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|کد مقاله||سال انتشار||تعداد صفحات مقاله انگلیسی||ترجمه فارسی|
|13170||2012||11 صفحه PDF||سفارش دهید|
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Publisher : Elsevier - Science Direct (الزویر - ساینس دایرکت)
Journal : International Review of Financial Analysis, Volume 24, September 2012, Pages 1–11
We test for departures from normal and independent and identically distributed (NIID) log returns, for log returns under the alternative hypothesis that are self-affine and either long-range dependent, or drawn randomly from an L-stable distribution with infinite higher-order moments. The finite sample performance of estimators of the two forms of self-affinity is explored in a simulation study. In contrast to rescaled range analysis and other conventional estimation methods, the variant of fluctuation analysis that considers finite sample moments only is able to identify both forms of self-affinity. When log returns are self-affine and long-range dependent under the alternative hypothesis, however, rescaled range analysis has higher power than fluctuation analysis. The techniques are illustrated by means of an analysis of the daily log returns for the indices of 11 stock markets of developed countries. Several of the smaller stock markets by capitalization exhibit evidence of long-range dependence in log returns.
Long-range dependence and stable laws in asset returns have been investigated by researchers in finance for several decades. Long-range dependence implies a power–law decay of the autocovariance function in the time domain (Banerjee & Urga, 2005). Stable laws accommodate departures from normality and the central-limit theorem for independent and identically distributed returns or log returns (Levy, 1925). Following the pioneering work of Mandelbrot, 1963, Mandelbrot, 1967 and Mandelbrot, 1971, models that accommodate long-range dependence and stable laws have been employed to describe stock market behavior. These models represent an application of fractal mathematics to financial economics, a topic that has attracted widespread interest, and controversy, in recent years. A fractal exhibits the properties of self-similarity or scale invariance. Stock returns may exhibit the weaker property of self-affinity. After the application of a suitable rescaling transformation, which takes the form of a single non-random contraction dependent upon the time scale only, a self-affine returns series exhibits the property of self-similarity. A self-affine returns series has the same distributional properties (after rescaling) when returns are measured at any frequency, and is said to be unifractal or monofractal. Two classes of process, in which log returns are either non-independent or non-normal or both, embody the properties of self-affinity and unifractality (Cont and Tankov, 2004 and Mandelbrot et al., 1997). First, if log returns are fractionally integrated, the log returns series measured at any frequency exhibits the property of long-range dependence, and the log price series is characterized as Fractional Brownian Motion (FBM).1 Second, the class of probability distributions known as Levy-stable, Pareto-Levy stable or L-stable (Levy, 1925, Mandelbrot, 1963 and Mandelbrot, 1967) includes several heavy-tailed distributions with infinite higher-order moments (including the variance).2 If log returns are either fractionally integrated and long-range dependent, or L-stable with infinite higher-order moments, then log returns are self-affine. The scaling behavior of the series with respect to variation in the frequency (the time scale over which returns are measured) is conveniently summarized by a parameter known as the Hurst exponent. This paper contributes to two strands of literature, on fractional integration and long-range dependence, and on L-stable distributions. We examine the performance of estimators of the Hurst exponent, in the case where there is long-range dependence, and in the case where the distribution of log returns is L-stable (and there is no long-range dependence). Hypothesis tests for departures from the NIID case are developed, based on the application to simulated NIID log returns data of two widely-used methods for estimating the Hurst exponent: rescaled range analysis (RRA),3 and fluctuation analysis (FA).4 The performance of these tests under the alternative hypothesis is examined by evaluating power functions, using simulated self-affine process characterized as either long-range dependent, or L-stable with infinite higher-order moments. Monte Carlo simulations are employed, because the asymptotic properties of the RRA and FA estimators are indeterminate (Banerjee and Urga, 2005). In addition, we draw comparisons with the performance of other tests widely employed to estimate long-range dependence (Geweke and Porter-Hudak, 1983 and Robinson, 1995), and the characteristic exponent of an L-stable distribution (de Haan and Resnick, 1980, Hill, 1975 and Pickands, 1975). In much of the previous literature, researchers have reported evidence concerning the fractal properties of financial returns series in the form of point estimates of the Hurst exponent, or graphical analysis of scaling behavior. In the absence of any basis for assessing the statistical significance of possible departures from the NIID case, however, much of this evidence is at best suggestive of the possibility that models based on fractal mathematics might provide a more satisfactory representation of the behavior of returns than models embodying the NIID assumption. This paper relocates several established but informal procedures within a conventional and formal hypothesis testing framework, enabling conclusions to be drawn based on the standard criteria of statistical inference. The principal findings are as follows. Tests for departure from the NIID case based on RRA and FA perform well when log returns are self-affine and long-range dependent under the alternative hypothesis. In this case, the test based on RRA has higher power than the tests based on the three variants of FA that are considered. However, the test based on RRA performs poorly when log returns are self-affine and L-stable with infinite higher-order moments under the alternative hypothesis. In this case, the choice of sample moments over which the FA is computed is crucial: the FA should not consider sample moments whose true values are infinite. As an estimator of the Hurst exponent, the variant of the FA that considers finite sample moments only is unique (among the estimators considered in this paper) in terms of its reliability under both of the long-range dependent and L-stable alternatives to the null hypothesis of NIID log returns. In the empirical application, the stock market returns of several smaller markets exhibit evidence of long-range dependence. While previous studies provide evidence on the relation between market efficiency and long-range dependence in international stock markets, they do not specifically investigate the link between market size and long-range dependence. Furthermore, most previous empirical contributions neglect the impact of short-range dependence and non-normality on the estimation of the long-range dependence parameter. For two stock markets (the US and Netherlands) we find evidence of self-affine scaling behavior that is consistent with log returns having been drawn from an L-stable distribution with infinite higher-order moments, rather than long-range dependence. This latter result is important, since it challenges the presumption that the variance of returns is finite. An infinite variance renders several of the tools of mainstream finance inapplicable, but it does not necessarily indicate a departure from the conditions for weak-form market efficiency. The remainder of the paper is structured as follows. Section 2 outlines the implications of self-affine alternatives to NIID log returns for the Efficient Markets Hypothesis (EMH), and several other tools of mainstream finance. Section 3 describes statistical tests for an NIID null hypothesis against self-affine alternatives. Section 4 reports critical values and power functions for these tests, based on Monte Carlo simulated data. Section 5 applies the tests to log returns data for 11 developed country stock market indices for the period 1987–2011. Finally, Section 6 summarizes and concludes.
نتیجه گیری انگلیسی
This study develops hypothesis tests for departures from the null hypothesis of NIID logarithmic returns for the case where log returns are self-affine under the alternative hypothesis. A self-affine returns series has the same distributional properties, after the application of a suitable rescaling transformation, for returns measured at any frequency (daily, weekly, monthly, and so on). If log returns are either long-range dependent, or L-stable with infinite higher-order moments including the variance, the scaling behavior with respect to variation in the frequency is conveniently summarized by a parameter known as the Hurst exponent. The tests for NIID log returns against self-affine alternatives that are examined in this study are based on widely-used estimation methods for the Hurst exponent known as rescaled range analysis (RRA) and fluctuation analysis (FA). The principal methodological findings are as follows. The performance of tests based on RRA and FA is satisfactory when log returns are self-affine and characterized by long-range dependence. In this case, the test based on RRA has higher power than tests based on FA. However, the test based on RRA performs poorly when log returns are self-affine and characterized as L-stable with infinite higher-order moments. In this case, the choice of sample moments over which the FA is computed is crucial: the FA should not consider moments whose true values are infinite. The use of RRA is inappropriate in this case, because RRA is based on an examination of the sample scaling behavior of the second moment, whose true value is infinite. As an estimator of the Hurst exponent, the variant of the FA that considers finite sample moments only is uniquely reliable (among the estimators considered in this study) under both of the fractionally integrated and L-stable alternatives to the NIID null hypothesis. The techniques are illustrated by means of an empirical analysis of the daily log returns for the indices of the stock markets of 11 developed countries. Three of these markets are classified as large in terms of market capitalization, and eight are classified as small. We find strong evidence of self-affine scaling behavior for four markets, Finland, Germany, Ireland and Sweden. In all four cases, long-range dependence appears to be the source of the self-affine scaling behavior. The results are consistent with the hypothesis of a negative relation between market size and long-range dependence. We find weak evidence of self-affine scaling behavior in two further cases, the US and the Netherlands, for which the results are consistent with log returns having been drawn from an L-stable distribution with infinite higher-order moments, rather than long-range dependence. While long-range dependence implies a violation of the conditions for weak-form market efficiency, an infinite population variance for returns is not necessarily a sign of market inefficiency. However, infinite variance is inconsistent with asset-pricing models and risk-management techniques that rely on the second moment of returns as a measure of risk.