Is it possible to beat the random walk model in exchange rate forecasting? More evidence for Brazilian case/E possivel bater o passeio aleatorio na previsao da taxa de cambio? Mais evidencia para o caso Brasileiro.
| Autor | Marcal, Emerson Fernandes |
| Cargo | Ensayo |
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Introduction
The seminal study of Meese & Rogoff (1983) on exchange rate forecastability had a great impact on international finance literature. The authors compared exchange rate projections obtained from structural models against a naive random walk. They used structural monetary models of the 80s.1 Their main result showed that it is not easy to outperform forecasts of a naive random walk model. Subsequently, an extensive literature emerged, but the result of Meese & Rogoff (1983) still holds. This is the so-called Meese-Rogoff (MR) puzzle.
In a recent paper, Rossi (2013) reviewed the literature that followed the work of Meese and Rogoff, aiming to confirm and explain their result. Rossi (2013) showed that it is still difficult to beat the random walk, particularly in an out-of-sample exercise. She ran a comprehensive exercise with different sets of fundamentals, econometric model specifications, samples, and countries. She showed that the MR puzzle still holds, particularly in an out-of-sample exercise. However, she did not include Brazil in her research.
The purpose of our paper is to run an exercise similar to Rossi (2013) using Brazilian data. We focus our analysis on multivariate econometric models with monetary fundamentals. In addition, we opt to run a forecast exercise using bias correction and forecasting combination techniques. We combine the forecasts of the models among themselves and with the random walk. We perform a pseudo real-time exercise to replicate, as closely as possible, the forecast that one could have carried out at a particular time in the past. We use the Model Confidence Set (MCS) algorithm developed by Hansen et al. (2011) to evaluate the predictive equivalence of the forecasts.
Our results suggest that the MR puzzle holds for Brazilian data. It is hard to beat the random walk without drift for almost all analysed horizons from one up to six quarters. Moreover, it is much easier to beat the random walk with drift than without drift.
The paper is divided into six sections. The first section is this introduction. The second section discusses the strategy for constructing forecasts using the fundamentals suggested by the model of the 80s. In the third section, the MCS algorithm is described. In the fourth section, we briefly discuss the results of Rossi (2013) and some key references regarding the MR puzzle. The fifth section presents the results of our empirical exercise and compares them with the literature. Finally, some concluding remarks are drawn.
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Constructing a strategy to forecast exchange rate
In this section, we briefly describe the equation used to construct forecasts based on the monetary exchange rate models of the 80s as well as on econometric models.
2.1 The random walk model
In this study, the goal is to compare forecasts obtained from the random walk models with and without drift against a wide array of econometric models. The random walk model with drift is given by
[y.sub.t] = [y.sub.t]-1 + a + [[epsilon].sub.t] (1)
where [[epsilon].sub.t] is a random variable with zero mean that is independent over time. The model without drift can be obtained by assuming that a = 0.
The k steps ahead forecast is given by
[E.sub.t][[y.sub.t] + k] = [y.sub.t] + a x k (2)
2.2 The structural models of the 80s
In addition to the aforementioned random walk models, this study uses vector autoregressive models with and without an error correction mechanism in order to construct forecasts. (2) The choice of which explanatory variables to include in the models is made based on the economic models of the 80s and 90s that served as a basis for the article of Meese and Rogoff. Some key references are Frenkel (1976), Bilson (1978), Dornbusch (1976), and Frankel (1979).
These models link the exchange rate to a set of fundamentals. The model of the 80s implies an equation similar to (3), with different restrictions imposed on the coefficients according to variants of the basic model:
[e.sub.t] = [[beta].sub.0] + [[beta].sub.1] ([y.sub.t]-[y.sup.*.sub.t]) + [[beta].sub.2] ([i.sub.t]-[i.sup.*.sub.t]) + [[beta].sub.3] ([m.sub.t]-[m.sup.*.sub.t]) + [[beta].sub.4] ([[pi].sub.t]-[[pi].sup.*.sub.t]) + [[beta].sub.5]([p.sub.t]-[p.sup.*.sub.t]) + [v.sub.t] (3)
where [e.sub.t] denotes an exchange rate between countries i and j, [y.sub.t]-[y.sup.*.sub.t] the difference in the real income, [m.sub.t]-[m.sup.*.sub.t] the difference in monetary aggregate, and [[pi].sub.t]-[[pi].sup.*.sub.t] the difference in inflation rates. [v.sub.t] is a random variable with zero mean.
2.3 Single-equation models
The first step in constructing a forecast based on (3) is to estimate the parameters using some econometric technique. Ordinary least square is one common choice in the literature, but others techniques can be used as well.
We calculate the expectations based on the information available at time t-1.
[E.sub.t-1] ([e.sub.t]) = [[beta] .sub.0] + [E.sub.t-1] [[[beta].sub.1]([y.sub.t]-[y.sup.*.sub.1]) + [[beta].sub.2] ([i.sub.t]-[i.sup.*.sub.t]) + [[beta].sub.3]([m.sub.t]-[m.sup.*.sub.t]) + [[beta].sub.4]([[pi].sub.t]- [[pi].sup.*.sub.t]) + [[beta].sub.5] ([p.sub.t]-[P.sup.*.sub.t])] (4)
Assuming that it is not possible to predict any change in the fundamental using the information available until t-1, the forecast for the exchange rate in t based on information t-1 is given by (5):
[E.sub.t-l]([e.sub.t]) = [[beta].sub.0] + [[beta].sub.1]([y.sub.t-l]-[y.sup.*.sub.t-l]) + [[beta].sub.2]([i.sub.t-1]- [i.sup.*.sub.t-1]) + [[beta].sub.3]([m.sub.t-l]-[m.sup.*.sub.t-l]) + [[beta].sub.4]([[pi].sub.t-l]-[[pi].sub.*.sub.t- 1]) + [[beta].sub.5]([p.sub.t-l]-[P.sup.*.sub.t-1])] (5)
The forecasts are constructed using (5). It is also possible to predict a change in fundamentals using past information. If this is the case, an econometric model can be formulated, which leads us to the multivariate equation approach.
2.4 Multiple-equations models
Two different econometrics models are used in this paper. The first is the vector autoregressive (VAR) model, and the second is the vector error correction (VEC) model.
2.4.1 VAR model
One possible way of modelling the exchange rate and the fundamental is to use a VAR model:
[Y.sub.t]=[[PI].sub.1][Y.sub.t-1] + ... + [[PI].sub.k-1][Y.sub.t-k + 1] + [tau] + [[epsilon].sub.t] (6)
where [[epsilon].sub.t] are random normal and uncorrelated errors, [OMEGA] denotes the variance and covariance matrix of the errors that do not vary with time, and [theta] = [[[PI].sub.1], ..., [[PI].sub.k], [tau]] contains the parameters of the model. The vector [Y.sub.t] contains the exchange rate and set of fundamentals chosen by the analyst.
2.4.2 VEC model
We assume that the local data generation process for the exchange rate and a set of fundamentals is given by the following VAR model:
[DELTA][Y.sub.t] = [[GAMMA].sub.1][DELTA][Y.sub.t-1] + ... + [[GAMMA].sub.k-1][DELTA][Y.sub.t-k + 1] + [alpha][beta]'[Y.sub.t-1] + [mu] + [[epsilon].sub.t] (7)
where [[epsilon].sub.t] are random normal and uncorrelated errors, [OMEGA] denotes the variance and covariance matrix of the errors that do not vary with time, and [theta] = [[[GAMMA].sub.1], ...[[GAMMA].sub.k-1], [alpha], [beta], [mu]] contains the parameters of the model. The vector [Y.sub.t] contains the exchange rate and set of fundamentals chosen by the analyst. [DELTA]denotes the first difference.
2.5 Bias correction approach
One way to improve the forecast performance of a particular model is the bias correction approach. If one model systematically forecasts in one wrong direction, the analyst can, ideally, correct the forecast by adding a term to avoid the bias.
Our approach is inspired by the paper of Issler & Lima (2009). Suppose that we want to forecast the exchange for t + 1 with information available until t. We compute forecasts for a window of length [tau] from t-[tau] to t and collect all errors of these forecasts. Using an average of these errors ([??]c) and under certain conditions, this simple average will provide a consistent estimate of the bias.
Our bias-corrected forecast is calculated by the following formula:
[sub.t][F.sup.BC.sub.t + h] = [tau] [F.sub.t + h]-[??]c (8)
where h>0 denotes the horizon of the forecast.
2.6 Combined forecast techniques
Granger & Ramanathan (1984) and Bates & Granger (1969) suggested that a combination of two forecasts can generate more precise forecasts. There is extensive literature discussing alternative methods for combining forecasts. In this paper, we opt to use a simple combination technique. We combine each pair of forecasts using a simple average. We aim to evaluate whether this simple technique pays off. In her empirical exercise, Rossi (2013) did not use any forecast combination; nor did the seminal paper of Meese & Rogoff (1983).
We explore two types of combinations. The first is a combination of all possible pairs of structural model forecasts. The second combines the random walk forecast with each structural model forecast. If any structural model contains relevant information regarding the future, it may not be able to beat the random walk; however, combined with it, the projection may outperform the random walk. We aim to investigate if it is possible to further improve the predictive power of the random walk.
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How to choose among different forecast models
In this section, we discuss two criteria used to compare the predictive forecasts of different models. The first is the classical Diebold-Mariano test Diebold & Mariano (1995). The second is the model confidence set developed by Hansen et al. (2011). The latter can be seen as a refinement of the former test.
3.1 Classical Diebold-Mariano test
In empirical applications, it is often the case that two or more time series models are available for forecasting a variable:
Define [THETA] = {[y.sub.[tau]] ; [tau] = 1, 2, ...., k} as the set with the actual values of a variable and [[THETA].sub.1] = {[y.sup.1.sub.[tau]]; [tau] = 1, 2...
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