1 Testing for Long Range Dependence in Banking Equity Indices Daniel O. Cajueiro and Benjamin M. Tabak Abstract — This paper presents empirical evidence of long range dependence in returns and volatility for banking in- dices for 41 different countries. We employ the Rescaled Hurst analysis and develop a formal statistical procedure to test for long range dependence. This procedure allows to rank these countries by relative inefficiency, which can provide guidance for investors and portfolio managers. Keywords — banking indices, fractional dynamics, long memory, developed economics, emerging markets. I. Introduction A lot of research has been undertaken in recent years focusing on whether stock returns and volatility possess long range dependence. If there is long range dependence in stock returns then one could improve forecasts in the long term on the dynamics of these time series. Further- more, the Black-Scholes model is not valid anymore, which is one of the pillars of modern finance. Basically, most of the financial theory relies on the hypothesis that returns do not present long range dependence. This explains why the topic has been a hot subject in the past years. If one can find evidence of long range dependence then we should incorporate this in our pricing models (options pricing) and also in portfolio and risk management (forecasting expected returns and volatility). Most of the literature has focused on aggregate indices for a variety of countries (See Cajueiro and Tabak (2004a, 2004b, 2004c, 2004d) and Barkoulas and Baum (1996, 1998)). Cheung and Lai (1995) study 18 countries and find little support for long memory in international stock returns. Hiemstra and Jones (1997) employ the rescaled range test (R/S) to the return series of 1,952 common stocks and found that long memory is not a widespread characteristic of these stocks and that there is persistent long memory in the returns of a small proportion of stocks. Koong et al.(1997) study Pacific Basin stock returns and found little support for long memory in these indices. On the other hand Crato and Lima (1994) show evidence of strong long memory in high-frequency data in the condi- tional variance of U.S. stock returns 1 . The literature that focuses on particular sectorial indices is relatively small. Evaluating whether long-range depen- dence exists in specific sectors of the economy is particu- larly useful and could help understand the origins of such Daniel Cajueiro is with Universidade Cat´olica de Bras´ ılia – Mestrado em Economia de Empresas. SGAN 916, M´odulo B – Asa Norte. DF 70790-160 Brazil. Benjamin Tabak is with Banco Central do Brasil SBS Quadra 3, Bloco B, 9 andar. DF 70074-900 1 See also Ding et al. (1993) that provides evidence suggesting that there is substantially more correlation between absolute returns than returns themselves. a phenomena, that seems to be a commonplace in econo- physics. This paper intends to fill this existing gap in the litera- ture by studying long range dependence in banking indices around the world. Our focus on the banking sector relies on the fact that this is certainly one of the most important sectors in the economy. We test for long range dependence for mean returns and volatility for 41 different countries, including developed and emerging economies. The rest of the paper proceed as follows. In the next section we provide a description of the methodology that is used throughout the paper. Section 3 discusses the data that is used, while section 4 presents empirical re- sults on long-range dependence measures. Finally, section 5 presents final remarks and suggestions for further research. II. Measures of Long-Range Dependence In this paper, our measure of long range dependence is the Hurst’s exponent provided by the R/S analysis. We present two different set of results: (1) the R/S analysis; (2) the R/S analysis with a shuffling procedure to purge for short term autocorrelation. The shuffling procedure intends to remove any extra long range dependence that may be presented in the data 2 . The R/S method (Hurst, 1951) due to its simplicity is the most popular way to detect long-range dependence and it is described in which follows. Let X(t) be the price of a stock on a time t and r(t) be the logarithmic return denoted by r(t) = ln  X(t+1) X(t)  . The R/S statistic is the range of partial sums of devi- ations of times series from its mean, rescaled by its stan- dard deviation. So, consider a sample of continuously com- pounded asset returns {r(1),r(2), ··· ,r(τ )} and let r τ de- note the sample mean 1 τ ∑ τ r(τ ) where τ is the time span considered. Then the R/S statistic is given by (R/S) τ ≡ 1 s τ  max 1≤t≤τ t  k=1 (r(k) - r τ ) - min 1≤t≤τ t  k=1 (r(k) - r τ )  (1) where s τ is the usual standard deviation estimator s τ ≡  1 τ  t (r(t) - r τ ) 2  1 2 (2) 2 So, in (2) and (4), we apply the given method to shuffled data in blocks of predetermined size, i.e., we pick a random permutation of the data series within each block of predetermined size and apply the R/S analysis to this shuffled data. The effect of random permutations in these small blocks is to destroy any particular structure of auto- correlation within these blocks [shuffled data was used, for instance, in the context of long range dependence in Erramili et al. (1996)].