A Markov-Switching Model for Indian Stock Price and Volume

Abstract

Using weekly data from the Indian stock market, we examine the relationship between stock price and trading volume. Our framework is Markov Switching-Vector Error Correction Model (MS-VECM). We justify the use of nonlinear model using the Brock, Dechert and Scheinkman (BDS) test and the information criteria. The long-run dynamics are characterised by one cointegrating vector relating the price to trading volume. We find that stock price is weakly exogenous only in the high volatility regime. The MS-VECM with two regimes provides a good characterisation of the Indian stock market and performs well relative to the other linear and nonlinear models.

JEL Classification: C32, G12, E32

Keywords

Markov-switching model impulse response India

Introduction

Following Engle and Granger (1987), vast amount of research has been devoted to the analysis of cointegrated financial variables, for example, price–dividend or return–volume relationship (Gallant, Rossi & Tauchen, 1992; Lamoureux & Lastrapes, 1990; Tauchen & Pitts 1983). Most studies assume the adjustment towards equilibrium to be linear. The linearity assumption may not be satisfied and therefore could lead to model mis-specification and unreliable inference. For example, Hiemstra and Jones (1994) point out significant nonlinear dynamics between stock trading volume and prices.

Three main factors could have an impact on the relationship between price and volume, namely rate of information flow and its diffusion to the market, size of the market and existence of short selling constraints. Price changes can be interpreted as the evaluation of new information, whereas volume acts as an indicator measuring the disagreement amongst the investors about this information. Two stylised facts exist: (a) volume is relatively heavy in bull market and light in bear market, implying positive correlation between volume and returns and (b) it takes volume to make movement in prices, implying a positive correlation between volume and the magnitude of return.¹ The mixture distribution models of Clark (1973) and Epps and Epps (1976) claim that stock returns and trading volumes are jointly dependent on the same underlying latent information. Campbell, Grossman and Wang (1993) propose a model where a set of ‘noise’ traders cause changes in trading volume, which market makers observe as if their expected return is higher.² The possibility of a feedback where price movements might cause further changes in volume may not be ruled out.³ Most of the above-mentioned studies focus almost exclusively on the well-developed financial markets, usually the US markets. Few exceptions exist: Basci, Ozyildirim and Aydogan (1996) in case of Turkey, and Saatcioglu and Starks (1998) in case of Latin American countries. The results obtained are of mixed in nature.⁴

The relationship between volume and return could differ depending on regime, that is, different in a declining market compared to a rising market. Peters (1994) noted that stock prices and returns are cyclical, imperfectly predictable in the short run, and unpredictable in the long run. The price–volume relationship could thus exhibit nonlinear and possibly chaotic behaviour related to a time-varying positive feedback. Therefore, one needs to accommodate structural/regime changes. Lamoureux and Lastrapes (1990) pointed out that failure to allow for regime shifts or structural changes leads to an overstatement of the persistence of the variances of a series. Given this, numerous studies have analysed the relationship between economic and financial variables using the Markov switching (MS) framework; however, such studies are absent in case of price–volume relationship.⁵

Our article is an attempt in this direction. We examine the dynamic relationship between return and volume using the MS-VECM framework. We introduce the possibility of switches in the long-run equilibrium in a cointegrated vector-autoregressive model (VAR) by allowing switch in both: the covariance matrix and weighting matrix in the error-correction term. We find that two-regime MS model with changing intercept and variance represents our data well. We find the presence of one cointegrating relation between the stock price and the trading volume and the evidence of price being weakly exogenous in the MS-VECM framework.

We choose India as our application using the data from the Bombay Stock Exchange (BSE) over the period 1996–2003. At least two reasons can be cited for our choice. The BSE is the oldest in Asia and has the greatest number of listed companies in the world. The stock markets in India have undergone various policy reforms, namely introduction of derivatives of both stocks and market index, various scams and crises, changes in terms of clearing and settlement rules. Therefore, it would be worthwhile to examine the price–volume relationship in a regime-switching framework using data from India. In Table 1 we report the chronology of events that had taken place in India in the stock market during the period 1996–2003.

Table 1.

Chronology of Major Events Related to Indian Stock Market

Date	Event
Nov. 1996	National Securities Depository Limited (NSDL) began operations.
Oct. 1997	The Securities and Exchange Board (SEBI) reduced the margin on carry-forward deliveries.
March 1999	Finance Minister’s announcement of removal of taxes gains on investment in new issues.
June 2000	Start of equity index futures trading.
Feb. 2001	The President of the Bombay Stock Exchange (BSE) resigned.
June 2001	Start of equity index options trading.
April 2001	Ban on short sales. Securities lending to be applicable to the transactions under the ALBM and BLESS facility of NSE and BSE.
July 2001	SEBI halted the practice of carry-forward trading. Introduction of compulsory rolling settlement (T + 5), start of stock options.
April 2003	Change in rolling settlement from T + 3 to T + 2.

Source: Authors’ Calculation.

The rest of the article is organised as follows. The second section outlines the data and its properties motivating the empirical methodology. We describe the methodology in the third section. The fourth section presents the estimation results. Finally, we present our conclusion in the fifth section.

Data and Summary Statistics

Our data is a weekly aggregate of time series of daily trades and price data for individual stocks listed in the BSE during the period 1996–2003. We focus on turnover calculated as cumulative daily turnover as a measure of trading activity as in Lo and Wang (2000). Figure 1 displays the time series of weekly value-weighted turnover (and, its first difference) and price (return) for our BSE portfolio. The value-weighted turnover had increased dramatically until mid-2001, with a fall following the policy change related to rolling settlement, followed by a slow increase again after mid-2001.⁶ The growth from 1996 through mid-2001 may have been partly due to technological innovations that had lowered transaction costs. The volume had declined in the first quarter of 2001–2002 following decisions affecting several structural changes in the market including a shift to rolling settlement (initially in respect of major securities on T + 5 basis in July 2001, later for all securities) and ban on short sales. Such changes are usually accompanied by a fall in volume. Our claim is that the data generating process for the trading volume and the price series would, thus, be characterised by changing means implying different regimes. In Table 2, we report various summary statistics for the series over the sample period. The empirical distributions of the turnover series and the price series are positively skewed and are not normal.⁷ The first 12 autocorrelations and the corresponding Ljung– Box Q-statistics for the 12th order reported in Table 2 exhibit persistent behaviour in both series.

Figure 1.

Source: Authors’ Calculation.

Markov-switching Model

Hamilton (1989) has used the MS model to analyse the growth rate in US GNP.⁸ In our empirical framework, we use a more generalised model where the autoregressive parameters and the mean are regime dependent and the error term is heteroskedastic. We claim that y_t is a 2x1 vector consisting of price and volume and model y_t as a pth order autoregression process:

y_{t} = ν + A_{1} y_{t - 1} + \dots + A_{p} y_{t - p} + u_{t}, u_{t} ~ I I D (0, Σ)

(1)

Table 2.

Summary Statistics for Weekly Turnover and Price Series of BSE

Statistics	BSE
Statistics	V_t	P_t
Mean	0.0033	898.9619
Std.dev.	0.002	561.3146
Skewness	1.2946‡	2.57‡
Kurtosis	1.68‡	8.1171‡
Jarque–Bera	157.1841‡	1523.0553‡
Autocorrelations:
ρ1	0.9481	0.9768
ρ2	0.9101	0.9558
ρ3	0.878	0.9273
ρ4	0.8548	0.8938
ρ5	0.8316	0.8757
ρ6	0.8082	0.8537
ρ7	0.7899	0.8363
ρ8	0.764	0.8148
ρ9	0.7441	0.7919
ρ10	0.7255	0.7709
ρ11	0.711	0.7483
ρ12	0.7065	0.7301
Ljung–Box Q12	3177.8308‡	3515.0256‡
ADF Statisics	– 2.4259	– 1.4518
Sample Period	Jan. 1996-July 2003	–

Source: Authors’ Calculation.

Note: ‡ indicates significance at 0.01 level of significance.

We denote A(L) = I_k – A₁L – . . . – A_pL^p as the (K × K)-dimensional lag polynomial. Assuming the characteristic roots of Equation (1) lying outside the unit circle, we can express equation (1) as a mean-adjusted form of a VAR model with normal distribution for the error term:

y_{t} - μ = A_{1} (y_{t - 1} - μ) + \dots + A_{p} (y_{t - p} - μ) + u_{t}

(2)

where $μ = {(I_{K} - \sum_{j = 1}^{p} A_{j})}^{- 1} ν$ is the (K × 1)-dimensional means of y_t. To accommodate the regime-switching framework, we assume that the parameters of y_t depend upon the unobservable variable s_t. The unobservable realisation of the regime s_t ∈ (1, . . ., M) is governed by a discrete time, discrete state Markov stochastic process as defined by the following transition probabilities:

p_{i j} = \Pr (s_{t + 1} = j | s_{t} = i), \sum_{j = 1}^{M} p_{i j} = 1 \forall i, j \in (1, \dots, M) .

(3)

We extend Equation (2) to allow for the existence of M regimes:

y_{t} - μ (s_{t}) = A_{1} (s_{t}) (y_{t - 1} - μ (s_{t - 1})) + \dots + A_{p} (s_{t}) (y_{t - p} - μ (s_{t - p})) + u_{t}

(4)

where $u_{t} ~ (0, Σ s_{t})$ and n(s_t), A₁(s_t),. . . ., A_p(s_t), $Σ s_{t}$ are shift functions describing the dependence of the parameters on the realised regimes s_t. For a smooth transition of the mean of the process, we modify Equation (4) with regime-dependent intercept term o(s_t):

y_{t} = ν (s_{t}) + A_{1} y_{t - 1} + \dots + A_{p} y_{t - p} + u_{t}

(5)

The most general specification of an MS–VAR model where all the parameters are conditioned on the state s_t of the Markov chain can be expressed as:

y_{t} = {\begin{matrix} ν_{1} + A_{11} y_{t - ​ 1} + .... + A_{p 1} y_{t - p} + Σ_{1}^{1 / 2} u_{t} & i f s_{t} = 1 \\ ν_{M} + A_{1 M} y_{t - ​ 1} + .... + A_{p m} y_{t - p} + Σ_{M}^{1 / 2} u_{t} & i f s_{t} = M \end{matrix}

Assuming the examined series are I(1), following Krolzig (1997), we consider the VECM for the I(1) variables:

Δ y_{t} = ν (s_{t}) + α (s_{t}) (β y_{t - ​ 1}) + \sum_{k ​ = ​ 1}^{p ​ - ​ 1} Γ_{k} (Δ y_{t - k}) + u_{t}

(6)

where Δy_t is an m-dimensional vector of differenced variables of interest, o(s_t) is a regime-dependent intercept term, Γ_k are parameter matrices. We allow the error variance to change across states $u_{t} ~ (0, Σ (s_{t}))$ . In our specification, a(s_t) denotes the matrix of adjustment parameters, whereas b is the matrix of long-run parameters (cointegrating vectors). The unknown parameters are estimated by the maximum log likelihood function via Expected Maximum (EM) algorithm.

Estimation Results

Cointegration Analysis

We employ a three-step procedure in our empirical analysis. First, we conduct unit root tests on price and volume series. Next, we perform level regressions to examine the long-run properties between price and volume. Finally, lagged values of the residuals from the level regression are utilised in the error correction models for price and volume series.

We use both: the Augmented Dickey Fuller (ADF) test, and the Kwaitkowski, Phillps, Schimdt and Shin (KPSS, 2002) test. The ADF test may not be reliable in the presence of structural breaks. Therefore, we also perform Zivot–Andrews (1992) unit root test (see Zivot & Andrews, 1992) allowing for a single break both in intercept and trend.⁹ The volume and price series are found to be non-stationary as reported in Table 2.

We report Johansen’s (Johansen, 1988, 1991) cointegration test results in Table 3. We accept the hypothesis of one cointegrating relationship at the 2.5 per cent level, thus providing support for the presence of long-run relationship between stock prices and trading volumes. The outcomes of the test of parameter instability for the linear VAR model due to Hansen (1992a, b), Andrews (1993) and Andrews and Ploberger (1994)¹⁰ yield evidence against it.¹¹

Table 3.

Johansen Cointegration Test (BSE)

HypothesisedNo. of CE(s)	Eigenvalue	Trace StatisticsStatistics	2.5%Critical Value	5%Critical Value
None†At most 1	0.03480.0106	18.074.18	17.524.95	15.413.76
	Instability Test
SupLR	28.6770 (0.0043)
ExpLR	9.4395 (0.0304)
AvgLR	8.3133 (0.2582)

Source: Authors’ Calculation.

Notes: † denotes rejection of the hypothesis at 2.5 per cent level of significance.

p-values reported in parenthesis are due to Andrews (1993).

Given the existence of one cointegrating relationship, we estimate the linear VECM model using the residuals from the Johansen model.¹² Table 4 reports the estimates of the linear VECM model. The coefficient of the error component term for the price equation is insignificant, implying that price seems to be weakly exogenous.¹³ A test of nonlinearity on the residuals of the linear VECM following Brock, Dechert and Sheinkman (1987; henceforth, BDS) confirms the presence of nonlinearity in the residuals.¹⁴

Table 4.

ML Estimation Results for the Linear VECM Parameters (BSE)

	Δv_t	Δp_t
Intercepts
v ₁	0.0089 (0.9367)	0.0001 (0.0268)
Short run dynamics
Δv_t_–1	–0.1452 (–2.8077)***	–0.0053 (–0.2972)
Δv_t_–2	–0.0924 (–1.7872)*	–0.0070 (–0.3908)
Δv_t_–3	–0.0953 (–1.8987)*	–0.0030 (–0.1742)
Δp_t_–1	0.1907 (1.2814)	0.0995 (1.9377)*
Δp_t_–2	0.0605 (0.4074)	0.1367 (2.6658)***
Δp_t_–3	–0.0449 (–0.3013)	–0.0071 (–0.1371)
Error correction
ecm_t _–1	–0.0694 (–3.2291)***	0.0093 (1.2520)
Standard errors
σ	0.1936	0.0654
Correlation
Δv_t	1.0000	0.1219
Δp_t	0.1219	1.0000
BDS Test	3.2804‡	8.0987‡
AIC/HQ	–3.1013	–3.025

Source: Authors’ Calculation.

Notes: ‡ indicates significance at 1 per cent level of significance.

t-statistics are given in parenthesis.

*** denotes significance at 1 per cent, ** at 5 per cent and * at 10 per cent level, respectively.

MS-VECM Model

We have used a combination of tests to find the correct specification of the MS model.¹⁵ We use information criteria (AIC/HQ) based tests to determine the number of regimes and select an MSIAH (Markov Switching Intercept Autoregressive Heteroskedastic)-VECM model with two regimes and three lags. We allow for shift in intercepts, in error variances, in coefficients of autoregressive parameters as well as in the error-correction terms.¹⁶

We present our results in Table 5. Regime 1 corresponds to the high volatility regime, whereas regime 2 to the tranquil regime. The error correction coefficient for the price equation is significant in regime 1 but insignificant in regime 2. The error correction coefficient for the volume equation is significant in regime 2, but not in regime 1. This implies that price is weakly exogenous in the high volatility regime, whereas volume turns out to be weakly exogenous in the low volatility regime. The coefficient of the error correction terms reflect speedy adjustment in the high volatility regime. We are unable to document this in the linear VECM model.

Table 5.

ML Estimation Results for the MSIAH(2)-VECM Parameters (BSE)

	Δv_t	Δp_t
Regime 1
v₁	–0.001376 (–0.052)	–0.018399 (–1.1801)
Δv_t_–1	0.399345 (3.0628)	–0.067638 (–0.8479)
Δv_t_–2	–0.192629 (–1.2476)	0.016537 (0.1941)
Δv_t_–3	0.041921 (0.3149)	0.066599 (0.856)
Δp_t_–1	–0.098098 (–0.4982)	0.070666 (0.5572)
Δp_t_–2	0.026273 (0.1353)	0.15648 (1.2216)
Δp_t_–3	0.083581 (0.413)	0.059296 (0.4493)
ecm_t _–1	–0.108515 (–2.2162)	–0.00406 (–0.1338)
σ ₁	0.174062	0.116719
Regime 2
v ₂	0.006299 (0.5638)	0.005056 (2.1283)
Δv_t_–1	–0.297387 (–4.2846)	0.009301 (0.7028)
Δv_t_–2	–0.158704 (–2.5475)	–0.007849 (–0.6104)
Δv_t_–3	–0.136988 (–2.3913)	–0.017224 (–1.4292)
Δp_t_–1	0.494591 (1.8855)	0.066085 (1.1552)
Δp_t_–2	0.112984 (0.4852)	0.102302 (1.9255)
Δp_t_–3	–0.271287 (–1.2463)	–0.099888 (–2.0713)
ecm_t _–1	–0.044704 (–1.5535)	0.010611 (1.8106)
σ ₂	0.17927	0.038741
Regime 1 Correlations
Δv_t	1.0000	0.1152
Δp_t	0.1152	1.0000
Regime 2 Correlations
Δv_t	1.0000	0.2258
Δp_t	0.2258	1.0000
Loglik	718.4086
AIC/HQ	–3.4613	–3.3007

Source: Authors’ Calculation.

Note: t-statistics are given in parenthesis.

The return volatility in regime 1 of the price equation is 0.117 and it is almost three times higher than that of regime 2. The first regime characterises the events or news that are disseminated to the public leading to a price change to reflect these news. However, little trading would occur and so volume is less volatile in the first regime. Current weeks’ return is not related to past weeks’ volumes in either of the two regimes. Current weeks’ return is positively and significantly related to past return (lag 2) and negatively and significantly related to past weeks’ return (lag 3) in regime 2. Current week’s volume is positively and significantly related to past weeks’ volume in regime 1. Negative and significant coefficients are associated with the past weeks’ volume in regime 2 (for all the lags). We also observe that in regime 2 past weeks’ return significantly and positively affects the current volume. The correlations between the variables conditional on the past differ across regimes.

The resulting regime probabilities are plotted in Figure 2. The filtered probability represents the conditional probability based on the information contained in the information set and observed up to date t. The smoothed probability, on the other hand, represents the conditional probability based on the information available throughout the whole sample at future date T. In MS models, the classification of the regimes and dating amounts to assigning every observation to a regime s_t. At each point in time, the smoothed regime probabilities are calculated. In case of two regimes, the classification rule simplifies to assigning the observation to the first regime if Pr(s_t = 1|Y_T) > 0.5 and to the second if Pr(s_t = 1|Y_T) < 0.5.

The regime 1 probabilities characterise the period of episodes (events) mainly like February–March 1999 (the then Finance Minister Mr Jaswant Sinha’s second budget announcement of reduction in capital tax gains and new incentives in mutual funds), September/October 1999 to October 2000 (consisting of major events like the general elections in 1999, rising oil prices, start of equity index future trading, resignation of the BSE president), April 2001 (ban on short sales), July–September 2001 (introduction of rolling settlement, lift of ban on short sales, World Trade Center Disaster), and March–April 2003 (changes in rolling settlement from T + 3 to T + 2, lowering of interest rates to the lowest level in 32 years by the Central Bank of India, and US-led attack on Iraq). Such domestic and international macroeconomic events are said to have important impact in regime 1.

Figure 2.

Source: Authors’ Calculation.

Table 6 reports the transition probability matrix, whereas in Table 7 we present the number of observations in each regime along with the unconditional probability and the half-life or expected durations for each regime. The ergodic probability of being in state s_t = 1 is given by:

π_{1} = \frac{1 - p_{11}}{2 - p_{11} - p_{22}}

(7)

Expected duration of the first regime is calculated as:

d_{1} = \frac{1}{1 - p_{11}}

(8)

Similarly, we have calculated the ergodic probability r₂ and expected duration d₂ for the second regime. We note that the unconditional probability of being in state s_t = 1 is 0.225 with an estimated durations of 13.9 weeks. Given the above findings, we have also performed the regime classification measure following Ang and Bekaert (2002) to ascertain the performance of the model. We define this for m regimes as:

R C M (m) = 100 m^{2} \times 1 / T \sum_{t ​ = 1}^{T} ​ \prod_{i ​ = 1}^{m} p_{i, t ​},

(9)

where p_i,t is the ex-ante smoothed regime probability. Good regime classification is associated with a low value of the measure. The regime classification measure stands at 13.9. In sum, we conclude that the use of MS-VECM is appropriate for the Indian stock market to examine the relationship between price and volume.

Table 6.

Matrix of Transition Probabilities

	Regime 1	Regime 2
Regime1	0.9281	0.0719
Regime2	0.0209	0.9791

Source: Authors’ Calculation.

p_ij = Pr (s_t_{+ 1} = j | s_t = i).

Table 7.

Regimes and Duration (BSE)

	No. of Obs.	Ergodic Probability	Duration
Regime 1	89.3	0.2251	13.9
Regime 2	302.7	0.7749	47.86

Source: Authors’ Calculation.

Note: Probability is the unconditional one.

Figure 3 displays the cumulative impulse response functions of the volume and return series following a one standard error shock. In both regimes, a shock to the volume would lower the return and the volume almost by the same magnitude. However, a shock to the return shows the opposite: both return and volume increas but the magnitude of the increase differs. In the high volatility regime (regime 1), return increases by 12 per cent whereas only by 4 per cent in the low volatility regime (regime 2). Response of the volume remains around 4 per cent in regime 1 compared to 1.5 per cent in regime 2. The impact of the shock persists over the period.

Figure 3.

Source: Authors’ Calculation.

Conclusion

Our article uses the MS-VECM model to examine the joint dynamics of stock price and trading volume. To test for the MS model, we have utilised a two-step approach which involves first testing for cointegration under the assumption of linear adjustment and then testing for MS behaviour in the dynamics of the error-correction model. Thus, we first investigated linear relationships between two variables using the standard VECM. Results of the Johansen’s technique support the evidence of one cointegrating vector. Using standard linear VECM, we find the evidence of stock price being weakly exogenous for both the stock markets. However, the test on residuals from the linear model supports the presence of nonlinear pattern in the two variables. To control for nonlinearity, we implement a regime switching VECM. The evidence of stock price being weakly exogenous is only true in the high volatility regime. Most investors in the Indian stock markets are late in the informational queue and only trade for some time after new information hits the market. This explanation is easily conceivable in the Indian stock market where the state of their development might not allow spontaneous information dissemination. In general, the information arrival here is likely to be sequential. Investors who based their investment strategies on momentum have to adjust their strategies when trading on Indian stock markets in the high volatility regime.

Footnotes

Acknowledgements

We thank Yongcheol Shin for helpful comments that have helped to improve the article substantially.

Notes

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