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Revista Brasileira de Finanças ISSN: Sociedade Brasileira de Finanças Brasil Fernandes Marçal, Emerson; Hadad Junio, Eli Is It Possible to Beat the Random Walk Model in Exchange

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Revista Brasileira de Finanças ISSN: Sociedade Brasileira de Finanças Brasil Fernandes Marçal, Emerson; Hadad Junio, Eli Is It Possible to Beat the Random Walk Model in Exchange Rate Forecasting? More Evidence for Brazilian Case Revista Brasileira de Finanças, vol. 14, núm. 1, marzo, 2016, pp Sociedade Brasileira de Finanças Rio de Janeiro, Brasil Available in: How to cite Complete issue More information about this article Journal's homepage in redalyc.org Scientific Information System Network of Scientific Journals from Latin America, the Caribbean, Spain and Portugal Non-profit academic project, developed under the open access initiative Is It Possible to Beat the Random Walk Model in Exchange Rate Forecasting? More Evidence for Brazilian Case (É Possível Bater o Passeio Aleatório na Previsão da Taxa de Câmbio? Mais Evidência para o Caso Brasileiro) Emerson Fernandes Marçal* Eli Hadad Junior** Abstract The seminal study of Meese and Rogoff on exchange rate forecastability had a great impact on the international finance literature. The authors showed that exchange rate forecasts based on structural models are worse than a naive random walk. This result is known as the Meese Rogoff (MR) puzzle. Although the validity of this result has been checked for many currencies, studies for the Brazilian currency are not common. In 1999, Brazil adopted the dirty floating exchange rate regime. Our goal is to run a pseudo real-time experiment to investigate whether forecasts based on econometric models that use the fundamentals suggested by the exchange rate monetary theory of the 80s can beat the random model for the case of the Brazilian currency. Our work has three main differences with respect to Rossi (2013). We use a bias correction technique and forecast combination in an attempt to improve the forecast accuracy of our projections. We also combine the random walk projections with the projections of the structural models to investigate if it is possible to further improve the accuracy of the random walk forecasts. However, our results are quite in line with her results. We show that it is not difficult to beat the forecasts generated by the random walk with drift using Brazilian data, but that it is quite difficult to beat the random walk without drift. Our results suggest that it is advisable to use the random walk without drift, not only the random walk with drift, as a benchmark in exercises that claim the MR result is not valid. Keywords: Meese-Rogoff puzzle, forecasting, exchange rate. JEL Codes: F31, F32, F41, C51. Submetido em 21 de fevereiro de Aceito em 24 de fevereiro de Publicado on-line em 21 de Abril de O artigo foi avaliado segundo o processo de duplo anonimato além de ser avaliado pelo editor. Editor responsável: Márcio Laurini. *Head of Center for Applied Macroeconomic Research at Sao Paulo School of Economics and CSSA-Mackenzie. **Mackenzie Presbyterian University Rev. Bras. Finanças (Online), Rio de Janeiro, 14, No. 1, March 2016, pp ISSN , ISSN online c 2016 Sociedade Brasileira de Finanças, under a Creative Commons Attribution 3.0 license - Marçal, E. F., Junior, E. H. Resumo O trabalho seminal de Meese e Rogoff sobre previsibilidade da taxa de câmbio teve grande impacto na literatura de finanças internacionais. Os autores mostraram que previsões baseadas em modelos econômicos estruturais tinham um desempenho pior que um passeio aleatório ingênuo. Este resultado é conhecido na literatura como o quebra-cabeça de Meese-Rogoff. Ainda que a validade deste resultado tenha sido checado para um número grande de moedas, estudos para a moeda brasileira ainda não são tão comuns pois o Brasil adotou o regime de câmbio flexível apenas a partir de Rossi (2013) realizou um estudo amplo do quebra-cabeça proposto pelos autores mas não fez a análise dos dados brasileiros. O objetivo deste trabalho é simular um exercício de tempo real para investigar se as previsões basedas em modelos econômicos de determinação de taxa de câmbio que usam os fundamentos dos modelos desenvolvidos nos anos oitenta tem desempenho melhor que o modelo de passeio aleatório. O trabalho tem três diferenças principais em relação ao feito por ela. Utiliza-se a técnica de correção de viés e de combinação de previsões na tentativa de melhorar a precisão das previsões. Também combinase as previsões do passeio aleatório com as dos modelos estruturais. Entretanto os resultados obtidos continuam em linha com da autora. O presente trabalho mostra que não é difícil gerar previsões com melhor desempenho que um passeio aleatório com tendência (drift) mas é extremamente difícil bater o desempenho do passeio aleatório ingênuo (sem tendência). O trabalho sugere que é fortemente recomendado utilizar o passeio aleatório sem tendência em exercícios que visem avaliar o quebra-cabeça de Meese e Rogoff. Palavras-chave: Meese-Rogoff puzzle, previsão, taxa de câmbio. 1. 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 1 See, for example, Frenkel (1976), Dornbusch (1976), Frankel (1979), and Hooper & Morton (1982). 66 Rev. Bras. Finanças (Online), Rio de Janeiro, V14, No. 1, March 2016 Is It Possible to Beat the Random Walk Model in Exchange Rate Forecasting? More Evidence for Brazilian Case 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. 2. 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 t = y t 1 + a + ε t (1) Rev. Bras. Finanças (Online), Rio de Janeiro, V14, No. 1, March Marçal, E. F., Junior, E. H. where ε 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 2.2 The structural models of the 80s E t [y t+k ] = y t + a k (2) 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 t=β 0 +β 1 (y t yt )+β 2(i t i t )+β 3(m t m t )+β 4(π t πt )+β 5(p t p t )+vt (3) where e t denotes an exchange rate between countries i and j, y t y t the difference in the real income, m t m t the difference in monetary aggregate, and π t π t the difference in inflation rates. v 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 t 1 (e t)=β 0 +E t 1 [β 1 (y t yt )+β 2(i t i t )+β 3(m t m t )+β 4(π t πt )+β 5(p t p 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): 2 One can see Enders (2008) for a textbook explanation. 68 Rev. Bras. Finanças (Online), Rio de Janeiro, V14, No. 1, March 2016 Is It Possible to Beat the Random Walk Model in Exchange Rate Forecasting? More Evidence for Brazilian Case E t 1(e t) = β 0 + β 1(y t 1 y t 1) + β 2(i t 1 i t 1) + β 3(m t 1 m t 1)+ β 4(π t 1 π t 1) + β 5(p t 1 p 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 VAR model One possible way of modelling the exchange rate and the fundamental is to use a VAR model: Y t = Π 1 Y t Π k 1 Y t k+1 + τ + ε t (6) where ε t are random normal and uncorrelated errors, Ω denotes the variance and covariance matrix of the errors that do not vary with time, and θ = [Π 1,..., Π k, τ] contains the parameters of the model. The vector Y t contains the exchange rate and set of fundamentals chosen by the analyst 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: Y t = Γ 1 Y t Γ k 1 Y t k+1 + αβ Y t 1 + µ + ε t (7) where ε t are random normal and uncorrelated errors, Ω denotes the variance and covariance matrix of the errors that do not vary with time, and θ = [Γ 1,...Γ k 1, α, β, µ] contains the parameters of the model. The vector Y t contains the exchange rate and set of fundamentals chosen by the analyst. denotes the first difference. Rev. Bras. Finanças (Online), Rio de Janeiro, V14, No. 1, March Marçal, E. F., Junior, E. H. 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 τ from t τ to t and collect all errors of these forecasts. Using an average of these errors ( ˆbc) 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: tf BC t+h = t F t+h ˆbc (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. 3. 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 70 Rev. Bras. Finanças (Online), Rio de Janeiro, V14, No. 1, March 2016 Is It Possible to Beat the Random Walk Model in Exchange Rate Forecasting? More Evidence for Brazilian Case 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 Θ = {y τ ; τ = 1, 2,..., k} as the set with the actual values of a variable and Θ 1 = {y 1 τ ; τ = 1, 2,..., k}, Θ 2 = {y 2 τ ; τ = 1, 2,..., k} as the set of predictions of models 1 and 2, respectively. Define the forecast error for model i as: e i τ = y τ y i τ (9) Then, choose some loss function g(e i τ ), with the difference given by: d i,j τ = g(e i τ ) g(e j τ ) (10) Let us state that the two models will have equal forecast accuracy if and only if the loss function has an expected value of zero for all τ. Diebold and Mariano formulated the following null hypothesis: H 0 : E(d i,j τ ) = 0 for all τ (11) against the alternative hypothesis that the models do not have the same level of accuracy: Now consider the following quantity: H 1 : E(d i,j τ ) 0 (12) d i,j = M τ=1 di,j τ M (13) Using a robust estimate of the variance of d i,j denoted by V AR( ˆ d i,j ), and providing that certain regularity conditions hold, the following statistic is proposed to test the null: Rev. Bras. Finanças (Online), Rio de Janeiro, V14, No. 1, March Marçal, E. F., Junior, E. H. DM = ˆ d i,j V AR( d i,j ) a N(0, 1) (14) One serious limitation of the Diebold-Mariano framework is that it is not designed to deal with many different competing models simultaneously. If there is a benchmark, all remaining models could be compared against the benchmark. However, if the analyst wants to rank the models and has no particular interest in choosing a benchmark, another framework should be tried. Hansen et al. (2011) tries to fill this gap. 3.2 The model confidence set The model confidence set (MCS) is a model selection technique developed by Hansen, Lunde, and Nason (2011). It consists of an algorithm that ranks the forecasts from models. M contains the best model(s) chosen from a collection of models, M 0, in which the best model is defined using criteria related to prediction quality. Definition 1: The set of superior objects is defined by: M {i ɛ M 0 : E(d i,j τ ) 0 for all j ɛ M 0 } In the following, we let M denote the complement to M. That is, M {i ɛ M 0 : E(d i,j τ ) 0 for all j ɛ M 0 } The MCS selects a model using an equivalence test, δ M, and an elimination rule, ϱ M. The equivalence test is applied to the set M = M 0. If the equivalence hypothesis is rejected, then there is evidence that there is a set of inferior models in terms of forecast accuracy. Therefore, the rule ϱ M is used to eliminate the models with poor predictive quality. The procedure is repeated until the equivalence test, δ M, is accepted. Then, the model ( M F ) is selected to be the set of the best final models. The null hypothesis of the test is: H 0 M : E(d i,j τ ) = 0 for all i, j ɛ M (15) where M M 0. The alternative hypothesis is: H 1 M : E(d i,j τ ) 0 for some i, j ɛ M (16) Note that there might be better models outside of the set of candidate models, M 0. The goal of the MCS is to determine M. 72 Rev. Bras. Finanças (Online), Rio de Janeiro, V14, No. 1, March 2016 Is It Possible to Beat the Random Walk Model in Exchange Rate Forecasting? More Evidence for Brazilian Case The null hypothesis can be tested using the following statistic: 3 T D = i ɛ M t 2 i (17) where t i = i V AR( ˆ d d and d i M 1 d i ) j ɛ M ij The statistic given by (17) has a non-standard distribution that can be simulated using bootstrap techniques. The elimination rule is: The algorithm ϱ M = arg max i (t i ) (18) The MCS algorithm takes the following steps: (i) Initially set M = M 0 ; (ii) Test HM 0 using δ M at level α; (iii) If HM 0 is not rejected then the procedure ends and the final set is ˆM 1 α = M, otherwise we use ϱ M to eliminate an object from M and repeat step (i). The authors show that the MCS has the following statistical properties: (i) lim n P (M ˆM 1 α ) 1 α and (ii) lim n P (i ɛ ˆM 1 α ) = 0 for all i ɛ M Ranking the models: MCS p-values The elimination rule, ϱ M, defines a sequence of random sets, M 0 = M 1 M 2... M m0, where M i = {ϱ i,..., ϱ m0 } and m 0 are the number of elements in M 0. ϱ M0 is the first to be eliminated, ϱ M1 is the second to be eliminated, and so forth. At the end, only one model survives. We set the p-value of this model as 1. We collect the p-value of the eliminated model if it is higher than the p-value of the previously eliminated model. If it is not, we opt to maintain the p-values of the previous rejections. The MCS p-values are convenient because they make it easier for the analyst to determine whether a particular object is in ˆM 1 α. 3 There are others possible choices. Rev. Bras. Finanças (Online), Rio de Janeiro, V14, No. 1, March Marçal, E. F., Junior, E. H. 3.3 Pseudo real-time exercise The data gathered for the countries is used to create many variants of the structural models in order to forecast the exchange rate. The sample is split into two parts. The first half of the sample is used to estimate the models, and the second half is used to evaluate the forecast performance of the models in various horizons. In the exercise, we attempt to simulate a real-time operation. We use an information set that reflects, as closely as possible, the one available to agents at the time of the forecast. In other words, the models are re-estimated at each poin

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