Uncovering bias in uncovered interest parity tests

In this paper, we re-examine the hypothesis of uncovered interest parity (UIP), which in its basic form implies that the (nominal) expected return to speculation in the forward foreign exchange market conditional on available information should be zero. This is an “efficient-markets hypothesis” (EMH) for foreign exchange markets: if all available information is used rationally by risk-neutral agents in determining the spot and forward exchange rates, then the expected rate of return to speculation will be zero and the foreign exchange market is said to be efficient. This is a joint hypothesis since it includes the assumption of rational expectations (REH) and the assumption that the risk premium for the forward rate is zero. In fact, rejection of the UIP hypothesis does not immediately translates into a rejection of the efficiency of the foreign exchange market, which could be due economic agents being risk averse. Still testing whether the UIP hypothesis holds has been and continue to be a topic of considerable interest from both theoretical and empirical perspectives.

Testing the rationality hypothesis and exchange market efficiency is embedded in the general problem of estimating the parameters of a k -step-ahead linear forecasting equation. When the sampling interval is finer than the interval over which forecasts are made (in this case the maturity time of the forward exchanges rates), the forecast error is serially correlated. As noted by Hansen and Hodrick (1980), under rational expectations (REH) the forecast error is serially correlated up to lag $k−1$ and OLS remains consistent but appropriate modifications in the estimation of the asymptotic covariance matrix are needed. However, if the REH is rejected and the forecast error is serially correlated beyond lag $k−1$⁠, OLS is no longer consistent when the regressors are not exogenous; see Perron and González-Coya (2022). Contrary to what is asserted in Hansen and Hodrick (1980), GLS is consistent when the regressors are pre-determined provided the roots of the MA polynomial are inside the unit circle, i.e. MA process is invertible, as shown in Perron and González-Coya (2022). Moreover, GLS remains consistent when the forecast error follows a linear invertible process.

The first contribution of this paper is to provide a consistent and efficient framework to estimate and perform tests of the parameters of a k -step-ahead linear forecasting equation that remains valid whether the REH holds or not. We apply the FGLS procedure developed in Perron and González-Coya (2022) which is consistent using non-exogenous regressors, provided the errors follows an stationary invertible linear process. The second contribution is to extend their FGLS procedure to cases with lagged dependent variables included as regressors. Following the work of Perron and Olivari (2023), we introduce an instrumental variable (IV)-based approach for this problem that requires pre-determined but not necessarily exogenous IVs for consistency. The third contribution is to apply our FGLS procedures to the two main regressions suggested in the literature to test the UIP. We use 30 years of data for three currencies, and we reconsider the framework and regressions used by Fama (1984) and Hansen and Hodrick (1980). We provide extensive simulation experiments to assess the finite sample performance of our FGLS procedure relative to OLS. We show that FGLS achieves important reductions in mean-squared error (MSE) and allow tests with much greater power.

The Fama regression assesses whether the current forward-spot differential, $ft,h−st$⁠, is a good predictor of the future change in the spot rate, $st+h−st$⁠. Most results available in the literature suggest a negative estimate of the relevant parameter, which instead should take value one if the UIP holds. This is often referred to as the “forward discount anomaly”, which refers to the widespread empirical finding that the returns on nominal exchange rates is negatively correlated with the lagged forward premium. It implies an appreciating currency for the high interest rate country. It is an “anomaly” as rational expectations would imply the opposite; if all currencies are equally risky, investors would demand higher interest rates on currencies expected to fall in value. Our results, in contrast, indicate positive values, sometimes not significantly different from one. This finding suggests that the “forward discount anomaly” might be a consequence of OLS providing an inconsistent estimate and our FGLS procedure being consistent and efficient under a broader range of possible scenarios.

The regression adopted by Hansen and Hodrick (1980) is to test whether past values of the forward-spot differential help predict the current value $ft,h−st$⁠, conditioning on some covariates involving the past forward-spot differentials from some other countries. Under the UIP and EMH, there should be no predictive power as all information contained in the information set at time t should already have been accounted for by the market in setting the forward rates. Here, contrary to most empirical results available in the literature, in particular those based on some OLS regression or GMM, our robust and efficient procedure cannot reject the null hypothesis that the UIP holds. Hence, overall, our results can be viewed as overturning the so-called “forward discount anomaly.”

The main methodological contribution of this paper is that it extends Perron and González-Coya (2022)FGLS procedure for the case in which lagged dependent variables are included as regressors (subsection 3.1). In subsection 4.2, we discuss an application of Perron and Olivari (2023) FGLS-IV method when the instrumental variable set includes lagged pre-determined regressors as instruments. These contributions, provide a consistent and efficient framework to estimate and perform tests of the parameters of a k-step-ahead linear forecasting equation that remains valid whether the REH holds or not. Empirically, the most significant contribution is that by applying the feasible GLS and GLS-IV methodologies described above to the estimation of the Fama and Hansen and Hodrick (1980) regressions, respectively, we obtain results that widely differ from the consensus in the literature. For the Fama regression, most results available in the literature suggest a negative estimate of the relevant parameter, often referred to as the “forward discount anomaly.” Our results in contrast, indicate positive values, sometimes not significantly different from one. This finding suggests that the “forward discount anomaly” might be a consequence of OLS providing an inconsistent estimate and our FGLS procedure being consistent and efficient under a broader range of possible scenarios. We also show statistically significant discrepancies between the OLS and GLS-IV estimates in the Hansen and Hodrick (1980) regression.

The remainder of this paper is as follows. Section 2 describes the formulations of the UIP hypothesis and reviews previous econometric tests. Section 3 describes the FGLS procedures and Section 4 introduces two instrumental variable (IV) based approaches for a setting in which lagged dependent variables are included as regressors. Section 5 studies the Fama (1984) regression and Section 6 is focused on the Hansen and Hodrick (1980) regression. For both specifications, we provide extensive Monte Carlo experiments to assess the finite sample performance of our FGLS procedures relative to OLS. We also provide empirical results for the sample period January 1993 to January 2023. In Section 7, we present a test for the hypothesis that the OLS residuals exhibits serial correlation of order greater than $k−1$ and discuss its empirical implications to understand the various conflicting results. In Section 8, we provide practical recommendations and brief concluding remarks are presented in Section 9. A supplement provide additional details.

Let $st=ln(St)$ and $ft,k=ln(Ft,k)$⁠, where $St$ and $Ft,k$ are the levels of the spot exchange rate and the k-period forward exchange rate at time t. With $it$⁠, the domestic nominal interest rate and $it∗$ the corresponding foreign interest rate, the theory of UIP implies that $E(st+k−st|Φt)=(it−it∗)$⁠. Hence, UIP requires the twin assumptions of rational expectations and a constant or zero risk premium. Given the no arbitrage condition, the covered interest parity (CIP) condition implies that $(it−it∗)=(ft,k−st)$ holds as an identity. Hence, the UIP condition is also frequently expressed as $E(st+k−st|Φt)=(ft,k−st)$⁠. Since $st+k−ft,k$ is an approximate measure of the rate of return to speculation, we can express the efficient-markets hypothesis as $ft,k=E(st+k|Φt)$⁠. This implies forecast errors $st+k−ft,k$ uncorrelated with information available at time t, $Φt$⁠.

As in Hansen and Hodrick (1980), we consider the general problem of estimating the parameters of a k-step-ahead linear forecasting equation, $E(yt+k|Φt)=xt′β$⁠. Then:

where rational expectations impose a specific structure on the forecast error $ut+k=yt+k−E(yt+k|Φt)$⁠. Due to the $k−1$ period overlap in the sequential k-step-ahead forecasts, $ut+k$ has an MA(⁠$k−1$⁠) representation. Thus:

and the OLS estimates of $α,β$ are consistent since the regressors are pre-determined via the rational expectations hypothesis. We shall label this as the “RE case”. Note, however, that we could well be faced with a model of the form:

where $ηt$ is some serially correlated process involving innovations dated before period t; e.g. an $AR(1)$ process of the form $ηt=ρηt−1+et$ for some sequence of i.i.d. innovations $(e1,…,eT)$⁠. In this case, if the regressors are non-exogenous with respect to past values of $ηt$⁠, OLS is no longer consistent since $E(xtηt) ≠ 0$⁠. We shall label this as the “general case”, as it encompasses the “RE case”. This motivates the necessity of an estimate that is consistent under the both the “RE case” and the “general case”, i.e. allowing the errors to be serially correlated beyond lag $k−1$⁠. Contrary to what is asserted in Hansen and Hodrick (1980), GLS is consistent in both cases, provided the errors $(ut+k+ηt)$ can be represented as some invertible linear process, as shown in Perron and González-Coya (2022).

Example 1. A k-step ahead forecast error process can be serially correlated beyond lags$k−1$under, e.g. Adaptive Expectations (AE) (seeMuth, 1960). An exponentially weighted moving average forecast arises from the following model of expectations adapting to changing conditions. Under AE, it is assumed that the forecast is changed from one period to the next by an amount proportional to the latest observed error:

As shown inMuth (1960), the solution of this difference equation is an exponentially weighted forecast $yte=κ∑i=1∞(1−κ)i−1yt−i$⁠. The forecast error is thus:

To characterize the process of the forecast error $ut$⁠, we shall impose a functional form on $yt$⁠. It is standard in the AE literature, to assume that $yt$ has a permanent and a transitory component, $yt=y¯t+ωt,$where the permanent component is defined by $y¯t=y¯t−1+εt=∑i=1tεi$with$εt∼i.i.d(0,σε2)$⁠, $ωt∼i.i.d(0,σω2)$ and $εt,ωt$ independent. In this case, the forecast error $ut$ follows an $AR(1)$ process. The details of this derivation are spelled in the  Appendix A.1.

Several tests of the UIP hypothesis have been proposed in the literature. Bilson (1981) and Fama (1984) analyzed regression (1) with $yt+k=st+k−st$ and $xt=ft,k−st$ with $k=1$⁠. In this setting a test of UIP is that $H0:α=0$ and $β=1$⁠. It has been noted by Fama (1984) and many subsequent studies that the estimated slope coefficient $β$ is frequently negative. This is known as the Forward Premium Anomaly: the country with the higher rate of interest has an appreciating currency rather than a depreciating currency; a violation of UIP. Hansen and Hodrick (1980) estimate the model (1) with $yt+k=st+k−ft,k$⁠, the forecast error, and uses lagged dependent variables as regressors; $xt=(yt,yt−1))$ with $k=13$⁠. The test is $H0:α=0$ and $β1=β2=0$⁠. A simplified version of this regression, with $xt=yt$ and test $H0:α=0$ and $β1=0$⁠, has been studied by Baillie et al. (2023). Hansen and Hodrick (1980) also proposed a test that includes the lagged forecast errors of four other currencies:

where $sti−ft−k,ki$ is the forecast error for country i and $stj−ft−k,kj$ is the forecast error for country $j ≠ i$⁠. We focus on regression (2) as Hansen and Hodrick (1980) concludes that the multicountry test appears to be more powerful.

We consider the linear regression:

where the error term follows a short-memory linear process:

where $εt∼i.i.d.(0,σ2)$⁠. The polynomial $θ(L)=∑j=0∞θjL j$ is assumed to satisfy $θ0=1$ (a normalization), $∑j=0∞ j|θj|<∞$ so that the process is short-memory and $θ(L)$ is invertible, i.e. we can write $θ(L)−1ut+k=εt+k$⁠. We use the FGLS procedure developed in Perron and González-Coya (2022) to estimate regression (3). The idea is to consistently approximate $ut+k$ using an autoregression of order $kT$⁠:

with $kT→∞$ and $kT3/T→0$ as $T→∞$⁠, to ensure consistent estimates; see Berk (1974). Replacing equation (3) and re-arranging terms we have the following:

with $δj=βρj$ for $j=1,…,kT$⁠. Equation (5) is often called the Durbin regression (see Durbin, 1970). The order of the autoregression, $kT∗$⁠, is determined via the minimization of the BIC suggested by Schwarz (1978) for $kT∈[0,kmax]$ where $kmax$ is such that $kmax3/T→0$ as $T→∞$⁠. We use the method suggested by Ng and Perron (2005) to ensure a proper comparison across models with different values of $kT$⁠, i.e. using the same effective number of observations. We estimate the Durbin equation (5) via OLS with $kt=kT∗$⁠. Using the OLS estimates of $ρj$⁠, $ρ^jD$⁠, we construct the quasi-differenced variables:

The FGLS estimate of $β$ is the OLS estimate of the quasi-differenced regression:

The resulting estimate $β^FGLS$ will be consistent provided the regressors $xt$ are pre-determined with respect to values $εt$ prior to period $t+k$⁠; see Perron and González-Coya (2022).

Remark 1. The consistency of our FGLS and GLS-IV estimators is unaffected by conditional heteroskedasticity. The key requirement for consistency is that the error process $ut+h$ admits a stationary invertible linear representation, so that the autoregressive filtering recovers serially uncorrelated errors and the quasi-differenced regressors remain pre-determined. Our variance estimators and the analogous IV-based expressions to be discussed below assume unconditional homoskedasticity of the quasi-differenced errors. Under conditional heteroskedasticity, these estimators remain consistent for the unconditional variance of the coefficient estimates.

Consider regression (3) with $xt′=(yt−h,wt′)$ for some $h>0$⁠, where $wt$ is a vector of $n+1$ pre-determined regressors that includes a constant term. For simplicity we omit the constant term without loss of generality. Write the regression as:

If $ut$ is autocorrelated beyond lags $i≥h−1$⁠, OLS applied to regression (8) is not consistent as $yt−h$ and $ut$ are not independent. Wallis (1967) and Malinvaud (1966) studied regression (8) with $h=1$ where the error terms follows an $AR(1)$ process. Expression for the asymptotic bias of the OLS estimates of $α$ and $β$ are given by Malinvaud (1966) and Griliches (1961). We can write the Durbin regression as follows:

where $γh=ρh+β$⁠, $γj+h=ρj+h−δj+h$⁠, for $j=1,…,kT∗−h$⁠; $δj+h=ρj+hβ$⁠, for $j=kT∗−h+1,…,kT∗$⁠; and $ψij=αiρj$ for $j=1,…,kT∗$⁠, $i∈1,…,n$⁠. For the case $h=kT∗=1$ and $wt$ a scalar (i.e. $n=1$⁠), Wallis (1967) and Malinvaud (1966) (p. 469) propose to estimate regression (9) using OLS and then estimate $ρj$ using $−ψj/α$⁠. For the general case $k>1$ and $kT∗>h$⁠, the same approach can be applied. Note that we need $kT∗≥h$⁠, otherwise the estimates are not consistent. But this condition will be satisfied, at least in large samples, when using the BIC to select the lag order. However, note that when $n>1$⁠, $ρj$ cannot be uniquely identified. We shall propose two estimation methods; one based on the Durbin regression (9) and one based on a first-stage instrumental variable (IV) estimate. These follow similar steps as in the feasible GLS procedure discussed above.

Remark 2. Estimating the autoregressive coefficients, $ρj$for$j=1,…,kT∗$⁠, from regression (9) using the Wallis (1967) and Malinvaud (1966) method does not allow us to uniquely identify $ρj$ in the general case with $k>1$⁠, $kT∗>kt$ and $n>1$⁠. However, we can obtain efficient estimates $ρ˜j∗$⁠, $j=1,…,kT∗$ by using a convex combination of the estimates $ρ˜ij=−ψ˜ij/α˜i$ (⁠$j=1,…,kT∗$⁠) for $i∈1,…,n$⁠. For ease of the exposition, suppose that $n=2$⁠. Then, we can construct efficient estimates $ρ˜j∗=λρ˜1j+(1−λ)ρ˜2j$⁠, where the optimal $λ$ that minimize $Var(ρ˜j∗)$ is (the details are in the  Appendix A.2):

In practice, we can estimate $Var(ρ˜ij)$ using a first order Taylor expansion:

The first order Taylor expansions of $Cov(ρ˜1j,ρ˜2j)$ are cumbersome. Note that if the instruments $wtj$ are independent of each other, $Cov(ρ˜1j,ρ˜2j)$ will be arbitrarily small in large samples. Hence, we set $Cov(ρ˜1j,ρ˜2j)=0$⁠.

While the adaptive expectations framework provides an analytically tractable illustration of how forecast errors can be serially correlated beyond lag $h−1$⁠, we emphasize that it is one instance of a broader class of departures from the rational expectations paradigm that generate such excess autocorrelation. Several economically motivated models of expectations formation share this property. Under sticky information models Mankiw and Reis (2002), agents update their information sets infrequently, so that aggregate expectations adjust sluggishly to new information. This generates forecast errors that inherit the persistence of the underlying fundamentals, producing autocorrelation well beyond the $h−1$ lags implied by overlapping forecasts alone. Similarly, noisy information models in the tradition of Sims (2003) and Woodford (2003), in which agents observe fundamentals with noise and must solve a signal extraction problem, produce forecast errors whose serial correlation structure reflects the dynamics of the signal-to-noise ratio rather than the forecast horizon alone. Models with heterogeneous expectations – where agents use different forecasting rules and the population weights evolve over time – also generate aggregate forecast errors with rich autocorrelation structures; see, e.g. De Grauwe and Grimaldi (2006). Finally, behavioral models featuring extrapolative or momentum-based expectations Barberis et al. (1998) can produce persistent forecast errors when agents systematically over- or under-react to recent exchange rate movements. The key insight is that our FGLS procedure does not require the econometrician to specify or identify which of these mechanisms is operative. It requires only that the error process be representable as a stationary invertible linear process. This agnosticism is a practical advantage: the procedure delivers consistent and efficient estimates regardless of whether the excess serial correlation originates from adaptive learning, rational inattention, heterogeneous beliefs or any other mechanism that generates a departure from the pure MA$(h−1)$ structure.

An alternative to estimate regression (8) with $n≥1$ is to use an instrumental variable procedure. If the regressors $wt$ are exogenous, then $wt−h$ are valid instruments for $yt−h$ and the two-stage least squares (2SLS) estimates will be consistent. Liviatan (1963) propose to use $wt−h$ as instruments for $yt−h$ and $wt$ as an instrument for itself, so that the instrument set is $Zt={wt−h,wt}$⁠. We can potentially select a larger set of instrumental variables for $yt−h$⁠; e.g. lags or order $i≥h$ of $wt$⁠. In this case $zt={wt,wt−i,p>i≥h}$⁠, for some $p>h$⁠. Note that p can be adaptively selected using appropriate tests for over-identifying restrictions (see Small, 2007). However, if the regressors $wt$ are not exogenous and $ut$ is autocorrelated beyond lags $i≥h−1$⁠, $wt−h$ are no longer valid instruments. Hence, the IV estimate of regression (8) using the instrument set $zt$ will not be consistent. This motivates the use of the GLS-IV procedure suggested by Perron and Olivari (2023). The idea is simple: first transform the model to have serially uncorrelated errors and then estimate the transformed model via IV using the set of transformed instruments. The resulting estimate will be consistent as the instrument set and regressors are pre-determined in the transformed model.

We first discuss two methods that are valid with exogenous instruments if $ut$ is autocorrelated beyond lags $h−1$⁠. One is the widely used so-called “optimal GMM” procedure. The other is akin to the GLS procedure discussed above but with the first-step using the GMM estimate to obtain estimate of the residuals and construct the autoregressive filtering.

Using the set of $n(p−h)$ exogenous instruments $zt$⁠, then, as shown in Hansen (1982), the best estimator of $β∗=(β,α)$ based on the instruments and the moment condition $E(Zu)=0$ is as follows:

where the $n(p−h)×n(p−h)$ matrix $Ω$ is given by $Ω=limT→∞T−1E[Z′uu′Z]$⁠. We can write $Ω=∑s=−∞∞Rv(s)$⁠, where $Rv(s)=E[Zh(t)utZh′(t−s)ut−s]$ and $Zh(t)={zt,…,zt−h}$⁠. Then $Ω$ can thus be consistently estimated by $Ω^=∑s=−TTλ(s,m)R^v(s)$⁠, where $R^v(s)=T−1∑t=1T−|s|vt′vt+|s|$⁠, with $vt=Zh′(t)ut$⁠, $λ(s,m)$ is some kernel or weight function and m is the bandwidth; see, e.g. Andrews (1991).

We also consider a FGLS method that does not rely on the Durbin regression (9). Instead, it uses a first-stage GMM estimate to obtain a consistent estimate of the residuals, $u˜t$⁠. We can thus identify the autoregressive parameters when the instruments are exogenous. The GMM-GLS-IV procedure to estimate regression (8) with exogenous instruments $wt$ is the following: 1) obtain the GMM estimator of regression (8), given by (10) using the set of instruments $zt={wt,wt−h,wt−h−j,j=1,…,h}$⁠. Compute the residuals, $u˜t=yt−α˜iv′wt−β˜ivyt−h$⁠; 2) select the order $kT∗$ of the autoregression:

via the minimization of the BIC for $kT∈[0,kmax]$ where $kmax$ is such that $kmax3/T→0$ as $T→∞$⁠; 3) estimate the autoregression (11) with $kT=kT∗$ to obtain consistent estimates $ρ˜j$ (⁠$j=1,…,k∗$⁠); 4) use $ρ˜j$ (⁠$j=1…,kT∗$⁠) to construct the quasi-differenced variables $yt∗=(yt−∑j=1kT∗ρ˜jyt−j)$⁠, $yt−h∗=(yt−h−∑j=1kT∗ρ˜jyt−h−j)$ and $wt∗=(wt−∑j=1kT∗ρ˜jwt−j)$⁠; 5) the GMM-GLS-IV estimate of $β$ is the IV estimate of the quasi-differenced regression:

using the set of quasi-differenced instruments $zt∗={wt∗,wt−h∗}$⁠.

We next describe the GLS-IV procedure to estimate regression (8), which is valid with non-exogenous instruments, provided they are pre-determined. The steps are the following: 1) Select the order of the Durbin regression (9), $kT∗$ via the minimization of BIC for $kT∈[0,kmax]$ where $kmax$ is such that $kmax3/T→0$ as $T→∞$⁠; 2) Estimate the Durbin regression (9) with the selected value $kT∗$ using OLS. The estimates the autoregressive coefficients, $ρ˜j∗$⁠, $j=1,…,kT∗$ are obtained using the efficient method described in Remark 2; 3) Use $ρ˜j$⁠, $j=1…,kT∗$ to construct the quasi-differenced variables $yt∗$⁠, $yt−k∗$ and $wt∗$⁠, as in Step 4 for GMM-GLS-IV; 4) The GLS-IV estimate of $β$ is the IV estimate of the quasi-differenced regression (12) using the set of quasi-differenced instruments $zt∗={wt∗,wt−h∗}$⁠.

Given that the OLS estimates from the Durbin regression (9) are consistent and that the transformed model (12) has serially uncorrelated errors, the GLS-IV estimates are consistent under the stated conditions. To the best of our knowledge, there is no other consistent estimation method requiring only pre-determined regressors for the general linear regression (8) without restricting the error process $ut$⁠. Maximum likelihood and non-linear least squares require an a priori known error process. In the simulation experiments reported in subsection 6.1.2, we show that the fact that $ρ˜j$ is not uniquely identified when we have more than one regressor (⁠$n>1$⁠), does not affect the efficiency of GLS-IV.

The Fama (1984) regression is as follows:

with $h=4$ for 1-month forward rates and $h=12$ for 3-month forward rates, when using weekly data. Estimates of (13) tell us whether the current forward-spot differential, $ft,h−st$⁠, has power to predict the future change in the spot rate, $st+h−st$⁠. Evidence that $β$ is significantly different from zero means that the forward rate observed at t has information about the spot rate to be observed at $t+h$⁠. Under the efficient-market hypothesis, we have $H0:α=0,β=1$⁠. Under the null hypothesis, the log of the forward rate provides an unbiased forecast of the log of the future spot exchange rate. Derivations from $β=1$ are sometimes interpreted as a measure of the variation of the premium in the forward rate.

Let $β¯$ be the OLS estimate of $β$ in regression (13). If the estimator is consistent, we have as follows:

If expectations are rational, then $st+1−st=Et(st+1)−st+εt+1$⁠, where $εt+1=st+1−Et(st+1)$ is the forecast error. In this case, $Cov(ft,h−st,st+1−st)=Cov(ft,h−st,Et(st+1)−st)$⁠. The foreign exchange risk premium when expectations are rational is defined as $rptre=ft,h−Et(st+1)$⁠. Under risk neutrality, expected profits from forward market speculation would be zero as agents would drive $ft,h$ into equality with $Et(st+1)$⁠. Write $Et(st+1)−st=ft,h−st−rptre$ and replace into equation (14), so that $plimT→∞(β¯)=1−βrp$⁠, where:

A negative estimate of $β$ in regression (13) is a robust finding in the literature (see Engel, 1996). This is known as the “forward discount anomaly”; it is a widespread empirical finding that the returns on nominal exchange rates appear to be negatively correlated with the lagged forward premium. Bilson (1981) and Fama (1984) provide evidence that the estimates of $β$ are less than zero. Many subsequent studies have confirmed that finding, for dollar exchange rates and a large number of exchange rates and time periods (see, for example, Bekaert and Hodrick, 1993; Backus et al., 1993; Hai et al., 1997). Froot (1990) notes that the average value of $β¯$ over 75 published estimates is −0.88. Only a few of the estimates are greater than zero, and none is greater than 1. The forward discount anomaly implies an appreciating currency for the high interest rate country. It is an “anomaly” as rational expectations would imply the opposite; if all currencies are equally risky, investors would demand higher interest rates on currencies expected to fall in value. The survey by Engel (1996) focuses on the possibility that $β¯rp ≠ 0$ among the possible explanations for finding $β¯<0$⁠. Other possible interpretations are that the forward rate is a biased predictor of the future spot rate, and/or that it is evidence of a time-varying risk premium. In this paper, we provide a novel insight on this empirical regularity. We argue that the OLS and GMM estimates of $β$ are not consistent, as the error term in regression (13) is serially correlated beyond lags $k−1$⁠. We find that the FGLS estimates, which are consistent under a broader range of conditions, are significantly non negative but in general smaller than 1.

Daily data were obtained for the spot exchange rates for the UK pound (US-UK), Canadian dollar (US-CAD) and Japanese yen (US-JP) as well as the one- and three-month forward exchange rates data for the three currencies. As in Hansen and Hodrick (1980), the data were sampled to form a weekly series constructed by taking observation on Tuesday of each week. If no Tuesday observation was available, we used the Wednesday observations. The source of the forward exchange data for US-UK and US-CAD is Barclays Bank PLC; the source for US-JP is the Bank of Tokyo Mitsubishi. For all the data sets, we use all the information available until January 17, 2023 but the starting date of the time series differ: a) US-UK: starting date of October 11, 1983, 2,050 observations; b) US-CAD: starting date of December 14, 1984, 1,989 observations; c) US-JP: starting date of September 1, 1993, 1,535 observations.

In this section, we provide simulation results related to the Fama (1984) regression (13) under the “RE case” or efficient market hypothesis (EMH), $H0:α=0,β=1$⁠. We consider $h=4$ for one-month forward rates and $h=12$ for three-month forward rate. As discussed in Section 2, the EMH implies that $ut+h$ has an $MA(h−1)$ representation. We simulate an $MA(h−1)$ process with parameters calibrated to replicate the observed autocorrelation function up to lag $h−1$ of the FGLS residuals of equation (13) using US-UK data. The details are presented in the  Appendix A.3. For $h=4$ we have an $MA(3)$ representation with:

For $h=12$ we have an $MA(11)$ representation with coefficients:

We simulate an error process based on equation (15), i.e. $ut+h=θ(L)εt$⁠, where $εt∼i.i.d.$$N(0,σε2)$ and $σε2$ is estimated using the residuals of an initial Fama FGLS regression (13) for US-UK. The data generating process (DGP) uses $xt=ft,h−st$ observed in the data for US-UK, and the stated simulated error process. We artificially generate $yt+4$ to satisfy the null hypothesis $H0:β=1$⁠. Thus, the DGP is as follows:

We also consider a departure from the “RE case” with errors $ut+h$ following the “general case” with serial correlation at lags $i≥h$⁠. We assume that $ut+h$ follows an $ARMA(1,h)$ process with MA coefficients given by (15) and (16) depending on $h=4$ and $h=12$⁠, respectively. The AR coefficient is set to $ρ=0.6$⁠. We consider two sampling periods; the complete sample from October 11, 1983 to January 17, 2023 with 2,050 observations and the one spanning November 1. 1989 to April 1, 2021, as considered in Baillie et al. (2023). We perform 5,000 replications. We consider the following estimators of $α,β$⁠, from regression (13): a) The OLS estimate with HAC standard errors based on the quadratic spectral window suggested by Andrews (1991) with automatic bandwidth selection using an $ARMA(1,1)$ approximation; b) The Durbin estimate based on the regression (5) with $yt=st+h−st$ and $xt={1,ft,h−st}$ and $kT∗$ selected using the BIC. For the complete sample, we consider $kmax=40$ and for the sub-sample we set $kmax=30$⁠; c) The FGLS estimate based on regression (7) with $yt=st+h−st$⁠, $xt={1,ft,h−st}$ and $kT∗$ selected using the BIC. The same values of $kmax$ are used. We estimate the sample variance of the FGLS estimator using $Var(γ^FGLS)=(X∗′X∗)−1σ^FGLS2$⁠, where $X∗$ is the $T×2$ matrix of quasi-differenced regressors (including a constant term) and $σ^FGLS2$ is the sample variance of the FGLS residuals, $u^FGLS=yt−γ^FGLS′xt$⁠. The confidence intervals at the 95% nominal level for the jth coefficient are obtained using $γ^j±z0.975Var(γ^j,FGLS)1/2$⁠, where $z0.975$ is the 0.975 quantile of the normal distribution.

The simulation results are presented in Table 1 for $h=4$ and Table 2 for $h=12$⁠. In line with the theory, the mean squared error (MSE) of OLS is small when the error term follows an $MA(h)$ process and deteriorates when the error is serially correlated at lags $i≥h$⁠. Clearly, FGLS outperforms OLS and Durbin in all cases. Even under the “RE case”, the MSE of OLS is on average 3.4 times larger than of FGLS, while the MSE of Durbin is close to that of OLS case. When the errors follow an $ARMA(1,h)$ process, OLS is no longer efficient, and its MSE is on average 27 times that of FGLS. As expected, the Durbin estimate remains consistent in this case, but its MSE is on average three times that of FGLS. FGLS also has the smallest variance. The variance of Durbin is on average three times the variance of FGLS for $h=4$ and two times the variance of FGLS for $h=12$⁠. The coverage rates of the confidence interval for FGLS are near the nominal 90% and have shortest lengths. OLS with HAC standard errors exhibits substantial size distortions with $ARMA(1,h)$ errors, especially when $h=12$ case. The OLS based HAC standard errors provides confidence intervals close to the nominal level in some cases, at the expense of a very large variance. For the $ARMA(1,h)$ case with $k=4$$(12)$ the variance of OLS is 30 (25) times the variance of FGLS. This results are in line with those from Perron and González-Coya (2022).

For the simulations related to power, we use US-UK observed data $xt=ft,h−st$ for the period November 1, 1989 to April 1, 2021 with one-month forward rates, $h=4$⁠. The DGP is as follows:

where $β∈{0,0.25,0.5,0.75,1,1.25,1.5,1.75,2}$⁠. The null value is $β=1$⁠. The error process $ut+4$ is simulated as before; under the “RE case” $ut+4$ follows an $MA(3)$ process, and under the “general case” $ut+4$ follows an $ARMA(1,3)$ process. Figures 1 and 2 present plots of the empirical rejection frequencies for t-test of $H0:β=1$ with nominal size 0.05 for the OLS estimates with HAC-$ARMA(1,1)$ standard errors, the FGLS estimates based on regression (7) and the Durbin estimates based on the regression (5). For the latter two, $kT∗$ is selected via BIC with $kmax=30$⁠. Figure 1 pertains to “RE case” with the errors an $MA(3)$ process, while Figure 2 pertains to the “general case” with errors following an $ARMA(1,3)$ process.

Note that in Figure 1, “RE case” with $MA(3)$ errors, OLS and Durbin has very small rejection frequencies even when $β$ is far from the null value 1, even though they are consistent, which can be attributed to their lack of efficiency in finite samples. As shown in Figure 2, the power functions are similar in the case with $ARMA(1,3)$ errors, with the exception that the power function of OLS decreases and flattens with an almost constant 10% rejection frequency for all values, despite having a more liberal size.

We present the estimation results of regression (13) using the three estimates considered before: OLS, Durbin estimates based on the regression (5) and FGLS based on regression (7). We use data for three currencies, US-UK, US-CAD and US-JP and we consider two sampling periods; the first one is the complete sample and the second one spans from November 1, 1989 to April 1, 2021, the sampling period considered in Baillie et al. (2023). For the Durbin and FGLS estimates we set $kmax=40$ for the full sample and $kmax=30$ for the sub-sample period. Tables 3 and 4 present the estimation results using one-month forward rates (⁠$h=4$⁠) and the three-month forward rates (⁠$h=12$⁠), respectively. In line with the empirical regularity in the literature, the OLS estimates of $β$ are not significantly positive in all cases. In contrast, the Durbin and FGLS estimates are positive in most cases and significantly non negative in some. In some cases, such as the US-JP exchange rate for three-month forward rates, we observe a negative significant OLS estimate and a positive significant FGLS estimate. Recall that the Durbin and FGLS estimates are consistent even when the error follows a general linear process. Section 7 present the outcome of the Cumby and Huizinga (1992) test (CH-test) for autocorrelation at lags $q>h−1$⁠. The outcome is a rejection in all cases. Hence, our finding suggests that the forward discount anomaly might be a consequence of OLS providing an inconsistent estimate of $β$ caused by non-exogenous regressors. We thus interpret the large differences between the OLS and the FGLS estimates as evidence of OLS being inconsistent.