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Forward Exchange Rate Puzzle: Joining the Missing Pieces in the Rand-US Dollar

Exchange Market

Lumengo Bonga-Bonga1

Working Paper Number 122

1 Senior Lecturer, University of Johannesburg. Email: [email protected]

mailto:[email protected]

Forward Exchange Rate Puzzle: Joining the Missing Piecesin the Rand-US Dollar Exchange Market

Lumengo Bonga-Bonga

April 3, 2009

Abstract

The Unbiased Forward Rate Hypothesis (UFRH) stipulates that the forward rates shouldbe a perfect predictor for the future spot rates. A number of studies have tested the UFRH andforeign market e ciency and concluded that the hypothesis does not hold. This phenomenonis known as the UFRH puzzle. A number of studies that reject the UFRH have made use ofordinary least square (OLS) methods and support a linear adjustment between spot and forwardexchange rates. This paper establishes that the use of a linear model in testing the UFRH canlead to a misspecication problem if indeed there is a nonlinear adjustment between the forwardand spot exchange rates. In order to overcome the problem of model misspecication, this paperapplies the nonlinear method of the class of the Smooth Transition Regression (STR) model inassessing the relationship between the Rand-US Dollar future spot and forward exchange rates.With the aid of a series of diagnostic tests, the paper shows that there is indeed a nonlinearadjustment process between the Rand-US Dollar spot and forward exchange rates and thatthere exists a regime in the STR model where the UFRH eventually holds. Furthermore, theout-of-sample forecast results show that the STR forecasting method outperforms the OLS andrandom walk methods in forecasting the future spot exchange rate.

1 Introduction

The E cient Market Hypothesis (EMH), also known as the informational e cient market, relieson the e cient exploitation of information by economic actors in predicting prices of securities orassets (Reilly and Brown, 2000). For example, foreign exchange market participants are e cient ifthey use information embedded in the forward exchange rate to predict the future spot rate. Thus,in an e cient forward exchange market, the forward rate is the unbiased predictor of the futurespot rate. This is known as the Unbiased Forward Rate Hypothesis (UFRH). A number of studieshave been conducted to test whether the forward exchange rate is indeed an unbiased forecast ofthe future spot rate. While some of the studies support the existence of the unbiased forwardrate hypothesis, a number of studies have contested the validity of such a hypothesis (see Engel,1996 for a comprehensive review of literature on UFRH). The rejection of the UFRH is also knownas the forward exchange rate puzzle. In attempting to explain why the UFRH is rejected, Fama(1984) and Nieuwland et al (2000) consider the possibility of risk premium in the foreign exchangemarket. They contend that risk-averse investors require compensation to assume risk and thereforethe forward exchange rate is not a rational predictor of the future spot exchange rate. A numberof reasons for the failure of the UFRH have been identied; these include irrational expectations byforeign exchange market participants, the assumption of adaptive learning by economic agents andthe measurement error or models misspecication (Razzak, 2002; Psaradakis et al, 2005).

Senior Lecturer, University of Johannesburg. Email: [email protected]

1

This paper focuses on the cause related to the misspecication of the model for the failure of theUFRH and shows that a linear model is inadequate in characterising the relationship between theRand-US Dollar future spot and forward exchange rates. Wesso (1999) investigated the empiricalissue of market e ciency for the South African currency from January 1987 to November 1998. Theresults of his study rejected the UFRH in the Rand-US Dollar exchange market. Wessos empiricalstudy was based on the constant coe cients assumption. Nonetheless, the author suggested thatfurther research be devoted to the analysis of the time-variant coe cients considering a numberof structural breaks or regime shifts present in the relationship between the Rand-US Dollar spotand forward exchange rates. Sarno, Valente and Leon (2004) conducted a number of Monte Carloexperiments to demonstrate that there is strong evidence that the relationship between spot andforward exchange rates is characterised by signicant nonlinearities. In addition, Baillie and Kili(2006) support the proposition that nonlinearity is an important aspect of the uncovered interestparity (UIP) and apply the technique of the logistic smooth transition dynamic regression (LSTR)to model the relationship between the rate of appreciation of nine currencies and interest ratedierentials. The results of their study indicate that the UIP has a high probability to hold in theouter regime of the LSTR model. As explained in Section two, the UIP is a pre-condition for theUFRH to hold. Expanding on the work of Ballie and Kili (2006), this paper aims to test whetherthe Rand-US Dollar forward exchange rates are unbiased predictors of future spot rates. Firstly,this paper aims to determine whether the relationship between the Rand-US Dollar forward andspot exchange rate is linear or nonlinear. Secondly, the paper selects and estimates a suitable classof smooth transition regression models to illustrate the relationship between the future spot andforward exchange rates in the Rand-US Dollar exchange market. Thirdly, this paper compares theout-of-sample forecasting performance of the STR, linear and random walk methods in predictingthe Rand-US Dollar future spot rates.The remainder of the paper is structured as follows. Section two provides a brief review of

related literature. Section three describes the nonlinear model of the type of the smooth transitionautoregressive (STR) model. An empirical analysis and the relevant discussion of the reportedresults are provided in section four. Lastly, section ve presents the conclusion and a summary ofthe main ndings of this paper.

1.1 Literature Review

In an e cient speculative market, prices should fully reect information available to market par-ticipants and it should be impossible for a trader to earn excess return on speculation (Sarno andTaylor, 2003). Any arbitrage opportunity that presents itself in the foreign exchange market willrapidly cancel out with the change in the conditions of supply and demand. In other words, in thelong term, the forward exchange rate should be a perfect predictor of the spot exchange rate.The UFRH is derived from two important parity conditions, namely the uncovered interest parity

(UIP) and the covered interest parity (CIP). The UIP is given as follow:

EtSt+kSt

=1 + r

1 + rf(1)

Where rand rf represent the domestic and foreign interest rate respectively prevailing at time t. Stis the spot exchange rate and EtSt+k is the expectation at time t of the spot price prevailing at timet+k. Equation (1) states that the risk-free return from a local investment is equal to the comparablereturn in a foreign instrument plus an expected appreciation rate of the foreign currency (Marston,1997). Since the outcome of the future spot rate is uncertain, investors with risk aversion prefer tosell the total proceeds of their investment in the forward market. As a result, a covered version ofinterest rate parity is expressed as follow:

FtSt

=1 + r

1 + rf(2)

2

Where Ft is the k periods forward rate at time t.Linking expression (1) and (2) and applying the natural logarithm expression, one would then

arrive at the conclusion that:

ft = Et (st+k) (3)

Where ft and st+k are the natural logarithm of Ft and St+k;respectively. Expression (3) impliesthat the forwad rate at time t should be equal to the market expectations of the future spot rate,given the information at time t:This expresion provides a basis for the testing of an unbiased forwardrate hypothesis.According to Siegel (1972) a paradox arises when Equation (3) is formulated in levels rather

than logarithm. Siegels paradox entails at least one currency to be biased for purely mathematicalreasons. It can be indicated that if Expression (3) is expressed in level, changing the numeraire willresult in the following:

1

Ft= Et

1

St+k

(4)

Equation (4) cannot imply to be the inverse of the level Equation (3). However, Mcculloch (1975)indicated that the Siegel paradox is unimportant for empirical work. Thus, most empirical workdenes the variables in logarithm form and the UFRH is mostly estimated with the use of thefollowing equation:

st+k = + ft + t+k (5)

t+k is the rational expectations forecast error with Et+k=It

= 0. In order to test the hypothesis

that the forward rate is an unbiased predictor of the spot rate, the restriction =1 is tested. Astrong form of an unbiased market e ciency hypothesis and no risk premium implies testing =0 (aconstant risk premium equals zero) and =1. Moreover, for the UFRH test the error terms, t+ kwith k = 0:::n, should be serially uncorrelated and homoskedastic.A number of estimations that have tested the unbiased forward rate hypothesis with the aid of

Equation (5) have rejected the hypothesis and many reasons have been advanced. For example,Fama (1984) argues that the failure of the UFRH is due to the fact that the risk premium on for-eign exchange rates is extremely variable. The OLS method is not designed to model time varyingparameters. However, McCallum (1994) provides a dierent explanation. McCallum explains thefailure of the UFRH as neglect to take into account the fact that monetary authorities pursue in-terest rate smoothing and avoid exchange rate chang