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# Stock returns predictability and the adaptive market hypothesis in emerging markets: evidence from India

- Gourishankar S Hiremath
^{1}Email author and - Jyoti Kumari
^{1}

**3**:428

https://doi.org/10.1186/2193-1801-3-428

© Hiremath and Kumari; licensee Springer. 2014

**Received:**20 July 2014**Accepted:**28 July 2014**Published:**12 August 2014

## Abstract

This study addresses the question of whether the adaptive market hypothesis provides a better description of the behaviour of emerging stock market like India. We employed linear and nonlinear methods to evaluate the hypothesis empirically. The linear tests show a cyclical pattern in linear dependence suggesting that the Indian stock market switched between periods of efficiency and inefficiency. In contrast, the results from nonlinear tests reveal a strong evidence of nonlinearity in returns throughout the sample period with a sign of tapering magnitude of nonlinear dependence in the recent period. The findings suggest that Indian stock market is moving towards efficiency. The results provide additional insights on association between financial crises, foreign portfolio investments and inefficiency.

### JEL codes

G14; G12; C12

- Adaptive market hypothesis
- Market efficiency
- Random walk
- Autocorrelation
- Nonlinearity
- Predictability
- Financial crisis
- Evolving efficiency
- Emerging markets

- G14
- G12
- C12

## 1 Background

There is no theory that has attracted volumes of research like efficient market hypothesis (EMH) over four decades. It is the well-known, yet highly controversial theory of Neoclassical School of Finance, which has influenced modern finance in both theory and practice. A market is efficient when prices always ‘fully reflect’ available information (Fama 1970)^{a}. In such an efficient market, when new information arrives, security prices quickly respond and incorporate all information at any point of time, and reach a new equilibrium. Moreover, collection of information is costly and there will be no extra returns on such actions in an informationally efficient market. The fundamental or technical analysis cannot outperform a simple strategy of buying and holding diversified securities. In other words, the EMH rules out any active portfolio management^{b}.

Despite a large body of research on EMH both from developed and developing markets, the consensus on this issue that whether markets are efficient or not, thus continues to be elusive. In recent years, although there is striking evidence that stock returns do not follow random walk and possess some components of predictability, there is a lack of strong alternative theoretical explanations to EMH. Nevertheless, using an evolutionary approach to economic interaction, Lo (2004) has proposed the adaptive market hypothesis (AMH) which can coexist with the EMH in an intellectually consistent manner. The emerging and developing markets have more tendencies to reject EMH because of several market frictions. Unlike EMH that assumes a frictionless market, AMH accommodates market frictions and asserts that markets evolve over a period. In light of this, the present article aims to determine whether AMH provides a better description of the Indian stock market, one of the emerging markets. To the best of our knowledge, there are no studies of this kind in India.

Lo (2004) offers an alternative market theory to EMH from a behavioural perspective, according to which, markets are adaptable and switch between efficiency and inefficiency at different points of time. Lo (2004) applies the evolutionary approach of biology to economic interactions and explains the adaptive nature of the agents and consequently how market becomes adaptive. According to Lo (2005), “degree of market efficiency is related to environmental factors characterizing market ecology such as the number of competitors, the magnitude of profit opportunities available, and the adaptability of the market participants. In contrast to EMH, which assumes a frictionless market, AMH asserts that the laws of natural selection or “survival of the richest” determines the evolution of markets and institutions in real world markets, which have frictions.

Unlike investors in efficient markets, investors do make mistakes and learn to adapt their behaviour accordingly in the framework of AMH. The AMH has a number of practical implications. First, the risk-reward relationship changes over time because of the preferences of the populations in the market. Second, the movement of past prices influences the current preferences because of the forces of natural selection and thus AMH contrasts the weak form of efficiency where history of prices is of no use. Third, in adaptive market, arbitrage opportunities do exist from time to time. From an evolutionary perspective, the profit opportunities are being constantly created and disappear. This calls for investment strategies according to the market environment. In other words, AMH implies “complex market dynamics” which necessitates the active portfolio management. Fourth, innovation is a key to survival and AMH suggest adapting to changing market conditions to ensure a consistent level of expected returns. Finally, market efficiency is not an all or none condition but a characteristic that varies continuously over time and across markets^{c}. Hence, a financial market may witness the periods of efficiency and inefficiency.

The AMH though still in its infancy is attracting attention from researchers. Ito and Sugiyama (2009) find time varying market inefficiency in the US. Charles et al. (2012) holds AMH true in case foreign exchange rates of developing countries in which they find episodes of return predictability depending on market conditions. Kim *et al*. (2011) tests whether the US stock market evolved over time in the US. They find market conditions as the driving factors of predictability and market is more efficient after 1980s than the previous periods. Exploring the relative efficiency, Noda (2012) concludes that TOPIX support AMH while TSE2 rejects AMH in case of Japan. Alvarez-Ramirez *et al.* (2012) provide evidence in favour of AMH and find the US market more efficient during 1973 to 2003. Urquhart and Hudson (2013) document mixed results for the US, the UK and Japan and conclude that the AMH provides a convincing description of these markets.

Given the importance of AMH, a fresh study of efficiency of Indian stock market is required for the following reasons. First, a limited number of studies empirically tested EMH in context of India but the findings are mixed (E.g. Rao and Mukherjee 1971; Sharma and Kennedy 1977; Barua 1981; Amanulla and Kamaiah 1998; Poshakwale 2002, Hiremath and Kamaiah 2010, 2012 among others). There are no studies on Indian stock market, which investigated adaptive behaviour of stock markets. Second, the economic reforms in India were introduced in early 1990s to infuse energy and vibrancy to the process of economic growth. In addition, capital market plans with setting up of National Stock Exchange (NSE) and changes in the market microstructure and trading practices from 1994 sought a transparent, fair and efficient market. As a result, India’s financial system grew by leaps and bounds in post liberalization era. As per the S & P Fact book (Standard and Poor’s 2012), Indian stock market now has the largest number of listed companies on its exchanges. The growing percentage of market capitalization to the GDP and the increasing integration of the Indian market with the global economy indicate the phenomenal growth of the Indian equity market and its growing importance in the economy. Therefore, it is reasonable to expect that emerging markets like India exhibit different characteristics, which distinguish it from developed stock markets and such features influence the nature of market efficiency. Third, the capital market of India emerged as one of the important destinations for investment. The keen interest of foreign institutional investors (FIIs) in Indian stock market for portfolio diversification and higher expected returns is evident from surging foreign investment into Indian capital market. The yield sensitive portfolio investments positively offer liquidity to local markets and sometime trigger panic in the market by reversing the investments. It is logical to expect influence of FIIs on efficiency. Finally, financial crises, both of domestic and foreign origin may affect efficiency of local financial markets.

In this light, departing from the previous studies on efficiency of Indian stock market, we make the following contributions. To the best of our knowledge, this is the first comprehensive work on Indian stock market, which examines the AMH. Thus, the present article complements literature on AMH and extends existing work that has examined efficiency of Indian stock market. Essentially, the available studies refer to the 1980s and early 1990s and hence could not capture the changes in the nature of stock market efficiency in the post policy reforms era. The present study covers the period (1991 to 2013) of such changes is in order. Further, the present study employs methods and techniques, which are first of their kind in the Indian context. Finally, the issue of nonlinearity in stock returns is addressed in this paper seldom received due attention in India. The remainder of the article is structured as follows. Section 2 describes data and econometric methods implemented for estimations. Section 3 discusses the main results and evaluates the relevance of AMH for India. Section 4 summarizes and concludes.

## 2 Methods

For empirical testing, we use daily values of Sensex and Nifty, the major indices traded in India and together constitute 99.9 percent of total market capitalization. The Sensex data is from January 1991 to March 2013 while Nifty data spans from January 1994 to March 2013. To capture changing efficiency or evolving nature of the market, we divide the whole sample into two yearly subsamples^{d}. The present study employs both linear and nonlinear tests for empirical testing of AMH. The sample characteristics and a set of the tests make the results of the present study robust and reduce the risk of overemphasizing the generality of the findings. The following subsections offer a brief description of these tests.

### 2.1 Linear Tests

#### 2.1.1 Autocorrelation Test

*iid*) of random variables. It helps to evaluate whether successive values of serial correlation are different from zero. To test the joint hypothesis that all autocorrelation coefficients

*ρ*

_{ k }are simultaneously equal to zero, we use Ljung and Box’s (1978) portmanteau Q statistic. The test statistic is

where *n* is the number of observations, *m* lag length. The test follows a chi-square (χ^{2}) distribution.

#### 2.1.2 Runs Test

where X is the total number of runs, c_{
i
} is the number of returns changes of each category of sign (*i =* 1, 2, 3). The ER in Equation (2) has an approximate normal distribution for large X. Hence, to test the null hypothesis, we use standard Z statistic.

#### 2.1.3 Variance Ratio Test

_{ t }, the variance of r

_{ t }+ r

_{t-1}are required to be twice the variance of r

_{ t }. Let the ratio of the variance of two period returns, r

_{ t }(2) ≡ r

_{ t }- r

_{t - 1}, to twice the variance of a one-period return r

_{ t }. Then variance ratio VR (2) is

*ρ*(1) is the first order autocorrelation coefficient of returns {r

_{ t }}. RWH which requires zero autocorrelations holds true when VR (2) =1. The VR (2) can be extended to any number of period returns,

*q*. Lo and MacKinlay (1988) showed that the

*q*period variance ratio satisfies the following relation:

_{ t }(

*k*) ≡ r

_{ t }+ r

_{t - 1}+ … + r

_{t - k + 1}and

*ρ*(

*k*) is the

*k*

^{th}order autocorrelation coefficient of {r

_{ t }} Equation (4) shows that at all

*q,*VR (

*q*) = 1. For random walk to hold, variance ratio is expected to be equal to unity. The test is based on standard asymptotic approximations. Lo-MacKinlay proposed Z (

*q*) standard normal test statistic under the null hypothesis of homoscedastic increments and VR (

*q*) =1. The rejection of RWH because of heteroscedasticity, a common feature of financial returns is not useful for any practical purpose. Hence, Lo-MacKinlay constructed a heteroscedastic robust test statistic Z* (

*q*) which can be defined as

which follows a standard normal distribution asymptotically. Thus, according to variance ratio test, the returns process is a random walk when the variance ratio at a holding period *q* is a unity. The variance ratio less than unity imply negative autocorrelation and greater than one indicates positive autocorrelation.

#### 2.1.4 Multiple Variance Ratio Test

*q*) = 1, but in multiple variance ratio test,

*M*

_{ r }= (

*q*

_{ i }) =

*VR*(

*q*) – 1 = 0 which is generalized to a set of

*m*variance ratio tests as

_{0i}is rejected. The heteroscedastic test statistic in Chow-Denning is:

where Z*(*q*_{
i
}) is defined as in Equation (5). Chow-Denning test follows studentized maximum modulus, SMM(α, *m*, T), distribution with *m* parameters and T degrees of freedom. The RWH is rejected, if the value of the standardized test statistic CD. is greater than the SMM critical values at the chosen significance level.

### 2.2 Nonlinear tests

To test the presence of nonlinear dependence, we have carried out a set of nonlinear tests to avoid sensitivity of empirical results to the test employed. Before performing these tests, the linear dependence is removed from the data through fitting AR (*p*). The optimal lag is selected so that no Ljung-Box (LB) Q statistic for residuals extracted from AR (*p*) model is significant at 1 per cent level. Besides, we corrected the financial returns for heteroscedasticity. Therefore, rejection of null for residuals implies presence of nonlinear dependence in returns and market inefficiency.

#### 2.2.1 McLeod-Li Test

where ${r}_{a}^{2}$ is the autocorrelation of the squared residuals and ${e}_{t}^{2}$ is obtained after fitting appropriate AR (*p*). McLeod-Li tests for 2nd order nonlinear dependence.

#### 2.2.2 Tsay Test

*K =*k (k-1)/2 column vector contains all the possible cross products of the form r

_{t-1}r

_{t-j}where

*ϵ*[i, k]. Thus, ${v}_{t,1}={r}_{t-1}^{2};\phantom{\rule{0.25em}{0ex}}{v}_{2}={r}_{t-1},{r}_{t-2};{v}_{t,3}={r}_{t-1}{r}_{t-3};{v}_{t\mathrm{K}+1}={r}_{t-2}{r}_{t-3};{v}_{t,k+2}={r}_{t-2}{r}_{t-4}\phantom{\rule{0.25em}{0ex}}\dots $ and ${v}_{t,k}={r}_{t-k}^{2}$. Further, let ${\widehat{v}}_{t,i}$ denote the projection of

*v*

_{t,i}on

*r*

_{t - 1}…,

*r*

_{t - k}, on the subspace orthogonal to

*r*

_{t - 1}, …

*r*

_{t - k}(the residuals from a regression of

*v*

_{t,i}on

*r*

_{t - 1}, …,

*r*

_{t - k}. Using following regression, the parameters

*γ*

_{1}, …

*γ*

_{ k }are estimated:

The Tsay F statistic is for testing the null hypothesis that *γ*_{1}, … *γ*_{
k
} are all zero.

#### 2.2.3 ARCH-LM test

^{2}of an auxiliary regression, is defined as

When the sample size is *n*, under the null hypothesis of a linear generating mechanism for {e_{t}}, the test statistic NR^{2} for this regression is asymptotically distributed, ${\chi}_{p}^{2}$.

#### 2.2.4 Hinich bicorrelation test

_{t}} are realizations of a stationary pure noise process that has zero bicorrelation (H). Thus, under the null, bicorrelations (H) are expected to be equal to zero. The alternative hypothesis is that the process has some non-zero bicorrelations (third order nonlinear dependence).

where $G\left(r,s\right)=\phantom{\rule{0.25em}{0ex}}{\mathrm{\Sigma}}_{k=1}^{T-S}\left[Z\left({t}_{k}\right)Z\left({t}_{k}+r\right)\left({t}_{k}+s\right)\right]$. Z (t_{k}) are standard observations at time t = k, and L = T^{c} with 0 < c < 0.5^{e}.

#### 2.2.5 BDS test

*et al.*(1996) developed a portmanteau test for time-based dependence in a series, popularly known as BDS (named after its authors). The BDS test uses the correlation dimension of Grassberger and Procaccia (1983). To perform the test for a sample of

*n*observations {x

_{1},..,x

_{n}}, an embedding dimension

*m*, and a distance

*ϵ*, the correlation integral C

_{m}(n, ϵ) is estimated by

*n*is sample size,

*m*is embedding dimension and

*ϵ*is the maximum difference between pairs of observations counted in estimating the correlation integral. The test statistic is:

The BDS considers the random variable √n(C_{m}(n, ϵ) – C_{1}(n, ϵ)^{m} which, for an *iid* process converges to the normal distribution as *n* increases. It has power against a variety of possible alternative specifications like nonlinear dependence and chaos. We use different *m*, and *ϵ* to estimate the BDS statistic.

## 3 Results and discussion

**Descriptive statistics**

Sample period | Mean | Minimum | Maximum | S.D | Skewness | Kurtosis | Jaqua-Bera |
---|---|---|---|---|---|---|---|

| |||||||

| 0.000553 | -0.136 | 0.159 | 0.017 | -0.042 | 5.893 | 7780.03 |

Jan 1991 – Dec 1993 | 0.001988 | -0.136 | 0.123 | 0.024 | -0.047 | 4.624 | 541.93 |

Jan 1994 - Dec 1996 | -0.000116 | -0.046 | 0.056 | 0.014 | 0.454 | 1.229 | 68.16 |

Jan 1997 – Dec 1999 | 0.000656 | -0.086 | 0.073 | 0.018 | -0.086 | 2.091 | 135.45 |

Jan 2000 – Dec 2002 | -0.000525 | -0.074 | 0.071 | 0.017 | -0.338 | 2.165 | 160.65 |

Jan 2003 – Dec 2005 | 0.001348 | -0.118 | 0.079 | 0.013 | -1.139 | 11.120 | 4075.26 |

Jan 2006 – Dec 2008 | 0.000035 | -0.116 | 0.079 | 0.021 | -0.344 | 2.584 | 222.00 |

Jan 2009 – Dec 2011 | 0.000635 | -0.075 | 0.159 | 0.016 | 1.291 | 14.567 | 6766.81 |

Jan 2012- Mar 2013 | 0.000699 | -0.027 | 0.026 | 0.009 | 0.077 | 0.580 | 5.014 |

| |||||||

| 0.00035 | -0.130 | 0.163 | 0.0162 | -0.122 | 6.428 | 8262.46 |

Jan 1994 - Dec 1996 | -0.0002 | -0.043 | 0.054 | 0.0139 | 0.498 | 1.456 | 92.78 |

Jan 1997 – Dec 1999 | 0.0006 | -0.088 | 0.099 | 0.0098 | 0.009 | 3.680 | 422.17 |

Jan 2000 – Dec 2002 | -0.0004 | -0.072 | 0.072 | 0.0160 | -0.244 | 2.652 | 227.11 |

Jan 2003 – Dec 2005 | 0.0012 | -0.130 | 0.079 | 0.0139 | -1.407 | 12.870 | 5488.81 |

Jan 2006 – Dec 2008 | 0.0001 | -0.130 | 0.067 | 0.0209 | -0.530 | 3.298 | 372.82 |

Jan 2009 – Dec 2011 | 0.0006 | -0.063 | 0.163 | 0.0156 | 1.403 | 15.912 | 8078.16 |

Jan 2012 – Mar 2013 | 0.0007 | -0.027 | 0.027 | 0.0091 | 0.079 | 0.643 | 6.09 |

### 3.1 Linear dependence

**LB Q and runs tests statistics**

Sample periods | LB (5) | LB (15) | LB (20) | Runs Z Statistics |
---|---|---|---|---|

| ||||

| -0.001 | 0.024 | -0.023 | -6.385* |

(46.45)* | (75.99)* | (96.84)* | ||

Jan 1991 – Dec 1993 | 0.086 | 0.113 | 0.055 | - 3.528* |

(21.04)* | (52.22)* | (60.94)* | ||

Jan 1994 - Dec 1996 | 0.015 | 0.011 | -0.052 | - 4.236 |

(38.39)* | (48.20)* | (50.89)* | ||

Jan 1997 – Dec 1999 | -0.050 | -0.020 | -0.046 | -1.842 |

(3.73) | (17.04) | (22.41) | ||

Jan 2000 – Dec 2002 | -0.022 | 0.006 | -0.094 | - 2.611* |

(6.34) | (14.42) | (31.77)** | ||

Jan 2003 – Dec 2005 | -0.032 | -0.056 | 0.010 | - 2.358* |

(26.58)* | (35.45)* | (44.28)* | ||

Jan 2006 – Dec 2008 | -0.017 | 0.011 | -0.049 | - 1.3356 |

(7.60) | (15.23) | |||

Jan 2009 – Dec 2011 | -0.055 | 0.002 | -0.081 | - 0.439 |

(6.65) | (17.41) | (31.47)** | ||

Jan 2012 – Mar 2013 | -0.008 | 0.009 | 0.024 | - 0.929 |

(2.15) | (12.78) | (22.54) | ||

| ||||

| -0.008 | 0.001 | -0.042 | - 5.765* |

(34.69)* | (60.71)* | (91.60)* | ||

Jan 1994 - Dec 1996 | 0.030 | 0.003 | -0.020 | - 5.161* |

(44.35)* | (57.73)* | (59.53)* | ||

Jan 1997 – Dec 1999 | 0.002 | -0.016 | 0.009 | - 0.052 |

(0.267) | (14.35) | (23.68) | ||

Jan 2000 – Dec 2002 | 0.016 | 0.013 | -0.107 | - 2.962* |

(12.74)** | 21.02 | 38.90* | ||

Jan 2003 – Dec 2005 | -0.037 | -0.059 | 0.013 | - 2.270** |

37.46* | 55.30* | 61.54* | ||

Jan 2006 – Dec 2008 | -0.011 | 0.026 | -0.066 | - 1.105 |

4.81 | 24.02 | 31.58** | ||

Jan 2009 – Dec 2011 | -0.060 | -0.006 | -0.006 | 0.0367 |

4.98 | 16.93 | 29.14 | ||

Jan 2012 – Mar 2013 | 0.000 | 0.005 | 0.017 | -0.874 |

2.69 | 14.66 | 21.51 |

The runs test results for the last three subsamples show no evidence of autocorrelation. We find no linear autocorrelations during those periods in which the major events namely, the East Asian financial crisis, dot.com bubble burst, and sub-prime mortgage crisis occurred. The autocorrelation and runs test results indicate that the Indian stock market is switching between efficiency and inefficiency. In other words, these results seem to support the view that Indian stock market is adaptive.

*q*) for Sensex and Nifty during the full sample are greater than unity and statistically significant at 1 percent level, indicating returns do not follow a random walk (Table 3). Nevertheless, variance ratio statistics at any investment horizon in all the subsamples are insignificant indicating independence of returns in these sub-periods. The sequential procedure of Lo and MacKinlay (1988) test leads to size distortions and the test ignores the joint nature of random walk. To overcome this problem, The Chow-Denning test, statistically superior to individual variable ratio test, indicates predictability of stock returns in India by rejecting null of random walk over the whole sample at 5 per cent level of significance. However, every subsample provides evidence of the independence of returns. The individual and multiple variance ratio results suggest that the Indian market is largely efficient surrounded by brief periods of predictability which disappear because information quickly begins to reflect in returns and market moves towards efficiency again.

**Variance ratio test statistics**

Sample periods | Lo-MacKinlay variance ratios for investment horizons (q) | Chow and Denning statistic | |||
---|---|---|---|---|---|

2 | 4 | 8 | 16 | ||

| |||||

| 1.08* | 1.12* | 1.12*** | 1.19** | 3.767** |

(3.767) | (2.878) | (1.868) | (2.071) | ||

Jan 1991 – Dec 1993 | 1.11 | 1.20 | 1.26 | 1.42 | 1.066 |

(1.066) | (1.083) | (0.910) | (1.001) | ||

Jan 1994 - Dec 1996 | 1.21 | 1.27 | 1.32 | 1.21 | 0.772 |

(0.772) | (0.567) | (0.424) | (0.196) | ||

Jan 1997 – Dec 1999 | 1.04 | 1.08 | 1.03*** | 1.06 | 0.291 |

(0.291) | (0.292) | (0.082) | (0.094) | ||

Jan 2000 – Dec 2002 | 1.06 | 1.09 | 1.10 | 1.11 | 0.391 |

(0.391) | (0.298) | (0.211) | (0.163) | ||

Jan 2003 – Dec 2005 | 1.08 | 1.02 | 1.08 | 1.15 | 0.271 |

(0.273) | (0.052) | (0.104) | (0.129) | ||

Jan 2006 – Dec 2008 | 1.07 | 1.07 | 0.985 | 1.05 | 0.653 |

(0.653) | (0.346) | (-0.045) | (0.124) | ||

Jan 2009 – Dec 2011 | 1.06 | 1.06 | 1.01 | 1.09 | 0.319 |

(0.319) | (0.172) | (0.022) | (0.117) | ||

Jan 2012 – Mar 2013 | 0.98 | 1.06 | 1.10 | 1.10 | 0.012 |

(-0.01) | (0.023) | (0.024) | (0.018) | ||

| |||||

| 1.07* | 1.08** | 1.06 | 1.10 | 3.180* |

(3.180) | *(1.896) | (1.071) | (1.121) | ||

Jan 1994 - Dec 1996 | 1.23 | 1.31 | 1.40 | 1.25 | 1.055 |

(1.055) | (0.789) | (0.673) | (0.304) | ||

Jan 1997 – Dec 1999 | 1.00 | 0.99 | 0.96 | 0.96 | 0.003 |

(0.003) | (-0.005) | (-0.107) | (-0.068) | ||

Jan 2000 – Dec 2002 | 1.09 | 1.08 | 1.11 | 1.15 | 0.602 |

(0.602) | (0.284) | (0.260) | (0.252) | ||

Jan 2003 – Dec 2005 | 1.11 | 1.06 | 1.12 | 1.16 | 0.587 |

(0.586) | (0.183) | (0.218) | (0.203) | ||

Jan 2006 – Dec 2008 | 1.06 | 1.06 | 0.99 | 1.07 | |

(0.677) | (0.395) | (-0.015) | (0.216) | 0.677 | |

Jan 2009 – Dec - 2011 | 1.04 | 1.05 | 1.00* | 1.07 | 0.288 |

(0.276) | (0.172) | (0.002) | (0.112) | ||

Jan 2012 – Mar 2013 | 0.97 | 1.06* | 1.10** | 1.12** | 0.021 |

(-0.021) | *(0.032) | (0.035) | (0.029) |

*et al*. (2011) who observed no predictability during stock market crashes (1929 and 1987).

### 3.2 Nonlinearity in stock returns

*ρ*) model so that any remaining dependence would be nonlinear. LB statistics for residuals extracted after filtering for linearity show no autocorrelation up to lag 20 for each subsample of Sensex and Nifty (Table 4).

**McLeod-Li test statistics**

Sample periods | AR (ρ) | LB (5) | LB (15) | LB (20) | McLeod-Li statistic | ||
---|---|---|---|---|---|---|---|

Lag 5 | Lag 15 | Lag 20 | |||||

| |||||||

| 9 | 0.043 | 0.748 | 26.32 | 988.6* | 2130.1* | 2415.5* |

(1.000) | (1.000) | (0.155) | (0.000) | (0.000) | (0.000) | ||

Jan 1991 – Dec 1993 | 7 | 0.196 | 24.06 | 29.57 | 81.7* | 238.0* | 255.3* |

(0.999) | (0.064) | (0.077) | (0.000) | (0.000) | (0.000) | ||

Jan 1994 - Dec 1996 | 2 | 4.745 | 16.94 | 19.14 | 47.17* | 97.49* | 130.53* |

(0.447) | (0.322) | (0.512) | (0.000) | (0.000) | (0.000) | ||

Jan 1997 – Dec 1999 | 1 | 3.64 | 16.86 | 22.38 | 30.19* | 41.84* | 52.99* |

(0.602) | (0.327) | (0.320) | (0.000) | (0.000) | (0.000) | ||

Jan 2000 – Dec 2002 | 2 | 3.161 | 11.00 | 26.48 | 187.69* | 296.07* | 329.56* |

(0.675) | (0.752) | (0.150) | (0.000) | (0.000) | (0.000) | ||

Jan 2003 – Dec 2005 | 2 | 8.673 | 20.459 | 23.306 | 245.29* | 263.26* | 264.19* |

(0.122) | (0.155) | (0.274) | (0.000) | (0.000) | (0.000) | ||

Jan 2006 – Dec 2008 | 2 | 2.349 | 18.209 | 23.168 | 277.27* | 590.73* | 671.05* |

(0.798) | (0.251) | (0.280) | (0.000) | (0.000) | (0.000) | ||

Jan 2009 – Dec - 2011 | 1 | 6.263 | 18.81 | 23.67 | 4.712 | 29.28** | 33.42** |

(0.281) | (0.222) | (0.296) | (0.451) | (0.014) | (0.030) | ||

Jan 2012 – Mar 2013 | 0 | 2.152 | 12.788 | 22.546 | 1.550 | 15.00 | 27.79 |

(0.827) | (0.618) | (0.311) | (0.907) | (0.451) | (0.114) | ||

| |||||||

| 11 | 0.028 | 6.371 | 26.939 | 550.20* | 964.60* | 1066.23* |

(1.000) | (0.972) | (0.137) | (0.000) | (0.000) | (0.000) | ||

Jan 1994 - Dec 1996 | 2 | 5.813 | 19.919 | 21.824 | 69.38* | 154.97* | 185.82* |

(0.324) | (0.175) | (0.350) | (0.000) | (0.000) | (0.000) | ||

Jan 1997 – Dec 1999 | 0 | 0.267 | 14.356 | 23.686 | 23.72* | 28.13** | 49.47* |

(0.998) | (0.498) | (0.256) | (0.000) | (0.020) | (0.000) | ||

Jan 2000 – Dec 2002 | 2 | 1.429 | 7.764 | 22.700 | 108.87* | 199.44* | 220.53* |

(0.921) | (0.932) | (0.303) | (0.000) | (0.000) | (0.000) | ||

Jan 2003 – Dec 2005 | 2 | 8.715 | 21.42 | 28.444 | 286.24* | 310.20* | 311.07* |

(0.157) | (0.321) | (0.099) | (0.000) | (0.000) | (0.000) | ||

Jan 2006 – Dec 2008 | 2 | 1.593 | 19.37 | 26.913 | 232.57* | 441.67* | 489.24* |

(0.902) | (0.197) | (0.137) | (0.000) | (0.000) | (0.000) | ||

Jan 2009 – Dec - 2011 | 2 | 3.379 | 14.305 | 26.676 | 2.177 | 18.701 | 22.491 |

(0.641) | (0.502) | (0.144) | (0.824) | (0.227) | (0.314) | ||

Jan 2012 – Mar 2013 | 0 | 2.697 | 14.663 | 21.516 | 2.667 | 16.101 | 30.169*** |

(0.746) | (0.475) | (0.367) | (0.751) | (0.375) | (0.067) |

**Tsay, Engle LM and H statistics**

Sample period | AR (ρ) | Tsay F statistic | Engle LM statistic | H statistic | ||||
---|---|---|---|---|---|---|---|---|

Lag 5 | Lag 15 | Lag 20 | Lag 5 | Lag 15 | Lag 20 | |||

| ||||||||

| 9 | 7.862* | 3.613* | 3.039* | 564.1* | 729.5* | 758.2* | 3760.9* |

(0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | ||

Jan 1991 – Dec 1993 | 7 | 2.837* | 1.907* | 1.786* | 54.8* | 101.2* | 110.5* | 405.6* |

(0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | ||

Jan 1994 - Dec 1996 | 2 | 1.858* | 1.273** | 1.282** | 32.4* | 50.5* | 74.9* | 139.7* |

(0.000) | (0.041) | (0.016) | (0.000) | (0.000) | (0.000) | (0.000) | ||

Jan 1997 – Dec 1999 | 1 | 2.436* | 1.686* | 1.457* | 28.86* | 37.5* | 47.7* | 183.9* |

(0.001) | (0.000) | (0.000) | (0.000) | (0.001) | (0.005) | (0.000) | ||

Jan 2000 – Dec 2002 | 2 | 2.396* | 2.433* | 2.168* | 110.67* | 138.8* | 148.9* | 364.8* |

(0.002) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | ||

Jan 2003 – Dec 2005 | 2 | 6.609* | 2.257* | 1.910* | 268.96* | 272.2* | 272.3* | 721.7* |

(0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | ||

Jan 2006 – Dec 2008 | 2 | 4.734* | 2.746* | 2.667* | 153.7* | 179.4* | 181.7* | 680.9* |

(0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | ||

Jan 2009 – Dec - 2011 | 1 | 1.24 | 2.50* | 2.483* | 4.9 | 22.8 | 24.3 | 242.9* |

(0.229) | (0.000) | (0.000) | (0.495) | (0.088) | (0.231) | (0.000) | ||

Jan 2012 – Mar 2013 | 0 | 0.560 | 1.558* | 1.073 | 1.6 | 13.3 | 25.8 | 52.8 |

(0.903) | (0.003) | (0.359) | (0.911) | (0.576) | (0.172) | (0.198) | ||

| ||||||||

| 9 | 6.240* | 2.877* | 2.427* | 352.20* | 425.36 | 437.38* | 1848.41* |

(0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | ||

Jan 1994 - Dec 1996 | 2 | 1.509 | 1.583* | 1.436* | 50.21* | 77.758 | 92.62* | 158.67* |

(0.095) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | ||

Jan 1997 – Dec 1999 | 0 | 2.842* | 1.687* | 1.295** | 24.21* | 27.85** | 51.658* | 157.77* |

(0.000) | (0.000) | (0.011) | (0.000) | (0.022) | (0.000) | (0.000) | ||

Jan 2000 – Dec 2002 | 2 | 1.852** | 2.173* | 1.949* | 79.97* | 126.46* | 130.54* | 380.80* |

(0.024) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | ||

Jan 2003 – Dec 2005 | 2 | 6.757* | 2.413* | 1.985* | 315.46* | 320.34* | 321.86* | 799.69* |

(0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | ||

Jan 2006 – Dec 2008 | 2 | 5.583* | 2.705* | 2.459* | 125.40* | 152.41* | 158.98* | 663.08* |

(0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | (0.000) | ||

Jan 2009 – Dec - 2011 | 2 | 0.873 | 2.313* | 2.268* | 2.023 | 15.292 | 16.87 | 195.63* |

(0.593) | (0.000) | (0.000) | (0.845) | (0.430) | (0.661) | (0.000) | ||

Jan 2012 – Mar 2013 | 0 | 0.489 | 1.577* | 1.181 | 2.931 | 14.672 | 27.786 | 56.255 |

(0.945) | (0.003) | (0.193) | (0.710) | (0.475) | (0.114) | (0.121) |

The H statistics reject null of pure noise for Sensex and Nifty in all the subsamples with the exception of subsample 2012–2013 (Table 5). This exposes nonlinear characteristics of Indian stock returns during these periods. Finally, the BDS statistics support evidence of nonlinear dependence during the subsamples and full sample for both the indices (Table 6). The rejection for residuals from AR (*ρ*) indicates presence of nonlinear dependence and implies the possible predictability of future returns using the history of returns.

The trends in McLeod-Li show stronger presence of nonlinear dependence in Sensex and Nifty returns during subsamples 2003–2005 and 2006–2008 (Figure 2). Again, the trends in Engle LM, Tsay, H and BDS test statistics are low indicating lesser magnitude of nonlinear dependency in stock returns till the year 2000 and thereafter returns exhibit increasing nonlinear tendency reaching peak during subsample, 2006–2008. The subsample 2006–2008 that possess pockets of strong presence of nonlinear dependence is associated with sub-prime mortgage and global financial crisis (2008). In post 2008 subsample, however, all the test statistics suggest relatively weaker presence of nonlinear dependence in returns (See, Figure 2).

**BDS test statistics**

Sample period | AR (ρ) | m = 2, ϵ = 0.75s | m = 4, ϵ = 1.0s | m = 8, ϵ =1.25S | m = 10, ϵ = 1.50s |
---|---|---|---|---|---|

| |||||

| 9 | 17.65* (0.000) | 27.88* (0.000) | 42.94*(0.000) | 44.24* (0.000) |

Jan 1991 – Dec 1993 | 7 | 3.74* (0.000) | 4.42*(0.000) | 6.33*(0.000) | 7.40*(0.000) |

Jan 1994 - Dec 1996 | 2 | 5.76* (0.000) | 8.24* (0.000) | 11.66* (0.000) | 12.21*(0.000) |

Jan 1997 – Dec 1999 | 1 | 3.00* (0.002) | 4.52* (0.000) | 6.30*(0.000) | 6.95* (0.000) |

Jan 2000 – Dec 2002 | 2 | 8.46*(0.000) | 13.08*(0.000) | 18.11*(0.000) | 19.20*(0.000) |

Jan 2003 – Dec 2005 | 2 | 3.85* (0.000) | 5.60* (0.000) | 9.01*(0.000) | 9.77* (0.000) |

Jan 2006 – Dec 2008 | 2 | 9.08* (0.000) | 14.26*(0.000) | 24.40*(0.000) | 23.56*(0.000) |

Jan 2009 – Dec - 2011 | 1 | 3.85* (0.000) | 7.40*(0.000) | 12.98* (0.000) | 14.11* (0.000) |

Jan 2012 – Mar 2013 | 0 | -0.71* (0.474) | 0.306* (0.759) | 1.893* (0.058) | 2.288* (0.022) |

| |||||

| 11 | 15.15* (0.000) | 23.89 (0.000) | 35.94 (0.000) | 37.65 (0.000) |

Jan 1994 - Dec 1996 | 2 | 5.322* (0.000) | 8.534* (0.000) | 11.33* (0.000) | 11.73* (0.000) |

Jan 1997 – Dec 1999 | 0 | 2.149* (0.031) | 4.081* (0.000) | 5.351* (0.000) | 5.808* (0.000) |

Jan 2000 – Dec 2002 | 2 | 8.08* (0.000) | 12.14* (0.000) | 15.28* (0.000) | 15.59* (0.000) |

Jan 2003 – Dec 2005 | 2 | 4.67* (0.000) | 6.43* (0.000) | 9.68* (0.000) | 10.89* (0.000) |

Jan 2006 – Dec 2008 | 2 | 8.63* (0.000) | 13.89* (0.000) | 23.71* (0.000) | 22.49* (0.000) |

Jan 2009 – Dec - 2011 | 2 | 3.73* (0.000) | 6.61* (0.000) | 12.02* (0.000) | 12.97* (0.000) |

Jan 2012 – Mar 2013 | 0 | -1.05 (0.292) | 0.288 (0.772) | 1.732 (0.083) | 2.905 (0.004) |

The linear test results support the proposition that Indian stock market switched between efficient and inefficient periods and this finding is consistent with Charles et al. (2012) Kim *et al.* (2011), and Alvarez-Ramirez *et al.* (2012). Nevertheless, the present evidence of strong presence of nonlinear dependence in stock returns throughout the sample suggests that the Indian stock market still inefficient and not experienced efficiency yet. Our finding of highest magnitude of nonlinearity during periods of financial crises in Indian stock returns is consistent with the findings of Urquhart and Hudson (2013) who found similar evidence for the US market. The evidence of nonlinearity during financial crises shows that the stock market crash and economic crises negatively affect the stock market efficiency. Furthermore, the present study finds outflow of FIIs during global financial crises. This evidence suggests that the increasing integration of Indian capital market has not only brought the benefits but also exposed the market to external shocks.

## 4 Summary and conclusion

The present paper has investigated the adaptive market hypothesis (AMH) in India, one of the fastest growing markets. The linear test results indicated a cyclical pattern in autocorrelations suggesting that the Indian stock market switched between periods of efficiency and inefficiency and market has become efficient from the year 2003. The findings from each of the nonlinear tests suggest a strong presence of nonlinear dependence in Indian stock returns throughout the sample period implying possible predictability of returns and consequent excess returns. The nonlinearity in stock returns was highest during various financial crises originated outside India and this finding shows association of informational inefficiency and financial crises. Furthermore, the vulnerability of Indian stock market to the external shocks in a financially liberalized economy is evident from the outflow of FIIs owing to external events. The present evidence of influence of financial crises and reversal of FIIs on efficiency of stock market should be interpreted as identifying an association rather than causality.

The reforms initiated have positive influence on stock market is evident from the fall in magnitude of nonlinear dependence in recent periods. The present study finds that Indian stock market is still evolving and not fully adaptive, as it has not gone through a single period of efficiency. The linear independence and weaker presence of nonlinear dependence in returns from 2009 is sufficient to conclude that Indian stock market is moving towards efficiency. The present finding of an increased possibility of predictability during financial crises and large outflow of investment calls for appropriate policy measures to address the external shocks and retain the confidence of foreign investors.

### Endnotes

^{a}The seminal work of Bachelier (1900) laid theoretical foundation for the theory of efficient market. The pioneering work of Samuelson (1965) added rigour to the theory of stock market efficiency.

^{b}Malkiel (1973) writes to the extent that ‘a blindfolded chimpanzee throwing darts at the Wall Street could select a portfolio that would do as well as the experts’.

^{c}Campbell *et al.* (1997) note that testing of market efficiency as a condition of all or nothing is not useful and such an efficient market is the economically unrealizable ideal market. They suggest relative efficiency because measuring efficiency provides more insights than testing it.

^{d}A rolling sample is an alternative method used in empirical literature to examine evolving nature. However, an event may squeeze or influence the overall results. Hinich and Patterson (1995) suggested a windowed approach. We did not find any optimization benefits in using rolling sample in the present context.

^{e}Hinich and Patterson in their unpublished work of 1995 recommend c = 0.4. The same is followed here.

## Declarations

### Acknowledgements

We thank the participants of Fourth International Conference on Applied Econometrics held on March 20 & 21, 2014 at IBS Hyderabad, India for the useful suggestions. Usual disclaimer applies.

## Authors’ Affiliations

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## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.