In our paper published in vol. 22, no. 1, of Econometric Theory (2006, pp. 69–97) the proof of (22) in Lemma 3.1 is incorrect as given. The problem in the proof occurs after the first display on p. 93 where we attempted to deduce (22) from the already established relation (21). The argument given there is incorrect, because an existence-quantor was mistaken for an all-quantor. We show how the problem can be repaired. In the course of this, we also note a useful variant of another result in Lemma 3.1.

This article studies the asymptotic properties of empirical nonparametric regressions that partially misspecify the relationships between nonstationary variables. In particular, we analyze nonparametric kernel regressions in which a potential nonlinear cointegrating regression is misspecified through the use of a proxy regressor in place of the true regressor. Such models occur in linear and nonlinear regressions where the regressor suffers from measurement error or where the true regressor is a latent or filtered variable as in mixed-data-sampling. The treatment allows for endogenous regressors as the latent variable and proxy variables that cointegrate asymptotically with the true latent variable, including correctly specified as well as misspecified systems, and is therefore intermediate between nonlinear nonparametric cointegrating regression and completely spurious nonparametric nonstationary regression. The results relate to recent work on dynamic misspecification in nonparametric nonstationary systems and the limit theory accommodates regressor variables with autoregressive roots that are local to unity and whose errors are driven by long memory and short memory innovations, thereby encompassing applications with a wide range of economic and financial time series. Some implications for forecasting under misspecification are also examined.

This paper introduces a new class of time-varying vector moving average processes of infinite order. These processes serve dual purposes: (1) they can be used to model time-varying dependence structures, and (2) they can be used to establish asymptotic theories for multivariate time series models. To illustrate these two points, we first establish some fundamental asymptotic properties and use them to infer the trending term of a vector moving average infinity process. We then investigate a class of time-varying VARX models. Finally, we demonstrate the empirical relevance of the theoretical results using extensive simulated and real data studies.

This paper is concerned with the general problem of testing one form of covariance structure against another in a normal linear regression. It is shown that all the point-optimal tests recently proposed by King and his associates can be interpreted as special cases of a Cox test for non-nested hypotheses. This provides a synthesis of a whole range of point-optimal tests as well as demonstrating that King and his associates have exposed a class of Cox tests which have an exact distribution.

The purpose of this paper is to provide a systematic treatment of the problem of estimation in systems of linear stochastic equations where some of the disturbances are uncorrelated and where the identifying equations are equivalent to a linear system.

This paper provides nonparametric specification tests for the commonly used homogeneous and stable coefficients structures in panel data models. We first obtain the augmented residuals by estimating the model under the null hypothesis and then run auxiliary time series regressions of augmented residuals on covariates with time-varying coefficients (TVCs) via sieve methods. The test statistic is then constructed by averaging the squared fitted values, which are close to zero under the null and deviate from zero under the alternatives. We show that the test statistic, after being appropriately standardized, is asymptotically normal under the null and under a sequence of Pitman local alternatives. A bootstrap procedure is proposed to improve the finite sample performance of our test. In addition, we extend the procedure to test other structures, such as the homogeneity of TVCs or the stability of heterogeneous coefficients. The joint test is extended to panel models with two-way fixed effects. Monte Carlo simulations indicate that our tests perform reasonably well in finite samples. We apply the tests to re-examine the environmental Kuznets curve in the United States, and find that the model with homogenous TVCs is more appropriate for this application.

In this paper, we consider subgeometric (specifically, polynomial) ergodicity of univariate nonlinear autoregressions with autoregressive conditional heteroskedasticity (ARCH). The notion of subgeometric ergodicity was introduced in the Markov chain literature in the 1980s, and it means that the transition probability measures converge to the stationary measure at a rate slower than geometric; this rate is also closely related to the convergence rate of $\beta $-mixing coefficients. While the existing literature on subgeometrically ergodic autoregressions assumes a homoskedastic error term, this paper provides an extension to the case of conditionally heteroskedastic ARCH-type errors, considerably widening the scope of potential applications. Specifically, we consider suitably defined higher-order nonlinear autoregressions with possibly nonlinear ARCH errors and show that they are, under appropriate conditions, subgeometrically ergodic at a polynomial rate. An empirical example using energy sector volatility index data illustrates the use of subgeometrically ergodic AR–ARCH models.

The convergence rates of the sums of α-mixing (or strongly mixing) triangular arrays of heterogeneous random variables are derived. We pay particular attention to the case where central limit theorems may fail to hold, due to relatively strong time-series dependence and/or the nonexistence of higher-order moments. Several previous studies have presented various versions of laws of large numbers for sequences/triangular arrays, but their convergence rates were not fully investigated. This study is the first to investigate the convergence rates of the sums of α-mixing triangular arrays whose mixing coefficients are permitted to decay arbitrarily slowly. We consider two kinds of asymptotic assumptions: one is that the time distance between adjacent observations is fixed for any sample size n; and the other, called the infill assumption, is that it shrinks to zero as n tends to infinity. Our convergence theorems indicate that an explicit trade-off exists between the rate of convergence and the degree of dependence. While the results under the infill assumption can be seen as a direct extension of those under the fixed-distance assumption, they are new and particularly useful for deriving sharper convergence rates of discretization biases in estimating continuous-time processes from discretely sampled observations. We also discuss some examples to which our results and techniques are useful and applicable: a moving-average process with long lasting past shocks, a continuous-time diffusion process with weak mean reversion, and a near-unit-root process.

This paper considers testing for unit roots in Gaussian panels with cross-sectional dependence generated by common factors. Within our setup, we can analyze restricted versions of the two prevalent approaches in the literature, that of Moon and Perron (2004, Journal of Econometrics 122, 81–126), who specify a factor model for the innovations, and the PANIC setup proposed in Bai and Ng (2004, Econometrica 72, 1127–1177), who test common factors and idiosyncratic deviations separately for unit roots. We show that both frameworks lead to locally asymptotically normal experiments with the same central sequence and Fisher information. Using Le Cam’s theory of statistical experiments, we obtain the local asymptotic power envelope for unit-root tests. We show that the popular Moon and Perron (2004, Journal of Econometrics 122, 81–126) and Bai and Ng (2010, Econometric Theory 26, 1088–1114) tests only attain the power envelope in case there is no heterogeneity in the long-run variance of the idiosyncratic components. We develop a new test which is asymptotically uniformly most powerful irrespective of possible heterogeneity in the long-run variance of the idiosyncratic components. Monte Carlo simulations corroborate our asymptotic results and document significant gains in finite-sample power if the variances of the idiosyncratic shocks differ substantially among the cross-sectional units.

We study semiparametric inference in a small-dimensional vector autoregressive (VAR) model of order p augmented by unobservable common factors with a dynamic described by a VAR process of order q. This state-space specification is useful to measure separately the direct causality effects and the responses to dynamic common factors. We show that the state-space parameters are identifiable from the autocovariance function of the observed process. We estimate the model by means of a multistep procedure in closed-form, which combines an eigenvalue–eigenvector matrix decomposition and a linear instrumental variable estimation allowing for Hansen–Sargan specification tests. We study the asymptotic and finite-sample properties of the parameter estimators and of rank tests for selecting the number of unobservable factors and VAR orders. In an empirical illustration, we investigate the dynamic common factors and the spillover effects that explain the co-movements among the log daily realized volatilities of four European stock market indices.

We study identification in nonparametric regression models with a misclassified and endogenous binary regressor when an instrument is correlated with misclassification error. We show that the regression function is nonparametrically identified if one binary instrument variable and one binary covariate satisfy the following conditions. The instrumental variable corrects endogeneity; the instrumental variable must be correlated with the unobserved true underlying binary variable, must be uncorrelated with the error term in the outcome equation, but is allowed to be correlated with the misclassification error. The covariate corrects misclassification; this variable can be one of the regressors in the outcome equation, must be correlated with the unobserved true underlying binary variable, and must be uncorrelated with the misclassification error. We also propose a mixture-based framework for modeling unobserved heterogeneous treatment effects with a misclassified and endogenous binary regressor and show that treatment effects can be identified if the true treatment effect is related to an observed regressor and another observable variable.