Volatility persistence in the presence of structural breaks in the Indian banking sector.

Kumar, Dilip ; Maheswaran, S.

1. INTRODUCTION

Modeling the volatility of asset returns has been a fertile research topic in the area of economics and finance for the last two decades because of its importance for capital market theories (Baillie et al., 1996). Volatility, in general, represents risk or uncertainty associated with the asset and, hence, exploring the behavior of volatility of asset returns is relevant for the pricing of financial assets, risk management, portfolio selection, trading strategies and the pricing of derivative instruments (Poon and Granger, 2003). The Indian banking sector has experienced significant growth in the last decade and has become an important investment target, by providing enormous investment opportunities to investors and portfolio managers. Like in the other sectors in India, investors investing in the banking sector in India face higher risk, as well. Hence, it is essential to study the behavior of the volatility of returns from the Indian banking sector. It is well known that the volatility of financial asset returns changes over time. The dynamic nature of the volatility is generally modeled by making use of the Generalized Autoregressive Conditional Heteroskedasticity (GARCH) class of models (Engle, 1982 and Bollerslev, 1986) by specifying the conditional mean and conditional variance equations, which are potentially helpful in forecasting the future volatility of asset prices. Numerous extensions of GARCH models have been proposed in the literature. For instance, Engle and Bollerslev (1986) propose the Integrated GARCH (IGARCH) model to capture the impact of a shock on the future volatility over an infinite horizon.

The volatility of the returns of financial assets may be affected substantially by infrequent structural breaks or regime shifts due to domestic and global macroeconomic and political events. The standard GARCH model does not incorporate sudden changes in the variance and hence, may be inappropriate for investigating volatility persistence and volatility forecasting. Lastrapes (1989) applies the Autoregressive Conditional Heteroscedasticity (ARCH) model to exchange rates and finds a significant reduction in the estimated volatility persistence when he accounts for monetary regime shifts. Lamoureux and Lastrapes (1990) investigate the persistence of volatility in the GARCH family of models when there are sudden changes in the variance and find that volatility persistence is overstated if structural breaks are ignored. Sudden changes in the variance can also influence the intensity or the direction of information flow among markets, stocks or portfolios as shown by Ross (1989).

Inclan and Tiao (1994) propose the Iterated Cumulative Sum of Squares (ICSS) algorithm which can help in detecting structural breaks in the volatility of a financial time series. The ICSS algorithm detects both a significant increase and decrease in volatility and, hence, can help in identifying both the beginning and the ending of volatility regimes. Aggarwal et al. (1999) apply the ICSS algorithm on some emerging market indices for the period from 1985 to 1995, and find that volatility shifts are impacted mainly by the local macroeconomic events and the only global event over the sample period that affected several emerging markets was the October 1987 stock market crash in the United States. Malik (2003) apply the ICSS algorithm in detecting time periods of sudden changes in the volatility of five major exchange rates, and find that volatility persistence is overstated if those sudden changes are ignored. Fernandez and Arago (2003) utilize the ICSS algorithm to detect structural changes in the variance for European stock indices and their findings are in confirmation with the findings of Aggarwal et al. (1999) that the markets not only react to local economic and political news, but also to news originating in other markets. Malik, Ewing, and Payne (2005) find that controlling for regime shifts in volatility dramatically reduces the persistence of volatility in the Canadian stock market. Hammoudeh and Li (2008) also obtain similar findings for the Gulf Cooperation Council (GCC) stock markets. Wang and Moore (2009) find that, with the new European Union members, the persistence in volatility is significantly reduced when the model incorporates regime changes.

The central aim of this paper is to examine the sudden changes in volatility in the Indian banking sector using the ICSS algorithm and investigate the impact of such sudden changes on the persistence of volatility, from the vantage point of volatility modeling and to assess the forecasting ability using the GARCH class of models for the Gaussian distribution, Student's i distribution and the generalized error distribution (GED) for the period from January 5, 2000 to November 30, 2011. In particular, we investigate whether or not the inclusion of regime shifts in the GARCH class of models reduces the persistence of volatility. In addition, we also compare the out-of-sample performance of the GARCH class of models with and without sudden changes by considering the one-step ahead forecasting ability. We find that incorporating regime shifts in the GARCH model provide better performance in terms of forecasting ability. The study of the impact of structural changes in volatility on the accuracy of volatility forecasts has largely been ignored in the context of the Indian market. Hence, our study can be considered as a contribution on this topic in the context of the Indian banking sector.

The remainder of this paper is organized as follows: Section 2 introduces the tests we will use in this study. Section 3 describes the data and discusses the computational details. Section 4 reports the empirical results and section 5 concludes with a summary of our main findings.

2. METHODOLOGY

2.1. Detecting points of sudden changes in variance

We employ the ICSS (iterated cumulative sum of squares) algorithm introduced by Inclan and Tiao (1994) to detect sudden changes in the variance of a given time series. According to the ICSS algorithm, the return series exhibits a stationary variance over the time period until a sudden change occurs in the variance and thereafter the variance becomes stationary again until another sudden change occurs. This process is repeated through time, and hence provides for a time series to have an unknown number of sudden changes in the variance.

Suppose [[epsilon].sub.t] is a time series with zero mean and with unconditional variance [[sigma].sup.2]. Suppose the variance within each interval is given by [[tau].sub.j.sup.2], where j = 0, 1, ..., [N.sub.T] and [N.sub.T] is the total number of variance changes in T observations, and 1 < [k.sub.1] < [k.sub.2] < ... < [k.sub.NT] < T are the change points.

[[sigma].sup.2.sub.t] = [[tau].sup.2.sub.0] for 1 < t < [k.sub.1] (1a)

[[sigma].sup.2.sub.t] = [[tau].sup.2.sub.1] for [k.sub.1] < t < [k.sub.2] (1b)

[[sigma].sup.2.sub.t] = [[tau].sup.2.sub.NT] for [k.sub.NT] < t < T (1c)

In order to estimate the number of changes in the variance and the time point of each variance shift, a cumulative sum of squares procedure is used. The cumulative sum of the squared observations from the start of the series to the kth point in time is given as:

[C.sub.k] = [k.summation over (t=1)] [[epsilon].sup.2.sub.t]

where k = 1, ..., T. The [D.sub.k] statistics are given as:

[D.sub.k] = ([C.sup.k]/[C.sup.T]) - [k/T] k = 1, ..., T with [D.sub.0] = D = 0

where [C.sub.T] is the sum of squared residuals from the whole sample period.

If there are no sudden changes in the variance of the series then the [D.sub.k] statistic oscillates around zero and when plotted against k, it looks like a horizontal line. On the other hand, if there are sudden changes in the variance of the time series, then the [D.sub.k] statistics values drift either above or below zero. Critical values obtained from the asymptotic distribution of [D.sub.k] can be used to assess the significance of changes in the variance under the null hypothesis of constant variance. The null hypothesis of constant variance is rejected if the maximum absolute value of [D.sub.k] is greater than the critical value. Hence, if [max.sub.k] [square root of (T/2)] [absolute value of [D.sub.k]] is more than the predetermined boundary, then [k.sup.*] is taken as an estimate of the variance change point. The 95th percentile critical value for the asymptotic distribution of [max.sub.k] [square root of (T/2)] [absolute value of [D.sub.k]] is 1.358 as given in Inclan and Tiao (1994) and Aggarwal et al. (1999) and hence the upper and the lower boundaries can be set at [+ or -] 1.3 58 in the Dk plot. If the value of the statistic falls outside of these boundaries, then a sudden change in variance is identified.

There exists a plethora of literature saying that the variance of financial data is time varying, going back to Engle (1982) and Bollerslev (1986). However, it needs to be noted that the ICSS algorithm assumes constant variance within each regime. Sanso, Arago and Carrion (2004) find certain drawbacks in the ICSS algorithm that invalidates its use for financial time series. In particular, the ICSS algorithm neglects the excess kurtosis properties of the process and also it does not take into consideration the conditional heteroskedasiticity that is well known to exist in financial time series. To deal with these drawbacks, they propose the critical value of 1.4058, which corrects for excess kurtosis and conditional heteroskedasticity, as estimated by Monte Carlo simulations. In this paper, we follow the recommendations of Sanso et al. (2004) and use a critical value of 1.4058 for [max.sub.k] [square root of (T/2)] [absolute value of [D.sub.k]].

Inclan and Tiao (1994) find that if the time series has multiple change points, then it is difficult for [D.sub.k] statistic to detect the correct change points at different intervals due to the masking effect. To overcome this problem of masking effect, Inclan and Tiao (1994) propose an algorithm that looks at different pieces of the time series for the identification of change points in the variance. The ICSS algorithm looks for one break point at a time by means of the [D.sub.k] statistic. Once a breakpoint is detected, then the sample series is further segmented to look for other break points. When all the breakpoints in the series have been identified, then the next step is to estimate the GARCH models with and without sudden changes in the variance.

2.2. GARCH model

The log returns are calculated from the stock price indices; i.e.

[Y.sub.t] = [l.sub.n] ([P.sub.t]/[P.sub.t] - 1) x 100

where [P.sub.t] is a value of the index at time t and In is the natural logarithm.

The autoregressive moving average model with order p and q (ARMA(p, q)) is given as below:

[y.sub.t] = [mu] + [[mu].summation over (j+1)] [[beta].sub.1][y.sub.t+1] + [[mu].summation over (j=1)] [[??].sub.1] [[epsilon].sub.t-j] + [[epsilon].sub.[upsilon]] (3)

where [[epsilon].sub.t] is serially uncorrected, but dependent to its lagged values.

The symmetric GARCH model (Bollerslev, 1986) deals with the symmetric feature of volatility. The standard generalized autoregressive conditional heteroskedasticity (GARCH) model is given as:

[[epsilon].sub.t] = [Z.sub.t][[sigma].sub.t] [Z.sub.t] - N(0.1)

[[sigma].sup.2.sub.t] = [omega] + [alpha](L[)[epsilon].sup.2.sub.t] +[beta](L) [[sigma].sup.2.sub.t] (4)

where [omega] > 0, and [alpha](L) and [beta](L) are polynomials in the backshift operator L (L' [x.sub.t] = [x.sub.t-i]) of order q and p, respectively. Equation (4) can be rewritten as infinite-order ARCH process (assuming that [[alpha].sub.i] [greater than or equal to] 0 and [[beta].sub.i] [greater than or equal to] 0 for all 0.

[phi](L) [[epsilon].sup.2.sub.t] - [[sigma].sup.2.sub.t] = [omega] + [1 - [beta](L)] [V.sub.[upsilon]] (5)

where vt = [[epsilon].sup.2.sub.t] - [[sigma].sup.2.sub.t] is interpreted as the innovation in the conditional variance, which has a zero mean and is serially uncorrelated and [theta](L) = [1 - [alpha](L) - [beta](L)] Here the sum of [alpha](L) and [beta](L) measures the persistence of volatility for a given shock. If ([alpha](L) + [beta](L)) is equal to 1, then the GARCH (p, q) process becomes an IGARCH (p,q) process and it may be recalled that the shocks to an IGARCH process have a permanent effect on the volatility of a return series. For the GARCH (1,1) process, equation (5) canbe reduced to:

[[epsilon].sup.2.sub.t] = [[sigma].sup.2] + [v.sub.t] + [[theta].sub.1][v.sub.t-1] + [[theta].sub.2][v.sub.t-2] + ... + [[theta].sub.k][v.sub.t-k] + ... (6)

where [[sigma].sup.2] is the unconditional variance and is equal to [omega]/(1 -[alpha] - [beta]). [[theta].sub.i] is a non-linear function of the ARCH and GARCH parameters. If [[theta].sub.k] (= [partial derivative][[epsilon].sup.2.sub.t]/[partial derivative][v.sub.t-k]) remains large as kincreases, then the shocks in the series show higher degree of persistence. We plot the dynamic impulse response function to study the relationship between [theta] and k.

2.3. Combined model of sudden changes with the GARCH model

The GARCH (p,q) model with sudden changes in variance can be expressed as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

where [D.sub.1], ..., [D.sub.n] are the dummy variables taking a

value of 1 from each point of sudden change in the variance onwards and 0 elsewhere.

Engle and Ng (1993) propose the sign bias, negative size bias, positive size bias and joint tests in the standardized residuals to determine the response of the asymmetric volatility models to news. From equation (7), [z.sub.t] = [[epsilon].sub.t]/[[alpha].sub.t]. Suppose [S.sub.t]--is a dummy variable which takes value 1 if [[epsilon].sub.t-1] is negative and [S.sub.t.sup.+] is a dummy variable that takes value 1 if [[epsilon].sub.t-1] is positive and zero otherwise. Hence, the regression equations for the sign bias, negative size bias, positivesize bias and joint tests are as follow:

Sign biastest: [z.sup.2.sub.t] = [alpha] + b [S.sup.-.sub.t] + [e.sub.t]

Negativesize biastest: [z.sup.2.sub.t] = [alpha] + b [S.sup.-.sub.t] [[epsilon].sub.t-1] + [e.sub.t]

Positive sizebiastest: [z.sup.2.sub.t] = [alpha] + b[S.sup.-.sub.t] [[epsilon].sub.t-1] + [e.sub.t]

Joint test: [z.sup.2.sub.t] = [alpha] + b[S.sup.-.sub.t] + c[s.sup.-.sub.t] [[epsilon].sub.t-1] + d[S.sup.-.sub.t] [[epsilon].sub.t-1] [[epsilon].sub.t]

where a, b, c and d are constants, et is the residual series of the regression equations.

3. DATA AND COMPUTATIONAL DETAILS

In order to investigate the impact of sudden changes in volatility on the volatility persistence of the banking sector in India, we use the weekly price data (3) of the CNX Bank Nifty Index (composed of 12 stocks from the banking sector) for the period from January 5, 2000 to November 30, 2011. All data are obtained from the website: www.nseindia.com. The weekly data are associated with Wednesday. If Wednesday is a holiday, Tuesday data points are used instead. Table 1 provides the descriptive statistics of the weekly returns of CNX Bank Nifty.

The return series of CNX Bank Nifty exhibits a leptokurtic distribution (fat tails) and is positively skewed. The Jarque-Bera statistic confirms the significant non-normality in the return series. The ARCHLM test supports the presence of conditional heteroskedasticity and Box-Pierce Q-test strongly rejects the null hypothesis of no significant autocorrelations in the first 20 lags in the return series. Insignificant KPSS and significant ADF test statistics confirm the non-rejection of the null hypothesis of stationarity and the rejection of the null hypothesis of a unit root in the series.

4. EMPIRICAL RESULTS

4.1. Sudden changes in variance

Table 2 presents the number of sudden changes in the variance, the time periods identified as when such sudden changes have occurred, the mean return during that time period and the standard deviation of the returns over the respective time periods between variance changes for the CNX Bank Nifty index.

We detect three change points in the CNX Bank Nifty index using the ICSS algorithm which represent the presence of four distinct volatility regimes in the time series of returns. The time points of the sudden change in the volatility of CNX Bank Nifty are related to various domestic and global economic events to a moderate degree. In May 2006, Indian stock market indices suffer a major decline of about 1100 points. The turbulence in the period from 2008 to 2009 was caused by the impact of the global financial crisis (sub-prime crisis) which adversely impacted the Indian stock market also. In 2009, the UPA election victory was a major event that impacted the Indian stock market in terms of reducing the uncertainty about the future of the Indian economy. These macroeconomic and political factors may have contributed to the increase in market return volatility which in turn contributed to the overall uncertainty in the Indian banking sector also. In addition, it may be noted that the sub-prime crisis adversely impacted Indian banks because of their exposure to international markets. It can be observed that the first volatility regime (from 12-Jan-2000 to 10-May-2006) and the last volatility regime (27-May-2009 to 3 O-Nov-2011) have nearly the same standard deviation but the mean return in first regime is about 3.44 times the mean return in the last regime. Furthermore, we observe a higher standard deviation during the period of the subprime crisis and this is also the only regime with a negative mean return.

[FIGURE 1 OMITTED]

Figure 1 presents a graphical representation of the sudden changes in the variance and the related volatility regimes for the CNX Bank Nifty index. The bands represent [+ or -] 3 standard deviations for the time point when sudden changes are experienced. Hence, the figure clearly displays where the regime begin and end, as identified by the ICSS algorithm.

4.2. ARMA (0,0)-GARCH (1,1) estimation with and without sudden changes

After identifying the time points of sudden changes in the variance using the ICSS algorithm, the next step is to introduce these sudden changes in the variance in the GARCH class of models. For each GARCH model, we consider three different distributions: the Normal distribution, the Student's t distribution and the GED distribution. The in-sample estimation period is from January 12, 2000 through December 15, 2010 with a total of 571 observations. The remaining 50 observations covering the period from December 22, 2010 through November 30, 2011 are set aside for the evaluation of the out-of-sample performance of all the GARCH models used in this study.

First, we determine the order of ARMA(p,q)-GARCH(1,1) model for the CNX Bank Nifty index based on the minimum value of the Schwarz Bayesian Information Criterion (SBIC). We find the ARMA (0,0) specification to be suitable for the mean equation over the given estimation period. Table 3 and 4 presents the results from the standard GARCH model with and without sudden changes in the variance for the Normal distribution, the Student's t distribution and the GED distribution. The set of dummy variables is included in the variance equation of the GARCH model accounting for different volatility regimes. ARCH-LM test statistic for 10 lags and Ljung-Box statistics for 20 lags are used as diagnostic tests for the standardized residuals and the squared standardized residuals from the GARCH model.

In the ARMA (0,0)-GARCH (1,1) model without a consideration for the sudden changes in the volatility, a and p are highly significant at 1 % level of significance and the persistence of shock (represented by (a + P)) is very high (0.976 for the Normal distribution, 0.972 for the Student t distribution and 0.974 for the GED distribution). This indicates that volatility shocks are highly persistent when we ignore the sudden changes in volatility in the model. On the other hand, ([alpha] + [beta]) is only (0.793 for the Normal distribution, 0.796 for the Student t distribution and 0.794 for the GED distribution) for the GARCH model which accounts for sudden changes in volatility, which indicates that the persistence of variance is drastically reduced when sudden changes are included in the model. These results are consistent with the earlier findings of Lamoureux and Lastrapes (1990), Aggarwal, Inclan and Leal (1999), Malik, Ewing and Payne (2005) and others, who have argued that the standard GARCH model overestimates volatility persistence when ignoring sudden changes in the unconditional variance. Our results also support the same notion in the context of the Indian banking sector that volatility persistence is significantly reduced when we incorporate regime shifts in the model. The model with the Gaussian distribution performs quite well when sudden changes are considered. On the other hand, the estimates of the models when sudden changes are ignored with the Student's t and GED innovations clearly suggest that the conditional distribution has fatter tails than the Normal distribution because n is significantly between 1 and 2.

We evaluate the accuracy of model specifications by mean of several diagnostic tests. The Ljung-Box test statistics for the standardized residuals, Q (20) and the squared standardized residuals, Qs(20), up to 20 lags are insignificant for all model specifications at the 5% level of significance, which indicates that the standardized residuals and the squared standardized residuals are independently and identically distributed (IID) series. The ARCH-LM statistic up to 10 lags also confirms the absence of heteroskedasticity in the residual series for all model specifications. Also, the estimated residuals from the models when we account for sudden changes in the variance follow a Gaussian distribution due to the Jarque-Bera statistics. This indicates that the GARCH model which considers sudden changes in the variance is well specified for examining the volatility persistence in Indian banking sector. Furthermore, we do not find any significant bias from the perspective of the sign bias, negative size bias, positive size bias and joint tests in the standardized residuals, as proposed by Engle and Ng (1993), for all the estimated GARCH models used in this study.

[FIGURE 2 OMITTED]

Figure 2 presents the plots of the dynamic impulse response functions derived from equation (6) for the CNX Bank Nifty index for the GARCH models, with innovations following Gaussian, Student's t and GED distribution, up to a forecast horizon of 30 weeks. The impulse response for the GARCH model controlling for the endogenously determined sudden changes in volatility is given by solid line and the impulse response for the GARCH model ignoring sudden changes in volatility is given by dashed line. The comparison of the impulse response functions further emphasizes the importance of incorporating sudden changes in the variance in the model. The response to a unit shock to nt exhibits rapid decay when regime shifts are accounted for in the GARCH model when compared to when the regime shifts are ignored. We find CNX Bank Nifty dynamic impulse response function shows that 5.86% of the impact of a unit shock on conditional variance persists after 15 weeks, in case the model does not account for regime control variables, but only 0.15% of unit shock persists when controlled for sudden changes in the variance.

4.3. Out-of-sample forecasts

In this section, we investigate the forecasting ability of the GARCH model with and without incorporating the sudden changes in the variance. We use the squared return as a volatility proxy for the out-of-sample evaluation. We calculate root mean squared error (RMSE), mean absolute error (MAE) and logarithmic loss errors (LLE) to measure the forecast accuracy of the models used.

If [[sigma].sup.2.sub.f,t] is a volatility forecast for day t and [[sigma].sup.2.sub.a,t] is the actual volatility on day t, then

RMSE = [[[1/T] [T.summation over (t=1)] [([[sigma].sup.2.sub.ft] - [[sigma].sup.2.sub.at]).sup.2]].sup.1/2]

MAE = [1/T] [T.summation over (t=1)] [absolute value of [[sigma].sup.2.sub.ft] [[sigma].sup.2.sub.at]]

LLE = [1/T] [T.summation over (t=1)] [ln ([[sigma].sup.2.sub.ft]/[[sigma].sup.2.sub.at])]

where T is the number of forecasting data points.

Table 5 presents the forecast evaluation of 50 one-step ahead forecasts generated from ARMA (0,0)-GARCH (1,1) model for the Normal distribution, Student's t distribution and the GED distribution with and without incorporating sudden changes in the variance.

The results indicate that the GARCH models which incorporate sudden changes in the variance provide relatively good forecasts of the Indian banking sector's volatility whereas the GARCH models without considering regime control variables seem to be a poor alternative. Hence, the results of the one-step-ahead forecast evaluation analysis suggest that the volatility models which account for sudden changes in the variance provide excellent out of--sample predictability.

5. CONCLUSION

In this study, we have detected sudden changes in the volatility of the Indian banking sector and investigated the impact of such sudden changes on volatility persistence, from the vantage point of volatility modeling in general and in particular, on the forecasting ability of the models. We have applied the Iterated Cumulative Sum of Squares (ICSS) algorithm to identify the regime shifts in the data series. We find that the sudden changes in volatility are largely associated with domestic and global macroeconomic and political events. In addition, when these regime shifts are incorporated in the volatility model (GARCH models), we find that the persistence in volatility comes down significantly. This suggests that ignoring sudden changes in volatility will lead to overestimating the persistence of volatility which in turn may lead to potential errors by risk managers to come up with the Value-at-Risk (VaR) measure. Risk managers, generally, set minimum capital requirements based on the VaR measure and an overestimation of the persistence of volatility leads to a misevaluation of the VaR and hence the minimum capital necessary for safety. Also, out-of-sample forecast evaluation analysis confirms that volatility models that incorporate regime shifts provide more accurate one-step-ahead volatility forecasts than their counterparts without regimeshifts. Hence, considering sudden changes in the variance may improve the accuracy of the estimation of the volatility persistence and consequently in the VaR and may help in the optimal allocation of funds.

Caption: Figure 1: Time plot for the returns with a band of [+ or -] 3 standard deviation for the CNX Bank Nifty index.

Caption: Figure 2: Dynamic impulse response function for CNX Bank Nifty index returns.

Note: The solid line corresponds to the GARCH model, controlling for the endogenously determined sudden changes in volatility and the dashed line corresponds to the GARCH model, ignoring sudden changes in volatility.

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Dilip Kumar (1) S. Maheswaran (2)

(1) Corresponding author, Research Scholar, Institute for Financial Management and Research, Chennai 600034 dilip.kumar@ifmr.ac.in

(2) Professor, Centre for Advanced Financial Studies, Institute for Financial Management and Research, Chennai 600034, mahesh(S).ifmr.ac.in.

(3) The reason we have preferred to make use of weekly data is that daily observations may be associated with biases due to non-trading, the bid-ask spread and asynchronous prices as explained in Lo and MacKinlay (1988).

(#) Means significant at 1% level. Where ARCH-LM (10) refers to the Lagrange multiplier test for conditional heteroskedasticity with 10 lags.

Table 1: Descriptive statistics of stock returns

 CNX Bank Nifty

Mean 0.003
Median 0.005
Standard deviation 0.050
Minimum -0.174
Maximum 0.294
Quartile 1 -0.024
Quartile 3 0.034
Skewness 0.240
Kurtosis 2.988
Jarque-Bera Statistics 240.011 (#)
ARCH-LM(IO) 34.502 (#)

Table 2: Sudden changes in the variance as identified by the
ICSS algorithm

Index Number of Time period between Mean Standard
 change variance changes deviation
 points
CNX Bank 3 12-Jan-2000 to 0.004726 0.040575
Nifty 10-May-2006
 17-May-2006 to 0.003577 0.054418
 21-May-2008
 28-May-2008 to -0.000610 0.095912
 20-May-2009
 27-May-2009 to 0.001373 0.039083
 30-Nov-2011

Table 3: ARMA (0,0)-GARCH (1,1) model with no sudden changes
in variance

Without ICSS

 GARCH_N GARCH_t GARCH_GED

[mu] 0.443 * 0.485 (#) 0.498 (#)
 (0.183) (0.185) (0.173)
[omega] 0.581 0.676 ([dagger]) 0.630 ([dagger])
 (0.358) (0.355) (0.358)
[[alpha].sub.1] 0.083 (#) 0.078 (#) 0.082 (#)
 (0.030) (0.026) (0.028)
[[beta].sub.1] 0.893 (#) 0.894 (#) 0.892 (#)
 (0.035) (0.031) (0.033)
1/v -- 0.087 * 0.641 (#)
 -- (0.038) (0.059)
Log-Likelihood -1690.542 -1687.361 -1686.611
SBIC 5.966 5.966 5.963
JB-stat 8.670 * 9.886 (#) 9.139*
 [0.013] [0.007] [0.010]
Q(20) 22.610 23.247 22.826
 [0.308] [0.277] [0.297]
Qs(20) 15.649 15.019 15.082
 [0.617] [0.661] [0.656]
ARCH-LM(IO) 0.607 0.568 0.577
 [0.808] [0.840] [0.833]
Sign bias 0.675 0.628 0.623
 [0.499] [0.530] [0.533]
Negative size bias 0.668 0.857 0.739
 [0.504] [0.392] [0.460]
Positive size bias 0.704 0.599 0.669
 [0.482] [0.549] [0.503]
Joint test 5.292 5.474 5.152
 [0.152] [0.140] [0.161]

Table 4: ARMA (0,0)-GARCH (1,1) model while controlling for
sudden changes in the variance

With ICSS

 GARCH_N GARCH_t GARCH_GED

[mu] 0.461 * 0.492 * 0.500 (#)
 (0.184) (0.187) (0.181)
[omega] 2.763 2.708 2.696
 (1.969) (2.177) (2.094)
[[alpha].sub.1] 0.049 0.049 ([dagger]) 0.050 ([dagger])
 ([dagger])
 (0.029) (0.029) (0.029)
[[beta].sub.1] 0.744 (#) 0.747 (#) 0.744 (#)
 (0.126) (0.140) (0.135)
1/v -- 0.048 0.589 (#)
 -- (0.034) (0.052)
Log-Likelihood -1678.485 -1677.763 -1676.976
SBIC 5.957 5.966 5.963
JB-stat 3.330 3.293 3.330
 [0.189] [0.193] [0.189]
Q(20) 23.728 23.667 23.603
 [0.254] [0.257] [0.260]
Qs(20) 18.237 17.699 17.756
 [0.440] [0.476] [0.472]
ARCH-LM(IO) 0.747 0.716 0.717
 [0.680] [0.710] [0.709]
Sign bias 1.465 1.426 1.422
 [0.143] [0.154] [0.155]
Negative 0.198 0.150 0.186
 size bias [0.843] [0.881] [0.853]
Positive 0.461 0.445 0.465
 size bias [0.645] [0.656] [0.642]
Joint test 5.869 5.697 5.634
 [0.118] [0.127] [0.131]

Table 5: Out-of-sample forecast evaluation

 Without With
 ICSS ICSS

GARCH (1,1)--Gaussian

RMSE 21.165 21.084
MAE 17.978 15.104
LLE 1.647 1.201

GARCH(1,1)--Student -t

RMSE 21.179 21.094
MAE 17.965 15.094
LLE 1.645 1.197

GARCH(1,1)--GED

RMSE 21.198 21.113
MAE 18.030 15.061
LLE 1.652 1.188