This paper describes an estimator of the additive components of a nonparametric additive model with an unknown link function. When the additive components and link function are twice differentiable with sufficiently smooth second derivatives, the estimator is asymptotically normally distributed with a rate of convergence in probability of n−2/5. This is true regardless of the (finite) dimension of the explanatory variable. Thus, the estimator has no curse of dimensionality. Moreover, the asymptotic distribution of the estimator of each additive component is the same as it would be if the link function and the other components were known with certainty. Thus, asymptotically there is no penalty for not knowing the link function or the other components.

We study the convergence in distribution of M-estimators over a convex kernel. Under convexity, the limit distribution of M-estimators can be obtained under minimal assumptions. We consider the case when the limit is arbitrary, not necessarily normal. If some Taylor expansions hold, the limit distribution is stable. As an application, we examine the limit distribution of M-estimators for the multivariate linear regression model. We obtain the distributional convergence of M-estimators for the multivariate linear regression model for a wide range of sequences of regressors and different types of conditions on the sequence of errors.

Important features of certain economic models may be revealed by studying positive eigenfunctions of appropriately chosen linear operators. Examples include long-run risk–return relationships in dynamic asset pricing models and components of marginal utility in external habit formation models. This paper provides identification conditions for positive eigenfunctions in nonparametric models. Identification is achieved if the operator satisfies two mild positivity conditions and a power compactness condition. Both existence and identification are achieved under a further nondegeneracy condition. The general results are applied to obtain new identification conditions for external habit formation models and for positive eigenfunctions of pricing operators in dynamic asset pricing models.

We consider the least-squares estimator in a strictly stationary first-order autoregression without an estimated intercept. We study its continuous time asymptotic distribution based on an asymptotic framework where the sampling interval converges to zero as the sample size increases. We derive a momentgenerating function which permits the calculation of percentage points and moments of this asymptotic distribution and assess the adequacy of the approximation to the finite sample distribution. In general, the approximation is excellent for values of the autoregressive parameter near one. We also consider the behavior of the power function of tests based on the normalized leastsquares estimator. Interesting nonmonotonic properties are uncovered. This analysis extends the study of Perron [15] and helps to provide explanations for the finite sample results established by Nankervis and Savin [13].

This note adds to the recent research project on treatment choice under ambiguity. I generalize the Manski (Journal of Econometrics, in press) analysis of minimax regret treatment choice by considering a more general setting and, more importantly, by solving for the treatment rule given finitely many (as opposed to two) treatments. The most interesting finding is that with three or more undominated treatments, the minimax regret treatment rule may assign the same treatment to all subjects; thus, the most salient feature of the two-treatment case does not generalize.I thank Chuck Manski, the co-editor, and especially an anonymous referee for helpful comments. Financial support through the Robert Eisner Memorial Fellowship, Department of Economics, Northwestern University, in addition to a Dissertation Year Fellowship, The Graduate School, Northwestern University, is gratefully acknowledged.

In this paper, we propose a simple nonparametric test for testing the null hypothesis of constant coefficients against nonparametric smooth coefficients in a semiparametric varying coefficient model with integrated time series. We establish the asymptotic distributions of the proposed test statistic under both null and alternative hypotheses. Moreover, we derive a central limit theorem for a degenerate second order U-statistic, which contains a mixture of stationary and nonstationary variables and is weighted locally on a stationary variable. This result is of independent interest and useful in other applications. Monte Carlo simulations are conducted to examine the finite sample performance of the proposed test.

This paper develops a theory of estimating parameters of a generated regressor model in which some explanatory variables in the equation of interest are the unknown conditional means of certain observable variables given other observable regressors. The paper imposes a weak nonparametric restriction on the form of the conditional means and maintains a single-index assumption on the distribution of the dependent variable in the equation of interest. The estimation method follows a two-step approach: The first step estimates the conditional means in the index nonparametrically, and the second step estimates the parameters by an analytically convenient weighted average derivative method. It is established that the two-step estimator is root-n-consistent and asymptotically normal. The asymptotic variance exceeds that of the one-step hypothetical estimator, which would be obtainable if the first-step regression were known.

Consider the model y = x′β0 + f*(z) + ε, where ε [d over =] N(0, σ02). We calculate the smallest asymptotic variance that n1/2 consistent regular (n1/2CR) estimators of β0 can have when the only information we possess about f* is that it has a certain shape. We focus on three particular cases: (i) when f* is homogeneous of degree r, (ii) when f* is concave, (iii) when f* is decreasing. Our results show that in the class of all n1/2CR estimators of β0, homogeneity of f* may lead to substantial asymptotic efficiency gains in estimating β0. In contrast, at least asymptotically, concavity and monotonicity of f* do not help in estimating β0 more efficiently, at least for n1/2CR estimators of β0.

This paper considers the conventional recursive (otherwise known as plug-in) and direct multistep forecasts in a panel vector autoregressive framework. We derive asymptotic expressions for the mean square prediction error (MSPE) of both forecasts as N (cross sections) and T (time periods) grow large. Both the bias and variance of the least squares fitting are manifest in the MSPE. Using these expressions, we consider the effect of model specification on predictor accuracy. When the fitted lag order (q) is equal to or exceeds the true lag order (p), the direct MSPE is larger than the recursive MSPE. On the other hand, when the fitted lag order is underspecified, the direct MSPE is smaller than the recursive MSPE. The recursive MSPE is increasing in q for all q ≥ p. In contrast, the direct MSPE is not monotonic in q within the permissible parameter space. Extensions to bias-corrected least squares estimators are considered.

This paper considers a nonparametric functional coefficient model with an unknown link function. The model gives flexibility to the standard interaction-variable model by allowing an arbitrary functional form of heterogeneous marginal effects. A local rank estimation procedure is proposed for the functional coefficients along with its asymptotic property.

This paper considers classical test statistics, namely, the likelihood ratio, efficient score, and Wald statistics, for econometric models under simulation estimation. The simulated likelihood ratio, simulated efficient score, and simulated Wald test statistics are shown to be asymptotically equivalent. Because the simulated score vector can be asymptotically biased, limiting distributions of these simulated statistics can be asymptotically noncentral χ2 distributed. This paper studies inference issues with various simulated test statistics. Monte Carlo results are also provided to compare and demonstrate finite sample properties of simulated test statistics.

This paper is concerned with the nonparametric estimation of regression quantiles of a response variable that is randomly censored. Using results on the strong uniform convergence rate of U-processes, we derive a global Bahadur representation for a class of locally weighted polynomial estimators, which is sufficiently accurate for many further theoretical analyses including inference. Implications of our results are demonstrated through the study of the asymptotic properties of the average derivative estimator of the average gradient vector and the estimator of the component functions in censored additive quantile regression models.

In this paper, we study the asymptotics of nonstationary fractional integrated time series, the long memory time series with d ≥ ½, with special attention focused on the cases when d = (2p + 1)/2 for integer n no less than 0. There is considerable empirical evidence showing long memory of this magnitude in many economic time series including the inflation rate and the stock market volatility. A study of the large-sample property is therefore both needed and useful. Also, we found the asymptotics of nonstationary fractional integrated time series useful in the study of the large-sample theory of the Kwiatkowski–Phillips–Schmidt–Shin test (1992, Journal of Econometrics 54, 159–178).

Fan and Yao (1998) proposed an efficient method to estimate the conditional variance of heteroskedastic regression models. Chen, Cheng, and Peng (2009) applied variance reduction techniques to the estimator of Fan and Yao (1998) and proposed a new estimator for conditional variance to account for the skewness of financial data. In this paper, we apply empirical likelihood methods to construct confidence intervals for the conditional variance based on the estimator of Fan and Yao (1998) and the reduced variance modification of Chen et al. (2009). Simulation studies and data analysis demonstrate the advantage of the empirical likelihood method over the normal approximation method.

As an alternative to quasi-maximum likelihood, targeting estimation is a much applied estimation method for univariate and multivariate GARCH models. In terms of variance targeting estimation, recent research has pointed out that at least finite fourth moments of the data generating process is required, if one wants to perform inference in GARCH models by relying on asymptotic normality of the estimator. Such moment conditions may not be satisfied in practice for financial returns, highlighting a potential drawback of variance targeting estimation. In this paper, we consider the large-sample properties of the variance targeting estimator for the multivariate extended constant conditional correlation GARCH model when the distribution of the data generating process has infinite fourth moments. Using nonstandard limit theory, we derive new results for the estimator stating that, under suitable conditions, its limiting distribution is multivariate stable. The rate of consistency of the estimator is slower than $\sqrt T$ and depends on the tail shape of the data generating process. A simulation study illustrates the derived properties of the targeting estimator.

We examine a higher-order spatial autoregressive model with stochastic, but exogenous, spatial weight matrices. Allowing a general spatial linear process form for the disturbances that permits many common types of error specifications as well as potential ‘long memory’, we provide sufficient conditions for consistency and asymptotic normality of instrumental variables, ordinary least squares, and pseudo maximum likelihood estimates. The implications of popular weight matrix normalizations and structures for our theoretical conditions are discussed. A set of Monte Carlo simulations examines the behaviour of the estimates in a variety of situations. Our results are especially pertinent in situations where spatial weights are functions of stochastic economic variables, and this type of setting is also studied in our simulations.

Simple methods are developed for deriving Edgeworth, saddlepoint, and related expansions for the estimators of multivariate and nonlinear models. Illustrations are provided. Simulations are reported indicating the methods work well compared to standard asymptotic and bootstrapped approaches.

We consider the cointegration tests of Johansen (1988, Journal of Economic Dynamics and Control 12, 231–254; 1991, Econometrica 59, 1551–1580) when a vector autoregressive (VAR) process of order k is used to approximate a more general linear process with a possibly infinite VAR representation. Traditional methods to select the lag order, such as Akaike's information criterion (AIC) or the Bayesian information criterion, often lead to too parsimonious a model with the implication that the cointegration tests suffer from substantial size distortions in finite samples. We extend the analysis of Ng and Perron (2001, Econometrica 69, 1519–1554) to derive a modified Akaike's information criterion (MAIC) in this multivariate setting. The idea is to use the information specified by the null hypothesis as it relates to restrictions on the parameters of the model to keep an extra term in the penalty function of the AIC. This MAIC takes a very simple form for which this extra term is simply the likelihood ratio test for testing the null hypothesis of r against more than r cointegrating vectors. We provide theoretical analyses of its validity and of the fact that cointegration tests constructed from a VAR whose lag order is selected using the MAIC have the same limit distribution as when the order is finite and known. We also provide theoretical and simulation analyses to show how the MAIC leads to VAR approximations that yield tests with drastically improved size properties with little loss of power.We are grateful to two referees for especially useful and constructive comments.

Nonstationary integrated time series may be fractionally cointegrated. Here we propose a test for the null hypothesis of no cointegration. It builds on the asymptotically normal Lagrange multiplier (LM) test against fractional alternatives applied to single equation residuals. It is shown that the LM test applied naively to residuals from a static level regression does not retain asymptotic normality. However, when the LM statistic is employed with residuals from a regression of differenced variables, then the test statistic is shown to have a standard normal limiting distribution. Monte Carlo experiments establish its relevance in finite samples.We thank Norbert Christopeit and three anonymous referees for corrections and very valuable comments.

In this paper we analyze bootstrap procedures for systems cointegration tests with a prior adjustment for deterministic terms suggested by Saikkonen et al. (2006, Econometric Theory 22, 15–68) and Saikkonen and Lütkepohl (2000, Journal of Time Series Analysis 21, 435–456). The asymptotic properties of the bootstrap test procedures are derived, and their small-sample properties are studied. The simulation study also considers the standard asymptotic test versions and the Johansen cointegration test for comparison.