 0.2514
min: -0.1315
max: 0.1199
mean: -2.1666e-004
median: 1.1847e-004
mode: -9.5846e-004
std: 0.0175
range: 0.2514
6.GARCH(3,2)
11,GJR(1,2)
16.GJR(3,3)
min: -0.1316
max: 0.1198
mean: -2.8590e-004
median: 4.9227e-005
mode: -0.0010
std: 0.0175
range: 0.2514
min: -0.1315
max: 0.1199
mean: -1.9268e-004
median: 1.4245e-004
mode: -9.3448e-004
std: 0.0175
range: 0.2514
min: -0.1315
max: 0.1199
mean: -2.4530e-004
median: 8.9833e-005
mode: -9.8709e-004
std: 0.0175
range: 0.2514
7.GARCH(3,3)
12.GJR(2,2)
17.GJR(1,3)
min: -0.1316
max: 0.1198
mean: -2.8600e-004
median: 4.9133e-005
mode: -0.0010
std: 0.0175
range: 0.2514
min: -0.1315
max: 0.1199
mean: -2.1665e-004
median: 1.1848e-004
mode: -9.5845e-004
std: 0.0175
range: 0.2514
min: -0.1315
max: 0.1199
mean: -1.9258e-004
median: 1.4255e-004
mode: -9.3438e-004
std: 0.0175
range: 0.2514
8.GARCH(1,3)
13.GJR(2,1)
18.GJR(2,3)
min: -0.1316
max: 0.1198
mean: -2.7259e-004
median: 6.2538e-005
mode: -0.0010
std: 0.0175
range: 0.2514
min: -0.1314
max: 0.1200
mean: -1.6369e-004
median: 1.7144e-004
mode: -9.0549e-004
std: 0.0175
range: 0.2514
min: -0.1315
max: 0.1199
mean: -2.4575e-004
median: 8.9375e-005
mode: -9.8755e-004
std: 0.0175
range: 0.2514
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 loglkelihood 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 (36.7438.55). 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.
[3] Bollerslev, T., R.Y. Chou, and K.F. Kroner, "ARCH Modeling in Finance: A Review of the
Theory and EmpiricalEvidence," Journal of GARCH, Vol.52, 1992, pp. 5–59.
[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 NonNested 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.
[9] Hamilton, J.D., Time Series Analysis, Princeton University Press, Princeton, NJ, 1994.
[10]Stock Prices data downloaded from www.nse-india.com
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
A time series is defined as a set of quantitative observations arranged in chronological order. We
generally assume that time is a discrete variable. Time series have always been used in the field
of econometrics. Already at the outset, JAN T INBERGEN (1939) constructed the first econometric
model for the United States and thus started the scientific research programme of empirical
econometrics. At that time, however, it was hardly taken into account that chronologically
ordered observations might depend on each other. The prevailing assumption was that, according
to the classical linear regression model, the residuals of the estimated equations are stochastically
independent from each other. For this reason, procedures were applied which are also suited for
cross section or experimental data without any time dependence.Economic time series
characterstics probability distributions for asset returns often exhibit fatter tails than the standard
normal, or Gaussian, distribution. The fat tail phenomenon is called excess kurtosis. Time series
that exhibit a fat tail distribution are often called leptokurtic. Volatility clustering (a type of
heteroscedasticity) accounts for some but not all of the fat tail effect (excess kurtosis) typically
observed in financial data.Finally, certain classes of asymmetric GARCH models can also capture
the leverage effect. This effect often results in observed asset returns being negatively correlated
with changes in volatility.
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
Closing stock prices
Figure 2
Returns over the period
Trading Days (4168) (Y axis)
Closing Stock Price (X axis)
min: 1
max: 4168
mean: 2.0845e+003
median: 2.0845e+003
mode: 1
std: 1.2033e+003
range: 4167
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)
min: -0.1305
max: 0.1209
mean: 7.4180e-004
median: 0.0011
mode: 0
std: 0.0175
range: 0.2514
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:
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
,
assigned to
,because the weights must sum to one ,we have :
is the weight
+ + =1 ------- (2)
d .GJR-GARCH Model (Glosten-Jagannathan-Runkle Model)
One of the primary restrictions of GARCH Models is that they enforce a symmetrical
response of volatility to positive and negative shocks .This phenomenon arises because
the conditional variance in the GARCH equation is a function of lagged residuals(Un1)and not their signs (by squaring them, the sign is lost ). However, it has been observed
that a negative shock to financial time series is likely to cause volatility to rise by more
than a positive shock of the same magnitude, however, this kind of behavior is generally
observed in equities returns where the conventional GARCH(1,1) model fails. The above
stated phenomenon is popularly known as “Leverage Effect “ in stock markets. So, in
order to handle the asymmetric news GJR-GARCH model was proposed by the three
eminent scientists Glosten, Jagannathan and Runkle in the year 1993[7], this model
incorporated asymmetricity which was earlier absent in GARCH model. The model can
be simply regarded as an extension of GARCH with an additional term added to account
for possible asymmetries. The conditional variance equation [8] of GJR Model is given
below:-
----- (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 model1 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 nonnegativity will be
>0, >0, >=0 & + >=0,the model is still admissible even if
=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)
2.GARCH(1,2)
3.GARCH(2,2)
min: -0.1316
max: 0.1198
mean: -2.7267e-004
median: 6.2456e-005
mode: -0.0010
std: 0.0175
range: 0.2514
min: -0.1316
max: 0.1198
mean: -2.7236e-004
median: 6.2771e-005
mode: -0.0010
std: 0.0175
range: 0.2514
min: -0.1316
max: 0.1198
mean: -2.8598e-004
median: 4.9150e-005
mode: -0.0010
std: 0.0175
range: 0.2514
4.GARCH(2,1)
9.GARCH(2,3)
14.GJR(3,1)
min: -0.1316
max: 0.1198
mean: -2.8010e-004
median:5.5028e-005
mode: -0.0010
std: 0.0175
range: 0.2514
min: -0.1316
max: 0.1198
mean: -2.8602e-004
median: 4.9106e-005
mode: -0.0010
std: 0.0175
range: 0.2514
min: -0.1314
max: 0.1200
mean: -1.6378e-004
median: 1.7135e-004
mode: -9.0557e-004
std: 0.0175
range: 0.2514
5.GARCH(3,1)
10.GJR(1,1)
15.GJR(3,2)
min: -0.1316
max: 0.1198
mean: -2.8047e-004
median: 5.4656e-005
mode: -0.0010
std: 0.0175
range: 0.2514
min: -0.1314
max: 0.1200
mean: -1.4618e-004
median: 1.8895e-004
mode: -8.8797e-004
std: 0.0175
range: 0.2514
min: -0.1315
max: 0.1199
mean: -2.1666e-004
median: 1.1847e-004
mode: -9.5846e-004
std: 0.0175
range: 0.2514
6.GARCH(3,2)
11,GJR(1,2)
16.GJR(3,3)
min: -0.1316
max: 0.1198
mean: -2.8590e-004
median: 4.9227e-005
mode: -0.0010
std: 0.0175
range: 0.2514
min: -0.1315
max: 0.1199
mean: -1.9268e-004
median: 1.4245e-004
mode: -9.3448e-004
std: 0.0175
range: 0.2514
min: -0.1315
max: 0.1199
mean: -2.4530e-004
median: 8.9833e-005
mode: -9.8709e-004
std: 0.0175
range: 0.2514
7.GARCH(3,3)
12.GJR(2,2)
17.GJR(1,3)
min: -0.1316
max: 0.1198
mean: -2.8600e-004
median: 4.9133e-005
mode: -0.0010
std: 0.0175
range: 0.2514
min: -0.1315
max: 0.1199
mean: -2.1665e-004
median: 1.1848e-004
mode: -9.5845e-004
std: 0.0175
range: 0.2514
min: -0.1315
max: 0.1199
mean: -1.9258e-004
median: 1.4255e-004
mode: -9.3438e-004
std: 0.0175
range: 0.2514
8.GARCH(1,3)
13.GJR(2,1)
18.GJR(2,3)
min: -0.1316
max: 0.1198
mean: -2.7259e-004
median: 6.2538e-005
mode: -0.0010
std: 0.0175
range: 0.2514
min: -0.1314
max: 0.1200
mean: -1.6369e-004
median: 1.7144e-004
mode: -9.0549e-004
std: 0.0175
range: 0.2514
min: -0.1315
max: 0.1199
mean: -2.4575e-004
median: 8.9375e-005
mode: -9.8755e-004
std: 0.0175
range: 0.2514
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 loglkelihood 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 (36.7438.55). 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
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[10]Stock Prices data downloaded from www.nse-india.com