Introduction
In economics and finance, whether or not asset prices display long-range dependence is still an important area of research because of its importance for capital market theories (see for instance, Mandelbrot (1971), Greene and Fielitz (1977), Cutland et al. (1995)). Analysis related to long-memory property can be realized through estimation of the fractional integration parameter or the Hurst exponent. The subject of detecting long memory in a given time series was first studied by Hurst (1951), an English hydrologist, who proposed the concept of the Hurst exponent based on Einstein’s contributions to Brownian motion in physics to deal with the obstacles related to the reservoir control near the Nile river dam. The Hurst exponent has characteristics that reflect facts having a bearing on market efficiency. Market inefficiency refers to the fact that the market does not react immediately as new information flows in, but responds to it gradually over a period of time. A violation of the efficient market hypothesis would support the presence of persistence (long memory) or anti-persistence (mean reversion) in the stock market. The presence of long memory in the evolution of asset prices describes the higher-order correlation structure in the series and supports the possibility of predicting its behavior over time. The analysis of long memory in asset prices is important for practitioners because its presence can significantly impact risk management, portfolio selection and trading strategies.
Two measures of long-range dependence are commonly used in finance literature. The first measure ‘H’, the Hurst exponent (or the selfsimilarity parameter), is a dimensionless parameter and diverse methodologies exist to estimate it. The Hurst exponent concept finds its applications in many research fields, including financial studies, due to the ground-breaking work of Mandelbrot (1963, 1997) and Peters (1991, 1994). The Hurst exponent lies in the range 0 ≤ H ≤ 1. If the Hurst exponent is 0.5, then the process is said to follow a random walk. When the Hurst exponent is greater than 0.5, it suggests positive long-range auto correlation in the return series or persistence in stock price series. On the other hand, when the Hurst exponent is smaller than 0.5, it suggests the presence of negative auto correlation in returns or mean reversion in stock price series. The second measure ‘d’ is the fractional integration parameter, which can be estimated from fitting an ARFIMA (p,d,q) model. The Hurst exponent ‘H’ and the fractional integration parameter ‘d’ are related by the formula H = d + 0.5. The Hurst exponent and the fractional integration parameter can be estimated through a variety of techniques, two of which have been adopted in this paper, viz., the wavelet analysis and the Local Whittle analysis.
Mandelbrot (1972) finds that rescaled-range (R/S) analysis shows superior properties over auto correlation and variance analysis (because it can work with distributions with infinite variance) and spectral analysis (because it can detect nonperiodic cycles). Greene and Fielitz (1977) utilize the Hurst R/S method and provide evidence in support of long memory in daily stock return series. Lo (1991) finds that the classical R/S test used by Mandelbrot and Green & Fielitz suffers from a drawback wherein it is unable to distinguish between long memory and shor t- range dependence. Lo (1991) proposes a modified test of the R/S statistic which can distinguish between short-term dependence and long memory, and finds that daily stock returns do not show long-range dependence properties. Willinger et. al. (1999) empirically finds that Lo’s modified R/S test leading to the acceptance of the null hypothesis of no longrange dependence for CSRP (Center for Research in Security Prices) data is less conclusive than it appears. This is so because of the conservative nature of the test statistic in rejecting the null hypothesis of no long-range dependence, by attributing what is found in the data to short-term dependence instead. Baillie et. al. (1995) investigated the long-range dependence properties in inflation time series and found positive results. Lo (2000) applies non-parametric tests to investigate the market efficiency of six Asian stock markets and finds that none of the markets are stationary or exhibit random behavior. Corazza and Malliaris (2002) find that the Hurst exponent does not remain fixed, but changes dynamically over time. They also provide evidence that foreign currency returns follow either a fractional Brownian motion or a Pareto-Levy stable distribution. Cajueiro and Tabak (2004) use the rolling sample approach to calculate Hurst exponents over October 1992 to October 1996 and provide evidence of long-range dependence in Asian markets. Cajueiro and Tabak (2005) study the possible sources of long-range dependence in returns of Brazilian stocks and find that firm specific variables can partially explain long-range dependence measures, such as the Hurst exponent. Karuppiah and Los (2005) apply the wavelet multi resolution analysis to examine longterm dependencies in currency markets of Germany, Japan, Hong Kong, Indonesia, Malaysia, Philippines, Singapore, Taiwan, and Thailand and find that the German Mark/Dollar and the Japanese Yen/Dollar rates exhibit anti-persistent characteristics. Kyaw et. al. (2006) examines longrange dependence in returns from Latin American stock and currency markets using Hurst exponents based on the wavelet multi resolution analysis and finds mixed results for different Latin American markets. Maghyereh (2007) applies the semiparametric Local Whittle analysis on financial returns and volatility of Middle East and North African markets and finds a strong degree of longrange dependence in their equity returns and volatility. Souza et. al. (2008) studies the evolution of long memory over time in returns and volatilities of British pound futures contracts by using the classic R/S approach, the detrended fluctuation analysis (DFA) approach and the generalized Hurst exponent (GHE) approach and finds a change in long-memory characteristics of the British pound around the time of the European financial crisis. Mabrouk et. al. (2008) investigates the long-range dependence property in stock prices and volatility of various emerging and developed markets by using wavelet analysis and finds that the longmemory property in stock returns is approximately associated with emerging markets in comparison to developed markets. Also found is strong evidence of long memory in all the volatility series. Serletis and Rosenberg (2009) use the detrending moving average (DMA) approach to calculate the Hurst exponent and find evidence in support of antipersistence (mean reversion) in the US stock market. They also estimate the Local Hurst exponent (on non-overlapping windows of 50 observations) to examine the evolution of efficiency characteristics of index returns over time. Kristoufek (2010) reexamines the results of Serletis and Rosenberg (2009) and finds that there are no signs of antipersistence in the US stock market.
Techniques like R/S analysis, DFA analysis, DMA analysis and GHE analysis estimate the Hurst exponent in the time domain. On the other hand, wavelet analysis and the Local Whittle analysis are frequency domain approaches to detect the presence of long memory in a given time series. The computation of the fractional differencing parameter from the Local Whittle estimator is based on the periodogram and the estimation procedure is based on the maximum likelihood approach. Wavelet analysis estimates the Hurst exponent based on varying degree of smoothness, which varies from low to high, in that order, for the Haar wavelet, Daubechies-4, Daubechies-12 and Daubechies-20 wavelets.
The core contribution of this study is twofold. First, we study the long-memory properties in stock prices in PIIGS economies using wavelets analysis and Local Whittle analysis (semi-parametric technique). Wavelet analysis (see Mulligan (2004); Gencay et. al. (2005)) and Local Whittle analysis (see Maghyereh (2007)) have many applications in the field of economics and finance including the analysis of long memory in asset prices. Second, we also study the evolution of market efficiency of their stock prices over time using a non-overlapping rolling sub-sample of 256 observations to estimate Hurst exponents over time, which is the same as in Serletis and Rosenberg (2009). To our knowledge, the estimation of the Hurst exponent using a nonoverlapping rolling sub-sample approach has not been implemented previously in stock prices of PIIGS economies. In particular, we would like to note that the time varying Hurst exponent approach is very important to understand the dynamic nature of the evolution of market efficiency. PIIGS economies are at the center of study worldwide due to the current European financial crisis. PIIGS countries have experienced similar economic conditions and financial problems over the past several years. They have common high levels of government debt, recently low GDP, high public-spending, high unemployment, high labor costs, financial sector problems and an inability to deal with debt. Despite facing identical problems, their equity markets are distinct, which is evident from their varying market capitalizations and style distributions. Hence, it is reasonable to conclude that stock markets of PIIGS e c o n o m i e s e x h i b it d i v e r s e e f f i c i e n c y characteristics. Furthermore, the constituents of indices belonging to PIIGS economies are different from country to country and news that hits the market may have different firm specific implications and hence have a diverse impact on different indices. Moreover, Figure 1 also indicates a difference in the behavior of prices and returns for the period under consideration for PIIGS economies.
The remainder of this paper is organized as follows: Section 2 introduces tests we will use in this study. Section 3 discusses the Monte Carlo simulations to examine their small sample properties. Section 4 describes data used in this study and discusses the computational details. Section 5 reports empirical results. Section 6 concludes with a summary of our primary findings.
