Nonlinear Mean Reversion across National Stock Markets: Evidence from Emerging Asian Markets
 Author: CHEN SHULING, KIM HYEONGWOO
 Publish: International Economic Journal Volume 25, Issue2, p239~250, June 2011

ABSTRACT
This paper seeks empirical evidence of nonlinear meanreversion in
relative national stock price indices for Emerging Asian countries. It is well known that conventional linear unit root tests suffer from low power against the stationary nonlinear alternative. Implementing the nonlinear unit root tests proposed by Kapetanioset al . (2003) and Cerratoet al . (2009) for the relative stock prices of Emerging Asian markets, we find strong evidence of nonlinear mean reversion, whereas linear tests fail to reject the unit root null for most cases. We also report some evidence that stock markets in China and Taiwan are highly localized.

KEYWORD
Linear unit root test , nonlinear unit root test , nonlinear panel unit root test , international relative stock prices

1. Introduction
In the field of finance, mean reversion properties of asset prices have been widely investigated to examine the validity of the contrarian investment strategy.1 Despite extensive studies, empirical evidence on mean reversion of stock prices is still mixed at best.2 A growing quantity of literature has also started investigating mean reversion among international stock price indices. Among others, Kasa (1992) reported cointegrating relations for the national stock indices of five developed countries, while Richards (1995) found no such relations when he used proper critical values.
More recently, Balvers
et al. (2000) employed a seemingly unrelated regression (SUR) technique for stock prices in 18 developed countries relative to a reference index, such as the US stock index. They reported strong evidence of mean reversion. Similar evidence has been reported by Chaudhuri and Wu (2004) for 17 emerging equity markets.It should be noted, however, that their technique is subject to the following problems. First, their SUR estimation imposes a homogeneity assumption that assumes all countries share identical speeds of mean reversion. This is a very strong assumption and contradicts common wisdom. Second, their panel unit root testmay have a serious size distortion problem in the presence of crosssection dependence, which was pointed out by Phillips and Sul (2003).
We note that SUR/panel unit root tests are not the only way of improving the power of unit root tests, and take a different approach by implementing a nonlinear unit root test proposed by Kapetanios
et al. (2003) for each of nine Emerging Asian countries to improve the power of unit root tests. In addition, we employ a nonlinear panel unit root test recently proposed by Cerratoet al. (2009). These tests allow different mean reversion rates across countries, thus do not require the homogeneity assumption.Unlike the conventional linear unit root test, their tests allowsmooth transition between the stationary regime and the nonstationary regime around the longrun equilibrium value, which can be justified by nonlinear adjustments of financial market variables in the presence of fixed transaction costs.3 Using the Morgan Stanley Capital International (MSCI) stock index data for these countries, we find very strong evidence of nonlinear meanreversion across these countries.We also find that only some, but not all, Emerging Asian countries possess nonlinear cointegrating relations with the US stock index as well as theWorld stock index, which provides less support for the homogeneity assumption of the SUR unit root test.
Similar work has been done by Lim and Liew (2007) and Hasanov (2007), who test the nonlinear mean reversion for individual Asian equity prices. It should be noted, however, that they test the unit root null hypothesis for the nominal equity prices without taking any economic fundamentals (e.g., priceearning ratio or dividend yield) into consideration.4 Our work is different from theirs, since we test the nonlinear mean reversion for the
relative prices or stock price deviations from a reference index. When a country shares a fundamental value, possibly a unit root process, with a reference country or index, the stock price deviation from the reference index should be meanreverting.To deal with the second issue, we implement an array of panel unit root tests, including a newly proposed nonlinear panel unit root test by Cerrato
et al . (2009). We do find that controlling crosssection dependence substantially lowers rejection probabilities of the panel unit root tests.5 However, we find much stronger evidence of nonlinear mean reversion relative to its linear counterpart irrespective of the treatment of crosssection dependence.The rest of the paper is organized as follows. Section 2 describes our baseline linear model of stock indices in two countries.We extend this model to a nonlinear adjustment model in Section 3 and to a nonlinear panel model in Section 4. Section 5 reports our main findings. Section 6 concludes.
1If asset prices are meanreverting, shortselling assets with relatively better performance and buying assets with poor performance in the past may create excess returns. See DeBondt and Thaler (1985). 2For example, Fama and French (1988) and Poterba and Summers (1988) found evidence that favors meanreversion in US stock prices.Yet, many others questioned the validity of meanreversion on the robustness issue with regards to the choice of sample period (Kim et al., 1991), the distributional assumptions (Kim et al., 1991; McQueen, 1992), and small sample bias (Richardson and Stock, 1989; Richardson, 1993). 3For example, in the presence of market frictions or transaction costs, arbitrages occur only when the deviations from the fundamental values are big (see, among others; Dumas, 1992; Michael et al., 1997). In other words, when the deviations are relatively small, asset prices may exhibit local nonstationarity around the longrun equilibrium values in the absence of any arbitrage. When dealing with an aggregate price index, a smooth transition model would make more sense, since the transaction costs might be different across the products. 4One may test the unit root null for the nominal price deviations from a fundamental variable such as the dividend yield and the priceearning ratio. 5Kim (2009) finds similar results for stock price indices of 18 countries with welldeveloped capital markets.
2. The Linear Cointegration Model
We first consider a linear model for the stock markets in two countries,
A andB . Letand
be the log of the stock index and the log of its fundamental value for country
i , respectively. Ifis meanreverting around
its stochastic process can be represented as the following error correction model,
where −1 <
λ < 0 is a common convergence rate parameter forA andB , andis an idiosyncratic meanzero i.i.d. process. The timevarying fundamental term
a possible unit root process, is not directly observable but is assumed to obey the following stochastic process,
where
is the common component for
and
b^{i} is a countryspecific constant, andis an idiosyncratic zeromean, possibly serially correlated stationary process.
Combining equations (1) and (2), we obtain
where
For notational simplicity, let rt denote the stock price deviations (or the relative stock price),
Lagging time subscript by one, we get
or equivalently,
where
ρ = 1 +λ is the persistence parameter of the deviation.Note that the error term
ε_{t} is serially correlated even whenis an i.i.d. process. In order to control this serial correlation, we augment the equation (4) as follows:
where
e_{t} is a martingale difference sequence that generatesε_{t} .Note that the regression equation (5) is a conventional augmented DickeyFuller (ADF) regression equation with a known cointegrating vector [1 –1] for the integrated processes
and
When
and
share acommonunit root process
in equation (2), the stock price deviation
r_{t} should be stationary (0 <ρ < 1), and the conventional ADF test applies to test such a linear cointegration relation across the stock markets inA andB .3. The Nonlinear Cointegration Model
We extend the regression model (5) to a nonlinear cointegration model that allows nonlinear adjustments of the stock price deviation
r_{t} . Stock prices may adjust to its longrun equilibrium only when the deviation is big enough in the presence of fixed a transaction cost. Then,r_{t} may follow a unit root process locally around the longrun equilibrium value, when the transaction cost is prohibitively high. Such a stochastic process can be represented by the following exponential smooth transition autoregressive process. Abstracting from a constant for notational simplicity,where
κ is a strictly positive scale parameter so thatand
d is a delay parameter.Note that when
r_{t−d} is very big, put differently, stock price indices significantly deviate from each other,becomes about zero, and the equation (6) reduces to a stationary
AR (1) process, where 1 +λ =ρ < 1. On the other hand, ifr_{t−d} is close to zero,is about unity, which leads to a unit root process.
Since
λ is not identified under the unit root null hypothesis,6 Kapetanioset al . (2003) transformed equation (6) toBy the Taylor approximation of equation (7), they obtained the following equation
They show that, under the unit root null, the least squares
t statistic forhas the following asymptotic distribution
where
W (z ) is the standard Brownian motion defined onz ∈ [0, 1].When error terms (
ε_{t} ) are serially correlated, equation (8) can be augmented as follows6This is the socalled Davies’ Problem.
4. The Nonlinear Panel Cointegration Model
Lastly, we consider a nonlinear panel cointegration test proposed by Cerrato
et al. (2009), which is an extension of Kapetanioset al. (2003) and Pesaran (2007). Their nonlinear test is more powerful than conventional linear panel unit root tests, such as the IPS test by Imet al. (2003), and can allow for crosssection dependence.For this purpose, rewrite equation (7) as follows.
where
δ_{i} is a countryspecific factor loading,f_{t} is a common factor, andu_{i,t} is a possibly serially correlated idiosyncratic shock.7 Cerratoet al. (2009) suggest the following nonlinear crosssection augmented IPStype statistics.where
t_{i} (N,T ) is the tstatistic forβ _{i,0} from the following least squares regression,for the serially uncorrelated error case and for the serially correlated error case, respectively, and
is the crosssection average at time t,which proxies thecommon factor component for
i = 1, . . . ,N .Note that, in the absence of crosssection dependence,
γ_{i,j} = 0 for alli andj and the test statistic is reduced to nonlinear IPStype statistic.7Recall, in Section 2, that we construct ri,t as a deviation of individual stock price from its fundamental value. Therefore, ft can be interpreted as any remaining common shock components that originate from the emerging Asian countries.
5. Empirical Results
We use the monthly data obtained from the Morgan Stanley Capital International (MSCI) for stock market indices of nine Emerging Market (EM) Asian countries, the US stock index, and the World stock index as well as two local reference indices, the EM–Asia and the EM–Far East indices. The data cover the period fromDecember 1987 through December 2007 with the exceptions of China, India, and Pakistan.8 The observations are endofperiod valueweighted stock prices of many companies in each market. The indices include reinvested gross dividends and are transformed to the US dollar terms using endofperiod foreign exchange rates.
Table 1 presents descriptive statistics for the logarithm of the stock price indices for Emerging Asian countries and reference indices.
Following Balvers et al. (2000), we begin our analysis by implementing the ADF test for the stock price deviations of EM–Asia indices relative to the US stock index and the World stock index. We choose the number of lags (
k ) by the GeneraltoSpecific Rule (Hall, 1994) as recommended by Ng and Perron (2001) and implement the tests when an intercept is included and when an intercept and time trend are included.9 As shown in Table 2, the ADF test rejects the unit root null for virtually no country. The only exception was the Taiwan index deviation relative to the World index when the trend term is included.In contrast to the results from the ADF test, our nonlinear unit root test rejects the null of unit root for four countries, Indonesia, Korea, Malaysia, and Pakistan, at the 5% significance level irrespective of the choice of the reference index. When we relax the significance level to 10%, the unit root null is rejected for two more countries, Taiwan and Thailand. Such findings imply that the stock price indices in many EM Asian markets exhibit socalled ‘coupling’ relations with these reference indices in the longrun. Our findings also suggest that there exist nonnegligible sources of market frictions in EMAsia markets. It is interesting to see that we find strong evidence of meanreversion for a subset of these countries. This finding implies that the homogeneity assumption by Balvers
et al . (2000) may be problematic.Next, we turn our attention to pairwise unit root tests across EM Asian countries (see Table 3). Again, the linear test hardly rejects the unit root null. The only exception is Korea, where the test rejects the null for a maximum of four out of eight local partners. Surprisingly, the nonlinear test with an intercept rejects the unit root null for 18 out of 36 pairs favoring nonlinear mean reversion (Table 4). By allowing trend stationarity, we obtain seven additional rejections totaling 25 rejections out of 36 pairs.
We also consider the cases when a local aggregate stock index such as the EM–Asia index or the EM–Far East index serves as a reference index. Again, we obtain very strong evidence of nonlinear mean reversion for the deviations of China, Indonesia, Korea, and Taiwan relative to these local reference indices. It is interesting to see that the stock indices of China and Taiwan exhibit very strong tendencies toward these local indices, whereas they have relatively weak longrun relations with the US stock index and theWorld stock index.We interpret this as the evidence of localized stock markets for those countries.
Lastly, we implement an array of panel unit root tests with four different reference indices, the US index, theWorld index, the EM–Asia index, and the EM–Far East index. Results are reported in Table 5.
We first test the null of a unit root with the linear stationarity alternative hypothesis using the IPS panel unit root test. The IPS test fails to reject the null hypothesis for all cases even at the 10% significance level. However, a nonlinear IPStype panel unit root test, based on equations (13) or (14) with a restriction of
γ_{i,j} = 0, rejects the null of a unit root at the 5% significance level for all cases.Next, we implement the crosssection augmented IPStype (CIPS) test by Pesaran (2007) as well as the nonlinear CIPS (NCIPS) test by Cerrato
et al . (2009). It should be noted that allowing for crosssection dependence reduces the probability of rejection of the null hypothesis substantially, which may be consistent with the findings of Phillips and Sul (2003).10 It should be also noted, however, that thep values of NCIPS statistics are uniformly less than their linear counterparts. Therefore, we find much stronger evidence in favor of nonlinear mean reversion no matter how we treat crosssection dependence.8For these countries, the observations span from December 1992 ending December 2007. 9Note that meanreversion property is more closely related to the ADF test with an intercept only, since rejecting the unit root null from the ADF test with both intercept and time trend implies that the series is trend stationary. Therefore, the ADF test with both deterministic terms should be understood as a supplementary test when the test with an intercept only does not reject the unit root null. 10This casts doubt on the results of Balvers et al. (2000) and Chaudhuri and Wu (2004).
6. Concluding Remarks
This paper investigates nonlinear mean reversion across international stock markets using Morgan Stanley Capital International monthly stock index data for nine Emerging Asian countries along with both the global and the local reference indices. As a preliminary analysis, we implement conventional linear unit root tests for the stock price deviations relative to reference indices. The linear test fails to reject the unit root null for most countries. Pairwise tests yield similar results.
As Taylor
et al . (2001) noted, such results may result from a lowpower problem of the ADF test when the true data generating process is a nonlinear transition autoregressive model,which can be theoretically justified by transaction cost arguments. By implementing univariate and panel nonlinear unit root tests, we find strong evidence of mean reversion favoring nonlinear adjustments of stock prices toward the fundamental values. Hence, our results imply that nonnegligible sources of market frictions exist such as strictly positive transaction costs. We also find some evidence of highly localized stock markets for China and Taiwan.

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[Table 1.] Descriptive statistics for the log stock price indices

[Table 2.] Unit root test for the log stock price deviations relative to reference indices

[Table 3.] Linear unit root test for the log stock price deviations across EMAsia countries

[Table 4.] Nonlinear unit root test for the log stock price deviations across EMAsia countries

[Table 5.] Panel unit root tests for EMAsia countries