Long memory and fractional integration in high frequency data on the US dollar/British pound spot exchange rate☆
Introduction
The Efficient Market Hypothesis (EMH) in its weak form rules out the possibility of abnormal systematic profits over and above transaction costs and risk premia, as prices should fully reflect available information (see Fama, 1970). The implication is that stock prices should follow a random walk process, which implies unpredictable returns (see Summers, 1986). Therefore, a finding of mean reversion in stock prices is seen as inconsistent with equilibrium asset pricing models (see, e.g., Fama and French, 1988, Poterba and Summers, 1988). A large number of studies have been carried out to establish whether (log-)prices are indeed I(1) and, consequently, stock market returns I(0) series, although business cycle variation and short-range dependence might also lead to a rejection of long memory in stock prices (see Lo, 1991). However, as we argued in Caporale and Gil-Alana (2002), the assumptions imposed by standard unit root tests might be too restrictive, and the possibility of fractional orders of integration with a slow rate of decay should be considered. Therefore that study performed tests allowing for fractional alternatives and incorporating the I(0) and the I(1) models as particular cases of interest, and found that US real stock returns are close to being I(0) (note that if shocks are weakly autocorrelated, markets will not be efficient).
A subsequent contribution (see Caporale & Gil-Alana, 2007) decomposed the stochastic process followed by US stock prices into a long-run component described by the fractional differencing parameter (d) and a short-run (ARMA) structure. Finally, in Caporale and Gil-Alana (2008) we introduced a more general model which, instead of considering exclusively the component affecting the long-run or zero frequency, also takes into account the cyclical structure. Specifically, a procedure was applied which allows to test simultaneously for unit roots with possibly fractional orders of integration at both the zero and the cyclical frequencies. Modelling simultaneously with long range dependence the zero and the cyclical frequencies can solve at least to some extent the problem of misspecification that might arise with respect to these two frequencies.
However, the fractional differencing parameter may be very sensitive to the data frequency used in the analysis. In fact, it has often been claimed that aggregation is behind fractional integration: Robinson (1978) and Granger (1980) showed that the aggregation of heterogeneous individual AR processes may produce fractional integration. On the other hand, it is well known that temporal aggregation leads to finite sample biases in the estimates of the fractional differencing parameter (see, e.g. Souza & Smith, 2002).1This is the main issue that will be investigated in the present study by using a high frequency dataset on the US dollar–British pound spot exchange rate collected every 1, 2, 3, 5, and 10 min. As in Caporale and Gil-Alana (2008), we start the analysis by using a general long memory model that incorporates poles at both the zero and a cyclical frequency; however, since the evidence clearly suggested an order of integration not significantly different from zero for the cyclical frequency, we then focus exclusively on the long run or zero frequency.
Fractional integration in exchange rates markets has been examined in various papers, many of them testing the Purchasing Power Parity (PPP) condition, which occupies a central place in international economics. Applying R/S techniques to daily rates for the British pound, French franc and Deutsche mark, Booth, Kaen, and Koveos (1982) found positive memory during the flexible exchange rate period (1973–1979) but negative one (i.e., anti-persistence) during the fixed exchange rate period (1965–1971). Later, Cheung (1993) also found evidence of long memory behaviour in foreign exchange markets during the managed floating regime. On the other hand, Baum, Barkoulas, and Caglayan (1999) estimated fractional ARIMA (ARFIMA) models for real exchange rates in the post-Bretton Woods era and found almost no evidence to support long run PPP. Additional studies on exchange rate dynamics using fractional integration are those by Crato and Ray (2000), Wang (2004), Dufrenot, Mathieu, Mignon, and Peguin-Feissolle (2006), Dufrenot, Lardic, Mathieu, Mignon, and Peguin-Feissolle (2008) and Aloy, Booutahar, Gente, and Peguin-Feissolle (2011) among others. All these papers, however, focus on low frequency (mainly quarterly) data, and do not examine the case of high frequency (intra-day) data.
The present study focuses on the case of spot exchange rates with the aim of gaining some insights into the interaction between fractional integration and high frequency data. The results suggest that lower degrees of memory are associated with lower data frequencies. The layout of the paper is as follows. Section 2 describes the econometric methodology used. Section 3 provides details of the data and discusses the empirical results. Section 4 summarises the main findings and offers some concluding remarks.
Section snippets
Methodology
There are two definitions of long memory, one in the frequency domain and the other in the time domain. Let us consider a zero-mean covariance stationary process {xt, t = 0, ± 1, …} with autocovariance function γu = E(xtxt + u). The time domain definition of long memory states that:
Assume that xt has an absolutely continuous spectral distribution, so that it has a spectral density function, f(λ); according to the frequency domain definition of long memory the spectral density function is
Data and empirical results
The data used for the analysis are taken from Reuters, and are intraday data at the 1, 2, 3, 5, 10-minute frequency. Specifically, the series examined is the spot nominal exchange rate of the US dollar pound vis-à-vis the British pound, for different sample periods with a duration of one and a half days. We only report the results for the period 13/05/2010 (11:47)–14/05/2010 (21:07), since those for other samples were very similar.3
Conclusions
Despite the existence of a very extensive literature, there is still lack of consensus on what is the most appropriate specification for many financial series. For instance, whether asset returns of asset prices are predictable or not is still controversial: the efficiency market hypothesis suggests that they should follow a random walk (see Fama, 1970), but mean reversion is often found (see, e.g., Poterba & Summers, 1988). More recently, it has become clear that it is essential to consider
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The second-named author gratefully acknowledges the financial support from the Ministry of Education of Spain (ECO2011-2014–28196-ECON Y FINANZAS, Spain) and from a Jeronimo de Ayanz project of the Government of Navarra. We are also grateful to the Editor and the two anonymous referees as well as Walter Kramer and the other participants in the Symposium on high frequency data in empirical finance, Technische Universitat Dortmund, Dortmund, Germany, 1–2 July 2010.