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.Forward exchange rate anomaly; nonlinearity; random field regression

Derek Bond

Edward J O’Brien

Michael J Harrison

Niall Hession

English

Research Papers in Economics

 
 
 
 
 
 
UCD CENTRE  FOR  ECONOMIC  RESEARCH 
 
 
 
WORKING  PAPER  SERIES  
 
 2008 
 
 
          Nonlinearity as an Explanation of the Forward 
      Exchange Rate Anomaly 
 
          Derek Bond and Niall Hession, University of Ulster, 
            Michael J Harrison, University College Dublin  
            and Edward J O’Brien, European Central Bank           
          
WP08/01 
 
January 2008 
 
 
 
 
 
 
UCD SCHOOL OF ECONOMICS 
UNIVERSITY COLLEGE DUBLIN 
BELFIELD  DUBLIN  4 
 
 
 
 
 
 
 
 
 Nonlinearity as an Explanation of the Forward
Exchange Rate Anomaly
D. Bonda M.J. Harrisonb,∗ N. Hessiona E.J. O’Brienc,†
December 30, 2007
a University of Ulster, Coleraine, Co. Londonderry, BT52 1SA, UK.
b School of Economics, University College Dublin, Belﬁeld, Dublin 4, Ireland.
c European Central Bank, Kaiserstraße 29, D-60311 Frankfurt am Main, Germany.
Abstract
This paper shows that nonlinearity can provide an explanation for the for-
ward 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, ob-
tain parameter estimates that conform with theory to a high degree of preci-
sion: the anomaly disappears.
Keywords: Forward exchange rate anomaly; nonlinearity; random ﬁeld
regression.
J.E.L. Classiﬁcation: C22, F31.
∗Corresponding author. Tel.: +35317168386; fax: +35312830068; email:
michael.harrison@ucd.ie.
†The views expressed in this paper do not necessarily reﬂect those of the European Central
Bank or its members.
11 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, in-
cluding 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 tran-
sition autoregression is now more popular; see, for example, Sarno et al. (2004) and
Baillie and Kili¸ c (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.
2 Models and methodology
The equations commonly used to test the forward rate unbiasedness (fru)h y p o t h -
esis are
∆kst+k = α1 + β1(ft,k − st)+u1,t+k (1)
and
st+k = α2 + β2ft,k + u2,t+k, (2)
2where st and ft,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 ui,t+k,
i =1 ,2, are hypothesized white noise disturbance terms. Theoretically, αi =0a n d
βi =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
∆kst+k = α1 + β1(ft,k − st)+γ1t + u1,t+k (3)
and
st+k = α2 + β2ft,k + γ2t + u2,t+k, (4)
and the disturbances as
ui,t+k = λim(xi,t)+ i,t+k, (5)
where, for i =1 ,2, m(xi,t) is a random ﬁeld intended to capture any nonlinearity, λi
is a parameter that measures the “weight” of the nonlinearity in the speciﬁcation,
xi,t = xi,t gi is the Hadamard product of the 2-vector of explanatory variables, xi,t,
and an associated vector of parameters, gi,a n d i,t+k ∼ N(0,σ 2). The individual
elements of gi, gi1 and gi2, indicate the extent to which each explanatory variable
contributes to any nonlinearity, gi2 being associated with the time variable for both
i =1a n di = 2. As before, αi =0a n dβi = 1 under fru theory, and γi =0 ,
i =1 ,2, is also expected.
Adopting a Gaussian random ﬁeld and the Hamilton (2001) speciﬁcation of its
variance-covariance matrix, (3) and (4) may be estimated by maximum likelihood
using standard algorithms. A simple check for nonlinearity in each equation is also
available by testing the null hypothesis H0 : λi =0 ,i =1 ,2, using the Lagrange
multiplier principle. Details of the estimation and testing procedures are given by
Hamilton (2001), and summarized in Bond et al. (2005).
33D a t a a n d r e s u l t s
Following Gil-Alana (2002), the case of the Canadian dollar was chosen to explore the
role of nonlinearity in the forward rate anomaly. Weekly series of sterling-Canadian
dollar rates for the period December 1994 to June 2005 were used, the source being
Thomson Financial Datastream. The computations were done using the Hamilton
(2001) Gauss code and the algorithm-switching optimization procedure proposed
by Bond et al. (2005).
The results from the random ﬁeld regression analysis are given in Table 1, where
the ﬁgures for ζ are estimates of λ
σ.
Table 1: Random ﬁeld estimates
linear nonlinear
Equation (3) cf t,k − st tσζ f t,k − st t
2.741
(6.090)
1.305
(0.578)
−0.246
(2.144)
0.128
(0.967)
84.867
(639.699)
−0.014
(0.036)
12.893
(0.599)
Equation (4) cf t,k tσζf t,k t
−0.001
(0.002)
1.004
(0.004)
0.0002
(0.0003)
0.001
(0.00002)
1.974
(0.245)
−0.0001
(0.064)
1.169
(0.102)
Note: c denotes the dummy unity associated with the intercepts in the equations.
Estimated standard errors are given in parentheses.
Estimating the models without constant terms produced very similar results to
those in Table 1. There is overwhelming evidence of nonlinearity, with the Hamilton
Lagrange multiplier test statistics for the two equations being 381.46 and 5925.76,
far in excess of the 5 per cent χ2
1 critical value of 3.84. Moreover, the estimates
show that the nonlinearity is associated with the time variable, which has a statis-
tically signiﬁcant coeﬃcient in the nonlinear component of both models. The high
signiﬁcance of the σ and ζ estimates in the exchange rates equation, (4), and the
contrasting lack of signiﬁcance of these estimates in the premium equation, (3), may
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
4diﬀerent 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 γi is also estimated to be statistically
insigniﬁcant in both equations, as expected.
4 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 nonlinear-
ity or parameter instability to the forward anomaly debate, and may have some
contribution to make to the puzzle that Maynard (2006) suggests remains.
5References
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7

Nonlinearity as an Explanation of the Forward Exchange Rate Anomaly

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