A. Exchange Rates

We begin with daily exchange rate data from Barclays and Reuters via Datastream. We convert daily spot exchange rates to a monthly frequency using end-of-month values (e.g., Burnside, Eichenbaum, Kleshchelski, and Rebelo (Reference Burnside, Eichenbaum, Kleshchelski and Rebelo2011a), Lustig et al. (Reference Lustig, Roussanov and Verdelhan2011)). Our sample consists of currencies for the Euro area and 13 countries: the United Kingdom, Switzerland, Japan, Canada, Australia, New Zealand, Sweden, Norway, Denmark, Germany, Italy, France, and the Netherlands. Our universe of exchange rates is similar to the sample of developed countries employed in other studies (e.g., Lustig et al. (Reference Lustig, Roussanov and Verdelhan2011), Menkhoff, Sarno, Schmeling, and Schrimpf (Reference Menkhoff, Sarno, Schmeling and Schrimpf2012a)). The Euro area replaces Germany, Italy, France, and the Netherlands after the Euro’s introduction in 1999. We use $ {S}_{i,t} $ to denote the month- $ t $ spot exchange rate for country $ i $ , expressed as the number of country- $ i $ currency units per U.S. dollar (e.g., Lustig et al. (Reference Lustig, Roussanov and Verdelhan2011), Menkhoff et al. (Reference Menkhoff, Sarno, Schmeling and Schrimpf2012a), Menkhoff, Sarno, Schmeling, and Schrimpf (Reference Menkhoff, Sarno, Schmeling and Schrimpf2012b)). An increase in $ {S}_{i,t} $ thus represents an appreciation of the U.S. dollar. The country- $ i $ log exchange rate change is denoted by $ \Delta {s}_{i,t}={s}_{i,t}-{s}_{i,t-1} $ , where $ {s}_{i,t}=\ln \left({S}_{i,t}\right) $ . Except for four countries, the sample ends in 2020:09; for France, Germany, Italy, and the Netherlands, the sample ends in 1998:12, the last month for which these countries had their own currencies. Based on data availability for the predictors, the sample begins in 1985:01 for all countries, except for the Euro area, where the sample begins in 1999:02, corresponding to the introduction of the Euro in January of 1999.Footnote ⁴

B. Country Characteristics

We consider a large set of country characteristics based on a variety of macroeconomic and financial variables, all of which are based on data available to FX market participants. Section A.1 of the Appendix provides more detailed descriptions of the predictors and the motivation for their inclusion. Specifically, we consider 10 monthly country characteristics: i) inflation differential ( $ {\mathrm{INF}}_{i,t} $ ), ii) unemployment rate gap differential ( $ {\mathrm{UN}}_{i,t} $ ), iii) bill yield differential ( $ {\mathrm{BILL}}_{i,t} $ ), iv) note yield differential ( $ {\mathrm{NOTE}}_{i,t} $ ), v) bond yield differential ( $ {\mathrm{BOND}}_{i,t} $ ), vi) dividend yield differential ( $ {\mathrm{DP}}_{i,t} $ ), vii) price–earnings ratio differential ( $ {\mathrm{PE}}_{i,t} $ ), viii) stock market time-series momentum differential ( $ \mathrm{SRET}{12}_{i,t} $ ), ix) idiosyncratic volatility ( $ {\mathrm{IV}}_{i,t} $ ), and x) idiosyncratic skewness ( $ {\mathrm{IS}}_{i,t} $ ). All the differentials are measured as the country- $ i $ value minus the U.S. value. IV and IS are computed using fitted residuals for the Lustig et al. (Reference Lustig, Roussanov and Verdelhan2011) 2-factor model estimated using daily data for month $ t $ for country- $ i $ log currency excess returns. The characteristics are based on popular predictors from the literature that relate to well-known theories, such as relative PPP, Taylor-rule fundamentals, UIP, and uncovered equity parity (UEP).

C. Global State Variables

Given that we are generating short-horizon forecasts, allowing for state variables that may influence short-term market sentiment is important. We term these global variables, although the first two relate mainly to the United States, because they all likely affect U.S. dollar exchange rates. We consider six monthly global variables: i) economic policy uncertainty ( $ {\mathrm{EPU}}_t $ ; Baker, Bloom, and Davis (Reference Baker, Bloom and Davis2016)), ii) monetary policy uncertainty ( $ {\mathrm{MPU}}_t $ ; Baker et al. (Reference Baker, Bloom and Davis2016)), iii) geopolitical risk ( $ {\mathrm{GR}}_t $ ; Caldara and Iacoviello (Reference Caldara and Iacoviello2022)), iv) global FX volatility ( $ {\mathrm{GVOL}}_t $ ; Menkhoff et al. (Reference Menkhoff, Sarno, Schmeling and Schrimpf2012a)), v) global FX illiquidity ( $ {\mathrm{GILL}}_t $ ; Menkhoff et al. (Reference Menkhoff, Sarno, Schmeling and Schrimpf2012a)), and vi) global FX correlation ( $ {\mathrm{GCOR}}_t $ ; Mueller, Stathopoulos, and Vedolin (Reference Mueller, Stathopoulos and Vedolin2017)).Footnote ⁵ Each captures general conditions that can affect the predictive ability of the country characteristics.

A. Baseline Specification

We denote the collection of characteristics in Section II for country $ i $ and month $ t $ by $ {\left\{{z}_{j,i,t}\right\}}_{j=1}^J $ , where $ J=10 $ , for $ i=1,\dots, N $ , and $ t=1,\dots, T $ .Footnote ⁶ The collection of global state variables in Section II is denoted by $ {\left\{{g}_{k,t}\right\}}_{k=1}^K $ , where $ K=6 $ . We interact each country characteristic with the set of global variables, yielding the set $ {\left\{{\left\{{z}_{j,i,t}\hskip0.1em {g}_{k,t}\right\}}_{k=1}^K\right\}}_{j=1}^J $ . After standardizing all the predictors, our baseline panel predictive regression is given by

(1) $$ \Delta {s}_{i,t+1}=\sum \limits_{j=1}^J{\beta}_j^z\underset{{\tilde{z}}_{j,i,t}}{\underbrace{\left[\frac{1}{\sigma_{j,i}^z}\left({z}_{j,i,t}-{\mu}_{j,i}^z\right)\right]}}+\sum \limits_{j=1}^J\sum \limits_{k=1}^K{\beta}_{j,k}^{z,g}\underset{\tilde{z_{j,i,t}\hskip0.1em {g}_{k,t}}}{\underbrace{\left[\frac{1}{\sigma_{j,i,k}^{z,g}}\left({z}_{j,i,t}\hskip0.1em {g}_{k,t}-{\mu}_{j,i,k}^{z,g}\right)\right]}}+{\varepsilon}_{i,t+1} $$

for $ i=1,\dots, N $ and $ t=1,\dots, T-1 $ , where $ {\tilde{z}}_{j,i,t} $ ( $ \tilde{z_{j,i,t}\hskip0.1em {g}_{k,t}} $ ) is the standardized country characteristic (interaction term), $ {\mu}_{j,i}^z $ ( $ {\mu}_{j,i,k}^{z,g} $ ) is the mean of $ {z}_{j,i,t} $ ( $ {z}_{j,i,t}\hskip0.1em {g}_{k,t} $ ), $ {\sigma}_{j,i}^z $ ( $ {\sigma}_{j,i,k}^{z,g} $ ) is the standard deviation of $ {z}_{j,i,t} $ ( $ {z}_{j,i,t}\hskip0.1em {g}_{k,t} $ ), and $ {\varepsilon}_{i,t+1} $ is a zero-mean disturbance term.

Equation (1) is high dimensional, as it includes $ J\left(K+1\right)=10\times 7=70 $ predictors. We include many more predictors than existing studies of exchange rate predictability, which typically consider only a limited number of country characteristics. The inclusion of a broad set of characteristics is germane, because we do not know a priori which are relevant for forecasting exchange rates. It also helps guard against data mining or snooping, as we include an extensive set of potentially relevant fundamentals in the model from the start. In addition to numerous country characteristics, we include their interactions with the set of global variables, which allows for the predictive relations between exchange rate changes and fundamentals to vary with global economic conditions. To the best of our knowledge, such interactions have not previously been considered in the out-of-sample exchange rate predictability literature.

Equation (1) also pools the data, thereby restricting the response of $ \Delta {s}_{i,t+1} $ to $ {\tilde{z}}_{j,i,t} $ and $ \tilde{z_{j,i,t}\hskip0.1em {g}_{k,t}} $ to be the same across $ i $ . These slope homogeneity restrictions reduce the parameter space to help guard against overfitting. Although the slope homogeneity restrictions are unlikely to hold exactly—potentially inducing bias in the parameter estimates—they also decrease the variance of the parameter estimates, which can improve out-of-sample performance in light of the bias–variance trade-off. Whether pooling is beneficial is ultimately an empirical issue determined by the bias–variance trade-off; for our application, we find that pooling leads to better time-series forecasts of individual exchange rates.Footnote ⁷

Furthermore, observe that equation (1) does not include an intercept term. Because the predictors are in deviation form, we effectively impose the restrictions that the country-specific means for $ \Delta {s}_{i,t} $ for $ i=1,\dots, N $ are 0. In other words, equation (1) is tantamount to a fixed-effects specification in which the country-specific means for the log exchange rate changes are all 0. These 0 restrictions are consistent with economic intuition and the data, and they further reduce the number of parameters that we need to estimate, again helping to guard against overfitting in light of the bias–variance trade-off.

1. Regime Switching

The inclusion of the interaction terms in equation (1) effectively permits a form of regime switching in the predictive relations between the country characteristics and exchange rate changes. To see this, note that the response of $ \Delta {s}_{i,t+1} $ to a change in $ {z}_{j,i,t} $ in equation (1) is given by

(2) $$ \frac{\mathrm{\partial \Delta }{s}_{i,t+1}}{\partial {z}_{j,i,t}}=\underset{\mathrm{slope}\ \mathrm{coefficient}\ \mathrm{for}\;{z}_{j,i,t}}{\underbrace{\beta_j^z\hskip0.1em \frac{1}{\sigma_{j,i}^z}+\sum \limits_{k=1}^K\;{\beta}_{j,k}^{z,g}\hskip0.1em \frac{1}{\sigma_{j,i,k}^{z,g}}\hskip0.1em {g}_{k,t}}} $$

for $ j=1,\dots, J $ and $ i=1,\dots, N $ . According to equation (2), the slope coefficient for $ {z}_{j,i,t} $ varies linearly with $ {g}_{k,t} $ for $ k=1,\dots, K $ . For example, if $ {\beta}_{j,k}^{z,g}>0 $ , then the response of $ \Delta {s}_{i,t+1} $ to a change in $ {z}_{j,i,t} $ increases linearly with $ {g}_{k,t} $ . We refer to equation (1) and the concomitant response in equation (2) as encapsulating continuous regime switching. Setting $ {\beta}_{j,k}^{z,g}=0 $ for $ j=1,\dots, J $ and $ k=1,\dots, K $ in equation (1) (i.e., excluding the interaction terms) produces a panel predictive regression based on the country characteristics alone:

(3) $$ \Delta {s}_{i,t+1}=\sum \limits_{j=1}^J{\beta}_j^z\hskip0.1em {\tilde{z}}_{j,i,t}+{\varepsilon}_{i,t+1}, $$

where the number of predictors is reduced to $ J=10 $ . For equation (3), the derivative in equation (2) simplifies to the first term on the right-hand side ( $ {\beta}_j^z\hskip0.1em (1/{\sigma}_{j,i}^z) $ ), so regime switching is absent.

2. Elastic Net

An out-of-sample forecast of $ \Delta {s}_{i,T+1} $ based on the baseline panel predictive regression in equation (1) and data available through $ T $ is given by

(4) $$ {\hat{\Delta s}}_{i,T+1\mid T}=\sum \limits_{j=1}^J{\hat{\beta}}_j^z\hskip0.1em {\tilde{z}}_{j,i,T}+\sum \limits_{j=1}^J\sum \limits_{k=1}^K{\hat{\beta}}_{j,k}^{z,g}\hskip0.1em \tilde{z_{j,i,T}\hskip0.1em {g}_{k,T}}, $$

where $ {\hat{\beta}}_j^z $ and $ {\hat{\beta}}_{j,k}^{z,g} $ are estimates of $ {\beta}_j^z $ and $ {\beta}_{j,k}^{z,g} $ , respectively, in equation (1) for $ j=1,\dots, J $ and $ k=1,\dots, K $ based on data available through $ T $ . The plugged-in predictors on the right-hand side of equation (4), $ {\tilde{z}}_{j,i,T} $ and $ \tilde{z_{j,i,T}\hskip0.1em {g}_{k,T}} $ , are standardized using data through $ T $ . By estimating the slope coefficients and standardizing the predictors using data available through $ T $ (i.e., the time of forecast formation), we avoid “look-ahead” bias in the forecast.

Even after imposing the slope homogeneity and zero-mean restrictions in equation (1), because the panel predictive regression is high dimensional and the unpredictable component in monthly exchange rate changes is inherently large, conventional OLS estimation of the parameters $ {\beta}_j^z $ and $ {\beta}_{j,k}^{z,g} $ for $ j=1,\dots, J $ and $ k=1,\dots, K $ is highly susceptible to overfitting. Thus, we consider penalized (or regularized) regression, which shrinks the parameter estimates toward 0 to reduce their variance and improve out-of-sample performance according to the bias–variance trade-off.

We employ penalized regression via the ENet (Zou and Hastie (Reference Zou and Hastie2005)), a popular and powerful machine-learning technique that mitigates overfitting by including a penalty term in the objective function for estimating equation (1). Collecting the parameters in the vector $ \underset{J\left(K+1\right)}{\boldsymbol{\beta}}={\left[{\beta}_1^z\hskip0.5em \cdots \hskip0.5em {\beta}_J^z\hskip0.5em {\beta}_{1,1}^{z,g}\hskip0.5em \cdots \hskip0.5em {\beta}_{1,K}^{z,g}\hskip0.5em \cdots \hskip0.5em {\beta}_{J,1}^{z,g}\hskip0.5em \cdots \hskip0.5em {\beta}_{J,K}^{z,g}\right]}^{\prime } $ , the ENet objective function can be expressed as

(5) $$ {\displaystyle \begin{array}{ll}& \underset{\boldsymbol{\beta} \in {\mathrm{\mathbb{R}}}^{J\left(K+1\right)}}{\arg \hskip0.22em \min}\hskip0.24em \frac{1}{2N\left(T-1\right)}\sum \limits_{i=1}^N\sum \limits_{t=1}^{T-1}{\left[\Delta {s}_{i,t+1}-\left(\sum \limits_{j=1}^J{\beta}_j^z\hskip0.1em {\tilde{z}}_{j,i,t}+\sum \limits_{j=1}^J\sum \limits_{k=1}^K{\beta}_{j,k}^{z,g}\hskip0.1em \tilde{z_{j,i,t}\hskip0.1em {g}_{k,t}}\right)\right]}^2+\\ {}& \hskip2em \underset{\mathrm{penalty}\ \mathrm{term}}{\underbrace{\lambda \left[0.5\left(1-\delta \right)\parallel \boldsymbol{\beta} {\parallel}_2^2+\delta \parallel \boldsymbol{\beta} {\parallel}_1\right]}},\end{array}} $$

where $ \lambda \ge 0 $ is a regularization hyperparameter that governs the degree of shrinkage; $ \parallel \cdot {\parallel}_1 $ and $ \parallel \cdot {\parallel}_2 $ are the $ {\mathrm{\ell}}_1 $ and $ {\mathrm{\ell}}_2 $ norms, respectively; and $ \delta $ is a blending hyperparameter for the $ {\mathrm{\ell}}_1 $ and $ {\mathrm{\ell}}_2 $ components in the penalty term.Footnote ⁸ When $ \lambda =0 $ , there is no shrinkage, so the ENet and OLS objective functions coincide; when $ \lambda >0 $ , the penalty term shrinks the parameter estimates toward 0 to mitigate overfitting. The ENet penalty term in equation (5) includes both $ {\mathrm{\ell}}_1 $ (least absolute shrinkage and selection operator or LASSO; Tibshirani (Reference Tibshirani1996)) and $ {\mathrm{\ell}}_2 $ (ridge; Hoerl and Kennard (Reference Hoerl and Kennard1970)) components. The former permits shrinkage to 0 (for sufficiently large $ \lambda $ ), so the ENet performs variable selection (i.e., can produce a sparse model).Footnote ⁹ We tune the hyperparameters $ \lambda $ and $ \delta $ via the extended regularized information criterion (ERIC; Hui, Warton, and Foster (Reference Hui, Warton and Foster2015)), which is a relatively stringent information criterion, because it tends to induce more shrinkage than cross-validation and other information criteria.Footnote ¹⁰ Considering grids of values for $ \lambda $ and $ \delta $ , we select the values that minimize the ERIC. Given our high-dimensional and noisy data environment, the ERIC appears effective for tuning the hyperparameters.

B. Deep Neural Network

The baseline specification in equation (1) permits nonlinearities in the predictors through the interaction terms. To allow for additional nonlinearities, we consider the following general panel predictive regression:

(6) $$ \Delta {s}_{i,t+1}=f\left({\tilde{\boldsymbol{x}}}_{i,t}\right)+{\varepsilon}_{i,t+1} $$

for $ i=1,\dots, N $ and $ t=1,\dots, T-1 $ , where $ \underset{J\left(K+1\right)}{{\tilde{\boldsymbol{x}}}_{i,t}}=\Big[{\displaystyle \begin{array}{ccccc}{\tilde{z}}_{1,i,t}& \cdots & {\tilde{z}}_{J,i,t}& \tilde{z_{1,i,t}\hskip0.1em {g}_{1,t}}& \cdots \end{array}} $ $ {\displaystyle \begin{array}{ccccc}\tilde{z_{1,i,t}\hskip0.1em {g}_{K,t}}& \cdots & \tilde{z_{J,i,t}\hskip0.1em {g}_{1,t}}& \cdots & \tilde{z_{J,i,t}\hskip0.1em {g}_{K,t}}\end{array}}\Big]{}^{\prime } $ and $ f\left({\tilde{\boldsymbol{x}}}_{i,t}\right) $ is a generic function constituting the conditional expectation function. We model $ f\left({\tilde{\boldsymbol{x}}}_{i,t}\right) $ in equation (6) via a DNN, a popular machine-learning device for flexibly modeling complex relations between predictors and a target variable. A feedforward neural network is composed of multiple layers. The first is the input layer, which is simply the set of predictors. One or more hidden layers are next, where each hidden layer contains multiple neurons with a nonlinear activation function. The input layer is densely connected to the first hidden layer, and each subsequent hidden layer is densely connected to its preceding hidden layer. The neurons produce signals that are passed forward through the hidden layers. Finally, an output layer translates the signals from the last hidden layer into a prediction.

It is well known that a neural network with a single hidden layer and a sufficiently large number of neurons can, theoretically, approximate any smooth function (e.g., Hornik, Stinchcombe, and White (Reference Hornik, Stinchcombe and White1989)). However, neural networks with multiple hidden layers (i.e., DNNs) and a more limited number of neurons in each layer often perform better in practice than neural networks with one hidden layer and a large number of neurons (e.g., Goodfellow, Bengio, and Courville (Reference Goodfellow, Bengio and Courville2016), Gu et al. (Reference Gu, Kelly and Xiu2020)). Our DNN has four hidden layers with 16, 8, 4, and 2 neurons, respectively, with dense connections between adjacent layers and a single output node producing the exchange rate change forecast. The specification of 16 neurons in the first hidden layer is a compromise between two popular rules of thumb: half or the square root of the number of predictors. Section IA.2 of the Supplementary Material provides further details on the architecture, activation function, and estimation strategy, including the steps that we take to guard against overfitting in our noisy data environment. As in the baseline panel predictive regression in equation (1), the DNN predicts a value of 0 for the exchange rate change when all of the (standardized) predictors are at their mean value of 0.

C. Ensemble

We also compute an ensemble forecast by taking the average of the ENet and DNN forecasts, which provides a convenient strategy for blending the different forecast strategies. An average of multiple individual forecasts is known as a “voting ensemble” in the machine-learning literature. An ensemble forecast recognizes that we do not know the best individual model a priori and allows us to take advantage of potential complementarities between models.

A. Forecast Construction

Mimicking the situation of a forecaster in real time, we generate exchange rate forecasts for the 1995:01–2020:09 out-of-sample period as follows: Reserving the first 10 years of data (1985:01–1994:12) for the initial training sample, we estimate the baseline panel predictive regression in equation (1) and the DNN in Section III.B using data available through 1994:12. We then use the fitted prediction models and 1994:12 predictor values for each country to compute forecasts for 1995:01. Next, we re-estimate the models using data available through 1995:01 and use the fitted models and 1995:01 predictor values for each country to generate forecasts for 1995:02. We continue in this fashion through the end of the out-of-sample period, providing us with a set of exchange rate forecasts for the available countries for each of the 309 months comprising the out-of-sample period. By retraining the prediction models each month as new data become available, we update the fitted models in a timely manner; similarly, we also regularly retune the hyperparameters.Footnote ¹¹ Note that there is no look-ahead bias in the forecasts, as we only use data available at the time of forecast formation when training the models (and standardizing the predictors).

For Germany, Italy, France, and the Netherlands, there are only 48 monthly forecasts (1995:01–1998:12) available for evaluation, due to these countries joining the Euro area in 1999:01. Thus, we do not include these four countries when evaluating forecast performance. However, we use data for these countries when training the prediction models, as this increases the number of training-sample observations, thereby improving estimation efficiency. After imposing a minimum requirement of 12 monthly observations before a currency is included in the panel predictive regression, there are 248 forecasts (2000:02–2020:09) available for the Euro area. For the remaining nine countries, forecasts are available for the entire 1995:01–2020:09 out-of-sample period (309 observations). In addition to reporting results for the individual countries, we report results for the entire collection of countries taken together (excluding Germany, Italy, France, and the Netherlands, resulting in $ 248+9\times 309=\mathrm{3,029} $ observations).

B. Test Statistics

We compare competing forecasts that incorporate information from economic fundamentals to the no-change benchmark, the most stringent benchmark for out-of-sample exchange rate prediction (Rossi (Reference Rossi2013)). For country $ i $ , we can conveniently compare the relative accuracy of a competing forecast vis-à-vis the no-change benchmark in terms of MSPE in the time-series dimension using the out-of-sample $ {R}^2 $ statistic (Fama and French (Reference Fama and French1989), Campbell and Thompson (Reference Campbell and Thompson2008)):

(7) $$ {R}_{i,\mathrm{OS}}^2=1-\frac{\sum_{t={T}_1+1}^{T_2}{\left({\hat{e}}_{i,t\mid t-1}^{\mathrm{Compete}}\right)}^2}{\sum_{t={T}_1+1}^{T_2}{\left({\hat{e}}_{i,t\mid t-1}^{\mathrm{Bench}}\right)}^2}, $$

where $ {\hat{e}}_{i,t\mid t-1}^j=\Delta {s}_{i,t}-{\hat{\Delta s}}_{i,t\mid t-1}^j $ for $ j=\mathrm{Bench},\mathrm{Compete} $ ; $ {\hat{\Delta s}}_{i,t\mid t-1}^{\mathrm{Bench}}=0 $ is the no-change benchmark forecast; $ {\hat{\Delta s}}_{i,t\mid t-1}^{\mathrm{Compete}} $ is a competing forecast; $ {T}_1 $ is the last observation for the initial in-sample period; and $ {T}_2 $ is the last available observation for the out-of-sample period. The $ {R}_{i,\mathrm{OS}}^2 $ statistic measures the proportional reduction in MSPE for a competing forecast vis-à-vis the benchmark. Because the predictable component in monthly exchange rate changes is intrinsically small, the $ {R}_{i,\mathrm{OS}}^2 $ statistic will necessarily be small. Nevertheless, even a seemingly small degree of monthly exchange rate predictability can be economically meaningful, as we show in Section VI and as is the case for stock return predictability (Campbell and Thompson (Reference Campbell and Thompson2008)).

To test whether the competing forecast provides a statistically significant improvement in MSPE relative to the benchmark when comparing nested models, we compute the Clark and West (Reference Clark and West2007) adjusted version of the popular Diebold and Mariano (Reference Diebold and Mariano1995) and West (Reference West1996) statistic. Specifically, we use the Clark and West (Reference Clark and West2007) statistic to test (in population) $ {H}_0 $ : $ {\mathrm{MSPE}}^{\mathrm{Bench}}\le {\mathrm{MSPE}}^{\mathrm{Compete}} $ ( $ {R}_{i,\mathrm{OS}}^2\le 0 $ ) against $ {H}_A $ : $ {\mathrm{MSPE}}^{\mathrm{Bench}}>{\mathrm{MSPE}}^{\mathrm{Compete}} $ ( $ {R}_{i,\mathrm{OS}}^2>0 $ ). For country $ i $ , the Clark and West (Reference Clark and West2007) statistic can be conveniently computed using the $ t $ -statistic corresponding to the intercept term ( $ {a}_{0,i} $ ) in the following regression:

(8) $$ \underset{l_{i,t}}{\underbrace{{\left({\hat{e}}_{i,t\mid t-1}^{\mathrm{Bench}}\right)}^2-{\left({\hat{e}}_{i,t\mid t-1}^{\mathrm{Compete}}\right)}^2+{\left({\hat{\Delta s}}_{i,t\mid t-1}^{\mathrm{Bench}}-{\hat{\Delta s}}_{i,t\mid t-1}^{\mathrm{Compete}}\right)}^2}}={a}_{0,i}+{\varepsilon}_{i,t} $$

for $ t={T}_1+1,\dots, {T}_2 $ , where $ {l}_{i,t} $ is the adjusted loss differential. Clark and West (Reference Clark and West2007) show that the $ t $ -statistic corresponding to $ {a}_{0,i} $ in equation (8) is well approximated by the standard normal distribution under the null hypothesis.

We also compute a composite out-of-sample $ {R}^2 $ statistic based on the entire collection of countries taken together:

(9) $$ {R}_{\mathrm{All},\mathrm{OS}}^2=1-\frac{\sum_{i=1}^N{\sum}_{t={T}_1+1}^{T_2}{\left({\hat{e}}_{i,t\mid t-1}^{\mathrm{Compete}}\right)}^2}{\sum_{i=1}^N{\sum}_{t={T}_1+1}^{T_2}{\left({\hat{e}}_{i,t\mid t-1}^{\mathrm{Bench}}\right)}^2}. $$

For testing $ {H}_0 $ : $ {R}_{\mathrm{All},\mathrm{OS}}^2\le 0 $ against $ {H}_A $ : $ {R}_{\mathrm{All},\mathrm{OS}}^2>0 $ , we compute a pooled version of the Clark and West (Reference Clark and West2007) statistic by calculating the $ t $ -statistic corresponding to the intercept term ( $ {a}_0 $ ) in the following regression:

(10) $$ {l}_{i,t}={a}_0+{\varepsilon}_{i,t} $$

for $ i=1,\dots, N $ and $ t={T}_1+1,\dots, {T}_2 $ , where we cluster the standard errors by country and time. A caveat to the $ {R}_{\mathrm{All},\mathrm{OS}}^2 $ statistic is that it can reflect cross-sectional as well as time-series predictability, whereas, in line with the exchange rate predictability puzzle, we are interested in the latter. With this caveat in mind, we report the $ {R}_{\mathrm{All},\mathrm{OS}}^2 $ statistic in equation (9) to provide an overall measure of the accuracy of the exchange rate forecasts for the entire group of countries taken together.

C. Forecast Accuracy: Outperforming the Random Walk

Table 1 reports the $ {R}_{i,\mathrm{OS}}^2 $ statistic in equation (7) for different forecasts and the 10 countries for which we have at least 248 monthly observations for the forecast evaluation period. It also reports the $ {R}_{\mathrm{All},\mathrm{OS}}^2 $ statistic in equation (9) for the 10 countries taken together. The fourth and fifth columns of the table report results for forecasts based on the baseline panel predictive regression in equation (1) estimated via OLS (baseline OLS) and the ENet (baseline ENet), respectively, whereas the sixth column provides results for the panel predictive regression without interaction terms in equation (3) estimated via the ENet (no-interaction ENet). Finally, the seventh and eighth columns report results for the DNN in Section III.B and the ensemble forecast in Section III.C (an average of the baseline ENet and DNN forecasts), respectively.Footnote ¹²

TABLE 1Out-of-Sample $R^2$ $ {R}^2 $ upper R 2 Statistics

TABLE 1 Long description

The table consists of 8 columns: Country, Out-of-Sample Period, Number of Observations, Baseline O L S, Baseline ENet, No-Interaction ENet, D N N, and Ensemble.

* The Out-of-Sample Period for most countries is 1995:01 to 2020:09 with 309 observations, except for the Euro area (2000:02 to 2020:09, 248 observations) and the ‘All’ aggregate (3,029 observations).

* Baseline O L S shows negative values for all entries, ranging from -3.21 (Norway) to -19.75 (Canada).

* Baseline ENet shows positive, significant results for the United Kingdom (3.24), Australia (1.47), New Zealand (2.15), Sweden (1.86), Norway (1.91), Denmark (2.23), and Euro area (2.38).

* No-Interaction ENet has lower performance, with several negative values and only small positive significant results for the United Kingdom, Australia, New Zealand, Norway, and Denmark.

* D N N (Deep Neural Network) shows consistently positive and significant results for most countries, notably 3.03 for the United Kingdom and 2.15 for Switzerland.

* Ensemble (average of ENet and D N N) provides the highest or near-highest R super 2 values for most rows, including 3.41 for the United Kingdom and 2.65 for the Euro area.

* The ‘All’ aggregate row shows significant R super 2 values for Baseline ENet (1.51), D N N (1.74), and Ensemble (1.86), while Baseline O L S (-8.74) and No-Interaction ENet (-0.29) are negative.

The $ {R}_{i,\mathrm{OS}}^2 $ and $ {R}_{\mathrm{All},\mathrm{OS}}^2 $ statistics are all negative in the fourth column of Table 1, so the baseline OLS forecast always fails to outperform the no-change benchmark. The negative $ {R}_{i,\mathrm{OS}}^2 $ statistics are often quite sizable in magnitude for the individual countries, and the $ {R}_{\mathrm{All},\mathrm{OS}}^2 $ statistic is $ -8.74\% $ for the 10 countries taken together in the last row. Its poor out-of-sample performance indicates that the baseline OLS forecast substantially overfits the data, which is not surprising in our high-dimensional and noisy data environment.

As reported in the fifth column of Table 1, employing shrinkage to mitigate overfitting proves highly beneficial, as the baseline ENet forecast evinces markedly better out-of-sample performance than the baseline OLS forecast. Specifically, the $ {R}_{i,\mathrm{OS}}^2 $ statistics are positive for 8 of the 10 countries for the baseline ENet forecast, and the reduction in MSPE relative to the no-change benchmark is significant at conventional levels for seven of the eight countries. Seven (four) of the $ {R}_{i,\mathrm{OS}}^2 $ statistics are above 1% (2%) and reach as high as 3.24% (the United Kingdom), so they are sizable in the context of the literature on out-of-sample exchange rate predictability. For the complete set of countries taken together, the $ {R}_{\mathrm{All},\mathrm{OS}}^2 $ statistic is 1.51% (significant at the 5% level).

The sixth column of Table 1 reports results for the no-interaction ENet forecast. This forecast is based on the panel predictive regression without interaction terms in equation (3), which, as discussed in Section III.A, is akin to restricting the slope coefficients for the country characteristics to be constant (i.e., no regime switching). The $ {R}_{i,\mathrm{OS}}^2 $ statistics are positive for 5 of the 10 countries, and all 5 are significant at conventional levels. Taking the 10 countries together, the $ {R}_{\mathrm{All},\mathrm{OS}}^2 $ statistic is negative ( $ -0.29\% $ ). Overall, there is some evidence that using country characteristics without regime switching outperforms the no-change benchmark. However, comparing the results in the fifth and sixth columns of Table 1, the $ {R}_{i,\mathrm{OS}}^2 $ and $ {R}_{\mathrm{All},\mathrm{OS}}^2 $ statistics are all larger for the baseline ENet forecast vis-à-vis the no-interaction ENet forecast, often by a substantial margin. Thus, we have strong evidence that allowing for regime switching in the responses of exchange rate changes to characteristics improves out-of-sample predictability.

According to the seventh column of Table 1, the DNN forecast performs well. The $ {R}_{i,\mathrm{OS}}^2 $ statistics are positive for all countries, with significant improvements in MSPE relative to the no-change benchmark for eight of them. Like those for the baseline ENet forecast, the $ {R}_{i,\mathrm{OS}}^2 $ statistics for the individual countries are sizable, and eight (four) are above 1% (2%), with a maximum of 3.03% (the United Kingdom). For the 10 countries considered jointly, the $ {R}_{\mathrm{All},\mathrm{OS}}^2 $ statistic is 1.74% for the DNN forecast (significant at the 1% level).

By combining the baseline ENet and DNN forecasts, the ensemble approach in the last column of Table 1 provides the best overall performance. The $ {R}_{i,\mathrm{OS}}^2 $ statistics are positive for all 10 countries, and the improvements in MSPE vis-à-vis the no-change benchmark are significant for eight of them. The $ {R}_{i,\mathrm{OS}}^2 $ statistics are also often quite sizable, with eight (four) above 1% (2%), and the maximum is 3.41% (the United Kingdom). For the 10 countries taken together, the $ {R}_{\mathrm{All},\mathrm{OS}}^2 $ statistic is 1.86% (significant at the 1% level). Note that for a number of countries, the $ {R}_{i,\mathrm{OS}}^2 $ statistic for the ensemble forecast is larger than the highest corresponding statistic in the fifth and seventh columns, pointing to meaningful complementarities in the baseline ENet and DNN forecasts.Footnote ¹³

To examine the performance of the baseline ENet, DNN, and ensemble forecasts over time, Figure 1 shows rolling 60-month $ {R}_{i,\mathrm{OS}}^2 $ and $ {R}_{\mathrm{All},\mathrm{OS}}^2 $ statistics for the baseline ENet, DNN, and ensemble forecasts (considered in turn). The horizontal axes in Figure 1 correspond to the end of the 60-month rolling window. Figure 1 demonstrates that our machine-learning approach consistently generates out-of-sample gains over time. Except for Japan, the clear majority of the rolling 60-month $ {R}_{i,\mathrm{OS}}^2 $ statistics are positive for the individual countries, and the $ {R}_{\mathrm{All},\mathrm{OS}}^2 $ statistics for the ensemble forecast are positive for nearly every 60-month window. The rolling $ {R}_{i,\mathrm{OS}}^2 $ and $ {R}_{\mathrm{All},\mathrm{OS}}^2 $ statistics are quite large for 60-month windows that include the nadir of the global financial crisis in late 2008, with values above 6% in many cases. Substantive out-of-sample gains are also evident outside of the crisis. For example, for numerous countries, including Switzerland, Australia, Sweden, Norway, and Denmark, the $ {R}_{i,\mathrm{OS}}^2 $ statistics are relatively large for windows ending in the early 2000s. In addition, $ {R}_{i,\mathrm{OS}}^2 $ statistics are sizable for windows ending in the later part of the out-of-sample period for Switzerland, Japan, Sweden, Norway, and Denmark. Along this line, the $ {R}_{i,\mathrm{OS}}^2 $ statistics for 60-month windows ending around 2004–2006 are substantive for countries such as the United Kingdom, Canada, Australia, and New Zealand.Footnote ¹⁴ Overall, Table 1 and Figure 1 indicate that the information in our large set of predictors and machine-learning methods can be used to generate sizable and consistent improvements in out-of-sample forecast accuracy.

FIGURE 1Rolling Out-of-Sample $R^2$ $ {R}^2 $ upper R 2 Statistics

Figure 1 depicts rolling 60-month $ {R}_{i,\mathrm{OS}}^2 $ and $ {R}_{\mathrm{All},\mathrm{OS}}^2 $ statistics in percentage. The horizontal axis gives the end of the 60-month period. Vertical bars delineate business-cycle recessions as dated by the National Bureau of Economic Research.

FIGURE 1 Long description

A multi-panel display of 11 line graphs arranged in a 4 by 3 grid, with the final row containing two panels. Each graph plots Percent on the y-axis against years from 2000 to 2020 on the x-axis. Vertical gray shaded bars indicate N B E R recession periods.

Individual panels are titled: United Kingdom, Switzerland, Japan, Canada, Australia, New Zealand, Sweden, Norway, Denmark, Euro area, and All.

Each panel contains three data series:

* Baseline E Net: A dark blue line showing significant volatility, often dipping below zero percent, particularly around 2008 and 2014.

* D N N: A magenta line that generally stays above the Baseline E Net in the early 2000s but fluctuates significantly across different regions.

* Ensemble: A yellow line that typically tracks between the other two models, showing a notable peak across almost all panels during the 2008 to 2010 period.

Common trends include a sharp spike in R super 2 values for all models around the 2009 global financial crisis, followed by a decline and subsequent smaller peaks around 2015. The Euro area panel begins its data series later, around 2005.

A legend at the bottom center identifies the lines: dark blue for Baseline E Net, magenta for D N N, and yellow for Ensemble.

Although our approach performs well when the U.S. dollar is used as the base currency, which is consistent with its dominant role as the primary settlement currency in international trade and global financial markets, we find less predictive power when other currencies serve as the base (such as the Japanese yen and Swiss franc, as examined in Section IA.3 of the Supplementary Material). Understanding the economic forces underlying this asymmetry is an interesting avenue for future research.

Section A.2 of the Appendix lists a variety of robustness checks regarding forecast accuracy (with detailed results reported in the Supplementary Material). Overall, alternative approaches do not perform as well as the baseline ENet, DNN, and ensemble forecasts in terms of forecast accuracy, as reported in Table 1.

D. Forecast Accuracy and Global FX Volatility

Figure 1 indicates that the machine-learning forecasts perform particularly well relative to the no-change benchmark during periods of increases in FX market turbulence, such as during the worst phase of the global financial crisis in late 2008. To investigate the relation between out-of-sample exchange rate predictability and FX market volatility more formally, we estimate an augmented version of equation (8):

(11) $$ {l}_{i,t}={a}_{i,0}+{a}_{i,1}{\mathrm{GVOL}}_t+{\varepsilon}_{i,t} $$

for $ i=1,\dots, N $ and $ t={T}_1+1,\dots, {T}_2 $ , where $ {l}_{i,t} $ is the Clark and West (Reference Clark and West2007) adjusted loss differential for country $ i $ for the no-change benchmark forecast vis-à-vis the competing forecast (baseline ENet, DNN, or ensemble), and $ {\mathrm{GVOL}}_t $ is our measure of global FX volatility. To test whether the relative accuracy of the competing forecast increases as global FX volatility rises, we test $ {H}_0 $ : $ {a}_{i,1}\le 0 $ against $ {H}_A $ : $ {a}_{i,1}>0 $ . We also consider a composite version of equation (11) for all the countries taken together, which is an augmented version of equation (10):

(12) $$ {l}_{i,t}={a}_0+{a}_1{\mathrm{GVOL}}_t+{\varepsilon}_{i,t} $$

for $ i=1,\dots, N $ and $ t={T}_1+1,\dots, {T}_2 $ . To again test whether the relative accuracy of the competing forecast increases with global FX volatility, we test $ {H}_0 $ : $ {a}_1\le 0 $ against $ {H}_A $ : $ {a}_1>0 $ . We cluster the standard errors by country and time for equation (12).

Table 2 reports estimates of $ {a}_{i,1} $ and $ {a}_1 $ in equations (11) and (12), respectively. For the baseline ENet forecast in the fourth column, the coefficient estimates are positive for all countries except Japan; apart from Japan, nine of the estimates are significant, eight of them at the 1% level. The coefficient estimates are positive for all countries for both the DNN and ensemble forecasts in the fifth and sixth columns, respectively; six (nine) of the estimates are significant for the DNN (ensemble) forecast, with nine of the estimates significant at the 1% level for the ensemble forecast. Taking the 10 countries together, the pooled coefficient estimates are positive and significant (at the 1% level) for the baseline ENet, DNN, and ensemble forecasts in the last row of Table 2. In sum, we find extensive evidence that the relative accuracy of the baseline ENet, DNN, and ensemble forecasts significantly increases as global FX volatility increases.

TABLE 2Global FX Volatility and Relative Forecast Accuracy

TABLE 2 Long description

The table consists of 6 columns and 12 rows. The columns are labeled: 1 Country, 2 Out-of-Sample Period, 3 No. of Obs., 4 Baseline ENet, 5 D N N, and 6 Ensemble.

* United Kingdom: 1995:01 to 2020:09, 309 observations, Baseline ENet 7.22***, D N N 0.71, Ensemble 3.96***.

* Switzerland: 1995:01 to 2020:09, 309 observations, Baseline ENet 3.76**, D N N 2.90**, Ensemble 3.33***.

* Japan: 1995:01 to 2020:09, 309 observations, Baseline ENet minus 0.88, D N N 2.76***, Ensemble 0.94.

* Canada: 1995:01 to 2020:09, 309 observations, Baseline ENet 8.15***, D N N 1.37, Ensemble 4.76***.

* Australia: 1995:01 to 2020:09, 309 observations, Baseline ENet 11.20***, D N N 3.72***, Ensemble 7.46***.

* New Zealand: 1995:01 to 2020:09, 309 observations, Baseline ENet 8.91***, D N N 1.16, Ensemble 5.04***.

* Sweden: 1995:01 to 2020:09, 309 observations, Baseline ENet 7.57***, D N N 0.95, Ensemble 4.26***.

* Norway: 1995:01 to 2020:09, 309 observations, Baseline ENet 9.89***, D N N 2.41*, Ensemble 6.15***.

* Denmark: 1995:01 to 2020:09, 309 observations, Baseline ENet 6.50***, D N N 3.37***, Ensemble 4.93***.

* Euro area: 2000:02 to 2020:09, 248 observations, Baseline ENet 20.87***, D N N 8.22***, Ensemble 14.55***.

* All (Pooled): 1995:01 to 2020:09, 3,029 observations, Baseline ENet 8.05***, D N N 2.64***, Ensemble 5.35***.

Asterisks indicate significance levels: * for 10 percent, ** for 5 percent, and *** for 1 percent.

A. Portfolio Construction

Consider a U.S. investor who goes long (short) the currency for country $ i $ (U.S. dollar). We can approximate the log excess return for the investment as $ {rx}_{i,t+1}\approx \left({r}_{i,t}-{r}_{\mathrm{US},t}\right)-\Delta {s}_{i,t+1} $ , where $ {rx}_{i,t} $ is the month- $ t $ log currency excess return for country $ i $ and $ {r}_{i,t} $ $ \left({r}_{\mathrm{US},t}\right) $ is the government bill yield for country $ i $ (the United States). Assume that the investor has mean–variance preferences. The investor allocates monthly across the U.S. dollar and all available foreign currencies. At the end of month $ T $ , the investor’s objective function is

(13) $$ \underset{{\boldsymbol{w}}_{T+1\mid T}}{\max}\;{\boldsymbol{w}}_{T+1\mid T}^{\prime}\hskip0.1em {\hat{\boldsymbol{\mu}}}_{T+1\mid T}-0.5\gamma \hskip0.1em {\boldsymbol{w}}_{T+1\mid T}^{\prime}\hskip0.1em {\hat{\Sigma}}_{T+1\mid T}\hskip0.1em {\boldsymbol{w}}_{T+1\mid T}, $$

where

(14) $$ {\hat{rx}}_{i,T+1\mid T}=\left({r}_{i,T}-{r}_{\mathrm{US},T}\right)-{\hat{\Delta s}}_{i,T+1\mid T}, $$

$ {\hat{\Delta s}}_{i,T+1\mid T} $ ( $ {\hat{rx}}_{i,T+1\mid T} $ ) is the investor’s exchange rate change (currency excess return) forecast for country $ i $ , $ {\hat{\boldsymbol{\mu}}}_{T+1\mid T}={\left[\begin{array}{ccc}{\hat{rx}}_{1,T+1\mid T}& \cdots & {\hat{rx}}_{N,T+1\mid T}\end{array}\right]}^{\prime } $ , $ {\hat{\Sigma}}_{T+1\mid T} $ is the investor’s estimate of the covariance matrix for the currency excess returns, $ {\boldsymbol{w}}_{T+1\mid T}={\left[\begin{array}{ccc}{w}_{1,T+1\mid T}& \cdots & {w}_{N,T+1\mid T}\end{array}\right]}^{\prime } $ is the $ N $ -vector of portfolio weights that will be in effect for month $ T+1 $ , $ {w}_{i,T+1\mid T} $ is the allocation to currency $ i $ (the U.S. dollar allocation is $ 1-{\boldsymbol{w}}_{T+1\mid T}^{\prime }{\boldsymbol{\iota}}_N $ , where $ {\boldsymbol{\iota}}_N $ is an $ N $ -vector of ones), and $ \gamma $ is the coefficient of relative risk aversion. As is common among practitioners, we assume that the investor uses an exponentially weighted moving average (EWMA) estimator for $ {\hat{\boldsymbol{\varSigma}}}_{T+1\mid T} $ (with a decay factor of 0.92). To keep the portfolio weights in plausible ranges when investing in multiple assets, we impose the restrictions that $ -0.5\le {w}_{i,T+1\mid T}\le 0.5 $ for $ i=1,\dots, N $ . We assume that $ \gamma =5 $ ; the results are similar for reasonable alternative $ \gamma $ values.

We measure the economic value of a machine-learning exchange rate forecast vis-à-vis the no-change benchmark by considering two cases. In the first case, the Bench-Opt portfolio, the investor uses the no-change benchmark forecast ( $ {\hat{\Delta s}}_{i,T+1\mid T}=0 $ ) in equation (14), so they ignore exchange rate predictability and set the expected exchange rate change to 0 (i.e., they use the stringent no-change benchmark forecast from the literature). In the second case, the ML-Opt portfolio, the investor uses the baseline ENet, DNN, or ensemble forecast for $ {\hat{\Delta s}}_{i,T+1\mid T} $ in equation (14), so they exploit exchange rate predictability by using the information in the predictors when allocating to foreign currencies. The Bench-Opt portfolio can be viewed as an optimal carry-trade portfolio, in the sense that it uses the bill yield differential to forecast the currency excess return ( $ {\hat{\Delta s}}_{i,T+1\mid T}=0 $ in equation (14)) in a mean–variance optimization framework (e.g., Daniel et al. (Reference Daniel, Hodrick and Lu2017)). The ML-Opt portfolio refines the Bench-Opt portfolio by augmenting the bill yield differential with a machine-learning exchange rate forecast to predict the currency excess return.

We construct out-of-sample portfolio weights as follows: We first use data through 1994:12 to compute the EWMA covariance matrix estimate and baseline ENet, DNN, or ensemble forecasts for 1995:01. We then solve equation (13) using the EWMA covariance estimate and baseline ENet, DNN, or ensemble (no-change) forecasts to compute the ML-Opt (Bench-Opt) portfolio weights for 1995:01. Next, we use data through 1995:01 to generate the EWMA covariance estimate and baseline ENet, DNN, or ensemble forecasts for 1995:02, and we compute the ML-Opt and Bench-Opt portfolio weights for 1995:02. We proceed in this manner through the end of the out-of-sample period, thereby mimicking the situation of an investor in real time. By comparing the performance of the ML-Opt portfolio to that of the Bench-Opt portfolio, we can gauge the economic value of the machine-learning forecasts to an investor.Footnote ²⁰

We compute the annualized average utility gain for the investor when they use the ML-Opt portfolio in place of the Bench-Opt portfolio:

(15) $$ \mathrm{Gain}=12\left[\left({\overline{rx}}_{\mathrm{ML}\hbox{-} \mathrm{Opt}}-0.5{\gamma \sigma}_{\mathrm{ML}\hbox{-} \mathrm{Opt}}^2\right)-\left({\overline{rx}}_{\mathrm{Bench}\hbox{-} \mathrm{Opt}}-0.5{\gamma \sigma}_{\mathrm{Bench}\hbox{-} \mathrm{Opt}}^2\right)\right], $$

where $ {\overline{rx}}_{\mathrm{ML}\hbox{-} \mathrm{Opt}} $ ( $ {\overline{rx}}_{\mathrm{Bench}\hbox{-} \mathrm{Opt}} $ ) and $ {\sigma}_{\mathrm{ML}\hbox{-} \mathrm{Opt}} $ ( $ {\sigma}_{\mathrm{Bench}\hbox{-} \mathrm{Opt}} $ ) are the mean and volatility, respectively, for the monthly ML-Opt (Bench-Opt) portfolio excess return over the out-of-sample period. Equation (15) is the annualized gain in certainty equivalent return, which can be interpreted as the maximum annualized portfolio management fee that the investor would be willing to pay to have access to the ML-Opt portfolio vis-à-vis the Bench-Opt portfolio. It measures the economic value of a machine-learning forecast to the investor.

B. Portfolio Performance

Table 3 reports portfolio performance metrics for the ML-Opt and Bench-Opt portfolios. In addition to the annualized average utility gain for the ML-Opt portfolio relative to the Bench-Opt portfolio, Table 3 reports annualized means, volatilities, and Sharpe ratios for the ML-Opt and Bench-Opt portfolio excess returns, as these are often of interest for analyzing foreign-currency (including carry-trade) portfolios.

TABLE 3Portfolio Performance: Allocating Across All Available Foreign Currencies

TABLE 3 Long description

The table consists of five columns: 1. Portfolio, 2. Annualized Average Utility Gain (%), 3. Annualized Mean (%), 4. Annualized Volatility (%), and 5. Annualized Sharpe Ratio.

Panel A. 1995:01 to 2020:09 Out-of-Sample Period:

* M L-Opt, baseline ENet: Utility Gain 4.50, Mean 12.89, Volatility 13.30, Sharpe Ratio 0.97.

* M L-Opt, D N N: Utility Gain 1.98, Mean 9.27, Volatility 11.54, Sharpe Ratio 0.80.

* M L-Opt, ensemble: Utility Gain 3.04, Mean 10.84, Volatility 12.39, Sharpe Ratio 0.87.

* Bench-Opt: Utility Gain not applicable, Mean 6.45, Volatility 12.39, Sharpe Ratio 0.64.

Panel B. 1995:01 to 2008:08 Out-of-Sample Period:

* M L-Opt, baseline ENet: Utility Gain 0.89, Mean 13.36, Volatility 12.93, Sharpe Ratio 1.03.

* M L-Opt, D N N: Utility Gain 0.95, Mean 12.36, Volatility 11.19, Sharpe Ratio 1.11.

* M L-Opt, ensemble: Utility Gain 0.82, Mean 12.60, Volatility 11.82, Sharpe Ratio 1.07.

* Bench-Opt: Utility Gain not applicable, Mean 10.93, Volatility 10.61, Sharpe Ratio 1.06.

Panel C. 2008:09 to 2020:09 Out-of-Sample Period:

* M L-Opt, baseline ENet: Utility Gain 8.46, Mean 12.35, Volatility 13.75, Sharpe Ratio 0.90.

* M L-Opt, D N N: Utility Gain 3.08, Mean 5.77, Volatility 11.88, Sharpe Ratio 0.49.

* M L-Opt, ensemble: Utility Gain 5.44, Mean 8.85, Volatility 13.03, Sharpe Ratio 0.68.

* Bench-Opt: Utility Gain not applicable, Mean 1.37, Volatility 9.39, Sharpe Ratio 0.15.

All Sharpe Ratios except Bench-Opt in Panel C are marked with asterisks indicating statistical significance.

Panel A of Table 3 reports results for the full 1995:01–2020:09 out-of-sample period. The ML-Opt portfolios based on the baseline ENet, DNN, and ensemble forecasts all perform very well, delivering annualized Sharpe ratios between 0.80 (DNN) and 0.97 (baseline ENet), all of which are significant at the 1% level. The Bench-Opt portfolio, which ignores exchange rate predictability, performs worse, with a Sharpe ratio of 0.65 (significant at the 1% level). The ML-Opt portfolios also provide substantive economic value to the investor vis-à-vis the Bench-Opt portfolio, with sizable average utility gains, ranging from 198 (DNN) to 450 (baseline ENet) basis points.

Although carry-trade portfolios perform well before the global financial crisis, they experience large losses in late 2008, and their performance generally deteriorates following the crisis. With this in mind, Panels B and C of Table 3 report results for the 1995:01–2008:08 and 2008:09–2020:09 subsamples, respectively. The start of the second subsample coincides with the bankruptcy of Lehman Brothers on Sept. 15, 2008, during the worst phase of the global financial crisis.

As discussed in Section VI.A, the Bench-Opt portfolio can be viewed as an optimal carry-trade portfolio. For the first subsample in Panel B of Table 3, the average return and Sharpe ratio for the Bench-Opt portfolio are considerably higher than their values for the full sample, with an annualized average return of 10.93% and a Sharpe ratio of 1.06 (significant at the 1% level). As reported in Panel C, beginning in September 2008, the average return and Sharpe ratio decline dramatically, with values of 1.37% and 0.15, respectively (the latter is insignificant at conventional levels). For comparison, we construct a standard (i.e., nonoptimized) long-short carry-trade portfolio that sorts currencies based on bill yield differentials into quintiles and goes long (short) the highest (lowest) interest-rate currencies each month, following the construction of the carry factor in Lustig et al. (Reference Lustig, Roussanov and Verdelhan2011). The conventional carry-trade portfolio generates annualized Sharpe ratios of 0.33, 0.77, and $ -0.08 $ for the 1995:01–2020:09, 1995:01–2008:08, and 2008:09–2020:09 out-of-sample periods, respectively, so it suffers a sharp deterioration in performance starting in September 2008.Footnote ²¹ Although the Bench-Opt portfolio performs better than the conventional carry-trade portfolio, like the latter, it still experiences a substantive decline in performance starting in late 2008.

Turning to the ML-Opt portfolios, in terms of the Sharpe ratio, the ML-Opt portfolios perform similarly to the Bench-Opt portfolio for the first subsample in Panel B of Table 3. Nevertheless, the ML-Opt portfolios still provide significant economic value to the investor relative to the Bench-Opt portfolio, with annualized average utility gains of 82 (ensemble) to 95 (DNN) basis points. Differences in performance between the Bench-Opt and ML-Opt portfolios become much more marked for the second subsample in Panel C. The Sharpe ratios for the ML-Opt portfolios remain relatively high in the second subsample. Furthermore, the annualized average utility gains accruing to the ML-Opt portfolios are especially sizable for the second subsample, ranging from 308 (DNN) to 846 (baseline ENet) basis points.

In sum, Table 3 reports that the machine-learning exchange rate forecasts deliver substantive economic value from an investment perspective. The economic value of the forecasts is especially evident beginning in late 2008 during the worst phase of the global financial crisis. Although the performance of optimal and conventional carry-trade portfolios declines precipitously starting in late 2008, by harnessing the information in economic fundamentals and their interactions with global variables (especially global FX volatility), the ML-Opt portfolios continue to generate substantial investment gains through the end of the out-of-sample period.

To provide additional perspective on the economic value of the machine-learning exchange rate forecasts, Figure 3 depicts log cumulative excess returns for the ML-Opt portfolios as well as the Bench-Opt and conventional carry-trade portfolios. The three ML-Opt portfolios and the Bench-Opt portfolio perform somewhat similarly through the summer of 2008, although the ML-Opt portfolios fare substantially better in the late 1990s and early 2000s in the wake of the Asian financial and Long-Term Capital Management crises. Although the ML-Opt and Bench-Opt portfolios experience losses in September of 2008 during the Lehman bankruptcy, their subsequent performances differ markedly, in line with Panel C of Table 3. The Bench-Opt portfolio suffers additional sizable losses later in 2008 in Figure 3, and its cumulative return essentially “flatlines” thereafter. The conventional carry-trade portfolio (which is not based on an optimization framework) suffers an even larger drop in late 2008 compared with the Bench-Opt portfolio and also flatlines subsequently. In sharp contrast, the ML-Opt portfolios make a strong recovery in late 2008 and continue to produce gains consistently thereafter, especially the ML-Opt portfolio based on the baseline ENet forecast. As indicated along the right-hand side of Figure 3, the log cumulative return profiles translate into sizable differences in terminal wealth. For an investor starting with $1 at the end of 1994:12, they would only have $4.61 ($2.02) at the end of 2020:09 based on the Bench-Opt (conventional carry-trade) portfolio; if they instead relied on an ML-Opt portfolio, they would have $21.76, $9.11, or $13.28 using the baseline ENet, DNN, or ensemble forecast, respectively.

FIGURE 3Log Cumulative Portfolio Excess Returns

Figure 3 depicts log cumulative excess returns for the ML-Opt, Bench-Opt, and conventional carry-trade portfolios. The evaluation period is 1995:01–2020:09. The numbers on the right are terminal wealth values, assuming that an investor begins with $1. Vertical bars delineate business-cycle recessions as dated by the National Bureau of Economic Research.

FIGURE 3 Long description

The x-axis represents time from 1995 to 2020. The y-axis represents log cumulative excess returns ranging from 0 to 3. A legend in the top-left identifies five lines: M L-Opt baseline E Net (dark blue), M L-Opt D N N (purple), M L-Opt ensemble (pink), Bench-Opt (orange), and Conventional carry trade (yellow).

All portfolios begin at 0 in 1995. The three M L-Opt portfolios show a consistent upward trend, significantly outperforming the Bench-Opt and Conventional carry trade lines. Vertical gray bars indicate recessions, with a notable sharp dip across all lines around 2008, marked by a dashed vertical line labeled Lehman bankruptcy.

Following the 2008 dip, the M L-Opt portfolios recover and continue to climb, while the Conventional carry trade remains relatively flat. The terminal wealth values at the far right of the graph are:

* M L-Opt baseline E Net: $21.76

* M L-Opt ensemble: $13.28

* M L-Opt D N N: $9.11

* Bench-Opt: $4.61

* Conventional carry trade: $2.02

Section A.3 of the Appendix lists a variety of robustness checks regarding portfolio performance (with detailed results reported in the Supplementary Material). Overall, the results show that the strong performance of the ML-Opt portfolios is robust along numerous dimensions.