We propose a new test against a change in correlation at an unknown point in time based on cumulated sums of empirical correlations. The test does not require that inputs are independent and identically distributed under the null. We derive its limiting null distribution using a new functional delta method argument, provide a formula for its local power for particular types of structural changes, give some Monte Carlo evidence on its finite-sample behavior, and apply it to recent stock returns.

The primary concern is to establish a fairly general framework in which estimators resulting from estimating equations gnθ = 0 are not consistent. This leads on to consistency by an intuitive route. Asymptotic distributions of consistent estimators are also touched upon, and the results are applied to various examples.

This paper proposes tests on semiparametric models based on the sum of squared residuals from a least-squares procedure. Smoothness conditions are imposed on the nonparametric portion of the model to obtain asymptotic normality of the sum of squared residuals. The approach yields tests of specification, significance, smoothness and concavity and allows for heteroskedastic residuals.

We consider the problem of estimating the unconditional distribution of a post-model-selection estimator. The notion of a post-model-selection estimator here refers to the combined procedure resulting from first selecting a model (e.g., by a model-selection criterion such as the Akaike information criterion [AIC] or by a hypothesis testing procedure) and then estimating the parameters in the selected model (e.g., by least squares or maximum likelihood), all based on the same data set. We show that it is impossible to estimate the unconditional distribution with reasonable accuracy even asymptotically. In particular, we show that no estimator for this distribution can be uniformly consistent (not even locally). This follows as a corollary to (local) minimax lower bounds on the performance of estimators for the distribution; performance is here measured by the probability that the estimation error exceeds a given threshold. These lower bounds are shown to approach ½ or even 1 in large samples, depending on the situation considered. Similar impossibility results are also obtained for the distribution of linear functions (e.g., predictors) of the post-model-selection estimator.The research of the first author was supported by the Max Kade Foundation and by the Austrian National Science Foundation (FWF), Grant no. P13868-MAT. A preliminary draft of the material in this paper was written in 1999.

In this paper, we develop tests for structural breaks of factor loadings in dynamic factor models. We focus on the joint null hypothesis that all factor loadings are constant over time. Because the number of factor loading parameters goes to infinity as the sample size grows, conventional tests cannot be used. Based on the fact that the presence of a structural change in factor loadings yields a structural change in second moments of factors obtained from the full sample principal component estimation, we reduce the infinite-dimensional problem into a finite-dimensional one and our statistic compares the pre- and postbreak subsample second moments of estimated factors. Our test is consistent under the alternative hypothesis in which a fraction of or all factor loadings have structural changes. The Monte Carlo results show that our test has good finite-sample size and power.

This paper proposes a nonparametric, kernel-based test of parametric quantile regression models. The test statistic has a limiting standard normal distribution if the parametric quantile model is correctly specified and diverges to infinity for any misspecification of the parametric model. Thus the test is consistent against any fixed alternative. The test also has asymptotic power 1 against local alternatives converging to the null at proper rates. A simulation study is provided to evaluate the finite-sample performance of the test.

This paper considers tests for structural change of the cointegrating vector and the adjustment vector in the error correction model with an unknown change point. This paper derives new tests for structural change, which are applicable to maximum likelihood estimation. Our tests for structural change of the cointegrating vector have the same nonstandard asymptotic distributions that have been found by Hansen (1992a, Journal of Business and Economic Statistics 10, 321–335). In contrast, the tests on the adjustment vector have the same asymptotic distributions that have been found by Andrews and Ploberger (1994, Econometrica 62, 1383–1414) for models with stationary variables. Asymptotic critical values are provided.

Recent advances in the application of game theory to the study of auctions have spawned a growing empirical literature involving both experimental and field data. In this paper, we focus on four different mechanisms (the Dutch, English, first-price sealed-bid, and Vickrey auctions) within one of the most commonly used theoretical models (the independent private values paradigm) to investigate issues of identification, estimation, and testing in parametric structural econometric models of auctions.

In Pötscher (1991, Econometric Theory 7, 163–185) the asymptotic distribution of a post-model-selection estimator, both unconditional and conditional on selecting a correct model (minimal or not), has been derived. Limitations of these results are (i) that they do not provide information on the distribution of the post-model-selection estimator conditional on selecting an incorrect model and (ii) that the quality of this asymptotic approximation to the finite-sample distribution is not uniform with respect to the underlying parameters. In the present paper we first obtain the unconditional and also the conditional finite-sample distribution of the post-model-selection estimator, which turn out to be complicated and difficult to interpret. Second, we obtain approximations to the finite-sample distributions that are as simple and easy to interpret as the asymptotic distributions obtained in Pötscher (1991) but at the same time are close to the finite-sample distributions uniformly with respect to the underlying parameters. As a by-product, we also obtain the asymptotic distribution conditional on selecting an incorrect model.We thank the co-editor Richard Smith and the two referees for helpful comments on a previous version of this paper. Hannes Leeb's research was supported by the Austrian Science Foundation (FWF), project P13868-MAT.

In this paper we bring together those properties of the Kronecker product, the vec operator, and 0-1 matrices which in our view are of interest to researchers and students in econometrics and statistics. The treatment of Kronecker products and the vec operator is fairly exhaustive; the treatment of 0–1 matrices is selective. In particular we study the “commutation” matrix K (defined implicitly by K vec A = vec A′ for any matrix A of the appropriate order), the idempotent matrix N = ½ (I + K), which plays a central role in normal distribution theory, and the “duplication” matrix D, which arises in the context of symmetry. We present an easy and elegant way (via differentials) to evaluate Jacobian matrices (first derivatives), Hessian matrices (second derivatives), and Jacobian determinants, even if symmetric matrix arguments are involved. Finally we deal with the computation of information matrices in situations where positive definite matrices are arguments of the likelihood function.

Data envelopment analysis (DEA) and free disposal hull (FDH) estimators are widely used to estimate efficiencies of production units. In applications, both efficiency scores for individual units as well as average efficiency scores are typically reported. While several bootstrap methods have been developed for making inference about the efficiencies of individual units, until now no methods have existed for making inference about mean efficiency levels. This paper shows that standard central limit theorems do not apply in the case of means of DEA or FDH efficiency scores due to the bias of the individual scores, which is of larger order than either the variance or covariances among individual scores. The main difficulty comes from the fact that such statistics depend on efficiency estimators evaluated at random points. Here, new central limit theorems are developed for means of DEA and FDH scores, and their efficacy for inference about mean efficiency levels is examined via Monte Carlo experiments.

The Kalman filter is used to derive updating equations for the Bayesian data density in discrete time linear regression models with stochastic regressors. The implied “Bayes model” has time varying parameters and conditionally heterogeneous error variances. A σ-finite Bayes model measure is given and used to produce a new-model-selection criterion (PIC) and objective posterior odds tests for sharp null hypotheses like the presence of a unit root. This extends earlier work by Phillips and Ploberger [18]. Autoregressive-moving average (ARMA) models are considered, and a general test of trend-stationarity versus difference stationarity is developed in ARMA models that allow for automatic order selection of the stochastic regressors and the degree of the deterministic trend. The tests are completely consistent in that both type I and type II errors tend to zero as the sample size tends to infinity. Simulation results and an empirical application are reported. The simulations show that the PIC works very well and is generally superior to the Schwarz BIC criterion, even in stationary systems. Empirical application of our methods to the Nelson-Plosser [11] series show that three series (unemployment, industrial production, and the money stock) are level- or trend-stationary. The other eleven series are found to be stochastically nonstationary.

We establish valid Edgeworth expansions for the distribution of smoothed nonparametric spectral estimates, and of studentized versions of linear statistics such as the sample mean, where the studentization employs such a nonparametric spectral estimate. Particular attention is paid to the spectral estimate at zero frequency and, correspondingly, the studentized sample mean, to reflect econometric interest in autocorrelation-consistent or long-run variance estimation. Our main focus is on stationary Gaussian series, though we discuss relaxation of the Gaussianity assumption. Only smoothness conditions on the spectral density that are local to the frequency of interest are imposed. We deduce empirical expansions from our Edgeworth expansions designed to improve on the normal approximation in practice and also deduce a feasible rule of bandwidth choice.

This book seeks to integrate research on cause and effect inference from cognitive science, econometrics, epidemiology, philosophy, and statistics. It puts forward the work of its author, his collaborators, and others over the past two decades as a new account of cause and effect inference that can aid practical researchers in many fields, including econometrics. Pearl adheres to several propositions on cause and effect inference. Though cause and effect relations are fundamentally deterministic (he explicitly excludes quantum mechanical phenomena from his concept of cause and effect), cause and effect analysis involves probability language. Probability language helps to convey uncertainty about cause and effect relations but is insufficient to fully express those relations. In addition to conditional probabilities of events, cause and effect analysis requires graphs or diagrams and a language that distinguishes intervention or manipulation from observation. Cause and effect analysis also requires counterfactual reasoning and causal assumptions in addition to observations and statistical assumptions.

A test for neglected nonlinearities in regression models is proposed. The test is of the Davidson-MacKinnon type against an increasingly rich set of non-nested alternatives, and is based on sieve estimation of the alternative model. For the case of a linear parametric model, the test statistic is shown to be asymptotically standard normal under the null, while rejecting with probability going to one if the linear model is misspecified. A small simulation study suggests that the test has adequate finite sample properties, but one must guard against over fitting the nonparametric alternative.

In this paper we develop a simple procedure that delivers tests for the presence of a broken trend in a univariate time series that do not require knowledge of the form of serial correlation in the data and are robust as to whether the shocks are generated by an I(0) or an I(1) process. Two trend break models are considered: the first holds the level fixed while allowing the trend to break, while the latter allows for a simultaneous break in level and trend. For the known break date case, our proposed tests are formed as a weighted average of the optimal tests appropriate for I(0) and I(1) shocks. The weighted statistics are shown to have standard normal limiting null distributions and to attain the Gaussian asymptotic local power envelope, in each case regardless of whether the shocks are I(0) or I(1). In the unknown break date case, we adopt the method of Andrews (1993) and take a weighted average of the statistics formed as the supremum over all possible break dates, subject to a trimming parameter, in both the I(0) and I(1) environments. Monte Carlo evidence suggests that our tests are in most cases more powerful, often substantially so, than the robust broken trend tests of Sayginsoy and Vogelsang (2004). An empirical application highlights the practical usefulness of our proposed tests.

In this paper, a necessary and sufficient condition for the existence of the unconditional fourth moment of the GARCH(p,q) process is given and also an expression for the moment itself. Furthermore, the autocorrelation function of the centered and squared observations of this process is derived. The statistical theory is further illustrated by a few special cases such as the GARCH(2,2) process and the ARCH(q) process.

Diffuse priors lead to pathological posterior behavior when used in Bayesian analyses of simultaneous equation models (SEM's). This results from the local nonidentification of certain parameters in SEM's. When this a priori known feature is not captured appropriately, it results in an a posteriori favoring of certain specific parameter values that is not the consequence of strong data information but of local nonidentification. We show that a proper consistent Bayesian analysis of a SEM explicitly has to consider the reduced form of the SEM as a standard linear model on which nonlinear (reduced rank) restrictions are imposed, which result from a singular value decomposition. The priors/posteriors of the parameters of the SEM are therefore proportional to the priors/posteriors of the parameters of the linear model under the condition that the restrictions hold. This leads to a framework for constructing priors and posteriors for the parameters of SEM's. The framework is used to construct priors and posteriors for one, two, and three structural equation SEM's. These examples together with a theorem, showing that the reduced forms of SEM's accord with sets of reduced rank restrictions on standard linear models, show how Bayesian analyses of generally specified SEM's can be conducted.

Models with single-index structures are among the many existing popular semiparametric approaches for either the conditional mean or the conditional variance. This paper focuses on a single-index model for the conditional quantile. We propose an adaptive estimation procedure and an iterative algorithm which, under mild regularity conditions, is proved to converge with probability 1. The resulted estimator of the single-index parametric vector is root-n consistent, asymptotically normal, and based on simulation study, is more efficient than the average derivative method in Chaudhuri, Doksum, and Samarov (1997, Annals of Statistics 19, 760–777). The estimator of the link function converges at the usual rate for nonparametric estimation of a univariate function. As an empirical study, we apply the single-index quantile regression model to Boston housing data. By considering different levels of quantile, we explore how the covariates, of either social or environmental nature, could have different effects on individuals targeting the low, the median, and the high end of the housing market.

This paper summarizes recent Bayesian research on unit roots for the applied macroeconomist in the way Campbell and Perron [8] summarized the classical unit roots perspective. The appropriate choice of a prior is discussed. In recognizing a consensus distaste for explosive roots, I find the popular Normal-Wishart priors centered at the unit root to be reasonable provided they are modified by concentrating the prior mass for the time trend coefficient toward zero as the largest root approaches unit from below. I discuss that the tails of the predictive density can be sensitive to the prior treatment of explosive roots. Because the focus of an investigation often is on a particular persistence property or medium-term forecasting property of the data, I conclude that Bayesian methods often deliver natural answers to macroeconomic questions.