In this paper we propose a new variance estimator for OLS as well as for nonlinear estimators such as logit, probit and GMM. This variance estimator enables cluster-robust inference when there is two-way or multi-way clustering that is nonnested. The variance estimator extends the standard cluster-robust variance estimator or sandwich estimator for one-way clustering (e.g. Liang and Zeger (1986), Arellano (1987)) and relies on similar relatively weak distributional assumptions. Our method is easily implemented in statistical packages, such as Stata and SAS, that already offer cluster-robust standard errors when there is one-way clustering. The method is demonstrated by a Monte Carlo analysis for a two-way random effects model; a Monte Carlo analysis of a placebo law that extends the state-year effects example of Bertrand et al. (2004) to two dimensions; and by application to two studies in the empirical public/labor literature where two-way clustering is present.

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Bilateral, product-level data exhibit a number of strong patterns that can be used to evaluate international trade theories, notably the spatial incidence of “export zeros ” (correlated with distance and importer size), and of export unit values (positively related to distance). We show that leading theoretical trade models fail to explain at least some of these facts, and propose a variant of the Melitz model that can account for all the facts. In our model, high quality firms are the most competitive, with heterogeneous quality increasing with firms ’ heterogeneous cost. (JEL F11, F14, F40) The gravity equation relates bilateral trade volumes to distance and country size. Countless gravity equations have been estimated, usually with “good ” results, and trade theorists have proposed various theoretical explanations for gravity’s success. However, the many potential explanations for the success of the gravity equation make it a problematic tool for discriminating among trade models. 1 As a matter of arithmetic, the value of trade depends on the number of goods

This paper presents a novel way to conduct inference using dependent data in time series, spatial, and panel data applications. Our method involves constructing t and Wald statistics utilizing a cluster covariance matrix estimator (CCE). We then use an approximation that takes the number of clusters/groups as fixed and the number of observations per group to be large and calculate limiting distributions of the t and Wald statistics. This approximation is analogous to ‘fixed-b’ asymptotics of Kiefer and Vogelsang (2002, 2005) (KV) for heteroskedasticity and autocorrelation consistent inference, but in our case yields standard t and F distributions where the number of groups essentially plays the role of sample size. We provide simulation evidence that demonstrates our procedure outperforms conventional inference procedures and performs well comparably to KV.

This paper develops a method for performing inference using spatially dependent data. We consider test statistics formed using nonparametric covariance matrix estimators that account for heteroskedasticity and spatial correlation (spatial HAC). We provide distributions of commonly used test statistics under “fixed-b” asymptotics, in which HAC smoothing parameters are proportional to the sample size. Under this sequence, spatial HAC estimators are not consistent but converge to non-degenerate limiting random variables that depend on the HAC smoothing parameters and kernel. We show that the limit distributions of commonly used test statistics are pivotal but non-standard, so critical values must be obtained by simulation. We provide a simple and general simulation procedure based on the i.i.d. bootstrap that can be used to obtain critical values. We illustrate the potential gains of the new approximation through simulations and an empirical example that examines the effect of unjust dismissal doctrine on temporary help services employment.

This paper studies robust inference for linear panel models with fixed effects in the presence of heteroskedasticity and spatiotemporal dependence of unknown forms. We propose a bivariate kernel covariance estimator, which is flexible to nest existing estimators as special cases with certain choices of bandwidths. For distributional approximations, we consider two different types of asymptotics. When the level of smoothing is assumed to increase with the sample size, the proposed estimator is consistent and the associated Wald statistic converges to a χ2 distribution. We show that our covariance estimator improves upon existing estimators in terms of robustness and efficiency. When we assume the level of smoothing to be held fixed, the covariance estimator has a random limit and we show by asymptotic expansion that the limiting distribution of the test statistic depends on the bandwidth parameters, the kernel function, and the number of restrictions being tested. As this distribution is nonstandard, we establish the validity of an F-approximation to this distribution, which greatly facilitates the test. For optimal bandwidth selection, we propose a procedure based on the upper bound of asymptotic mean square error criterion. The flexibility of our estimator and proposed bandwidth selection procedure make our estimator adaptive to the dependence structure in data. This adaptiveness automates the selection of covariance estimator. That is, our estimator reduces to the existing estimators which are designed to cope with the particular dependence structures. Simulation results show that the F-approximation and the adaptiveness work reasonably well.

moving blocks bootstrap for panel linear regression models with

This paper surveys methods that have been added to the microeconometrician’s toolkit over the past twenty-five years, and some recent developments in these newer methods. These methods include GMM, empirical likelihood, simulation-based estimation, quantile regression, semiparametric estimation, robust inference, and bootstrap. The paper also considers estimation of marginal effects that can be given a causative interpretation, notably treatment effects, unobserved heterogeneity, and common data complications of sampling and missing and mismeasured data.

The literature on training has pointed out that macroeconomic fluctuations can have a positive or a negative effect on training decisions. On the one hand, the opportunity cost to train is lower during downturns, and thus training should be counter-cyclical. On the other hand, a positive shock may be related to the adoption of new technologies and increased returns to skill, making training incidence pro-cyclical. The first contribution of this paper is to document, using the Canadian panel of Workplace and Employee Survey, that (i) training moves counter-cyclically with aggregate output fluctuations (more training in downturns), while at the same time (ii) the relative position of sectoral output has a positive impact on training decisions (more training in sectors doing relatively better). This second fact is novel and unexplored. Overall, the results show that the firms’ decisions to train are quite complex − in order to fully understand them, one needs to take into account not only the change in aggregates, but also the relative position of each sector in the economy. The second contribution of the paper is to illustrate the mechanisms at work by incorporating training decisions into a standard Mortensen-Pissarides

In this paper we survey methods to control for regression model error that is correlated within groups or clusters, but is uncorrelated across groups or clusters. Then failure to control for the clustering can lead to understatement of standard errors and overstatement of statistical significance, as emphasized most notably in empirical studies by Moulton (1990) and Bertrand, Duflo and Mullainathan (2004). We emphasize OLS estimation with statistical inference based on minimal assumptions regarding the error correlation process. Complications we consider include cluster-specific fixed effects, few clusters, multi-way clustering, more efficient feasible GLS estimation, and adaptation to nonlinear and instrumental variables estimators.