The Random Walk Hypothesis
-Evidence from the Japanese Stock Market-
Madalina-Anda Corneanu
Student ID: 10258256
Supervisor: Prof. Dr. Peter Boswijk
Bachelor Thesis Economics and Business Specialization Economics and Finance Faculty of Economics and Business University of Amsterdam
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1. Introduction
The importance of the Random Walk Hypothesis for the finance world is most easily understood in the broader context of market efficiency. When markets are truly efficient, and the stock prices accurately reflect all available information, it becomes impossible for investors to consistently outperform the market; any attempt in trying to predict future movements based on past performance would be hopeless. In that sense, the random walk model claims that the behavior of stock prices is not more predictable than the weather, since the best estimate of tomorrow’s price is simply the price realized today.
Historically, the French mathematician Louis Bachelier was the first one to tackle the subject of random walk behavior in stock prices, back in 1900. Forgotten for almost half a century, the topic regained attention in the 1950s when Kendall (1953) and Fama (1965), among others, attested the random walk behavior of stock prices (Dimson & Mussavian, 1988). The hypothesis became especially popular in 1973 with the publication of Burton Malkiel’s book “A Random Walk Down Wall Street”, but it began receiving severe criticism soon after that, most notably from Andrew Lo and Craig MacKinlay’s “Stock Market Prices Do Not Follow Random Walks: Evidence from a Simple Specification Test” (1988). Later studies trying to assess the presence of random walk behavior in stock markets yielded mixed results.
The Japanese market, surprisingly, has not been the subject of extensive study, and the various papers approaching the subject did not reach a clear consensus. Therefore, our research aims at providing an answer to the question of whether stock prices in Japan follow the Random Walk Hypothesis. The paper focuses on the period following the First Oil Crisis of 1973, and takes into consideration the various shifts in the Japanese economy by dividing the data into four sub-periods. As opposed to most of the previous studies, we also employ tests that allow for heteroskedasticity in order to obtain more robust results.
The paper is structured as follows. In Section 2 we provide an overview of some of the most notable events affecting the Japanese economy in the period under consideration. Section 3 will discuss the theoretical concepts most relevant to our research, namely The Efficient Market Hypothesis, The Martingale and The Random Walk Hypothesis. Then, Section 4 will summarize the existing empirical literature internationally and with focus on the Japanese market. In Section 5 we introduce our sample data and describe some of its characteristics. Section 6 contains a description of the two methods we employ to test for the presence of a random walk: a Modified Box-Pierce test and a Variance Ratio test. The results are presented and interpreted in Section 7, while Section 8 summarizes the paper and presents the answer to our research question.
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2. Economic History
After the post-WWII recovery of 1945-1949, the Japanese economy entered an era of high and sustained growth and international re-integration. Between 1950 and 1973 the average annual growth of the GDP was approximately 9.5%, greater than in any subsequent period (Angus, 2003). During this period Japan also joined the IMF, World Bank, GATT, the United Nations and OECD, and entered a phase of trade liberalization (Ohno, 2006). The Japanese economy, which at the time depended greatly on foreign oil, was severely hit by the Oil Crisis of 1973, and by 1974 the growth rate of the GDP turned to a negative 1.23% (Angus, 2003). The country had entered a period of recession combined with inflation.
Starting with the year 1975, Japan began recovering and until 1985 it had reached an average annual GDP growth rate of 3.75%. Although present, the Second Oil Crisis of 1979-1980 had a milder impact on the economy. At the same time, the late 1970s marked the start of a period of financial liberalization. According to the World Federation of Exchanges, the domestic market capitalization of Japan rose from approximately 150 billion USD in 1975 to almost 650 billion USD in 1984.
Stock and land prices continued to grow from the early to the late 1980s at an impressive rate, causing the rise of an asset price bubble. The bubble reached its peak in 1989, with the Japanese stock market capitalization at a record 4 trillion USD (Stone & Ziemba, 1993), while its subsequent burst brought a long lived recession. Two possible causes for the appearance of the bubble have been proposed, namely bank deregulation and monetary expansion due to low short-term interest rates (Ohno, 2006).
The period following the burst of the asset price bubble is often referred to as the Lost Decade. During this period the stock market contracted, annual GDP growth slowed down to an average of 1.42%. The economy seemed to recover a couple of times during the period, but each time the recovery was short-lived. Stock and land priced declined drastically, and by the end of 1997 fears of a potential banking crisis had materialized. In an effort to stimulate the economy, in 1999 the Bank of Japan adopted the drastic measure of lowering short-term interest rates to zero (Ohno, 2006).
The subsequent period has been characterized by an even lower annual growth of the GDP, averaging only 0.81% for the years 2001-2013 and a troubled economy. In 2006, the Bank of Japan finally put an end to the zero interest rate policy by rising its lending rate to 0.25 (Obstfeld, 2009). When the global financial crisis of 2007-2008 hit, Japan’s exposure to subprime mortgages was limited, so the economy was not directly affected by the collapse. However, by the end of 2008 foreign demand declined and the yen appreciated, hurting the country’s exports (Iwaisako, 2010).
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3. Theoretical Concepts
3.1 The Efficient Market Hypothesis
As Fama (1965) mentions, the discussion of the random walk theory usually starts from the premise of market efficiency. The Efficient Market Hypothesis states that the interactions of a large number of rational, profit-maximizing investors give rise to markets where at any point in time prices correctly reflect all available relevant information. Fama argues that these prices are, in fact, good estimates of intrinsic value. Although deviations from or changes in the intrinsic value do occur, they have to be random, otherwise rational investors would be able to predict the path of the stock price and make a profit out of it, instantly eliminating the mispricing (Fama, 1965).
The consequences of the hypothesis are, therefore, that past stock prices are uninformative of future performance and that investors cannot persistently generate abnormal returns. This, in turn, suggests that in order to evaluate whether a market is efficient, a model for normal returns has to be specified. For this reason, any test of market efficiency suffers from the joint hypothesis problem, and in case of a rejection of the hypothesis it is unclear whether the market is truly inefficient or the wrong asset pricing model has been used.
In the next pages we will focus on the issue of the predictability of asset prices, and we begin by discussing the simplest model of asset pricing – The Martingale.
3.2 The Martingale Model
The Martingale Model can be traced back to Georlamo Cardono’s 1565’s “Book on Games of Chance”, one of the first works to discuss probability and game theory (Campbell, Lo, MacKinlay & Whitelaw, 1998). In his manuscript, Cardono touches on the subject of fair games, which, as Campbell et al. argue, lays the foundation for the martingale model.
A martingale is defined as a stochastic process where the expectation of a variable at time t+1, conditional on all its previous values, simply equals the value at time t. Formally, the model can be written as : . For financial markets, this implies
that changes in asset prices, conditional on the price history, should have an expected value of 0. Because price changes are all uncorrelated, similar to the Efficient Market Hypothesis, the martingale theory calls attention to the impossibility of predicting price movements from historical information alone. (Campbell et al., 1998).
One possible issue with the martingale is the fact that the model, as presented, does not account for risk in asset prices. When compensation for risk is taken into consideration, the expected change in the price of a stock can be thought of being positive. In fact, LeRoy (1973) demonstrates that under risk-aversion, the exact martingale property does not hold at all. Nonetheless, the martingale model sets the basis for a better understanding of the random walk model presented in the next section.
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3.3 The Random Walk Hypothesis
The first remarks on random walk behavior can be traced back to the French mathematician Louis Bachelier, who in 1900 published his PhD thesis on the topic “The Theory of Speculation”. In the introduction, Bachelier observes that although market prices are formed based on past, present and discounted future events, these events seem to have no significant relation to future price changes (Bachelier, 1900).
Building on the martingale model, the random walk model for stock prices usually takes the following form: , where represents the natural logarithm of the price at time t, is the expected log price change, or trend, and is the error term or white noise. According to Fama (1965) two aspects are central to the random walk model: the independence and probability distribution of the price changes.
Based on the different assumptions that can be made about the increments, Campbell et al. (1998) differentiate between three types of random walk models. The first version, abbreviated RW1, requires the error terms to be independent and identically distributed (IID) with mean 0 and variance . Commonly used tests that are compatible with this model are tests for the frequency of sequences and reversals, and runs tests.
Although RW1 is the simplest model of The Random Walk Hypothesis, it is also the most restrictive. The identical distribution assumption is particularly problematic, as it has been shown that stock prices tend to exhibit heteroskedasticity (French, Schwert & Stambaugh, 1987). For this reason, the second version of the model relaxes this assumption while maintaining the independence assumption of RW1, as such allowing for unconditional heteroskedasticity. At the same time, RW2 poses some difficulties that are not encountered in RW1. As Campbell et al. (1998) mention, without the identical distribution assumption, tests of independence become increasingly difficult. Nonetheless two types of tests can still be performed: technical analysis and filter rules.
The third and weakest form of The Random Walk Hypothesis (RW3) does not require independence, but only uncorrelated increments, and accounts for most of the research on the random walk behavior of stock prices (Campbell et al., 1988). This version allows for both conditional and unconditional forms of heteroskedasticity. Usual approaches in testing the hypothesis involve tests on the autocorrelation coefficients and variance ratio tests.
With respect to the probability distribution of the log price changes, the theory of random walks requires the price changes to follow a certain probability distribution, but it does not specify a required form or shape for this distribution (Fama, 1965). A number of alternatives have been proposed over time. Osborne (1959), for example, suggested that changes should simply follow a normal Gaussian distribution, yet later studies found that the data did not seem to support Osborne’s hypothesis (Fama, 1965; Praetz, 1972). Some researchers noted that leptokurtosis (resulting in a more peaked and heavy-tailed distribution) seems to be a common phenomenon in the empirical distributions of price changes (Moore, 1962; Cootner, 1962). A more extreme approach was suggested by Mandelbrot (1963), who advocated that price changes should rather follow a stable
5 Paretian distribution with infinite variance. His tests on cotton prices seemed to confirm the hypothesis and Fama’s (1965) own studies indicated that the stable Paretian distribution might indeed be a better alternative to the normal distribution. Nonetheless, Fama (1965) and Campbell et al. (1998) argue that assuming such a distribution would make the use of common statistical techniques more difficult, especially since most financial models require finite variances.
4. Empirical evidence
4.1 International Tests of Random Walk
As Cooper (1982) points out, studies performed on the United States and United Kingdom markets were, until that date at least, generally in support of The Random Walk Hypothesis, while evidence from less developed markets supported the hypothesis to a lesser extent.
Kendall (1953) was among the first ones to test The Random Walk Hypothesis on asset prices over long periods of time. In his study, he used 22 series of weekly data: 19 British industrial share prices from 1928 to 1938, two series related to spot prices of wheat on the Chicago exchange market, and a series composed of spot prices of cotton listed on the New York exchange. For each of these indices Kendall computed the correlation coefficients. Overall, he found little serial correlation in the series, which would suggest that the prices he analyzed did exhibit a random walk behavior.
Another well documented study was Fama’s “Behavior of Stock Market Prices” (1965) on the 30 individual stocks of the Dow Jones Industrial Average index. Fama used daily prices from 1957 to 1962 and performed three types of random walk tests: a serial correlation test, a runs test and a test based on filter techniques. With minor exceptions, the three tests did not find evidence against the The Random Walk Hypothesis.
Yet another prominent study was Lo and MacKinlay’s “Stock Market Prices Do Not Follow Random Walks: Evidence from a Simple Specification Test” (1988). The authors developed a variance ratio test that is robust to heteroskedasticity and non-normality and applied it to observations from the NYSE-Amex index for the years 1962 to 1985. They tested The Random Walk Hypothesis on equal-weighted and value-weighted NYSE-Amex indices, five size-based portfolios and individual securities. The results rejected The Random Walk Hypothesis for the indices and the portfolios. The individual securities, however, seemed to be slightly negatively correlated, but not at a significant level. Lo and MacKinlay argued that this might be because of the firm specific risk that would otherwise disappear when securities are combined into portfolios.
Although the above mentioned papers are of great importance to the theory of random walks, our focus in this papers is going to be the study of the Japanese market. Therefore, the next section will discuss the empirical results of random walk tests applied to this specific market.
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4.2 Tests of Random Walk on Japan
Compared to tests on The United States or Europe, the empirical literature of random walk behavior on the Japanese market is not as extensive. Although most of the studies yielded results in favor of the hypothesis, the difference in the types of tests, period and frequency of observation impede us from drawing a definitive conclusion. Some of the results available in the literature have been discussed below.
Ang and Pohlman (1978) were among the first to study the behavior of Japanese markets from a random walk perspective. Their research was conducted on weekly data of individual stocks from 1970 to 1974, and based on the computed correlation coefficients they concluded that the Japanese market did not exhibit significant deviations from the random walk model. At the same time, Ang and Pohlman acknowledge the limitations of their study due to the restricted number of stocks and observations.
Similar results were obtained by Hong (1978) and Cooper (1982). Hong (1978) used daily and weekly data of the Nikkei Dow Jones Stock Average index from 1973 to 1976 and performed serial correlation and runs tests, while Cooper (1982) added a spectral analysis to the two methods already mentioned. Although the time span of the observations is not explicitly mentioned, Cooper’s (1982) analysis seems to refer to stock prices observed during the 60s and 70s.
Huang’s (1995) analysis uses daily data from the period 1988-1992 and, as opposed to previous studies, employs variance ratio and Dickey-Fuller tests. Chang et al. (2004) also apply multivariate variance ratio tests, this time to daily prices on two sub-periods: 1998-2000 and 1998-2000-2002. Their results are similar in that neither of them is able to reject The Random Walk Hypothesis. Ko and Lee’s (1991) contribution, on the other hand, contradicts earlier studies. The results of their serial correlation, runs and Markov processes tests on the Tokyo Stock Exchange price index for four two-year sub-periods between 1981 and 1988 strongly rejected The Random Walk Hypothesis.
Worthington and Higg’s study (2005) is particularly interesting due to its mixed results. The prices of the value-weighted Japanese market index between 1987 and 2003 seem to depart significantly from the random walk model when serial correlation and unit root tests are performed. When the multiple variance ration test is employed, however, The Random Walk Hypothesis is only rejected for the case where homoskedasticity is assumed, but not for the case of heteroskedasticity. Here it should be noted that the according to Campbell et al. (1998), despite the common null hypothesis they share with random walk tests, Dickey-Fuller and other unit root tests are not designed to assess the predictability of asset returns. Thus, the conclusions derived from performing these tests might be problematic.
Overall, we can see that the evidence of random walk behavior on the Japanese market is mixed. Apart from the differences in the tests employed, it should be noted that the choice of sample period greatly influences the results of the previously mentioned studies.
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5. Data and Properties
5.1 The Series
In order to obtain an accurate representation of the Japanese stock market, we have decided to base our analysis on indices containing the largest, most liquid stocks traded on Tokyo Stock Exchange (TSE). For this reason, out of the three sections of TSE: First Section- including only blue chip stocks, Second Section- including smaller companies, and Mothers- including high growth and emerging stocks, this paper is focused on stocks listed on the First Section only . Due to the different methodologies employed by each index, we have chosen to rely on both Nikkei Stock Average (Nikkei or Nikkei225) and Tokyo Stock Price Index (TOPIX), the two most commonly cited indices in Japan. Apart from the broad market indices, we also examine the 33 TOPIX sector indices, which jointly amount for the entire First Section of the TSE and might give a good understanding of stock behavior in specific industries. A more detailed description of the indices is presented in Section 5.2.
Our analysis is, therefore, based on daily closing prices for the following 35 series: the Nikkei225 and TOPIX indices, with data covering the period from 1 January 1975 to 30 May 2014, and the 33 TOPIX sector indices with data ranging from 4 January 1983 to 30 May 2014. The choice of using high frequency data is motivated by Shiller and Perron’s study (1985), which concludes that when testing for the presence of a random walk, the power of the test approaches one only when both the span and the frequency of observations are increased. It should however be acknowledged that the use of closing daily prices might induce biases due to the non-trading days and the bid-ask spread (Lo & MacKinlay, 1988).
The data is divided into four sub-periods:
(1) before the start of the Japanese Asset Price Bubble (1 January 1975-31 December 1984)
(2) during the bubble (1 January 1985- 31 December 1990) (3) during The Lost Decade (1 January 1991- 29 December 2000) (4) after the Lost Decade (1 January 2001 -30 May 2014)
In the case of the 33 TOPIX sector indices, the pre-bubble period will only contain data from 4 January 1983 to 31 December 1984. We intend to perform tests of Random Walk behavior on both the individual sub-periods, as well as on the entire sample.
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5.2 The Indices
Nikkei225 is a price-weighted index composed of the 225 most prominent stocks listed on the First Section of the Tokyo Stock Exchange (TSE). The index was provided by TSE from 1950 up to 1970, after which the business newspaper Nihon Keizai Shimbun took over computing and publishing rights.
According to the Nikkei Stock Average Index Guidebook (2011), constituents are chosen based on two criteria: market liquidity and sector balance. The first refers to the practice of including only the most liquid stocks available on the market, as defined by their trading volume and the magnitude of price fluctuations by trading volume, while the latter corresponds to trying to maintain an accurate representation of the market in terms of the number of constituents per sector. The index composition is subject to an annual Periodic Review, but delistings due to events such as the bankruptcy of a constituent company are handled when needed through Extraordinary Replacement. The index is then computed using the Dow method by summing the prices of its component stocks, adjusted by presumed par value, and then dividing this sum by the a denominator called the Divisor.
TOPIX, on the other hand, is a market capitalization-weighted index computed and published by TSE ever since 1969, and it covers all domestic common stocks on the First Section of the exchange. For this reason the composition of the index is not subject to review, and companies are added or deleted only with new listings or delistings on the First Section. As stated in the Tokyo Stock Exchange Index Guidebook (2014), the index is computed by dividing the current free-float adjusted market value by the base market value and then multiplying the result by the base point (100 for TOPIX and 1,000 for TOPIX Sector Indices).
Apart from TOPIX, TSE is also computing a series of sector and size-based sub- indices which jointly amount for the entire First Section. TOPIX Sector Indices consists of 33 different categories, each representing a certain industry as defined by the Securities Identification Code Committee. The Size-based Sub-Indices Core30, Large70, Mid400 and Small represent categories of stocks sorted by their market capitalization.
Although both TOPIX and Nikkei are widely used as a benchmark of the overall Japanese market performance, their distinctive methodologies might give rise to slight variations in the results of our analysis. It is, therefore, useful to describe and compare some of the properties of the two. The sector weights of each index, as well as their exposure to various sizes of companies for the year 2014 are summarized below. The results have been obtained by cross-referencing the companies listed on the First Section of TSE, together with their classification, with the component weights announced by TSE and Nikkei. To obtain an approximation of the sector weights of TOPIX for each of the four periods, we regressed its continuously-compounded returns on the same type of returns of each of the 33 TOPIX Sector series. The results are presented in Figure 5.2.1. The 33 industry classes listed by the TSE have been grouped in six sector classes following the Nikkei methodology. The complete list of industries can be found in Table 1 of the Appendix.
9 From Figure 5.2.1 we can see significant differences in the sector weights of the two indices. While TOPIX is by construction fairly balanced in terms of exposure to various sectors, with weights ranging from 7.04% to 24.81%, Nikkei appears to be overexposed to Technology and Consumer Goods and underexposed to the other classes. The distribution of its weights is also more spread, with values ranging from 3.28% to 38.4%. These differences should not come as a surprise. Even though one of Nikkei’s criteria for selecting component stocks is linked to maintaining a certain sector balance similar to that of TSE, the policy refers to the number of companies included from each industry, and has no implication for the corresponding weights.
With respect to the composition by company size, the two indices seem to have similar exposures. As expected, Nikkei is less influenced by small companies and more exposed to medium or large companies. However, as it can be seen from Figure 5.2.2. , its exposure to the largest 30 stocks on the TSE is lower than the one of TOPIX. This might be due to Nikkei’s price-weighted policy, where even though only the most liquid stocks are included in the index, greater weight is attached to the highest priced stocks.
Figure 5.2.3 suggests that TOPIX’s exposure to the six sectors varies over time. Generally, the index seems to be the least impacted by the Transportation and Utilities sector, with weights 8.39% and 11.08%, and more sensitive to Finance and Technology. The highest exposures are of 32.4% to the financial sector for the second period, and of 28.31% and 26.12% to the technology category for periods four and one, respectively.
38.40% 19.40% 19.40% 16.01% 3.52% 3.28% 24.81% 23.67% 10.84% 19.67% 13.97% 7.04% Technology Capital goods Consumer Goods Materials Finance Transporation & Utilities
Figure 5.2.1 Composition of the Japanese Stock Market Indices TOPIX and Nikkei by Sector
TOPIX Nikkei
Results obtained by matching the officially announced component weights of all individual stocks included in TOPIX and Nikkei with their respective classifications as defined by TSE.
Source: Listed Company Search - Tokyo Stock Exchange, Inc.
TOPIX Component Stocks Weight and Free Float Weight - Tokyo Stock Echange, Inc. Nikkei Stock Average Weight - Nikkei, Inc.
10 32.10% 36.46% 30.33% 1.07% 35.66% 25.64% 29.56% 9.14% Core30 Large70 Mid400 Small
Figure 5.2.2 Composition of the Japanese Stock Market Indices TOPIX and Nikkei by Company Size
TOPIX Nikkei
Results obtained by matching the officially announced component weights of all individual stocks included in TOPIX and Nikkei with their respective classifications as defined by TSE.
Source: Listed Company Search - Tokyo Stock Exchange, Inc.
TOPIX Component Stocks Weight and Free Float Weight - Tokyo Stock Echange, Inc. Nikkei Stock Average Weight - Nikkei, Inc
8.39% 22.50% 20.54% 5.92% 16.53% 26.12% 11.08% 32.40% 20.24% 7.48% 12.57% 16.22% 9.75% 22.82% 16.05% 16.79% 10.95% 23.64% 8.57% 15.24% 17.06% 10.59% 20.23% 28.31% Transportation & Utilities Finance Materials Consumer Goods Capital Goods Technology
Figure 5.2.3 Estimated Sector Exposure of the Japanese Marker Index TOPIX
Period 4 Period 3 Period 2 Period 1
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5.3 Descriptive Statistics
Overall, we have obtained approximately 9,720 observations for Nikkei225 and TOPIX, and approximately 7,730 observations for each of the 33 sector indices. The characteristics of the two broad indices are included in Table 5.3. The annual average return for the entire period is somewhat low for both indices, at only 3.51% for Nikkei and 3.81% for TOPIX. As expected, the first and the second sub-periods have the highest average return of the entire sample. They differ, however, in their other moments. The returns pertaining to the asset price bubble are more volatile, have a slightly more negative skew and possess an extremely high kurtosis, suggesting that some extreme values are present. After the burst of the bubble, the average return turns negative in the third sub-period and then rises slightly in the fourth. The standard deviations of these two last periods are the highest of the entire sample.
Table 5.3 Descriptive statistics for Nikkei and TOPIX Index No. obs . Average daily Average annual Min Max Std. Dev. Skewness Kurtosis Overall Nikkei 9719 0.01% 3.51% -16.14% 13.23% 1.31% -0.35 9.76 TOPIX 9718 0.02% 3.81% -15.81% 12.86% 1.17% -0.40 10.99 1975-1985 Nikkei 2478 0.04% 10.86% -4.53% 4.45% 0.66% -0.08 3.92 TOPIX 2473 0.05% 11.97% -4.45% 3.37% 0.56% -0.06 3.81 1985-1991 Nikkei 1478 0.05% 12.03% -16.14% 12.43% 1.25% -0.94 29.36 TOPIX 1478 0.04% 10.62% -15.81% 9.12% 1.20% -1.29 27.55 1991-2001 Nikkei 2462 -0.02% -5.11% -7.23% 7.66% 1.45% 0.15 2.46 TOPIX 2467 -0.01% -2.99% -6.32% 7.28% 1.22% 0.21 3.50 2001-2014 Nikkei 3301 0.00% 0.61% -12.11% 13.23% 1.58% -0.42 6.34 TOPIX 3300 0.00% -0.27% -10.01% 12.86% 1.43% -0.36 6.16
6. Methodology
As there is a consensus in the literature regarding the presence of heteroskedasticity in economic time series (Lo and MacKinlay, 1988), for the purpose of this paper we are going to focus on testing the third version of The Random Walk Hypothesis, as described by Campbell et al. (1997). The only requirement of the RW3 model is that the increments be uncorrelated at all leads and lags. Therefore, the most straightforward methods of testing the RW3 hypothesis involve analyzing the autocorrelation coefficients either individually or jointly. Because of their higher power, we chose to employ two widely used, heteroskedasticity-robust tests of random walk behavior: a modified version of the Box-Pierce test, and Lo and MacKinlay’s Variance Ratio test. While the first test is constructed on the sum of squared autocorrelations, the second one is based on linear combinations of the coefficients. The tests will be described in the following sections.
12 The Random Walk model we are considering is:
Here, is the natural logarithm of the price . The continuously compounded returns are computed using the following formula:
( ) (
)
One advantage of using this transformation instead of regular returns is time-additivity. Computations are simplified and time series properties are more easily derived due to the fact that the multi-period log return equals the sum of single-period log returns:
( ) ∑
A second advantage of using continuously compounded returns is that the method avoids violating the limited liability property of financial assets. When regular returns are used, and log price changes are normally distributed, then there is a positive probability that at some point , implying that an investor could lose more than his initial investment (Campbell et al. ,1997). If the natural logarithm of prices is used instead, limited liability is preserved, as the price can no longer be negative.
6.1 Modified Box-Pierce Test
To verify whether the increments of returns are serially correlated, we consider the following estimate for the -th order autocorrelation coefficient ̂ , where ̂ is the estimate of the -th order auto-covariance:
̂ ̂
̂ ∑
( ̂)( ̂)
∑ ( ̂)
The condition of the Random Walk Model that all autocorrelation coefficients be zero can be summarized by the null and alternative hypotheses:
13 In order to test the hypothesis that the first n autocorrelation coefficients are jointly equal to zero, a Portmanteau Box-Pierce test can be applied, with the following statistic:
∑ ̂
The issue with the Box-Pierce test (as well as with the improved Ljung-Box test) is that its asymptotic distribution was derived under the independence assumption. Because of this, when heteroskedasticity is present the test might yield biased results due to the underestimation of the standard errors (Hsieh, 1989). Over the time, two solutions have been proposed to circumvent this problem: adjusting the test statistic to allow for nonlinear dependence, or bootstrapping the test to estimate the appropriate critical values (Escanciano & Lobato, 2009). Because of its simplicity, we are going to focus on the first method. We chose to follow the method described in Escanciano and Lobato (2009) due to the similarities to Lo and MacKinlay’s (1988) approach.
Under the null hypothesis, the conditional heteroskedasticity robust statistic is computed as follows: ∑ ̂ ∑ ̂ ̂ ̂ ∑ ( ̂) ( ̂)
We have chosen to apply the Modified Box-Pierce test for values of n of 1, 2, 5 and 20. That is because we wanted to observe the serial correlation between stock returns at very low lags- day by day and during the first two days, at medium lags- up to after one week, and at larger lags- up to one month.
6.2 Variance Ratio Test
The second test we are employing is the Variance Ratio (VR) test of Lo and MacKinlay (1988). The method is based on the property of uncorrelated variables given by the Bienaymé formula: the variance of a sum of uncorrelated variables is equal to the sum of the variances. It then follows that under homoskedasticity, the variance of the return over consecutive periods approaches multiplied by the variance of the return over one period, when the sample size is large enough. Therefore, testing for the presence of a random walk becomes in essence a test of whether (∑ )
( ) equals one. Of course,
as previously discussed, the homoskedasticity assumption for returns is generally not fulfilled in reality, and returns tend to follow a distribution that is more peaked and heavy tailed than the normal one. For this reason, Lo and MacKinlay designed a statistic that can
14 be applied even under some general forms of heteroskedasticity, and which does not require a Gaussian distribution of the increments, but only finite variances. The test is described below.
Lo and MacKinlay define the general -period variance ratio statistic as:
( ) ∑ ( )
( )
Under the assumption of heteroskedastic increments, Lo and MacKinley (1988) impose four conditions on the null hypothesis. The first one is that the increments should be uncorrelated. This is in essence the hypothesis we want to test. Conditions two and three provide a description of the maximum amount of dependence and heterogeneity acceptable such that the Law of Large Numbers and the Central Limit Theorem still hold to some degree. Condition four implies that the error terms’ sample autocorrelation coefficients are uncorrelated asymptotically.
The corresponding test statistic for overlapping -th increments is:
̅ ( ) ∑ ( ) ̂( ) ̂( ) ∑ ( ̂)( ̂) ∑ ( ̂)
Here, “ ” means asymptotically equal and ̂( ) is the -th order autocorrelation coefficient estimator. In order to standardize the test statistic, Lo and MacKinlay also computed the asymptotic variances of ̅ ( ) and ̂( ) as follows:
̂( ) ∑ [ ( )] ̂( ) ̂( ) ∑ ( ∑ ( ̂) ( ̂) ̂)
The obtained standardized test statistic ( ) is asymptotically standard normally distributed:
15 ( ) √ ̅ ( )
√ ̂( ) ( )
Because the Variance Ratio statistic tests only the first -1 lagged autocorrelations we have decided to apply the test to values of q equal to 2, 3, 6, and 21. In this manner both BP and VR would use the same lagged autocorrelation coefficients and in turn generate more comparable results. In fact, the tests are asymptotically equivalent for n=q-1=1, as they are both based on the first order autocorrelation coefficients.
7. Results
The results of the modified Box-Pierce and the Variance Ratio tests at lags n and q-1 up to 1, 2, 5, and 20 for Nikkei 225 and TOPIX are summarized below, in Table 7.1. The complete set of results, including the test statistics for the 33 TOPIX sector indices can be found in Tables 1-6 of the Appendix. The asterisks *, ** and *** represent test statistics that are significant at 10%, 5% and 1%, respectively. The Appendix results can also be visualized in Figure 7.2, where the color of the cells is increasing in brightness for decreasing significance levels.
Table 7.1 Test statistics for Nikkei and TOPIX
Modified Box-Pierce Variance Ratio
n=1 n=2 n=5 n=20 q-1=1 q-1=2 q-1=5 q-1=20 Overall Nikkei 225 0.23 4.21 4.92 14.92 -0.48 -1.50 -1.61 -0.83 TOPIX 8.07** 11.59** 14.88 28.63* 2.84*** 0.84 -0.23 0.18 1975-1985 Nikkei 225 15.62*** 15.98*** 19.04** 31.38* 3.95*** 3.83*** 2.82*** 0.98 TOPIX 30.94*** 31.95*** 33.93*** 44.28*** 5.55*** 5.46*** 4.45*** 2.93*** 1985-1991 Nikkei 225 0.79 9.79* 14.22 28.59* 0.89 -0.09 -0.39 0.1 TOPIX 2.47 7.39 11.25 24.97 1.57 0.91 0.64 0.77 1991-2001 Nikkei 225 1.3 6.59 7.05 17.82 -1.15 -2.05** -1.7* -0.89 TOPIX 12.12*** 20.02*** 21.16** 35.87** 3.48*** 1.91* 0.55 0.17 2001-2014 Nikkei 225 1.46 1.52 2.97 9.83 -1.21 -1.17 -1.36 -0.86 TOPIX 0.1 0.16 2.59 9.32 0.32 0.17 -0.60 -0.69
16 2 .9 2 * * *significant at 1% 2 .3 8 * * significant at 5% 1 .7 9 * significant at 10% Nikkei 0.23 4.21 4.92 14.92 -0.48 -1.5 -1.61 -0.8315.62*** 15.98***19.04**31.38* 3.95*** 3.83*** 2.82*** 0.98 0.799.79* 14.2228.59* 0.89 -0.09 -0.39 0.1 1.3 6.59 7.05 17.82 -1.15-2.05** -1.7* -0.89 1.46 1.52 2.97 9.83 -1.21 -1.17 -1.36 -0.86 TOPIX 8.07** 11.59** 14.8828.63* 2.84*** 0.84 -0.23 0.1830.94*** 31.95*** 33.93*** 44.28*** 5.55*** 5.46*** 4.45*** 2.93*** 2.47 7.39 11.25 24.97 1.57 0.91 0.64 0.7712.12*** 20.02***21.16** 35.87**3.48***1.91* 0.55 0.17 0.1 0.16 2.59 9.32 0.32 0.17 -0.6 -0.69 Banks 36.07***36.33***45.52***65.47***6.01*** 3.49*** 1.31 1.1210.84***11.51**11.74 23.573.25*** 3.26*** 2.79*** 2.76*** 14.6***14.62** 18.34** 36.77**3.79*** 3.53*** 3.45*** 3.85*** 19*** 23.39***27.94***44.85***4.36*** 3.08*** 1.18 0.37 5.64* 6.05 13.48 24.12 2.37** 2.41** 0.66 -0.85 Insurance 7.93** 9.01 16.28*34.93**2.82*** 0.83 -0.62 -0.99 8.44** 8.48 10.2636.79**2.9***2.46** 2.31** 1.1115.32***16.15***18.32**39.33***3.91*** 3.3*** 2.7*** 0.96 7.76**15.96***18.02*32.73**2.78*** 1.33 -0.01 0.01 0.03 0.03 7.14 19.49 -0.17 -0.14 -1.25 -1.8* Other Financing Business26.31***26.74***29.35***42.59***5.12*** 3.05*** 1.32 1.24 5.12* 5.28 6.84 18.46 2.22** 2.2** 2.04** 0.82 3.73 3.75 5.19 14.4 1.93* 1.84* 1.96** 2.43**22.9***24.87***25.58***37.26**4.78*** 3.74***2.44** 1.76*10.04***10.12* 14.35 24.693.16*** 2.68*** 1.14 0.01 Sec. & Comm.Futures 61.64***62.98***67.49***81.06***7.85*** 4.35*** 2.72*** 2.59***5.53* 5.64 6.52 21.19 2.34** 1.97** 1.38 0.6 27.86***29.21***30.31***40.98***5.25*** 4.31*** 3.5*** 3.02***44.19***48.87***52.24***64.5*** 6.64*** 5.2*** 3.08***2.19** 7.34** 7.53 11.78 19.65 2.7*** 2.6*** 0.96 0.32 Elecrtric Appliances 15.46***18.36***26.05***34.91**3.93*** 1.06 -0.44 -0.4 8.93** 9.26* 10.06 31.14*2.98*** 2.93*** 2** -0.17 1.65 5.72 11.63 19.54 1.28 0.5 -0.41 -0.4534.59***36.88***41.42***53.48***5.87*** 4.59***2.21** 0.65 1.22 1.47 3.89 11.95 1.10 0.70 -0.37 -0.58 Info. & Communication 7.18** 12.56**13.8532.29**2.67*** 0.51 -0.1 0.09 0.73 0.83 1.94 11.83 0.81 0.85 1.17 0.74 1.16 5.61 8.06 26.11 1.07 0.12 0.35 1.15 8.61** 10.68* 15.1238.19***2.93***2.12** 0.57 -0.22 0 1.87 5.01 21.93 -0.03 -0.61 -1.46 -1.72* Pharmaceutical 0.77 2.58 4.11 25.28 0.88 -0.03 -0.51 -0.7 1.39 8.49 14 43.35***1.17 2.02** 2.12** 1.61 5.79* 7.1 12.2934.71** 2.4** 1.88* 1 -0.01 2.25 3.18 6.05 13.61 1.5 0.95 1.15 0.33 2.82 3.84 4.87 19.64 -1.69*-1.97** -2.02**-1.69* Precision Instruments 2.84 7.06 11.05 21.87 1.68* -0.54 -1.16 -0.49 3.81 3.97 5.35 16.71 1.92* 1.92* 1.62 0.25 1.26 3.77 7.04 16.2 1.12 0.59 0.13 0.09 7.36** 12.11**16.63*36.73**2.71*** 1.47 -0.24 -0.27 0.05 0.75 3.17 9.97 -0.23 -0.61 -1.35 -0.61 Fish., Agr. & Forest. 1.9 6.8 11.46 28.6* 1.37 -0.03 -0.71 -0.82 0.36 0.36 6.32 20.4 0.47 0.27 -0.61 -1.17 4.21 8.81 13.08 27.53 2.05** 1.15 0.48 -0.02 0.18 2.96 7.44 17.81 -0.42 -1.12 -1.29 -1.01 0.15 0.23 6.23 16.14 -0.41 -0.28 -0.41 -0.80 Foods 0.17 3.3 5.68 13.98 -0.42 -0.97 -1.07 -0.38 2.57 2.71 5.77 29.73* 1.47 1.15 0.63 -0.86 3.06 6.14 7.93 24.69 1.75* 1.09 0.72 1.14 3.48 4.12 6.45 15.92 -1.87*-2.04** -1.37 -0.58 2.93 3.32 7.14 13.59 -1.73* -1.83* -1.73* -1.33 Retail Trade 17.13***19.03***21.29**40.18***4.14***2.01** 1.45 1.95* 8.14** 8.52 10.99 24.842.84*** 2.79*** 2.85***2.09**20.88***23.54***25.46***48.14***4.56*** 3.59*** 2.59***2.31** 5.44* 6.5 11.4337.67***2.33** 1.68* 0.67 1.22 1.76 1.9 3.66 15.89 1.32 1.05 0.25 0.18 Service 43.34***43.47***47.17***78.04***6.58*** 4.17*** 2.76*** 2.95*** 3.25 3.26 5.63 16.03 1.74* 1.54 1.49 0.98 6.88** 7.75 10.0638.57***2.62***2.05** 1.55 2.42**50.99***51.03***54.41***106.91***7.13*** 6.21*** 3.8***2.12** 2.45 2.46 6.99 17.06 1.56 1.46 0.45 0.34 Machinery 19.75***20.94***21.04** 35.02**4.44***2.41** 1.13 1.1 27.29***27.5***30.85***41.39***5.22*** 4.82*** 4.49*** 3.34*** 4.86* 6.02 11.9138.64***2.2** 1.73* 1.37 0.94 24.56***25.45***28.03***41.89***4.95*** 4.09*** 3.67*** 3.06*** 4.3 4.68 5.6 16.2 2.07** 1.45 0.42 -0.31 Construction 13.32***14.49**15.34 29.25*3.65***2.29** 1.45 1.31 4.83* 4.9 5.62 30.63* 2.18** 2.06** 1.72* 0.18 1.91 3.56 5.54 22.19 1.38 0.85 0.58 0.18 31*** 31.48***33.13***42.65***5.57*** 4.81*** 4.11*** 3.51*** 0.44 0.54 1.84 9.32 0.65 0.42 -0.22 -0.74 Real Estate 9.59***20.78***25.78***42.71***3.09*** 0.77 -0.49 -0.47 0.99 1.68 5.71 17.52 0.97 0.56 1.1 0.66 0.25 1.14 2.4 16.45 0.5 0.29 0.18 -0.4516.15***31.55***36.11***45.57***4.02***2.01** -0.09 -0.56 6.69** 8.61 12.8 22.232.58***1.69* 0.38 -0.10 Transportation Equip. 1.33 8.14 14.14 29.96* 1.15 -0.78 -1.73* -1.31 0.41 0.46 2.92 7.73 0.63 0.65 0.62 -0.66 0.14 7.38 14.11 29.37* 0.38 -0.35 -0.58 -0.3 0.34 9.55* 11.85 28.81* 0.58 -0.77 -1.54 -1.42 0.85 1.54 4.16 10.03 0.92 0.44 -0.53 -0.82 Other Products 7.18** 8.74 15.1241.35***2.68*** 0.39 -0.84 -0.14 2.84 3.76 4.59 27.95 1.65* 1.86* 1.59 0.28 2.58 3.3 7.83 28.08 1.6 1.22 0.19 0.24 3.42 5.19 7.46 23.32 1.85* 1.05 -0.16 -0.19 2.08 2.51 6.39 23.34 1.44 0.96 -0.23 -0.37 Air Transport 0.37 1.71 9.77 25.03 0.6 0.07 -0.76 0.07 1.41 2 3.87 20.23 -1.2 -0.83 -0.78 -0.76 1.23 2.33 3.49 15.27 1.11 0.66 0.39 0.4 0.37 0.51 6.84 14.34 -0.62 -0.71 -0.57 -0.14 0.41 1.31 10.74 27.53 0.64 0.28 -0.38 -0.39 Land Transport 5.69* 7.19 9.3945.48*** 1.41 0.66 1.32 8.15** 9.27* 13.09 22.552.83*** 2.99*** 3.1***2.39**15.31***15.83***17.49*42.46*** 3.9*** 3.28*** 2.9*** 2.48** 0.42 5.19 5.68 19.54 -0.65 -1.52 -1.47 -0.22 1.06 1.1 10.4 23.08 -1.03 -1.03 -1.55 -1.48 Marine Transport 8.58** 10.42* 11.89 20.93 0.87 0.01 0.98 3.79 17.28 -0.23 0.14 -0.13 -0.07 1.65 2.03 2.87 29.59* 1.27 0.95 0.51 0.54 4.53 8.27 10.37 24.72 2.11** 1.18 0.88 0.93 3.77 4.17 6.19 17.75 1.94* 1.41 0.20 0.29 Whsg. & Harbor Trsp. Srv.9.71***12.42**15.3132.86**3.12*** 1.06 1.32 2.28 2.7 5.83 16.1 1.5 1.16 1.53 0.58 19.3***19.62***21.61**46.89***4.39*** 3.68*** 3.28***2.48** 1.21 4.28 6.69 16.88 1.1 0.31 0.15 0.22 0.14 0.53 6.26 21.53 0.34 0.04 -0.52 -0.51 Electric Power & Gas 2.92 3.87 6.15 16.33 1.71* 1.3 0.62 0.44 2.6 2.82 5.28 20.22 1.59 1.02 1.07 0.85 1.73 1.73 3.03 14.45 1.3 1.2 1.22 0.67 0.24 11.31**13.7443.83***0.46 -0.73 -1.09 -1.7* 0.56 0.59 3.67 18.7 0.75 0.76 0.73 0.26 Iron & Steel 5.68* 6.79 9.64 20.76 2.38** 0.69 -0.02 0.28 0.36 4.35 7.8 25.11 0.59 -0.28 0.14 -0.36 0.2 4.38 7.93 28.67* 0.45 -0.14 -0.21 0.01 19.57***21.62***24.09***46.6*** 4.41*** 3.47***1.78* 1.41 0.53 0.55 4.49 10.38 0.73 0.71 -0.16 -0.35 Metal Products 0.9 2.87 3.5 22.59 0.95 -0.17 -0.22 0.3911.89***12.37**15.64 22.383.44*** 3.37***2.25** 0.67 5.04* 6.57 7.69 24.22 2.24** 1.49 0.95 1.29 3.82 5.1 5.61 18.96 1.95* 1.27 0.79 1.28 1.19 1.72 2.67 11.7 -1.10 -1.31 -1.36 -1.03 Non-Ferrous Metals 8.42** 9.25* 10.93 19.87 2.9*** 1.03 0.03 0.2 4.86* 6.31 7.57 40.38***-2.21** -1.48 -0.62 -0.11 1.9 6.76 9.97 25.11 1.37 0.52 0.03 -0.2710.33***11.94**12.1432.52**3.21***2.36** 1.55 1.09 3.05 3.05 5.62 12.79 1.74* 1.51 0.31 -0.15 Chemical 5.04* 9.1 10.2 18.73 2.24** 0.47 -0.2 0.15 8.72** 8.73 9.54 20.492.92*** 2.6*** 2.26** 0.22 5.05* 7.52 11.09 31.39*2.25** 1.64 1.02 0.81 7.43** 12.52**14.35 20.052.72*** 1.44 1.16 1.03 0.09 0.57 2.63 8.99 -0.31 -0.60 -1.19 -0.90 Glass & Ceramics Prds. 2.89 7.23 7.46 19.13 1.69* -0.06 -0.62 0.04 7.65** 7.65 13.64 29.73*2.68***2.35** 1.56 -0.19 0.21 5.93 8.34 17.59 0.45 -0.26 -0.42 -0.02 0 1.37 3.13 16.29 -0.04 -0.48 -0.09 0.3 3.17 4.32 6.9 15.58 1.78* 0.99 -0.16 -0.23 Pulp & Paper 6.52** 7.05 10.79 30.65*2.55** 1.16 0.16 0.36 0.51 0.84 3.21 21 0.66 0.86 0.65 0.1 2.93 2.93 6.69 29.08* 1.7* 1.61 0.67 0.52 9.06** 9.57* 12.3 28.083.01***2.42** 0.97 0.49 0.06 0.38 2.4 17.88 0.23 -0.03 -0.18 -0.52 Rubber Products 0.15 7.5 15.5134.93** 0.39 -2.05** -2.43** -1.8* 2.4 3.27 6.41 22.64 -1.58 -1.85* -1.89* -1.87* 1.12 3.6 5.66 17.19 1.06 0.37 -0.05 -0.24 0 3.84 7.99 28.41* -0.08 -0.72 -1.64 -0.98 0.03 1.55 7.6 27.6 0.16 -0.39 -1.78* -1.57 Textiles & Apparels 3.34 6.26 7.43 19.7 1.83* 0.43 -0.19 -0.05 4.19 4.25 4.42 18.95 2.02** 1.71* 1.24 -0.36 3.29 6.87 12.2436.87**1.81* 1.15 0.6 0.06 3.9 3.98 5.59 14.59 1.98** 1.66* 1.37 1.31 0.01 1.29 2.48 11.19 0.08 -0.43 -0.93 -0.92 Wholesale Trade 29.25***29.61***32.8*** 42.3*** 5.41*** 2.85*** 1.9* 1.62 4.18 4.28 5.05 14.44 2.04** 1.94* 1.32 0.19 3.31 6.41 7.23 29.26* 1.82* 1.19 0.59 0.27 31.26***31.52***33.71***62.05***5.59*** 5.29*** 3.75*** 2.87*** 3.56 3.9 5.48 16.43 1.88* 1.29 0.11 -0.21 Mining 1.51 5.62 6.87 24.34 1.23 -0.47 -0.63 0.3 0 0.32 0.61 17.69 0.04 0.28 0.48 1.14 1.42 3.77 5.26 21.48 1.19 0.39 -0.26 0.36 3.32 5.15 10.86 22.27 1.82* 1.04 1.25 1.88* 0 1.4 3 12.48 -0.05 -0.58 -1.25 -1.01 Oil & Coal Products 0.23 8.53 12.64 24.88 0.46 -1.87* -1.61 -0.69 0.57 2.25 2.62 8.81 -0.84 -1.58 -1.52 -1.01 0.69 1.62 5.48 14.17 0.82 0.46 -0.05 0.44 1.33 5.44 7.39 17.85 -1.17 -1.86* -1.86* -0.63 1.04 4.37 7.95 17.92 1.02 0.04 -1.18 -0.67
Figure 7.2 Test statistics for TOPIX, Nikkei and the 33 TOPIX Sector Indices
Variance Ratio Modified Box Pierce
Variance Ratio Modified Box Pierce
Modified Box Pierce Modified Box Pierce
Modified Box Pierce Variance Ratio Variance Ratio Variance Ratio
Period 4 Period 2 Period 3 B r o a d M k t . T r a n s p o r t a t io n & U t il it ie s M a t e r ia ls Overall Period 1 F in a n c e T e c h n o lo g y C o n s u m e r G o o d s C a p it a l G o o d s
17 When the entire period is considered, the null hypothesis is not rejected for the Nikkei 225 index. However, the negative VR statistic for q-1=2 suggests that consecutive returns are generally slightly negatively correlated. In the case of TOPIX, the model is rejected at a significance level of 5% only at lower lags. The index displaying positive first order serial correlation, as indicated by the significant VR statistic at q-1=2.
The BP test strongly rejects The Random Walk Hypothesis for the Finance Sector, except for the Insurance Industry, but when the VR test is employed, the rejection occurs mostly at lower values of q-1. This divergence might be the result of the latter attaching declining weights to higher autocorrelations, and therefore being less sensitive to those coefficients. Other industries for which the model is generally rejected are Electric Appliances, Information and Communication, Retail Trade, Wholesale Trade, Service, Machinery and Real Estate.
The pre-asset price bubble period strongly rejects The Random Walk Hypothesis for both Nikkei and TOPIX, due to positive serial correlation. Nikkei is, however, not rejected at higher orders of n and q-1. Surprisingly, although the broad market indices mostly reject the null, most of the sector indices are in support of The Random Walk Hypothesis. The Machinery Industry is the only series that rejects the null at 1% at all lags. The Finance Sector rejects the null more often when the VR test is employed, which might indicate serial correlation only at lower lags. Electric Appliances, Retail Trade, Land Transport, Metal and Chemical products all exhibit low order serial correlation, as suggested by the stronger rejections of the VR test.
The second sub-period, corresponding to the rise and burst of the asset price bubble, closely resembles a random walk. Only the BP test statistics of Nikkei for n=2 and n=20 are slightly significant at a level of 10%. This might suggest that although the period was characterized by a rapid growth and decline of prices, consecutive returns were generally not serially correlated. The same low rate of rejection applies for most of the Sector Indices. Again, we notice that the Finance Sector generally exhibits significant serial correlation. Retail Trade, Land Transport and Warehousing & Harbor industries are also rejected, although the same does not hold for the rest of their respective sectors.
The data belonging to the Lost Decade period exhibits a couple of departures from the random walk model, especially under the BP test and under the lower lags of the VR. Nikkei returns exhibit first order negative autocorrelation, but are generally in support of the random walk. TOPIX, however, displays a significant first order autocorrelation coefficient and rejects the null under the BP test at all lags. Once again, the Finance Sector exhibits significantly correlated returns at all lags. With the exception of Transportation & Utilities sectors, all other series persistently reject the null for at least some of their component industries.
For the last sub-period, neither Nikkei nor TOPIX are rejected. With the exception of Banks, Other Financing Business and Securities & Commodity Futures series which exhibit some serial correlation at lower lags, most industries are in favor of the Random Walk Model.
18 From Table 7.1 it can be noticed that The Random Walk Hypothesis seems to be rejected slightly more often for TOPIX than for Nikkei 225. For all sample periods, the VR test statistic of TOPIX at q-1=1 is positive, suggesting that the index returns are generally positively correlated. The same does not hold for Nikkei, which exhibits negatively correlated returns for the entire sample and for the last two sub-periods. The differences between Nikkei and TOPIX most probably arise from their different sector weights, as suggested in Section 5.2.
Overall, with the exception of Finance, most sectors are not necessarily consistent in their rejection of the random walk model. The clearest results are obtained for the last two sub-periods. The fact that the returns of the recent 13 years particularly support the random walk hypothesis might be due to either a good functioning of the market and higher liquidity, or the random nature of shocks that hit the economy in that particular period.
8. Conclusion
Although The Random Walk Hypothesis has received increased attention over the time, the model has not been often tested on the Japanese market, and the existing research did not reach a clear consensus. For this reason, in an attempt to provide a more extensive analysis we have chosen to analyze the random walk model during a 40 year long period and four sub-periods, using two broad market indices and 33 sector series, and by employing tests that are robust to heteroskedasticity.
As previously discussed in Section 3, the evolution of the Japanese economy after 1975 can be described as following four general trends. The first sub-period, from 1975 to 1985, has been characterized by high growth (although not as high as the pre-Oil Crisis period), and gradual financial liberalization. The second sub-period, from 1985 to 1991, is often associated with the rise and fall of the Japanese asset price bubble. In 1989, during its peak, the market capitalization reached 4 trillion USD, while stock and land prices were at a record high. The burst of the bubble was followed by a period of slow growth and banking crisis named the Lost Decade. The slow growth declined even further in the years between 2001 and 2014.
Due to the changing economic conditions, as well as our use of a large number of indices, the results of our analysis vary greatly. As explained in Section 7, the Random Walk Hypothesis is generally rejected for the Finance sector. As for the other sectors and industries, we have noticed that while the fourth sub-period is the most supportive of the random walk model, the third appears to be rejecting the null more often. The results for the first and second periods are not as clear, with industries such as Retail Trade, Machinery, Land Transport and Warehousing & Harbor Transportation often rejecting the null without coordination with their respective sectors.
19 The two broad market indices Nikkei and TOPIX have also yielded slightly different results, mainly due to their distinctive methodologies, as explained in Section 5.2. Although the two rejected the random walk model persistently during the first period under consideration and jointly supported the model for the second and fourth, TOPIX was rejected more often during the third period, as well as when the entire sample was considered.
20
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22
Appendix
Table 1. Sector
Sector Code Industry
Finance TOPIX 3 Banks
TOPIX 12 Insurance
TOPIX 20 Other Financial Business
TOPIX 28 Securities and Commodity Futures Technology TOPIX 7 Electric Appliances
TOPIX 23 Precision Instruments
TOPIX 11 Information & Communication TOPIX 22 Pharmaceutical
Materials TOPIX 13 Iron & Steel TOPIX 16 Metal Products TOPIX 18 Non-Ferrous Metal TOPIX 4 Chemical
TOPIX 10 Glass & Ceramics Products TOPIX 24 Pulp & Paper
TOPIX 27 Rubber Products TOPIX 30 Textiles & Apparels TOPIX 32 Wholesale Trade TOPIX 17 Mining
TOPIX 19 Oil & Coal Products Transporation
& Utilities
TOPIX 2 Air Transport TOPIX 14 Land Transport TOPIX 15 Marine Transport
TOPIX 33 Warehouseing & Harbor Transporation Services TOPIX 6 Electric Power&Gas
Consumer Goods TOPIX 8 Fishery, Agriculture & Forestry TOPIX 9 Foods
TOPIX 26 Retail Trade TOPIX 29 Services Capital goods TOPIX 1 Machinery
TOPIX 5 Construction TOPIX 21 Other Products TOPIX 25 Real Estate
23
Table 2. Test statistics for the entire sample period
Modified Box-Pierce Variance Ratio
n=1 n=2 n=5 n=20 q-1=1 q-1=2 q-1=5 q-1=20 NIKKEI 0.23 4.21 4.92 14.92 -0.48 -1.50 -1.61 -0.83 TOPIX 8.07** 11.59** 14.88 28.63* 2.84*** 0.84 -0.23 0.18 TOPIX1 19.75*** 20.94*** 21.04** 35.02** 4.44*** 2.41** 1.13 1.1 TOPIX2 0.37 1.71 9.77 25.03 0.6 0.07 -0.76 0.07 TOPIX3 36.07*** 36.33*** 45.52*** 65.47*** 6.01*** 3.49*** 1.31 1.12 TOPIX4 5.04* 9.1 10.2 18.73 2.24** 0.47 -0.2 0.15 TOPIX5 13.32*** 14.49** 15.34 29.25* 3.65*** 2.29** 1.45 1.31 TOPIX6 2.92 3.87 6.15 16.33 1.71* 1.3 0.62 0.44 TOPIX7 15.46*** 18.36*** 26.05*** 34.91** 3.93*** 1.06 -0.44 -0.4 TOPIX8 1.9 6.8 11.46 28.6* 1.37 -0.03 -0.71 -0.82 TOPIX9 0.17 3.3 5.68 13.98 -0.42 -0.97 -1.07 -0.38 TOPIX10 2.89 7.23 7.46 19.13 1.69* -0.06 -0.62 0.04 TOPIX11 7.18** 12.56** 13.85 32.29** 2.67*** 0.51 -0.1 0.09 TOPIX12 7.93** 9.01 16.28* 34.93** 2.82*** 0.83 -0.62 -0.99 TOPIX13 5.68* 6.79 9.64 20.76 2.38** 0.69 -0.02 0.28 TOPIX14 5.69* 7.19 9.39 45.48*** 2.38** 1.41 0.66 1.32 TOPIX15 8.58** 10.42* 11.89 20.93 2.92*** 0.96 0.52 0.87 TOPIX16 0.9 2.87 3.5 22.59 0.95 -0.17 -0.22 0.39 TOPIX17 1.51 5.62 6.87 24.34 1.23 -0.47 -0.63 0.3 TOPIX18 8.42** 9.25* 10.93 19.87 2.9*** 1.03 0.03 0.2 TOPIX19 0.23 8.53 12.64 24.88 0.46 -1.87* -1.61 -0.69 TOPIX20 26.31*** 26.74*** 29.35*** 42.59*** 5.12*** 3.05*** 1.32 1.24 TOPIX21 7.18** 8.74 15.12 41.35*** 2.68*** 0.39 -0.84 -0.14 TOPIX22 0.77 2.58 4.11 25.28 0.88 -0.03 -0.51 -0.7 TOPIX23 2.84 7.06 11.05 21.87 1.68* -0.54 -1.16 -0.49 TOPIX24 6.52** 7.05 10.79 30.65* 2.55** 1.16 0.16 0.36 TOPIX25 9.59*** 20.78*** 25.78*** 42.71*** 3.09*** 0.77 -0.49 -0.47 TOPIX26 17.13*** 19.03*** 21.29** 40.18*** 4.14*** 2.01** 1.45 1.95* TOPIX27 0.15 7.5 15.51 34.93** 0.39 -2.05** -2.43** -1.8* TOPIX28 61.64*** 62.98*** 67.49*** 81.06*** 7.85*** 4.35*** 2.72*** 2.59*** TOPIX29 43.34*** 43.47*** 47.17*** 78.04*** 6.58*** 4.17*** 2.76*** 2.95*** TOPIX30 3.34 6.26 7.43 19.7 1.83* 0.43 -0.19 -0.05 TOPIX31 1.33 8.14 14.14 29.96* 1.15 -0.78 -1.73* -1.31 TOPIX32 29.25*** 29.61*** 32.8*** 42.3*** 5.41*** 2.85*** 1.9* 1.62 TOPIX33 9.71*** 12.42** 15.31 32.86** 3.12*** 1.79* 1.06 1.32
24
Table 3. Test statistics for period -1985
Modified Box-Pierce Variance Ratio
n=1 n=2 n=5 n=20 q-1=1 q-1=2 q-1=5 q-1=20 NIKKEI 15.62*** 15.98*** 19.04** 31.38* 3.95*** 3.83*** 2.82*** 0.98 TOPIX 30.94*** 31.95*** 33.93*** 44.28*** 5.55*** 5.46*** 4.45*** 2.93*** TOPIX1 27.29*** 27.5*** 30.85*** 41.39*** 5.22*** 4.82*** 4.49*** 3.34*** TOPIX2 1.41 2 3.87 20.23 -1.2 -0.83 -0.78 -0.76 TOPIX3 10.84*** 11.51** 11.74 23.57 3.25*** 3.26*** 2.79*** 2.76*** TOPIX4 8.72** 8.73 9.54 20.49 2.92*** 2.6*** 2.26** 0.22 TOPIX5 4.83* 4.9 5.62 30.63* 2.18** 2.06** 1.72* 0.18 TOPIX6 2.6 2.82 5.28 20.22 1.59 1.02 1.07 0.85 TOPIX7 8.93** 9.26* 10.06 31.14* 2.98*** 2.93*** 2** -0.17 TOPIX8 0.36 0.36 6.32 20.4 0.47 0.27 -0.61 -1.17 TOPIX9 2.57 2.71 5.77 29.73* 1.47 1.15 0.63 -0.86 TOPIX10 7.65** 7.65 13.64 29.73* 2.68*** 2.35** 1.56 -0.19 TOPIX11 0.73 0.83 1.94 11.83 0.81 0.85 1.17 0.74 TOPIX12 8.44** 8.48 10.26 36.79** 2.9*** 2.46** 2.31** 1.11 TOPIX13 0.36 4.35 7.8 25.11 0.59 -0.28 0.14 -0.36 TOPIX14 8.15** 9.27* 13.09 22.55 2.83*** 2.99*** 3.1*** 2.39** TOPIX15 0.01 0.98 3.79 17.28 -0.23 0.14 -0.13 -0.07 TOPIX16 11.89*** 12.37** 15.64 22.38 3.44*** 3.37*** 2.25** 0.67 TOPIX17 0 0.32 0.61 17.69 0.04 0.28 0.48 1.14 TOPIX18 4.86* 6.31 7.57 40.38*** -2.21** -1.48 -0.62 -0.11 TOPIX19 0.57 2.25 2.62 8.81 -0.84 -1.58 -1.52 -1.01 TOPIX20 5.12* 5.28 6.84 18.46 2.22** 2.2** 2.04** 0.82 TOPIX21 2.84 3.76 4.59 27.95 1.65* 1.86* 1.59 0.28 TOPIX22 1.39 8.49 14 43.35*** 1.17 2.02** 2.12** 1.61 TOPIX23 3.81 3.97 5.35 16.71 1.92* 1.92* 1.62 0.25 TOPIX24 0.51 0.84 3.21 21 0.66 0.86 0.65 0.1 TOPIX25 0.99 1.68 5.71 17.52 0.97 0.56 1.1 0.66 TOPIX26 8.14** 8.52 10.99 24.84 2.84*** 2.79*** 2.85*** 2.09** TOPIX27 2.4 3.27 6.41 22.64 -1.58 -1.85* -1.89* -1.87* TOPIX28 5.53* 5.64 6.52 21.19 2.34** 1.97** 1.38 0.6 TOPIX29 3.25 3.26 5.63 16.03 1.74* 1.54 1.49 0.98 TOPIX30 4.19 4.25 4.42 18.95 2.02** 1.71* 1.24 -0.36 TOPIX31 0.41 0.46 2.92 7.73 0.63 0.65 0.62 -0.66 TOPIX32 4.18 4.28 5.05 14.44 2.04** 1.94* 1.32 0.19 TOPIX33 2.28 2.7 5.83 16.1 1.5 1.16 1.53 0.58
25
Table 4. Test statistics for period 1985-1991
Modified Box-Pierce Variance Ratio
n=1 n=2 n=5 n=20 q-1=1 q-1=2 q-1=5 q-1=20 NIKKEI 0.79 9.79* 14.22 28.59* 0.89 -0.09 -0.39 0.1 TOPIX 2.47 7.39 11.25 24.97 1.57 0.91 0.64 0.77 TOPIX1 4.86* 6.02 11.91 38.64*** 2.2** 1.73* 1.37 0.94 TOPIX2 1.23 2.33 3.49 15.27 1.11 0.66 0.39 0.4 TOPIX3 14.6*** 14.62** 18.34** 36.77** 3.79*** 3.53*** 3.45*** 3.85*** TOPIX4 5.05* 7.52 11.09 31.39* 2.25** 1.64 1.02 0.81 TOPIX5 1.91 3.56 5.54 22.19 1.38 0.85 0.58 0.18 TOPIX6 1.73 1.73 3.03 14.45 1.3 1.2 1.22 0.67 TOPIX7 1.65 5.72 11.63 19.54 1.28 0.5 -0.41 -0.45 TOPIX8 4.21 8.81 13.08 27.53 2.05** 1.15 0.48 -0.02 TOPIX9 3.06 6.14 7.93 24.69 1.75* 1.09 0.72 1.14 TOPIX10 0.21 5.93 8.34 17.59 0.45 -0.26 -0.42 -0.02 TOPIX11 1.16 5.61 8.06 26.11 1.07 0.12 0.35 1.15 TOPIX12 15.32*** 16.15*** 18.32** 39.33*** 3.91*** 3.3*** 2.7*** 0.96 TOPIX13 0.2 4.38 7.93 28.67* 0.45 -0.14 -0.21 0.01 TOPIX14 15.31*** 15.83*** 17.49* 42.46*** 3.9*** 3.28*** 2.9*** 2.48** TOPIX15 1.65 2.03 2.87 29.59* 1.27 0.95 0.51 0.54 TOPIX16 5.04* 6.57 7.69 24.22 2.24** 1.49 0.95 1.29 TOPIX17 1.42 3.77 5.26 21.48 1.19 0.39 -0.26 0.36 TOPIX18 1.9 6.76 9.97 25.11 1.37 0.52 0.03 -0.27 TOPIX19 0.69 1.62 5.48 14.17 0.82 0.46 -0.05 0.44 TOPIX20 3.73 3.75 5.19 14.4 1.93* 1.84* 1.96** 2.43** TOPIX21 2.58 3.3 7.83 28.08 1.6 1.22 0.19 0.24 TOPIX22 5.79* 7.1 12.29 34.71** 2.4** 1.88* 1 -0.01 TOPIX23 1.26 3.77 7.04 16.2 1.12 0.59 0.13 0.09 TOPIX24 2.93 2.93 6.69 29.08* 1.7* 1.61 0.67 0.52 TOPIX25 0.25 1.14 2.4 16.45 0.5 0.29 0.18 -0.45 TOPIX26 20.88*** 23.54*** 25.46*** 48.14*** 4.56*** 3.59*** 2.59*** 2.31** TOPIX27 1.12 3.6 5.66 17.19 1.06 0.37 -0.05 -0.24 TOPIX28 27.86*** 29.21*** 30.31*** 40.98*** 5.25*** 4.31*** 3.5*** 3.02*** TOPIX29 6.88** 7.75 10.06 38.57*** 2.62*** 2.05** 1.55 2.42** TOPIX30 3.29 6.87 12.24 36.87** 1.81* 1.15 0.6 0.06 TOPIX31 0.14 7.38 14.11 29.37* 0.38 -0.35 -0.58 -0.3 TOPIX32 3.31 6.41 7.23 29.26* 1.82* 1.19 0.59 0.27 TOPIX33 19.3*** 19.62*** 21.61** 46.89*** 4.39*** 3.68*** 3.28*** 2.48**
26
Table 5. Test statistics for period 1991-2001
Modified Box-Pierce Variance Ratio
n=1 n=2 n=5 n=20 q-1=1 q-1=2 q-1=5 q-1=20 NIKKEI 1.3 6.59 7.05 17.82 -1.15 -2.05** -1.7* -0.89 TOPIX 12.12*** 20.02*** 21.16** 35.87** 3.48*** 1.91* 0.55 0.17 TOPIX1 24.56*** 25.45*** 28.03*** 41.89*** 4.95*** 4.09*** 3.67*** 3.06*** TOPIX2 0.37 0.51 6.84 14.34 -0.62 -0.71 -0.57 -0.14 TOPIX3 19*** 23.39*** 27.94*** 44.85*** 4.36*** 3.08*** 1.18 0.37 TOPIX4 7.43** 12.52** 14.35 20.05 2.72*** 1.44 1.16 1.03 TOPIX5 31*** 31.48*** 33.13*** 42.65*** 5.57*** 4.81*** 4.11*** 3.51*** TOPIX6 0.24 11.31** 13.74 43.83*** 0.46 -0.73 -1.09 -1.7* TOPIX7 34.59*** 36.88*** 41.42*** 53.48*** 5.87*** 4.59*** 2.21** 0.65 TOPIX8 0.18 2.96 7.44 17.81 -0.42 -1.12 -1.29 -1.01 TOPIX9 3.48 4.12 6.45 15.92 -1.87* -2.04** -1.37 -0.58 TOPIX10 0 1.37 3.13 16.29 -0.04 -0.48 -0.09 0.3 TOPIX11 8.61** 10.68* 15.12 38.19*** 2.93*** 2.12** 0.57 -0.22 TOPIX12 7.76** 15.96*** 18.02* 32.73** 2.78*** 1.33 -0.01 0.01 TOPIX13 19.57*** 21.62*** 24.09*** 46.6*** 4.41*** 3.47*** 1.78* 1.41 TOPIX14 0.42 5.19 5.68 19.54 -0.65 -1.52 -1.47 -0.22 TOPIX15 4.53 8.27 10.37 24.72 2.11** 1.18 0.88 0.93 TOPIX16 3.82 5.1 5.61 18.96 1.95* 1.27 0.79 1.28 TOPIX17 3.32 5.15 10.86 22.27 1.82* 1.04 1.25 1.88* TOPIX18 10.33*** 11.94** 12.14 32.52** 3.21*** 2.36** 1.55 1.09 TOPIX19 1.33 5.44 7.39 17.85 -1.17 -1.86* -1.86* -0.63 TOPIX20 22.9*** 24.87*** 25.58*** 37.26** 4.78*** 3.74*** 2.44** 1.76* TOPIX21 3.42 5.19 7.46 23.32 1.85* 1.05 -0.16 -0.19 TOPIX22 2.25 3.18 6.05 13.61 1.5 0.95 1.15 0.33 TOPIX23 7.36** 12.11** 16.63* 36.73** 2.71*** 1.47 -0.24 -0.27 TOPIX24 9.06** 9.57* 12.3 28.08 3.01*** 2.42** 0.97 0.49 TOPIX25 16.15*** 31.55*** 36.11*** 45.57*** 4.02*** 2.01** -0.09 -0.56 TOPIX26 5.44* 6.5 11.43 37.67*** 2.33** 1.68* 0.67 1.22 TOPIX27 0 3.84 7.99 28.41* -0.08 -0.72 -1.64 -0.98 TOPIX28 44.19*** 48.87*** 52.24*** 64.5*** 6.64*** 5.2*** 3.08*** 2.19** TOPIX29 50.99*** 51.03*** 54.41*** 106.91*** 7.13*** 6.21*** 3.8*** 2.12** TOPIX30 3.9 3.98 5.59 14.59 1.98** 1.66* 1.37 1.31 TOPIX31 0.34 9.55* 11.85 28.81* 0.58 -0.77 -1.54 -1.42 TOPIX32 31.26*** 31.52*** 33.71*** 62.05*** 5.59*** 5.29*** 3.75*** 2.87*** TOPIX33 1.21 4.28 6.69 16.88 1.1 0.31 0.15 0.22
27
Table 6. Test statistics for period 2001-2014
Modified Box-Pierce Variance Ratio
n=1 n=2 n=5 n=20 q-1=1 q-1=2 q-1=5 q-1=20 NIKKEI 1.46 1.52 2.97 9.83 -1.21 -1.17 -1.36 -0.86 TOPIX 0.1 0.16 2.59 9.32 0.32 0.17 -0.60 -0.69 TOPIX1 4.3 4.68 5.6 16.2 2.07** 1.45 0.42 -0.31 TOPIX2 0.41 1.31 10.74 27.53 0.64 0.28 -0.38 -0.39 TOPIX3 5.64* 6.05 13.48 24.12 2.37** 2.41** 0.66 -0.85 TOPIX4 0.09 0.57 2.63 8.99 -0.31 -0.60 -1.19 -0.90 TOPIX5 0.44 0.54 1.84 9.32 0.65 0.42 -0.22 -0.74 TOPIX6 0.56 0.59 3.67 18.7 0.75 0.76 0.73 0.26 TOPIX7 1.22 1.47 3.89 11.95 1.10 0.70 -0.37 -0.58 TOPIX8 0.15 0.23 6.23 16.14 -0.41 -0.28 -0.41 -0.80 TOPIX9 2.93 3.32 7.14 13.59 -1.73* -1.83* -1.73* -1.33 TOPIX10 3.17 4.32 6.9 15.58 1.78* 0.99 -0.16 -0.23 TOPIX11 0 1.87 5.01 21.93 -0.03 -0.61 -1.46 -1.72* TOPIX12 0.03 0.03 7.14 19.49 -0.17 -0.14 -1.25 -1.8* TOPIX13 0.53 0.55 4.49 10.38 0.73 0.71 -0.16 -0.35 TOPIX14 1.06 1.1 10.4 23.08 -1.03 -1.03 -1.55 -1.48 TOPIX15 3.77 4.17 6.19 17.75 1.94* 1.41 0.20 0.29 TOPIX16 1.19 1.72 2.67 11.7 -1.10 -1.31 -1.36 -1.03 TOPIX17 0 1.4 3 12.48 -0.05 -0.58 -1.25 -1.01 TOPIX18 3.05 3.05 5.62 12.79 1.74* 1.51 0.31 -0.15 TOPIX19 1.04 4.37 7.95 17.92 1.02 0.04 -1.18 -0.67 TOPIX20 10.04*** 10.12* 14.35 24.69 3.16*** 2.68*** 1.14 0.01 TOPIX21 2.08 2.51 6.39 23.34 1.44 0.96 -0.23 -0.37 TOPIX22 2.82 3.84 4.87 19.64 -1.69* -1.97** -2.02** -1.69* TOPIX23 0.05 0.75 3.17 9.97 -0.23 -0.61 -1.35 -0.61 TOPIX24 0.06 0.38 2.4 17.88 0.23 -0.03 -0.18 -0.52 TOPIX25 6.69** 8.61 12.8 22.23 2.58*** 1.69* 0.38 -0.10 TOPIX26 1.76 1.9 3.66 15.89 1.32 1.05 0.25 0.18 TOPIX27 0.03 1.55 7.6 27.6 0.16 -0.39 -1.78* -1.57 TOPIX28 7.34** 7.53 11.78 19.65 2.7*** 2.6*** 0.96 0.32 TOPIX29 2.45 2.46 6.99 17.06 1.56 1.46 0.45 0.34 TOPIX30 0.01 1.29 2.48 11.19 0.08 -0.43 -0.93 -0.92 TOPIX31 0.85 1.54 4.16 10.03 0.92 0.44 -0.53 -0.82 TOPIX32 3.56 3.9 5.48 16.43 1.88* 1.29 0.11 -0.21 TOPIX33 0.14 0.53 6.26 21.53 0.34 0.04 -0.52 -0.51
