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

Bond, D. (Derek)

Harrison, Michael J.

Hession, Niall

O'Brien, Edward J. (Edward Joseph)

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

O\u27Brien, Edward J. (Edward Joseph)

Irish Universities

Nonlinearity as an explanation of the forward
exchange rate anomaly

Research Repository UCD

Title Nonlinearity as an explanation of the forward exchange rate anomalyAuthors(s) Bond, D. (Derek), Hession, Niall, Harrison, Michael J., O'Brien, Edward J. (Edward Joseph)Publication date 2008-01Publication information Bond, D. (Derek), Niall Hession, Michael J. Harrison, and Edward J. (Edward Joseph) O’Brien. “Nonlinearity as an Explanation of the Forward Exchange Rate Anomaly” (January, 2008).Series UCD Centre for Economic Research Working Paper Series, WP08/01Publisher University College Dublin. School of EconomicsItem record/more informationhttp://hdl.handle.net/10197/1272Downloaded 2024-01-25T04:02:17ZThe UCD community has made this article openly available. Please share how this accessbenefits you. Your story matters! (@ucd_oa)© Some rights reserved. For more information      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 ForwardExchange Rate AnomalyD. Bonda M.J. Harrisonb,∗ N. Hessiona E.J. O’Brienc,†December 30, 2007a 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.AbstractThis paper shows that nonlinearity can provide an explanation for the for-ward exchange rate anomaly (Fama, 1984). Using sterling-Canadian dollardata, and modelling nonlinearity of unspecified form by means of a randomfield, 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 fieldregression.J.E.L. Classification: C22, F31.∗Corresponding author. Tel.: +353 1 7168386; fax: +353 1 2830068; email:michael.harrison@ucd.ie.†The views expressed in this paper do not necessarily reflect those of the European CentralBank or its members.11 IntroductionThe fact that the forward exchange rate does not provide an unbiased forecast ofthe future spot exchange rate has generated an extensive literature since the seminalpaper of Fama (1984); see the surveys of Hodrick (1987) and Engel (1996). Somerecent work, such as that by Gil-Alana (2002), has suggested that this conflict withtheory may be due to fractional integration of the forward rate; see also Baillieand Bollerslev (2000) and Maynard and Phillips (2001). However, the findings ofMaynard (2006), using tests robust to persistence in conditioning variables, suggestthat a substantial economic puzzle remains. In light of this, another approach thatfocuses 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 limitsto speculation; see Taylor (2006). Markov-switching models have been used in anattempt to handle such nonlinearity (Engel and Hamilton, 1990) but smooth tran-sition autoregression is now more popular; see, for example, Sarno et al. (2004) andBaillie and Kiliç (2005). The aim of this paper is to report on the use of random fieldregression (Hamilton, 2001), which does not require the specification of a particularnonlinear functional form nor the choice of any transition variable. The outline ofthe 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 caseconsidered, the anomaly disappears.2 Models and methodologyThe equations commonly used to test the forward rate unbiasedness (fru) hypoth-esis are∆kst+k = α1 + β1(ft,k − st) + u1,t+k (1)andst+k = α2 + β2ft,k + u2,t+k, (2)2where st and ft,k are the (log) spot and forward exchange rates, respectively, at timet, 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 = 0 andβi = 1, i = 1, 2, but numerous empirical studies, based on a wide variety of exchangerates and time periods, have generally failed to corroborate this.The approach here is to allow for the possibility of nonlinearity over time byre-specifying (1) and (2) as∆kst+k = α1 + β1(ft,k − st) + γ1t + u1,t+k (3)andst+k = α2 + β2ft,k + γ2t + u2,t+k, (4)and the disturbances asui,t+k = λim(xi,t) + εi,t+k, (5)where, for i = 1, 2, m(xi,t) is a random field intended to capture any nonlinearity, λiis a parameter that measures the “weight” of the nonlinearity in the specification,xi,t = xi,tgi is the Hadamard product of the 2-vector of explanatory variables, xi,t,and an associated vector of parameters, gi, and εi,t+k ∼ N(0, σ2). The individualelements of gi, gi1 and gi2, indicate the extent to which each explanatory variablecontributes to any nonlinearity, gi2 being associated with the time variable for bothi = 1 and i = 2. As before, αi = 0 and βi = 1 under fru theory, and γi = 0,i = 1, 2, is also expected.Adopting a Gaussian random field and the Hamilton (2001) specification of itsvariance-covariance matrix, (3) and (4) may be estimated by maximum likelihoodusing standard algorithms. A simple check for nonlinearity in each equation is alsoavailable by testing the null hypothesis H0 : λi = 0, i = 1, 2, using the Lagrangemultiplier principle. Details of the estimation and testing procedures are given byHamilton (2001), and summarized in Bond et al. (2005).33 Data and resultsFollowing Gil-Alana (2002), the case of the Canadian dollar was chosen to explore therole of nonlinearity in the forward rate anomaly. Weekly series of sterling-Canadiandollar rates for the period December 1994 to June 2005 were used, the source beingThomson Financial Datastream. The computations were done using the Hamilton(2001) Gauss code and the algorithm-switching optimization procedure proposedby Bond et al. (2005).The results from the random field regression analysis are given in Table 1, wherethe figures for ζ are estimates of λσ.Table 1: Random field estimateslinear nonlinearEquation (3) c ft,k − st t σ ζ ft,k − st t2.741(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) c ft,k t σ ζ ft,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 tothose in Table 1. There is overwhelming evidence of nonlinearity, with the HamiltonLagrange multiplier test statistics for the two equations being 381.46 and 5925.76,far in excess of the 5 per cent χ21 critical value of 3.84. Moreover, the estimatesshow that the nonlinearity is associated with the time variable, which has a statis-tically significant coefficient in the nonlinear component of both models. The highsignificance of the σ and ζ estimates in the exchange rates equation, (4), and thecontrasting lack of significance of these estimates in the premium equation, (3), mayreasonably be assumed to stem from what, in the time series literature, is known asthe “pile up” phenomenon; see Hamilton (2005).The most significant aspect of these results is that when nonlinearity is modelledwith a random field, the intercept in the exchange rates equation is not significantly4different from zero and the slope coefficient is estimated, with great precision, to beunity, in accordance with exchange rate theory. Similarly, in the exchange premiumequation the intercept and slope are not significantly different from zero and unity,respectively, though as the estimated standard errors are larger in this case, theresult is not quite as striking. The coefficient γi is also estimated to be statisticallyinsignificant in both equations, as expected.4 Concluding remarksIn this paper we have investigated nonlinearity as a possible explanation of thewell-known foreign exchange rate anomaly. Using the Hamilton (2001) approach tononlinear inference and sterling-Canadian dollar data, we find strong evidence oftime-dependent nonlinearity and that when the nonlinearity is modelled by meansof a random field, exchange rate theory is confirmed and the anomaly removed.This result provides an alternative explanation of the rejection of the hypothesis ofunbiasedness of the forward rate as a predictor of the future spot rate to that offractional integration of the forward rate proposed by Gil-Alana (2002), and unitroots 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 somecontribution to make to the puzzle that Maynard (2006) suggests remains.5ReferencesBaillie, R.T. and Bollerslev, T., 2000, The forward premium anomaly is not as badas you think, Journal of International Money and Finance 19, 471-488.Baillie, R.T. and Kiliç, R., 2005, Do asymmetric and nonlinear adjustments ex-plain the forward premium anomaly?, Working Paper No. 543, Department ofEconomics, Queen Mary College, University of London.Bond, D., Harrison, M.J. and O’Brien, E.J., 2005, Investigating nonlinearity: anote on the estimation of Hamilton’s random field regression model, Studies inNonlinear Dynamics and Econometrics 9, Article 2.Crowder, W.J., 1995, Covered interest parity and international capital market effi-ciency, International Review of Economics and Finance 4, 115-132.Engel, C., 1996, The forward discount anomaly and the risk premium: a survey ofrecent evidence, Journal of Empirical Finance 3, 123-192.Engel, C. and Hamilton, J.D., 1990, Long swings in the dollar: are they in the dataand do markets know it?, American Economic Review 80, 689-713.Fama, E., 1984, Forward and spot exchange rates, Journal of Monetary Economics14, 319-338.Gil-Alana, L.A., 2002, Empirical evidence of the spot and the forward exchangerates in Canada, Economics Letters 77, 405-409.Hai, W., Mark, N.C. and Wu, Y., 1997, Understanding spot and forward exchangerate regressions, Journal of Applied Econometrics 12, 715-734.Hamilton, J.D., 2001, A parametric approach to flexible nonlinear inference, Econo-metrica 69, 537-573.Hamilton, J.D., 2005, Comment on Investigating Nonlinearity, Studies in NonlinearDynamics and Econometrics 9, Article 3.6Hodrick, R.J., 1987, The Empirical Evidence on the Efficiency of Forward and Fu-tures Foreign Exchange Markets. (Harwood, London).Maynard, A. and Phillips, P.C.B., 2001, Rethinking an old empirical puzzle: econo-metric evidence on the forward discount anomaly, Journal of Applied Economet-rics 16, 671-708.Maynard, A., 2006, The forward premium anomaly: statistical artefact or economicpuzzle? new evidence from robust tests, Canadian Journal of Economics 39, 1244-1281.Sarno, L., Valente, G. and Hyginus, L., 2004, The forward bias puzzle and nonlin-earity in deviations from uncovered interest parity: new perspectives, Universityof Warwick and International Monetary Fund Mimeograph.Taylor, M.P., 2006, Real exchange rates and purchasing power parity: mean-reversion in economic thought, Applied Financial Economics, 16, 1-17.7

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