A Variance-Ratio Test of Random Walk in International Stock Markets

Transcription

1 The Empirical Economics Letters, 11(8): (August 2012) ISSN A Variance-Ratio Test of Random Walk in International Stock Markets Bin Li * and Benjamin Liu Griffith Business School, Griffith University, Australia Abstract: Over the past several decades, researchers in finance, economics and other related fields conduct extensive studies to examine whether stock prices follow random walk. They, using various statistical techniques, often document mixed findings. In this paper, we test random walk hypothesis using the relatively fresh data, 34 MSCI country indexes from January 5, 1988 to December 28, We find that majority of these markets (25 out of 34 markets) follow random walk. Results, however, indicate that 4 out of 9 emerging and developing countries follow non-random walk. Keywords: International Stock Markets, Market Efficiency, Variance Ratio Test JEL Classification Number: G14 1. Introduction Over years, there are many researchers from both the industry and academic in finance, economics and other related fields who have conducted extensive studies that test whether stock prices follow random walk as the topic interests a range of readers such as investors, and concern whether stock prices are predictable based on historical patterns. In general, there are two distinct groups of research, one of which theoretically and empirically supports the random walk hypothesis (RWH) in the context of Efficient Market Hypothesis (EMH) (see detailed discussions in Fama, 1965, 1995). Fama (1995) notes that random walk hypothesis implies that a series of stock price changes has no memory, namely, the past history of a series cannot be utilized to predict its price in the future in terms of independence. A number of empirical studies (e.g., Cootner, 1962; Fama, 1965; Karemera, Ojah and Cole, 1999) support random walk hypothesis. More interestingly, Malkiel (1973) published a well-known book, A Random Walk down Wall Street that highlights stock price randomness (it is the tenth edition in 2011). Further, Karemera, Ojah and Cole (1999), using the data of 15 emerging markets, find that investors are unable to make systematic nonzero profit by using past stock price information in most markets. In contrast, many research casts doubt on RWH, including Lucas (1978), Grossman and Stigltz (1980), Lo and MacKinlay (1988), and Lim and Brooks (2010). They find that the stock prices follow nonrandom walk. In the book, A Non-Random Walk down Wall Street, Lo and MacKinlay (1999), using various techniques, present extensive evidence on a non-random walk behavior of stock prices. * Corresponding author. b.li@griffith.edu.au

2 The Empirical Economics Letters, 11(8): (August 2012) 776 As to whether the stock price behavior follows random walk is an ongoing debatable issue, it inspires us to use the latest data for further testing RWH. In this paper, we use 34 MSCI country index daily data from January 5, 1988 to December 28, 2010, covering the period of the GFC. We find that 25 out of the 34 markets have random walk debut or just 9 out of the 34 markets do not behave randomly. In addition, 4 out of 9 emerging and developing markets, 2 out of the G7 markets and 3 out of 14 other developed markets follow nonrandom walk. The remainder of the paper is organized as follows. Section 2 describes the data and summary statistics. Section 3 presents the empirical approach and results. Section 4 draws conclusions. 2. Data We test random walk behavior of stock market returns on 34 MSCI country indexes and the MSCI world index. Based on the categorization of the World Economic Outlook Database 2010 of the IMF, we divide the 34 countries into four groups: Group One is the G7 countries (Canada, France, Germany, Italy, Japan, UK, and USA); Group Two is the A4 (four Asian countries: Hong Kong, Korea, Singapore and Taiwan); Group Three is the ODE14 (14 Other Developed Economies: Australia, Austria, Belgium, Denmark, Finland, Greece, Ireland, Netherlands, New Zealand, Norway, Portugal, Spain, Sweden, and Switzerland); and Group Four is the EDE9 (9 Emerging and Developing Economies: Argentina, Chile, Indonesia, Jordan, Malaysia, Mexico, the Philippines, Thailand, and Turkey). Our data are weekly closing prices of the MSCI indices from January 5, 1988 to December 28, The prices are adjusted by dividend distribution, new equity issuance and share buyback. The data are sourced from DataStream. The market weekly return for week t is calculated as: R = log( P / Pit, 1) (1) where P is the price of stock market i on Tuesday of week t, and Pit, 1 is the price of stock market i on Tuesday of week t-1. Table 1 presents summary statistics of the weekly returns. It reports the sample means, standard deviations, medians, minimums, maximums, skewness, kurtosis, Jacque-Bera statistics, and the first-order autocorrelation coefficients. The mean weekly returns vary across countries, with the largest (1.03%) for Norway and the lowest (-0.05%) for Taiwan. Compared to other groups, the weekly data suggest that the emerging markets (EDE9 Group) do not exhibit higher return and higher volatility, which contradicts Harvey s

3 The Empirical Economics Letters, 11(8): (August 2012) 777 (1995) finding. The returns and the risks (standard deviations) are highest for the ODE14 group. The Jarque-Bera statistics suggest non-normal return distributions except for Italy. Most of the returns are skewed to the left and exhibit excess kurtosis. The first-order autocorrelation coefficients are negative and differ across countries. But most of them are small with the absolute values less than The MSCI world index weekly return is 0.39%, with a negative first-order autocorrelation coefficient. Table 1: Summary Statistics Country Code Country Mean Std. Dev. Median ( 100) ( 100) ( 100) Min ( 100) Max Jarque- ( 100) Skewness Kurtosis Bera G7 Markets CAN Canada FRA France GER Germany ITA Italy JAP Japan UKM UK USA USA Asian Markets HKG Hong Kong KOR Korea SIN Singapore TAI Taiwan Other Developed Markets AUS Australia AUT Austria BEL Belgium DEN Denmark FIN Finland GRE Greece IRE Ireland NET Netherlands NZD New Zealand NOR Norway POR Portugal SPA Spain SWE Sweden SWI Switzerland ρ(1)

4 The Empirical Economics Letters, 11(8): (August 2012) 778 Table 1 continued Emerging and Developing Markets ARG Argentina CHI Chile INO Indonesia JOR Jordan MAL Malaysia MEX Mexico PHI Philippines THA Thailand TUR Turkey World WRD MSCI World Note: The country categorization is based on the 2010 World Economic Outlook Database. All Jarque-Bera statistics for normality are significant at the 5% level. The samples are weekly from January 5, 1988 to December 28, Empirical Results We use variance ratios to test the Random Walk 3 model (or RW3) in Campbell, Lo and MacKinlay (1997, p.33). In RW3, the null hypothesis is: H R = µ + ε, (2) 0 : t and the error term ε t satisfies: Cov[ εt, εt k] = 0 for all k 0, and Cov[ εt, εt k] 0for all k = 0. A general q-period variance ratio statistic VR(q) can be constructed as (See Campbell et al., 1997, p.49): q 1 Var[ Rit, ( q)] k VR( q) = 1+ 2 (1 ) ( k) q Var[ R ] q ρ (3) k = 1 R ( k) R + R R, and ( k) where 1 k+ 1 ρ is the k th order autocorrelation coefficient of R series, and ε is an error term. Lo and MacKinlay (1988) show that a heteroskedasticity-consistent estimator of the asymptotic variance of VR( q )(denoted as θ ( q) ) is:

5 The Empirical Economics Letters, 11(8): (August 2012) 779 and ˆ k θ ( q) 4 (1 ) q 1 2 ˆ δ (4) k k = 1 q nq 2 2 nq ( p ˆ ˆ ij, pij, 1 µ ) ( pij, k pij, k 1 µ ) ˆ j= k+ 1 δk = (5) 2 nq 2 ( p ˆ ij, pij, 1 µ ) j= 1 Thus, a standardized test statistic ψ ( q) can be used to test the null hypothesis: ( VR( q) 1) jq Ψ ( q) = ~ N( 0,1) (6) ˆ θ Table 2 displays variance ratios for weekly stock market index returns on the 34 stock indices with an aggregate value q = 2, 4, 8, and 16. The heteroskedasticity-consistent test statistics are given in parentheses immediately on the right-hand side columns next to the variance ratios. The null hypothesis of the random walk is: The variance ratio is one and the test statistics are distributed standard normally at the asymptotical level. Table 2: Variance Ratios for Weekly Stock Market Index Returns Country Code Number q of base observations aggregated to form variance ratio T- Varianc T- Variance T- Varianc statistic e ratio statistic ratio statistic e ratio Variance ratio T- statistic G7 Markets CAN 0.90** (-2.00) 0.90 (-1.21) 0.87 (-0.99) 0.86 (-0.77) FRA 1.01 (0.20) 1.15 (1.24) (1.34) (1.94) 1.52** (2.25) GER 0.98 (-0.37) 1.06 (0.51) 1.17 (0.97) 1.28 (1.12) ITA 0.98 (-0.55) 0.98 (-0.30) 1.01 (0.09) 1.12 (0.65) JAP 0.96 (-0.54) 0.98 (-0.20) 1.03 (0.18) 1.10 (0.44) UKM 0.95 (-1.23) 1.01 (0.08) 1.14 (1.24) 1.23 (1.34) USA 0.92 (-1.89) 0.92 (-1.06) 0.92 (-0.67) 0.96 (-0.21)

6 The Empirical Economics Letters, 11(8): (August 2012) 780 Table 2 continued Asian Markets HKG 0.94 (-1.50) 0.95 (-0.65) 0.98 (-0.19) 1.05 (0.27) KOR 0.95 (-0.91) 0.99 (-0.13) 1.00 (0.01) 1.03 (0.16) SIN 0.97 (-0.71) 1.02 (0.27) 1.00 (0.04) 0.98 (-0.15) TAI 0.94 (-1.43) 0.93 (-0.90) 0.97 (-0.22) 1.06 (0.35) Other Developed Markets AUS 0.95 (-1.02) 1.00 (0.02) 1.02 (0.13) 1.08 (0.41) AUT 0.94 (-1.10) 0.93 (-0.76) 0.89 (-0.90) 0.81 (-1.11) BEL 0.98 (-0.41) 1.04 (0.36) 1.12 (0.82) 1.25 (1.21) DEN 1.02 (0.39) 1.06 (0.82) 1.18 (1.51) 1.34** (2.00) FIN 0.97 (-0.57) 1.02 (0.17) 1.08 (0.54) 1.07 (0.38) GRE 0.95 (-1.13) 0.98 (-0.24) 1.09 (0.79) 1.24 (1.46) IRE 0.94 (-1.09) 0.96 (-0.40) 1.02 (0.12) 1.05 (0.26) NET 0.93 (-1.62) 0.92 (-0.99) 0.88 (-1.01) 0.79 (-1.21) NZD 0.90** (-2.29) 0.88 (-1.43) 0.87 (-1.03) 0.90 (-0.55) NOR 1.14** (2.08) 1.40** (3.35) 1.84** (4.63) 2.53** (6.05) POR 1.04 (1.09) 1.15** (2.17) 1.21** (2.08) 1.25 (1.68) SPA 0.99 (-0.23) 1.05 (0.65) 1.18 (1.58) 1.34** (2.03) SWE 0.96 (-0.47) 1.00 (0.00) 1.11 (0.55) 1.06 (0.27) SWI 0.95 (-0.89) 0.95 (-0.54) 1.09 (0.59) 1.24 (1.11) Emerging and Developing Markets ARG 1.02 (0.48) 1.09 (0.96) 1.27 (1.79) 1.53** (2.45) CHI 0.96 (-1.12) 0.99 (-0.15) 1.04 (0.39) 1.09 (0.51) INO 0.98 (-0.28) 1.03 (0.26) 1.14 (0.87) 1.28 (1.23) JOR 0.97 (-0.78) 1.03 (0.36) 1.09 (0.75) 1.08 (0.49) MAL 0.98 (-0.57) 1.06 (0.77) 1.16 (1.39) 1.27 (1.67) MEX 1.01 (0.29) 1.14 (1.61) 1.28** (2.11) 1.47** (2.59) PHI 1.04 (0.96) 1.13 (1.81) 1.24** (2.08) 1.41** (2.39) THA 0.99 (-0.25) 1.08 (0.97) 1.14 (1.13) 1.22 (1.26) TUR 1.06 (1.73) 1.15** (2.35) 1.20** (2.01) 1.23 (1.56) World WRD 0.95 (-1.03) 0.99 (-0.14) 1.03 (0.23) 1.10 (0.51) Note: Variance-ratio test of RW3 for stock market indices. The country categorization is based on the 2010 World Economic outlook Database. The variance ratios ( VR( q ) ) are reported with the heteroskedasticity-consistent test statistics ( ψ ( q) ) given in parentheses immediately on the right-hand side columns. The null hypothesis of the random walk is that the variance ratio is one and the test statistics are standard normally distributed asymptotically. The samples are weekly from January 5, 1988 to December 28, The variance ratios that are statistically significant at the 1%, 5%, and 10% levels are denoted with ***, ** and * respectively.

7 The Empirical Economics Letters, 11(8): (August 2012) 781 Table 2 clearly shows that the random walk null hypothesis RW3 is not rejected for the stock markets in all the Asian 4 countries and the G7 countries except for Canada where the null hypothesis is rejected at the 5% significance level at q=2, and for France where the null hypothesis is rejected at the 5% significance level at q=16. For the US market, the variance ratios decline with q, and the autocorrelation is negative. The test statistics for the US market are statistically insignificant and therefore the null hypothesis is not rejected for any q. Our finding is consistent with Campbell et al. s (1997, p.69) finding using the CRSP value-weighted index for the sample Above all, the results suggest that these markets are quite competitive and generally weak-form efficient. For the markets in the ODE14 group, the random walk hypothesis cannot be rejected for most markets at any q except for Denmark (at q=16), New Zealand (at q=1), Norway (at any q), Portugal (at q=4, 8) and Spain (at q=16). It is worth noting that the variance ratios for Norway monotonically increase with q: the variance ratio rises from 1.14 (q=2) to 2.53 (q=16). Campbell et al. (1997, p.69) interpret that rising variance ratios suggest a positive serial correlation in multi-period returns. In fact, as Table 1 shows, the weekly stock market return for Norway has an autocorrelation coefficient of 0.14, the highest among the 34 countries. This also applies to Portugal, which has an autocorrelation coefficient of By contrast, the variance ratios for Canada decline with q and it has autocorrelation coefficient of One would expect that stock markets in the emerging and developing countries are less competitive and efficient than those in more developed markets. In fact, as Table 2 shows, the random walk hypothesis can be rejected for four countries (Argentina, Mexico, the Philippines, and Turkey) out of the nine countries in the EDE9 Group at some q. However, it is puzzling to see that stock markets in five countries (Denmark, New Zealand, Norway, Portugal, and Spain) out of the fourteen countries in the ODE14 Group still exhibit nonrandom walk behaviour to some extent. These markets are generally believed to be more efficient than the market in the EDE Group. In sum, the variance ratio test provides evidence against the random walk hypothesis of stock returns for most markets in the 34 countries. For the MSCI World Index, the test also suggests that the returns are random walk. 4. Concluding Remarks Whether stock prices behave randomly or not has attracted attention of a wide range of researchers from both industry and academic institutions that conduct extensive research on the random walk of stock price and document mixed findings. As it concerns whether the stock prices are predictable using their past historical data, the topic has become one of the hottest research over many decades. In this paper, we test random walk hypothesis

8 The Empirical Economics Letters, 11(8): (August 2012) 782 using fresh data, 34 MSCI country indexes from January 5, 1988 to December 28, We find that majority of these markets (25 out of 34 markets) follow random walk. Results, however, indicate that 4 out of 9 emerging and developing counties follow nonrandom walk. Our results are generally consistent with Karemera, Ojah and Cole (1999). Acknowledgements Bin Li would like to thank Mr. Hao Zhang for his excellent research assistance. He is also grateful for financial support from Griffith University under the 2011 Griffith University New Researcher Grant, Project Number: References Campbell, J.Y., A.W. Lo and A.C. MacKinlay, 1997, The Econometrics of Financial Markets, Princeton University Press. Cootner, P.H., 1962, Stock Price Random vs Systematic Changes, Industrial Management Review, 3, Fama, E.F., 1965, The Behavior of Stock Market Prices, Journal of Business, 38, Fama, E.F., 1995, Random Walk in Stock Market Prices, Financial Analysts Journal, 51, Grossman, S. and J. Stiglitz, 1980, On the Impossibility of Informationally Efficient Markets, American Economic Review, 70, Harvey, C. R., 1995, Predictable Risk and Returns in Emerging Markets, Review of Financial Studies, 8, Karemera, D., K. Ojah and J. A. Cole, 1999, Random Walks and Market Efficiency Tests: Evidence from Emerging Equity Markets, Review of Quantitative Finance and Accounting, 13, Lim, K.P. and R.D. Brooks, 2010, Why Do Emerging Stock Markets Experience More Persistent Price Deviations from a Radom Walk over Time? A Country Level Analysis, Macroeconomic Dynamics, 14, Lo, A.W. and A.C. MacKinlay, 1999, A Non-Random Walk Down Wall Street, Princeton University Press. Lo, A.W. and A.C. MacKinlay, 1988, Stock Market Prices Do Not Follow Random Walks: Evidence from a Simple Specification Test, Review of Financial Studies, 1, Lucas, R.E., 1978, Asset Prices in an Exchange Economy, Econometrica, 46, Malkiel, B.G., 1973, A Random Walk Down Wall Street: The Time-Tested Strategy for Successful Investing, W.W. Norton.