Testing for White Noise Hypothesis of Stock Returns

Transcription

1 Testing for White Noise Hypothesis of Stock Returns Jonathan B. Hill University of North Carolina Kaiji Motegi Kobe University This draft: July 8, 27 Abstract Weak form efficiency of stock markets is tested predominantly under an independence or martingale difference assumption. Since these properties rule out weak dependence that may exist in stock returns, it is of interest to test whether returns are white noise. We perform multiple white noise tests assisted by Shao s (2) dependent wild bootstrap. We reveal that, in rolling windows, the block structure inscribes an artificial periodicity in bootstrapped confidence bands. We eliminate the periodicity by randomizing a block size. In crisis periods, returns of FTSE and S&P have negative autocorrelations that are large enough to reject the white noise hypothesis. JEL classifications: C2, C58, G4. Keywords: dependent wild bootstrap, maximum correlation test, randomized block size, serial correlation, weak form efficiency, white noise test. We thank Eric Ghysels, Peter R. Hansen, Yoshihiko Nishiyama, and Ke Zhu for helpful comments. We also thank participants at 24th Kansai Keiryo Keizaigaku Kenkyukai, th Spring Meeting of the Japan Statistical Society, 5th anniversary seminar of the Department of Statistics and Actuarial Science, the University of Hong Kong, st Conference of Econometrics and Statistics, and 4th Annual Conference of the International Association for Applied Econometrics for helpful comments. The second author is grateful for financial supports from JSPS KAKENHI (Grant Number 6K74), Kikawada Foundation, Mitsubishi UFJ Trust Scholarship Foundation, Nomura Foundation, and Suntory Foundation. Department of Economics, University of North Carolina, Chapel Hill. jbhill@ .unc.edu Corresponding author. Graduate School of Economics, Kobe University. 2- Rokkodai-cho, Nada, Kobe, Hyogo Japan. motegi@econ.kobe-u.ac.jp

2 Introduction We perform a variety of tests of stock market efficiency over rolling data sample windows, and make new contributions to the study of white noise tests. A stock market is weak form efficient if stock prices fully reflect historical price information (Fama, 97). Empirical results have been mixed, with substantial debate between advocates of the efficient market hypothesis (EMH) and proponents of behavioral finance. More recently, the adaptive market hypothesis (AMH) proposed by Lo (24, 25) attempts to reconcile the two opposing schools, arguing that the degree of stock market efficiency varies over time. In line with these trends, a number of recent applications perform rolling window analysis in order to investigate the dynamic evolution of stock market efficiency. See Kim and Shamsuddin (28), Lim, Brooks, and Kim (28), Kim, Shamsuddin, and Lim (2), Lim, Luo, and Kim (23), Verheyden, De Moor, and Van den Bossche (25), Anagnostidis, Varsakelis, and Emmanouilides (26), and Urquhart and McGroarty (26). An advantage of rolling window analysis relative to non-overlapping subsample analysis is that the former does not require a subjective choice of the first and last dates of major events, including financial crises. A common perception in the literature is that a financial crisis augments investor panic and thus lowers market efficiency. Lim, Brooks, and Kim (28) and Anagnostidis, Varsakelis, and Emmanouilides (26) find that market efficiency is indeed adversely affected by financial crises. By comparison, Kim and Shamsuddin (28) and Verheyden, De Moor, and Van den Bossche (25) find relatively mixed results. In terms of methodology, many early contributions use Lo and MacKinlay s (988) variance ratio test and its variants developed by Chow and Denning (993), Choi (999), Chen and Deo (26), Kim (29), and Charles, Darné, and Kim (2) among others. 2 Other statistical tests include Hinich s (996) bicorrelation test, Hong s (999) spectral test, Wright s (2) nonparametric sign/rank test, Escanciano and Valasco s (26) generalized spectral test, and Es- See Yen and Lee (28) and Lim and Brooks (29) for an extensive survey of stock market efficiency. 2 See Charles and Darné (29) for a survey of variance ratio tests. 2

3 canciano and Lobato s (29) robust automatic portmanteau test. Note that the implicit null hypothesis of all tests above is either that returns are iid, or a martingale difference sequence (mds) because the utilized asymptotic theory requires such assumptions under the null. These properties rule out higher forms of dependence that may exist in stock returns, while the mds property is generally not sufficient for a Gaussian central limit theory (e.g. Billingsley, 96). Chen and Deo (26), for example, impose a martingale difference property on returns, and an eighth order unconditional cumulant condition. These are only shown to apply to GARCH and stochastic volatility processes with iid innovations, which ignores higher order dependence properties that arise under temporal aggregation (Drost and Nijman, 993). Moreover, their test is not a true white noise test since it does not cover serial correlations at all lags asymptotically. A natural alternative is simply a white noise test with only serial uncorrelatedness under the null, as well as standard higher moment and weak dependence properties to push through standard asymptotics. A rejection of the white noise hypothesis might serve as a helpful signal for arbitragers, since a rejection indicates the existence of non-zero autocorrelation at some lags. We place the present study in the literature that is strongly interested in whether asset returns are white noise, a useful albeit weak measure of market efficiency. 3 Formal white noise tests with little more than serial uncorrelatedness under the null have not been available until recently. See Hill and Motegi (27a) for many detailed references, some of which are discussed below. Conventional portmanteau or Q-tests bound the maximum lag and therefore are not true white noise tests, although weak dependence, automatic lag selection, and a pivotal structure irrespective of model filter are allowed. See Romano and Thombs (996), Lobato (2), Lobato, Nankervis, and Savin (22), Horowitz, Lobato, Nankervis, and Savin (26), Escanciano and Lobato (29), Delgado and Velasco (2), Guay, Guerre, and Lazarova (23), 3 Another interpretation of the present analysis is testing whether stock prices follow what Campbell, Lo, and MacKinlay (997) call Random Walk 3 (i.e. random walk with uncorrelated increment) in a strict sense. 3

4 Zhu and Li (25), and Zhang (26). 4 Hong (996, 2) standardizes a portmanteau statistic, allowing for an increasing number of serial correlations and standard asymptotics. See also Hong and Lee (23). Spectral tests operate on the maximum (and therefore increasing) number of serial correlations as the sample size increases, with variations due to Durlauf (99), Hong (996), Deo (2), and Delgado, Hidalgo, and Velasco (25). Durlauf (99) and Deo (2) apply their tests to stock returns. Cramér-von Mises and Kolmogorov-Smirnov variants can be found in Shao (2) with a dependent wild bootstrap procedure that allows for weak dependence under the null. A weighted sum of serial correlations also arises in the white noise test of Andrews and Ploberger (996), cf. Nankervis and Savin (2). Hill and Motegi (27a) develop a new theory for the maximum correlation test over an increasing maximum lag. They allow for a very broad class of dependent and heterogeneous data, and verify that Shao s (2) dependent wild bootstrap is valid in this general setting. They compare the relative performance of multiple test statistics, using the dependent wild bootstrap to ensure correctly sized tests asymptotically. They find that the max-correlation test, Andrews and Ploberger s (996) sup-lm statistics and the Cramér-von Mises statistics used by Shao (2) control for size fairly well. Those tests have comparable power for small lag lengths. An advantage of the max-correlation test emerges when lag length is large relative to sample size. The max-correlation test is designed to be sensitive to dependence at large displacements, and relative to the above tests is best at detecting distant non-zero autocorrelations. This paper uses Shao s (2) Cramér-von Mises test, Andrews and Ploberger s (996) sup- LM test, and Hill and Motegi s (27a) max-correlation test, each assisted by the dependent wild bootstrap, in order to test whether stock returns are white noise. We analyze daily stock price indices from China, Japan, the U.K., and the U.S. The entire sample period spans January 23 through October 25. We perform a rolling window analysis in order to capture subsample non- 4 Lobato (2), Zhu and Li (25), and Zhang (26) analyze stock returns, while Horowitz, Lobato, Nankervis, and Savin (26) and Escanciano and Lobato (29) analyze foreign exchange rates. 4

5 stationarity and therefore time-varying market efficiency. 5 The degree of stock market efficiency may well be time-dependent given the empirical evidence from the previous literature on the adaptive market hypothesis. It is of particular interest to see how market efficiency is affected by financial turbulence like the subprime mortgage crisis around 28. We are not aware of any applications of the dependent wild bootstrap, except for Shao (2, 2) who analyzes temperature data and stock returns in a full sample framework. The present study is therefore the first use of the dependent wild bootstrap in a rolling window framework, in which we found and corrected a key shortcoming. In rolling window sub-samples, the block structure inscribes an artificial periodicity in the bootstrapped data, and therefore in computed p-values or confidence bands. A similar periodicity occurs in the block bootstrap for dependent data, which Politis and Romano (994) correct by randomizing block size. We take the same approach to eliminate dependent wild bootstrap periodicity. See Section 2 for key details, and see the supplemental material Hill and Motegi (27b) for complete details. We find that the degree of market efficiency varies across countries and sample periods. Chinese and Japanese markets exhibit a high degree of efficiency since we generally cannot reject the white noise hypothesis. The same goes for the U.K. and the U.S. during non-crisis periods. When the U.K. and U.S. face greater uncertainty, for example during the Iraq War and the subprime mortgage crisis, we tend to observe negative autocorrelations that are large enough to reject the white noise hypothesis. A negative correlation, in particular at low lags, signifies rapid changes in market trading, which is corroborated with high volatility during these times. The appearance of negative autocorrelations in short (e.g. daily, weakly) and long (e.g., 3 or 5-year) horizon returns has been documented extensively. Evidence for positive or negative correlations depends heavily on the market, return horizon (daily, weekly, etc.) and the presence of crisis periods. See, e.g., Fama and French (988), who argue that predictable price variation due to mean-reversion in returns accounts for the negative correlation at short and long horizons. The remainder of the paper is organized as follows. In Section 2 we explain the white noise 5 Full sample analysis is performed in the supplemental material Hill and Motegi (27b) for completeness. 5

6 tests and the dependent wild bootstrap. Section 3 describes our data, and we report the empirical results in Section 4. Concluding remarks are provided in Section 5. 2 Methodology 2. White Noise Tests Let P t be the stock price index at day t {, 2,..., n}, then r t = ln(p t /P t ) is the log return. We assume returns are stationary in order to ensure that the various tests used in this paper all have their intended asymptotic properties. 6 Define the mean µ = E[r t ], autocovariances γ(h) = E[(r t µ)(r t h µ)], and autocorrelations ρ(h) = γ(h)/γ() for h. We wish to test weak form efficiency: H : ρ(h) = for all h against H : ρ(h) for some h. Similarly, write the sample mean ˆµ n = /n n t= r t, autocovariance ˆγ n (h) = /n n t=h+ (r t ˆµ n )(r t h ˆµ n ), and autocorrelation ˆρ n (h) = ˆγ n (h)/ˆγ n () for h. In order to ensure a valid white noise test and therefore capture all serial correlations asymptotically, we formulate test statistics based on the serial correlation sequence {ˆρ n (h)} Ln h= with sample-size dependent lag length L n as n. We use tests by Andrews and Ploberger (996), Shao (2), and Hill and Motegi (27a) due to their comparable size and power (cf. Hill and Motegi, 27a). The first of the three tests is the sup-lm test proposed by Andrews and Ploberger (996). 6 We performed the Phillips and Perron (988) test on market levels and differences: we fail to reject the unit root null hypothesis at any conventional level for levels, and reject the unit root null at the % level for differences. We performed three tests in each case: without a constant, with a constant, and with a constant and linear time trend. The test statistics require a nonparametric variance estimator. We use a Bartlett kernel variance estimator with Newey and West s (994) automatic lag selection. P-values are computed using MacKinnon s (996) (one-sided) p-values. 6

7 The test statistic has the equivalent representation (see Nankervis and Savin, 2): AP n = sup λ Λ n( λ2 ) ( Ln ) 2 λ h ˆρ n (h) where L n = n, h= where Λ is a compact subset of (, ). The latter ensures a non-degenerate test that obtains, under suitable regularity conditions, an asymptotic power of one when there is serial correlation at some horizon. Andrews and Ploberger (996) use L n = n for computing the test statistic, but truncate a Gaussian series that arises in the limit distribution in order to simulate critical values. Nankervis and Savin (2, 22) generalize the sup-lm test to account for data dependence, and truncate the maximum lag both during computation (hence L n < n ), and for the sake of simulating critical values. The truncated value used, however, does not satisfy L n as n, hence their version of the test is not consistent (it does not achieve a power of one asymptotically when the null is false). To control for possible dependence under the null, and allow for a better approximation of the small sample distribution, we bootstrap the test with Shao s (2) dependent wild bootstrap, discussed below. The second test is based on the following Cramér-von Mises [CvM] statistic used by Shao (2), which is based on the sample spectral density: { π n 2 C n = n ˆγ n (h)ψ h (λ)} dλ where ψ h (λ) = (hπ) sin(hλ). h= By construction all n possible lags are used. The test statistic has a non-standard limit distribution under the null, and Shao (2) demonstrates that a version of the dependent wild bootstrap proposed in Shao (2) is valid under certain conditions on moments and dependence. Third, the bootstrap max-correlation test proposed by Hill and Motegi (27a) is based on the test statistic: ˆT n = n max h L n ˆρ n (h). 7

8 In this case L n /n is required such that ˆρ n (h) is Fisher consistent for ρ(h) for each h L n. If the sequence of serial correlations were asymptotically iid Gaussian under the null, then the limit law of a suitably normalized ˆT n under the null is a Type I extreme value, or Gumbel, distribution. That result extends to dependent data under the null (see Xiao and Wu, 24, for theory and references). The non-standard limit law can be bootstrapped, as in Xiao and Wu (24), although they do not prove their double blocks-of-blocks bootstrap is valid asymptotically. Hill and Motegi (27a) sidestep an extreme value theoretic argument, and directly prove Shao s (2) dependent wild bootstrap is valid without requiring the null limit law of the maxcorrelation. They also sidestep Gaussian approximation theory exploited in, amongst others, Chernozhukov, Chetverikov, and Kato (23), allowing for a very general setting and filtered data. See Hill and Motegi (27a) for a broad literature review and discussion. Hill and Motegi (27a) find that the above three tests have comparable size and power in finite samples. When lag length L n is large relative to sample size, the max-correlation test dominates the others in terms of size and power Dependent Wild Bootstrap Each of the sup-lm, CvM, and max-correlation test statistics has a non-standard limit distribution under the null. We therefore use Shao s (2) dependent wild bootstrap in order to perform each test. The dependent wild bootstrap for the max-correlation test is executed as follows (sup-lm and CvM tests follow similarly). Set a block size b n such that b n < n. Generate iid random numbers {ξ,..., ξ n/bn } with E[ξ i ] =, E[ξi 2 ] =, and E[ξi 4 ] <. Assume for simplicity that the number of blocks n/b n is an integer. Standard normal ξ i satisfies these properties, and is used in the empirical application below. Define an auxiliary variable ω t block-wise as follows: 7 Other well-known test statistics include a standardized periodogram statistic of Hong (996), which is effectively a standardized portmanteau statistic with a maximum lag L n = n. The test statistic has a standard normal limit under the null, but Hill and Motegi (27a) show that an asymptotic test yields large size distortions. They also show that a bootstrap version, which is arithmetically equivalent to a bootstrapped portmanteau test, is often too conservative relative to the tests used in this study. We therefore do not include Hong s (996) test here. 8

9 {ω,..., ω bn } = ξ, {ω bn+,..., ω 2bn } = ξ 2,..., {ω (n/bn )b n +,..., ω n } = ξ n/bn. Thus, ω t is iid across blocks, but perfectly dependent within blocks. Compute ˆρ n (dw) (h) = ˆγ n () n n t=h+ ω t {(r t ˆµ n )(r t h ˆµ n ) ˆγ n (h)} for h =,..., L n, () and a bootstrapped test statistic { ˆT (dw) n = n max h Ln ˆρ (dw) n (h). Repeat M times, resulting in (dw) ˆT n,i } M i=. The approximate p-value is ˆp (dw) n,m (/M) M (dw) i= I( ˆT n,i ˆT n ). Now let the number of bootstrap samples satisfy M = M n as n. If ˆp (dw) n,m < α, then we reject the null hypothesis of white noise at significance level α. Otherwise we do not reject the null. The dependent wild bootstrapped CvM test and max-correlation tests are asymptotically valid and consistent, for large classes of processes that may be dependent under the null: the asymptotic probability of rejection at level α is exactly α, and the asymptotic probability of rejection is one if the series is not white noise. 8 It is also straightforward to show that the bootstrapped sup-lm test is asymptotically valid when the maximum lag is fixed for a similarly large class of dependent processes. We are not aware of a result in the literature that proves validity when the maximum lag is L n. Andrews and Ploberger (996) and Nankervis and Savin (2) use a simulation method based on a fixed maximum lag L in order to approximate an asymptotic critical value. Simulations in Hill and Motegi (27a), however, demonstrate that the bootstrapped test works well with L n increasing with n. In some applications it is of interest to test for ρ(h) = for a specific h. In that case () can be used to construct a bootstrapped confidence band (i.e. critical values) under ρ(h) =. Compute {ˆρ (dw) n,i (h)} M i= and sort them as ˆρ (dw) (h) ˆρ(dw) (h) ˆρ(dw) (h). The 95% band n,[] for lag h is then [ˆρ (dw) n,[.25 M] (h), ˆρ(dw) n,[.975 M] (h)]. Below we compute those bands with h =. Evidently, the dependent wild bootstrap has not been studied in a rolling window environment. Using the same auxiliary variable ω t throughout one window of size b n, with b n = o(n), is n,[2] n,[m] 8 Besides the dependent wild bootstrap, Zhu and Li s (25) block-wise random weighting bootstrap can be applied to the CvM test statistic. Hill and Motegi (27a) find that both bootstrap procedures are comparable in terms of empirical size and power. 9

10 key toward allowing for general dependence under the null. Unfortunately, it is easily shown that in a rolling window setting, the result is a periodically fluctuating p-value or confidence band, irrespective of the true data generating process (e.g. periodic fluctuations arise even for iid data). Thus, the dependent wild bootstrap does not generate stationary bootstrap samples in rolling windows of stationary data: an artificial seasonality across windows is present. The reason for the periodicity is that we have similar blocking structures every b n windows. Consider two windows that are apart from each other by b n windows. A block in one window is a scalar multiplication of a block in the other window, resulting in similar bootstrapped autocorrelations from the two windows. See the supplemental material Hill and Motegi (27b) for complete details and additional simulations that demonstrate the problem and solution, discussed below. The problem is well known in the context of the block bootstrap (e.g. Politis and Romano, 994). As in Politis and Romano (994), cf. Lahiri (999), we solve the problem by randomizing the block size for each bootstrap sample and window. Randomness across windows ensures that different windows have different blocking structures, and are therefore not multiples of each other. This removes the artificial nonstationarity successfully. Randomness across bootstrap samples makes the confidence bands less volatile, which is desired in terms of visual inspection. In a full sample environment, Shao (2) finds that block sizes b n = c n with c {.5,, 2} perform comparably. Hence we pick a middle value c = and add a certain amount of randomness to that. We draw a uniform random variable c on [ ι, + ι] with ι =.5 for each bootstrap sample and window, and use b n = c n. It remains as an open question how to pick ι in practice. There is a trade-off that a small ι does not fully eliminate periodicity due to a lack of randomness, while a large ι results in more volatility in confidence bands. In this paper we simply choose ι =.5, and verify via controlled experiments and empirical analysis that our choice yields sufficiently non-periodic, smooth confidence bands.

11 3 Data We analyze log returns of the daily closing values from the Chinese Shanghai Composite Index (Shanghai), the Japanese Nikkei 225 index (Nikkei), the U.K. FTSE Index (FTSE), and the U.S. S&P 5 index (SP5), all in local currencies from January, 23 through October 29, The Shanghai index is selected as a representative of emerging markets, while the latter three are known as some of the most liquid, mature, and influential markets. The sample size differs across countries due to different trading days, holidays, and other market closures: 3 days for Shanghai, 349 days for Nikkei, 3243 for FTSE, and 323 for SP5. Market closures are simply ignored, hence the sequence of returns are treated as daily for each observation. Figure plots each stock price index and the log return. The subprime mortgage crisis in caused a dramatic decline in the stock prices. FTSE and SP5 experienced relatively fast recovery from the stock price plummet in 28, while Shanghai and Nikkei experienced a longer period of stagnation. Each return series shows clear volatility clustering, especially during the crisis. Besides the subprime mortgage crisis, there are a few episodes of financial turbulence that may affect the autocorrelation structure of each market. First, in 24-25, the Shanghai stock market experienced an apparent bubble (and collapse) the magnitude of which is only slightly smaller than the subprime mortgage crisis. Second, Nikkei experienced a large negative log return of -.2 on March 5, 2, which is two business days after the Great East Japan Earthquake. Nuclear power plants in Fukushima were destroyed by a resulting tsunami, and there emerged pessimistic sentiment among investors on electricity supply. Third, in 22 and 23, the FTSE stock market faced a period of great uncertainty due to an economic recession, soaring oil prices, and the Iraq War. FTSE fell for nine consecutive trading days in January 23 losing 2.4% of its value. On March 3, 23 the FTSE experiences a rebound log return of.59, highlighting an unstable market condition. Fourth, the SP5 index generated a log return of -.69 on August 9 The data were retrieved from Bloomberg.com.

12 8, 2 because Standard & Poor s downgraded the federal government credit rating from AAA to AA+ on August 5. Insert Figure here Table lists sample statistics of return series. Each series has a positive mean, but it is not significant at the 5% level according to a bootstrapped confidence band. Shanghai returns have the largest standard deviation, but Nikkei returns have the greatest range: it has the largest minimum and maximum in absolute value. Each series displays negative skewness and large kurtosis, all stylized traits. Due to the negative skewness and excess kurtosis, the p-values of the Kolmogorov-Smirnov and Anderson-Darling tests of normality are well below % for all series, strong evidence against normality. Insert Table here 4 Empirical Results We now present the main empirical findings. We set the window size to be n = 24 trading days (roughly a year), which is similar to the window size in Verheyden, De Moor, and Van den Bossche (25). 4. Analysis of Serial Correlation Figure 2 presents first order sample autocorrelations over rolling windows. For each window, the 95% confidence band based on the dependent wild bootstrap is constructed under the null hypothesis of white noise. Window size is n = 24, and we begin with a conventional block size b n = [ n] = 5. The number of bootstrap samples is, for each window. Insert Figure 2 here 2

13 A striking result from Figure 2 is that the confidence bands exhibit periodic fluctuations, with the appearance of veritable seasonal highs and lows. Moreover, the zigzag movement repeats itself in every b n = 5 windows. The confidence bands are particularly volatile for Nikkei and SP5 in 2, reflecting the tsunami disaster and S&P securities downgrade shock. Note, however, that periodic confidence bands appear in all series and periods universally. In the supplemental material Hill and Motegi (27b), we elaborate on this phenomenon with computational details and magnified plots for ease of viewing. As discussed in Section 2, the periodicity arises because there are similar blocking structures in every b n windows. Data blocks from two windows separated by b n windows are scalar multiples of each other, resulting in similar bootstrapped autocorrelations from the two windows. A similar issue arises in samples generated by block bootstrap (see, e.g., Lahiri, 999). The solution proposed by Politis and Romano (994) is to randomize block size, which we follow. For each of the, bootstrap samples, and each window, we independently draw c from a uniform distribution on [.5,.5], and use b n = c n. A comparison of Figures 2 and 3 highlights the substantial impact of block size randomization. In Figure 3, randomization has clearly removed the periodic fluctuations. We still observe volatile bands for Nikkei right after the tsunami disaster and for SP5 right after the U.S. securities downgrade shock. These are not surprising results since the standard deviation, autocorrelation, and kurtosis of the log returns all spike in those periods. In what follows, our discussions focus on results with the randomized block size. Insert Figure 3 here For Shanghai, the confidence bands are roughly [-.,.] and they contain the sample correlation in most windows. Interestingly, the subprime mortgage crisis around 28 did not have a If we only randomize b n for each window (using the same randomized b n across all, bootstrap samples), then there exists volatility in the resulting confidence bands that largely exceeds the volatility of the observed data. By randomizing b n for each bootstrap draw and each window, both artificial nonstationarity and excess volatility are eradicated. See also the supplemental material for related plots from controlled experiments. In particular, we present magnified plots of bootstrapped confidence bands from simulated data, with and without randomized block size. 3

14 substantial impact on the correlation structure of the Shanghai market, although the stock price itself responded with a massive drop and volatility burst around 28 (Figure ). In 24-25, the confidence bands are slightly wider, reflecting the Shanghai stock market collapse. The correlations sometimes go beyond. and outside the confidence bands. Hence, conditional on lag h =, there is a possibility that the Shanghai collapse had an adverse impact on the market efficiency. The first order correlation for Nikkei generally lies in [-.,.] and they are insignificant in most windows. The correlation goes beyond. in only one out of 29 windows, which is window #772 (March 24, 2 - March 5, 2). This is the first window that contains the tsunami shock. In terms of negative correlations, we have ˆρ n () <. in 2 windows (approximately 7.2% of all windows). The zero hypothesis is rejected for 25 out of the 2 windows. Hence Nikkei rarely exhibits a significantly negative autocorrelation given our sample period. The tendency for negative correlations is much more prominent in FTSE. The correlation for FTSE goes below -. for 954 windows out of 34. A rejection occurs in 478 out of the 954 windows. The correlation goes even below -.2 for 6 windows, and a rejection occurs in 66 windows. Similar patterns are present in SP5. The correlation for SP5 goes below -. for 9 windows out of 299. A rejection occurs in 675 out of the 9 windows. The correlation goes even below -.2 for 3 windows, and a rejection occurs in 8 windows. It is interesting that the more mature and liquid markets have a stronger tendency for having negative correlations. Negative serial correlations may be evidence of mean reversion, and therefore long run stationarity (see Fama and French, 988), which fits our maintained assumption that log-prices are first difference stationary. Another implication from Figure 3 is that the significantly negative correlations concentrate on the period of financial turmoil: Iraq-War regime in 23 for FTSE and SP5 and the subprime mortgage crisis in 28 for SP5. In view of the extant evidence for positive serial correlation in market returns, this suggests that negative correlations may be indicative of trading turmoil, 4

15 due ostensibly to the rapid evolution of information. Evidence for negative correlations during crisis periods is not new. See, for example, Campbell, Grossman, and Wang (993) who find a negative relationship between trading volume and serial correlation: high volume days are associated with lower or negative correlations, due, they argue, to the presence of noninformational traders. 4.2 White Noise Tests We now perform the max-correlation, sup-lm, and CvM tests for each window. In the supplemental material Hill and Motegi (27b), lag length is variously L n = max{5, [δ n/ ln(n)]} with δ {.,.2,.4,.5,.}, which implies L n {5, 8, 7, 2, 43}, for the max-correlation and sup-lm tests. The largest possible lag length L n = n = 239 is also considered for the sup-lm and CvM tests. Note that, under suitable regularity conditions, as long as L n then the sup-lm and CvM tests will have their intended limit properties under the null and alternative hypotheses, even if L n = o(n). It turns out that our empirical results are not very sensitive to lag length. Unless otherwise specified, this paper focuses on results with δ =. (i.e. L n = 5) for the max-correlation and sup-lm tests and L n = 239 for the CvM test in order to save space. See the supplemental material Hill and Motegi (27b) for complete results. See Figures 4, 5, and 6 for the results of max-correlation test, sup-lm test, and CvM tests, respectively. In each figure we plot rolling window test statistics, critical values, and p-values. Also see Table 2 for summary results covering all lag lengths. Our results suggest that the stock markets of Shanghai and Nikkei are likely weak form efficient throughout the whole sample period. Based on the max-correlation test with lag 5, for example, a rejection happens in only 3.7% of all windows for Shanghai and 3.3% for Nikkei. We observe similar results based on the sup-lm and CvM tests. Recall that Shanghai has positive correlations that are barely significant at the 5% level in 5

16 24-25 (Figure 3). These are only first order correlations, and statistical significance of each tests declines when lags L n 5 are considered jointly. Insert Figures 4, 5, and 6 here The FTSE and SP5 market have more periods of inefficiency than Shanghai and Nikkei. Based on the CvM test, for example, a rejection happens in as many as 4.5% of all windows for FTSE and 2.% for SP5. The sup-lm test with any lag selection yields similar results, while the max-correlation test produces lower rejection frequencies (less than %). This difference is reasonable since the non-zero correlation at lag should be better captured by the sup-lm and CvM tests than the max-correlation test (cf. simulation experiments in Hill and Motegi, 27a,b). As seen in Figures 5 and 6, rejections occur continuously during the Iraq War and the subprime mortgage crisis. The former has a longer impact than the latter for FTSE, while the latter has a longer impact for SP5. This result is consistent with Figure 3. It is also consistent with the notion that high volatility is associated with lower or negative correlations, since these periods are marked by increased volatility. Insert Table 2 here In summary, the degree of stock market efficiency differs noticeably across countries and sample periods, as asserted by the adaptive market hypothesis. The Shanghai and Nikkei stock markets have a high degree of efficiency so that we often accept the white noise hypothesis of stock returns. The same goes for FTSE and SP5 during non-crises periods. When they are in unstable periods like the Iraq War and the subprime mortgage crisis, we tend to observe negative autocorrelations that are large enough to reject the white noise hypothesis. 6

17 5 Conclusion Much of the previous literature on testing for weak form efficiency of stock markets imposes an iid or mds assumption under the null hypothesis. The iid property rules out any form of conditional heteroskedasticity, and the mds property rules out higher level forms of dependence. It is thus of interest to perform white noise tests with little more than serial uncorrelatedness under the null hypothesis. A rejection of the white noise hypothesis might serve as a helpful signal of an arbitrage opportunity for investors, since it indicates the presence of non-zero autocorrelation at some lags. Using theory developed in Hill and Motegi (27a) that extends Shao s (2) dependent wild bootstrap to max-correlation and sup-lm tests, as well as Shao s (2) proposed Cramér-von Mises test, we test for weak form efficiency of Chinese, Japanese, U.K., and U.S. stock markets. We perform rolling window analysis in order to capture time-varying market efficiency. The present study is apparently the first use of the dependent wild bootstrap in a rolling window environment. The block structure inscribes an artificial periodicity in computed p-values or confidence bands over rolling windows, which we have removed by randomizing block sizes. Shanghai and Nikkei exhibit a high degree of efficiency such that we generally cannot reject the white noise hypothesis. The same goes for FTSE and SP5 during non-crisis periods. When FTSE and SP5 face greater uncertainty, for example during the Iraq War and the subprime mortgage crisis, we tend to observe negative autocorrelations that are large enough to reject the white noise hypothesis. A negative correlation, in particular at low lags, signifies rapid changes in market trading, which is corroborated with high volatility due to noninformational traders. References Anagnostidis, P., C. Varsakelis, and C. J. Emmanouilides (26): Has the 28 Financial Crisis Affected Stock Market Efficiency? The Case of Eurozone, Physica A, 447, Andrews, D. W. K., and W. Ploberger (996): Testing for Serial Correlation against an ARMA(,) Process, Journal of the American Statistical Association, 9,

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20 7, Lim, K.-P., W. Luo, and J. H. Kim (23): Are US Stock Index Returns Predictable? Evidence from Automatic Autocorrelation-Based Tests, Applied Economics, 45, Lo, A. W. (24): The Adaptive Markets Hypothesis: Market Efficiency from an Evolutionary Perspective, Journal of Portfolio Management, 3, (25): Reconciling Efficient Markets with Behavioral Finance: The Adaptive Markets Hypothesis, Journal of Investment Consulting, 7, Lo, A. W., and A. C. MacKinlay (988): Stock Market Prices Do Not Follow Random Walks: Evidence from a Simple Specification Test, Review of Financial Studies,, Lobato, I. N. (2): Testing that a Dependent Process Is Uncorrelated, Journal of the American Statistical Association, 96, Lobato, I. N., J. C. Nankervis, and N. E. Savin (22): Testing for Zero Autocorrelation in the Presence of Statistical Dependence, Econometric Theory, 8, MacKinnon, J. G. (996): Numerical Distribution Functions for Unit Root and Cointegration Tests, Journal of Applied Econometrics,, Nankervis, J. C., and N. E. Savin (2): Testing for Serial Correlation: Generalized Andrews-Ploberger Tests, Journal of Business and Economic Statistics, 28, (22): Testing for Uncorrelated Errors in ARMA Models: Non-Standard Andrews- Ploberger Tests, Econometrics Journal, 5, Newey, W. K., and K. D. West (994): Automatic Lag Selection in Covariance Matrix Estimation, Review of Economic Studies, 6, Phillips, P. C. B., and P. Perron (988): Testing for a Unit Root in Time Series Regression, Biometrika, 75, Politis, D. N., and J. P. Romano (994): The Stationary Bootstrap, Journal of the American Statistical Association, 89, Romano, J. P., and L. A. Thombs (996): Inference for Autocorrelations under Weak Assumptions, Journal of the American Statistical Association, 9, Shao, X. (2): The Dependent Wild Bootstrap, Journal of the American Statistical Association, 5, (2): A Bootstrap-Assisted Spectral Test of White Noise under Unknown Dependence, Journal of Econometrics, 62, Urquhart, A., and F. McGroarty (26): Are Stock Markets Really Efficient? Evidence of the Adaptive Market Hypothesis, International Review of Financial Analysis, 47, Verheyden, T., L. De Moor, and F. Van den Bossche (25): Towards a New Framework on Efficient Markets, Research in International Business and Finance, 34,

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22 Table : Sample Statistics of Log Returns of Stock Price Indices (//23 - /29/25) # Obs. Mean 95% Band Med. Stdev. Min. Max. Skew. Kurt. p-ks p-ad Shanghai [ 7.4, 7.2] Nikkei [ 5., 5.] FTSE [ 2.7, 2.6] SP [ 3.8, 3.7] % Band is a bootstrapped 95% confidence band for the sample mean. It is constructed under the null hypothesis of zero-mean white noise, using the dependent wild bootstrap with block size bn = n. The number of bootstrap samples is M =,. p-ks signifies a p-value of the Kolmogorov-Smirnov test, while p-ad signifies a p-value of the Anderson-Darling test. 22

23 Table 2: Rejection Ratio of White Noise Tests over Rolling Windows Max-Correlation Andrews-Ploberger CvM δ =. δ =.2 δ =.4 δ =.5 δ =. δ =. δ =.2 δ =.4 δ =.5 δ =. Ln = 5 Ln = 8 Ln = 7 Ln = 2 Ln = 43 Ln = 5 Ln = 8 Ln = 7 Ln = 2 Ln = 43 Ln = 239 Ln = 239 Shanghai Nikkei FTSE SP The ratio of rolling windows where the null hypothesis of white noise is rejected at the 5% level. Test statistics are Hill and Motegi s (27a) max-correlation statistic, Andrews and Ploberger s (996) sup-lm statistic, and the Cramér-von Mises statistic. We use Shao s (2) dependent wild bootstrap with block size bn = [c n]. We draw c U(.5,.5) independently across M = 5, bootstrap samples and rolling windows. The maximum lag lengths for the max-correlation and sup-lm tests are Ln = max{5, [δ n/ ln(n)]} with δ {.,.2,.4,.5,.} and window size n = 24 trading days. We also cover Ln = n for the sup-lm and CvM tests. 23

24 Figure : Daily Stock Prices and Log Returns (//23 - /29/25) Shanghai. 3 2 Subprime Jan5 Jan Jan5 Shanghai (Level) Subprime Shanghai Jan5 Jan Jan5 Shanghai (Return) Shanghai Subprime Tsunami Jan5 Jan Jan5 Nikkei (Level) Subprime Subprime Tsunami Shanghai -.2 Jan5 Jan Jan5.2. Nikkei (Return) Shanghai Iraq Jan5 Jan Jan5 FTSE (Level) Iraq Shanghai Subprime Jan5 Jan Jan5 FTSE (Return) Iraq Shanghai. Iraq AA+ Subprime Jan5 Jan Jan5 SP5 (Level) Subprime AA+ Shanghai Jan5 Jan Jan5 SP5 (Return) Left panels depict stock price indices in local currencies (standardized at on //23). Note that the maximum value of the vertical axis is 5 for Shanghai and 3 for the other series. Right panels depict log returns. Subprime signifies the subprime mortgage crisis around 28; Shanghai signifies the bubble and burst in Shanghai; Tsunami signifies the tsunami disaster caused by the Great East Japan Earthquake in March 2; Iraq signifies Iraq War; AA+ signifies the downgrade of the U.S. federal government credit rating from AAA to AA+. 24

25 Figure 2: Rolling Window Autocorrelations at Lag (Fixed Block Size) Jan5 Jan Shanghai -.5 Jan5 Jan Nikkei Jan5 Jan FTSE -.5 Jan5 Jan SP5 The solid black line depicts sample autocorrelations at lag. The dotted red line depicts the 95% confidence band constructed with Shao s (2) dependent wild bootstrap under the null hypothesis of white noise. Window size is n = 24 trading days, and block size is b n = [ n] = 5. The number of bootstrap samples is, for each window. Each point on the horizontal axis represents the initial date of each window. 25

26 Figure 3: Rolling Window Autocorrelations at Lag (Randomized Block Size).5.5 Shanghai -.5 Jan5 Jan Shanghai -.5 Tsunami Jan5 Jan Nikkei Iraq Jan5 Subprime Jan FTSE -.5 Iraq Subprime Jan5 Jan SP5 AA+ The solid black line depicts sample autocorrelations at lag. The dotted red line depicts the 95% confidence band constructed with Shao s (2) dependent wild bootstrap under the null hypothesis of white noise. Window size is n = 24 trading days. Block size is b n = [c n]. We draw c U(.5,.5) independently across M =, bootstrap samples and rolling windows. Each point on the horizontal axis represents the initial date of each window. Shanghai signifies the bubble and burst in Shanghai; Tsunami signifies the tsunami disaster caused by the Great East Japan Earthquake in March 2; Iraq signifies Iraq War; Subprime signifies the subprime mortgage crisis; AA+ signifies the downgrade of the U.S. federal government credit rating from AAA to AA+. 26

27 Figure 4: Max-Correlation Test in Rolling Window (Ln = 5) Jan5 Jan Bands / Shanghai Jan5 Jan Bands / Nikkei Jan5 Jan Bands / FTSE Jan5 Jan Bands / SP Jan5 Jan P-Values / Shanghai Jan5 Jan P-Values / Nikkei Jan5 Jan P-Values / FTSE Jan5 Jan P-Values / SP5 Hill and Motegi s (27a) max-correlation test with lag length Ln = 5 and window size n = 24 trading days. We use the dependent wild bootstrap with 5, replications for each window. Block size is bn = [c n], where c U(.5,.5) is drawn independently across bootstrap samples and windows. Each point on the horizontal axis represents the initial date of each window. In the upper panels the solid black line is the test statistic, and the dotted red line is the 5% critical value. In the lower panels p-values are plotted with p =.5 denoted. 27

28 Figure 5: Andrews-Ploberger Sup-LM Test in Rolling Window (Ln = 5) Jan5 Jan Bands / Shanghai Jan5 Jan Bands / Nikkei Jan5 Jan Bands / FTSE Jan5 Jan Bands / SP Jan5 Jan P-Values / Shanghai Jan5 Jan P-Values / Nikkei Jan5 Jan P-Values / FTSE Jan5 Jan P-Values / SP5 Andrews and Ploberger s (996) sup-lm test with lag length Ln = 5 and window size n = 24 trading days. We use the dependent wild bootstrap with 5, replications for each window. Block size is bn = [c n], where c U(.5,.5) is drawn independently across bootstrap samples and windows. Each point on the horizontal axis represents the initial date of each window. In the upper panels the solid black line is the test statistic, and the dotted red line is the 5% critical value. In the lower panels p-values are plotted with p =.5 denoted. 28