آیا بازارهای سرمایه اروپا کارآمدند؟ شواهد جدید از تجزیه و تحلیل فراکتال
|کد مقاله||سال انتشار||مقاله انگلیسی||ترجمه فارسی||تعداد کلمات|
|12624||2014||9 صفحه PDF||سفارش دهید||9206 کلمه|
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Publisher : Elsevier - Science Direct (الزویر - ساینس دایرکت)
Journal : International Review of Financial Analysis, Volume 20, Issue 2, April 2011, Pages 59–67
We report an empirical analysis of long-range dependence in the returns of eight stock market indices, using the Rescaled Range Analysis (RRA) to estimate the Hurst exponent. Monte Carlo and bootstrap simulations are used to construct critical values for the null hypothesis of no long-range dependence. The issue of disentangling short-range and long-range dependence is examined. Pre-filtering by fitting a (short-range) autoregressive model eliminates part of the long-range dependence when the latter is present, while failure to pre-filter leaves open the possibility of conflating short-range and long-range dependence. There is a strong evidence of long-range dependence for the small central European Czech stock market index PX-glob, and a weaker evidence for two smaller western European stock market indices, MSE (Spain) and SWX (Switzerland). There is little or no evidence of long-range dependence for the other five indices, including those with the largest capitalizations among those considered, DJIA (US) and FTSE350 (UK). These results are generally consistent with prior expectations concerning the relative efficiency of the stock markets examined.
Traditional capital markets theory relies on the assumption that logarithmic prices are martingales. Accordingly, logarithmic returns are temporally independent. Temporal dependence in returns is inconsistent with the Efficient Market Hypothesis (EMH) (Fama, 1970), mean-variance portfolio theory (Markowitz, 1952 and Markowitz, 1959), and the capital asset pricing model (CAPM) (Sharpe, 1964 and Lintner, 1965). While short-range dependence is unlikely to provide a basis for the development of trading strategies that deliver positive abnormal returns consistently, long-range dependence, under certain conditions, implies that trading strategies based on historical prices may be systematically profitable (Mandelbrot, 1971 and Rogers, 1997). In order to allow for temporal dependence in returns, and other stylized facts concerning stock market behavior, Peters (1994) introduces the Fractal Markets Hypothesis (FMH). The FMH does not reject a priori the assumption that returns are independent, but it does allow for a broader range of behavior. Accordingly, the FMH does not necessarily constitute an alternative to the EMH, but rather a generalization. The FMH derives from the theory of fractals ( Mandelbrot, 1982). A fractal is an object whose parts resemble the whole. Peters (1994) suggests that financial markets have a fractal structure: when markets are stable, returns calculated over different time scales (daily, weekly, monthly, and so on) exhibit the same autocovariance structure. For instance, if daily returns exhibit positive temporal dependence, so do weekly and monthly returns. This feature is called self-affinity. In the case where the variance and higher-order moments of the returns series are finite, the autocovariance structure of a self-affine time series is represented by the Hurst exponent. A Hurst exponent of 0.5 indicates that returns measured over any time scale are random. A Hurst exponent larger (smaller) than 0.5 indicates positive (negative) long-range dependence in returns measured over any time scale. In the case of positive long-range dependence, the autocorrelation function decays slowly. The Rescaled Range Analysis (RRA) has been employed widely to calculate the Hurst exponent, interpreted as an indicator of long-range dependence. Empirical studies of long-range dependence in financial returns series precede the FMH (Peters, 1994). Greene and Fielitz, 1977 and Peters, 1991 report evidence of long-range dependence in US stock market returns. Subsequent refinements of the methodology used to measure long-range dependence produce results consistent with the EMH (Lo, 1991). Recently, Serletis and Rosenberg (2009) fail to find evidence of long-range dependence for four US stock market indices. A number of studies have examined international stock markets (Cheung and Lai, 1995, Jacobsen, 1996, Opong et al., 1999, Howe et al., 1999, McKenzie, 2001, Costa and Vasconcelos, 2003, Kim and Yoon, 2004, Zhuang et al., 2004, Norouzzadeh and Jafari, 2005 and Onali and Goddard, 2009); commodities markets (Cheung and Lai, 1993, Alvarez-Ramirez et al., 2002 and Serletis and Rosenberg, 2007); and exchange rates (Mulligan, 2000, Kim and Yoon, 2004 and Da Silva et al., 2007). Grech and Mazur, 2004 and Grech and Pamula, 2008 examine the connection between the Hurst exponent and stock market crashes. In a cross-country analysis, Cajueiro and Tabak, 2004 and Cajueiro and Tabak, 2005 interpret estimated Hurst exponents for either stock returns or volatility as indicators of stock market efficiency. There is a lack of consensus in this literature as to the most appropriate method for measuring long-range dependence. Moreover, in the absence of a sound methodology for statistical inference, many studies base their conclusions solely on point estimates of the Hurst exponent. This paper reports tests for the validity of the EMH for eight stock markets that are believed to be at different stages of development. We examine stock market indices comprising of large numbers of stocks. Predictability of returns owing to thin trading is unlikely. Rejection of the null hypothesis of no long-range independence would suggest the possible existence of arbitrage opportunities. We contribute to the extant literature in several ways. First, we provide strong evidence of departure from random walk behavior in logarithmic prices, in the form of long-range dependence in logarithmic returns, for the Czech stock market index PX-Glob. Weaker evidence of long-range dependence is found for the Spanish and Swiss market indices, MSE and SWX. For the other five market indices examined, there is little or no evidence of long-range dependence in returns. Second, unlike most of the extant literature, we employ Monte Carlo simulations to construct critical values for the null hypothesis that returns are normal and independent and identically distributed (NIID), and bootstrap simulations to construct critical values for the null hypothesis that returns are independent and identically distributed (IID). The Monte Carlo simulations require a normality assumption, while the bootstrap simulations assume that the observed values of each returns series are representative of its underlying distributional properties. Third, it is widely recognized that the identification of long-range dependence in the presence of short-range dependence is challenging, owing to difficulties in disentangling the short and long memory components (Smith, Taylor, & Yadav, 1997). In some previous studies, the RRA is applied to the residuals of a fitted autoregressive model for the returns series, to eliminate any short-range dependence that may be present by pre-filtering before testing for long-range dependence (Peters, 1994, Jacobsen, 1996 and Opong et al., 1999). Alternative methods to control for short-range dependence are suggested by Lo (1991) and Fillol and Tripier (2004). We run the RRA on the original series both with and without pre-filtering. We compare the estimated Hurst exponent for pre-filtered returns with critical values constructed using NIID Monte Carlo and IID bootstrap simulations; and we compare the estimated Hurst exponent for unfiltered returns with critical values constructed using recursive Monte Carlo and recursive bootstrap simulations, in which the simulated series have a short-range dependence structure that corresponds to a fitted autoregressive model for the actual returns series for each index. The rest of the paper is structured as follows. Section 2 reviews the properties of self-affinity, long-range dependence, and the generalized Central Limit Theorem. Section 3 describes the methodology and data. Section 4 reports the empirical results. Section 5 concludes.
نتیجه گیری انگلیسی
This paper presents an empirical analysis of long-range dependence in the returns of eight indices for stock markets that are believed to be at different stages of development and market efficiency. Preliminary tests show evidence of non-normality in the probability distribution function of returns measured at various time scales. However, when temporal dependence is eliminated through a shuffle procedure, the distribution of the surrogate returns series tends towards normality at the longer time scales. We test for evidence of long-range dependence in the returns series for the eight indices, using the Rescaled Range Analysis (RRA) to estimate the Hurst exponent. We employ Monte Carlo and bootstrap simulation techniques to construct critical values for the null hypothesis of no long-range dependence, taking into account the well-known upward bias in the RRA estimator of the Hurst exponent. The Monte Carlo simulations are based on an assumption of normality. The bootstrap simulations make no normality assumption, but they assume that the observed values of each returns series are representative of its underlying distributional properties. The RRA is known to be robust to departures from normality, and this property is reflected in the fact that the Monte Carlo and bootstrap sampling distributions of the RRA estimator are generally similar. The issue of adjustment for the possible presence of short-range dependence is addressed by comparing estimated Hurst exponents for pre-filtered returns series with critical values based on simulated series that are independent and identically distributed (IID), and by comparing estimated Hurst exponents for unfiltered returns series with critical values obtained from simulations of series with patterns of short-range dependence imposed corresponding to the observed series. Pre-filtering by fitting a (short-range) autoregressive model tends to eliminate a portion of the long-range dependence when the latter is present. An estimated Hurst exponent for a pre-filtered series that is ‘significant’ when compared with IID critical values should therefore constitute a strong evidence of long-range dependence. Conversely, failure to pre-filter leaves open the possibility of conflating short-range and long-range dependence. Critical values for the test for long-range dependence based on simulated series with short-range dependence imposed using the coefficients from the fitted (short-range) autoregressive model for the original series may tend to be inflated, owing to a tendency to overstate the magnitudes of the true autoregressive coefficients if long-range dependence is also present. Therefore an estimated Hurst exponent for an unfiltered series that is ‘significant’ when compared with recursive critical values should also constitute strong evidence of long-range dependence. Among the eight stock market indices, there is a strong evidence of long-range dependence for the small central European Czech stock market index PX-glob, for which the departure from random walk behavior is highly significant; and there is weaker evidence of long-range dependence for two smaller western European stock market indices, MSE and SWX, for which the departure from random walk behavior is borderline significant. Based on Hurst exponents estimated over all available frequencies, there is no evidence of long-range dependence for the other five indices, including those for the stock markets with the largest capitalization among those considered, DJIA and FTSE350. If the estimated Hurst exponents are allowed to differ between the higher and lower frequencies, however, there is some evidence of departure from random walk behavior at the higher frequencies for DJIA and FTSE350, as well as for MSE, SWX and PX-glob. These results are generally consistent with our prior expectations, and with previous empirical evidence obtained using different methods, concerning the relative efficiency of the eight stock markets that are examined.