This paper examines the problem of identification and inference on a conditional moment condition model with missing data, with special focus on the case when the conditioning covariates are missing. We impose no assumption on the distribution of the missing data and we confront the missing data problem by using a worst case scenario approach.

We characterize the sharp identified set and argue that this set is usually too complex to compute or to use for inference. Given this difficulty, we consider the construction of outer identified sets (i.e. supersets of the identified set) that are easier to compute and can still characterize the parameter of interest. Two different outer identification strategies are proposed. Both of these strategies are shown to have nontrivial identifying power and are relatively easy to use and combine for inferential purposes.

We consider the method of moments estimation of a structural equation in a panel dynamic simultaneous equations model under different sample size combinations of cross-sectional dimension, N, and time series dimension, T. Two types of linear transformation to remove the individual-specific effects from the model, first difference and forward orthogonal demeaning, are considered. We show that the Alvarez and Arellano (2003) type GMM estimator under both transformations is consistent only if ${T \over N} \to 0$ as $\left( {N,T} \right) \to \infty $. However, it is asymptotically biased if ${{{T^3}} \over N} \to \kappa \ne 0 < \infty$ as $\left( {N,T} \right) \to \infty $. Since the validity of statistical inference depends critically on whether an estimator is asymptotically unbiased, we suggest a jackknife bias reduction method and derive its limiting distribution. Monte Carlo studies are conducted to demonstrate the importance of using an asymptotically unbiased estimator to obtain valid statistical inference.

In this study, we explore the partial identification of nonseparable models with continuous endogenous and binary instrumental variables. We show that the structural function is partially identified when it is monotone or concave in the explanatory variable. D’Haultfœuille and Février (2015, Econometrica 83(3), 1199–1210) and Torgovitsky (2015, Econometrica 83(3), 1185–1197) prove the point identification of the structural function under a key assumption that the conditional distribution functions of the endogenous variable for different values of the instrumental variables have intersections. We demonstrate that, even if this assumption does not hold, monotonicity and concavity provide identification power. Point identification is achieved when the structural function is flat or linear with respect to the explanatory variable over a given interval. We compute the bounds using real data and show that our bounds are informative.

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.

We consider a multivariate continuous-time process, generated by a system of linear stochastic differential equations, driven by white noise, and involving coefficients that possibly vary over time. The process is observable only at discrete, but not necessarily equally-spaced, time points (though equal spacing significantly simplifies matters). Such settings represent partial extensions of ones studied extensively by A.R. Bergstrom. A model for the observed time series is deduced. Initially we focus on a first-order model, but higher-order models are discussed in the case of equally-spaced observations. Some discussion of issues of statistical inference is included.

The paper examines the effect of conditional heteroskedasticity on least squares inference in stochastic regression models of unknown integration order and proposes an inference procedure that is robust to models within the (near) I(0)–(near) I(1) range with GARCH innovations. We show that a regressor signal of exact order $O_{p}\left ( n\kappa _{n}\right ) $ for arbitrary $\,\kappa _{n}\rightarrow \infty $ is sufficient to eliminate stationary GARCH effects from the limit distributions of least squares based estimators and self-normalized test statistics. The above order dominates the $O_{p}\left ( n\right ) $ signal of stationary regressors but may be dominated by the $O_{p}\left ( n^{2}\right ) $ signal of I(1) regressors, thereby showing that least squares invariance to GARCH effects is not an exclusively I(1) phenomenon but extends to processes with persistence degree arbitrarily close to stationarity. The theory validates standard inference for self normalized test statistics based on the ordinary least squares estimator when $\kappa _{n}\rightarrow \infty $ and $\kappa _{n}/n\rightarrow 0$ and the IVX estimator (Phillips and Magdalinos (2009a), Econometric Inference in the Vicinity of Unity. Working paper, Singapore Management University; Kostakis, Magdalinos, and Stamatogiannis, 2015a, Review of Financial Studies 28(5), 1506–1553.) when $\kappa _{n}\rightarrow \infty $ and the innovation sequence of the system is a covariance stationary vec-GARCH process. An adjusted version of the IVX–Wald test is shown to also accommodate GARCH effects in purely stationary regressors, thereby extending the procedure’s validity over the entire (near) I(0)–(near) I(1) range of regressors under conditional heteroskedasticity in the innovations. It is hoped that the wide range of applicability of this adjusted IVX–Wald test, established in Theorem 4.4, presents an advantage for the procedure’s suitability as a tool for applied research.

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

This paper studies the weak convergence of renorming volatilities in a family of GARCH(1,1) models from a functional point of view. After suitable renormalization, it is shown that the limiting distribution is a geometric Brownian motion when the associated top Lyapunov exponent γ > 0 and is an exponential functional of the maximum process of a Brownian motion when γ = 0. This indicates that the volatility of the GARCH(1,1)-type model has a completely different random structure according to the sign of γ. The obtained results further strengthen our understanding of volatilities in GARCH-type models. Simulation studies are carried out to assess our findings.

It is common for an applied researcher to use filtered data, like seasonally adjusted series, for instance, to estimate the parameters of a dynamic regression model. In this paper, we study the effect of (linear) filters on the distribution of parameters of a dynamic regression model with a lagged dependent variable and a set of exogenous regressors. So far, only asymptotic results are available. Our main interest is to investigate the effect of filtering on the small sample bias and mean squared error. In general, these results entail a numerical integration of derivatives of the joint moment generating function of two quadratic forms in normal variables. The computation of these integrals is quite involved. However, we take advantage of the Laplace approximations to the bias and mean squared error, which substantially reduce the computational burden, as they yield relatively simple analytic expressions. We obtain analytic formulae for approximating the effect of filtering on the finite sample bias and mean squared error. We evaluate the adequacy of the approximations by comparison with Monte Carlo simulations, using the Census X-11 filter as a specific example

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.

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.

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.

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.