CHOOSING THE BEST PERFORMING GARCH MODEL FOR SRI LANKA STOCK MARKET BY NON-PARAMETRIC SPECIFICATION TEST

This paper examines the performance of different kind of GARCH models with Gaussian, Student-t and generalized error distribution for Colombo Stock Exchange (CSE), in Sri Lanka. Analyzing the daily closing price index of CSE from January 02, 2007 to March 10, 2013. It was found that the Asymmetric GARCH models give better result than symmetric GARCH model. According to distributional assumption these models under Student-t as well as generalized error provided better fit than normal distributional assumption. The Non-Parametric Specification test suggest that the GARCH, EGARCH, TARCH and APARCH models with Student-t distributional assumption are the most successful model for CSE.


Introduction
The various well-known characteristics are common to many financial time series.Volatility clustering is often observed (i.e.large changes tend to be followed by large changes and small changes tend to be followed by small changes; Mandelbrot (1963) for early evidence).Second, financial time series often exhibit leptokurtosis, meaning that the distribution of their returns is fat-tailed see Fama (1965).Moreover, the so-called "leverage effect", first noted in Black (1976), refers to the fact that changes in stock prices tend to be negatively correlated with changes in volatility.
to model the third stylized fact, namely the "leverage effect".To solve this problem, many nonlinear extensions of the GARCH models have been proposed.Among the most widely spread are the Exponential GARCH (EGARCH) of Nelson (1991), The Threshold GARCH (TGARCH) model of Zakoian (1994), Glosten et al. (1993) and the Asymmetric Power ARCH (APARCH) of Ding et al. (1993).Unfortunately, GARCH models often do not fully capture the thick tails property of high frequency financial time series.This has naturally led to the use of non-normal distributions to better model this excess kurtosis.Bollerslev (1986), Baillie and Bollerslev (1989), Kaiser (1996) and Beine, et al. (2000) among others used Student-t distribution while Nelson (1991) and Kaiser (1996) suggested the Generalized Error Distribution (GED).Other propositions include mixture distributions such as the normal-lognormal (Hsieh 1989 or the Bernoulli-normal Vlaar and Palm 1993).Peters (2001) in his working paper showed the forecasting performance of different kinds of asymmetric GARCH model with normal, Student-t and skewed Student-t distributional assumption for two major European stock indices.Forecasting conditional variance with asymmetric GARCH models has been studied in several papers Pagan and Schwert (1990), Brailsford and Faff (1996), Franseset et al. (1998), Loudon et al. (2000).On the other hand, comparing normal density with non-normal ones also has been explored in many occasions Hsieh (1989), Baillie and Bollerslev (1989), Peters (2000) and Lambert and Laurent (2000).Marcucci (2003) compared a number of different GARCH models in terms of their ability to describe and forecast the behavior of volatility of financial time series.He estimated assuming both Gaussian innovations and fat-tailed distributions, such as student-and Generalized Error distribution.Hong and Li (2004) studied the Nonparametric Specification Tests of Discrete Time Spot Interest Rate Models in China.They examined a wide variety of popular spot rate models in China, including the single-factor diffusion models, GARCH models, Markov regimeswitching models and jump-diffusion models and the specification of these models.Hong and Li (2005) conducted a study on Nonparametric Specification Test for Continuous-Time Models with Applications to Spot Interest Rates.Batra (2004) examined the time varying pattern of stock return volatility in Indian stock market.In this study the daily closing index of CSE in Sri Lanka is used.The structure of the assignment is as follows: Section 2 presents the different GARCH models used in this study, densities are described in Section 3 and Section 4 present non-parametric specification test, Section 5 reports the empirical findings while Section 6 present the conclusions.

Models of the Study
The models considered in this paper are all ARCH-type.They differ on the functional form of 2 t  but the basic logic is the same.

GARCH Model
The GARCH model of Bollerslev (1986) can be expressed as Using the lag or backshift operator L , the GARCH (p, q) model is with . Based on Eq. ( 5), it is straightforward to show that the GARCH model is based on an infinite ARCH specification.If all the roots of the polynomial 0 5) lie outside the unit circle, we have , which may be seen as an ARCH(  ) process since the conditional variance linearly depends on all previous squared residuals.

EGARCH Model
The EGARCH or Exponential GARCH model was proposed by Nelson (1991).The specification for conditional variance is: The left hand side of Eq. ( 6) is the log of the conditional variance.This implies that the leverage effect is exponential rather than quadratic and that forecasts of the conditional variance are guaranteed to be non-negative.The presence of the leverage effects can be tested by the hypothesis 0

TGARCH Model
TARCH or Threshold ARCH and Threshold GARCH were introduced independently by Zakoian (1994) and Glosten et al. (1993).The generalized specification for the conditional variance is given by: In this model 0 increases volatility and we say that there is a leverage effect for the Taylor (1986) and Schwert (1990) introduced the standard deviation GARCH model where the standard deviation is modeled rather than the variance.Ding, et al. (1993) introduced the Asymmetric Power ARCH model.The APARCH (p, q) model can be expressed as:

APARCH Model
. This model is quite interesting since it couples the flexibility of a varying exponent with the asymmetry coefficient (to take the "leverage effect" into account).Moreover, the APARCH model includes seven other ARCH extensions as a special case: vii) The log-ARCH of Geweke (1986) and Pentula (1986), whenever 0   .

Distributional Assumptions
It may be expected that excess kurtosis and skewness displayed by the residuals of conditional heteroscedasticity model will be reduced if a more appropriate distribution is used.In this study, three distributions such as Gaussian, Student-t and Generalized Error are considered.Given a distributional assumption, ARCH models are typically estimated by the method of maximum likelihood.

Gaussian
The Normal distribution is by far the most widely used distribution when estimating and forecasting GARCH models.If we express the mean equation as in Eq. ( 1) and where T is the number of observations.

Student t-Distribution
For the Student-t distribution, the log-likelihood contributions are of the form: where  is the degree of freedom     , we have the normal distribution.So that the lower  the fatter the tails.

Generalized Error
For the GED, we have: (11) where the tail parameter , and fat-tailed if 2  r .

Non Parametric Specification Test
Recently Hong and Li (2005) proposed two new nonparametric transition density-based specification tests for continuous-time series models.These tests share the appealing features of both Ait-Sahalia (1996) and Gallant and Tauchen (1996) nonparametric approaches and have many additional nice properties.First, unlike Ait-Sahalia (1996) marginal density-based test, the tests are based on the transition density which captures the full dynamics of a continuous-time process.Second, to achieve robustness to persistent dependence, the data is transformed via a dynamic probability integral transform using the model transition density which is well known in statistics (e.g., Rosenblatt 1952) and is more recently used to evaluate out-of-sample density forecasts in discrete-time analysis (e.g.Diebold, et al. 1998, Hong et al. 2004).The transformed sequence is i.i.d.U[0,1] under correct model specification irrespective of the dependence structure of the original data.Third, to eliminate the well-known "boundary bias" of kernel estimators as documented in Chapman and Pearson (2000), a boundary-modified kernel is introduced.Fourth, to reduce the impact of parameter estimation uncertainty, a test based on the Hollinger metric is proposed.Fifth, the regularity conditions for asymptotic analysis are based on the model transition density rather than the stochastic differential equation of the underlying process.As a consequence, the tests are applicable to a vast variety of continuous-time and discrete-time dynamic models such as GARCH/stochastic volatility models, regime-switching models, jump-diffusion models and multi-factor diffusion models.The nonparametric specification test proposed recently by Hong and Li (2005) to evaluate different spot rate models.Assuming the underlying process {Xt} follows the following data generating process: If the model is correctly specified, the exists some almost surely for all ∆ >0 .
Consequently, the transformed series  where,  (16) and the first test statistic is a properly centered and scaled version of where the non-stochastic centering and scale factors To alleviate this problem they proposed a second test based on the square Hellinger metric, (20) which is a quadratic form between where 0 h A and 0 V are as in Eq. ( 18) and Eq. ( 19).Under correct model specification this test has the same asymptotic distribution as It can be summarized the omnibus evaluation procedures following Hong and Li (2005): (i) estimate the parameters of discrete spot rate models using maximum likelihood estimation (MLE) method to yield a n -consistent estimator  ˆ; (ii) compute the model generalized residual  14) for a pre-specified lag j , using a kernel in Eq. ( 15) and the bandwidth

Empirical Study
The data analyzed here is the daily closing price index for CSE in Sri Lanka from January 02, 2007 to March 10, 2013.The parameter estimation process choice here is MLE.The indices prices are transformed into their returns to obtain stationary series.The transformation is: , where y is the return index at time t .Refer table 1.
Some of the descriptive statistics for daily returns are displayed in Table 1.Mean returns of the CSE is 0.067.Volatility (measured as a standard deviation) is 1.6012.The return of CSE market is leptokurtic in the sense that kurtosis exceeds positive three and the return series also display significant skewness.According to Jarque and Bera (1987) statistics normality is rejected for the return series.The results from ARCH test can not reject the ARCH effect which indicate that the series have ARCH effect.Overall these results clarify support the rejection of the hypothesis that CSE time series of daily returns are time series with independent daily values.Moreover, the statistics justify use of the GARCH specification in modeling the volatility of Sri Lanka stock market.

Parameter Estimation
In comparison of the performance of GARCH models the simple mean equation is used:  Table 2 presents the maximum likelihood estimate (MLE) of the parameters and their corresponding t-statistic.The use of GARCH set of models seems to be justified.All coefficients are significant at 5% level of significance except the coefficient of  in case of Student-t and generalized error densities.Table 3 reports some useful statistics such as Box-Pierce statistics for both residuals and the squared residuals, Akaike Information Criteria (AIC) and Log Likelihood value.distributions clearly outperform the Gaussian.In the case of Student-t distribution the AIC value for the model GARCH, EGARCH, TGARCH and APARCH are less than the densities of Gaussian density.The log likelihood value is strictly increasing in case of Student-t densities comparing with the generalized error and Gaussian densities.All the models seem to do a good job in describing the dynamic of the first two moments of the series as shown by the Box-Pierce statistics for the residuals and the squared residuals which are all non-significant at 5% level.

Conclusions
This research paper examined a wide variety of popular GARCH models for CSE index return, including GARCH, EGARCH, TGARCH and APARCH model with Gaussian, Student-t and generalized error distribution.The comparison focused on two different aspects: the difference between symmetric and asymmetric GARCH (i.e., GARCH versus EGARCH, TGARCH and APARCH) and the difference among Gaussian, Student-t and Generalized error densities.The results show that noticeable improvements were made by using an asymmetric GARCH model in the conditional variance.According to distributional assumption these models with Student-t distribution give better results.The Non Parametric Specification test also suggests that the following model under studentt distribution can pass the original premise on 5% as well as 1% level.It means the GARCH, EGARCH, TGARCH and APARCH models under Studentt distributional assumption are significant models for CSE.Finally the study suggested that these four models were very suitable for the stock market of Sri Lanka.As a result, this study would be of great benefit to investors and policy makers at home and abroad.


and quite accurate when the true parameter 0 was known might be adversely affected by any imprecise estimate for  ˆ in finite samples.

Table 1 :
Descriptive Statistics of CSE daily returns.

Table 2 :
MLE of the parameters and their corresponding t-statistic

Table 3 :
Estimation Statistics-Model comparisonThe Akaike Information Criteria (AIC) and the log-likelihood values suggest the fact that TGARCH, EGARCH or APARCH models fit the data well than the traditional GARCH.On the basis of AIC and log-likelihood value the APARCH model outperforms than TGARCH and EGARCH model.EGARCH model gives better result than TGARCH model in case of Gaussian and Generalized error densities.Regarding the densities, the Student-t and Generalized error

Table 4 :
Models with Gaussian distributional assumption

Table 5 :
Models with student t-distributional assumption

Table 6 :
Models with generalized error distributional assumption

Table 4
statistics are overwhelmingly significant, suggesting that all four models are severely mis-specified at any reasonable significance level.statisticsfor the following four models are given in Table5ranges from -1.52 to 10.42 which can pass the original premise on 5% as well as 1% level.It means that GARCH, EGARCH, TGARCH and APARCH models under the student-t distributional assumption are the most successful models.Table6reports these models with Generalized Error distribution.The Q  (j) statistics for these models are range from 30.58 to 96.47.Adding Generalized Error distribution into these models reduces the Q (j) statistics are overwhelmingly significant, suggesting that all four models are not significance at any reasonable significance level.