Heteroskedastic behavior of the Indian stock market: evidence and explanation.

ABSTRACT

This paper investigates the heteroskedastic behavior of the Indian stock market using 'vanilla' GARCH (1, 1) model for a period of about 24 years from January 1980 to June 2003. The study reports an evidence of time-varying volatility which exhibits clustering, high persistence and predictability. Conditional volatility shows a clear evidence of volatility shifting over the period where the level of volatility for the decade Nineties is considerably higher than that of the decade Eighties and violent changes in share prices cluster around the boom of 1992, surpassing all previous records. Though the gradual shift of volatility started in response to strong economic fundamentals, the real cause for abrupt movement appears to be the imperfection of the market.

1. INTRODUCTION

The behavior of speculative price series has attracted the attention of researchers for nearly 100 years. Mandelbrot and Fama provided the first generally accepted evidence that suggests that the distribution of such asset prices are characterized by a number of stylized 'facts' such as kurtosis and heteroskadasticity (Mandelbrot, 1963 and Fama, 1965). Most importantly, asset returns are approximately uncorrelated but not independent through time as large (small) price changes tend to follow large (small) price changes. This temporal concentration of volatility is commonly referred to as 'volatility clustering' and it was not fully exploited for modeling purposes until the introduction of the Auto Regressive Conditional Heteroskedasticity (ARCH) model by Engle (1982). The ARCH model was unique in that it specified the variance of the error term in a regression equation as conditional on squared past errors. Hence, volatility in the ARCH model will exhibit periods of relative tranquility and volatility effectively capturing this volatility clustering characteristic so common to economic and financial time series data.

Bollerslev extended this idea into Generalized Autoregressive Conditional Heteroskedastic (GARCH) models which give more parsimonious results than ARCH models (Bollerslev, 1986). ARCH and GARCH models have become widespread tools for dealing with time series heteroskedastic models. The goal of such models is to provide a volatility measure--like a standard deviation--that can be used in financial decision concerning risk analysis, portfolio selection and derivative pricing.

Considering the importance of the models, the ARCH literature has developed so rapidly that there currently exists a veritable family of ARCH models incorporating the original ARCH model of Engle, GARCH model of Bollerslev as well as a host of other suitably acronymed models (see Bollerslev et al., 1994, or Bera and Higgins, 1993 for a survey). Curiously all of these models have been developed based on economic and financial time series data mostly taken from developed countries, where each of these subsequent contributions to the ARCH family have concentrated on refining both the mean and variance equations to better capture the stylized characteristics of the data. ARCH and GARCH literature on emerging markets is, however, scanty though comparatively higher reported returns of many of these markets have recently attracted increased interest of global portfolio investors in the developed countries. Indian stock market too promoted an accelerated growth and development both in qualitative and quantitative measures, particularly after the structural changes and financial liberalization policies initiated in 1991 and has opened a new vista to foreign institutional investors to diversify their global portfolios. Surprisingly enough in my knowledge there is no study identifying stochastic behavior particularly concerning volatility, using ARCH and GARCH methods on Indian data. Hence an attempt is made in the present study to fill up the gap in this direction. The objective of the study is simply to examine heteroskedastic behavior of the Indian stock market using plain GARCH model which allows for changing conditional volatility. We focus our attention on the following questions:

1. Does stock return volatility change over time? If so, are volatility changes predictable? To address these issues we will try to fit an appropriate GARCH model which may help to forecast the conditional variance of the daily price change.

2. What are the reasons behind volatility shifting? We will try to provide here only subjective explanation on the basis of available information.

We find strong evidence of time-varying volatility. Our results resemble those of many studies on developed markets: periods of high / low volatility tend to cluster, volatility shows high persistence and is predictable. High volatility though partially is explained by fundamental economic factors, a considerable part of the excessive movement may be attributed to 'fads' or 'bubble'.

We do hope the findings of the study would help investors for derivative pricing, VaR calculation and portfolio diversification. The findings may also be interest to policy makers interested in stock market movements, since internationalization of market could represent significant capital inflows or outflows, and this influences saving and consumption decision.

The remaining of the paper is organized as follows. Section II introduces the basic forms of ARCH and GARCH models. Section III provides a discussion of the empirical evidence. An attempt is also made here to explain volatility shifting over the period. Finally we conclude the study in Section IV.

2. ARCH AND GARCH MODELS:

Very many different types of GARCH models have been proposed in academic literature, but not all have found good practical applications. This section gives an introduction of a relatively basic form of GARCH models that are mostly used by the practitioners. The first autoregressive conditional heteroskedasticity (ARCH) model, introduced by Engle (1982), was applied to economic data. For financial data it is more appropriate to use a generalization of this model, the symmetric GARCH introduced by Bollerslev (1986).

Any GARCH model consists of two equations. The first is the conditional mean equation. This can be anything, but since the focus of GARCH is on the conditional variance equation, it is usual to have a very simple conditional mean equation. The simple GARCH models used in practice take the simple possible conditional equation

(1) [r.sub.t] = c + [[epsilon].sub.t]

where [r.sub.t] is the return series, constant return c is the average of returns over the data period, and [[epsilon].sub.t] is the unexpected return which is just the mean deviation return. The second equation in a GARCH model is the conditional variance equation. Different GARCH models arise because the conditional variance equations are specified in different forms.

2.1 Arch

The ARCH (q) process captures the conditional heteroskedasticity of financial returns by assuming that today's conditional variance is a weighted average of past squared unexpected returns:

(2) [h.sub.t] = [[alpha].sub.0] + [[alpha].sub.1] [[epsilon].sub.t-1.sup.2] + [[alpha].sub.2] [[epsilon].sub.t-2.sup.2] + ... + [[alpha].sub.q][[epsilon].sub.t-q.sup.2]

[[alpha].sub.0] > 0, [[alpha].sub.1], [[alpha].sub.2] ..., [[alpha].sub.q] [greater than or equal to] 0 and [e.sub.t] / [I.sub.t-1] ~ N(0, [h.sub.t])

If a major market movement occurred yesterday, the day before or up to q days ago, the effect will be to increase today's conditional variance because all parameters are constrained to be non-negative (and [[alpha].sub.0] is constrained to be strictly positive).

ARCH models are not often used in financial markets because the simple GARCH models perform so much better. In fact the ARCH model with exponentially declining lag coefficients is equivalent to a GARCH(1,1) model, as is shown below, so the GARCH process actually models an infinite ARCH process, with sensible constraints on coefficients and using only very few parameters.

2.2 Garch

The full GARCH (p,q) model adds p autoregressive terms to the ARCH (q) specification, and the conditional variance equation takes the form

(3) [h.sub.t] = [[alpha].sub.0] + [[alpha].sub.1] [[epsilon].sub.t-1.sup.2] + [[alpha].sub.2] [[epsilon].sub.t-2.sup.2] + .... + [[alpha].sub.q] [[epsilon].sub.t-q.sup.2] + [[beta].sub.1][h.sub.t-1] + [[beta].sub.2][h.sub.t-2] + .... [[beta].sub.p][h.sub.t-p]

[[alpha].sub.0] > 0, [[alpha].sub.1], [[alpha].sub.2], .... [[alpha].sub.q] [greater than or equal to] 0, [[beta].sub.1], [[beta].sub.2], ..., [[beta].sub.p] [greater than or equal to] 0

In the ARCH (q) process the conditional variance is specified as a linear function of past squared observations only, whereas the GARCH (p,q) process allows lagged conditional variances to enter as well. The GARCH (p,q) process defined above is stationary when

([[alpha].sub.1] + [[alpha].sub.2] + .... +, [[alpha].sub.q]) + ([[beta].sub.1] + [[beta].sub.2] + .... + [[beta].sub.p]) < 1

The simplest but often very useful GARCH process is the GARCH (1, 1), process which is also called the generic or 'vanilla' GARCH model given by

(4) [h.sub.t] = [[alpha].sub.0] + [[alpha].sub.1] [[epsilon].sub.t-1.sup.2] + [[beta].sub.1][h.sub.t-1] [[alpha].sub.0] > 0, [[alpha].sub.1] [greater than or equal to] 0, [[beta].sub.1] [greater than or equal to] 0.

The stationary condition for GARCH (1, 1) is [[alpha].sub.1] + [[beta].sub.1] < 1

Note that this model may also be written

(5) [h.sub.t] = [[alpha].sub.0] + [[alpha].sub.1] [e.sub.t-1.sup.2] + [[beta].sub.1][h.sub.t-1] = [[alpha].sub.0] + [[alpha].sub.1] [e.sub.t-1.sup.2] + [[beta].sub.1]([[alpha].sub.0] + [[alpha].sub.1] [e.sub.t-2.sup.2] + [[beta].sub.1][h.sub.t-2]) = [[alpha].sub.0] + [[alpha].sub.1] [e.sub.t-1.sup.2] + [[beta].sub.1][[alpha].sub.0] + [[beta].sub.1][[alpha].sub.1] [e.sub.t-2.sup.2] + [[beta].sub.1.sup.2][h.sub.t-2]

= [[alpha].sub.0] / (1 - [[beta].sub.1] + [[alpha].sub.1]([e.sub.t-1.sup.2] + [[beta].sub.1] [e.sub.t-2.sup.2] + [[beta].sub.1.sup.2][e.sub.t-3.sup.2] + ....)

so the GARCH(1, 1) model is equivalent to an infinite ARCH model with exponentially declining weights, as mentioned earlier.

3. EMPIRICAL EVIDENCE

In this section we first describe the data set and then try to fit an appropriate GARCH model to predict the time-varying volatility of daily returns. We also try to explain volatility shifting, if any, over the given period.

3.1 Data Description:

The sample data to be used here consist two sets. The first set comprises of the series of index number of share prices, namely, "The Economic Times Index Numbers of Ordinary Share Prices", compiled and published by The Economic Times on daily basis for a period of 2nd January 1980 to 24th December 1990. The second one is the S &P CNX Nifty compiled and published by NSE India for the period from 2nd January 1991 to 10th June 2003.

We have analyzed volatility using the combined data set of the Economic Times Index and the NIFTY together for a longer period from 1980 to 2003. Choice of the combined data sets is primarily guided by the availability of the share price index. To the best of our knowledge, no reliable single share price index was readily available for the period under study. S&P CNX Nifty and BSE Sensex are available only from 1990. Similarly the Economic Times Index is not available after 1995. Since none of the price indices was solely available for the entire period, the above two series have been considered successively (1980-1990 and 1991-2003) to cover a relatively longer period of time of nearly 24 years.

Simply to measure volatility for a longer period, the validity of using two indices comprising of different portfolios may be questioned. At least there are two points to defend the approach of using two indices in our study. First, no index maintain the same portfolio for a longer period of time. The portfolio is often reshuffled replacing inactive by active shares. Hence even the use of a single index provides no guarantee that the volatility estimate would be based on the same set of shares for the entire period. Second, there is no evidence of significant variation in volatility estimates even when several alternative indices comprising of different portfolios have been used simultaneously to measure it. (See Appendix where Figures 3 and 4 show the conditional volatility of two indices: S&P CNX Nifty and BSE Sensex respectively for the period from 2nd January 1991 to 10th June 2003. Also see Schwert, 1989)

[FIGURE 3 & 4 OMITTED]

Volatility has been estimated on return (q) which is defined as

[r.sub.t] = log ([P.sub.t+1]/[P.sub.t]),

where [r.sub.t] is logarithmic daily return at time t and [P.sub.t-1] and [P.sub.t] are daily price index at two successive days t-1 and t respectively.

Some summary statistics of the [r.sub.t] are shown in Table 1. The average of the returns [r.sub.t] is positive implying the fact that price series has increased over the period. The statistics show that returns are positively skewed although the skewness statistics are not large. The positive skewness implies that the return distributions of the shares traded in our markets have a higher probability of earning positive returns. The value of the kurtosis is greater than 3, meaning that it has a heavier tail than the standard normal distribution. The Jarque-Bera test statistic provides clear evidence to reject the null hypothesis of normality for the unconditional distribution of daily return.

3.2 Volatility Clustering

Figure 1 exhibits the return series of the combined data for the period 2nd January 1980 to 10th June 2003. From the figure it appears that there are stretches of time where the volatility is relatively high and stretches of time where the volatility is relatively low which suggests an apparent volatility clustering in some periods. Statistically volatility clustering implies a strong autocorrelation in squared returns, so a simple method for detecting volatility clustering is to calculate the first-order autocorrelation coefficient in squared returns. To test the joint hypothesis that all the serial correlations of the returns for lags 1 through k, are simultaneously equal to zero, one can use modified Box-Pierce (Ljung-Box-Pierce or simply Ljung-Box) statistic (Q), developed by Ljung and Box, which is defined as Q = n (n + 2) [SIGMA] [r.sup.2.sub.k] / (n-k), where n = sample size and k = lag length (Ljung and Box, 1978). The Q statistic is approximately (i.e., in large samples) distributed as the Chi-square distribution with k d.f. In an application, if the computed Q exceeds the critical Q value from the Chi-square table at the chosen level of significance, one can reject the null hypothesis that all [r.sub.k] are zero; at least some of them must be nonzero. The value of [Q.sup.2](24) test statistic (reported in Table 1), rejects the joint hypothesis that all the serial correlations of the squared returns for lags 1 through k are simultaneously equal to zero, and thereby suggests the presence of volatility clustering in the return series.

[FIGURE 1 OMITTED]

One may be curious to know about the reasons for this volatility clustering. Engle, Ito, and Lin provide two possible explanations for volatility clustering. First, if information arrives in clusters, returns may exhibit clustering. Nominal interest rate, dividend yield, money supply, oil price, margin requirement, business cycles, and information patterns are the sources of volatility clustering. Second, if participants have different prior beliefs and take time to digest the information shocks and to resolves their expectation differences, market dynamics can lead to volatility clustering (Engle et al., 1990)

3.3 Fitting GARCH (1, 1) Models

Once volatility clustering is confirmed, our focus is on determining the fitted GARCH model applicable to the return series. We first estimate the parameters, namely [[alpha].sub.0], [[alpha].sub.1], and [[beta].sub.1], for the GARCH(1, 1) model, then compute the series [h.sub.t] that is plotted in Figure 2.

[FIGURE 2 OMITTED]

While running GARCH(1, 1) process, we get the following estimated conditional variance equation

(6) [h.sub.t] = 0.015 + 0.10 [[epsilon].sup.2.sub.t-1] + 0.89 [h.sub.t-1] (6.317) (13.88) (136.3)

where the [[epsilon].sub.t] are the residuals. The standardized residuals are defined as [[epsilon].sub.t] / [([h.sub.t]).sup.1/2]. Figures in parentheses are t-statistics. The coefficients of the variance equation are highly significant.

Figure 2 shows the time series plot for this estimated series of conditional variance. It is clear that the volatility behavior in Figure 2 behaves qualitatively like the apparent volatility variation in the returns of the Figure 1. Notice that the estimated volatility is high for some periods and low for other periods. Recall that [[beta].sub.1] is close to one and [[alpha].sub.0] and [[alpha].sub.1] are small. Since [h.sub.t] = [[alpha].sub.0] + [[alpha].sub.1][e.sup.2.sub.t-1] + [[beta].sub.1][h.sub.t-1], we see that [h.sub.t] tends to [h.sub.t-1]. In other words, large values of [h.sub.t] are clustered together and so the small values of [h.sub.t] (Figure 2). Large value of lag coefficient [[beta].sub.1] indicates that shocks to conditional variance take a long time to die out, so volatility is 'persistent'. The relatively small value of error coefficient [[alpha].sub.1] implies that large market surprises induce relatively small revisions in future volatility. The total value of reaction coefficients ([[alpha].sub.1]) and the persistence coefficients ([[beta].sub.1]), measured by [[alpha].sub.1] + [[beta].sub.1] is around 0.99 which is though less then 1 but close to the Integrated GARCH model of Engle and Bollerslev (1986). This implies that current information is relevant in predicting future volatility, also at very long horizons. The presence of the near-integrated GARCH being close to but slightly less than unity is found in several financial markets series (Bollerslev, 1986).

Another characteristic of the estimated model is its ability to capture leptokurtosis in the data, even at the conditional level. The high value of excess kurtosis reported in table 3 implies that the market is likely to be affected by big surprises, conditional on the information available at any point in time.

3.4 Diagnostics for the GARCH (1, 1) Models:

After we have fitted the model, it is appropriate to examine how well the GARCH (1, 1) model fits the data. If the GARCH (1, 1) model describes the data then standardized residuals should have zero mean and unit variance and be independently and identically distributed. The mean and variance shown in table 2 are found to be -0.0099 and 0.99786, which suggest that the GARCH (1, 1) model nearly describes the data. Table 2 reports the test statistics of GARCH (1, 1) model based on the standardized residuals.

We estimate the standardized residuals [[epsilon].sub.t] / [([h.sub.t]).sup.1/2] and the squared standardized residuals and compute the Ljung-Box (Q) statistic to test the null hypothesis of no autocorrelation up to order twenty four. This test is an alternative to the Lagrange Multiplier test proposed by Engle (1982) to evaluate the specification of a GARCH process. Bollerslev and Mikkelsen show that this test has considerably more power in detecting model misspecifications (Bollerslev and Mikkelsen, 1994).

Though Q(24) statistic indicates serial correlation in the standardized residuals, the [Q.sup.2] (24) statistic suggests no serial correlation in the squared standardized residuals. This suggests that the GARCH (1, 1) model has been success in explaining the data.

3.5 Volatility Shifting

As mentioned earlier Figure 2 shows the combined time series plot for the estimated series of conditional variance. Conditional volatility of the combined series of around 24 years shows a clear evidence of volatility shifting over the period where violent changes in share prices cluster around the boom of 1992.

During the beginning of the decade eighties, share price fluctuation was modest and there was a tired look of share price movement up to 1994. From the beginning of the second half of eighties, however, there were indications of change in the mood of the market. Volatility took a momentum from 1985, and from the beginning of Nineties it took a new turn and in the year 1992, it surpassed all previous records. This period experienced the highest volatility in the history of Indian stock market (Roy and Karmakar, 1995) and this coincided with initial years of liberalization of the Indian economy after a long era of control. Experts and policy makers were bewildered by this phenomenon and particularly after the meteoric fall in share prices in 1992 when the ever largest security scam was unearthed in the Indian Stock market, doubts were expressed about the capabilities of the market to support the liberalization policy of the government. Violent fluctuation of 1992 was followed by a tranquil period of around 4 years and volatility again continued to increase till the end of the decade when a series of security scam was revealed once again in the Indian stock market.

3.6 Explanation

Why the stock market volatility changes over time? What are the reasons for higher volatility in some periods and lower for other periods? The present value concept defined below may be helpful to get the answers of the above questions.

The theoretical value of a share ([P.sub.t]) may be defined as the discounted present value of expected future cash flows to stockholders as:

(7) [P.sub.t] = [E.sub.t-1] [SIGMA] [D.sub.t+k] / [(1 + [R.sub.t+k]).sup.k]

where [D.sub.t+k] is the capital gain pus dividend paid to stockholders in period t+k and 1/(1 + [R.sub.t+k]) is the discount rate for period t+k based on information available at time t-1. ([E.sub.t-1] denotes the conditional expectation.) The conditional variance of the stock price at time t-1, [var.sub.t-1]([P.sub.t]), depends on the conditional variances of expected future cash flows and of future discount rates, and on the conditional co-variances between these series (Schwert, 1989).

At the aggregate level, the value of corporate equity clearly depends on the health of the economy. If discount rate is constant over time in (7), the conditional variance of security prices is proportional to the conditional variance of the expected future cash flows. Thus, it is plausible that a change in the level of uncertainty about future macroeconomic conditions would cause a proportional change in the stock market volatility. Hence if the fundamental economic factors change over time, stock market volatility changes over time. "Fads" or "bubble" in stock prices would introduce additional sources of volatility.

What are the factors that contributed to gradual rise in volatility since the beginning of the second half of eighties and the highest volatility in 1992? Are they simply fundamental economic factors or a number of imperfections which are responsible for the vociferous movement in share prices?

One of the intuitively appealing explanations combines both fundamental factors and irrational behavior of investors while interpreting the gradual rise in volatility. The initial boost up of share prices and fluctuation apparently owed much to the strong fundamentals of the decade eighties which were supplemented by a number of liberalizing policies and procedures in financial sector when the new economic policy was launched in July 1991. During this phase, "trend chasing" investors seemed to be further encouraged by the optimism expressed by the government, the media, and leading financial advisers and the general public entered the market herds. Obviously, they joined in clusters which resulted in a gradual shift in the demand for shares in a basically thin market. When investors were in frenzy, speculators armed with outside money entered the market adding fuel to the fire. Prices started moving undeviatingly, fluctuations were high, and it culminated in a record high in March 1992 the year when the Indian economy was in deep crisis. Following the rule of the market, eclipse followed illumination and in April 1992, the "bubbles" burst and price started its downward journey. The formation and eventual burst of the bubble was a period of extreme volatility of the Indian stock market. After a stable period of 4 years stock market again witnessed excessive fluctuation of share price at the end of the last decade. Scandal once again unveiled shady nexus between speculators and outside money particularly public money to strengthen activities of noise traders. The market is still dominated by the speculators and noise traders who often manipulate the prices at the cost of general investors and drives the price away from the fundamental level causing excessive movement in share prices.

Social cost associated with the high volatility is heavy. Investors think that the security market is the province of the speculators only. They have got duped and withdrawn en masse from the market. In many stock exchanges including the Calcutta Stock Exchange, the second largest exchange in the country, trading has almost stopped. As reflected in the primary market, the fund mobilization through IPOs has almost dried up.

4. CONCLUSION

In this paper we have shown how some recently developed models for time series, particularly applicable to financial time series are used. The special feature of the models is that the series volatility is modeled as a function of the previous values of the variable. The simpler form of GARCH model has been tried on a combined series of the Economic Times Index and the Nifty for a successive period of about 24 years. The 'vanilla' GARCH (1, 1) model has been fitted to the series. Once the model has been fitted to return series, it can be used to forecast volatility. The conditional volatility for the series has been plotted in Figure 2 over the period from January 1980 to June 2003. In the figure, we find strong evidence of time-varying volatility. We also find that periods of high and low volatility tend to cluster. Also, volatility show high persistence and is predictable. Evidence in support of a fat-tailed conditional distribution of returns is found, implying that large changes in speculative prices are expected relatively often in our country.

From figure 2 it appears that the level of volatility for the decade Nineties is considerably higher than that of Eighties and the volatility level of the year 1992 is the highest in the history of Indian stock market. Truly higher price movement started in response to strong economic fundamentals. But the real cause for abrupt movement is the imperfection of the market. Information that is available so far particularly after unearthing of a series of security scams reveals that outside finance often used by the noise traders for speculative business, along with other market imperfections, played significant role behind the excess volatility. Blessed with public money, the noise traders, in fact dominated market in most of the time, and over-powered information traders when trading against them belying the hope of Friedman on arbitrage operation (Friedman, 1953). The outweighing noise traders thus simply destabilize the market and move security prices away from fundamentals, resulting in 'fads' or 'bubbles' as the natural outcomes in the price formation process.

TABLE 1: SUMMARY STATISTICS OF [r.sub.t]

                     Statistic   P-value

Mean                 0.0515
Standard Deviation   1.44671
Skewness             0.0374
Kurtosis             6.6828
Jarque-Bera test     10161.9     <.005
[Q.sup.2] (24)       2996.11     <.005
Maximum              12.09
Minimum              -12.52
N                    5472

TABLE 2: GARCH (1, 1) STATISTICS

                   Statistic   P-value

Mean               -0.0099
Variance           0.99786
Skewness           0.2276
Kurtosis           4.1460
Jarque-Bera test   3957.21     <.005
Q (24)             216.57      <.005
Q2(24)             34.47       0.077

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Dr. Madhusudan Karmakar earned his Ph. D. at North Bengal University in 1994. Currently he is an Assistant Professor of Finance at Indian Institute of Management Lucknow.