Often researchers find parametric models restrictive and sensitive to deviations from the parametric specifications; semi-nonparametric models are more flexible and robust, but lead to other complications such as introducing infinite dimensional parameter spaces that may not be compact. The method of sieves provides one way to tackle such complexities by optimizing an empirical criterion function over a sequence of approximating parameter spaces, called sieves, which are significantly less complex than the original parameter space. With different choices of criteria and sieves, the method of sieves is very flexible in estimating complicated econometric models. For example, it can simultaneously estimate the parametric and nonparametric components in semi-nonparametric models with or without constraints. It can easily incorporate prior information, often derived from economic theory, such as monotonicity, convexity, additivity, multiplicity, exclusion and non-negativity. This chapter describes estimation of semi-nonparametric econometric models via the method of sieves. We present some general results on the large sample properties of the sieve estimates, including consistency of the sieve extremum estimates, convergence rates of the sieve M-estimates, pointwise normality of series estimates of regression functions, root-n asymptotic normality and efficiency of sieve estimates of smooth functionals of infinite dimensional parameters. Examples are used to illustrate the general results.

We argue that the current framework for predictive ability testing (e.g.,West, 1996) is not necessarily useful for real-time forecast selection, i.e., for assessing which of two competing forecasting methods will perform better in the future. We propose an alternative framework for out-of-sample comparison of predictive ability which delivers more practically relevant conclusions. Our approach is based on inference about conditional expectations of forecasts and forecast errors rather than the unconditional expectations that are the focus of the existing literature. We capture important determinants of forecast performance that are neglected in the existing literature by evaluating what we call the forecasting method (the model and the parameter estimation procedure), rather than just the forecasting model. Compared to previous approaches, our tests are valid under more general data assumptions (heterogeneity rather than stationarity) and estimation methods, and they can handle comparison of both nested and non-nested models, which is not currently possible. To illustrate the usefulness of the proposed tests, we compare the forecast performance of three leading parameter-reduction methods for macroeconomic forecasting using a large number of predictors: a sequential model selection approach,

Abstract. The problem of testing for linearity and the number of regimes in the context of self-exciting threshold autoregressive (SETAR) models is reviewed. We describe least-squares methods of estimation and inference. The primary complication is that the testing problem is non-standard, due to the presence of parameters which are only defined under the alternative, so the asymptotic distribution of the test statistics is non-standard. Simulation methods to calculate asymptotic and bootstrap distributions are presented. As the sampling distributions are quite sensitive to conditional heteroskedasticity in the error, careful modeling of the conditional variance is necessary for accurate inference on the conditional mean. We illustrate these methods with two applications Ð annual sunspot means and monthly U.S. industrial production. We find that annual sunspots and monthly industrial production are SETAR(2) processes. Keywords. SETAR models; Thresholds; Non-standard asymptotic theory; Bootstrap

Barron (1993) obtained a deterministic approximation rate (in L2-norm) of r-m for a class of single hid- den layer feedforward artificial neural networks (ANN) with r hidden units and sigmoid activation func- tions when the target function satisfies certain smoothness conditions. Hornik, Stinchcombe, White, and Auer (HSWA, 1994) extended Barron's result to a class of ANNs with possibly non-sigmoid activation approximating the target function and its derivatives simultaneously. Recently Makovoz (1996) obtained an improved degree of approximation rate r-o+/a for Barron's ANNs with sigmoid activation func- tion where d is the dimension of the domain of the target function.

We review key aspects of forecasting using nonlinear models. Because economic models are typically misspecified, the resulting forecasts provide only an approximation to the best possible forecast. Although it is in principle possible to obtain superior approximations to the optimal forecast using nonlinear methods, there are some potentially serious practical challenges. Primary among these are computational difficulties, the dangers of overfit, and potential difficulties of interpretation. In this chapter we discuss these issues in detail. Then we propose and illustrate the use of a new family of methods (QuickNet) that achieves the benefits of using a forecasting model that is nonlinear in the predictors while avoiding or mitigating the other challenges to the use of nonlinear forecasting methods. 1.

We introduce block bootstrap techniques that are (first order) valid in recursive estimation frameworks. There-after, we present two examples where predictive accuracy tests are made operational using our new bootstrap proce-dures. 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. In a Monte Carlo investigation, we compare the finite sample properties of our block bootstrap procedures with the parametric bootstrap due to Kilian (1999); within the context of encompassing and predictive accuracy tests. In the empirical illustration, it is found that unemployment has nonlinear marginal predictive content for inflation.

This paper considers the problem of consistent model specification tests using series estimation methods. The null models we consider in this paper all contain some nonparametric components. A leading case we consider is to test for an additive partially linear model. The null distribution of the test statistic is derived using a central limit theorem for Hilbert valued random arrays. The test statistic is shown to be able to detect local alternatives that approach the null models at the order of O p (n -1/2 ). We suggest to use the wild bootstrap method to approximate the critical values of the test. A small Monte Carlo simulation is reported to examine the finite sample performance of the proposed test. We also show

Abstract This paper builds on the labor econometrics and classical treatment effects literatures to provide a framework supporting causal concepts and methods for estimating effects of natural experiments operating over time in an explicitly dynamic time-series context. We examine conditions for the construction of covariates instrumental in identifying effects of interest that lead to new tests for unconfoundedness, a key condition for the identification of causal effects that we link to the concept of Granger non-causality. Our new tests for unconfoundedness are useful in both cross-section and dynamic time-series settings. Acknowledgments: The author is grateful for the comments and suggestions of the editor, two anony-

This paper unifies three complementary approaches to defining, identifying, and estimating causal effects: the classical structural equations approach of the Cowles Commision; the treatment effects framework of Rubin (1974) and Rosenbaum and Rubin (1983); and the Directed Acyclic Graph (DAG) approach of Pearl. The settable system framework nests these prior approaches, while affording significant improvements to each. For example, the settable system approach permits identification and estimation of causal effects without requiring exogenous instruments, generalizing the classical structural equations approach; it relaxes the stable unit treatment value assumption of the treatment effect approach and provides significant insight into the selection of covariates; and it accommodates mutual causality, generalizing the DAG approach. We provide necessary and sufficient conditions for identification of covariate-conditioned average causal effects, parametric and nonparametric estimation results, and new tests for unconfoundedness.

Forecasters and applied econometricians are often interested in comparing the predictive accuracy of nested competing models. A leading example of a context in which competing models are nested is when predictive ability is equated with “out-of-sample Granger causality”. In particular, it is often of interest to assess whether historical data from one variable are useful when constructing a forecasting model for another variable, and hence our use of terminology such as “out-of-sample Granger causality ” (see e.g. Ashley, Granger and Schmalensee (1980)). In this paper we examine and discuss three key issues one is faced with when constructing predictive accuracy tests, namely: the contribution of parameter estimation error, the choice of linear versus nonlinear models, and the issue of (dynamic) misspecification, with primary focus on the latter of these issues. One of our main conclusions is that there are a number of easy to apply statistics constructed using out of sample conditional moment conditions which are robust to the presence of dynamic misspecification under both hypothesis. We provide some new Monte Carlo findings and empirical evidence based on the use of such tests. In particular, we analyze the finite sample properties of the consistent out of sample test of Corradi and Swanson (2002) using data generating processes calibrated with