Valentina Corradi

A consistent test for nonlinear out of sample predictive accuracy

In this paper, we draw on both the consistent specification testing and the predictive ability testing literatures and propose a test for predictive accuracy which is consistent against generic nonlinear alternatives. Broadly speaking, given a particular reference model, assume that the objective is to test whether there exists any alternative model, among an infinite number of alternatives, that has better predictive accuracy than the reference model, for a given loss function. A typical example is the case in which the reference model is a simple autoregressive model and the objective is to check whether a more accurate forecasting model can be constructed by including possibly unknown (non)linear functions of the past of the process or of the past of some other process(es). We propose a statistic which is similar in spirit to that of White (2000), although our approach differs from his as we allow for an infinite number of competing models that may be nested. In addition, we allow for non vanishing parameter estimation error. In order to construct valid asymptotic critical values, we implement a conditional p-value procedure which extends the work of Inoue (1999) by allowing for non vanishing parameter estimation error.

Strong rules for detecting the number of breaks in a time series

This paper proposes a new approach for detecting the number of structural breaks in a time series when estimation of the breaks is performed one at the time. We consider the case of shifts in the mean of a possibly nonlinear process, allowing for dependent and heterogeneous observations. This is accomplished through a simple, sequential, almost sure rule ensuring that, in large samples, both the probabilities of overestimating and underestimating the number of breaks are zero. A new estimator for the long run variance which is consistent also in the presence of neglected breaks is proposed. The finite sample behavior is investigated via a simulation exercise. The sequential procedure, applied to the weekly Eurodollar interest rate, detects multiple breaks over the period 1973-1995.

NONPARAMETRIC BOOTSTRAP PROCEDURES FOR PREDICTIVE INFERENCE BASED ON RECURSIVE ESTIMATION SCHEMES

Our objectives in this paper are twofold. First, we introduce block bootstrap techniques that are (first order) valid in recursive estimation frameworks. Thereafter, we present two examples where predictive accuracy tests are made operational using our new bootstrap procedures. In one application, we outline a consistent test for out-of-sample nonlinear Granger causality, and in the other we outline a test for selecting amongst multiple alternative forecasting models, all of which are possibly misspecified. More specifically, our examples extend the White (2000) reality check to the case of non vanishing parameter estimation error, and extend the integrated conditional moment tests of Bierens (1982, 1990) and Bierens and Ploberger (1997) to the case of out-of-sample prediction. In both examples, appropriate re-centering of the bootstrap score is required in order to ensure that the tests have asymptotically correct size, and the need for such re-centering is shown to arise quite naturally when testing hypotheses of predictive accuracy. In a Monte Carlo investigation, we compare the finite sample properties of our block bootstrap procedures with the parametric bootstrap due to Kilian (1999); all within the context of various encompassing and predictive accuracy tests. An empirical illustration is also discussed, in which it is found that unemployment appears to have nonlinear marginal predictive content for inflation.

Bootstrap Conditional Distribution Tests in the Presence of Dynamic Misspecification

In this paper, we show the first order validity of the block bootstrap in the context of Kolmogorov type conditional distribution tests when there is dynamic misspecification and parameter estimation error. Our approach di®ers from the literature to date because we construct a bootstrap statistic that allows for dynamic misspecification under both hypotheses. We consider two test statistics; one is the CK test of Andrews (1997), and the other is in the spirit of Diebold, Gunther and Tay (1998). The limiting distribution of both tests is a Gaussian process with a covariance kernel that reflects dynamic misspecification and parameter estimation error. In order to provide valid asymptotic critical values we suggest an extention of the empirical process version of the block bootstrap to the case of non vanishing parameter estimation error. The findings from Monte Carlo experiments show that both statistics have good finite sample properties for samples as small as 500 observations.

Chapter 5 Predictive Density Evaluation

Abstract This chapter discusses estimation, specification testing, and model selection of predictive density models. In particular, predictive density estimation is briefly discussed, and a variety of different specification and model evaluation tests due to various authors including Christoffersen and Diebold [Christoffersen, P., Diebold, F.X. (2000). “How relevant is volatility forecasting for financial risk management?”. Review of Economics and Statistics 82, 12–22], Diebold, Gunther and Tay [Diebold, F.X., Gunther, T., Tay, A.S. (1998). “Evaluating density forecasts with applications to finance and management”. International Economic Review 39, 863–883], Diebold, Hahn and Tay [Diebold, F.X., Hahn, J., Tay, A.S. (1999). “Multivariate density forecast evaluation and calibration in financial risk management: High frequency returns on foreign exchange”. Review of Economics and Statistics 81, 661–673], White [White, H. (2000). “A reality check for data snooping”. Econometrica 68, 1097–1126], Bai [Bai, J. (2003). “Testing parametric conditional distributions of dynamic models”. Review of Economics and Statistics 85, 531–549], Corradi and Swanson [Corradi, V., Swanson, N.R. (2005a). “A test for comparing multiple misspecified conditional distributions”. Econometric Theory 21, 991–1016; Corradi, V., Swanson, N.R. (2005b). “Nonparametric bootstrap procedures for predictive inference based on recursive estimation schemes”. Working Paper, Rutgers University; Corradi, V., Swanson, N.R. (2006a). “Bootstrap conditional distribution tests in the presence of dynamic misspecification”. Journal of Econometrics, in press; Corradi, V., Swanson, N.R. (2006b). “Predictive density and conditional confidence interval accuracy tests”. Journal of Econometrics, in press], Hong and Li [Hong, Y.M., Li, H.F. (2003). “Nonparametric specification testing for continuous time models with applications to term structure of interest rates”. Review of Financial Studies, 18, 37–84], and others are reviewed. Extensions of some existing techniques to the case of out-of-sample evaluation are also provided, and asymptotic results associated with these extensions are outlined.

Reconsidering the continuous time limit of the GARCH(1, 1) process

In this note we reconsider the continuous time limit of the GARCH(1, 1) process. Let > k and p2 denote, respectively, the cumulative returns and the volatility processes. We consider the continuous time approximation of the couple (> k , p2 ). We show that, by choosing di!erent parameterizations, as a function of the discrete interval h, we can obtain either a degenerate or a non-degenerate di!usion limit. We then show that GARCH(1, 1) processes can be obtained as Euler approximations of degenerate di!usions, while any Euler approximation of a non-degenerate di!usion is a stochastic volatility process. ( 2000 Elsevier Science S.A. All rights reserved.

Predictive ability with cointegrated variables

In this paper we outline conditions under which the Diebold and Mariano (DM) (J. Bus. Econom. Statist. 13 (1995) 253) test for predictive ability can be extended to the case of two forecasting models, each of which may include cointegrating relations, when allowing for parameter estimation error. We show that in the cases where either the loss function is quadratic or the length of the prediction period, P, grows at a slower rate than the length of the regression period, R, the standard DM test can be used. On the other hand, in the case of a generic loss function, if P/R→π as T→∞, 0<π<∞, then the asymptotic normality result of West (Econometrica 64 (1996) 1067) no longer holds. We also extend the “data snooping” technique of White (Econometrica 68 (2000) 1097) for comparing the predictive ability of multiple forecasting models to the case of cointegrated variables. In a series of Monte Carlo experiments, we examine the impact of both short run and cointegrating vector parameter estimation error on DM, data snooping, and related tests. Our results suggest that size is reasonable for R and P greater than 50, and power improves with P, as expected. Furthermore, the additional cost, in terms of size distortion, due to the estimation of the cointegrating relations is not substantial. We illustrate the use of the tests in a nonnested cointegration framework by forming prediction models for industrial production which include two interest rate variables, prices, and either M1, M2, or M3.

Information in the Revision Process of Real-Time Datasets ∗

Rationality of early release data is typically tested using linear regressions. Thus, failure to reject the null does not rule out the possibility of nonlinear dependence. This paper proposes two tests which instead have power against generic nonlinear alternatives. A Monte Carlo study shows that the suggested tests have good finite sample properties. Additionally, we carry out an empirical illustration using a real-time dataset for money, output, and prices. Overall, we find strong evidence against data rationality. Interestingly, for money stock the null is not rejected by linear tests but is rejected by our tests.

Testing for Stationarity-Ergodicity and for Comovements Between Nonlinear Discrete Time Markov Processes

In this paper we introduce a class of nonlinear data generating processes (DGPs) that ara first order Markov and can be represented as the sum of a linear plus a bounded nonlinear component.

Continuous Approximations of Stochastic Evolutionary Game Dynamics

Continuous approximations that are ordinary differential equations (ODEs) or stochastic differential equations (SDEs) are often used to study the properties of discrete stochastic processes. We show that different ways of taking the continuous limit of the same model may result in either an ODE or a SDE and study the manner in which each approximates the discrete model. We compare the asymptotic properties of the continuous equations with those of the discrete dynamics and show that they tend to provide a better approximation when a greater amount of variance of the discrete model is preserved in the continuous limit. Journal of Economic Literature Classification numbers: C6, C7, D8