Shibata (1981, Biometrika 68, 45–54) considers data-generating mechanisms belonging to a certain class of linear regressions with errors that are independent and identically normally distributed. He compares the variable selection criteria AIC (Akaike information criterion) and BIC (Bayesian information criterion) using the following type of comparison. For each fixed possible data–generating mechanism, these criteria are compared as the data length increases. The results of this comparison have been interpreted as meaning that, in the context of the data-generating mechanisms considered by Shibata, AIC is better than BIC for large data lengths. Shibata's comparison is pointwise in the space of data–generating mechanisms (as the data length increases). Such comparisons are potentially misleading. We consider a simple class of data-generating mechanisms satisfying Shibata's assumptions and carry out a different type of comparison. For each fixed data length (possibly large) we compare the variable selection criteria for every possible data-generating mechanism in this class. According to this comparison, for this class of data-generating mechanisms no matter how large the data length AIC is not better than BIC.

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.

It is well known that a one-step scoring estimator that starts from any N1/2-consistent estimator has the same first-order asymptotic efficiency as the maximum likelihood estimator. This paper extends this result to k-step estimators and test statistics for k ≥ 1, higher order asymptotic efficiency, and general extremum estimators and test statistics.

The paper shows that a k-step estimator has the same higher order asymptotic efficiency, to any given order, as the extremum estimator toward which it is stepping, provided (i) k is sufficiently large, (ii) some smoothness and moment conditions hold, and (iii) a condition on the initial estimator holds.

For example, for the Newton–Raphson k-step estimator based on an initial estimator in a wide class, we obtain asymptotic equivalence to integer order s provided 2k ≥ s + 1. Thus, for k = 1, 2, and 3, one obtains asymptotic equivalence to first, third, and seventh orders, respectively. This means that the maximum differences between the probabilities that the (N1/2-normalized) k-step and extremum estimators lie in any convex set are o(1), o(N−3/2), and o(N−3), respectively.

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].

The method of moments procedure proposed by Carrasco and Florens (2000) permits full exploitation of the information contained in the characteristic function and yields an estimator which is asymptotically as efficient as the maximum likelihood estimator. However, this estimation procedure depends on a regularization or tuning parameter α that needs to be selected. The aim of the present paper is to provide a way to optimally choose α by minimizing the approximate mean square error (AMSE) of the estimator. Following an approach similar to that of Donald and Newey (2001), we derive a higher-order expansion of the estimator from which we characterize the finite sample dependence of the AMSE on α. We propose to select the regularization parameter by minimizing an estimate of the AMSE. We show that this procedure delivers a consistent estimator of α. Moreover, the data-driven selection of the regularization parameter preserves the consistency, asymptotic normality, and efficiency of the CGMM estimator. Simulation experiments based on a CIR model show the relevance of the proposed approach.

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.

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.

This paper considers nonparametric instrumental variable regression when the endogenous variable is contaminated with classical measurement error. Existing methods are inconsistent in the presence of measurement error. We propose a wavelet deconvolution estimator for the structural function that modifies the generalized Fourier coefficients of the orthogonal series estimator to take into account the measurement error. We establish the convergence rates of our estimator for the cases of mildly/severely ill-posed models and ordinary/super smooth measurement errors. We characterize how the presence of measurement error slows down the convergence rates of the estimator. We also study the case where the measurement error density is unknown and needs to be estimated, and show that the estimation error of the measurement error density is negligible under mild conditions as far as the measurement error density is symmetric.

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 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.

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.

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.

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.