A. The Forward Bias Puzzle

In an efficient speculative market, prices should fully reflect information available to market participants and it should be impossible for a trader to earn excess returns to speculation. Under foreign exchange market efficiency and risk neutrality, UIP holds:

where s_t denotes the logarithm of the spot exchange rate (domestic price of foreign currency) at time t; i_t,k and $i_{t.k}^{*}$ are the nominal interest rates available on similar domestic and foreign securities, respectively (with k periods to maturity); Δ_ks_t+k ≡ s_t+k – s_t; and the superscript e denotes the market expectation based on information at time t. Testing UIP in its form as given by Equation (1) is tantamount to testing the joint hypothesis that foreign exchange market participants are. in an aggregate sense, i) endowed with rational expectations and ii) risk neutral.

Most often, however, empirical analyses of UIP have taken place in the context of the relationship between spot and forward exchange rates under the assumption that covered interest parity (CIP) holds: $f_t^k-s_t=i_{t.k}-i_{t.k}^{*}$, where $f_t^k$ is the logarithm of the k-period forward rate (i.e., the rate agreed now for an exchange of currencies k periods ahead). Indeed, CIP is a reasonably mild assumption, given the extensive empirical evidence suggesting that CIP holds (Frenkel and Levich, 1975, 1977: Taylor, 1987; for a survey of this evidence, see, e.g.. Sarno and Taylor, 2003, Ch, 2). Note that, unlike CIP, UIP is not an arbitrage condition because one of the terms in the UIP equation, namely the exchange rate at time t + k, is unknown at time t and, therefore, non-zero deviations from UIP ex ante do not necessarily imply the existence of arbitrage profits ex post due to the foreign exchange risk associated with future exchange rate movements.

Using CIP and replacing the interest rate differential $i_{t,k}-i_{t,k}^{*}$ with the forward premium (or forward discount) $f_t^k-s_t$. a number of researchers have tested UIP by estimating a regression of the form:

where we have assumed k = 1 for simplicity, and v_t+1 is a disturbance term. Under UIP, α = 0, the slope parameter β = 1, and the disturbance term v_t+1 (the rational expectations forecast error under the null hypothesis) must be uncorrelated with information available at time t (e.g. Fama. 1984).

Empirical studies based on the estimation of Equation (2) for a large variety of currencies and time periods, generally report results which reject UIP (e.g.. see the references in the survey of Hodrick, 1987; Lewis, 1995; Taylor, 1995; Engel, 1996; Sarno. 2005). Indeed, it constitutes a stylized fact that estimates of β using exchange rates against the U.S. dollar are often statistically insignificantly different from zero and generally closer to minus unity than to plus unity (Froot and Thaler, 1990). The stylized fact of a negative β coefficient in this regression implies that the more the foreign currency is at a premium in the forward market, the less the home currency is predicted to depreciate.¹ The negative value of β is the central feature of the forward bias puzzle, and following much previous literature, we refer to Equation (2) as the “Fama regression.”^{2 3}

The literature has also investigated the predictability of U1P deviations (or foreign exchange excess returns) using the forward premium as a predictor variable in a linear model obtained from reparameterizing Equation (2) as follows:

where the excess returns $E{R}_{t+1}\equiv {\Delta }{s}_{t+1}-(f_t^1-s_t)\equiv s_{t+1}-f_t^1$ This regression was investigated, for example, by Biison (1981). Fama (1984), and Backus, Gregory, and Telmer (1993) and was shown to generate strong predictability of excess returns (deviations from UIP) on the basis of the lagged forward premium. Specifically, while β^τ should be zero under UIP, the evidence, consistent with a negative estimate of β in Equation (2), is that β^τ is negative and statistically significantly different from zero. Clearly, given that Equation (3) is obtained simply from reparameterizing the Fama regression (2), the forward bias puzzle arising from Equation (2) and the predictability of excess returns documented on the basis of Equation (3) must be linked, and any explanation of the forward bias puzzle (β ≠ 1) ought to be able to explain also the finding of a non-zero value of β^τ in Equation (3). We shall return to the link between the forward bias puzzle and the predictability of excess returns in Section V, where we will show that both these stylized facts can indeed be matched using a nonlinear model of regime-dependent UIP deviations which is rationalized by the existence of limits to foreign exchange speculation.

It is also worth noting that the framework described in this section (as well as our nonlinear framework described later) rely on the notion that spot and forward exchange rates are cointegrated and that the forward premium should, therefore, be stationary, in a similar fashion to Brenner and Kroner (1995) and Clarida and others (2003). Furthermore, another finding of the relevant literature which should be noted here and will be recalled later as appropriate is the evidence that estimates of β display significant time variation across different sample periods over the recent float.

The range of estimates of β over time is large and includes both negative and positive values (e.g.. Baillie and Bollerslev, 2000).

B. The Rationale for Nonlinearity in the Fama Regression

The idea that there may be nonlinearities in the spot-forward relationship is not novel. For example, the work of Dumas (1992) on general equilibrium models of exchange rate determination in a spatially separated world with international trade costs generated a variety of exchange rate equations where nominal exchange rates are shown to depend nonlinearly on their fundamentals in a way that reversion towards international parity conditions is a function of the size of the deviation from the parity conditions themselves—e.g., see Dumas (1992, p. 174, Equation 23); see also Baldwin (1990). Hollifield and Uppal (1997) and Obstfeld and Rogoff (2000). Sercu and Wu (2000) derive, in a partial equilibrium model, an expression for the spot-forward relationship where, in the presence of transaction costs, expected exchange rate changes and forward premia are imperfectly aligned, inducing nonlinearity in the spot-forward relationship. Mark and Moh (2002) and Moh (2002) study continuous-time models where UIP is a stochastic differential equation which has a solution where the exchange rate is a nonlinear function of the interest differential, modelled according to a jump-diffusion process regulated by occasional central bank intervention. This model records some success in matching some of the moments in the data and is capable of shedding some light on the forward bias puzzle when central bank intervention is not announced and takes the market by surprise.

A related, albeit different, rationalization of nonlinearity in the spot-forward relationship stems from the limits-to-speculation hypothesis. A rich account of the implications of limits to speculation for market efficiency tests and the nonlinear behavior of deviations from UIP is provided by Lyons (2001, Ch. 7, pp. 209-20). The line of reasoning is that financial institutions will only take up a currency trading strategy if the strategy yields a Sharpe ratio at least equal to an alternative investment strategy, say a buy-and-hold equity strategy. As it is well known, the Sharpe ratio is commonly defined as (E[R_s] – R_f)/σ_s, where E[R_S] is the expected return on the strategy, R_f is the risk-free interest rate, and σ_s is the standard deviation of the returns to the strategy. In essence, the Sharpe ratio may be seen as the expected excess return from speculation per unit of risk. Given that the realized Sharpe ratio for a buy-and-hold equity strategy has averaged about 0.4 on an annual basis for the United States over the last 50 years or so,⁴ a buy-and-hold currency trading strategy yielding a Sharpe ratio lower than 0.4 would not be worth taking up. Although the specific example in Lyons (2001) is based on comparisons of unconditional buy-and-hold Sharpe ratios (static strategies) with currency and equity strategies, the same logic is applicable to comparisons between two (or more) dynamic, conditional strategies, where one would examine conditional Sharpe ratios at a point in time and engage in market-timing activities.

Noting that under the null hypothesis that UIP holds (i.e., foreign exchange market efficiency). α = 0 and β = 1 in Equation (2) and the Sharpe ratio of currency strategies is zero, then it is only when β departs from unity that the numerator of the Sharpe ratio takes nonzero values.⁵ Indeed, according to Lyons’s calculations, it is only when β < –1 or β ≥ 3 that the Sharpe ratio for currency strategies is about the same as the average from a buy-and-hold equity strategy. i.e., 0.4 (see Lyons, 2001, p. 210). This argument effectively defines a band of inaction for the forward bias such that if –1 < β < β financial institutions would have no incentive to take up the currency strategy since a buy-and-hold equity strategy would have a higher return per unit of risk; within this band of inaction the forward bias and expected deviations from UIP are too small to attract speculative capital and, therefore, do not imply any glaring profit opportunity.

In its essence, the limits-to-speculation argument implies that, within a certain band of β (and. consequently of the Sharpe ratio), the forward bias does not attract capital and hence may potentially persist for a long time. In some sense, this argument suggests that limits to speculation and the existence of an opportunity cost of speculative capital create a band for the deviations from UIP where the marginal cost of taking up a currency strategy exceeds the marginal benefit.⁶ The crucial implication of the above analysis is that when limits to speculation of the kind described by Lyons (2001) are taken into account, the spot exchange rate and forward exchange rate need not move together at all times and, indeed, they may even move in opposite directions within a bounded interval without giving rise to any glaring profit opportunities. This argument is also consistent with the evidence that β (and hence the forward bias) is in fact highly time-varying, inducing parameter instability in standard linear regression models which assume a constant value of β over time (Baillie and Bollerslev, 2000). Arguments of this sort may be used to motivate the adoption of threshold-type models of the type originally proposed by Tong (1990) to empirically characterize the spot-forward relationship or the behavior of deviations from UIP: these threshold models would allow for a band within which β may differ from unity and may be positive, zero, or even negative, while outside the band the process switches abruptly to become exactly consistent with UIP and β = 1. Strictly speaking, assuming instantaneous allocation of speculative capital to currencies at the edges of the band of inaction then implies that the thresholds become reflecting barriers.

Nevertheless, while threshold-type models are appealing in this context, various arguments can be made to rationalize multiple-threshold or smooth, rather than single-threshold or discrete, nonlinear adjustment in deviations from UIP. First, the thresholds may be interpreted more broadly to reflect the opportunity cost of speculative capital, proportional transactions costs, and the tendency of traders or financial institutions to wait for sufficiently large Sharpe ratios before entering the market and trading (see, for example, Sofianos, 1993, Neal, 1996, and Dumas, 1992).

Second, one may argue that the assumption of instantaneous trade at the edges of the band of inaction should be replaced with the presumption that it takes some time to observe a profitable trading opportunity and execute transactions and that trade is infrequent (Dumas. 1992) and characterized by “limited participation” due to the fact that information costs may limit the participation of some classes of traders in derivatives markets (Grossman and Weiss, 1983 Hirshleifer, 1988). Essentially, limited participation models assume that agents adjust their portfolios infrequently, with a different subset of agents adjusting in each period. Limited participation in the foreign exchange market by nonfinancial corporations and unleveraged investors⁷ implies that their portfolio shifts will be gradual, rather than abrupt (Lyons, 2001, p. 218).

Third, in a market with heterogeneous agents who face different levels of position limits, agents essentially face bands of different sizes. For relatively small deviations of β from the edges of the band of inaction, only some traders or institutions will be able or willing to effect trades. As deviations from the edges of the band get larger, then progressively more agents will enter the market to effect trades. Thus, the forces pushing β toward the band of inaction will increase as the deviations from the edges of the band increase since an increasing number of agents face profitable opportunities. Consequently, we should observe a smooth transition toward UIP, with the speed of reversion of the deviations from UIP toward zero increasing with the degree of violation of the band of inaction itself (see Dumas, 1992).⁸

Overall, the arguments discussed above suggest that limits to speculation create a band within which UIP does not hold and where spot and forward rates may be unrelated or even move in opposite directions; further, deviations from UIP can stray beyond the values implied by the edges of the band. Once Sharpe ratios are large enough to attract speculative capital, deviations from UIP become increasingly mean reverting with the size of the Sharpe ratios. Under certain restrictive conditions (including, inter alia, identical limits to speculation and position limits, and homogeneity of agents), the reversion to UIP (β = 1) may be discrete, but in general it is smooth, and Teräsvirta (1994) and Granger and Lee (1999) suggest that even in the former case, time aggregation will tend to smooth the transition between regimes. Hence, smooth rather than discrete adjustment may be appropriate in the present context, and time aggregation and nonsynchronous adjustment by heterogeneous agents are likely to result in smooth aggregate regime switching. This is indeed the kind of behavior we shall try to capture in our empirical framework, as discussed in the next section. Also, note that this behavior is consistent with the evidence that spot and forward exchange rates are cointegrated (Brenner and Kroner, 1995; Clarida and others, 2003) since the wedge between exchange rate movements and the forward premium cannot grow to infinity without agents regarding departures from UIP as economically important and exploiting them. It is this speculative process that binds together spot and forward rates in the long run, in the same spirit as the general arguments in Brenner and Kroner (1995). This modeling strategy is also consistent with the evidence of parameter instability reported in Baillie and Bollerslev (2000), providing a rationale for the result of instability of the parameter β over time: under the working hypothesis of nonlinearity induced by limits to speculation. β is a function of time-varying Sharpe ratios and, therefore, is expected to vary over time.

A. The Empirical Framework

One characterization of nonlinear adjustment in the Fama regression which allows for smooth adjustment is in terms of a smooth transition regression (STR) model (Granger and Teräsvirta, 1993; Teräsvirta, 1994, 1998). An STR model of spot and forward rates may be written as follows:

where ε_t+1 is a disturbance term. The transition function $G[E{R}_t^e,{\gamma }]$ determines the degree of reversion to zero of the deviations from UIP and is itself governed by the parameter γ, which effectively determines the speed of reversion to UIP, and the transition variable. The transition variable is assumed to be the expected excess return $E{R}_t^e$ specifically. $E{R}_t^e$ is the deviation from UIP at time t + 1 that is expected by market participants conditional on information available at time t. Therefore, all terms on the right-hand side of Equation (4), namely $f_t^1-s_t$ and $E{R}_t^e$ are known at time t.

A simple transition function suggested by Granger and Teräsvirta (1993) and Teräsvirta (1994, 1998) is the exponential function:

in which case (4) would be termed an exponential STR or ESTR model. The exponential transition function G is bounded between zero and unity, i.e., G: ℜ → [0,1], has the properties G[0] = 0 and lim_x→±∞ G[x] = 1, and is symmetrically inverse-bell shaped around zero. These properties of the ESTR model are particularly attractive in the present context because they allow a smooth transition between regimes and symmetric adjustment of the deviations from UIP above and below the equilibrium level, consistent with the limits to speculation hypothesis. Note also that, given the definition in Equation (5), which relates G to H linearly, H: R → [0,1], has the properties H[0] = 1 and lim_x→±∞ H[x] = 0, and is symmetrically bell shaped around zero. Obviously. G = 0 when H = 1, and G = 1 when H = 0.

The transition parameter γ determines the speed of transition between the two extreme regimes, with lower absolute values of γ implying slower transitions. The “lower” and “upper” regimes are defined as the regimes corresponding to the two extreme values of the transition function, where G(•) = 0 and G(•) = 1, respectively. Investigation of the properties of the model in these two extreme regimes sheds light on the stability and dynamic properties of the STR model. The arguments in the spirit of limits to speculation suggest the restrictions α₂ = –α₁ and β₂ = 1 – β₁. Under these restrictions (which we test formally in our empirical work), the lower regime corresponds to $E{R}_t^e=0$ where G(•) = 0 and Equation (4) becomes a standard linear Fama regression of the form:

The upper regime corresponds, for a given γ. to ${\lim }_{E{R}_{t\to \pm \infty }^e}G[E{R}_t^e.{\gamma }]$ where (4) becomes a

different linear Fama regression with parameters exactly consistent with UIP:

This formulation of the nonlinear Fama regression has several virtues. First, the model nests the standard linear Fama regression, to which it would collapse in the absence of nonlinearity. Second, under the restrictions α₂ = –α₁ and β₂ = 1 – β₁. which are formally testable using standard statistical inference, this specification captures the behavior of the deviations from UIP which is implied by the theoretical considerations discussed in Section II. Deviations from UIP may be persistent and consistent with the well-known forward bias when they are in the neighborhood of UIP, that is, when expected excess returns are too small to attract speculative capital. In this lower regime. UIP does not hold but departures from UIP are economically small. However, for larger expected deviations from UIP (of either sign) that move exchange rates toward the upper regime, financial institutions would take up the glaring profit opportunities provided by currency trading strategies and induce reversion towards the UIP condition.

In short, the model allows for departures from UIP at all points where the transition function G (•) = 1 – H (•) ≠ 1. There is only one value of the transition function and one set of parameter restrictions that allow us to conclude that UIP holds (rational expectations and risk neutrality are both valid). In essence, the model may be seen as a characterization of the relation between spot and forward rates that allows for departures from UIP, while yielding UIP under specific restrictions. The conditions α₂ = – α₁ and β₂ = 1 – β₁ guarantee global stability, as exchange rate changes are moving in response to the lagged forward premium in a way to restore UIP under those conditions. Put another way, the model is consistent with the evidence of cointegration between spot and forward exchange rates, because the long-run tendency of the model is the UIP condition. where ${\Delta }{s}_{t+1}-(f_t^1-s_t)=0$. Because Δs_t+1 is stationary, and each of $f_t^1$ and s_t is nonstationary, the only way in which ${\Delta }{s}_{t+1}-(f_t^1-s_t)$ can be stationary is if $f_t^1$ and s_t co-move in the long run and cointegrate with a long-run parameter of unity (e.g., Brenner and Kroner, 1995).

Note that the transition function, G(•) is defined as a function of the conditional expected excess return, $E{R}_t^e$ Strictly speaking, the arguments presented in Section II.B are in terms of (conditional or unconditional) Sharpe ratios, which would require us to use the expected excess return divided by the standard deviation of the excess return. We use survey data on exchange rate expectations to construct a proxy for the conditional expected excess return. As described in the data section below, survey data to proxy a conditional standard deviation are not available. Consequently, in the empirical work, we standardize the excess return by its unconditional standard deviation (see Section IV.B). This proxy is, however, imperfect in that the Sharpe ratio is calculated using the conditional expected excess return in the numerator and the unconditional standard deviation in the denominator. We shall test for robustness of our core results using an alternative proxy for the risk factor in the denominator of the Sharpe ratio, namely, the implied volatility from currency options.

We note that, if the true DGP of the spot-forward relationship is indeed nonlinear of the form (4), then β, as given in Equation (2), will lie in the interval between β₁ and (β₁+ β₂) = 1. It seems plausible that if the distribution of UIP deviations is consistent with the majority of observations being in the lower regime (where β may be negative and the forward bias is expected to be persistent), one may well find negative values of β from estimating the linear Fama regression (2). We shall investigate exactly this issue in Section V using Monte Carlo methods.

It is also instructive to reparameterize the nonlinear Fama regression (4) in terms of deviations from UIP by subtracting the forward premium. $(f_t^1-s_t)$ from both sides of Equation (4) as follows:

The discussion on the effects of limits to speculation in the previous section suggests that the larger the expected deviation from UIP. the larger the effect of speculative forces in generating reversion toward UIP. This implies that while β* < 0 is admissible in Equation (8), one must have β₂> 0 and β* + β₂ = 0 for the forward premium to have no predictive power on future excess returns in the upper regime.

Note that Equation (8) may be seen as the nonlinear analogue of (and indeed nests) the predictability regression (3), exactly like Equation (4) is the nonlinear analogue of the Fama regression (2). Hence, Equation (8) also has implications for conventional tests of predictability of excess returns using the forward premium as a predictor variable based on a linear model obtained from repara-meterizing the Fama regression (2). Clearly, if the true DGP of UIP deviations is indeed nonlinear of the form (4) (or (8)). then β^τ as given in Equation (3) will lie in the interval between β* and (β* + β₂) = 0. Whether β^τ is closer to β* or to (β* + β₂) will depend on the distribution of UIP deviations, but it seems at least possible that if the distribution of UIP deviations is consistent with the majority of observations being in the lower regime one may find negative and statistically significant estimates of β^τ from estimating Equation (3). Again, we shall investigate this issue in Section V using Monte Carlo methods.

Granger and Teräsvirta (1993) and Teräsvirta (1994) also suggest the logistic function as a plausible transition function for some applications, resulting in a logistic STR (LSTR) model, which implies asymmetric behavior of the deviations from UIP according to whether they are positive or negative, which could potentially arise in the context of this paper due to, for example, short-sale restrictions. Hence, we do test for nonlinearities arising from the LSTR formulation as a test of specification of the estimated models in the section discussing the empirical analysis. Also, as a preliminary to our estimation of a nonlinear Fama regression, we evaluate the adequateness of the linear Fama regression by performing tests of linearity against nonlinearity of smooth-transition type (for both ESTR and LSTR formulations) and by testing the hypothesis of symmetry directly.

However, we wish to emphasize that, while our empirical analysis is inspired by the limits to speculation hypothesis, we do not claim to provide a direct test of this specific hypothesis, but rather a test of its general predictions for the relationship between spot and forward exchange rates. Our approach is best interpreted as an empirical characterization of the spot-forward relationship motivated by the limits to speculation hypothesis or simply as an empirical investigation of parsimonious models of foreign exchange excess returns. In particular, although we have focused on a specific nonlinear formulation of the relationship between exchange rates and forward premia capable of capturing some of the key predictions of the limits to speculation hypothesis, we cannot discard the possibility that the source of nonlinearities we document below is a factor other than limits to speculation.

B. The Solution of the Model When the Market Expectation Equals the Mathematical Expectation

In this subsection, we further clarify the properties of the model and the treatment of market (subjective) expectations and mathematical expectations. Taking expectations of Equation (8) and defining the mathematical expectation conditional on information at time t as $E_t(y_{t+1})=y_t^m$ yields

Then, noting that the market expectation needs not be equal to the mathematical expectation, we can define

with η_t = 0 if rational expectations hold. Substituting (10) into (9) gives

We can now formally derive the conditions under which there is a unique solution for $E{R}_t^e$ in this model. To this end, let us impose the conditions under which the theoretical implications of the limits to speculation hypothesis hold, i.e., α₂ = –α₁ and β₂ = 1 – β₁, and with expectations formed rationally, i.e., $E{R}_t^e=E{R}_t^m$ and η_t = 0. Equation (12) then becomes:

where $H[E{R}_t^m,\gamma ]=1-G[E{R}_t^m,\gamma ]=\text{exp}[-\gamma {(E{R}_t^m)}^2]$

Building on the results of Peel and Venetis (2005), the solution with respect to $E{R}_t^m$ is as follows:

where W(x) is the Lambert W function (also called Omega function), defined implicitly as the solution of W(x)e^w(x) =x; see Appendix I for a derivation of this equation. The solution captures the expectations-consistent process and embodies the notion that expectations are a monotonic function of the current forward premium, $(f_t^1-s_t)$ The expectations-consistent process implies that adjustment to UIP is faster the greater the expected deviation from UIP; essentially, on the basis of information on the forward premium, $(f_t^1-s_t)$ as well as the parameters α₁ and (β₁ – 1), agents form expectations of UIP deviations next period, which may be seen as underlying a carry-trade type strategy of the kind commonly followed by foreign exchange speculators. Speculative forces would therefore be increasing in the current forward bias, related to the expected deviation from UIP according to Equation (13), which is the underlying rationale of the arguments described in Section II.B—see also the discussion in Peel and Venetis (2005) on arbitrage and expectations consistency in the context of arbitrage in international goods markets and purchasing power parity.

The Lambert W function does not admit a closed-form expression, and so it would have to be approximated in empirical modeling if one wants to take to the data Equation (13) (see Corless and others, 1996, and the references therein). However, this is not a problem in the context of this paper because in our empirical model we employ survey data on exchange rate expectations to proxy the transition variable. $E{R}_t^e$ As discussed later, these expectations data are well known to be correlated with the forward premium, a fact that is consistent with the above theoretical result that expectations of UIP deviations are a monotonic function of current departures from UIP.

A. Data, Summary Statistics, and the Fama Regression

Our data set comprises weekly observations of spot and 4- and 13-week (or one- and three-month) forward U.S. dollar exchange rates against the Japanese yen, the U.K. sterling, the German mark, the euro, and the Swiss franc. Due to data availability considerations and the advent of the euro on January 1, 1999, the sample period spans from January 4, 1985 to December 31, 2002, for all exchange rates except for the German mark (January 7, 1986, to December 31, 1998) and the euro (January 5, 1999, to December 31, 2002). Following previous literature (e.g., Hansen and Hodrick, 1980, p. 852), data are taken on Tuesdays of every week, from Datastream. To keep the notation simple, a four-week change in a variable is stated as a change from t to t + 1, and $f_t^j$ is the forward rate for a contract with j months (or j months to maturity). From this data set, we constructed the time series of interest, namely the logarithm of the spot exchange rate, s_t and the logarithm of the one- and three-month forward exchange rates, $f_t^1$ and $f_t^3$ respectively, both at the weekly and monthly frequency. The core of the empirical work is based on s_t and $f_t^1$ at the weekly frequency, whereas we use the weekly $f_t^3$ as well as monthly data for s_t,$f_t^1$ and $f_t^3$ in our robustness analysis.

In order to construct a proxy for the expected excess return, $E{R}_t^e$ we use survey data on exchange rate expectations from Money Market Services (MMS). MMS reports the median forecasts of survey respondents. The survey provides data on a weekly basis for both one-month and three-month expectations of the dollar exchange rates examined. Participants to the survey include exchange rate dealers, banking and corporate economists, as well as market economists. Because MMS reports the median forecasts, this measure masks individual heterogeneity. However, to analyze issues related to market efficiency of the kind studied in this paper, one has to resort to an aggregate expectation of the exchange rate and, therefore, the MMS measure seems appropriate. Unfortunately, MMS does not report a measure of dispersion of the MMS forecasts nor does it report data on forecasts of the exchange rate volatility. This prevents us from constructing the ideal Sharpe ratio, namely the conditional expected excess return divided by the conditional expected standard deviation. Essentially, MMS only provides us with the numerator of the Sharpe ratio, which we shall standardize by dividing it by the unconditional standard deviation in the core results and by the conditional implied volatility in our robustness checks.

The advantage of using the MMS forecasts is that the properties of these exchange rate expectations are well-documented in the literature (see the references in Takagi, 1991; Osterberg, 2000; Sarno and Taylor, 2001). Specifically, expected changes in exchange rates have been found to have a tendency to underpredict actual exchange rate movements, implying that part of the actual exchange rate changes are unexpected. The evidence also suggests that “unbiasedness”—the notion that survey measures are unbiased forecasts of actual future outcomes—is rejected in time series evidence but is not rejected in cross-sectional evidence (Chinn and Frankel, 2002). Another feature of some importance in relating survey measures to the rational expectations hypothesis is that MMS forecasts are not consistent with the “orthogonality” hypothesis, implying that survey data do not fully incorporate all available information, in contrast with the rational expectations paradigm (see Takagi, 1991; Osterberg, 2000). Finally, exchange rate expectations are correlated with the forward premium, suggesting that at least some of the variation in the forward premium is due to expected depreciation—or. in other words, the variation in the forward premium cannot be solely due to risk premia (Chirm and Frankel, 2002).⁹

In Table 1, we report sample moments for several combinations of weekly spot and one-month forward exchange rates, including the forward premium $f_t^1-s_t$ (Panel A), the depreciation rate s_t+1 – s_t (Panel B), and the excess return $s_{t+1}-f_t^1$ (Panel C). The summary statistics confirm the stylized facts that each of $f_t^1-s_t,s_{t+1}-s_t$ and $s_{t+1}-f_t^1$ have a mean close to zero with a large standard deviation. However, while the first-order autocorrelation coefficient of the depreciation rate is very small in size (never higher than 0.09) and generally statistically insignificantly different from zero, the first-order autocorrelation coefficient of the forward premium is generally large (in the range between 0.439 for Germany and 0.761 for the United Kingdom) and statistically significantly different from zero, and the corresponding first-order autocorrelation coefficient of the excess return is small (in the range between 0.053 for Germany and 0.131 for the euro) but often statistically significant. These results are consistent with the stylized facts that the forward premium is a highly persistent process, and the depreciation rate shows weak serial correlation (or the exchange rate is a near random walk process).¹⁰

Notes: One-month log-forward and log-spot exchange rates, $f_t^1$ and s_t, are expressed as dollars per unit of foreign currency. Data are Tuesdays of every week, taken from Datastream. The sample period spans from January 4, 1985, to December 31, 2002, for all exchange rates except for the German mark (January 7, 1986, to December 31, 1998) and the euro (January 5, 1999, to December 31, 2002). Figures in parentheses are standard errors calculated by using an autocorrelation and heteroskedasticity consistent matrix of residuals, with three lags (Newey and West. 1987).

Panel C. Return from currency speculation (excess return), $s_{t+1}-f_t^1$
	mean		standard deviation		AR(1)
Japan	0.0002	(0.0007)	0.0159	(0.0049)	0.117	(0.028)
United Kingdom	-0.0025	(0.0061)	0.0151	(0.0051)	0.057	(0.031)
Germany	-0.0001	(0.0007)	0.0152	(0.0045)	0.053	(0.033)
Euro	0.0010	(0.0014)	0.0139	(0.0048)	0.131	(0.053)
Switzerland	0.0007	(0.0007)	0.0168	(0.0045)	0.085	(0.026)

Notes: One-month log-forward and log-spot exchange rates, $f_t^1$ and s_t, are expressed as dollars per unit of foreign currency. Data are Tuesdays of every week, taken from Datastream. The sample period spans from January 4, 1985, to December 31, 2002, for all exchange rates except for the German mark (January 7, 1986, to December 31, 1998) and the euro (January 5, 1999, to December 31, 2002). Figures in parentheses are standard errors calculated by using an autocorrelation and heteroskedasticity consistent matrix of residuals, with three lags (Newey and West. 1987).

View Table

Panel C. Return from currency speculation (excess return), $s_{t+1}-f_t^1$
	mean		standard deviation		AR(1)
Japan	0.0002	(0.0007)	0.0159	(0.0049)	0.117	(0.028)
United Kingdom	-0.0025	(0.0061)	0.0151	(0.0051)	0.057	(0.031)
Germany	-0.0001	(0.0007)	0.0152	(0.0045)	0.053	(0.033)
Euro	0.0010	(0.0014)	0.0139	(0.0048)	0.131	(0.053)
Switzerland	0.0007	(0.0007)	0.0168	(0.0045)	0.085	(0.026)

Notes: One-month log-forward and log-spot exchange rates, $f_t^1$ and s_t, are expressed as dollars per unit of foreign currency. Data are Tuesdays of every week, taken from Datastream. The sample period spans from January 4, 1985, to December 31, 2002, for all exchange rates except for the German mark (January 7, 1986, to December 31, 1998) and the euro (January 5, 1999, to December 31, 2002). Figures in parentheses are standard errors calculated by using an autocorrelation and heteroskedasticity consistent matrix of residuals, with three lags (Newey and West. 1987).

As a preliminary exercise, we estimated the conventional Fama regression (2) for each exchange rate examined. The results, reported in Table 2, are consistent with the existence of forward bias in that, while the constant term α is very close to zero and often statistically insignificant, β is estimated to be negative for all but one of the exchange rate regressions estimated, and it is often statistically insignificantly different from zero. The only exception is Germany, where the estimate of β is positive (about 0.32) and statistically significant, but this estimate does not comprise the theoretical value of unity that is implied by UIP when examining the standard errors. In the last column of Table 2 we also report the t-statistics for the significance of the parameter associated with the forward premium—namely β^τ—in a predictability regression of the form (3). Consistent with a large literature (e.g., Fama, 1984; Backus, Gregory, and Telmer, 1993), we find that, for each exchange rate, β^τ is statistically significantly different from zero, indicating a departure from market efficiency (under which β^τ = 0) and that the forward premium, which is an element of the market participants’ information set, can be used to predict foreign exchange excess returns.

Table 2.

Forward Premium (Fama) Regressions

Notes: The table shows the results from estimating, by ordinary least squares, the conventional forward premium (Fama) regression (2): ${\Delta }{s}_{t+1}=\alpha +\beta (f_t^1-s_t)+{\upsilon }_{t+1}$ Values in parentheses (SE(α) and SE(β)) are asymptotic standard errors calculated using an autocorrelation and heteroskedas-ticity consistent matrix of residuals (Newey and West, 1987). s.e. is the standard deviation of the residual υ_t+1; and T is the number of usable observations. The last column reports the t-statistic (namely t(β^τ)) for the parameter β^τ in regression (3): $E{R}_{t+1}=\alpha +{\beta }^{\tau }(f_t^1-s_t)+{\upsilon }_{t+1}$ where $E{R}_{t+1}\equiv {\Delta }{s}_{t+1}-(f_t^1-s_t)\equiv s_{t+1}-f_t^1$ and β^τ =β – 1.

Table 2.

Forward Premium (Fama) Regressions

	α	SE(α)	β	SE(β)	s.e.	T	t (βτ)
Japan	0.0015	(0.0005)	-0.2865	(0.1586)	0.015	939	-6.742
United Kingdom	-0.0003	(0.0004)	-0.3098	(0.2588)	0.014	939	-6.075
Germany	0.0004	(0.0006)	0.3212	(0.1495)	0.015	677	-4.712
Euro	-0.0001	(0.0008)	-0.8883	(0.4422)	0.013	208	-3.963
Switzerland	0.0012	(0.0006)	-0.3786	(0.1645)	0.016	939	-7.036

Notes: The table shows the results from estimating, by ordinary least squares, the conventional forward premium (Fama) regression (2): ${\Delta }{s}_{t+1}=\alpha +\beta (f_t^1-s_t)+{\upsilon }_{t+1}$ Values in parentheses (SE(α) and SE(β)) are asymptotic standard errors calculated using an autocorrelation and heteroskedas-ticity consistent matrix of residuals (Newey and West, 1987). s.e. is the standard deviation of the residual υ_t+1; and T is the number of usable observations. The last column reports the t-statistic (namely t(β^τ)) for the parameter β^τ in regression (3): $E{R}_{t+1}=\alpha +{\beta }^{\tau }(f_t^1-s_t)+{\upsilon }_{t+1}$ where $E{R}_{t+1}\equiv {\Delta }{s}_{t+1}-(f_t^1-s_t)\equiv s_{t+1}-f_t^1$ and β^τ =β – 1.

View Table

Table 2.

Forward Premium (Fama) Regressions

	α	SE(α)	β	SE(β)	s.e.	T	t (βτ)
Japan	0.0015	(0.0005)	-0.2865	(0.1586)	0.015	939	-6.742
United Kingdom	-0.0003	(0.0004)	-0.3098	(0.2588)	0.014	939	-6.075
Germany	0.0004	(0.0006)	0.3212	(0.1495)	0.015	677	-4.712
Euro	-0.0001	(0.0008)	-0.8883	(0.4422)	0.013	208	-3.963
Switzerland	0.0012	(0.0006)	-0.3786	(0.1645)	0.016	939	-7.036

Notes: The table shows the results from estimating, by ordinary least squares, the conventional forward premium (Fama) regression (2): ${\Delta }{s}_{t+1}=\alpha +\beta (f_t^1-s_t)+{\upsilon }_{t+1}$ Values in parentheses (SE(α) and SE(β)) are asymptotic standard errors calculated using an autocorrelation and heteroskedas-ticity consistent matrix of residuals (Newey and West, 1987). s.e. is the standard deviation of the residual υ_t+1; and T is the number of usable observations. The last column reports the t-statistic (namely t(β^τ)) for the parameter β^τ in regression (3): $E{R}_{t+1}=\alpha +{\beta }^{\tau }(f_t^1-s_t)+{\upsilon }_{t+1}$ where $E{R}_{t+1}\equiv {\Delta }{s}_{t+1}-(f_t^1-s_t)\equiv s_{t+1}-f_t^1$ and β^τ =β – 1.

B. The Nonlinear Fama Regression

In order to evaluate the validity of the assumption of linearity in the conventional Fama regressions reported in Table 2, we performed tests of linearity against the alternative of smooth transition nonlinearity, using the expected annualized excess return, $E{R}_t^e$ as the transition variable. We followed the Teräsvirta (1994, 1998) decision rule to select the most adequate transition function for modeling nonlinearity in the present context. This is a testing procedure designed to test the hypothesis of linearity and to select the most adequate nonlinear function between a logistic and an exponential function. As shown by the results in Table 3, the general linearity test F_L strongly rejects the null hypothesis of linearity. Employing the Teräsvirta rule to discriminate between exponential and logistic formulations led us to conclude that an exponential function (ESTR model) is the most adequate parametric formulation (given that F₂ yields the lowest p-value). This finding is consistent with our priors and the limits-to-speculation hypothesis, discussed in Section II.

Table 3.

Linearity Tests on the Fama Regression

Notes: The table reports the p-values from applying the linearity testing procedure suggested by Teräsvirta (1994, 1998). Assuming that a plausible, generic transition variable is q_t, the appropriate auxiliary regression for the linearity tests against a STR alternative is the following: ${\widehat{e}}_{t+1}={\mathbf{\vartheta }\prime }_0{\mathbf{A}}_{t+1}+{\mathbf{\vartheta }\prime }_1{\mathbf{A}}_{t+1}(q_t)+{\mathbf{\vartheta }\prime }_2{\mathbf{A}}_{t+1}(q_t{)}^2+{\mathbf{\vartheta }\prime }_3{\mathbf{A}}_{t+1}(q_t{)}^3$ + innovations, where ${\widehat{e}}_{t+1}$ is the estimated disturbance retrieved from the linear model being tested for linearity (in the present context it is the residual from each of the Fama regression models reported in Table 2), and A_t denotes the vector of explanatory variables in the model being tested, which in our case simply amounts to the lagged forward premium. The transition variable, q_t used in our tests is the expected excess return for one month ahead based on information at time t, $E{R}_t^e$ The general test for linearity against STR is then the ordinary F-test of the null hypothesis: $H_{0L}:{\mathbf{\vartheta }\prime }_1={\mathbf{\vartheta }\prime }_2={\mathbf{\vartheta }\prime }_3=\mathbf{0}$ The choice between a LSTR and an ESTR model is based on a sequence of nested tests within H_0l. First, the null hypothesis H_0l must be rejected using an ordinary F-test (F_l). Then the following three hypotheses are tested sequentially: $H_{03}:{\mathbf{\vartheta }\prime }_3=\mathbf{0};H_{02}:{\mathbf{\vartheta }\prime }_2|{\mathbf{\vartheta }\prime }_3=\mathbf{0};H_{01}:{\mathbf{\vartheta }\prime }_1|{\mathbf{\vartheta }\prime }_2={\mathbf{\vartheta }\prime }_3=\mathbf{0}$. Again, an F-test is used, with the corresponding test statistics denoted F_3, F_2, and F_1, respectively. The decision rule is as follows: if the test of H₀₂ has the smallest p-value, an ESTR is chosen, otherwise an LSTR is selected. The p-values were calculated using the appropriate F distribution.

Table 3.

Linearity Tests on the Fama Regression

	F L	F 3	F 2	F 1
Japan	0.020	0.077	0.006	0.142
United Kingdom	5.29×l0−16	0.219	0.038	0.365
Germany	3.10×l0−17	0.087	0.044	0.061
Euro	5.09×l0−5	0.268	0.027	0.224
Switzerland	0.011	0.983	0.001	0.546

Notes: The table reports the p-values from applying the linearity testing procedure suggested by Teräsvirta (1994, 1998). Assuming that a plausible, generic transition variable is q_t, the appropriate auxiliary regression for the linearity tests against a STR alternative is the following: ${\widehat{e}}_{t+1}={\mathbf{\vartheta }\prime }_0{\mathbf{A}}_{t+1}+{\mathbf{\vartheta }\prime }_1{\mathbf{A}}_{t+1}(q_t)+{\mathbf{\vartheta }\prime }_2{\mathbf{A}}_{t+1}(q_t{)}^2+{\mathbf{\vartheta }\prime }_3{\mathbf{A}}_{t+1}(q_t{)}^3$ + innovations, where ${\widehat{e}}_{t+1}$ is the estimated disturbance retrieved from the linear model being tested for linearity (in the present context it is the residual from each of the Fama regression models reported in Table 2), and A_t denotes the vector of explanatory variables in the model being tested, which in our case simply amounts to the lagged forward premium. The transition variable, q_t used in our tests is the expected excess return for one month ahead based on information at time t, $E{R}_t^e$ The general test for linearity against STR is then the ordinary F-test of the null hypothesis: $H_{0L}:{\mathbf{\vartheta }\prime }_1={\mathbf{\vartheta }\prime }_2={\mathbf{\vartheta }\prime }_3=\mathbf{0}$ The choice between a LSTR and an ESTR model is based on a sequence of nested tests within H_0l. First, the null hypothesis H_0l must be rejected using an ordinary F-test (F_l). Then the following three hypotheses are tested sequentially: $H_{03}:{\mathbf{\vartheta }\prime }_3=\mathbf{0};H_{02}:{\mathbf{\vartheta }\prime }_2|{\mathbf{\vartheta }\prime }_3=\mathbf{0};H_{01}:{\mathbf{\vartheta }\prime }_1|{\mathbf{\vartheta }\prime }_2={\mathbf{\vartheta }\prime }_3=\mathbf{0}$. Again, an F-test is used, with the corresponding test statistics denoted F_3, F_2, and F_1, respectively. The decision rule is as follows: if the test of H₀₂ has the smallest p-value, an ESTR is chosen, otherwise an LSTR is selected. The p-values were calculated using the appropriate F distribution.

View Table

Table 3.

Linearity Tests on the Fama Regression

	F L	F 3	F 2	F 1
Japan	0.020	0.077	0.006	0.142
United Kingdom	5.29×l0−16	0.219	0.038	0.365
Germany	3.10×l0−17	0.087	0.044	0.061
Euro	5.09×l0−5	0.268	0.027	0.224
Switzerland	0.011	0.983	0.001	0.546

Notes: The table reports the p-values from applying the linearity testing procedure suggested by Teräsvirta (1994, 1998). Assuming that a plausible, generic transition variable is q_t, the appropriate auxiliary regression for the linearity tests against a STR alternative is the following: ${\widehat{e}}_{t+1}={\mathbf{\vartheta }\prime }_0{\mathbf{A}}_{t+1}+{\mathbf{\vartheta }\prime }_1{\mathbf{A}}_{t+1}(q_t)+{\mathbf{\vartheta }\prime }_2{\mathbf{A}}_{t+1}(q_t{)}^2+{\mathbf{\vartheta }\prime }_3{\mathbf{A}}_{t+1}(q_t{)}^3$ + innovations, where ${\widehat{e}}_{t+1}$ is the estimated disturbance retrieved from the linear model being tested for linearity (in the present context it is the residual from each of the Fama regression models reported in Table 2), and A_t denotes the vector of explanatory variables in the model being tested, which in our case simply amounts to the lagged forward premium. The transition variable, q_t used in our tests is the expected excess return for one month ahead based on information at time t, $E{R}_t^e$ The general test for linearity against STR is then the ordinary F-test of the null hypothesis: $H_{0L}:{\mathbf{\vartheta }\prime }_1={\mathbf{\vartheta }\prime }_2={\mathbf{\vartheta }\prime }_3=\mathbf{0}$ The choice between a LSTR and an ESTR model is based on a sequence of nested tests within H_0l. First, the null hypothesis H_0l must be rejected using an ordinary F-test (F_l). Then the following three hypotheses are tested sequentially: $H_{03}:{\mathbf{\vartheta }\prime }_3=\mathbf{0};H_{02}:{\mathbf{\vartheta }\prime }_2|{\mathbf{\vartheta }\prime }_3=\mathbf{0};H_{01}:{\mathbf{\vartheta }\prime }_1|{\mathbf{\vartheta }\prime }_2={\mathbf{\vartheta }\prime }_3=\mathbf{0}$. Again, an F-test is used, with the corresponding test statistics denoted F_3, F_2, and F_1, respectively. The decision rule is as follows: if the test of H₀₂ has the smallest p-value, an ESTR is chosen, otherwise an LSTR is selected. The p-values were calculated using the appropriate F distribution.

Given the results from the linearity tests, we estimate the nonlinear Fama regression (4) by nonlinear least squares under the restrictions α₂ =–α₁ and β₂ = 1 – β₁ (which we test formally below). In estimation, we followed the recommendation of Granger and Teräsvirta (1993) and Teräsvirta (1994, 1998) of standardizing the transition variable $E{R}_t^e$ by dividing it by the sample standard deviation of the transition variable, ${\widehat{\sigma }}_{E{R}^e}$ and using a starting value of unity for the estimation algorithm. Also, since this standardization applies to the transition variable, which now becomes $(E{R}_t^e)/{\widehat{\sigma }}_{E{R}^e}$ the transition variable has a natural interpretation in terms of annualized Sharpe ratio, which tightens the link between the empirical framework and the limits-to-speculation hypothesis.

The results, reported in Panel A of Table 4, indicate that the Fama regression is indeed highly nonlinear. The estimated transition parameter appears to be significantly different from zero, in each equation, both on the basis of the individual asymptotic standard errors as well as on the basis of the Skalin’s (1998) parametric bootstrap likelihood ratio test (see the p-values in square brackets in the last column of Panel A of Table 4).¹¹ The estimates of the slope parameters β₁ and β₂ are correctly signed according to our priors based on the limits to speculation hypothesis, namely, we find a negative estimated value of β₁ and a large positive value of β₂ such that UIP holds exactly when the transition function G (•) = 1. The only exception is Germany, where we record a positively signed estimate of β₁ equal to about 0.15, which seems reasonable given that in estimation of the linear Fama regression we found that Germany was the only country for which a statistically significant positive value of β was found.¹² In turn, these values imply that, for small Sharpe ratios (the transition variable), UIP does not hold and we observe a forward bias, while for increasingly large Sharpe ratios (which are likely to attract speculative capital), reversion to UIP can occur rapidly. These findings also imply that, since reversion to UIP occurs rapidly for large Sharpe ratios, the bulk of the observations of the deviations from UIP is in the lower regime, potentially generating substantial persistence in the forward bias, as predicted by the limits to speculation hypothesis. We shall return to the analysis of the distribution of deviations from UIP in Section V.

Notes: Panel A. The table shows the results from the nonlinear forward premium regression${\Delta }{s}_{t+1}=\left[{\alpha }_1+{\beta }_1(f_t^1-s_t)\right]+\left[{\alpha }_2+{\beta }_2(f_t^1-s_t)\right]G\left[E{R}_t^e,\gamma \right]+{\epsilon }_{t+1}$ where α₂ = −α₁, β₂ = 1 − β₁ and $G\left[E{R}_t^e,\gamma \right]=\{1-\exp \left[-\gamma {\left(E{R}_t^e\right)}^2/{\widehat{\sigma }}^{2}E{R}^e\right]\}$. Values in parentheses (SE_x) are asymptotic standard errors for the parameter x, calculated using an autocorrelation and heteroskedasticity consistent matrix of residuals (Newey and West, 1987). Values in brackets are p-values for the null hypothesis that γ = 0, calculated by the parametric bootstrap procedure suggested by Skalin (1998) using 5, 000 replications. 0 denotes p-values lower than 10⁻⁵. Panel B. LR1 is the likelihood ratio test for the joint null hypothesis that ${\beta }_1^{+}={\beta }_1^{-}$ and ${\beta }_2^{+}={\beta }_2^{-}$ from the unrestricted nonlinear model: ${\Delta }{s}_{t+1}=\left[{\alpha }_1+{\beta }_1^{+}{\left(f_t^1-s_t\right)}^{+}+{\beta }_1^{-}{\left(f_t^1-s_t\right)}^{-}\right]+\left[{\alpha }_2+{\beta }_2^{+}{\left(f_t^1-s_t\right)}^{+}+{\beta }_2^{-}{\left(f_t^1-s_t\right)}^{-}\right]G\left[E{R}_t^e,\gamma \right]+{\epsilon }_{t+1}$ where ${\left(f_t^1-s_t\right)}^{+}$ and ${(f_t^1-s_t)}^{-}$ are forward premia $(f_t^1-s_t>0)$ and forward discounts $(f_t^1-s_t<0)$ respectively. LR2 is the likelihood ratio test for the null hypothesis that α₂ =–α_1, β₂ = 1 – β_1• F_nrn is the test for the null hypothesis of no remaining nonlinearity, constructed as in Eitrheim and Teräsvirta (1996). F_NSB is a test for the null hypothesis of no structural break in the model’s parameters, constructed as in Eitrheim and Teräsvirta’s (1996). For all test statistics, we report p-values, calculated using an autocorrelation and heteroskedasticity consistent matrix of residuals.

Panel B. Diagnostic tests
	LR1	LR2	F NRN	F NSB
Japan	0.115	0.901	0.903	0.547
United Kingdom	0.341	0.914	0.969	0.495
Germany	0.713	0.937	0.940	0.466
Euro	0.134	0.897	0.998	0.502
Switzerland	0.304	0.449	0.994	0.593

Notes: Panel A. The table shows the results from the nonlinear forward premium regression${\Delta }{s}_{t+1}=\left[{\alpha }_1+{\beta }_1(f_t^1-s_t)\right]+\left[{\alpha }_2+{\beta }_2(f_t^1-s_t)\right]G\left[E{R}_t^e,\gamma \right]+{\epsilon }_{t+1}$ where α₂ = −α₁, β₂ = 1 − β₁ and $G\left[E{R}_t^e,\gamma \right]=\{1-\exp \left[-\gamma {\left(E{R}_t^e\right)}^2/{\widehat{\sigma }}^{2}E{R}^e\right]\}$. Values in parentheses (SE_x) are asymptotic standard errors for the parameter x, calculated using an autocorrelation and heteroskedasticity consistent matrix of residuals (Newey and West, 1987). Values in brackets are p-values for the null hypothesis that γ = 0, calculated by the parametric bootstrap procedure suggested by Skalin (1998) using 5, 000 replications. 0 denotes p-values lower than 10⁻⁵. Panel B. LR1 is the likelihood ratio test for the joint null hypothesis that ${\beta }_1^{+}={\beta }_1^{-}$ and ${\beta }_2^{+}={\beta }_2^{-}$ from the unrestricted nonlinear model: ${\Delta }{s}_{t+1}=\left[{\alpha }_1+{\beta }_1^{+}{\left(f_t^1-s_t\right)}^{+}+{\beta }_1^{-}{\left(f_t^1-s_t\right)}^{-}\right]+\left[{\alpha }_2+{\beta }_2^{+}{\left(f_t^1-s_t\right)}^{+}+{\beta }_2^{-}{\left(f_t^1-s_t\right)}^{-}\right]G\left[E{R}_t^e,\gamma \right]+{\epsilon }_{t+1}$ where ${\left(f_t^1-s_t\right)}^{+}$ and ${(f_t^1-s_t)}^{-}$ are forward premia $(f_t^1-s_t>0)$ and forward discounts $(f_t^1-s_t<0)$ respectively. LR2 is the likelihood ratio test for the null hypothesis that α₂ =–α_1, β₂ = 1 – β_1• F_nrn is the test for the null hypothesis of no remaining nonlinearity, constructed as in Eitrheim and Teräsvirta (1996). F_NSB is a test for the null hypothesis of no structural break in the model’s parameters, constructed as in Eitrheim and Teräsvirta’s (1996). For all test statistics, we report p-values, calculated using an autocorrelation and heteroskedasticity consistent matrix of residuals.

View Table

Panel B. Diagnostic tests
	LR1	LR2	F NRN	F NSB
Japan	0.115	0.901	0.903	0.547
United Kingdom	0.341	0.914	0.969	0.495
Germany	0.713	0.937	0.940	0.466
Euro	0.134	0.897	0.998	0.502
Switzerland	0.304	0.449	0.994	0.593

Notes: Panel A. The table shows the results from the nonlinear forward premium regression${\Delta }{s}_{t+1}=\left[{\alpha }_1+{\beta }_1(f_t^1-s_t)\right]+\left[{\alpha }_2+{\beta }_2(f_t^1-s_t)\right]G\left[E{R}_t^e,\gamma \right]+{\epsilon }_{t+1}$ where α₂ = −α₁, β₂ = 1 − β₁ and $G\left[E{R}_t^e,\gamma \right]=\{1-\exp \left[-\gamma {\left(E{R}_t^e\right)}^2/{\widehat{\sigma }}^{2}E{R}^e\right]\}$. Values in parentheses (SE_x) are asymptotic standard errors for the parameter x, calculated using an autocorrelation and heteroskedasticity consistent matrix of residuals (Newey and West, 1987). Values in brackets are p-values for the null hypothesis that γ = 0, calculated by the parametric bootstrap procedure suggested by Skalin (1998) using 5, 000 replications. 0 denotes p-values lower than 10⁻⁵. Panel B. LR1 is the likelihood ratio test for the joint null hypothesis that ${\beta }_1^{+}={\beta }_1^{-}$ and ${\beta }_2^{+}={\beta }_2^{-}$ from the unrestricted nonlinear model: ${\Delta }{s}_{t+1}=\left[{\alpha }_1+{\beta }_1^{+}{\left(f_t^1-s_t\right)}^{+}+{\beta }_1^{-}{\left(f_t^1-s_t\right)}^{-}\right]+\left[{\alpha }_2+{\beta }_2^{+}{\left(f_t^1-s_t\right)}^{+}+{\beta }_2^{-}{\left(f_t^1-s_t\right)}^{-}\right]G\left[E{R}_t^e,\gamma \right]+{\epsilon }_{t+1}$ where ${\left(f_t^1-s_t\right)}^{+}$ and ${(f_t^1-s_t)}^{-}$ are forward premia $(f_t^1-s_t>0)$ and forward discounts $(f_t^1-s_t<0)$ respectively. LR2 is the likelihood ratio test for the null hypothesis that α₂ =–α_1, β₂ = 1 – β_1• F_nrn is the test for the null hypothesis of no remaining nonlinearity, constructed as in Eitrheim and Teräsvirta (1996). F_NSB is a test for the null hypothesis of no structural break in the model’s parameters, constructed as in Eitrheim and Teräsvirta’s (1996). For all test statistics, we report p-values, calculated using an autocorrelation and heteroskedasticity consistent matrix of residuals.