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Can affine term structure models help us predict exchange rates?

Article Excerpt
THE OBJECTIVE OF this paper is to evaluate whether imposing no-arbitrage restrictions on the temporal evolution of interest and exchange rates can help predict exchange rates out of sample. In our model, prediction is based on the information embedded in interest rate differentials between two countries. In fact, Clarida and Taylor (1997), using a linear vector-error-correction model (VECM) for the term structure of forward premiums (interest rate differentials), are able to beat the long-standing and devastating result found by Meese and Rogoff (1983a, 1983b) that standard empirical exchange rate models cannot outperform a simple random-walk (RW) forecast. Thus, interest rates in these two countries contain information that is useful to predict exchange rates.

We make use of the literature on international term structure modeling to estimate an arbitrage-free model. Papers in this literature include, for example, Saa-Requejo (1993), Frachot (1996), Backus, Foresi, and Telmer (2001), Dewachter and Maes (2001), Ahn (2004), Brennan and Xia (2006), Dong (2006), and Leippold and Wu (2007). These papers exploit the fact that the same factors that determine the risk premium in the term structure of interest rates in each country might also determine the risk premium in exchange rate returns. The innovation here is to evaluate the out-of-sample exchange rate predictability of this class of models using the methodology developed in Clark and West (2006, 2007).

In particular, our focus is on internationally affine term structure models, that is, models where not only interest rates (bond yields) are known affine functions of a set of state variables, but also the expected rate of depreciation (over any arbitrary period of time) satisfies this property. The main benefit of this class of models is that one avoids the use of Monte Carlo methods to compute the expected rate of depreciation. Although simulation methods have been used elsewhere (Dong 2006), they can be computationally costly because the model is reestimated at each point in time (of the out-of-sample period) in order to compute the corresponding dynamic forecasts. The conditions needed to obtain an expected rate of depreciation that is affine on the set of state variables can be found in Diez de los Rios (2007). (1) In addition, the expression for the expected rate of depreciation is exact and, therefore, it is not subject to any discretization biases.

The main disadvantage of an internationally affine model is that its tractability comes at the price of imposing more restrictions on top of the assumption of no-arbitrage. In fact, we find that the affine model used in this paper does a poor job in forecasting the German mark/euro and the Swiss franc exchange rates, and that these negative results can be explained by a rejection of the additional restrictions that tractability imposes on the temporal evolution of exchange rates. On the other hand, we find that the use of this term structure model reduces the root mean square error (RMSE) in forecasting the spot U.S. dollar-pounds sterling rate by around 35% at the l-year forecast horizon, and by around 10% in the U.S. dollar-Canadian dollar, both relative to a RW forecast. Moreover, we find that this RMSE reduction is statistically significant.

This paper is organized as follows. Section 1 describes the empirical model and its estimation. Section 2 presents the empirical results. Section 3 concludes. A not-for-publication appendix with additional results omitted for the sake of brevity can be obtained upon request.

1. EMPIRICAL MODEL AND ESTIMATION

The analysis is based on a two-country world where assets can be denominated in either domestic currency j : 1 (e.g., "U.S. dollars") or foreign currency j = 2 (e.g., "British pounds"). We start by assuming that the vector of state variables in this global economy is given by [x.sub.t] = ([r.sup.(1).sub.t], [r.sup.(2).sub.t], where [r.sup.(j).sub.t] is the instantaneous interest rate in country j (also known as the short rate). In addition, we assume that [x.sub.t] follows a VAR(1) model in continuous time:

d[x.sub.t] = [PHI]([theta] - [x.sub.t])dt + [[summation].sup.1/2]d[W.sub.t], (1)

where [W.sub.t] is a 2 x l vector of independent Brownian motions that describes the shocks in this economy.

We also consider, based on a no-arbitrage argument, the existence of a (strictly positive) discount factor (SDF), [M.sup.(j).sub.t], for each country. This SDF is assumed to follow the standard law of motion:

d[M.sup.(j).sub.t]/ [M.sup.(j).sub.t] = -[r.sup.(j).sub.t]dt - [[LAMBDA].sup.(j)t.sub.t]d[W.sub.t] j = 1,2, (2)

where [[LAMBDA].sup.(j).sub.t] is a 2 x 1 vector that is usually called the market price of risk, because it describes how the SDF responds to the shocks given by [W.sub.t]. In particular, we follow Dai and Singleton (2002) and Ang and Piazzesi (2003) in assuming that the market prices of risk are affine functions of [x.sub.t]:

[[LAMBDA].sup.(j).sub.t] = [[lambda].sup.(j).sub.0] + [[lambda].sup.(j).sub.1][x.sub.t], (3)

where [[lambda].sup.(j).sub.0] is a 2 x 1 vector, and [[lambda].sup.(j).sub.1] is a 2 x 2 matrix.

Under this setup, it is possible...

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