The Forward Rate as a Predictor of the Future Spot Exchange Rate - Essay Example

al Affiliation) THE FORWARD RATE IS AN UNBIASED PREDICTOR OF THE FUTURE SPOT EXCHANGE RATE Under conditions of rational expectations and risk neutrality in the foreign exchange market, it is expected that there should be a direct relationship between the forward rate and the corresponding future spot rate. Despite this assertion, cointegration-based tests of the unbiasedness hypothesis of the forward rate have generated mixed results. As such, in order to make use of important cross-sectional dependencies, this paper will test the unbiased hypothesis by applying a new panel unit-root test, the Johansen likelihood ration (JLR) test. This test provides vital methodological advantages over alternative standard panel unit-root tests. When used in a data set of 7 major currencies in the post-Bretton Woods age, the JLR test offers strong and credible evidence in support of a unitary cointegrating vector in between forward and corresponding future spot rates. However, the orthogonality condition is met only for 3 major currencies. Introduction and Discussion According to the forward rate unbiasedness hypothesis (FRUH), “under conditions of risk neutrality and rational expectations on the part of market agents, the forward rate is an unbiased predictor of the corresponding future spot rate”. Assuming the absence of a risk premium in the foreign exchange market, it must hold true that Et (St +k ) = f t (1) where f t is the log forward rate at time t for delivery k periods later, St +k is the corresponding log spot rate at time t + k , and Et (⋅) is the mathematical expectations operator conditioned on the information set available at time t .1 Assuming the formation of rational expectations (Muth (1960)), St +k = Et (St +k ) + ut +k Where ut +k , the rational expectations realized forecast error, must have a conditional expected value of zero and be uncorrelated with any information available at time t (the orthogonality condition). Substituting (1) into (2) yields St +k = f t + ut +k . (3) The hypothesis in (3) is usuallyally tested by running the "levels" regression2 St +k = α 0 +α1 f t + ut +k . (4) Given that St +k and f t are first-order integrated, or I(1), processes, the FRUH requires that St +k and f t be cointegrated with the cointegrating vector (1, − 1), that is, α 0 = 0 and α1 = 1 in (4).3 Under these restrictions, the forward rate does n o t systematically under- or over-predict the future spot rate, that is, the forward rate is a conditionally unbiased predictor of the corresponding future spot rate. In order for the FRUH to be empirically supported, St +k and f t should share one common stochastic trend and the realized forecast error St +k − f t should be a stationary. The empirical evidence on the existence of cointegration between St +k and f t is decidedly mixed. The Johansen Likelihood Ratio (JLR) Test Johansen (1992) puts forward a maximum likelihood technique to establish the number of common trends in a system of unit-root variables. Without any generality being lost, a p -dimensional vector autoregressive (VAR) process of k -th order can be written as: ∆X t = µ + Θ1 ∆X t −1 + ... + Θk −1 ∆X t −k +1 + ΠX t −k + ε t (5) where ∆ is the first-difference lag operator, µ is a ( p × 1) matrix of constants, Χt is a ( p × 1) random vector of time-series variables with order of integration of at most one denoted by I (1) , ε t is a sequence of zero-mean p -dimensional white noise vectors, Θi are ( p × p) matrices of parameters, and Π is a ( p × p) matrix of parameters, the rank of which contains information about long-run relationships among the variables in the VAR .6 Expression (5) is known as the vector error correction model (VECM). If Π has full rank, that is, rank (Π)= p , then all variables in the system are stationary. If the rank of Π is zero, then no cointegrating vectors exist. In the case of 0< r < p , r cointegrating vectors exist. In this case, there exist ( p × r) matrices α and β such that Π= αβ ′ . β is the matrix of cointegrating vectors and has the property that β ′ X t is stationary even though Χt may be individually I (1) processes. The null Hypothesis that one or more of the system processes are non-stationary can take the form of H 0 :rank(Π) < p (6) and be tested against the alternative that all system processes are stationary, that is, H1:rank(Π) = p . (7) To test the hypothesis in (6), it suffices to test that the smallest of the characteristic roots of Π is zero, as a rejection necessarily implies that all characteristic roots of Π are nonzero and therefore Π possesses full rank. Such a test can be constructed on the basis of the following test statistic, referred to as the Johansen likelihood ratio (JLR) test statistic: JLR = −T ln(1 − λ p), where λ p is the smallest eigenvalue of the generalized eigenvalue problem λ Skk − Sk 0 S00 −1 S0k = 0. (9) The Sij matrices are residual moment matrices from the VECM in (5). Usually, the JLR test statistic in (8) is asymptotically distributed as χ 2 (1) in the null hypothesis. Data The paper analyzed U.S. dollar spot and 90-day forward rates for 8 major currencies: Canadian dollar (CD), Deutsche mark (DM), British pound (BP), French franc (FF), Swiss franc (SF), Netherlands guilder (NG), and the Italian lira (IL). The sample period is from 1974:3 to 1996:4 at a quarterly frequency. Since the maturity date of the forward contract and the sampling frequency are similar, problems emanating from the use of overlapping data are bypassed. The 90-day forward rates are matched with the corresponding future spot rates and empirical analysis is conducted on the achieved forecast-error series St +k − f t , where k is a 3-month period. That is, we impose the vector Results (1, − 1) on the (St +k , f t Table 1 Table 2. Johansen Likelihood Ratio (JLR) Test Statistics Currencies Test Statistics CD, DM, UK, FF, SF, NG, IL 19.5084 DM, UK, FF, SF, NG, IL 21.7371 The null hypothesis in the JLR test is that at least one of the system variables under consideration is a unit-root process with the alternative being that all system variables are stationary processes. T h e asymptotic distribution of the JLR test is  2 1 ; consequently, the 5% asymptotic critical value i s 3.84. The 5% critical value adjusted for finite-sample bias is given by  2 1 T T  pk , where T is the number of observations, p is the number of system variables (dimension of the system), and k is lag order in the VECM. At the 5% level, the adjusted critical vaue is 4.6702 (4.5473) for t h e system of eight (seven) currencies. Table 3. Dependence Test Results for the Forecast Error Series Test Statistics CD DM BP FF S F N G IL j 1  0 1 0.0715 0.1145 0.2631** 0.1574 0.0933 0.1559 0.1692  2 -0.0288 -0.1350 -0.2001* -0.0979 -0.1137 -0.1513 -0.1196 Adjusted R2 -0.0177 0.0065 0.0627 0.0075 -0.0025 0.0190 0.0142 Model: St  k  f t   0  1St  f t   ut  k  0 1 1.5879*** 1.2288 1.8990** 0.6115 1.7560** 1.9507** 0.3934 Adjusted R2 0.0729 0.0142 0.0480 0.0035 0.0414 0.0460 -0.0043 8 i i j j i Model: St  k  f t   0    j St  f t   ut  k j 1 0 2 3 4 5 6 8 Read More