Testing for Weak Form E ciency of Stock Marketsmotegi/max_corr_EMPIRICS_v48.pdf · Testing for Weak...
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Testing for Weak Form Efficiency of Stock Markets * Jonathan B. Hill † – University of North Carolina Kaiji Motegi ‡ – Kobe University This draft: March 16, 2017 Abstract We perform a variety of white noise tests on daily stock market log returns over rolling windows of subsamples. Tests of weak form efficiency in stock returns are performed pre- dominantly under the null hypothesis of independence or a martingale difference property. These properties rule out higher forms of dependence that may exist in stock returns that are otherwise serially uncorrelated. It is therefore of interest to test whether returns are white noise, allowing for a wider range of conditionally heteroskedastic time series, but also for non-martingale difference white noise. Assisted by the dependent wild bootstrap (Shao, 2010, 2011), we use sup-Lagrange Multiplier, Cram´ er-von Mises, and max-correlation statis- tics in order to test the white noise hypothesis. Evidently the dependent wild bootstrap has only been used in a full data sample, hence a key shortcoming has gone unnoticed: in rolling window sub-samples, the block structure unintentionally inscribes an artificial periodicity in computed p-values or confidence bands. We eliminate periodicity by randomizing the block size across bootstrap samples and windows. We find that the degree of market efficiency varies across countries and sample periods. In the case of Chinese and Japanese markets we cannot reject the white noise hypothesis, suggesting a high degree of efficiency. The same goes for markets in the U.K. and the U.S., provided trading occurs during non-crisis periods. When U.K. and U.S. markets face greater uncertainty, we tend to observe negative autocorrelations that are large enough to reject the white noise hypothesis. JEL classifications: C12, C58, G14. Keywords: dependent wild bootstrap, randomized block size, serial correlation, weak form efficiency, white noise test, maximum correlation test. * We thank Yoshihiko Nishiyama and participants at the Econometric Seminar of the Institute of Economic Re- search, Kyoto University, 2016 Summer Workshop of Economic Theory, and 2016 Asian Meeting of the Econometric Society. † Department of Economics, University of North Carolina, Chapel Hill. E-mail: [email protected] ‡ Graduate School of Economics, Kobe University. E-mail: [email protected] 1
Transcript of Testing for Weak Form E ciency of Stock Marketsmotegi/max_corr_EMPIRICS_v48.pdf · Testing for Weak...
Testing for Weak Form Efficiency of Stock Markets∗
Jonathan B. Hill†– University of North CarolinaKaiji Motegi‡– Kobe University
This draft: March 16, 2017
Abstract
We perform a variety of white noise tests on daily stock market log returns over rollingwindows of subsamples. Tests of weak form efficiency in stock returns are performed pre-dominantly under the null hypothesis of independence or a martingale difference property.These properties rule out higher forms of dependence that may exist in stock returns thatare otherwise serially uncorrelated. It is therefore of interest to test whether returns arewhite noise, allowing for a wider range of conditionally heteroskedastic time series, but alsofor non-martingale difference white noise. Assisted by the dependent wild bootstrap (Shao,2010, 2011), we use sup-Lagrange Multiplier, Cramer-von Mises, and max-correlation statis-tics in order to test the white noise hypothesis. Evidently the dependent wild bootstrap hasonly been used in a full data sample, hence a key shortcoming has gone unnoticed: in rollingwindow sub-samples, the block structure unintentionally inscribes an artificial periodicity incomputed p-values or confidence bands. We eliminate periodicity by randomizing the blocksize across bootstrap samples and windows. We find that the degree of market efficiencyvaries across countries and sample periods. In the case of Chinese and Japanese marketswe cannot reject the white noise hypothesis, suggesting a high degree of efficiency. Thesame goes for markets in the U.K. and the U.S., provided trading occurs during non-crisisperiods. When U.K. and U.S. markets face greater uncertainty, we tend to observe negativeautocorrelations that are large enough to reject the white noise hypothesis.
JEL classifications: C12, C58, G14.Keywords: dependent wild bootstrap, randomized block size, serial correlation, weak formefficiency, white noise test, maximum correlation test.
∗We thank Yoshihiko Nishiyama and participants at the Econometric Seminar of the Institute of Economic Re-search, Kyoto University, 2016 Summer Workshop of Economic Theory, and 2016 Asian Meeting of the EconometricSociety.†Department of Economics, University of North Carolina, Chapel Hill. E-mail: [email protected]‡Graduate School of Economics, Kobe University. E-mail: [email protected]
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1 Introduction
We perform a variety of tests of weak 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, 1970). Empirical results have
been mixed, with substantial debate between advocates of the efficient market hypothesis (EMH)
and proponents of behavioral finance.1 More recently, the adaptive market hypothesis (AMH)
proposed by Lo (2004, 2005) attempts to reconcile the two opposing schools, arguing that the
degree of stock market efficiency varies over time.
In line with these trends, recent applications perform either non-overlapping subsample anal-
ysis or rolling window analysis in order to investigate the dynamic evolution of stock market
efficiency. See Cheong, Nor, and Isa (2007), Hoque, Kim, and Pyun (2007), Lim (2008), and
Urquhart and Hudson (2013) for use of non-overlapping subsamples, and for rolling windows see
Kim and Shamsuddin (2008), Lim, Brooks, and Kim (2008), Kim, Shamsuddin, and Lim (2011),
Lim, Luo, and Kim (2013), Verheyden, De Moor, and Van den Bossche (2015), Anagnostidis,
Varsakelis, and Emmanouilides (2016), and Urquhart and McGroarty (2016). An advantage of
rolling windows is that it does not require a subjective choice of the first and last dates of major
events, including financial crises.
A common perception in the recent literature is that a financial crisis augments investor panic
and hence lowers market efficiency. Cheong, Nor, and Isa (2007), Lim (2008), Lim, Brooks, and
Kim (2008), and Anagnostidis, Varsakelis, and Emmanouilides (2016) find that market efficiency is
indeed adversely affected by financial crises. By comparison, Hoque, Kim, and Pyun (2007), Kim
and Shamsuddin (2008), and Verheyden, De Moor, and Van den Bossche (2015) find relatively
mixed results.
In terms of methodology, many early contributions use Lo and MacKinlay’s (1988) variance
ratio test and its variants developed by Chow and Denning (1993), Choi (1999), Chen and Deo
(2006), Kim (2009), and Charles, Darne, and Kim (2011) among others.2 Other statistical tests
include Hinich’s (1996) bicorrelation test, Wright’s (2000) non-parametric sign/rank test, Escan-
ciano and Valasco’s (2006) generalized spectral test, and Escanciano and Lobato’s (2009) 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 as-
sumptions 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
1 See Yen and Lee (2008) and Lim and Brooks (2009) for an extensive survey of stock market efficiency.2 See Charles and Darne (2009) for a survey of variance ratio tests.
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theory (e.g. Billingsley, 1961). Chen and Deo (2006), 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, 1993).
Moreover, their test is not a true white-noise test since it does not test (asymptotically) that all
serial correlations are zero.
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.
Formal white noise tests with little more than serial uncorrelatedness under the null have not
been available until recently. See Hill and Motegi (2016a) 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 (e.g. Romano and Thombs, 1996,
Lobato, 2001, Lobato, Nankervis, and Savin, 2002, Horowitz, Lobato, Nankervis, and Savin, 2006,
Escanciano and Lobato, 2009, Delgado and Velasco, 2011, Guay, Guerre, and Lazarova, 2013, Zhu
and Li, 2015, Zhang, 2016).3 Hong (1996, 2001) standardizes a portmanteau statistic, allowing
for an increasing number of serial correlations and standard asymptotics. See also Hong and Lee
(2003).
Spectral tests operate on an increasing number of serial correlations as the sample size in-
creases, with variations due to Durlauf (1991), Hong (1996), Deo (2000), and Delgado, Hidalgo,
and Velasco (2005). Durlauf (1991) and Deo (2000) apply their tests to stock returns. Cramer-
von Mises and Kolmogorov-Smirnov variants can be found in Shao (2011) 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 (1996), cf. Nankervis
and Savin (2010).
Hong (1999) generalizes the definition of the spectral density to allow for a test of independence
(see also Hong and Lee, 2003). This is a powerful tool, but often in financial engineering and
portfolio analysis serial independence is too strong a benchmark for studying market performance
and determining efficiency. 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.
Hill and Motegi (2016a) develop a new theory for the maximum correlation test over an
3 Lobato (2001), Zhu and Li (2015), and Zhang (2016) analyze stock returns, while Horowitz, Lobato, Nankervis,and Savin (2006) and Escanciano and Lobato (2009) analyze foreign exchange rates.
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increasing maximum lag. They allow for a very broad class of dependent and heterogeneous data,
and verify that Shao’s (2011) 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 Andrews and Ploberger’s (1996) sup-
LM statistics and the Cramer-von Mises statistics used by Shao (2011) control for size fairly well.
The max-correlation test, however, results in the sharpest empirical size according to their broad
study. Overall, the max-correlation test yields either the highest power, or competitive power,
and does not experience as sharp a decline in power when testing at very large lags. Indeed, the
max-correlation test is specifically 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 (2011) Cramer-von Mises test, Andrews and Ploberger’s (1996) sup-
LM test, and Hill and Motegi’s (2016a) max-correlation test, assisted by the dependent wild
bootstrap, in order to test whether stock returns are white noise. This analysis can be interpreted
as testing whether stock prices follow what Campbell, Lo, and MacKinlay (1997) call Random
Walk 3 (i.e. random walk with uncorrelated increment) in a strict sense.
We analyze daily stock price indices from China, Japan, the U.K., and the U.S. The entire
sample period spans January 2003 through October 2015. As a preliminary analysis, we implement
white noise tests for the full sample period. We then perform a rolling window analysis in order
to capture subsample non-stationarity and therefore time-varying market efficiency. 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
2008.
We are not aware of any applications of the dependent wild bootstrap, except for Shao (2010,
2011) 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 (1994) 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 (2016b) 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 these
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markets 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. 1, 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 (1988), 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
tests that we use, and Section 3 describes our data. We report the empirical results for full sample
and rolling sample analyses in Section 4. Concluding remarks are provided in Section 5.
2 Methodology
Let Pt be the stock market index at day t ∈ {1, 2, . . . , n}, and rt = ln(Pt/Pt−1) is the log-return.
We assume returns are stationary in order to ensure that the various tests used this paper all have
their intended asymptotic properties.4 Define the mean return µ = E[rt], and autocovariances
γ(h) = E[(rt − µ)(rt−h − µ)] and autocorrelations ρ(h) = γ(h)/γ(0) for h ≥ 0. We wish to test
weak form efficiency:
H0 : ρ(h) = 0 for all h ≥ 1 against H1 : ρ(h) 6= 0 for some h ≥ 1.
Similarly, write the sample mean return µn = 1/n∑n
t=1 rt, autocovariance γn(h) = 1/n∑n
t=h+1(rt−µn)(rt−h − µn) and autocorrelation ρn(h) = γn(h)/γn(0) for h ≥ 0. 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 {ρn(h)}Lnh=1 with sample-size dependent lag
length Ln → ∞ as n → ∞. We use tests by Andrews and Ploberger (1996), Shao (2011), and
Hill and Motegi (2016a) due to their comparable size and power (cf. Hill and Motegi, 2016a).
The first of the three tests is the sup-LM test proposed by Andrews and Ploberger (1996).
4 We performed the Phillips and Perron (1988) test on market levels and differences: we fail to reject theunit root null hypothesis at any conventional level for levels, and reject the unit root null at the 1% level fordifferences. We performed three tests in each case: without a constant, with a constant and with a constant andlinear time trend. The test statistics require a nonparametric variance estimator. We use a Bartlett kernel varianceestimator with Newey and West’s (1994) automatic lag selection. P-values are computed using MacKinnon’s (1996)(one-sided) p-values.
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The test statistic has the equivalent representation (see Nankervis and Savin, 2010):
APn = supλ∈Λ
n(1− λ2)
(Ln∑h=1
λh−1ρn(h)
)2 where Ln = n− 1,
where Λ is a compact subset of (−1, 1). 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 (1996) use Ln = n− 1 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 (2010, 2012) generalize the sup-LM test to account for data dependence, and truncate
the maximum lag both during computation (hence Ln < n − 1), and for the sake of simulating
critical values. The truncated value used, however, does not satisfy Ln → ∞ 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 (2011)
dependent wild bootstrap, discussed below.
The second test is based on the following Cramer-von Mises [CvM] statistic used by Shao
(2011):
Cn = n
∫ π
0
{n−1∑h=1
γn(h)ψh(λ)
}2
dλ where ψh(λ) = (hπ)−1 sin(hλ).
By construction all n − 1 possible lags are used. The test statistic has a non-standard limit
distribution under the null, and Shao (2011) demonstrates that a version of the dependent wild
bootstrap proposed in Shao (2010) is valid under certain conditions on moments and dependence.
Third, the bootstrap max-correlation test proposed by Hill and Motegi (2016a) is based on
the test statistic:
Tn =√n max
1≤h≤Ln|ρn(h)|.
In this case Ln/n→ 0 is required such that ρn(h) is Fisher consistent for ρ(h) for each 1 ≤ h ≤ Ln.
If the sequence of serial correlations were asymptotically iid Gaussian under the null then the limit
law of a suitably normalized Tn under the null is a Type I extreme value distribution, or Gumbel.
That result extends to dependent data under the null (see Xiao and Wu, 2014, for theory and
references). The non-standard limit law can be bootstrapped, as in Xiao and Wu (2014), although
they do not prove their double blocks-of-blocks bootstrap is valid asymptotically. Hill and Motegi
(2016a) sidestep an extreme value theoretic argument, and directly prove Shao’s (2011) dependent
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wild bootstrap is valid without requiring the null limit law of the max-correlation.
Hill and Motegi (2016a) find that the above three tests have comparable size and power in
finite samples, although the bootstrap max-correlation test yields the sharpest size in general,
and in many cases obtains the highest power. Further, the max-correlation test is particularly
sensitive to non-zero correlations at distant lags.5
Each of the above three test statistics has a non-standard limit distribution under the null.
We therefore use Shao’s (2011) dependent wild bootstrap in order to perform each test. Shao
(2011) proves that the CvM test with the dependent wild bootstrap is asymptotically valid for
processes that may exhibit a wide range of dependence under the null. Hill and Motegi (2016a)
show that the dependent wild bootstrap is valid for the max-correlation test under an even larger
class of dependent processes. The types of dependence allowed include various non-linear ARMA-
GARCH processes with geometric (fast) or hyperbolic (slow) memory decay, and cover processes
with or without a mixing property. Shao’s (2011) environment is only known to hold for geometric
memory decay (see Shao, 2011, Hill and Motegi, 2016a).
The dependent wild bootstrap for the max-correlation test is executed as follows (sup-LM and
CvM tests follow similarly). Set a block size bn such that 1 ≤ bn < n. Generate iid random
numbers {ξ1, . . . , ξn/bn} with E[ξi] = 0, E[ξ2i ] = 1, and E[ξ4
i ] < ∞. Assume for simplicity that
the number of blocks n/bn is an integer. Standard normal ξi satisfies these properties, and is
used in the empirical application below. Define an auxiliary variable ωt block-wise as follows:
{ω1, ..., ωbn} = ξ1, {ωbn+1, ..., ω2bn} = ξ2, ..., {ω(n/bn−1)bn+1, ..., ωn} = ξn/bn . Thus, ωt is iid across
blocks, but perfectly dependent within blocks. Compute:
ρ(dw)n (h) =
1
γn(0)
1
n
n∑t=h+1
ωt {(rt − µn)(rt−h − µn)− γn(h)} for h = 1, . . . ,Ln, (1)
and a bootstrapped test statistic T (dw)n =
√nmax1≤h≤Ln |ρ
(dw)n (h)|. Repeat M times, resulting in
{T (dw)n,i }Mi=1. The approximate p-value is p
(dw)n,M ≡ (1/M)
∑Mi=1 I(T (dw)
n,i ≥ Tn). Now let the number
of bootstrap samples satisfy M = Mn → ∞ as n → ∞. If p(dw)n,M < α, then we reject the null
hypothesis of white noise at significance level α. Otherwise we do not reject the null. In our
empirical study we use Mn = 5, 000.
The dependent wild bootstrapped CvM test and max-correlation tests are asymptotically valid
5 Other well-known test statistics for testing the white noise hypothesis include a standardized periodogramstatistic of Hong (1996), which is effectively a standardized portmanteau statistic with a maximum lag Ln = n−1.The test statistic has a standard normal limit under the null, but Hill and Motegi (2016a) show that an asymptotictest yields large size distortions. They also show that a bootstrap version, which is arithmetically equivalent to abootstrapped portmanteau test, is often too conservative relative to the tests used in this study. We therefore donot include this test here.
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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.6 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 Ln → ∞. Andrews and Ploberger (1996) and Nankervis and Savin (2010) use a simulation
method based on a fixed maximum lag L in order to approximate an asymptotic critical value.
Simulations in Hill and Motegi (2016a), however, demonstrate that the bootstrapped test works
well with Ln increasing with n.
In some applications it is of interest to test for ρ(h) = 0 for a specific h. In that case (1) can
be used to construct a bootstrapped confidence band under ρ(h) = 0. Compute {ρ(dw)n,i (h)}Mi=1
and sort them as ρ(dw)n,[1](h) ≤ ρ
(dw)n,[2](h) ≤ · · · ≤ ρ
(dw)n,[M ](h). The 95% band for lag h is then
[ρ(dw)n,[.025∗M ](h), ρ
(dw)n,[.975∗M ](h)]. Below we compute those bands with h = 1 in the full sample and
rolling window frameworks.
Evidently the dependent wild bootstrap has not been studied in a rolling window environment.
Using the same auxiliary variable ωt throughout one window of size bn → ∞, with bn = o(n), is
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 bn windows.
Consider two windows that are apart from each other by bn windows. A block in one window
is a scalar multiplication of a block in the other window, resulting in similar bootstrapped auto-
correlations from the two windows. See the supplemental material Hill and Motegi (2016b) for
complete details and additional simulations that demonstrate the problem and solution, discussed
below.
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, conditional
on the sample. Randomness across bootstrap samples makes the confidence bands less volatile,
which is desired in terms of visual inspection. In a full sample we find using values considered in
Shao (2011), bn = c√n with c ∈ {1/2, 1, 2}, to be comparable, hence we simply use c = 1 for all
6 Besides the dependent wild bootstrap, Zhu and Li’s (2015) block-wise random weighting bootstrap can beapplied to the CvM test statistic. Hill and Motegi (2016a) find that both bootstrap procedures are comparable interms of empirical size and power.
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bootstrap samples and windows. In rolling windows we therefore draw a uniform random variable
c on [.5, 1.5] for each bootstrap sample and window, and use bn = c√n.
Randomizing a bootstrap block size to ensure stationary bootstrap draws (conditional on the
sample) is not new. Politis and Romano (1994) use randomized block sizes in their stationary
bootstrap subsampling method, a block bootstrap procedure that ensures a stationary bootstrap
draw. See Lahiri (1999) for enhanced theoretical properties. In a similar sense, randomizing block
size for the wild bootstrap removes artificially introduced non-stationarity in the bootstrap draws
over rolling windows, conditional on the sample.
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 100 Index (FTSE), and the
U.S. S&P 500 index (SP500), all in local currencies from January 1, 2003 through October 29,
2015.7 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: 3110
days for Shanghai, 3149 days for Nikkei, 3243 for FTSE, and 3230 for SP500. Market closures
are simply ignored, hence the sequence of returns are treated as daily for each observation.
Figure 1 plots each stock price index and the log return. The subprime mortgage crisis in
2007-2008 caused a dramatic decline in the stock prices. Comparing the first and last trading
days in 2008, the growth rate of stock price level is -65.5% for Shanghai, -39.7% for Nikkei, -30.9%
for FTSE, and -37.6% for SP500. The latter two countries experienced relatively fast recovery
from the stock price plummet in 2008, 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 2014-2015, the Shanghai stock
market experienced an apparent “bubble” (and collapse) the magnitude of which is only slightly
smaller than the subprime mortgage crisis. In the first half of 2014, the Shanghai index fluctuated
steadily around 2050 points. It then soared for a year to the peak of 5166.35 on June 12, 2015,
and then fell down sharply to 2927.29 on August 26, 2015.
Second, Nikkei experienced a large negative log return of -0.112 on March 15, 2011, 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
7The data were retrieved from Bloomberg.com.
9
on electricity supply.
Third, in 2002 and 2003, 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 2003 losing 12.4% of its value. On March 13, 2003 the FTSE experiences a
rebound gain of .059, highlighting an unstable market condition.
Fourth, the SP500 index generated a log return of -0.069 on August 8, 2011 because Standard
& Poor’s downgraded the SP500 federal government credit rating from AAA to AA+ on August
5.
Insert Figure 1 here
Table 1 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 has a large
kurtosis, all stylized traits. The kurtoses are 6.831 for Shanghai, 10.68 for Nikkei, 11.14 for FTSE,
and 14.02 for SP500. Shanghai has much smaller kurtosis than the other countries, but it is still
much larger than the kurtosis of the normal distribution. Due to the negative skewness and excess
kurtosis, the p-values of the Kolmogorov-Smirnov and Anderson-Darling tests of normality are
well below 1% for all countries, strong evidence against normality.
Insert Table 1 here
4 Empirical Results
We now present the main empirical findings.
4.1 Full Sample Analysis
Figure 2 shows sample autocorrelations of the daily return series from January 1, 2003 through
October 29, 2015. Lags h = 1, . . . , 25 trading days are considered. The 95% confidence bands
are constructed with Shao’s (2011) dependent wild bootstrap under the null hypothesis of white
noise. Hence a sample autocorrelation lying outside the confidence band can be thought of as an
evidence against the white noise hypothesis, conditional on each lag h. The number of bootstrap
samples is 5,000, and in all cases we draw from the standard normal distribution.
Insert Figure 2 here
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While most Shanghai sample correlations lie inside the 95% bands, there are some marginal
cases. Correlations at lags 3 and 4, for example, are respectively 0.036 and 0.064, which are on or
near the upper bounds. Further, Nikkei, FTSE, and SP500 have significantly negative correlations
at lag 1. The sample correlation with accompanied confidence band is -0.036 with [-0.031, 0.031]
for Nikkei; -0.054 with [-0.042, 0.042] for FTSE; -0.102 with [-0.079, 0.075] for SP500. Thus,
returns on each of these indices are unlikely to be white noise. Negative correlations at lag 1
suggest that traders switch between short and long positions actively on a daily basis, and as a
result a sharp price hike tends to be followed by a sharp price drop and vice versa.
In general, we observe insignificant autocorrelations at large lags (e.g. h > 15) for each series,
which suggests that there does not exist a noticeable seasonal effect.8
Table 2 compiles bootstrapped p-values from the max-correlation, sup-LM, and CvM white
noise tests. The max-correlation test requires the maximum lag length Ln = o(n), while the
sup-LM and CvM tests use Ln = n − 1. Nankervis and Savin (2010) truncate the correlation
series in the sup-LM statistic, using Ln = 20 for each n, which fails to deliver a consistent test.
We use each Ln = max{5, [δ × n/ ln(n)]} with δ ∈ {0.0, 0.2, 0.4, 0.5, 1.0} for the max-correlation
and sup-LM tests for comparability, as well as Ln = n− 1 for sup-LM and CvM tests. Note that,
under suitable regularity conditions, as long as Ln → ∞ then the sup-LM and CvM tests will
have their intended limit properties under the null and alternative hypotheses, even if Ln = o(n).
In the case of Shanghai, for example, we have n = 3110 and hence Ln ∈ {5, 77, 154, 193, 386}.The bootstrap block size is set to be bn = [
√n] as in Hill and Motegi (2016a), but values like
bn = [(1/2)√n] or bn = [2
√n] lead to similar results, cf. Shao (2011).
Insert Table 2 here
The strongest evidence for correlation in Shanghai comes from the max-correlation test at
maximum lag 5. As more lags are incorporated into the max-correlation and sup-LM statistics,
the bootstrapped p-values are progressively larger, and the CvM test with all possible lags likewise
fails to reject the null hypothesis. The evidence therefore favors a rejection of weak form efficiency
in the Shanghai market.
Contrary to the case of Shanghai, for Nikkei, FTSE, and SP500 the max-correlation test leads
to a non-rejection and the sup-LM and CvM tests lead to a rejection. The max-correlation p-value
is exactly .10 at lag 5 for SP500, suggesting weak evidence against white noise. There are several
explanations. First, this may be because Nikkei, FTSE, and S&P 500 have barely significant (and
small) negative autocorrelations at lag 1, as we saw in Figure 2, with significance at the 10% level
for Nikkei and FTSE, and 5% for the SP500. Second, the max-correlation test has the sharpest
8 Results with larger lags are available upon request.
11
empirical size amongst these tests, while sup-LM and CvM tests tend to be over-sized. Thus, we
may be viewing false rejections from sup-LM and CvM tests.
Each of the four series, however, has fairly weak serial correlation over each lag, with any
evidence of correlation coming at low lags. sup-LM and CvM tests result in greater sampling
error and therefore lower power as lags are added, and tend to be over-sized even at low lags.
Thus, if there is evidence against weak form efficiency in these markets it will generally come at
low lags. With that in mind, the max-correlation test, with generally sharper size and greater
power9, suggests the Shanghai index fails to be weak form efficient over the full sample, with
evidence against efficiency in the SP500.
4.2 Rolling Window Analysis
A potential drawback of the full sample analysis is that the degree of stock market efficiency
may not be constant throughout the sample period in reality, as pointed out by Lo (2004, 2005).
We therefore perform a rolling window analysis in order to capture the dynamic nature of market
efficiency. An advantage of this approach is that we can examine whether negative autocorrelations
arising in Nikkei, FTSE, and SP500 is a persistent phenomenon. We set the window size to be
240 trading days (roughly a year), which is similar to the window size in Verheyden, De Moor,
and Van den Bossche (2015).
4.2.1 Preliminary Analysis
As a preliminary analysis, Figure 3 plots the standard deviation, minimum, maximum, skew-
ness, and kurtosis of each return series in each rolling window. As expected, the standard deviation
for each country blows up around 2008, reflecting the subprime mortgage crisis. Shanghai’s stan-
dard deviation in 2015 is roughly as large as what it used to be in 2008, reflecting the Shanghai
stock market collapse.
There is a dramatic change in skewness and kurtosis for Nikkei and SP500 in 2010-2011.
Skewness suddenly reaches a large negative value of about -2, and kurtosis skyrockets to 20 for
Nikkei and to 14 for the SP500. These results stem from the tsunami disaster and S&P securities
downgrade shock explained in Section 3.
In early 2003, the FTSE had a large positive skewness of around 0.8 and a large kurtosis of
around 7, reflecting the Iraq War and the recession. The positive skewness stems from a large log
9Simulation experiments performed in Hill and Motegi (2016a) reveal that the dependent wild bootstrappedmax-correlation test over most cases considered has comparatively sharper size and higher power than the sup-LMand CvM tests, as well as a variant of Hong’s (1996) tests and the bootstrapped Q-test.
12
return of 0.059 realized on March 13, 2003, when it was a bear market.
Insert Figure 3 here
4.2.2 Analysis of Serial Correlation
Figure 4 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 = 240, and we begin with a conventional block size
bn = [√n] = 15. The number of bootstrap samples is 10,000 for each window.
Insert Figure 4 here
A striking result from Figure 4 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 bn = 15 windows. The confidence bands are particularly volatile for Nikkei and SP500 in
2011, 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 (2016b) 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 every bn windows. Data blocks
from two windows separated by bn 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, 1999). The solution proposed by Politis and Romano (1994)
is to randomize block size, which we follow. For each of the 10,000 bootstrap samples, and each
window, we independently draw c from a uniform distribution on [0.5, 1.5], and use bn = c√n.10
A comparison of Figures 4 and 5 highlights the substantial impact of block size randomization:
in Figure 5 randomization has clearly removed the periodic fluctuations.11 We still observe volatile
bands for Nikkei right after the tsunami disaster and for SP500 right after the U.S. securities
downgrade shock. These are not surprising results since the correlations themselves spike in those
periods. In what follows, we focus the discussion on results based on block randomization, and
10If we only randomize bn for each window (using the same randomized bn across all 10,000 bootstrap samples),then there exists volatility in the resulting confidence bands that largely exceeds the volatility of the observed data.By randomizing bn for each bootstrap draw and each window, both artificial nonstationarity and excess volatilityare eradicated.
11See also the supplemental material for related plots from controlled experiments. In particular, we presentmagnified plots of bootstrapped confidence bands from simulated data, with and without randomized block size.
13
therefore Figure 5.
Insert Figure 5 here
For Shanghai, the confidence bands are roughly [-0.1, 0.1] and they contain the sample corre-
lation in most windows. Interestingly, the subprime mortgage crisis around 2008 did not have a
substantial impact on the correlation structure of the Shanghai market, although the stock price
itself responded with a massive drop and volatility burst around 2008 (see Figures 1 and 3). In
2014-2015, the confidence bands are slightly wider, reflecting the Shanghai stock market collapse.
The correlations sometimes go beyond 0.1 and outside the confidence bands. Hence, conditional
on lag h = 1, there is a possibility that the Shanghai collapse had an adverse impact on the market
efficiency. This is clearly a richer implication than what we saw from the full sample analysis,
highlighting an advantage of rolling window analysis.
The first order correlation for Nikkei generally lies in [-0.1, 0.1] and they are insignificant
in most windows. The correlation goes beyond 0.1 in only one out of 2910 windows, which is
window #1772 (March 24, 2010 - March 15, 2011). This is the first window that contains the
tsunami shock. The correlation is 0.142 and the confidence band is [−0.216, 0.208] there, hence
the zero hypothesis is not rejected. In terms of negative correlations, we have ρn(1) < −0.1 in
210 windows (approximately 7.2% of all windows). The zero hypothesis is rejected for 25 out of
the 210 windows. This result is consistent with the negative correlation at lag 1 observed in the
full sample analysis.
The tendency for negative correlations is much more prominent in FTSE. The correlation for
FTSE goes below -0.1 for 954 windows out of 3004. A rejection occurs in 478 out of the 954
windows. The correlation goes even below -0.2 for 106 windows, and a rejection occurs in 66
windows.
Similarly, SP500 often has significantly negative correlations. The correlation for SP500 goes
below -0.1 for 1109 windows out of 2991. A rejection occurs in 675 out of the 1109 windows. The
correlation goes even below -0.2 for 31 windows, and a rejection occurs in 18 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 there-
fore long run stationarity (see Fama and French, 1988), which fits our maintained assumption
that log-prices are first difference stationary.
Another implication from Figure 5 is that the significantly negative correlations concentrate
on the period of financial turmoil: Iraq-War regime in 2003 for FTSE and SP500 and the subprime
mortgage crisis in 2008 for SP500. In view of the extant evidence for positive serial correlation
in market returns, this suggests that negative correlations may be indicative of trading turmoil,
due ostensibly to the rapid evolution of information.
14
Evidence for negative correlations during crisis periods is not new. See, for example,
Campbell, Grossman, and Wang (1993) who find a negative relationship between trading vol-
ume and serial correlation: high volume days are associated with lower or negative correlations,
due, they argue, to the presence of ”noninformational” traders.
4.2.3 White Noise Tests
We now perform the max-correlation, sup-LM, and CvM tests for each window. As in Section 4.1,
lag length is Ln = max{5, [δ × n/ ln(n)]} with δ ∈ {0.0, 0.2, 0.4, 0.5, 1.0} for the max-correlation
and sup-LM tests. Since the sample size is constant at window size n = 240, we have that
Ln = 5, 8, 17, 21, 43. We also cover Ln = n− 1 = 239 for the sup-LM and CvM tests. Block size
is bn = c√n, and we draw c from the uniform distribution between 0.5 and 1.5 independently
across M = 5, 000 bootstrap samples and rolling windows.
See Figures 6-8 for rolling window test statistics, critical values, and p-values as well as Table
3 for summary results. 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, however,
that Shanghai has positive correlations that are barely significant at the 5% level in 2014-2015
(see Figure 5). These are only first order correlations, and their magnitude is not too large, while
statistical significance of each tests declines when lags Ln ≥ 5 are considered jointly.
Insert Figures 6-8 here
The FTSE and SP500 market have more periods of inefficiency than Shanghai and Nikkei.
Based on the CvM test, for example, a rejection happens in as many as 14.5% of all windows for
FTSE and 21.0% for SP500. The sup-LM test with any lag selection yields similar results, while
the max-correlation test produces lower rejection frequencies (less than 10%). This difference
is reasonable since the negative correlation at lag 1 should be better captured by the sup-LM
and CvM tests than the max-correlation test (cf. simulation experiments in Hill and Motegi,
2016a,b). As seen in Figures 7-8, 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 SP500. This result is consistent with Figure 5. 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 3 here
15
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 cannot reject the white noise hypothesis of
stock returns. The same goes for FTSE and SP500 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. Thus, conditional on
being in a non-crisis period, the latter markets are efficient, but fare less well in terms of efficiency
during crisis periods, relative to Asian markets.
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 tests of the white noise hypothesis 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 (2016a) that extends Shao’s (2011) dependent wild
bootstrap to max-correlation and sup-LM tests, as well as Shao’s (2011) proposed Cramer-von
Mises test, we analyze weak form efficiency of Chinese, Japanese, U.K., and U.S. stock markets.
We perform both full sample and rolling window analyses, where the latter allows us 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 is removed by randomizing
block sizes.
We find that the degree of market efficiency varies across countries and sample periods. The
Shanghai and Nikkei returns exhibit a high degree of efficiency such that we generally cannot
reject the white noise hypothesis. The same goes for the FTSE and SP500 during non-crisis
periods. When those markets 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.
16
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20
Tab
le1:
Sam
ple
Sta
tist
ics
ofL
ogR
eturn
sof
Sto
ckP
rice
Indic
es(0
1/01
/200
3-
10/2
9/20
15)
#O
bs.
Mea
n95
%B
and
Med
.Std
ev.
Min
.M
ax.
Ske
w.
Kurt
.p-K
Sp-A
DShan
ghai
3110
2.9×
10−
4[−
7.4,
7.2]×
10−
40.
001
0.01
7-0
.093
0.09
0-0
.425
6.83
10.
000
0.00
1N
ikke
i31
492.
5×
10−
4[−
5.0,
5.1]×
10−
40.
001
0.01
5-0
.121
0.13
2-0
.532
10.6
80.
000
0.00
1F
TSE
3243
1.5×
10−
4[−
2.7,
2.6]×
10−
40.
001
0.01
2-0
.093
0.09
4-0
.133
11.1
40.
000
0.00
1SP
500
3230
2.7×
10−
4[−
3.8,
3.7]×
10−
40.
001
0.01
2-0
.095
0.11
0-0
.319
14.0
20.
000
0.00
1
”95%
Ban
d”
isa
boots
trap
ped
95%
con
fid
ence
ban
dfo
rth
esa
mp
lem
ean
.It
isco
nst
ruct
edu
nd
erth
enu
llhyp
oth
esis
of
zero
-mea
n
wh
ite
noi
se,
usi
ng
the
dep
end
ent
wil
db
oot
stra
pw
ith
blo
cksi
zeb n
=√n
.T
he
nu
mb
erof
boots
trap
sam
ple
sisM
=10,0
00.
”p-K
S”
sign
ifies
ap
-valu
eof
the
Kol
mogor
ov-S
mir
nov
test
,w
hil
e”p
-AD
”si
gnifi
esa
p-v
alu
eof
the
An
der
son
-Darl
ing
test
.
21
Tab
le2:
P-V
alues
ofW
hit
eN
oise
Tes
tsov
erth
eF
ull
Sam
ple
(01/
01/2
003
-10
/29/
2015
)
Sh
an
gh
ai
(3110
Day
s)
Max
-Cor
rela
tion
An
dre
ws-
Plo
ber
ger
CvM
δ=
0.0
δ=
0.2
δ=
0.4
δ=
0.5
δ=
1.0
δ=
0.0
δ=
0.2
δ=
0.4
δ=
0.5
δ=
1.0
−−
Ln
=5Ln
=77
Ln
=15
4Ln
=19
3Ln
=386Ln
=5Ln
=77Ln
=154Ln
=193Ln
=386Ln
=3109Ln
=3109
.053
.314
.380
.398
.447
.182
.149
.148
.157
.153
.161
.154
Nik
kei
(3149
Day
s)
Max
-Cor
rela
tion
An
dre
ws-
Plo
ber
ger
CvM
δ=
0.0
δ=
0.2
δ=
0.4
δ=
0.5
δ=
1.0
δ=
0.0
δ=
0.2
δ=
0.4
δ=
0.5
δ=
1.0
−−
Ln
=5Ln
=78
Ln
=15
6Ln
=19
5Ln
=390Ln
=5Ln
=78Ln
=156Ln
=195Ln
=390Ln
=3148Ln
=3148
.327
.612
.724
.681
.671
.083
.145
.142
.156
.145
.147
.060
FT
SE
(3243
Day
s)
Max
-Cor
rela
tion
An
dre
ws-
Plo
ber
ger
CvM
δ=
0.0
δ=
0.2
δ=
0.4
δ=
0.5
δ=
1.0
δ=
0.0
δ=
0.2
δ=
0.4
δ=
0.5
δ=
1.0
−−
Ln
=5Ln
=80
Ln
=16
0Ln
=20
0Ln
=401Ln
=5Ln
=80Ln
=160Ln
=200Ln
=401Ln
=3242Ln
=3242
.425
.517
.563
.571
.576
.065
.079
.074
.075
.076
.073
.068
SP
500
(3230
Day
s)
Max
-Cor
rela
tion
An
dre
ws-
Plo
ber
ger
CvM
δ=
0.0
δ=
0.2
δ=
0.4
δ=
0.5
δ=
1.0
δ=
0.0
δ=
0.2
δ=
0.4
δ=
0.5
δ=
1.0
−−
Ln
=5Ln
=79
Ln
=15
9Ln
=19
9Ln
=399Ln
=5Ln
=79Ln
=159Ln
=199Ln
=399Ln
=3229Ln
=3229
.094
.162
.169
.177
.171
.034
.032
.032
.030
.029
.028
.034
Boots
trapp
edp-v
alu
esof
Hill
and
Mote
gi’s
(2016a)
max-c
orr
elati
on
whit
enois
ete
st,
Andre
ws
and
Plo
ber
ger
’s(1
996)
sup-L
Mte
st,
and
the
Cra
mer
-von
Mis
es
test
.Shao’s
(2011)
dep
enden
tw
ild
boots
trap
wit
h5000
replica
tions
isuse
dfo
rea
chte
st.
The
maxim
um
lag
length
sfo
rth
em
ax-c
orr
elati
on
and
sup-L
Mte
sts
areL
n=
max{5,[δ×n/
ln(n
)]}
wit
hδ∈{0.0,0.2,0.4,0.5,1.0}.
The
sup-L
Mte
stis
als
oco
mpute
dw
ith
the
maxim
um
poss
iblen−
1la
gs.
The
CvM
test
use
s
Ln
=n−
1.
22
Tab
le3:
Rej
ecti
onR
atio
ofW
hit
eN
oise
Tes
tsov
erR
olling
Win
dow
s
Max
-Cor
rela
tion
An
dre
ws-
Plo
ber
ger
CvM
δ=
0.0
δ=
0.2
δ=
0.4
δ=
0.5
δ=
1.0
δ=
0.0
δ=
0.2
δ=
0.4
δ=
0.5
δ=
1.0
−−
Ln
=5L
n=
8L
n=
17L
n=
21L
n=
43L
n=
5L
n=
8L
n=
17L
n=
21L
n=
43L
n=
239
Ln
=23
9
Sh
an
ghai
0.03
70.0
25
0.00
40.
002
0.00
00.
002
0.00
60.
003
0.003
0.0
02
0.0
03
0.058
Nik
kei
0.03
30.0
39
0.00
00.
000
0.00
10.
016
0.01
60.
020
0.021
0.0
21
0.0
21
0.002
FT
SE
0.07
20.0
43
0.01
60.
015
0.04
50.
134
0.14
20.
131
0.129
0.1
31
0.1
32
0.145
SP
500
0.08
80.0
31
0.01
60.
013
0.00
70.
161
0.16
40.
158
0.157
0.1
58
0.1
57
0.210
The
rati
oof
rollin
gw
indow
sw
her
eth
enull
hyp
oth
esis
of
whit
enois
eis
reje
cted
at
the
5%
level
.T
est
stati
stic
sare
Hill
and
Mote
gi’s
(2016a)
max-c
orr
elati
on
stati
stic
,A
ndre
ws
and
Plo
ber
ger
’s(1
996)
sup-L
Mst
ati
stic
,and
the
Cra
mer
-von
Mis
esst
ati
stic
.W
euse
Shao’s
(2011)
dep
enden
tw
ild
boots
trap
wit
hblo
ck
sizeb n
=[c×√n
].W
edra
wc∼U
(0.5,1.5
)in
dep
enden
tly
acr
ossM
=5,0
00
boots
trap
sam
ple
sand
rollin
gw
indow
s.T
he
maxim
um
lag
length
sfo
rth
e
max-c
orr
elati
on
and
sup-L
Mte
sts
areL
n=
max{5,[δ×n/
ln(n
)]}
wit
hδ∈{0.0,0.2,0.4,0.5,1.0}
and
win
dow
size
n=
240
tradin
gday
s.W
eals
oco
ver
Ln
=n−
1fo
rth
esu
p-L
Mand
CvM
test
s.
23
Figure 1: Daily Stock Prices and Log Returns (01/01/2003 - 10/29/2015)
Jan05 Jan10 Jan150
100
200
300
400
500
Subprime
Shanghai
Shanghai (Level)
Jan05 Jan10 Jan15-0.2
-0.1
0
0.1
0.2
Subprime Shanghai
Shanghai (Return)
Jan05 Jan10 Jan150
100
200
300
Subprime
Shanghai
Tsunami
Nikkei (Level)
Jan05 Jan10 Jan15-0.2
-0.1
0
0.1
0.2
Subprime TsunamiShanghai
Nikkei (Return)
Jan05 Jan10 Jan150
100
200
300
Iraq
Subprime
Shanghai
FTSE (Level)
Jan05 Jan10 Jan15-0.2
-0.1
0
0.1
0.2
SubprimeIraq Shanghai
FTSE (Return)
Jan05 Jan10 Jan150
100
200
300
Subprime
Shanghai
AA+
SP500 (Level)
Jan05 Jan10 Jan15-0.2
-0.1
0
0.1
0.2
Subprime AA+Shanghai
SP500 (Return)
Left panels depict stock price indices in local currencies (standardized at 100 on 01/01/2003). Note that the maximum value
of the vertical axis is 500 for Shanghai and 300 for the other series. Right panels depict log returns. ”Subprime” signifies
the subprime mortgage crisis around 2008; ”Shanghai” signifies the 2014-2015 bubble and burst in Shanghai; ”Tsunami”
signifies the tsunami disaster caused by the Great East Japan Earthquake in March 2011; ”Iraq” signifies Iraq War (plus
economic stagnation) threatening the United Kingdom; ”AA+” signifies the downgrade of the U.S. federal government
credit rating from AAA to AA+. 24
Figure 2: Sample Autocorrelations of Log Returns of Stock Price Indices
5 10 15 20 25-0.2
-0.1
0
0.1
0.2
Shanghai
5 10 15 20 25-0.2
-0.1
0
0.1
0.2
Nikkei
5 10 15 20 25-0.2
-0.1
0
0.1
0.2
FTSE
5 10 15 20 25-0.2
-0.1
0
0.1
0.2
SP500
This figure plots sample autocorrelations at lags 1, . . . , 25. The 95% confidence bands are constructed with Shao’s
(2011) dependent wild bootstrap under the null hypothesis of white noise. The number of bootstrap samples is
5,000.
25
Figure 3: Sample Statistics over Rolling Windows
Jan05 Jan100.00
0.01
0.02
0.03
Std / Shanghai
Jan05 Jan100.00
0.01
0.02
0.03
Std / Nikkei
Jan05 Jan100.00
0.01
0.02
0.03
Std / FTSE
Jan05 Jan100.00
0.01
0.02
0.03
Std / SP500
Jan05 Jan10
-0.10
-0.05
0.00
Min / Shanghai
Jan05 Jan10
-0.10
-0.05
0.00
Min / Nikkei
Jan05 Jan10
-0.10
-0.05
0.00
Min / FTSE
Jan05 Jan10
-0.10
-0.05
0.00
Min / SP500
Jan05 Jan100.00
0.05
0.10
Max / Shanghai
Jan05 Jan100.00
0.05
0.10
Max / Nikkei
Jan05 Jan100.00
0.05
0.10
Max / FTSE
Jan05 Jan100.00
0.05
0.10
Max / SP500
Jan05 Jan10
-2.0
-1.0
0.0
1.0
2.0
Skew / Shanghai
Jan05 Jan10
-2.0
-1.0
0.0
1.0
2.0
Skew / Nikkei
Jan05 Jan10
-2.0
-1.0
0.0
1.0
2.0
Skew / FTSE
Jan05 Jan10
-2.0
-1.0
0.0
1.0
2.0
Skew / SP500
Jan05 Jan10
5.0
10.0
15.0
20.0
Kurt / Shanghai
Jan05 Jan10
5.0
10.0
15.0
20.0
Kurt / Nikkei
Jan05 Jan10
5.0
10.0
15.0
20.0
Kurt / FTSE
Jan05 Jan10
5.0
10.0
15.0
20.0
Kurt / SP500
Standard deviation (std), minimum (min), maximum (max), skewness (skew), and kurtosis (kurt) of stock returns
of Shanghai, Nikkei, FTSE, and SP500 in each rolling window with size 240 trading days. Each point on the
horizontal axis represents the initial date of each window.
26
Figure 4: Rolling Window Autocorrelations at Lag 1 (Fixed Block Size)
Jan05 Jan10-0.5
0
0.5
Shanghai
Jan05 Jan10-0.5
0
0.5
Nikkei
Jan05 Jan10-0.5
0
0.5
FTSE
Jan05 Jan10-0.5
0
0.5
SP500
The solid line depicts sample autocorrelations at lag 1. The dotted line depicts the 95% confidence band constructed
with Shao’s (2011) dependent wild bootstrap under the null hypothesis of white noise. Window size is n = 240
trading days, and block size is bn = [√n] = 15. The number of bootstrap samples is 10,000 for each window. Each
point on the horizontal axis represents the initial date of each window.
27
Figure 5: Rolling Window Autocorrelations at Lag 1 (Randomized Block Size)
Jan05 Jan10-0.5
0
0.5
Shanghai
Shanghai
Jan05 Jan10-0.5
0
0.5
Tsunami
Nikkei
Jan05 Jan10-0.5
0
0.5
Iraq Subprime
FTSE
Jan05 Jan10-0.5
0
0.5
Iraq Subprime AA+
SP500
The solid line depicts sample autocorrelations at lag 1. The dotted line depicts the 95% confidence band con-
structed with Shao’s (2011) dependent wild bootstrap under the null hypothesis of white noise. Window size is
n = 240 trading days. Block size is bn = [c√n]. We draw c ∼ U(0.5, 1.5) independently across M = 10, 000
bootstrap samples and rolling windows. Each point on the horizontal axis represents the initial date of each win-
dow. ”Shanghai” signifies the 2014-2015 bubble and burst in Shanghai; ”Tsunami” signifies the tsunami disaster
caused by the Great East Japan Earthquake in March 2011; ”Iraq” signifies Iraq War (plus economic stagnation)
threatening the United Kingdom; ”Subprime” signifies the subprime mortgage crisis around 2008; ”AA+” signifies
the downgrade of the U.S. federal government credit rating from AAA to AA+.
28
Fig
ure
6:M
ax-C
orre
lati
onT
est
inR
olling
Win
dow
(Ln
=5)
Jan0
5Ja
n10
0510
Ban
ds
/S
han
gh
ai
Jan0
5Ja
n10
0510
Ban
ds
/N
ikke
i
Jan0
5Ja
n10
0510
Ban
ds
/F
TS
E
Jan0
5Ja
n10
0510
Ban
ds
/S
P50
0
Jan0
5Ja
n10
0
0.51
P-V
alu
es/
Sh
an
ghai
Jan0
5Ja
n10
0
0.51
P-V
alu
es/
Nik
kei
Jan0
5Ja
n10
0
0.51
P-V
alu
es/
FT
SE
Jan0
5Ja
n10
0
0.51
P-V
alu
es/
SP
500
Hil
lan
dM
oteg
i’s
(201
6a)
max
-cor
rela
tion
test
.L
ag
len
gth
isLn
=m
ax{5,[δ×n/
ln(n
)]}
wit
hδ∈{0.0,0.2,0.4,0.5,1.0}
an
dw
ind
owsi
zen
=240
trad
ing
day
s,h
enceLn
=5,8,1
7,2
1,43
.W
eu
seth
ed
epen
den
tw
ild
boots
trap
wit
h5,0
00
rep
lica
tion
sfo
rea
chw
ind
ow.
Blo
cksi
zeisb n
=[c√n
],
wh
erec∼U
(0.5,1.5
)is
dra
wn
ind
epen
den
tly
acro
ssb
oots
trap
sam
ple
san
dw
ind
ows.
Each
poin
ton
the
hori
zonta
laxis
rep
rese
nts
the
init
ial
date
ofea
chw
ind
ow.
Inth
eu
pp
erp
anel
sth
eso
lid
bla
ckli
ne
isth
ete
stst
ati
stic
,an
dth
ed
ott
edre
dli
ne
isth
e5%
crit
ical
valu
e.In
the
low
erp
an
els
p-v
alu
esar
ep
lott
edw
ith
0.05
den
oted
.
29
Fig
ure
6:M
ax-C
orre
lati
onT
est
inR
olling
Win
dow
(Ln
=8)
Jan0
5Ja
n10
0510
Ban
ds
/S
han
gh
ai
Jan0
5Ja
n10
0510
Ban
ds
/N
ikke
i
Jan0
5Ja
n10
0510
Ban
ds
/F
TS
E
Jan0
5Ja
n10
0510
Ban
ds
/S
P50
0
Jan0
5Ja
n10
0
0.51
P-V
alu
es/
Sh
an
ghai
Jan0
5Ja
n10
0
0.51
P-V
alu
es/
Nik
kei
Jan0
5Ja
n10
0
0.51
P-V
alu
es/
FT
SE
Jan0
5Ja
n10
0
0.51
P-V
alu
es/
SP
500
30
Fig
ure
6:M
ax-C
orre
lati
onT
est
inR
olling
Win
dow
(Ln
=17
)
Jan0
5Ja
n10
0510
Ban
ds
/S
han
gh
ai
Jan0
5Ja
n10
0510
Ban
ds
/N
ikke
i
Jan0
5Ja
n10
0510
Ban
ds
/F
TS
E
Jan0
5Ja
n10
0510
Ban
ds
/S
P50
0
Jan0
5Ja
n10
0
0.51
P-V
alu
es/
Sh
an
ghai
Jan0
5Ja
n10
0
0.51
P-V
alu
es/
Nik
kei
Jan0
5Ja
n10
0
0.51
P-V
alu
es/
FT
SE
Jan0
5Ja
n10
0
0.51
P-V
alu
es/
SP
500
31
Fig
ure
6:M
ax-C
orre
lati
onT
est
inR
olling
Win
dow
(Ln
=21
)
Jan0
5Ja
n10
0510
Ban
ds
/S
han
gh
ai
Jan0
5Ja
n10
0510
Ban
ds
/N
ikke
i
Jan0
5Ja
n10
0510
Ban
ds
/F
TS
E
Jan0
5Ja
n10
0510
Ban
ds
/S
P50
0
Jan0
5Ja
n10
0
0.51
P-V
alu
es/
Sh
an
ghai
Jan0
5Ja
n10
0
0.51
P-V
alu
es/
Nik
kei
Jan0
5Ja
n10
0
0.51
P-V
alu
es/
FT
SE
Jan0
5Ja
n10
0
0.51
P-V
alu
es/
SP
500
32
Fig
ure
6:M
ax-C
orre
lati
onT
est
inR
olling
Win
dow
(Ln
=43
)
Jan0
5Ja
n10
0510
Ban
ds
/S
han
gh
ai
Jan0
5Ja
n10
0510
Ban
ds
/N
ikke
i
Jan0
5Ja
n10
0510
Ban
ds
/F
TS
E
Jan0
5Ja
n10
0510
Ban
ds
/S
P50
0
Jan0
5Ja
n10
0
0.51
P-V
alu
es/
Sh
an
ghai
Jan0
5Ja
n10
0
0.51
P-V
alu
es/
Nik
kei
Jan0
5Ja
n10
0
0.51
P-V
alu
es/
FT
SE
Jan0
5Ja
n10
0
0.51
P-V
alu
es/
SP
500
33
Fig
ure
7:A
ndre
ws-
Plo
ber
ger
Sup-L
MT
est
inR
olling
Win
dow
(Ln
=5)
Jan0
5Ja
n10
0102030
Ban
ds
/S
han
gh
ai
Jan0
5Ja
n10
0102030
Ban
ds
/N
ikke
i
Jan0
5Ja
n10
0102030
Ban
ds
/F
TS
E
Jan0
5Ja
n10
050100
150
200
Ban
ds
/S
P50
0
Jan0
5Ja
n10
0
0.51
P-V
alu
es/
Sh
angh
ai
Jan0
5Ja
n10
0
0.51
P-V
alu
es/
Nik
kei
Jan0
5Ja
n10
0
0.51
P-V
alu
es/
FT
SE
Jan0
5Ja
n10
0
0.51
P-V
alu
es/
SP
500
An
dre
ws
and
Plo
ber
ger’
s(1
996)
sup
-LM
test
.L
ag
len
gth
isLn
=m
ax{5,[δ×n/
ln(n
)]}
wit
hδ∈{0.0,0.2,0.4,0.5,1.0}
an
dw
ind
owsi
zen
=240
trad
ing
day
s,hen
ceLn
=5,
8,1
7,2
1,4
3.W
eals
oco
verLn
=n−
1=
239.
We
use
the
dep
end
ent
wil
db
oots
trap
wit
h5,0
00
rep
lica
tion
sfo
r
each
win
dow
.B
lock
size
isb n
=[c√n
],w
her
ec∼U
(0.5,1.5
)is
dra
wn
ind
epen
den
tly
acr
oss
boots
trap
sam
ple
san
dw
ind
ows.
Each
poin
ton
the
hor
izon
tal
axis
rep
rese
nts
the
init
ial
dat
eof
each
win
dow
.In
the
up
per
pan
els
the
soli
db
lack
lin
eis
the
test
stati
stic
,an
dth
ed
ott
edre
dli
ne
is
the
5%cr
itic
alva
lue.
Inth
elo
wer
pan
els
p-v
alu
esare
plo
tted
wit
h0.0
5d
enote
d.
34
Fig
ure
7:A
ndre
ws-
Plo
ber
ger
Sup-L
MT
est
inR
olling
Win
dow
(Ln
=8)
Jan0
5Ja
n10
010203040
Ban
ds
/S
han
gh
ai
Jan0
5Ja
n10
0102030
Ban
ds
/N
ikke
i
Jan0
5Ja
n10
0102030
Ban
ds
/F
TS
E
Jan0
5Ja
n10
0
100
200
300
Ban
ds
/S
P50
0
Jan0
5Ja
n10
0
0.51
P-V
alu
es/
Sh
angh
ai
Jan0
5Ja
n10
0
0.51
P-V
alu
es/
Nik
kei
Jan0
5Ja
n10
0
0.51
P-V
alu
es/
FT
SE
Jan0
5Ja
n10
0
0.51
P-V
alu
es/
SP
500
35
Fig
ure
7:A
ndre
ws-
Plo
ber
ger
Sup-L
MT
est
inR
olling
Win
dow
(Ln
=17
)
Jan0
5Ja
n10
010203040
Ban
ds
/S
han
gh
ai
Jan0
5Ja
n10
0102030
Ban
ds
/N
ikke
i
Jan0
5Ja
n10
0102030
Ban
ds
/F
TS
E
Jan0
5Ja
n10
0
100
200
300
Ban
ds
/S
P50
0
Jan0
5Ja
n10
0
0.51
P-V
alu
es/
Sh
angh
ai
Jan0
5Ja
n10
0
0.51
P-V
alu
es/
Nik
kei
Jan0
5Ja
n10
0
0.51
P-V
alu
es/
FT
SE
Jan0
5Ja
n10
0
0.51
P-V
alu
es/
SP
500
36
Fig
ure
7:A
ndre
ws-
Plo
ber
ger
Sup-L
MT
est
inR
olling
Win
dow
(Ln
=21
)
Jan0
5Ja
n10
010203040
Ban
ds
/S
han
gh
ai
Jan0
5Ja
n10
0102030
Ban
ds
/N
ikke
i
Jan0
5Ja
n10
0102030
Ban
ds
/F
TS
E
Jan0
5Ja
n10
0
100
200
300
Ban
ds
/S
P50
0
Jan0
5Ja
n10
0
0.51
P-V
alu
es/
Sh
angh
ai
Jan0
5Ja
n10
0
0.51
P-V
alu
es/
Nik
kei
Jan0
5Ja
n10
0
0.51
P-V
alu
es/
FT
SE
Jan0
5Ja
n10
0
0.51
P-V
alu
es/
SP
500
37
Fig
ure
7:A
ndre
ws-
Plo
ber
ger
Sup-L
MT
est
inR
olling
Win
dow
(Ln
=43
)
Jan0
5Ja
n10
010203040
Ban
ds
/S
han
gh
ai
Jan0
5Ja
n10
0102030
Ban
ds
/N
ikke
i
Jan0
5Ja
n10
0102030
Ban
ds
/F
TS
E
Jan0
5Ja
n10
0
100
200
300
Ban
ds
/S
P50
0
Jan0
5Ja
n10
0
0.51
P-V
alu
es/
Sh
angh
ai
Jan0
5Ja
n10
0
0.51
P-V
alu
es/
Nik
kei
Jan0
5Ja
n10
0
0.51
P-V
alu
es/
FT
SE
Jan0
5Ja
n10
0
0.51
P-V
alu
es/
SP
500
38
Fig
ure
7:A
ndre
ws-
Plo
ber
ger
Sup-L
MT
est
inR
olling
Win
dow
(Ln
=23
9)
Jan0
5Ja
n10
010203040
Ban
ds
/S
han
gh
ai
Jan0
5Ja
n10
0102030
Ban
ds
/N
ikke
i
Jan0
5Ja
n10
0102030
Ban
ds
/F
TS
E
Jan0
5Ja
n10
0
100
200
300
Ban
ds
/S
P50
0
Jan0
5Ja
n10
0
0.51
P-V
alu
es/
Sh
angh
ai
Jan0
5Ja
n10
0
0.51
P-V
alu
es/
Nik
kei
Jan0
5Ja
n10
0
0.51
P-V
alu
es/
FT
SE
Jan0
5Ja
n10
0
0.51
P-V
alu
es/
SP
500
39
Fig
ure
8:C
ram
er-v
onM
ises
Tes
tin
Rol
ling
Win
dow
(Ln
=23
9)
Jan
05
Jan
10
024681
0-5 B
an
ds
/S
han
gh
ai
Jan0
5Ja
n10
0
0.51
1.5
10-4 B
and
s/
Nik
kei
Jan
05
Jan
10
02461
0-5
Ban
ds
/F
TS
E
Jan0
5Ja
n10
0
0.51
10-4 B
and
s/
SP
500
Jan0
5Ja
n10
0
0.51
P-V
alu
es/
Sh
an
ghai
Jan0
5Ja
n10
0
0.51
P-V
alu
es/
Nik
kei
Jan0
5Ja
n10
0
0.51
P-V
alu
es/
FT
SE
Jan0
5Ja
n10
0
0.51
P-V
alu
es/
SP
500
Sh
ao’s
(201
1)C
ram
er-v
onM
ises
test
.L
agle
ngt
hisLn
=n−
1=
239.
We
use
the
dep
end
ent
wil
db
oots
trap
wit
h5,0
00
rep
lica
tion
sfo
rea
chw
ind
ow.
Blo
cksi
zeisb n
=[c√n
],w
her
ec∼U
(0.5,1.5
)is
dra
wn
ind
epen
den
tly
acr
oss
boots
trap
sam
ple
san
dw
ind
ows.
Each
poin
ton
the
hori
zonta
laxis
rep
rese
nts
the
init
ial
dat
eof
each
win
dow
.In
the
upp
erp
an
els
the
soli
db
lack
line
isth
ete
stst
ati
stic
,an
dth
ed
ott
edre
dli
ne
isth
e5%
crit
ical
valu
e.In
the
low
erp
anel
sp
-val
ues
are
plo
tted
wit
h0.0
5d
enote
d.
40
