Out-of-Sample Forecast Tests Robust to the Window Size Choice
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1 Out-of-Sample Forecast Tests Robust to the Window Size Choice Barbara Rossi and Atsushi Inoue Duke University and NC State January 6, 20 Abstract This paper proposes new methodologies for evaluating out-of-sample forecasting performance that are robust to the choice of the estimation window size. The methodologies involve evaluating the predictive ability of forecasting models over a wide range of window sizes. We show that the tests proposed in the literature may lack power to detect predictive ability, and might be subject to data snooping across di erent window sizes if used repeatedly. An empirical application shows the usefulness of the methodologies for evaluating exchange rate models forecasting ability. Keywords: Predictive Ability Testing, Forecast Evaluation, Estimation Window. Acknowledgments: We thank M. McCracken, J. Nason, A. Patton, K. Sill, D. Thornton and seminar participants at the Econometrics Workshop at the St. Louis Fed, Bocconi University and the Triangle Econometrics Conference for useful comments and suggestions. J.E.L. Codes: C22, C52, C53
2 Introduction This paper proposes new methodologies for evaluating the out-of-sample forecasting performance of economic models. The novelty of the methodologies that we propose is that they are robust to the choice of the estimation and evaluation window size. The choice of the estimation window size has always been a concern for practitioners, since the use of di erent window sizes may lead to di erent empirical results in practice. In addition, arbitrary choices of window sizes have consequences about how the sample is split into in-sample and out-of-sample portions. Notwithstanding the importance of the problem, no satisfactory solution has been proposed so far, 2 and in the forecasting literature it is common to only report empirical results for one window size. 3 This common practice raises two concerns. A rst concern is that the ad hoc window size used by the researcher may not detect signi cant predictive ability even if there would be signi cant predictive ability for some other window size choices. A second concern is the possibility that satisfactory results were obtained simply by chance, after data snooping over window sizes. That is, the successful evidence in favor of predictive ability might have been found after trying many window sizes, although only the results for the successful window size were reported and the search process was not taken into account when evaluating their statistical signi cance. 4 Ultimately, however, the size of the estimation window is not a parameter of interest for the researcher: the objective is rather to test predictive ability and, ideally, researchers would like to reach empirical conclusions that are robust to the choice of the estimation window size. This paper views the estimation window as a "nuisance parameter": we are not interested in selecting the "best" window; rather we would like to propose predictive ability tests The problem we address a ects not only rolling window forecasting schemes, which we focus on, but also recursive window forecasting schemes, which we discuss in the Appendix. 2 As discussed in Pesaran and Timmermann (2005), the choice of the window size depends on the nature of the possible model instability and the timing of the possible breaks. In particular, a large window is preferable if the data generating process is stationary, but comes at the cost of lower power since there are fewer observations in the evaluation window. Similarly, a shorter window may be more robust to structural breaks although may not provide as precise estimation as larger windows if the data are stationary. The empirical evidence shows that instabilities are widespread; e.g. Stock and Watson (2003a) for macroeconomic data, Paye and Timmermann (2006) for asset returns and Rossi (2006) for exchange rate models. 3 For example, see Meese and Rogo (983a), Chinn (99), Qi and Wu (2003), Cheung et al. (2005), van Dijk and Frances (2005), Clark and West (2006, 2007), Gourinchas and Rey (2007), and Molodtsova and Papell (2009). 4 Only rarely do researchers check the robustness of the empirical results to the choice of the window size by reporting results for a selected choice of window sizes. 2
3 that are "robust" to the choice of the estimation window size. The procedures that we propose ensure that this is the case by evaluating the models forecasting performance for a variety of estimation window sizes, and then taking summary statistics of this sequence. Our methodology can be applied to most tests of predictive ability that have been proposed in the literature, such as Diebold and Mariano (995), West (996), Clark and McCracken (200) and Clark and West (2007). 5 We also propose methodologies that can be applied to Mincer and Zarnowitz s (969) tests of forecast e ciency, as well as more general tests of forecast optimality. Although the paper focuses on rolling window forecast, similar results are developed for recursive window forecasts, and are provided in the Appendix. This paper is closely related to the works by Pesaran and Timmermann (2007) and Clark and McCracken (2009), and more distantly related to Pesaran, Pettenuzzo and Timmermann (2006) and Giacomini and Rossi (200). Pesaran and Timmermann (2007) propose cross validation and forecast combination methods that identify the "ideal" window size using sample information. In other words, Pesaran and Timmermann (2007) extend forecast averaging procedures to deal with the uncertainty over the size of the estimation window, for example, by averaging forecasts computed from the same model but over various estimation window sizes. Their objective is to improve the model s forecast. Similarly, Clark and McCracken (2009) combine rolling and recursive forecasts in the attempt of improving the forecasting model. Our paper instead proposes to take a summary statistics of tests of predictive ability computed over several estimation window sizes. Our objective is not to improve the forecasting model nor to estimate the ideal window size. Rather, our objective is to assess the robustness of conclusions of predictive ability tests to the choice of the estimation window size. Pesaran, Pettenuzzo and Timmermann (2006) have exploited the existence of multiple breaks to improve forecasting ability; in order to do so, they need to estimate the process driving the instability in the data. An attractive feature of the procedure we propose is that it does not need to know nor determine when the structural breaks have happened. Giacomini and Rossi (200) propose techniques to evaluate the relative performance of competing forecasting models in unstable environments, assuming a "given" estimation window size. In this paper, our goal is instead to ensure that forecasting ability tests be robust to the choice of the estimation window size. Finally, this paper is linked to the literature on data snooping: if researchers report empirical results for just one window size (or a couple of them) when they actually considered many possible window sizes prior to reporting their results, their inference will be incorrect. This paper provides a way to account for data snooping 5 The only assumption we make is that the window size be large relative to the sample size. 3
4 over several window sizes, and removes the arbitrary decision of the choice of the window length. 6 We show the usefulness of our methods in an empirical analysis. The analysis re-evaluates the predictive ability of models of exchange rate determination by verifying the robustness of the recent empirical evidence in favor of models of exchange rate determination (e.g. Molodtsova and Papell, 2009, and Engel, Mark and West, 2007) to the choice of the window size. Our results reveal that the forecast improvements found in the literature are much stronger when allowing for a search over several window sizes. The paper is organized as follows. Section 2 presents the econometric methodology. Section 3 shows some Monte Carlo evidence on the performance of our procedures in small samples, and Section 4 presents the empirical results. Section 5 concludes. 2 Econometric methodology Let h denote the ( nite) forecast horizon. We assume that the researcher is interested in evaluating the performance of h steps ahead direct forecasts for the scalar variable y t+h using a vector of predictors x t using a rolling window forecast scheme. The case of the recursive window forecast scheme will be considered in Appendix B. We assume that the researcher has P out-of-sample predictions available, where the rst prediction is made based on an estimate from a sample ; 2; :::; R, such that the last out-of-sample prediction is made based on an estimate from a sample of T R + ; :::; R + P = T where R + P + h = T + h is the size of the available sample. The methods proposed in this paper can be applied to out of sample tests of equal predictive ability, forecast rationality and unbiasedness. Let researchers bee interested in evaluating the forecasting performance of two competing models: Model, involving parameters, and Model 2, involving parameters. In the rolling window forecast method, the true but unknown model s parameters and are estimated by b t;r and b t;r, which are calculated using samples of R observations dated t R + ; :::; t, for n o T n t = R; R + ; :::; T. Let L () bt;r t+h and L (2) t+h b o T t;r denote the sequence of loss t=r t=r functions n of models and o 2 evaluating h steps ahead relative out-of-sample forecast errors, T and let L bt;r t+h ; b t;r denote their di erence. 7 t=r 6 See White (2000) or Clark and McCracken (200) for techniques robust to data snooping in a very di erent context, namely multiple model comparisons. 7 The rolling scheme case involves in principle the choice of two nuisance parameters: the size of the estimation window, and the rst estimation window, which could in principle be di erent. In practice, 4
5 We start by discussing results pertaining to widely used measures of relative forecasting performance, where the loss function is the di erence of the forecast error losses of two competing models. We consider two separate cases, depending on whether the models are nested or non-nested. Subsequently we present results for regression-based tests of predictive ability, such as Mincer and Zarnowitz (969) regressions, among others. 2. Non-Nested Model Comparisons Let the researcher evaluate the two models using the sample average of the sequence of standardized out-of-sample loss di erences: L T (R) b R T =2 TX L t+h ( b t;r ; b t;r ); where b 2 R is a consistent estimate of the long run variance matrix of the out-of-sample loss di erences. For example, this is the strategy adopted by Diebold and Mariano (995), West (996) and McCracken (2000). 8 The test statistics that each of these papers propose di er in the estimate of the variance. Since our approach is valid no matter which test statistic is considered, we will defer details on how the variance estimate is constructed. t=r We make the following high level assumption. 9 Assumption : E[L t+h ( ; )]g obeys a func- (a) The partial sum ^ R T P =2 [T ] t=[t ] fl t+h(^ t;r ; ^ t;r ) tional central limit theorem: ^ R T =2 [T ] X t=[t ] n o L t+h (^ t;r ; ^ t;r ) E[L t+h ( ; )] ) B() B(); () where [x] denotes the integer part of x, ) denotes weak convergence on the space of cadlag functions on (0; ), D(0; ) equipped with the Skorohod topology, and B() denotes the standard Brownian motion (see Billingsley (968) for the de nitions of weak convergence, space D and the Skorohod topology); however, researchers choose the rst estimation window to be the same as the size of the rolling window, the case we focus on. Appendix B shows that our proposed tests can be extended to the recursive scheme in a straightforward manner. We focus on the rolling scheme in the main text because it is the leading case in which the value of R is manipulated in practice. 8 Note that T P =2 LT (R) would be exactly the test statistic proposed by Diebold and Mariano (995), West (996) and McCracken (2000). 9 See Rossi and Sekhposyan (200) for more primitive assumptions. 5
6 (b) lim T;R! R=T = 2 (0; ). Assumption implies b R P =2 P T t=r L t+h( b t;r ; b t;r ) is asymptotically normally distributed for a given estimation window R, and it is therefore satis ed for the Diebold and Mariano (995), West (996) and McCracken (2000) tests, under their assumptions. Proposition describes our proposed procedure for non-nested models. We consider two appealing and intuitive types of weighting schemes over the window sizes. The rst scheme is to choose the largest value of the L T (R) test sequence, and corresponds to the test labeled R T. This mimics to the case of a researcher experimenting with a variety of window sizes, and reporting only the empirical results corresponding to the best evidence in favor of predictive ability. The second scheme is to take a weighted average of the L T (R) tests, giving equal weight to each test. This would correspond to the test labeled A T. Proposition (Out-of-sample Robust test for Non-nested Models) Suppose Assumption holds. Let and where R T = sup jl T (R) j; R = R; :::R; (2) R A T = R R + L T (R) b R T =2 RX jl T (R)j ; (3) R=R TX L T ( b t;r ; b t;r ); R = [T ] ; R = [T ]; R = [T ]; and ^ 2 R is a consistent estimator of 2. Under the null hypothesis H 0 : lim T! E [L T (R)] = 0 for all R; and t=r R T =) sup jb () B () j; (4) 2[;] A T =) Z jb () B ()j d; (5) where B () is a standard univariate Brownian motion. The null hypothesis for the R T test is rejected at the signi cance level when R T > k R whereas the null hypothesis for the A T test is rejected when A T > k A ; where the critical values ; k R and ; k A for various values of and = are reported in Table. 6
7 The proof of Proposition is provided in Appendix A. Note that the critical values for signi cance level are, respectively, k R and k A, where k R and k A solve ( ) P sup 2[;] jb () ( Z P jb () B () j > k R B ()j d > k A and are computed using Monte Carlo simulation methods. ) = (6) = ; (7) Note also that the practical implementation of (2) and (3) requires researchers to choose R and R. To avoid data snooping over the choices of R and R, we recommend researchers to x R = T R, and to use R = [0:5T ] in practice. A consistent estimate of 2 for non nested model comparisons in rolling windows is provided by McCracken (2000, p. 203, eqs. 5 and 6). For example, a consistent estimator when parameter estimation error is negligible is: ^ 2 R = q(p ) X i= q(p )+ TX ( ji=q(p )j)p L d bt;r t+h ; b t;r L d bt t+h i i;r ; b t i;r ; (8) t=r where L d bt;r t+h ; b t;r L bt;r t+h ; b t;r P P T t=r L bt;r t+h ; b t;r and q(p ) is a bandwidth that grows with P (e.g., Newey and West, 987). In particular, a leading case where (8) can be used is when the same loss function is used for estimation and evaluation. INSERT TABLE HERE 2.2 Nested Models Comparison For the case of nested models comparison, we follow Clark and McCracken (200) and Clark and West (2007), and focus on the empirically leading case of a quadratic loss function, which implies that the predictive ability is evaluated according to the Mean Squared Forecast Error. 0 Let Model be the parsimonious model, and Model 2 be the larger model that nests Model. Let y t+h the period t forecasts of y t+h from the two models be denoted by by t;t+h and by 2t;t+h. We consider two separate test statistics, proposed by Clark and McCracken (200) and Clark and West (2007), respectively. 0 The reason we focus on quadratic loss functions is that this is a crucial assumption in Clark and West (2007), which allows them to obtain asymptotically normal test statistics. Since the methods proposed in this paper rely on the asymptotic normality result, we maintain their assumption. 7
8 For the latter, de ne the adjusted mean square prediction error as: L adj t+h (b t;r ; b t;r ) = (y t+h by t;t+h ) 2 (yt+h by 2t;t+h ) 2 (by t;t+h by 2t;t+h ) 2 ; so that the rescaled Clark and West (2007) test statistic is: L adj T (R) b R T =2 TX t=r L adj t+h (b t;r ; b t;r ); (9) where b 2 R is a consistent estimate of the long run variance matrix of the adjusted out-ofsample loss di erences, L adj t+h (b t;r ; b t;r ). Note that, since the models are nested, the Clark and West s (2007) is one sided; as a consequence, our test procedure will be one sided too. Clark and West (2007) show that the test statistic (9) is "approximately normal"; if we extend the Clark and West s (2007) argument of "approximate normality" to partial sums of rescaled adjusted out-of-sample loss di erences, under their assumptions a logical approximation can be obtained as follows. We make the following high level assumption. o E[L adj t+h ( ; )] obeys a func- Assumption 2: (a) The partial sum ^ R T P n =2 [T ] t=[t ] L adj t+h (^ t;r ; ^ t;r ) tional central limit theorem: [T ] X ^ T =2 t=[t ] n L adj t+h (^ t;r ; ^ t;r ) (b) lim T;R! R=T = 2 (0; ). o E[L adj t+h ( ; )] ) B() B(); (0) Assumption 2 implies b R P =2 P T t=r Ladj t+h (b t;r ; b t;r ) is approximately asymptotically normally distributed for a given estimation window R, as in Clark and West (2007). 2 Proposition 2 presents our robust test for the case of nested models comparisons. The original Clark and West s (2007) statistic is b R P t+h (b t;r ; b t;r ): 2 More precisely, Clark and West (2007) numerically show that the tail of the asymptotic distribution =2 P T t=r Ladj of their test statistic can be bounded by that of the standard normal distribution. Because normal approximations to the distribution of their test statistic yields a conservative test for given R, our test is also conservative which is con rmed by the Monte Carlo experiment. Note that Clark and McCracken s (200) ENCNEW test would not satisfy this assumption, since it does not have an asymptotically Normal distribution. 8
9 Proposition 2 (Out-of-sample Robust test for Nested Models) Suppose Assumption 2 holds. Let and where R T = sup L adj T (R) ; R = R; :::R; () R A T = R R + L adj T (R) b R T =2 RX R=R TX t=r L adj T (R) ; (2) L adj t+h (b t;r ; b t;r ) and ^ 2 R is a consistent estimator of 2. Under the null hypothesis H 0 : lim T! E[L adj T (R)] = 0 for all R; and R T =) sup [B () B ()] ; (3) 2[;] A T =) Z [B () B ()] d; (4) where R = [T ] ; R = [T ]; R = [T ] ; and B () is a standard univariate Brownian motion. The null hypothesis for the R T test is rejected at the signi cance level when R T > k R whereas the null hypothesis for the A T test is rejected when A T > k A ; where the critical values ; k R and ; k A for various values of and = are reported in Table 2(a). The proof of Proposition 2 is provided in Appendix A. Note that the critical values for a signi cance level are, respectively, k R and k A, where k R and k A solve ( ) P sup 2[;] [B () B ()] > k R ( Z ) P [B () B ()] d > k A The critical values are obtained via Monte Carlo simulation methods. = (5) = : (6) A consistent estimator of 2 when parameter estimation error is negligible is ^ 2 R = q(p ) X i= q(p )+ ( ji=q(p )j)p TX t=r L adj;d bt;r t+h ; b t;r L adj;d bt t+h i i;r ; b t i;r ; (7) where L adj;d bt;r t+h ; b t;r L adj bt;r t+h ; b t;r P P T t=r Ladj bt;r t+h ; b t;r and q(p ) is a bandwidth that grows with P (e.g., Newey and West, 987) 9
10 INSERT TABLE 2a HERE Clark and West s (2007) test is only approximately asymptotically normal, and it may be conservative. Thus, we also consider statistics based on the Clark and McCracken s (200) ENCNEW test, which may have better small sample properties. statistic as: L E T (R) P P P T t=r De ne the ENCNEW (yt+h by t;t+h ) 2 (y t+h by t;t+h ) (y t+h by 2t;t+h ) P P T t=r (y t+h by 2t;t+h ) 2 ; (8) where P is the number of out-of-sample predictions available, and by t;t+h ; by 2t;t+h depend on the parameter estimates ^ t;r ; ^ t;r. As in Clark and McCracken (200), we make the following assumptions.. 3 Assumption 3: (a) The parameter estimates ^ t;r satis es ^ t;r = B (t) (t) where B (t) (t) = R P t j=t R+ ;j, and similarly for ^ t;r = B 2 (t) 2 (t) : R P t j=t R+ q ;j (b) Let U t = h u t ; x 0 2;t Ex 0 2;t; 0 2;t; vec 2;t 0 2;t E 2;t 0 2;t 0 ; vec (q2;t Eq 2;t ) 0i 0 : Then EU t = 0; Eq 2;t < is p.d.; for some r > 4, U t is uniformly L r bounded; for all t, Eu 2 t = 2 < ; for some r > d > 2, U t is strong mixing with coe cients of size rd= (r d) ; letting eu t denote the vector of non-redundant elements of U t ; lim T! T E PT t= e U t PT t= e U t 0 = < is p.d. (c) E 2;t 0 2;t = 2 Eq 2;t and E 2;t j 2;t j ; q 2;t j; j = ; 2; ::: = 0: (d) lim T;R! R=T = 2 (0; ) and the forecast horizon is one. Note that Assumption 3 imposes conditional homoskedasticity and one-step ahead forecast horizons; for situations where conditional heteroskedasticity and multi-step predictions are important, see Clark and McCracken (2005b). Proposition 3 presents our robust test for the case of nested models comparisons. The proof of Proposition 3 is provided in Appendix A. Proposition 3 (Out-of-sample Robust test for Nested Models II) Suppose Assumption 3 holds. Let R E T = sup L E T (R) ; R = R; :::R; (9) R 3 See Clark and McCracken (200) for a discussion of these assumptions. 0
11 and A E T = R R + RX L E T (R) ; (20) where L E T (R) is de ned in (8). Under the null hypothesis H 0 : lim T! E[L E T (R)] = 0 for all R; and Z R E T =) sup 2[;] A E T =) Z Z R=R [B k (s) B k (s )] 0 db k (s) ; (2) [B k (s) B k (s )] 0 db k (s) ; (22) where R = [T ]; R = [T ] ; and B k () is a standard k-variate Brownian motion and k is the number of parameters in the larger model in excess of the parameters in the smaller model. The null hypothesis for the R E T test is rejected at the signi cance level when RE T > kr whereas the null hypothesis for the A E T test is rejected when AE T > ka ; where the critical values ; k R and ; k A for various values of and = are reported in Table 2(b). 4 INSERT TABLE 2(b) HERE 2.3 Regression-Based Tests of Predictive Ability Under the widely used Mean Squared Forecast Error loss, optimal forecasts have a variety of properties. They should be unbiased, one step ahead forecast errors should be serially uncorrelated, h-steps ahead forecast errors should be correlated at most of order h (see Granger and Newbold, 996, and Diebold and Lopez, 996). It is therefore interesting to test such properties. We do so in the same framework of West and McCracken (998). We assume one is interested in the relationship between the prediction error and a vector of variables. Let the forecast error be v t+h ( ) v t+h, and its estimated value be v t+h bt;r bv t+h. The properties of interest involve the linear relationship between v t+h and a (p ) vector function of data at time t. 4 Note that the critical values n for a signi cance level are, respectively, k R and k A, where k R and k A solve P sup 2[;] R o [B k (s) B k (s )] 0 db k (s) > k R = and n R n P R o o [B k (s) B k (s )] 0 db k (s) > k A =, respectively. The critical values are obtained via Monte Carlo simulation methods.
12 For the purposes of this section, let us de ne the loss function of interest to be L t+h, whose estimated counterpart is L t+h ( b t;r ) b L t+h. To be more speci c: (i) Forecast Unbiasedness Tests: b Lt+h = bv t+h : (ii) Mincer-Zarnowitz s (969) Tests (or E ciency Tests): b L t+h = bv t+h X t, where X t is a vector of predictors known at time t (see also Chao, Corradi and Swanson, 200). (iii) Forecast Encompassing Tests (Clements and Hendry, 993, Harvey, Leybourne and Newbold, 998): b Lt+h = bv t+h f t ; where f t is the forecast of the encompassed model. (iv) Serial Uncorrelation Tests: b Lt+h = bv t+h bv t : More generally, let the loss function of interest to be the (p ) vector L t+h = v t+h g t, whose estimated counterpart is b L t+h = bv t+h bg t, where g t ( ) g t denotes the function describing the linear relationship between v t+h and a (p ) vector function of data at time t, with g t ( b t ) bg t. 5 The null hypothesis of interest is: E (L t+h ( )) = 0: (23) In order to test (23), one simply tests whether b L t+h has zero mean by a standard Wald test in a regression of b L t+h onto a constant (i.e. testing whether the constant is zero). That is, W T (R) = P TX t=r bl t+h ( b t;r ) 0 b R TX bl t+h ( b t;r ); where b R is a consistent estimate of the long run variance matrix of the adjusted out-ofsample losses discussed in details below. We make the following high level assumption. Assumption 4: (a) The partial sum ^ 2 T P n o =2 [T ] t=[t ] L t+h (^ t;r ) E[L t+h ( )] obeys a functional central limit theorem: ^ 2 R T =2 [T ] X t=[t ] t=r n o L t+h (^ t;r ) E[L t+h ( )] ) B p () B p () (24) where B p () denotes the p-dimensional standard Brownian motion; (b) lim T;R! R=T = 2 (0; ). A. 6 We have the following proposition. The proof of Proposition 4 is provided in Appendix 5 In the examples above, we have: (i) g t = ; (ii) g t = X t ; (iii) g t = f t ; (iv) g t = v t. 6 Note that the statistics proposed in this paper build upon: T P T b t=r L t+h ( b t;r ) 0 b P T b R t=r L t+h ( b t;r ); whereas the traditional Wald tests is: W T (R) = P P T b t=r L t+h ( b t;r ) 0 b P T b R t=r L t+h ( b t;r ): 2
13 Proposition 4 (Robust Regression-Based Tests) Suppose Assumption 4 holds. Let and where R W T A W T = = sup[l T (R) 0 b R L T (R)]; R = R; :::R; (25) R R R + RX [L T (R) 0 b R L T (R)]; (26) R=R L T (R) T =2 TX bl t+h ( b t;r ); and b R is a consistent estimator of. Under the null hypothesis H 0 : lim T! E (L t+h ( )) = 0 for all R; and R W T A W T =) t=r =) sup [B p () B p ()] 0 [B p () B p ()] ; (27) 2[;] Z [B p () B p ()] 0 [B p () B p ()] d; (28) where R = [T ] ; R = [T ]; R = [T ] ; and B p () is a standard p-dimensional Brownian motion. The null hypothesis for the R W T test is rejected at the signi cance level when R W T > kr whereas the null hypothesis for the A W T test is rejected when AW T > ka;w ;p ; where the critical values ; k;p R;W and ; k A;W ;p for various values of and = are reported in Table 3. 7 INSERT TABLE 3 HERE A simple, consistent estimator for b that ignores parameter estimation uncertainty is: b R = q(p ) X i= q(p )+ ( ji=q(p )j)p TX t=r bl (d) t+h (b t;r ) b L (d) t+h i (b t i;r ) 0 ; where L b(d) t+h (b t;r ) L b t+h ( b t;r ) P P T b t=r L t+h ( b t;r ) and q(p ) is a bandwidth that grows with P (e.g., Newey and West, 987). West and McCracken (998) have instead proposed a general variance estimator that takes into account estimation uncertainty. See West and McCracken (998) for conditions under which parameter estimation uncertainty is irrelevant. 7 Again, the critical values for a n signi cance level are, respectively, k R;W o and k A;W, where k R;W and k A;W solve P sup 2[;] [B p () B p ()] 0 [B p () B p ()] > k R;W =, n R o P [B p () B p ()] 0 [B p () B p ()] d > k A;W = : 3
14 Historically, researchers have estimated the alternative regression: bv t+h = bg tb 0 (R)+b t+h, where b (R) = P P T t=r bg tbg t 0 P P T t=r bg tbv t+h and b t+h is the tted error of the regression, and tested whether the coe cients equal zero. It is clear that under the additional assumption that E (g t g 0 t) is full rank, a maintained assumption in that literature, then the two procedures share the same null hypothesis and are therefore equivalent. However, in this case it is convenient to de ne the following re-scaled Wald test: 8 W (r) T (R) = P T b (R)0 b V (R)b (R) ; where b V (R) is a consistent estimate of the asymptotic variance of b (R) ; V : Proposition 5 presents results for this statistic. The proof of Proposition 5 is provided in Appendix A. Proposition 5 (Robust Regression-Based Tests II) Suppose Assumption 4 holds and E (g t g 0 t) is full rank. Let and where R W T A W T = = sup R P T b (R)0 b V (R)b (R) ; R = R; :::R; (29) R R + RX R=R TX b (R) = P bg t bg t 0 t=r P T b (R)0 b V (R)b (R) ; (30)! P! TX bg t bv t+h ; and b V (R) is a consistent estimator of V. Under the null hypothesis H 0 : lim T! E [b (R)] = 0 for all R; and R W T A W T =) t=r =) sup [B p () B p ()] 0 [B p () B p ()] ; (3) 2[;] Z [B p () B p ()] 0 [B p () B p ()] d; (32) where R = [T ] ; R = [T ]; R = [T ] ; and B p () is a standard p-dimensional Brownian motion. The null hypothesis for the R W T test is rejected when R T > kr;w ;p whereas the null hypothesis for the A W T test is rejected when A T > ka;w ;p : Simulated values of ; k;p R;W and ; k;p A;W for various values of ; = and p are reported in Table The traditional Wald test would be: b (R) 0 V b (R)b (R) : 9 The critical values for a signi cance n level are, respectively, k R;W o and k A;W, where k R;W and k A;W solve P sup 2[;] [B p () B p ()] 0 [B p () B p ()] > k R;W =, n R o P [B p () B p ()] 0 [B p () B p ()] d > k A;W = : 4
15 A consistent estimate of the asymptotic variance of b(r), when parameter estimation error is not relevant, is: V b (R) = P Sgg (R) b R Sgg (R), where Sgg = P P T t=r bg tbg t 0 and b R = P q(p ) i= q(p )+ ( ji=q(p )j)p P T t=r bg tbg 0 t ibv t+h bv t+h i. See West and McCracken (998) for alternative and more general variance estimators. Under more general speci cations for the loss function, these properties may not hold. In those situations, Patton and Timmermann (2007) show that a "generalized forecast error" does satisfy the same properties. In such situations, the procedures that we propose can be applied to the generalized forecast error. 2.4 Robust Point Forecasts The techniques discussed in this paper can also be used to construct robust con dence intervals for point forecasts. Let by t;t+h (R) denote the forecast of variable y t+h made at time t with a direct forecast method, using data estimated over a window of size R. It is practically convenient to assume that the data are stationary, the forecast model is correctly speci ed, and the forecast errors are optimal and normally distributed with zero mean and variance 2 conditional on the regressors (cfr. Stock and Watson, 2003b, p. 45). In such cases, a ( )00% forecast interval is given by by t;t+h (R)z b yt+h by t;t+h (R) where z is the critical value for the standard normal distribution at the 00% signi cance level (e.g..96 for a 5% signi cance level), and b yt+h by t;t+h (R) is an estimator of the root mean squared forecast error; for example, one could use the square root of a HAC robust estimator of the variance of the pseudo-out of sample forecasts or the standard error of the regression. However, again such forecasts are obtained conditional upon the choice of the rolling window size. We propose a robust point forecast obtained by averaging the forecasts obtained over all possible window sizes, that is: by t;t+h A = R R + RX by t;t+h (R) Assuming that parameter estimation error is asymptotically negligible, the estimate of the root mean squared forecast error will be consistent no matter which window size it is constructed with. We propose the researcher to use the estimate of the root mean squared forecast error based on the largest window size, b yt+h by t;t+h R. Then, our proposed 00% forecast interval is: by t;t+h A z b yt+h by t;t+h R. R=R 5
16 3 Monte Carlo evidence In this section, we evaluate the small sample properties of the methods that we propose, and compare them with the methods existing in the literature. We consider both nested and non-nested models forecast comparisons. For the nested models comparison, we consider two Data Generating Processes (DGPs). Let the rst (labeled "DGP") be: y t+ = t x t + t z t + " t+ ; t = ; :::; T, where x t ; " t+ and z t are all i.i.d. standard normals and independent of each other. We compare the following two nested models forecasts for y t : Model forecast : b t x t (33) Model 2 forecast : b 0 tw t ; where w t = [x t ; z t ] 0 ; and both models parameters are estimated by OLS in rolling windows of size R: b t (R) = t j=t R x jy j+ t j=t R x2 j and b t (R) = t j=t R w jwj 0 t j=t R w jy j+ for t = R; :::; T: We also consider a second and more realistic DGP (labeled "DGP2") that follows Clark and McCracken (2005a) and Pesaran and Timmermann (2007). Let!!!! y t+ x t+ = 0:3 0:5d t 0 0:5 y t x t + u y;t u x;t ; t = ; :::; T, where y 0 = x 0 = 0, and u y;t and u x;t are both i.i.d. standard normal and independent of each other. d = if t and 0 otherwise. By varying, we evaluate cases ranging from when there is no break ( = 0) to cases when there is a break (0 < < T ). We compare the following two nested models forecasts for y t+ : Model forecast : b 0 + b y t (34) Model 2 forecast : b 0 + b y t + b 2 x t ; and both models s parameters are estimated by OLS in rolling windows of size R: For the non-nested models comparison, we consider a third DGP (labeled "DGP3"): y t+ = t x t + t z t + " t ; t = ; :::; T, where x t and z t are standard normals independent of each other. We compare the following two non-nested models forecasts for y t : Model forecast : b t x t (35) Model 2 forecast : b t z t ; 6
17 and both models s parameters are estimated by OLS in rolling windows of size R. 3. Size properties The size properties of our test procedures in small samples are evaluated in a series of Monte Carlo experiments. The null hypothesis discussed in Propositions (2) and (3) holds in DGP if t = ; t = 0 for t = ; 2; :::T; and the same null hypothesis is satis ed in DGP2 if d t = 0 for t = ; 2; :::T: 20 The null hypothesis discussed in Proposition () holds in DGP3 holds if t = t = for t = ; 2; :::T: We investigate sample sizes where T = 50; 00; 200 and 500: In all experiments, we set = 0:5. The number of Monte Carlo replications is 5,000. We report empirical rejection probabilities of the tests R T and A T at the 0%, 5%, and % nominal levels. Table 4(a) reports results for the nested models comparison for DGP; results for DGP2 are reported in Table 4(b) whereas Table 4(c) reports results for the nonnested models comparison (DGP3). All tables show that each test has good size properties, although they may slightly under/over-reject for small sample sizes. INSERT TABLES 4(a,b,c) 3.2 Power properties The scope of this sub-section is three-fold. First, we evaluate the power properties of our proposed procedure to departures from the null hypothesis in small samples. Second, we show that the existing methods, which rely on an ad-hoc window size choice, may have no power at all to detect predictive ability. Third, we demonstrate that existing methods are subject to data mining if they are applied to many window sizes without correcting the appropriate critical values. For the nested models comparison, in DGP we let t = ; t = (t )+0 (t > ) ; for as in Table 5(a). For DGP2, we let d t = (t ) + 0 (t > ) ; for as in Table 5(b). For the non-nested models comparison, we let the DGP3 be such that t = ; t = 0 (t ) + (t > ) ; for as in Table 5(c). The column labeled "R T test" reports empirical rejection rates for (2) and the column labeled "A T test" reports empirical rejection rates for (3). In addition, in Tables 5(a,b) the columns labeled "Fixed R" report empirical rejection rates for the Clark and West s (2007) test implemented with a speci c value of R, for which the one-sided critical value is.645 for 20 Note that this corresponds to setting = 0. 7
18 a 5% nominal level test. The "Fixed R" results would correspond to the case of a researcher who has chosen one "ad-hoc" window size R, has not experimented with other choices, and thus might have missed predictive ability associated with alternative values of R. Similarly, in Table 5(c), the columns labeled "Fixed R" report rejection rates for the Diebold and Mariano s (995), West s (996) and McCracken s (200) test implemented with a speci c value of R, for which the two-sided critical value is.96 for a 5% nominal level test. 2 Finally, the columns labeled "Data Mining" report empirical rejection rates incurred by a researcher who is searching across all values of R 2 [0:5T; 0:85T ] but using the Clark and West (2007) test s critical value in Tables 5(a,b) and the Diebold and Mariano (995), West (996) and McCracken (200) test s critical value in Table 5(c) (that is, without taking into account the search procedure). The latter columns show how data snooping distorts inference when it is not properly taken into account. The empirical rejection rates are shown for various values of the break point,. The rst row in each table reports results for size. Table 5(a) shows that all tests have approximately the correct size except the "Data mining" procedure, which has size distortions, and leads to too many rejections. This implies that the empirical evidence in favor of the superior predictive ability of a model can be spurious if evaluated with the incorrect critical values. The remaining rows show the power of the tests. They demonstrate that, in general, in the presence of a structural break our proposed tests have better power than the tests based on a speci c rolling window. The latter may have negligible power for some combinations of the location of the break () and the size of the estimation window (R). Similar results hold for Tables 5(b) and 5(c). INSERT TABLES 5(a,b,c) 4 Empirical evidence The poor forecasting ability of economic models of exchange rate determination has been recognized since the works by Meese and Rogo (983a,b), who established that a random walk forecasts exchange rates better than any economic models in the short run. The Meese 2 The three tests di er because a di erent estimate of the variance is suggested. Diebold and Mariano (995) suggest no parameter estimation correction, and West (996) proposed an estimate that includes parameter estimation correction. McCracken (200) extended West s (996) results to rolling windows and more general loss functions. In our simulations, these tests are the same since parameter estimation uncertainty is asymptotically irrelevant. 8
19 and Rogo s (983a,b) nding has been con rmed by several researchers and the random walk is now the yardstick of comparison for the evaluation of exchange rate models. Recently, Engel, Mark and West (2007) and Molodtsova and Papell (2009) documented empirical evidence in favor of the out-of-sample predictability of some economic models, especially those based on the Taylor rule. However, the out-of-sample predictability that they report depends on certain parameters, among which the choice of the in-sample and out-of-sample periods, and the size of the rolling window used for estimation. The choice of such parameters may a ect the outcome of out-of-sample tests of forecasting ability in the presence of structural breaks. Rossi (2006) found empirical evidence of instabilities in models of exchange rate determination; Giacomini and Rossi (2009) evaluated the consequences of instabilities in the forecasting performance of the models over time; and Rogo and Stavrakeva (2008) also question the robustness of these results to the choice of the starting out-of-sample period. In this section, we test the robustness of these results to the choice of the rolling window size. It is important to notice that it is not clear a-priori whether our test would nd more or less empirical evidence in favor of predictive ability. In fact, there are two opposite forces at play. By considering a wide variety of window sizes, our tests might be more likely to nd empirical evidence in favor of predictive ability, as our Monte Carlo results have shown. However, by correcting statistical inference to take into account the search process across multiple window sizes, our tests might at the same time be less likely to nd empirical evidence in favor of predictive ability. Let s t denote the logarithm of the bilateral nominal exchange rate. 22 The rate of growth of the exchange rate depends on its deviation from the current level of a macroeconomic fundamental. Let f t denote the long-run equilibrium level of the nominal exchange rate as determined by the macroeconomic fundamental, and z t = f t where " t+ is an unforecastable error term. model, s t. Then, s t+ s t = + z t + " t+ (36) The rst model we consider is the Uncovered Interest Rate Parity (UIRP). In the UIRP f UIRP t = (i t i t ) + s t ; (37) where (i t i t ) is the short-term interest di erential between the home and the foreign countries. 22 The exchange rate is de ned as the domestic price of foreign currency. 9
20 The second model we consider is a model with Taylor rule fundamentals, as in Molodtsova and Papell (2007) and Engel, Mark and West (2007). Let t denote the in ation rate in the home country, t denote the in ation rate in the foreign country, denote the target level of in ation in each country, y gap t denote the output gap in the home country, 23 y gap t denote the output gap in the foreign country. Since the di erence in the Taylor rule of the home and foreign countries implies i t i t = ( t t ) + (y gap t determines the long run equilibrium level of the nominal exchange rate: y gap t ), we have that the latter T AY LOR ft = ( t t ) + (y gap t y gap t ) + s t : (38) The benchmark model, against which the forecasts of both models (37 and 38) are evaluated, is the random walk, according to which the exchange rate changes are forecasted to be zero. 24 We use monthly data from the International Financial Statistics database (IMF) and from the Federal Reserve Bank of St. Louis from 973:3 to 2008: for Japan, Switzerland, Canada, Great Britain, Sweden, Germany, France, Italy, the Netherlands, and Portugal. 25 The former database provides the seasonally adjusted industrial production index for output, and the 2-month di erence of the CPI for the annual in ation rate, and the interest rates. The latter provides the exchange rate series. The two models rolling forecasts (based on rolling windows calculated over an out-of-sample portion of the data starting in 983:2) are compared to the forecasts of the random walk, as in Meese and Rogo (983a,b). We focus on the methodologies in Proposition 2 for comparability with Molodtsova and Papell (2009), who use the Clark and West s (2007) test. In our exercise, = 0:5, which implies R = T and R= ( )T ; the total sample size T depends on the country, and the values of R and R are shown in Figures and 2, o ering a relatively large range of window sizes, all of which are su ciently large for asymptotic theory to provide a good approximation. Empirical results are shown in Table 6, and Figures and 2. The column labeled "Fixed R Test" in Table 6 reports the empirical results in the literature based on a window size R equal to 20, the same window size used in Molodtsova and Papell (2007). According to the "Fixed R test", the Taylor model is signi cantly outperforming a random walk for Canada 23 The output gap is the percentage di erence between actual and potential output at time t, where the potential output is the linear time trend in output. 24 We chose the random walk without drift to be the benchmark model because it is the toughest benchmark to beat (see Meese and Rogo, 983a,b). 25 Data on interest rates were incomplete for Portugal and the Netherlands, so we do not report UIRP results for these countries. 20
21 at 5% and for Japan at 0%, whereas the UIRP model outperforms the random walk for Canada at 5% and both Japan and the U.K. at the 0% signi cance level. According to our tests, instead, the empirical evidence in favor of predictive ability is much more favorable. Figures and 2 report the estimated Clark and West s (2007) test statistic for the window sizes that we consider. In particular, the predictive ability of the economic models tend to show up at smaller window sizes, as the gures show. INSERT TABLE 6 AND FIGURES AND 2 5 Conclusions This paper proposes new methodologies for evaluating economic models forecasting performance that are robust to the choice of the estimation window size. These methodologies are noteworthy since they allow researchers to reach empirical conclusions that do not depend on a speci c estimation window size. We show that tests traditionally used by forecasters su er from size distortions if researchers report, in reality, the best empirical result over various window sizes, but without taking into account the search procedure when doing inference in practice. Finally, our empirical results demonstrate that the recent empirical evidence in favor of exchange rate predictability is even stronger when allowing a wider search over window sizes. 2
22 References [] Billingsley, P. (968), Convergence of Probability Measures, John Wiley & Sons: New York, NY. [2] Chao, J.C., V. Corradi and N.R. Swanson (200), An Out-of-Sample Test for Granger Causality, Macroeconomic Dynamics. [3] Cheung, Y., M.D. Chinn and A.G. Pascual (2005), Empirical Exchange Rate Models of the Nineties: Are Any Fit to Survive?, Journal of International Money and Finance 24, [4] Chinn, M. (99), Some Linear and Nonlinear Thoughts on Exchange Rates, Journal of International Money and Finance 0, [5] Clark, T. and M. McCracken (2000), Not-for-Publication Appendix to Tests of Equal Forecast Accuracy and Encompassing for Nested Models, mimeo, Kansas City Fed. [6] Clark, T. and M. McCracken (200), Tests of Equal Forecast Accuracy and Encompassing for Nested Models, Journal of Econometrics 05(), [7] Clark, T. and M. McCracken (2005a), The Power of Tests of Predictive Ability in the Presence of Structural Breaks, Journal of Econometrics 24, -3. [8] Clark, T. and M. McCracken (2005b), Evaluating Direct Multistep Forecasts, Econometric Reviews 24(3), [9] Clark, T. and M. McCracken (2009), "Improving Forecast Accuracy by Combining Recursive and Rolling Forecasts," International Economic Review, 50(2), [0] Clark, T. and M. McCracken (200), "Reality Checks and Nested Forecast Model Comparisons," mimeo, St. Louis Fed. [] Clark, T. and K.D. West (2006), Using Out-of-Sample Mean Squared Prediction Errors to Test the Martingale Di erence Hypothesis, Journal of Econometrics 35, [2] Clark, T. and K.D. West (2007), Approximately Normal Tests for Equal Predictive Accuracy in Nested Models, Journal of Econometrics 38, [3] Clements, M.P. and D.F. Hendry (993)," On the Limitations of Comparing Mean Square Forecast Errors," Journal of Forecasting 2,
23 [4] Diebold, F.X. and J. Lopez (996), Forecast Evaluation and Combination, in Handbook of Statistics, G.S. Maddala and C.R. Rao eds., North-Holland, [5] Diebold, F.X. and R.S. Mariano (995), Comparing Predictive Accuracy, Journal of Business and Economic Statistics, 3, [6] Engel, C., N. Mark and K.D. West, Exchange Rate Models Are Not as Bad as You Think, in: NBER Macroeconomics Annual, Daron Acemoglu, Kenneth S. Rogo and Michael Woodford, eds. (Cambridge, MA: MIT Press, 2007). [7] Giacomini, R. and B. Rossi (200), Model Comparisons in Unstable Environments", Journal of Applied Econometrics 25(4), [8] Gourinchas, P.O., and H. Rey (2007), International Financial Adjustment, The Journal of Political Economy 5(4), [9] Granger, C.W.J. and P. Newbold (986), Forecasting Economic Time Series (2nd ed.), New York: Academic Press. [20] Harvey, D.I., S.J. Leybourne and P. Newbold (998), Tests for Forecast Encompassing, Journal of Business and Economic Statistics 6 (2), [2] McCracken, M. (2000), Robust Out-of-Sample Inference, Journal of Econometrics, 99, [22] Meese, R. and K.S. Rogo (983a), Exchange Rate Models of the Seventies. Do They Fit Out of Sample?, The Journal of International Economics 4, [23] Meese, R. and K.S. Rogo (983b), The Out of Sample Failure of Empirical Exchange Rate Models, in Jacob Frankel (ed.), Exchange Rates and International Macroeconomics, Chicago: University of Chicago Press for NBER. [24] Mincer, J. and V. Zarnowitz (969), The Evaluation of Economic Forecasts, in Economic Forecasts and Expectations, ed. J. Mincer, New York: National Bureau of Economic Research, pp. 8. [25] Molodtsova, T. and D.H. Papell (2009), Out-of-Sample Exchange Rate Predictability with Taylor Rule Fundamentals, Journal of International Economics 77(2). 23
24 [26] Newey, W. and K.D. West (987), A Simple, Positive Semi-De nite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix, Econometrica 55, [27] Patton, A.J. and A. Timmermann (2007), Properties of Optimal Forecasts Under Asymmetric Loss and Nonlinearity, Journal of Econometrics 40, [28] Paye, B. and A. Timmermann (2006), "Instability of Return Prediction Models". Journal of Empirical Finance 3(3), [29] Pesaran, M.H., D. Pettenuzzo and A. Timmermann (2006), "Forecasting Time Series Subject to Multiple Structural Breaks", Review of Economic Studies 73, [30] Pesaran, M.H. and A. Timmermann (2005), "Real-Time Econometrics," Econometric Theory 2(), pages [3] Pesaran, M.H. and A. Timmermann (2007), "Selection of estimation window in the presence of breaks", Journal of Econometrics 37(), [32] Qi, M. and Y. Wu (2003), Nonlinear Prediction of Exchange Rates with Monetary Fundamentals, Journal of Empirical Finance 0, [33] Rogo, K.S. and V. Stavrakeva, The Continuing Puzzle of Short Horizon Exchange Rate Forecasting," NBER Working paper No. 407, [34] Rossi, B. (2006), Are Exchange Rates Really Random Walks? Some Evidence Robust to Parameter Instability, Macroeconomic Dynamics 0(), [35] Rossi, B. and T. Sekhposyan (200), Understanding Models Forecasting Performance, mimeo, Duke University. [36] Stock, J.H. and M.W. Watson (2003a), Forecasting Output and In ation: The Role of Asset Prices, Journal of Economic Literature. [37] Stock, J.H. and M.W. Watson (2003b), Introduction to Econometrics, Addison Wesley. [38] Van Dijk, D. and P.H. Franses (2003), Selecting a Nonlinear Time Series Model using Weighted Tests of Equal Forecast Accuracy, Oxford Bulletin of Economics and Statistics 65, [39] West, K.D. (996), Asymptotic Inference about Predictive Ability, Econometrica, 64,
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