Bayesian Model Averaging and exchange rate forecasts

Abstract

Exchange rate forecasting is hard and the seminal result of Meese and Rogoff [Meese, R., Rogoff, K., 1983. Empirical exchange rate models of the seventies: Do they fit out of sample? Journal of International Economics 14, 3–24] that the exchange rate is well approximated by a driftless random walk, at least for prediction purposes, still stands despite much effort at constructing other forecasting models. However, in several other macro and financial forecasting applications, researchers in recent years have considered methods for forecasting that effectively combine the information in a large number of time series. In this paper, I apply one such method for pooling forecasts from several different models, Bayesian Model Averaging, to the problem of pseudo out-of-sample exchange rate predictions. For most currency–horizon pairs, the Bayesian Model Averaging forecasts using a sufficiently high degree of shrinkage, give slightly smaller out-of-sample mean square prediction error than the random walk benchmark. The forecasts generated by this model averaging methodology are however very close to, but not identical to, those from the random walk forecast.

Introduction

Out-of-sample forecasting of exchange rates is hard. Meese and Rogoff (1983) argued that all exchange rate models do less well in out-of-sample forecasting exercises than a simple driftless random walk. Although this finding was heresy to many at the time that Meese and Rogoff wrote their paper, it has now become the conventional wisdom. Mark (1995) claimed that a monetary fundamentals model can generate better out-of-sample forecasting performance at long horizons, but that result has been found to be very sensitive to the sample period (Groen, 1999, Faust et al., 2003). Claims that a particular variable has predictive power for exchange rates crop up frequently, but these results typically apply just to a particular exchange rate and a particular subsample. As such, they are by now met with justifiable skepticism and are thought of by many as the result of data-mining exercises.

However, in many contexts, researchers have recently made substantial progress in the econometrics of forecasting using large datasets (i.e. a large number of predictors). The trick is to combine the information in these different variables in a judicious way that avoids the estimation of a large number of unrestricted parameters. Bayesian VARs have been found to be useful in forecasting: these often use many time series, but impose a prior that many of the coefficients in the VAR are close to zero. Approaches in which the researcher estimates a small number of factors from a large dataset and forecasts using these estimated factors have also been shown to be capable of superior predictive performance (see for example Stock and Watson (2002) and Bernanke and Boivin (2003)). Stock and Watson, 2001, Stock and Watson, 2004 obtained good results in out-of-sample prediction of international output growth and inflation by taking forecasts from a large number of different simple models and just averaging them. They found the good performance of simple model averaging to be remarkably consistent across subperiods and across countries. The basic idea that forecast combination outperforms any individual forecast is part of the folklore of economic forecasting, going back to Bates and Granger (1969). It is of course crucial to the result that the researcher just averages the forecasts (or takes a median or trimmed mean). It is, in particular, tempting to run a forecast evaluation regression in which the weights on the different forecasts are estimated as free parameters. While this leads to a better in-sample fit, it gives less good out-of-sample prediction.

Bayesian Model Averaging (BMA) is another method for forecasting with large datasets that has received considerable recent attention in both the statistics and econometrics literature. The idea is to take forecasts from many different models, and to assume that one of them is the true model, but that the researcher does not know which this is. The researcher starts from a prior about which model is true and computes the posterior probabilities that each model is the true one. The forecasts from all the models are then weighted by these posterior probabilities. It has been used in a number of econometric applications, including output growth forecasting (Min and Zellner, 1993, Koop and Potter, 2003), cross-country growth regressions (Doppelhofer et al., 2000, Fernandez et al., 2001) and stock return prediction (Avramov, 2002, Cremers, 2002). Avarmov and Cremers both report improved pseudo-out-of-sample predictive performance from BMA.

The contribution of this paper is to argue that BMA is useful for out-of-sample forecasting of exchange rates in the last fifteen years although the magnitude of the improvement that it offers relative to the random walk benchmark is quite small.

One does not have to be a subjectivist Bayesian to believe in the usefulness of BMA, or of Bayesian shrinkage techniques more generally. A frequentist econometrician can interpret these methods as pragmatic devices that may be useful for out-of-sample forecasting in the face of model and parameter uncertainty.¹ The plan for the remainder of the paper is as follows. Section 2 describes the idea of BMA. The out-of-sample exchange rate prediction exercise is described in Section 3. Section 4 concludes.