Using the theory of multiple decisions, this paper conducts an analysis of structural stability in location and variance for the linear-regression model. It shows that both cusum and cusum-of-squares techniques are in certain senses optimal. However, the same tests are optimal for changes in location and variance. It is also shown that there is an inherent contradiction between the use of recursive residuals and cusum techniques. The concept of a localized Bayes rule is introduced.

On Hadamard product of square roots of correlation matrices—solution.

In a recent Econometric Theory problem, it was demonstrated that in a conditionally heteroskedastic time series regression with martingale difference errors the use of lagged values of regressors as instruments may not increase the efficiency of estimation relative to ordinary least squares. We provide an example of an analogous phenomenon in a model with serially correlated errors, where the optimal instrumental variables estimator is asymptotically as efficient as the instrumental variables estimator constructed as optimal when ignoring the presence of conditional heteroskedasticity.I thank a referee for providing useful comments that improved the presentation and the co-editor Paolo Paruolo for his patience.

For many time series in empirical macro and finance, it is assumed that the logarithm of the series is a unit root process. Since we may want to assume a stable growth rate for the macroeconomics time series, it seems natural to potentially model such a series as a unit root process with drift. This assumption implies that the level of such a time series is the exponential of a unit root process with drift and therefore, it is of substantial interest to investigate analytically the behavior of the exponential of a unit root process with drift. This paper shows that the sum of the exponential of a random walk with drift converges in distribution, after rescaling by the exponential of the maximum value of the random walk process. A similar result was established in earlier work for unit root processes without drift. The results derived here suggest the conjecture that also in the case when the Dickey-Fuller test or the KPSS statistic is applied to the exponential of a unit root process with drift, these tests will asymptotically indicate stationarity.

In this paper, we derive the asymptotic properties of the density-weighted average derivative estimator when a regressor is contaminated with classical measurement error and the density of this error must be estimated. Average derivatives of conditional mean functions are used extensively in economics and statistics, most notably in semiparametric index models. As well as ordinary smooth measurement error, we provide results for supersmooth error distributions. This is a particularly important class of error distribution as it includes the Gaussian density. We show that under either type of measurement error, despite using nonparametric deconvolution techniques and an estimated error characteristic function, we are able to achieve a $\sqrt {n}$ -rate of convergence for the average derivative estimator. Interestingly, if the measurement error density is symmetric, the asymptotic variance of the average derivative estimator is the same irrespective of whether the error density is estimated or not. The promising finite sample performance of the estimator is shown through a Monte Carlo simulation.

In time series regression models with “short-memory” residual processes, the Durbin–Watson statistic (DW) has been used for the problem of testing for independence of the residuals. In this paper we elucidate the asymptotics of DW for “long-memory” residual processes. A standardized Durbin–Watson statistic (SDW) is proposed. Then we derive the asymptotic distributions of SDW under both the null and local alternative hypotheses. Based on this result we evaluate the local power of SDW. Numerical studies for DW and SDW are given.

We develop a nonparametric Bayesian analysis of regression discontinuity (RD) designs, allowing for covariates, in which we model and estimate the unknown functions of the forcing variable by basis expansion methods. In a departure from current methods, we use the entire data on the forcing variable, but we emphasize the data near the threshold by placing some knots at and near the threshold, a technique we refer to as soft-windowing. To handle the nonequally spaced knots that emerge from soft-windowing, we construct a prior on the spline coefficients, from a second-order Ornstein–Uhlenbeck process, which is hyperparameter light, and satisfies the Kullback–Leibler support property. In the fuzzy RD design, we explain the divergence between the treatment implied by the forcing variable, and the actual intake, by a discrete confounder variable, taking three values, complier, never-taker, and always-taker, and a model with four potential outcomes. Choice of the soft-window, and the number of knots, is determined by marginal likelihoods, computed by the method of Chib [Journal of the American Statistical Association, 1995, 90, 1313–1321] as a by-product of the Markov chain Monte Carlo (MCMC)-based estimation. Importantly, in each case, we allow for covariates, incorporated nonparametrically by additive natural cubic splines. The potential outcome error distributions are modeled as student-t, with an extension to Dirichlet process mixtures. We derive the large sample posterior consistency, and posterior contraction rate, of the RD average treatment effect (ATE) (in the sharp case) and RD ATE for compliers (in the fuzzy case), as the number of basis parameters increases with sample size. The excellent performance of the methods is documented in simulation experiments, and in an application to educational attainment of women from Meyersson [Econometrica, 2014, 82, 229–269].

For the problem of testing the hypothesis that all m coefficients of the right-hand-side endogenous variables in an instrumental variables (IV) regression are zero, the likelihood ratio (LR) test can, if the reduced form covariance matrix is known, be rendered similar by a conditioning argument. To exploit this fact requires knowledge of the relevant conditional cumulative distribution function (c.d.f.) of the LR statistic, but the statistic is a function of the smallest characteristic root of an (m + 1)-square matrix and is therefore analytically difficult to deal with when m > 1. We show in this paper that an iterative conditioning argument used by Hillier (2009) and Andrews, Moreira, and Stock (2007 Journal of Econometrics 139, 116–132) to evaluate the c.d.f. in the case m = 1 can be generalized to the case of arbitrary m. This means that we can completely bypass the difficulty of dealing with the smallest characteristic root. Analytic results are obtained for the case m = 2, and a simple and efficient simulation approach to evaluating the c.d.f. is suggested for larger values of m.

A Taylor series approach is used to derive approximations to the approximate slope of LR, W, and LM test statistics. These can be useful when the approximate slopes themselves are complicated. The results are applied to the test of linear coefficient restrictions in the linear model with a nonscalar covariance matrix. They can be used to find cases where the Wald test may badly over-reject in this model. The results shed some new light on the over-rejection that has been noted in the special cases of AR(1) disturbances and the SURE model.

This paper considers the estimation of panel data models with interactive fixed effects where the idiosyncratic errors are subject to conditional quantile restrictions. An easy-to-implement two-step estimator is proposed for the coefficients of the observed regressors. In the first step, the principal component analysis is applied to the cross-sectional averages of the regressors to estimate the latent factors. In the second step, the smoothed quantile regression is used to estimate the coefficients of the observed regressors and the factor loadings jointly. The consistency and asymptotic normality of the estimator are established under large $N,T$ asymptotics. It is found that the asymptotic distribution of the estimator suffers from asymptotic biases, and this paper shows how to correct the biases using both analytical and split-panel jackknife bias corrections. Simulation studies confirm that the proposed estimator performs well with moderate sample sizes.

Combining cross-sectional and time-series data is a long and well-established practice in empirical economics. We develop a central limit theory that explicitly accounts for possible dependence between the two datasets. We focus on common factors as the mechanism behind this dependence. Using our central limit theorem (CLT), we establish the asymptotic properties of parameter estimates of a general class of models based on a combination of cross-sectional and time-series data, recognizing the interdependence between the two data sources in the presence of aggregate shocks. Despite the complicated nature of the analysis required to formulate the joint CLT, it is straightforward to implement the resulting parameter limiting distributions due to a formal similarity of our approximations with Murphy and Topel’s (1985, Journal of Business and Economic Statistics 3, 370–379) formula.

It is shown that in a first-order mixed autoregressive moving average model, a Lagrange multiplier test for the autoregressive unit-root hypothesis can be inconsistent against stationary alternatives.

This paper investigates Granger noncausality and the cointegrating relation between two time series in the Hilbert space framework. This framework allows us to analyze the relationship between cointegration and distance between two information sets. In particular, we prove that if two variables, X and Y, are cointegrated, then the distance between two information sets, concerning the differenced series ΔX and ΔY, must be less than the standard deviation of ΔX.