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Nonlinearity as an Explanation of the Forward Exchange Rate Anomaly

This paper shows that nonlinearity can provide an explanation for the forward exchange rate anomaly (Fama, 1984). Using sterling-Canadian dollar data, and modelling nonlinearity of unspecified form by means of a random field, we find strong evidence of time-wise nonlinearity and, significantly, obtain parameter estimates that conform with theory to a high degree of precision: the anomaly disappears.


Introduction
The fact that the forward exchange rate does not provide an unbiased forecast of the future spot exchange rate has generated an extensive literature since the seminal paper of Fama (1984); see the surveys of Hodrick (1987) and Engel (1996). Some recent work, such as that by Gil-Alana (2002), has suggested that this conflict with theory may be due to fractional integration of the forward rate; see also Baillie and Bollerslev (2000) and Maynard and Phillips (2001). However, the findings of Maynard (2006), using tests robust to persistence in conditioning variables, suggest that a substantial economic puzzle remains. In light of this, another approach that focuses on nonlinearity as a way of explaining the anomaly is gathering interest.
Nonlinearity may arise in exchange rate data for several economic reasons, including transactions costs, central bank interventions and the existence of limits to speculation; see Taylor (2006). Markov-switching models have been used in an attempt to handle such nonlinearity (Engel and Hamilton, 1990) but smooth transition autoregression is now more popular; see, for example, Sarno et al. (2004) and Baillie and Kiliç (2005). The aim of this paper is to report on the use of random field regression (Hamilton, 2001), which does not require the specification of a particular nonlinear functional form nor the choice of any transition variable. The outline of the paper is as follows: Section 2 summarizes the models and methodology used; Section 3 describes the data and results; and Section 4 concludes that in the case considered, the anomaly disappears.

Models and methodology
The equations commonly used to test the forward rate unbiasedness (fru) hypothesis are and where s t and f t,k are the (log) spot and forward exchange rates, respectively, at time t, k is the length of the forward contract, ∆ k is the k-period change, and the u i,t+k , i = 1, 2, are hypothesized white noise disturbance terms. Theoretically, α i = 0 and β i = 1, i = 1, 2, but numerous empirical studies, based on a wide variety of exchange rates and time periods, have generally failed to corroborate this.
The approach here is to allow for the possibility of nonlinearity over time by re-specifying (1) and (2) as and and the disturbances as where, for i = 1, 2, m(x i,t ) is a random field intended to capture any nonlinearity, λ i is a parameter that measures the "weight" of the nonlinearity in the specification, and an associated vector of parameters, g i , and i,t+k ∼ N(0, σ 2 ). The individual elements of g i , g i1 and g i2 , indicate the extent to which each explanatory variable contributes to any nonlinearity, g i2 being associated with the time variable for both i = 1 and i = 2. As before, α i = 0 and β i = 1 under fru theory, and γ i = 0, i = 1, 2, is also expected.
Adopting a Gaussian random field and the Hamilton (2001)  The results from the random field regression analysis are given in Table 1, where the figures for ζ are estimates of λ σ .  Estimating the models without constant terms produced very similar results to those in Table 1 reasonably be assumed to stem from what, in the time series literature, is known as the "pile up" phenomenon; see Hamilton (2005).
The most significant aspect of these results is that when nonlinearity is modelled with a random field, the intercept in the exchange rates equation is not significantly different from zero and the slope coefficient is estimated, with great precision, to be unity, in accordance with exchange rate theory. Similarly, in the exchange premium equation the intercept and slope are not significantly different from zero and unity, respectively, though as the estimated standard errors are larger in this case, the result is not quite as striking. The coefficient γ i is also estimated to be statistically insignificant in both equations, as expected.

Concluding remarks
In this paper we have investigated nonlinearity as a possible explanation of the well-known foreign exchange rate anomaly. Using the Hamilton (2001) approach to nonlinear inference and sterling-Canadian dollar data, we find strong evidence of time-dependent nonlinearity and that when the nonlinearity is modelled by means of a random field, exchange rate theory is confirmed and the anomaly removed.
This result provides an alternative explanation of the rejection of the hypothesis of unbiasedness of the forward rate as a predictor of the future spot rate to that of fractional integration of the forward rate proposed by Gil-Alana (2002), and unit roots as suggested in earlier work such as that by Crowder (1995) and Hai et al. (1997). It therefore adds weight to the earlier work on the relevance of nonlinearity or parameter instability to the forward anomaly debate, and may have some contribution to make to the puzzle that Maynard (2006) suggests remains.