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lag structure and microstructure effects in the Indian stock market
Int. Fin. Markets, Inst. and Money 14 (2004) 385–400
Market capitalisation, cross-correlations, the lead/lag structure and microstructure effects in the Indian stock market Sunil Poshakwale∗ , Michael Theobald Department of Accounting and Finance, The Birmingham Business School, The University of Birmingham, Edgbaston, Birmingham B15 2TT, UK Accepted 6 December 2003 Available online 18 May 2004
Abstract The lead/lag relationship between portfolios comprising large and small cap firms is analytically derived in terms of their speeds of adjustment and degrees of thin trading. Using a number of Indian equity index series that differ in their market capitalization characteristics, large cap indices are found to lead small cap indices and to have higher speeds of adjustment towards intrinsic values. However, pure thin trading effects and an interaction effect between thin trading and speeds of adjustment are found to make significant contributions to the lead/lag effect. An empirical analysis of the underlying intrinsic value process using a reduced form model developed in the paper indicates that a small degree of overreaction is present in intrinsic values series. © 2004 Elsevier B.V. All rights reserved. JEL classification: G14; G15; C51 Keywords: Lead/lag structure; Thin trading; Large and small capitalisation stocks
1. Introduction The impacts of market capitalization upon a variety of phenomena have been investigated since the size effect was first documented by Banz (1981). In this paper we investigate size related impacts upon the price adjustment process in an emerging market, India, to establish ∗
Corresponding author. Tel.: +44-121-414-7792; fax: +44-121-414-6238. E-mail address: [email protected] (S. Poshakwale).
1042-4431/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.intfin.2003.12.001
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the generality of the results documented in developed markets to less developed markets. A number of empirical studies have established that lead/lag relationships can occur across portfolios, particularly when formed on a size-related basis. That is, portfolios comprising large market capitalization stocks tend to lead the corresponding small market capitalization portfolios (see Campbell et al. (1997) for a summary and overview of this literature). One prime candidate for explaining the size-related lead/lag or cross-correlation structures is thin trading, since, in general, smaller market capitalization stocks are more thinly traded than larger market capitalization stocks. The differential in the degree of thin trading will cause smaller stock prices to lag behind those of their larger counterparts. Since portfolios are used in these studies and since thin trading effects are more prevalent in portfolios than for individual stocks, the rationale appears more compelling. However, Campbell et al. (1997) conclude that “. . . the recent empirical evidence provides little support for non-trading as an important source of spurious correlation in the returns of common stock over daily and long frequencies” (Campbell et al. (1997), p. 134). While many papers confirm that thin trading will not be the sole source of autocorrelations, there is some disagreement with regard to the extent to which thin trading leads to such phenomena. For example, Boudoukh et al. (1994, p542) argue that “. . . the effect of non-synchronous trading has most probably been understated in the literature”. In essence, then, the econometrician must determine the extent to which the lead/lag relationship derives from sluggish relative price adjustments, per se, and the extent to which this phenomenon reflects thin trading effects. We return to this issue in this paper by formally modelling the extent to which prices under or over-react relative to their intrinsic values using a partial-adjustment with noise model (Amihud and Mendelson (1987)). By modelling both the thin trading and the partial adjustment processes we are able to establish the extent to which cross-correlations derive from partial adjustments and the extent to which they derive from thin trading. Since the speed of adjustment and thin trading effects are distinct phenomena with differing implications it is important to be able to distinguish between them. Within the partial adjustment modelling framework market prices are assumed to partially adjust towards their intrinsic values. To develop a reduced form equation in terms of observable variables, a stochastic process has to be assumed for the intrinsic values; in general, the process is assumed to be a random walk. However, the intrinsic value process itself may vary with market capitalization and, indeed, as between developed and emerging markets. The determination of the actual process is, of course, an empirical issue and in this paper we develop a modelling structure for price adjustments with a more general underlying intrinsic value process. The resultant closed form equation that obtains affords the means of estimating both the speed of adjustment and the form of the intrinsic value process. In this fashion we are able to establish the sensitivity of the estimates of adjustment speeds to assumptions regarding the underlying intrinsic value adjustment process. The lead/lag and cross-correlation effect deriving from differing market capitalizations is investigated by using Indian equity market indices that differ in their market capitalization characteristics. The Nifty senior and the SENSEX indices which are employed in this study reflect more strongly large cap properties than the Nifty junior and BSENI indices. On a priori grounds, and given the empirical results established in other markets (e.g. Campbell et al. (1997)) the former indices would be anticipated to lead the latter indices and to have more rapid adjustments towards intrinsic values. However, the latter indices would be
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anticipated to be subject to greater degrees of thin trading. The procedures developed in this paper provide the means of establishing whether the lead/lag effects documented in these indices derive from speed of adjustment effects or thin trading effects. The plan of this paper is as follows. In Section 2 the modelling relationships that underpin the empirical work are developed. Specifically, the relative contributions of speed of adjustment and thin trading effects to the lead/lag relationship across differing portfolios are analytically developed. The impacts of differing intrinsic value processes upon the observed price adjustment process are also analytically established. Section 3 presents the empirical results that are obtained from using the data set of Indian equity market indices, while conclusions are presented in Section 4. 2. Modelling relationships 2.1. Thin trading effects Thin trading and less than full price adjustments towards intrinsic value will both induce autocorrelations in return series—and in a relatively similar fashion (see, for example, Theobald and Yallup (2001)). Earlier estimators of partial adjustment factors (such as Damodaran (1993) as corrected by Brisley and Theobald (1996)) did not adjust for thin trading; in Theobald and Yallup (1998, 2004), estimators were developed that incorporate thin trading. A na¨ıve approach to evaluating the differential impacts of these two phenomena would be to estimate the partial adjustment factors with and without correcting for thin trading1 (perhaps establishing the statistical significance of differences) and then evaluating the cross-covariance predictions of both estimates. However, as will be demonstrated in this paper there is an interaction between these two effects which means that such an approach would be misspecified. Theobald and Yallup (1998) demonstrate that, in the absence of thin trading, an estimator of the speed of adjustment factor, g(i) is given by (1 − g(i)) =
cov{R(i, t), R(ic , t − 1)} cov{R(i, t), R(ic , t)}
(1)
where R(i, t) is the return on stock (or instrument) “i” in period “t”; the “c” superscript denotes a complement (i.e. other stock or instrument) and “cov” is the covariance operator. The partial adjustment coefficient, g(i), represents the speed of adjustment of the natural logarithm of prices, P(i, t), towards the natural log of intrinsic values, V(i, t) as P(i, t) − P(i, t − 1) = g(i){V(i, t) − P(i, t − 1)} + u(i, t)
(2)
where u(i, t) is an i.i.d. noise term.2 Using Eq. (1) in the context of the small/large cap lead/lag relationship, we have 1 By for example using a thin trading “error purger” approach such as that employed in Stoll and Whaley (1990). 2 Intrinsic values are assumed to follow a logarithmic random walk with drift in Amihud and Mendelson (1987). Full adjustment towards intrinsic values would correspond to the case where g(i) = 1, underreactions arise where g(i) < 1 and overreactions occur with g(i) > 1.
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cov{R(L, t − 1), R(S, t)} = (1 − g(S))[cov{R(L, t), R(S, t)}]
(3)
i.e. large cap stocks or portfolios, with returns R(L, t), will lead small cap stocks or portfolios, with returns R(S, t), when g(S) < 1 where g(S) is the speed of adjustment of small cap stocks, i.e. small stocks do not fully adjust towards intrinsic values,3 as per Eq. (2). However, this relationship is predicated on the absence of thin trading. If we model the thin trading process as ∅(i)R(L, t − i) + v(m, L, t) (4a) R(m, L, t) = i=0
and R(m, S, t) =
θ(j)R(S, t − j) + v(m, S, t)
(4b)
i=0
see, for example, Cohen et al. (1986); Theobald and Price (1984); Stoll and Whaley (1990); Miller et al. (1994), with “m” indicating a measured return (i.e. subject to thin trading and its absence denoting a “true” return; that is, one not subject to thin trading), ∅(i) and θ(i) variables such that when ∅(o) and θ(o) equal one there is no thin trading and {v(m, L, t), v(m, S, t)} are noise terms, then cov{R(m, L, t − 1), R(m, S, t)} = ∅(i)θ(i) cov{R(L, t − 1), R(S, t)} (i = 0) + ∅(i)θ(i + 1) cov{R(L, t), R(S, t)} (i = 0) + ∅(i + 1)θ(i) cov{R(L, t − 2), R(S, t)} (i = 0) + ∅(i + 2)θ(i) cov{R(L, t − 3), R(S, t)} (i = 0) + · · · + ∅(i)q(i + 2) cov{R(L, t), R(S, t − 1)} (i = 0) + · · · (5) If we assume that stocks trade in consecutive periods (Scholes and Williams (1977); Miller et al. (1994)), with a stationary cross-covariance structure, Eq. (5) can be simplified as, cov{R(m, L, t − 1), R(m, S, t)} = {∅(0)θ(0) + ∅(1)θ(1)}[cov{R(L, t − 1), R(S, t)}] + ∅(0)θ(1)cov{R(L, t − 1), R(S, t − 1)} + ∅(1)θ(0)cov{R(L, t − 2), R(S, t)} (6) If there are no leads and lags in the “true” returns processes (e.g. true prices fully adjust towards their intrinsic values), then cov{R(m, L, t − 1), R(m, S, t)} = ∅(0)θ(1)cov{R(L, t − 1), R(S, t − 1)}
(6a)
That is, the lead/lag cross-covariances induced by thin trading are summarised at Eq. (6a). In the presence of small cap speeds of adjustments less than complete (i.e. g(S) < 1), there will be cross-covariances induced in the true series. That is, using the relationship at 3 The adjustment speed of the large cap stocks may also be reflected in Eq. (3) by substituting for cov{R(L,t), R(S,t)} as cov{R(L, t − 1), R(S, t)} = (1 − g(S))(1 − g(F))−1 cov{R(L,t), R(S,t − 1)}. Since g(L) > g(S) and could be equal to one, we work with Eq. (3) to avoid non-determinancies due to zero denominations.
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Eq. (3) in Eq. (6), above, we have cov{R(m, L, t − 1), R(m, S, t)} = {(∅(0)θ(0) + ∅(1)θ(1))(1 − g(S)) + ∅(0)θ(1) + ∅(1)θ(0)(1 − g(S))2 } × cov{R(L, t), R(S, t)}
(7)
That is, the joint impacts of thin trading and tardy speeds of adjustments comprise three terms, two of which entail interactions between these two phenomena. Eq. (7) still contains the thin trading term at Eq. (6a). Note that when the thin trading is concentrated in small stocks only, Eq. (7) becomes cov{R(m, L, t − 1), R(m, S, t)} = {θ(0)(1 − g(S)) + θ(1)}cov{R(L, t), R(S, t)} = {1 − θ(0)g(S)}cov{R(L, t), R(S, t)}
(8)
2.2. Intrinsic value process The partial adjustment with noise process assumes that market prices only partially adjust towards their intrinsic or full information values. The intrinsic value process is commonly assumed to be a random walk (Amihud and Mendelson (1987)). There is, however, empirical support for stock prices following, for example, a mean-reversion process (see, for example, Poterba and Summers (1988), Lo and MacKinlay (1989) and Richardson and Stock (1989)). While this mean-reversion could be a manifestation of microstructure effects such as thin trading (see, for example, Campbell et al. (1997), p89), the intrinsic value process could still differ across small and large cap stocks/portfolios due to other intrinsic differences between these classes of stocks. The “behavioral” literature provides rationales for non-random walk adjustment processes (for example, Barberis et al. (1998); Daniel et al. (1998)) where price series may manifest both over and under-reaction characteristics over differing return horizons. While such model types may provide explanations for incomplete or over adjustments to intrinsic values, they also indicate that the underlying process itself may not necessarily follow a random walk. Differential information availability across market capitalisation fractiles may also lead to differing underlying processes. Irrespective of causes and rationales, the nature of the underlying process is an empirical question and it is investigated by positing a process of the form V(i, t) = γ(i)V(i, t − 1) + e(i, t)
(9)
where γ(i) is the process parameter and e(i,t) the process disturbance.4 When γ(i) = 1, the intrinsic value process is a random walk, while with γ(i) < 1 the process is mean reverting, for example. The reduced form that is obtained by combining Eqs. (2) and (9) is P(i, t) − P(i, t − 1) = (γ(i) − g(i))[P(i, t − 1) − P(i, t − 2)] − g(i)(1 − γ(i))P(i, t − 2) + g(i)e(i, t) + u(i, t) − γ(i)u(i, t − 1) 4
A drift term was omitted as its inclusion had no substantive impact upon the analytical structure.
(10)
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Thus, by estimating {g(i), γ(i)} across large and small cap deciles, insights into the impacts of the intrinsic value process upon parameter estimates can be gleaned and differences in the intrinsic value processes across differing market capitalizations established. The parameters in Eq. (10) arise in a non-linear fashion and, as a consequence, Eq. (10) is estimated by non-linear least squares.5
3. Empirical results 3.1. Sample Return data on four Indian stock market indices were used—the Nifty senior and junior indices (from January 2, 1997 to February 9, 2001) and the SENSEX and BSENI indices (from January 2, 1990 to February 9, 2001). Daily closing prices for the indices have been collected from Datastream International. The Nifty senior index is a well diversified index and comprises 50 stocks which represent 25 sectors of the economy. The base period for the index is November 3, 1995 with a base value of 1000 and base capital of Rs.2.06 trillion (US$46bn approx.). Market participants use this index for benchmarking fund portfolios, index based derivatives and index funds. The index is computed using market capitalization weights. All companies to be included in the Nifty senior index should have a minimum market capitalization of Rs.5 billion (US$112m approx.) and its stock should have traded on 85% of the available trading days. The Nifty senior index represents 45% of the total market capitalization. The Nifty junior index was introduced on January 1 1997, with a base value of 1000, a base capital of Rs. 0.43 trillion (US$9bn approx.) and includes 50 companies. A listed company needs to have a minimum market capitalization of Rs. 2 billion (US$45m approx.) and its stock should have traded on 85% of the trading days. The index represents about 7% of total market capitalization. Given the characteristics of these two indices, the Nifty senior index would be anticipated to manifest larger capitalization properties, the Nifty junior index smaller capitalization properties. The SENSEX includes equity shares of the top 30 companies selected on the basis of market capitalization, liquidity, depth, trading frequency and industry representation. The selected stock should figure in the top 100 companies listed by market capitalization. Also, the market capitalization of the stock should be more than 0.5% of the total market capitalization of the index i.e. the minimum weighting should be 0.5%. Since the SENSEX is a market capitalization weighted index, this is one of the primary criteria for stock selection. The index was launched on 2 January 1986 with 1978–1979 as the base year. The BSENI index is comprised of 100 shares listed on India’s five major stock exchanges at Mumbai (Bombay), Calcutta, Delhi, Ahmedabad and Chennai (Madras). There is no minimum market capitalization requirement for inclusion. However, the selected stock should 5 In testing for the statistical significance of γ(i) from one in Eq. (10) we do not face the stationary/non-stationary stochastic process problems that are inherent in the unit root literature. That is, the reduced form of the estimating equation at Eq. (10) is stationary whether or not γ(i) is equal to unity. As such, then, in our estimations we do not introduce “Dickey Fuller” type modifications to the standard errors deriving from the non-linear least squares estimator.
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Table 1 Descriptive statistics for the indices (daily log price relatives) Descriptive statistic
Mean Std. Dev. Skewness Kurtosis Jarque Bera
Index Nifty senior
Nifty junior
0.038 1.77 −0.11 5.67 321.42 (0.00)
0.077 2.14 −0.29 4.86 170.04 (0.00)
SENSEX
BSENI
0.05 1.93 0.34 13.70 11131.88 (0.00)
0.05 1.65 0.48 13.28 1032.30 (0.00)
P-values for the Jarque–Bera statistic are contained in parentheses.
have been traded on every trading day for the last one year and should be among the top 150 companies listed by the average number of trades per day for the previous year. These criteria may be relaxed in exceptional cases such as trading suspensions, etc. Due consideration is also given to the representation of various industry-groups and the stock’s trading record on major stock exchanges. The BSENI index used to take prices of certain stocks from other exchanges in order to reflect market movements at the national level. However, changes in trading technology, longer trading periods and almost instantaneous availability of information across the country have ensured that there is little or no difference in the prices of the index stocks. Therefore, the Exchange administration decided to redesignate the BSENI as the ‘BSE 100’ index and since October 14, 1996, only prices on the Bombay Stock Exchange (BSE) are taken to calculate the index. The index was started in January 1989 with 1983–1984 as the base year. The SENSEX index would be anticipated to reflect larger cap stock characteristics than the BSENI index as a result of the previously described compositions for the indices. The descriptive statistics of each of the index return series are contained in Table 1, where it can be seen from the Jarque–Bera statistics that all series are statistically significantly non-normal and that for those series reflecting the latter part of the 1990s the return distributions are negatively skewed as found in most international stock markets during that period. The Nifty junior index is the most volatile in terms of standard deviations. 3.2. Cross-correlations The contemporaneous and one period leading and lagging return cross-correlations between the Senior and Junior Nifty indices and the SENSEX and BSENI indices are contained in Table 2 for the whole dataset for each index pair and for a number of 1 year calendar periods. Prior empirical research in the US (e.g. Campbell et al. (1997)) would suggest that from the descriptions of the indices the Nifty senior series should lead the Nifty junior and that the SENSEX should lead the BSENI.6 As can be seen in Table 2 the contemporaneous cross-correlations are all relatively high, indicative of the presence of common factors un6
The use of stock market indices with differing market capitalization characteristics to investigate size related phenomena goes back to Roll (1981).
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Table 2 Cross-correlation structures across differing indices Sample frame
Correlation type
Cross-correlation between Nifty senior and Nifty junior
Whole data
seta
SENSEX and BSENI
Contemporaneous Lead Lag
0.7891 0.0750 0.0200
0.7801 0.1641 0.1483
2000–2001
Contemporaneous Lead Lag
0.7639 0.1085 0.0339
0.9297 0.0306 0.0208
1999
Contemporaneous Lead Lag
0.7951 0.0471 −0.260
0.9678 0.0081 −0.0184
1998
Contemporaneous Lead Lag
0.8176 0.0120 0.0080
0.9185 0.0335 0.0346
1997
Contemporaneous Lead Lag
0.8862 0.1192 0.0539
0.9185 0.0351 0.0351
1996
Contemporaneous Lead Lag
– – –
0.9235 0.1357 0.1808
1995
Contemporaneous Lead Lag
– – –
0.8497 0.3461 0.1056
1994
Contemporaneous Lead Lag
– – –
0.8027 0.2989 0.1531
1993
Contemporaneous Lead Lag
– – –
0.7716 0.2837 0.1686
1990–1992
Contemporaneous Lead Lag
– – –
0.7349 0.1465 0.1656
a The whole sample frame comprises the period 1997–2001 for the Nifty series and 1990–2001 for the SENSEX and BSENI series.
derpinning the return generating processes for each of these indices. In the majority of cases investigated, the index reflecting the larger capitalization stocks leads the lesser capitalization index more strongly than lagging it. This is more particularly the case for the Nifty index series where potentially, the market capitalization differences are more apparent and consistent. In conclusion, then, evidence of leading cross-correlations driven by capitalization differentials is demonstrated to be present in the Indian stock market series analysed in this paper and having empirically established this phenomenon we will now proceed to investigate the relationship further in terms of speeds of adjustment and thin trading effects.
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3.3. Speeds of adjustments estimates A number of differing estimators were proposed and discussed in Section 2 of this paper. In this section we utilise a number of these estimators to estimate the speeds of adjustments of the four indices that comprise our sample. Four differing estimators are used:7 (i) the cross-covariance ratio estimator (Theobald and Yallup (1998)) where the speed of adjustment, g(i), i = S, L, is given by g(S) = cov[R(L, t − 1), R(S, t)]{cov[R(L, t), R(S, t)]}−1
(11)
g(L) = cov[R(L, t), R(S, t − 1)]{cov[R(L, t), R(S, t)}−1 ;
(12)
and
(ii) an ARMA(1,1) model ((Theobald and Yallup (2004)) where the stochastic process underpinning the speed of adjustment estimator is R(i, t) = (1 − g(i))R(i, t − 1) + g(i)e(i, t) − u(i, t) − u(i, t − 1)
(13)
where e(i, t) is the innovation in the intrinsic value process and u(i, t) the noise in the speed of adjustment process; (iii) an ARMA (1, X) model (Theobald and Yallup (2004)), where X, the optimal moving average order which reflects thin trading effects is determined by the Schwartz Information Criterion, and; (iv) an AR(1) model8 which assumes an absence of spread and noise effects in the ARMA (1,1) model.9 If the capitalization induced lead/lag relationship derives purely from large cap stocks (indices) adjusting more rapidly to information than smaller cap stocks (indices), then the speeds of adjustment for the former stock types should be larger than for the latter stock types. If thin trading has an impact upon this lead/lag relationship, then controlling for this effect should narrow the speed of adjustment differential and lead to speeds of adjustment closer to full (i.e. unity). The results that derive from using these four estimators are contained in Table 3. As can be seen in that table the results broadly conform to the two predictions made in the previous paragraph. That is, for all estimators the speed of adjustment for the larger cap index (the Nifty senior and SENSEX) is more commonly larger than for the index that reflects smaller capitalization stocks10 (the Nifty junior and BSENI); furthermore, the occurrence of adjustments statistically significantly different from one is more frequent for the smaller 7 A fifth estimator which does not assume that the underlying process follows a random walk (i.e. Eq. (10)) is also used, but we defer discussion of this model’s results to Section 3.4. 8 This model will relate directly to the cross-covariance ratio estimator when second moments are stationary. 9 Damodaran (1993) and Brisley and Theobald (1996) estimator is not used as this has poor sample properties as demonstrated in Theobald and Yallup (1998, 2004). 10 Potential sectoral impacts were investigated by estimating speeds of adjustments for two large cap sectors—banks and petrochemicals—and two small cap sectors—engineering and hotels. The speeds of adjustments for the large cap sectors estimated from the ARMA(1,1) specification for the period 1/2/1995 to 2/9/2001 were larger than the corresponding estimates for the two small cap sectors, consistent with the results for the broad based market indices.
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Table 3 Estimates of adjustment speeds and underlying process Sample frame
Speed of adjustment
Gamma
AR(1)
ARMA(1, 1)
ARMA(1, X)
Cross-covar. ratio
Eq. (10)
Eq. (10)
Whole data set Nifty senior Nifty junior SENSEX BSENI
0.955 0.913* 0.966 0.892*
0.604 0.597 0.397 0.527*
0.956 0.920* 0.967 0.891*
0.905 0.975 0.839 0.858
0.876 0.788** 0.983* 0.947*
0.923 0.916** 1.002* 1.005*
2000–2001 Nifty senior Nifty junior SENSEX BSENI
0.962 0.881 0.994 0.919
0.220* 0.513 0.244* 0.284*
0.960 0.891 0.996 0.926
0.861 0.958 0.967 0.978
0.975* 0.955* 0.984* 0.945*
1.002* 1.002* 1.000* 1.004*
1999 Nifty senior Nifty junior SENSEX BSENI
0.946 0.925 0.998 0.976
0.843 0.077* 0.933 0.850
0.944 0.936 0.990 0.980
0.942 0.966 0.992 1.019
0.983* 0.963* 0.985* 0.949*
1.000* 1.009* 1.002* 1.008*
1998 Nifty senior Nifty junior SENSEX BSENI
0.973 0.968 0.964 0.999
0.332* 0.474 0.567 0.515
0.933 0.973 0.966 0.980
0.980 0.987 0.962 0.962
0.980* 0.963* 0.982* 0.946*
1.001* 1.005* 1.000* 1.002*
1997 Nifty senior Nifty junior SENSEX BSENI
0.934 0.897 0.885 0.840
0.880 0.825 0.194* 0.358*
0.957 0.896 0.887 0.840
0.867 0.939 0.853 0.804
0.980* 0.960* 0.979* 0.944*
1.001* 1.002* 1.001* 1.004*
1996 Nifty senior Nifty junior SENSEX BSENI
– – 0.862* 0.814*
– – 0.756 0.237*
– – 0.849* 0.816*
– – 0.592 0.876
– – 0.975* 0.940*
– – 1.000* 1.001*
1995 Nifty senior Nifty junior SENSEX BSENI
– – 0.943 0.683*
– – 0.930 0.997
– – 0.943 0.682*
– – 0.626 0.809
– – 0.979* 0.940*
– – 0.999* 0.999*
1994 Nifty senior Nifty junior SENSEX BSENI
– – 0.873 0.726*
– – 0.476 0.978
– – 0.858* 0.719*
– – 0.632 0.782
– – 0.976* 0.945*
– – 1.000* 1.002*
1993 Nifty senior Nifty junior
– –
– –
– –
– –
– –
– –
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Table 3 (Continued) Sample frame
Speed of adjustment
Gamma
AR(1)
ARMA(1, 1)
ARMA(1, X)
Cross-covar. ratio
Eq. (10)
Eq. (10)
SENSEX BSENI
0.986 0.754*
0.041* 0.624
0.986 0.753*
0.875 0.745
0.985* 0.946*
1.002* 1.005*
1990–1992 Nifty Senior Nifty Junior SENSEX BSENI
– – 0.986 0.904
– – 0.214* 0.955
– – 0.983 0.898
– – 0.785 0.788
– – 0.986* 0.952*
– – 1.003* 1.008*
X, the optimal MA component, is determined via the Schwartz information criterion; *, statistically significant from one at the 5% significance level.
cap indices. Reflecting thin trading effects in the estimators (i.e. moving from the ARMA (1,1) model to the ARMA (1,X) model)11 leads to estimates that are closer to unity as argued previously on a priori grounds. When both thin trading and lagged price adjustment effects are present within the data set, the ARMA (1,X) model will provide the most appropriate estimates of the speeds of adjustment coefficients since the higher order moving average component will be a manifestation of thin trading effects. As can be seen in Table 3, for the whole dataset, the speeds of adjustment estimates are higher for the indices reflecting the larger market capitalizations (i.e. the Nifty senior and SENSEX indices) and the speeds of adjustment are statistically significantly less than one for the indices reflecting the smaller market capitalization stocks (i.e. the Nifty junior and BSENI indices). Within this specification, then, the adjustments of larger cap stocks (indices) are, effectively, fully efficient (i.e. less than full adjustment does not occur in a statistically significant fashion). Subperiod speeds of adjustment are generally higher for the larger cap indices (with the exception of 1998 for both index series), although underadjustments are not always statistically significantly different from one for the small cap indices (and, indeed, for two subperiods—1996 and 1994—the underreactions for the larger cap index, SENSEX, are statistically significant from one as well as for the small cap index). Institutions and investors outside Bombay did not have the same access as those in Bombay prior to 1994 (see, for example, Shah and Thomas (1997)). Since there is a strong link between price adjustment and volume (Karpoff (1987)) this restriction to access would be anticipated to lead to less rapid speeds of adjustment. Effectively, such restricted investors would not participate in the price formation and price discovery (Leach and Madhavan (1992, 1993)) processes. When the speeds of adjustment deriving from the ARMA (1,X) model for the SENSEX and BSENI indices are compared over the sample periods pre and post-1994 it can be seen in Table 3 that the adjustment speeds for the larger cap SENSEX index are broadly comparable over the two sample partitions. That is, the unequal access 11 The higher order moving average component picking up thin trading effects. Effectively, the ARMA(1,1) specification assumes an absence of thin trading effects. By incorporating thin trading processes such as those summarized in Eqs. (4a) and (4b) within the modelling structure an ARMA (1,X) specification arises withX > 1.
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had no significant impact upon price formation and adjustment in large cap stocks. However, in the case of the BSENI index the ARMA (1,X) adjustment speeds are usually, but not always, larger post-1994, particularly for the period from 1998 onwards. The overall impact of this restricted access is, then, not wholly unambiguous. However, any effect that arises appears to be concentrated in the lower volume, low cap stocks with no discernible effect in large cap stocks. 3.4. Speeds of adjustment and the underlying process The price adjustment process summarised at Eq. (10) modelled the partial adjustment towards an underlying process which was not necessarily a random walk. All estimators in Section 3.3 assumed that the underlying process did follow a random walk; that is, prices were fully informationally efficient. Effectively, if the underlying process does follow a random walk, the gamma at Eq. (10) will be unity and the model will become the ARMA (1, 1) process derived in Theobald and Yallup (2004). The underlying process may differ with market capitalisation due, for example, to differences in analyst coverages, information availability, etc. The estimated results using non-linear least squares are contained in the last two columns of Table 3 and as can be seen the results again differ across the indices in a size related fashion. As in the cases of the estimators reported in Section 3.3 above, the speeds of adjustment are higher for the larger capitalization indices. Indeed, they are higher for the whole dataset and in each of the subperiods for both the Nifty senior/junior indices and the SENSEX/BSENI indices. However, they differ from the prior estimation results in that the speeds of adjustment in all cases (i.e. all index series) are statistically significantly less than one (i.e. full adjustment). Essentially, then, the results reported are sensitive to the specification of the underlying, intrinsic value process. The fuller specification at Eq. (10) indicates, then, that while the larger cap indices have higher speeds of adjustment than lower cap stocks/indices, both under-react in a statistically significant fashion to the intrinsic value process. When the estimates of gamma, the coefficient in the intrinsic value process, are assessed they indicate a statistically significant tendency to overreact for all indices (with the exception of SENSEX & BSENI in 1995). When the underlying intrinsic value process is characterised by overreaction, estimates for the restricted process (i.e. a random walk) will be higher than when estimates are generated relative to the overreacting intrinsic value process. In general, the degrees of overreaction are very small economically, but the overreactions are always greater for the smaller cap indices. Intrinsic values, then, are characterised by overreactions in this setting and are consistent with the ‘overconfidence’ model posited via Daniel et al. (1998) wherein overreactions to private signals occur in the shorter term. 3.5. Relative impacts of speeds of adjustment and thin trading In Section 2 of this paper it was argued that both speed of adjustment and thin trading effects could have impacts on the lead/lag relationship between large and small capitalization stocks and that the decomposition of the lead/lag relationship into the elements deriving from each of these effects was not straightforward (Theobald and Yallup (2001) demonstrate these impacts within a stochastic process framework). This was because, as demonstrated
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Table 4 Estimates of Thin Trading Parameters Sample frames
Whole data set 2000–2001 1999 1998 1997 1996 1995 1994 1993 1990–1992
Thin trading parameters Nifty senior and Nifty junior
SENSEX and BSENI
θ(0)
∅(0)
θ(0)
∅(0)
0.926 0.931 0.982 0.900 0.884 – – – – –
0.962 0.923 1.000 0.973 0.963 – – – – –
0.919 1.037 1.044 0.999 0.915 0.740 0.732 0.728 1.020 0.888
0.933 1.078 1.087 0.933 0.887 0.860 0.895 0.800 0.880 0.894
at Eq. (7) in Section 2, there were a number of interaction effects deriving from these two distinct phenomena. To gain insights into the relative magnitudes of these effects it is necessary to estimate a number of parameters—that is, g(S), {∅(i),θ(i): i = 0, 1}, the speed of adjustment coefficients and the parameters of the thin trading process. Now g(S) has already been estimated in Section 3.3, above. The ∅(i) and θ(i) can be estimated from the equation system (8), (12) and (13) in Theobald and Yallup (1998) and are provided in Table 4.12 The results reported in Table 4 indicate that the degree of thin trading is usually greater for the small cap indices (i.e. ∅(0) > θ(0)). That is, more of the contemporaneous “true” return is reflected in the large cap indices as reflected by the higher contemporaneous parameter, ∅(0). In eleven of the fifteen cases reported, we find that∅(0) > θ(0); if fifteen independent Bernouilli trials are conducted with equal probability of success (∅(0) > θ(0)) or failure (∅(0) < θ(0)), the probability of observing this result is 0.021. Effectively, then, as has been documented elsewhere (e.g. Dimson (1979)) size or market capitalization does serve as an instrumental variable for thin trading. The results indicate that, in general, the indices are characterised by relatively frequent trading (which would be anticipated given the index composition requirements discussed in Section 3) since the {∅(0), θ(0)} are generally near one, though in the 1994, 1995 and 1996 subperiods for the BSENI index, the values for θ(0) fall below 0.800.13 There are a small number of subperiod instances where {θ(0), ∅(0)} are greater than one—we interpret these results as indicating an absence of thin trading in such instances (i.e. {θ(0), ∅(0)} = 1).14 Effectively, the three equations can be rearranged to express ∅(0) and θ(0) as direct functions of the cross-covariance terms. 13 The greater degrees of measured thin trading for these years is somewhat contrary to expectations given the greater access to investors outside Bombay post 1994 (Shah and Thomas (1997)). However, as in the case of speeds of adjustment, the degrees of thin trading for both indices decrease through time, particularly for the period from 1998 onwards. Also, see Poshakwale (2002). 14 Using the approximate asymptotic standard errors conditional upon speed of adjustment factors derived in Theobald and Yallup (1998), these estimates were not statistically significantly greater than unity. 12
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The relative contribution of thin trading and sluggish price adjustments (i.e. speeds of adjustment less than unity) are assessed by using Eq. (7) which was developed in Section 2 of this paper. In Eq. (7) the first term ([∅(0)θ(0) + ∅(1)θ(1)]{1 − g(S)}) reflects joint thin trading and adjustment effects, while the second term (∅(0)θ(1)) reflects purely thin trading effects. The third term, ∅(1) θ(0) (1−g(S))2 , incorporates both effects, but as it contains both ∅(1)—predicted low—and the square of a small number (1 − g(S))2 —its effect is predicted to be small on a priori grounds. To provide a parsimonious and focussed discussion of the results we will analyse only the results pertaining to the whole dataset. The speeds of adjustment estimates used will be those from the ARMA(1,X) model (since this model does provide speeds of adjustment estimates in the presence of thin trading) and Eq. (10), where both the speed of adjustment and the intrinsic value process parameters are estimated. In the case of the Nifty index series, using the ARMA(1,X) estimates, 50.2% of the lead/lag effect comes from the first term, 49.8% from the second term and 0.1% from the third (which was, as predicted above, low). Thus almost half of the lead/lag effect comes from pure thin trading effects (the second term), the remaining 50% coming from joint thin trading/adjustment effects. When the estimated speeds of adjustment are obtained from Eq. (10), again the third term has virtually no impact; however, in this case 66% of the lead/lag effect derives from pure thin trading effects. Similar results are obtained for the SENSEX & BSENI index series. That is, the contribution of the third term is similarly small (around 0.1% again) while in the case of the ARMA(1,X) estimates 44.6% of the lead/lag effect derives from pure thin trading impacts (the second term in Eq. (7)) and with estimates deriving from Eq. (10) 62.3% emanates from the thin trading effects. Overall, then, thin trading effects alone contribute significantly to the lead/lag effects; however, a mixed adjustment/thin trading effect contributes between 34% and 55% of this lead/lag effect, depending upon the index series and estimates used. Thus, the thin trading effect is a strong contributory factor to the large cap/small cap lead-lag relationship in the Indian market in contrast to the situation in the US as summarised in Campbell et al. (1997). This result is potentially a reflection of the fact that the extent of thin trading in a less developed market such as the Indian market is somewhat larger than that encountered in a more developed market such as the US. Since the stocks underpinning the indices used in this study have, in emerging market terms, a relatively high liquidity ratio (Endo (1998)) the contribution of thin trading effects to the lead/lag relationship in other emerging markets would be anticipated to be greater.
4. Conclusions In this paper the lead/lag relationship between large market capitalization stocks and small market capitalization stocks is investigated in both analytical and empirical terms. The analytics indicated that the lead/lag relationship derives from differential speeds of adjustment across size categories. Thin trading effects further contribute to this lead/lag relationship. The lead/lag relationship between large and small cap stocks is empirically established using four Indian stock market index series. Consistent with the analytics, the
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speeds of adjustment for large cap indices are found to be higher than the speeds of adjustment in small cap indices. In general, the speeds of adjustment reflect underreactions rather than overreactions, particularly for the small cap indices. While the latter indices will be anticipated to be subject to greater degrees of thin trading (as confirmed empirically in this paper), the lower speeds of adjustment for small cap stocks still persist after the estimates are corrected for thin trading effects by using the ARMA (1,X) model. The analytics indicated that lead/lag effects derived from pure thin trading effects and by interaction effects between thin trading and speeds of adjustment. Empirical analyses employing the analytic relationships indicated that between 45 and 66% of the lead/lag effect derives purely from thin trading effects, the remainder emanating from an interaction effect between thin trading and adjustment speeds. These effects are larger than those reported in the US and are a reflection of the greater incidence of thin trading within an emerging market. When the underlying intrinsic value process is investigated, it is found to be characterised by small degrees of overreactions, the overreaction being greater for the small cap indices. Overall, this paper has demonstrated the importance of assessing both differential speeds of adjustment and thin trading effects when evaluating the lead/lag relationships between large and small cap stocks. This will be particularly pertinent in emerging markets where thin trading effects may be more prevalent. The analytics provide a framework for analysing these effects in other markets. While the empirical results confirm the lead/lag relationships established elsewhere in the literature, a significant source of this effect in India derives from thin trading rather than sluggish adjustment, per se.
Acknowledgements We would like to thank Ike Mathur and an anonymous referee for useful comments and suggestions. Any remaining errors are the sole responsibility of the authors.
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Dimson, E., 1979. Risk measurement when shares are subject to infrequent trading. Journal of Financial Economics 7 (2), 197–226. Endo, T., 1998. The Indian Securities Market—A Guide for Foreign and Domestic Investors. Vision Books, India. Karpoff, J., 1987. The relation between price change and trading volume: a survey. Journal of Financial and Quantitative Analysis 22, 109–126. Leach, J., Madhavan, A., 1992. Intertemporal price discovery by market makers. Journal of Financial Intermediation 2, 207–235. Leach, J., Madhavan, A., 1993. Price experimentation and security market structure. Review of Financial Studies 6, 375–404. Lo, A., MacKinlay, C., 1989. The size and power of the variance ratio test in finite samples. Journal of Econometrics 40, 203–238. Miller, M., Muthuswamy, J., Whaley, R., 1994. Mean reversion of Standard and Poor’s 500 index basis changes: arbitrage induced or statistical illusion? Journal of Finance 49, 479–514. Poshakwale, S., 2002. The random walk hypothesis in the emerging Indian stock market. Journal of Business Finance and Accounting 29 (9–10), 1275–1299. Poterba, J., Summers, L., 1988. Mean reversion in stock returns: evidence and implications. Journal of Financial Economics 22, 27–60. Richardson, M., Stock, J., 1989. Drawing inferences from statistics based on multi-year asset returns. Journal of Financial Economics 25, 323–348. Roll, R., 1981. A possible explanation of the small-firm effect. Journal of Finance 36, 879–888. Scholes, M., Williams, J., 1977. Estimating betas from non-synchronous data. Journal of Financial Economics 5, 309–328. Shah, A., Thomas, S., 1997. Securities markets—towards greater efficiency, In: Parikh, K. (Ed.), India Development Report. Oxford University Press, UK. Stoll, H., Whaley, R., 1990. The dynamics of stock index and stock index futures returns. Journal of Financial and Quantitative Analysis 25, 441–468. Theobald, M., Price, V., 1984. Seasonality estimation in thin markets. Journal of Finance 39, 377–392. Theobald, M., Yallup, P., 1998. Measuring cash-futures temporal effects in the UK using partial adjustment factors. Journal of Banking & Finance 22, 221–243. Theobald, M., Yallup, P., 2001. Mean reversion and basis dynamics. Journal of Futures Markets 21, 797–818. Theobald, M., Yallup, P., 2004. Determining security speed of adjustment coefficients. Journal of Financial Markets 7, 75–96.
Market capitalisation, cross-correlations, the lead/lag structure and microstructure effects in the Indian stock market Sunil Poshakwale∗ , Michael Theobald Department of Accounting and Finance, The Birmingham Business School, The University of Birmingham, Edgbaston, Birmingham B15 2TT, UK Accepted 6 December 2003 Available online 18 May 2004
Abstract The lead/lag relationship between portfolios comprising large and small cap firms is analytically derived in terms of their speeds of adjustment and degrees of thin trading. Using a number of Indian equity index series that differ in their market capitalization characteristics, large cap indices are found to lead small cap indices and to have higher speeds of adjustment towards intrinsic values. However, pure thin trading effects and an interaction effect between thin trading and speeds of adjustment are found to make significant contributions to the lead/lag effect. An empirical analysis of the underlying intrinsic value process using a reduced form model developed in the paper indicates that a small degree of overreaction is present in intrinsic values series. © 2004 Elsevier B.V. All rights reserved. JEL classification: G14; G15; C51 Keywords: Lead/lag structure; Thin trading; Large and small capitalisation stocks
1. Introduction The impacts of market capitalization upon a variety of phenomena have been investigated since the size effect was first documented by Banz (1981). In this paper we investigate size related impacts upon the price adjustment process in an emerging market, India, to establish ∗
Corresponding author. Tel.: +44-121-414-7792; fax: +44-121-414-6238. E-mail address: [email protected] (S. Poshakwale).
1042-4431/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.intfin.2003.12.001
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the generality of the results documented in developed markets to less developed markets. A number of empirical studies have established that lead/lag relationships can occur across portfolios, particularly when formed on a size-related basis. That is, portfolios comprising large market capitalization stocks tend to lead the corresponding small market capitalization portfolios (see Campbell et al. (1997) for a summary and overview of this literature). One prime candidate for explaining the size-related lead/lag or cross-correlation structures is thin trading, since, in general, smaller market capitalization stocks are more thinly traded than larger market capitalization stocks. The differential in the degree of thin trading will cause smaller stock prices to lag behind those of their larger counterparts. Since portfolios are used in these studies and since thin trading effects are more prevalent in portfolios than for individual stocks, the rationale appears more compelling. However, Campbell et al. (1997) conclude that “. . . the recent empirical evidence provides little support for non-trading as an important source of spurious correlation in the returns of common stock over daily and long frequencies” (Campbell et al. (1997), p. 134). While many papers confirm that thin trading will not be the sole source of autocorrelations, there is some disagreement with regard to the extent to which thin trading leads to such phenomena. For example, Boudoukh et al. (1994, p542) argue that “. . . the effect of non-synchronous trading has most probably been understated in the literature”. In essence, then, the econometrician must determine the extent to which the lead/lag relationship derives from sluggish relative price adjustments, per se, and the extent to which this phenomenon reflects thin trading effects. We return to this issue in this paper by formally modelling the extent to which prices under or over-react relative to their intrinsic values using a partial-adjustment with noise model (Amihud and Mendelson (1987)). By modelling both the thin trading and the partial adjustment processes we are able to establish the extent to which cross-correlations derive from partial adjustments and the extent to which they derive from thin trading. Since the speed of adjustment and thin trading effects are distinct phenomena with differing implications it is important to be able to distinguish between them. Within the partial adjustment modelling framework market prices are assumed to partially adjust towards their intrinsic values. To develop a reduced form equation in terms of observable variables, a stochastic process has to be assumed for the intrinsic values; in general, the process is assumed to be a random walk. However, the intrinsic value process itself may vary with market capitalization and, indeed, as between developed and emerging markets. The determination of the actual process is, of course, an empirical issue and in this paper we develop a modelling structure for price adjustments with a more general underlying intrinsic value process. The resultant closed form equation that obtains affords the means of estimating both the speed of adjustment and the form of the intrinsic value process. In this fashion we are able to establish the sensitivity of the estimates of adjustment speeds to assumptions regarding the underlying intrinsic value adjustment process. The lead/lag and cross-correlation effect deriving from differing market capitalizations is investigated by using Indian equity market indices that differ in their market capitalization characteristics. The Nifty senior and the SENSEX indices which are employed in this study reflect more strongly large cap properties than the Nifty junior and BSENI indices. On a priori grounds, and given the empirical results established in other markets (e.g. Campbell et al. (1997)) the former indices would be anticipated to lead the latter indices and to have more rapid adjustments towards intrinsic values. However, the latter indices would be
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anticipated to be subject to greater degrees of thin trading. The procedures developed in this paper provide the means of establishing whether the lead/lag effects documented in these indices derive from speed of adjustment effects or thin trading effects. The plan of this paper is as follows. In Section 2 the modelling relationships that underpin the empirical work are developed. Specifically, the relative contributions of speed of adjustment and thin trading effects to the lead/lag relationship across differing portfolios are analytically developed. The impacts of differing intrinsic value processes upon the observed price adjustment process are also analytically established. Section 3 presents the empirical results that are obtained from using the data set of Indian equity market indices, while conclusions are presented in Section 4. 2. Modelling relationships 2.1. Thin trading effects Thin trading and less than full price adjustments towards intrinsic value will both induce autocorrelations in return series—and in a relatively similar fashion (see, for example, Theobald and Yallup (2001)). Earlier estimators of partial adjustment factors (such as Damodaran (1993) as corrected by Brisley and Theobald (1996)) did not adjust for thin trading; in Theobald and Yallup (1998, 2004), estimators were developed that incorporate thin trading. A na¨ıve approach to evaluating the differential impacts of these two phenomena would be to estimate the partial adjustment factors with and without correcting for thin trading1 (perhaps establishing the statistical significance of differences) and then evaluating the cross-covariance predictions of both estimates. However, as will be demonstrated in this paper there is an interaction between these two effects which means that such an approach would be misspecified. Theobald and Yallup (1998) demonstrate that, in the absence of thin trading, an estimator of the speed of adjustment factor, g(i) is given by (1 − g(i)) =
cov{R(i, t), R(ic , t − 1)} cov{R(i, t), R(ic , t)}
(1)
where R(i, t) is the return on stock (or instrument) “i” in period “t”; the “c” superscript denotes a complement (i.e. other stock or instrument) and “cov” is the covariance operator. The partial adjustment coefficient, g(i), represents the speed of adjustment of the natural logarithm of prices, P(i, t), towards the natural log of intrinsic values, V(i, t) as P(i, t) − P(i, t − 1) = g(i){V(i, t) − P(i, t − 1)} + u(i, t)
(2)
where u(i, t) is an i.i.d. noise term.2 Using Eq. (1) in the context of the small/large cap lead/lag relationship, we have 1 By for example using a thin trading “error purger” approach such as that employed in Stoll and Whaley (1990). 2 Intrinsic values are assumed to follow a logarithmic random walk with drift in Amihud and Mendelson (1987). Full adjustment towards intrinsic values would correspond to the case where g(i) = 1, underreactions arise where g(i) < 1 and overreactions occur with g(i) > 1.
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cov{R(L, t − 1), R(S, t)} = (1 − g(S))[cov{R(L, t), R(S, t)}]
(3)
i.e. large cap stocks or portfolios, with returns R(L, t), will lead small cap stocks or portfolios, with returns R(S, t), when g(S) < 1 where g(S) is the speed of adjustment of small cap stocks, i.e. small stocks do not fully adjust towards intrinsic values,3 as per Eq. (2). However, this relationship is predicated on the absence of thin trading. If we model the thin trading process as ∅(i)R(L, t − i) + v(m, L, t) (4a) R(m, L, t) = i=0
and R(m, S, t) =
θ(j)R(S, t − j) + v(m, S, t)
(4b)
i=0
see, for example, Cohen et al. (1986); Theobald and Price (1984); Stoll and Whaley (1990); Miller et al. (1994), with “m” indicating a measured return (i.e. subject to thin trading and its absence denoting a “true” return; that is, one not subject to thin trading), ∅(i) and θ(i) variables such that when ∅(o) and θ(o) equal one there is no thin trading and {v(m, L, t), v(m, S, t)} are noise terms, then cov{R(m, L, t − 1), R(m, S, t)} = ∅(i)θ(i) cov{R(L, t − 1), R(S, t)} (i = 0) + ∅(i)θ(i + 1) cov{R(L, t), R(S, t)} (i = 0) + ∅(i + 1)θ(i) cov{R(L, t − 2), R(S, t)} (i = 0) + ∅(i + 2)θ(i) cov{R(L, t − 3), R(S, t)} (i = 0) + · · · + ∅(i)q(i + 2) cov{R(L, t), R(S, t − 1)} (i = 0) + · · · (5) If we assume that stocks trade in consecutive periods (Scholes and Williams (1977); Miller et al. (1994)), with a stationary cross-covariance structure, Eq. (5) can be simplified as, cov{R(m, L, t − 1), R(m, S, t)} = {∅(0)θ(0) + ∅(1)θ(1)}[cov{R(L, t − 1), R(S, t)}] + ∅(0)θ(1)cov{R(L, t − 1), R(S, t − 1)} + ∅(1)θ(0)cov{R(L, t − 2), R(S, t)} (6) If there are no leads and lags in the “true” returns processes (e.g. true prices fully adjust towards their intrinsic values), then cov{R(m, L, t − 1), R(m, S, t)} = ∅(0)θ(1)cov{R(L, t − 1), R(S, t − 1)}
(6a)
That is, the lead/lag cross-covariances induced by thin trading are summarised at Eq. (6a). In the presence of small cap speeds of adjustments less than complete (i.e. g(S) < 1), there will be cross-covariances induced in the true series. That is, using the relationship at 3 The adjustment speed of the large cap stocks may also be reflected in Eq. (3) by substituting for cov{R(L,t), R(S,t)} as cov{R(L, t − 1), R(S, t)} = (1 − g(S))(1 − g(F))−1 cov{R(L,t), R(S,t − 1)}. Since g(L) > g(S) and could be equal to one, we work with Eq. (3) to avoid non-determinancies due to zero denominations.
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Eq. (3) in Eq. (6), above, we have cov{R(m, L, t − 1), R(m, S, t)} = {(∅(0)θ(0) + ∅(1)θ(1))(1 − g(S)) + ∅(0)θ(1) + ∅(1)θ(0)(1 − g(S))2 } × cov{R(L, t), R(S, t)}
(7)
That is, the joint impacts of thin trading and tardy speeds of adjustments comprise three terms, two of which entail interactions between these two phenomena. Eq. (7) still contains the thin trading term at Eq. (6a). Note that when the thin trading is concentrated in small stocks only, Eq. (7) becomes cov{R(m, L, t − 1), R(m, S, t)} = {θ(0)(1 − g(S)) + θ(1)}cov{R(L, t), R(S, t)} = {1 − θ(0)g(S)}cov{R(L, t), R(S, t)}
(8)
2.2. Intrinsic value process The partial adjustment with noise process assumes that market prices only partially adjust towards their intrinsic or full information values. The intrinsic value process is commonly assumed to be a random walk (Amihud and Mendelson (1987)). There is, however, empirical support for stock prices following, for example, a mean-reversion process (see, for example, Poterba and Summers (1988), Lo and MacKinlay (1989) and Richardson and Stock (1989)). While this mean-reversion could be a manifestation of microstructure effects such as thin trading (see, for example, Campbell et al. (1997), p89), the intrinsic value process could still differ across small and large cap stocks/portfolios due to other intrinsic differences between these classes of stocks. The “behavioral” literature provides rationales for non-random walk adjustment processes (for example, Barberis et al. (1998); Daniel et al. (1998)) where price series may manifest both over and under-reaction characteristics over differing return horizons. While such model types may provide explanations for incomplete or over adjustments to intrinsic values, they also indicate that the underlying process itself may not necessarily follow a random walk. Differential information availability across market capitalisation fractiles may also lead to differing underlying processes. Irrespective of causes and rationales, the nature of the underlying process is an empirical question and it is investigated by positing a process of the form V(i, t) = γ(i)V(i, t − 1) + e(i, t)
(9)
where γ(i) is the process parameter and e(i,t) the process disturbance.4 When γ(i) = 1, the intrinsic value process is a random walk, while with γ(i) < 1 the process is mean reverting, for example. The reduced form that is obtained by combining Eqs. (2) and (9) is P(i, t) − P(i, t − 1) = (γ(i) − g(i))[P(i, t − 1) − P(i, t − 2)] − g(i)(1 − γ(i))P(i, t − 2) + g(i)e(i, t) + u(i, t) − γ(i)u(i, t − 1) 4
A drift term was omitted as its inclusion had no substantive impact upon the analytical structure.
(10)
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Thus, by estimating {g(i), γ(i)} across large and small cap deciles, insights into the impacts of the intrinsic value process upon parameter estimates can be gleaned and differences in the intrinsic value processes across differing market capitalizations established. The parameters in Eq. (10) arise in a non-linear fashion and, as a consequence, Eq. (10) is estimated by non-linear least squares.5
3. Empirical results 3.1. Sample Return data on four Indian stock market indices were used—the Nifty senior and junior indices (from January 2, 1997 to February 9, 2001) and the SENSEX and BSENI indices (from January 2, 1990 to February 9, 2001). Daily closing prices for the indices have been collected from Datastream International. The Nifty senior index is a well diversified index and comprises 50 stocks which represent 25 sectors of the economy. The base period for the index is November 3, 1995 with a base value of 1000 and base capital of Rs.2.06 trillion (US$46bn approx.). Market participants use this index for benchmarking fund portfolios, index based derivatives and index funds. The index is computed using market capitalization weights. All companies to be included in the Nifty senior index should have a minimum market capitalization of Rs.5 billion (US$112m approx.) and its stock should have traded on 85% of the available trading days. The Nifty senior index represents 45% of the total market capitalization. The Nifty junior index was introduced on January 1 1997, with a base value of 1000, a base capital of Rs. 0.43 trillion (US$9bn approx.) and includes 50 companies. A listed company needs to have a minimum market capitalization of Rs. 2 billion (US$45m approx.) and its stock should have traded on 85% of the trading days. The index represents about 7% of total market capitalization. Given the characteristics of these two indices, the Nifty senior index would be anticipated to manifest larger capitalization properties, the Nifty junior index smaller capitalization properties. The SENSEX includes equity shares of the top 30 companies selected on the basis of market capitalization, liquidity, depth, trading frequency and industry representation. The selected stock should figure in the top 100 companies listed by market capitalization. Also, the market capitalization of the stock should be more than 0.5% of the total market capitalization of the index i.e. the minimum weighting should be 0.5%. Since the SENSEX is a market capitalization weighted index, this is one of the primary criteria for stock selection. The index was launched on 2 January 1986 with 1978–1979 as the base year. The BSENI index is comprised of 100 shares listed on India’s five major stock exchanges at Mumbai (Bombay), Calcutta, Delhi, Ahmedabad and Chennai (Madras). There is no minimum market capitalization requirement for inclusion. However, the selected stock should 5 In testing for the statistical significance of γ(i) from one in Eq. (10) we do not face the stationary/non-stationary stochastic process problems that are inherent in the unit root literature. That is, the reduced form of the estimating equation at Eq. (10) is stationary whether or not γ(i) is equal to unity. As such, then, in our estimations we do not introduce “Dickey Fuller” type modifications to the standard errors deriving from the non-linear least squares estimator.
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Table 1 Descriptive statistics for the indices (daily log price relatives) Descriptive statistic
Mean Std. Dev. Skewness Kurtosis Jarque Bera
Index Nifty senior
Nifty junior
0.038 1.77 −0.11 5.67 321.42 (0.00)
0.077 2.14 −0.29 4.86 170.04 (0.00)
SENSEX
BSENI
0.05 1.93 0.34 13.70 11131.88 (0.00)
0.05 1.65 0.48 13.28 1032.30 (0.00)
P-values for the Jarque–Bera statistic are contained in parentheses.
have been traded on every trading day for the last one year and should be among the top 150 companies listed by the average number of trades per day for the previous year. These criteria may be relaxed in exceptional cases such as trading suspensions, etc. Due consideration is also given to the representation of various industry-groups and the stock’s trading record on major stock exchanges. The BSENI index used to take prices of certain stocks from other exchanges in order to reflect market movements at the national level. However, changes in trading technology, longer trading periods and almost instantaneous availability of information across the country have ensured that there is little or no difference in the prices of the index stocks. Therefore, the Exchange administration decided to redesignate the BSENI as the ‘BSE 100’ index and since October 14, 1996, only prices on the Bombay Stock Exchange (BSE) are taken to calculate the index. The index was started in January 1989 with 1983–1984 as the base year. The SENSEX index would be anticipated to reflect larger cap stock characteristics than the BSENI index as a result of the previously described compositions for the indices. The descriptive statistics of each of the index return series are contained in Table 1, where it can be seen from the Jarque–Bera statistics that all series are statistically significantly non-normal and that for those series reflecting the latter part of the 1990s the return distributions are negatively skewed as found in most international stock markets during that period. The Nifty junior index is the most volatile in terms of standard deviations. 3.2. Cross-correlations The contemporaneous and one period leading and lagging return cross-correlations between the Senior and Junior Nifty indices and the SENSEX and BSENI indices are contained in Table 2 for the whole dataset for each index pair and for a number of 1 year calendar periods. Prior empirical research in the US (e.g. Campbell et al. (1997)) would suggest that from the descriptions of the indices the Nifty senior series should lead the Nifty junior and that the SENSEX should lead the BSENI.6 As can be seen in Table 2 the contemporaneous cross-correlations are all relatively high, indicative of the presence of common factors un6
The use of stock market indices with differing market capitalization characteristics to investigate size related phenomena goes back to Roll (1981).
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Table 2 Cross-correlation structures across differing indices Sample frame
Correlation type
Cross-correlation between Nifty senior and Nifty junior
Whole data
seta
SENSEX and BSENI
Contemporaneous Lead Lag
0.7891 0.0750 0.0200
0.7801 0.1641 0.1483
2000–2001
Contemporaneous Lead Lag
0.7639 0.1085 0.0339
0.9297 0.0306 0.0208
1999
Contemporaneous Lead Lag
0.7951 0.0471 −0.260
0.9678 0.0081 −0.0184
1998
Contemporaneous Lead Lag
0.8176 0.0120 0.0080
0.9185 0.0335 0.0346
1997
Contemporaneous Lead Lag
0.8862 0.1192 0.0539
0.9185 0.0351 0.0351
1996
Contemporaneous Lead Lag
– – –
0.9235 0.1357 0.1808
1995
Contemporaneous Lead Lag
– – –
0.8497 0.3461 0.1056
1994
Contemporaneous Lead Lag
– – –
0.8027 0.2989 0.1531
1993
Contemporaneous Lead Lag
– – –
0.7716 0.2837 0.1686
1990–1992
Contemporaneous Lead Lag
– – –
0.7349 0.1465 0.1656
a The whole sample frame comprises the period 1997–2001 for the Nifty series and 1990–2001 for the SENSEX and BSENI series.
derpinning the return generating processes for each of these indices. In the majority of cases investigated, the index reflecting the larger capitalization stocks leads the lesser capitalization index more strongly than lagging it. This is more particularly the case for the Nifty index series where potentially, the market capitalization differences are more apparent and consistent. In conclusion, then, evidence of leading cross-correlations driven by capitalization differentials is demonstrated to be present in the Indian stock market series analysed in this paper and having empirically established this phenomenon we will now proceed to investigate the relationship further in terms of speeds of adjustment and thin trading effects.
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3.3. Speeds of adjustments estimates A number of differing estimators were proposed and discussed in Section 2 of this paper. In this section we utilise a number of these estimators to estimate the speeds of adjustments of the four indices that comprise our sample. Four differing estimators are used:7 (i) the cross-covariance ratio estimator (Theobald and Yallup (1998)) where the speed of adjustment, g(i), i = S, L, is given by g(S) = cov[R(L, t − 1), R(S, t)]{cov[R(L, t), R(S, t)]}−1
(11)
g(L) = cov[R(L, t), R(S, t − 1)]{cov[R(L, t), R(S, t)}−1 ;
(12)
and
(ii) an ARMA(1,1) model ((Theobald and Yallup (2004)) where the stochastic process underpinning the speed of adjustment estimator is R(i, t) = (1 − g(i))R(i, t − 1) + g(i)e(i, t) − u(i, t) − u(i, t − 1)
(13)
where e(i, t) is the innovation in the intrinsic value process and u(i, t) the noise in the speed of adjustment process; (iii) an ARMA (1, X) model (Theobald and Yallup (2004)), where X, the optimal moving average order which reflects thin trading effects is determined by the Schwartz Information Criterion, and; (iv) an AR(1) model8 which assumes an absence of spread and noise effects in the ARMA (1,1) model.9 If the capitalization induced lead/lag relationship derives purely from large cap stocks (indices) adjusting more rapidly to information than smaller cap stocks (indices), then the speeds of adjustment for the former stock types should be larger than for the latter stock types. If thin trading has an impact upon this lead/lag relationship, then controlling for this effect should narrow the speed of adjustment differential and lead to speeds of adjustment closer to full (i.e. unity). The results that derive from using these four estimators are contained in Table 3. As can be seen in that table the results broadly conform to the two predictions made in the previous paragraph. That is, for all estimators the speed of adjustment for the larger cap index (the Nifty senior and SENSEX) is more commonly larger than for the index that reflects smaller capitalization stocks10 (the Nifty junior and BSENI); furthermore, the occurrence of adjustments statistically significantly different from one is more frequent for the smaller 7 A fifth estimator which does not assume that the underlying process follows a random walk (i.e. Eq. (10)) is also used, but we defer discussion of this model’s results to Section 3.4. 8 This model will relate directly to the cross-covariance ratio estimator when second moments are stationary. 9 Damodaran (1993) and Brisley and Theobald (1996) estimator is not used as this has poor sample properties as demonstrated in Theobald and Yallup (1998, 2004). 10 Potential sectoral impacts were investigated by estimating speeds of adjustments for two large cap sectors—banks and petrochemicals—and two small cap sectors—engineering and hotels. The speeds of adjustments for the large cap sectors estimated from the ARMA(1,1) specification for the period 1/2/1995 to 2/9/2001 were larger than the corresponding estimates for the two small cap sectors, consistent with the results for the broad based market indices.
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Table 3 Estimates of adjustment speeds and underlying process Sample frame
Speed of adjustment
Gamma
AR(1)
ARMA(1, 1)
ARMA(1, X)
Cross-covar. ratio
Eq. (10)
Eq. (10)
Whole data set Nifty senior Nifty junior SENSEX BSENI
0.955 0.913* 0.966 0.892*
0.604 0.597 0.397 0.527*
0.956 0.920* 0.967 0.891*
0.905 0.975 0.839 0.858
0.876 0.788** 0.983* 0.947*
0.923 0.916** 1.002* 1.005*
2000–2001 Nifty senior Nifty junior SENSEX BSENI
0.962 0.881 0.994 0.919
0.220* 0.513 0.244* 0.284*
0.960 0.891 0.996 0.926
0.861 0.958 0.967 0.978
0.975* 0.955* 0.984* 0.945*
1.002* 1.002* 1.000* 1.004*
1999 Nifty senior Nifty junior SENSEX BSENI
0.946 0.925 0.998 0.976
0.843 0.077* 0.933 0.850
0.944 0.936 0.990 0.980
0.942 0.966 0.992 1.019
0.983* 0.963* 0.985* 0.949*
1.000* 1.009* 1.002* 1.008*
1998 Nifty senior Nifty junior SENSEX BSENI
0.973 0.968 0.964 0.999
0.332* 0.474 0.567 0.515
0.933 0.973 0.966 0.980
0.980 0.987 0.962 0.962
0.980* 0.963* 0.982* 0.946*
1.001* 1.005* 1.000* 1.002*
1997 Nifty senior Nifty junior SENSEX BSENI
0.934 0.897 0.885 0.840
0.880 0.825 0.194* 0.358*
0.957 0.896 0.887 0.840
0.867 0.939 0.853 0.804
0.980* 0.960* 0.979* 0.944*
1.001* 1.002* 1.001* 1.004*
1996 Nifty senior Nifty junior SENSEX BSENI
– – 0.862* 0.814*
– – 0.756 0.237*
– – 0.849* 0.816*
– – 0.592 0.876
– – 0.975* 0.940*
– – 1.000* 1.001*
1995 Nifty senior Nifty junior SENSEX BSENI
– – 0.943 0.683*
– – 0.930 0.997
– – 0.943 0.682*
– – 0.626 0.809
– – 0.979* 0.940*
– – 0.999* 0.999*
1994 Nifty senior Nifty junior SENSEX BSENI
– – 0.873 0.726*
– – 0.476 0.978
– – 0.858* 0.719*
– – 0.632 0.782
– – 0.976* 0.945*
– – 1.000* 1.002*
1993 Nifty senior Nifty junior
– –
– –
– –
– –
– –
– –
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Table 3 (Continued) Sample frame
Speed of adjustment
Gamma
AR(1)
ARMA(1, 1)
ARMA(1, X)
Cross-covar. ratio
Eq. (10)
Eq. (10)
SENSEX BSENI
0.986 0.754*
0.041* 0.624
0.986 0.753*
0.875 0.745
0.985* 0.946*
1.002* 1.005*
1990–1992 Nifty Senior Nifty Junior SENSEX BSENI
– – 0.986 0.904
– – 0.214* 0.955
– – 0.983 0.898
– – 0.785 0.788
– – 0.986* 0.952*
– – 1.003* 1.008*
X, the optimal MA component, is determined via the Schwartz information criterion; *, statistically significant from one at the 5% significance level.
cap indices. Reflecting thin trading effects in the estimators (i.e. moving from the ARMA (1,1) model to the ARMA (1,X) model)11 leads to estimates that are closer to unity as argued previously on a priori grounds. When both thin trading and lagged price adjustment effects are present within the data set, the ARMA (1,X) model will provide the most appropriate estimates of the speeds of adjustment coefficients since the higher order moving average component will be a manifestation of thin trading effects. As can be seen in Table 3, for the whole dataset, the speeds of adjustment estimates are higher for the indices reflecting the larger market capitalizations (i.e. the Nifty senior and SENSEX indices) and the speeds of adjustment are statistically significantly less than one for the indices reflecting the smaller market capitalization stocks (i.e. the Nifty junior and BSENI indices). Within this specification, then, the adjustments of larger cap stocks (indices) are, effectively, fully efficient (i.e. less than full adjustment does not occur in a statistically significant fashion). Subperiod speeds of adjustment are generally higher for the larger cap indices (with the exception of 1998 for both index series), although underadjustments are not always statistically significantly different from one for the small cap indices (and, indeed, for two subperiods—1996 and 1994—the underreactions for the larger cap index, SENSEX, are statistically significant from one as well as for the small cap index). Institutions and investors outside Bombay did not have the same access as those in Bombay prior to 1994 (see, for example, Shah and Thomas (1997)). Since there is a strong link between price adjustment and volume (Karpoff (1987)) this restriction to access would be anticipated to lead to less rapid speeds of adjustment. Effectively, such restricted investors would not participate in the price formation and price discovery (Leach and Madhavan (1992, 1993)) processes. When the speeds of adjustment deriving from the ARMA (1,X) model for the SENSEX and BSENI indices are compared over the sample periods pre and post-1994 it can be seen in Table 3 that the adjustment speeds for the larger cap SENSEX index are broadly comparable over the two sample partitions. That is, the unequal access 11 The higher order moving average component picking up thin trading effects. Effectively, the ARMA(1,1) specification assumes an absence of thin trading effects. By incorporating thin trading processes such as those summarized in Eqs. (4a) and (4b) within the modelling structure an ARMA (1,X) specification arises withX > 1.
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had no significant impact upon price formation and adjustment in large cap stocks. However, in the case of the BSENI index the ARMA (1,X) adjustment speeds are usually, but not always, larger post-1994, particularly for the period from 1998 onwards. The overall impact of this restricted access is, then, not wholly unambiguous. However, any effect that arises appears to be concentrated in the lower volume, low cap stocks with no discernible effect in large cap stocks. 3.4. Speeds of adjustment and the underlying process The price adjustment process summarised at Eq. (10) modelled the partial adjustment towards an underlying process which was not necessarily a random walk. All estimators in Section 3.3 assumed that the underlying process did follow a random walk; that is, prices were fully informationally efficient. Effectively, if the underlying process does follow a random walk, the gamma at Eq. (10) will be unity and the model will become the ARMA (1, 1) process derived in Theobald and Yallup (2004). The underlying process may differ with market capitalisation due, for example, to differences in analyst coverages, information availability, etc. The estimated results using non-linear least squares are contained in the last two columns of Table 3 and as can be seen the results again differ across the indices in a size related fashion. As in the cases of the estimators reported in Section 3.3 above, the speeds of adjustment are higher for the larger capitalization indices. Indeed, they are higher for the whole dataset and in each of the subperiods for both the Nifty senior/junior indices and the SENSEX/BSENI indices. However, they differ from the prior estimation results in that the speeds of adjustment in all cases (i.e. all index series) are statistically significantly less than one (i.e. full adjustment). Essentially, then, the results reported are sensitive to the specification of the underlying, intrinsic value process. The fuller specification at Eq. (10) indicates, then, that while the larger cap indices have higher speeds of adjustment than lower cap stocks/indices, both under-react in a statistically significant fashion to the intrinsic value process. When the estimates of gamma, the coefficient in the intrinsic value process, are assessed they indicate a statistically significant tendency to overreact for all indices (with the exception of SENSEX & BSENI in 1995). When the underlying intrinsic value process is characterised by overreaction, estimates for the restricted process (i.e. a random walk) will be higher than when estimates are generated relative to the overreacting intrinsic value process. In general, the degrees of overreaction are very small economically, but the overreactions are always greater for the smaller cap indices. Intrinsic values, then, are characterised by overreactions in this setting and are consistent with the ‘overconfidence’ model posited via Daniel et al. (1998) wherein overreactions to private signals occur in the shorter term. 3.5. Relative impacts of speeds of adjustment and thin trading In Section 2 of this paper it was argued that both speed of adjustment and thin trading effects could have impacts on the lead/lag relationship between large and small capitalization stocks and that the decomposition of the lead/lag relationship into the elements deriving from each of these effects was not straightforward (Theobald and Yallup (2001) demonstrate these impacts within a stochastic process framework). This was because, as demonstrated
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Table 4 Estimates of Thin Trading Parameters Sample frames
Whole data set 2000–2001 1999 1998 1997 1996 1995 1994 1993 1990–1992
Thin trading parameters Nifty senior and Nifty junior
SENSEX and BSENI
θ(0)
∅(0)
θ(0)
∅(0)
0.926 0.931 0.982 0.900 0.884 – – – – –
0.962 0.923 1.000 0.973 0.963 – – – – –
0.919 1.037 1.044 0.999 0.915 0.740 0.732 0.728 1.020 0.888
0.933 1.078 1.087 0.933 0.887 0.860 0.895 0.800 0.880 0.894
at Eq. (7) in Section 2, there were a number of interaction effects deriving from these two distinct phenomena. To gain insights into the relative magnitudes of these effects it is necessary to estimate a number of parameters—that is, g(S), {∅(i),θ(i): i = 0, 1}, the speed of adjustment coefficients and the parameters of the thin trading process. Now g(S) has already been estimated in Section 3.3, above. The ∅(i) and θ(i) can be estimated from the equation system (8), (12) and (13) in Theobald and Yallup (1998) and are provided in Table 4.12 The results reported in Table 4 indicate that the degree of thin trading is usually greater for the small cap indices (i.e. ∅(0) > θ(0)). That is, more of the contemporaneous “true” return is reflected in the large cap indices as reflected by the higher contemporaneous parameter, ∅(0). In eleven of the fifteen cases reported, we find that∅(0) > θ(0); if fifteen independent Bernouilli trials are conducted with equal probability of success (∅(0) > θ(0)) or failure (∅(0) < θ(0)), the probability of observing this result is 0.021. Effectively, then, as has been documented elsewhere (e.g. Dimson (1979)) size or market capitalization does serve as an instrumental variable for thin trading. The results indicate that, in general, the indices are characterised by relatively frequent trading (which would be anticipated given the index composition requirements discussed in Section 3) since the {∅(0), θ(0)} are generally near one, though in the 1994, 1995 and 1996 subperiods for the BSENI index, the values for θ(0) fall below 0.800.13 There are a small number of subperiod instances where {θ(0), ∅(0)} are greater than one—we interpret these results as indicating an absence of thin trading in such instances (i.e. {θ(0), ∅(0)} = 1).14 Effectively, the three equations can be rearranged to express ∅(0) and θ(0) as direct functions of the cross-covariance terms. 13 The greater degrees of measured thin trading for these years is somewhat contrary to expectations given the greater access to investors outside Bombay post 1994 (Shah and Thomas (1997)). However, as in the case of speeds of adjustment, the degrees of thin trading for both indices decrease through time, particularly for the period from 1998 onwards. Also, see Poshakwale (2002). 14 Using the approximate asymptotic standard errors conditional upon speed of adjustment factors derived in Theobald and Yallup (1998), these estimates were not statistically significantly greater than unity. 12
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The relative contribution of thin trading and sluggish price adjustments (i.e. speeds of adjustment less than unity) are assessed by using Eq. (7) which was developed in Section 2 of this paper. In Eq. (7) the first term ([∅(0)θ(0) + ∅(1)θ(1)]{1 − g(S)}) reflects joint thin trading and adjustment effects, while the second term (∅(0)θ(1)) reflects purely thin trading effects. The third term, ∅(1) θ(0) (1−g(S))2 , incorporates both effects, but as it contains both ∅(1)—predicted low—and the square of a small number (1 − g(S))2 —its effect is predicted to be small on a priori grounds. To provide a parsimonious and focussed discussion of the results we will analyse only the results pertaining to the whole dataset. The speeds of adjustment estimates used will be those from the ARMA(1,X) model (since this model does provide speeds of adjustment estimates in the presence of thin trading) and Eq. (10), where both the speed of adjustment and the intrinsic value process parameters are estimated. In the case of the Nifty index series, using the ARMA(1,X) estimates, 50.2% of the lead/lag effect comes from the first term, 49.8% from the second term and 0.1% from the third (which was, as predicted above, low). Thus almost half of the lead/lag effect comes from pure thin trading effects (the second term), the remaining 50% coming from joint thin trading/adjustment effects. When the estimated speeds of adjustment are obtained from Eq. (10), again the third term has virtually no impact; however, in this case 66% of the lead/lag effect derives from pure thin trading effects. Similar results are obtained for the SENSEX & BSENI index series. That is, the contribution of the third term is similarly small (around 0.1% again) while in the case of the ARMA(1,X) estimates 44.6% of the lead/lag effect derives from pure thin trading impacts (the second term in Eq. (7)) and with estimates deriving from Eq. (10) 62.3% emanates from the thin trading effects. Overall, then, thin trading effects alone contribute significantly to the lead/lag effects; however, a mixed adjustment/thin trading effect contributes between 34% and 55% of this lead/lag effect, depending upon the index series and estimates used. Thus, the thin trading effect is a strong contributory factor to the large cap/small cap lead-lag relationship in the Indian market in contrast to the situation in the US as summarised in Campbell et al. (1997). This result is potentially a reflection of the fact that the extent of thin trading in a less developed market such as the Indian market is somewhat larger than that encountered in a more developed market such as the US. Since the stocks underpinning the indices used in this study have, in emerging market terms, a relatively high liquidity ratio (Endo (1998)) the contribution of thin trading effects to the lead/lag relationship in other emerging markets would be anticipated to be greater.
4. Conclusions In this paper the lead/lag relationship between large market capitalization stocks and small market capitalization stocks is investigated in both analytical and empirical terms. The analytics indicated that the lead/lag relationship derives from differential speeds of adjustment across size categories. Thin trading effects further contribute to this lead/lag relationship. The lead/lag relationship between large and small cap stocks is empirically established using four Indian stock market index series. Consistent with the analytics, the
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speeds of adjustment for large cap indices are found to be higher than the speeds of adjustment in small cap indices. In general, the speeds of adjustment reflect underreactions rather than overreactions, particularly for the small cap indices. While the latter indices will be anticipated to be subject to greater degrees of thin trading (as confirmed empirically in this paper), the lower speeds of adjustment for small cap stocks still persist after the estimates are corrected for thin trading effects by using the ARMA (1,X) model. The analytics indicated that lead/lag effects derived from pure thin trading effects and by interaction effects between thin trading and speeds of adjustment. Empirical analyses employing the analytic relationships indicated that between 45 and 66% of the lead/lag effect derives purely from thin trading effects, the remainder emanating from an interaction effect between thin trading and adjustment speeds. These effects are larger than those reported in the US and are a reflection of the greater incidence of thin trading within an emerging market. When the underlying intrinsic value process is investigated, it is found to be characterised by small degrees of overreactions, the overreaction being greater for the small cap indices. Overall, this paper has demonstrated the importance of assessing both differential speeds of adjustment and thin trading effects when evaluating the lead/lag relationships between large and small cap stocks. This will be particularly pertinent in emerging markets where thin trading effects may be more prevalent. The analytics provide a framework for analysing these effects in other markets. While the empirical results confirm the lead/lag relationships established elsewhere in the literature, a significant source of this effect in India derives from thin trading rather than sluggish adjustment, per se.
Acknowledgements We would like to thank Ike Mathur and an anonymous referee for useful comments and suggestions. Any remaining errors are the sole responsibility of the authors.
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