Academic journal article Atlantic Economic Journal

# Stochastic Behavior of Nominal Exchange Rates

Academic journal article Atlantic Economic Journal

# Stochastic Behavior of Nominal Exchange Rates

## Article excerpt

Introduction

This paper examines the daily structure of the nominal exchange rates in the U.K., Germany, and Japan in relation to the U.S. dollar by means of using fractionally integrated techniques. For this purpose, the author uses a version of the tests of Robinson [1994a] that permits us to test I(d) statistical models including weakly autocorrelated disturbances. In doing so, a greater degree of flexibility in the dynamic behavior of the series is allowed, not achieved by the classical representations based on I(0) and I(1) processes.

An I(0) process {[u.sub.t], t = 0, [+ or -] 1, ...} is defined as a covariance stationary process with spectral density function that is positive and finite at the zero frequency. Then, {[x.sub.t], t = 0, [+ or -] 1, ...} is I(d) if:

[(1 - L).sup.d] [x.sub.t] = [u.sub.t], t = 1,2, ... (1)

[x.sub.t] = 0, t [less than or equal to] 0, (2)

where the polynomial in (1) can be expanded in terms of its binomial expansion such that for all real d:

[(1 - L).sup.d] = [summation over ([infinity]/j=0)] (d/j) [(-1).sup.j] [L.sup.j] = 1 -dL + d(d-1)/2 [L.sup.2] - ...

Thus, the parameter d plays a crucial role when describing the dependence between the observations. If d = 0 in (1), [x.sub.t] = [u.sub.t], and a weakly autocorrelated [u.sub.t] is allowed for. Then, [x.sub.t] is short memory as opposed to the concept of long memory when d > 0. Here, the unit root (d = 1) appears as the most widely application used in the empirical work. However, as was argued by Adenstedt [1974] and Taqqu [1975], d can also be a real number. If d [member of] (0, 0.5), [x.sub.t] is still covariance stationary, and if d [member of] (0.5, 1), [x.sub.t] is no longer stationary but mean reverting, with the effect of the shocks dying away in the long run. Finally, if d [greater than or equal to] 1, the series is non-stationary and non-mean-reverting. These processes were initially proposed by Granger [1980, 1981], Granger and Joyeux [1980] and Hosking [1981] and were theoretically justified in terms of aggregation of ARMA series by Robinson [1978], Granger [1980]. Similarly, Croczek-Georges and Ma ndelbrot [1995], Taqqu et al. [1997], Chambers [1998], and Lippi and Zaffaroni [1999] also use aggregation to motivate long memory processes, while Parke [1999] uses a closely related discrete time error duration model.

Traditionally, it has been assumed that the exchange rates have a unit root implying that shocks have permanent effects on the series. [for example, Taylor, 1995; Breuer, 1996; Rogoff," 1996], though other authors [for example, Abuaf and Jorion, 1990; Glen, 1992; Lothian and Taylor, 1996] argue in the opposite direction suggesting mean reverting behavior. Most of these articles concentrate on real rather than nominal exchange rates though Cheung [1993] reports evidence of long memory in the nominal exchange rates. Other articles, suggesting that the exchange rates are mean reverting and, in particular, that they can be specified in terms of I(d) statistical models are Diebold et al. [1991], Cheung and Lai [1993] and, more recently, Gil-Alana [2000a].

This article investigates the nominal exchange rates and their corresponding returns in the U.K., Germany, and Japan in relation to the U.S. dollar by means of using fractionally integrated techniques. A motivation for this is as follows: Given interest and inflation rates in two countries and a constant risk premium, it could be argued that the nominal exchange rate should follow a pure random walk on the grounds that the market would immediately react to incorporate any expected future appreciation of the exchange rate. This would imply that future returns are unforecastable, that is, a martingale difference sequence. However, and more generally, the exchange rate return ([E.sub.t][DELTA][y.sub.t]) can be written as the sum of the interest rate differential (or forward premium [i. …

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