Publications

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Uniform Critical Values for Likelihood Ratio Tests in Boundary Problems (with Giuseppe Cavaliere, Rasmus Pedersen and Anders Rahbek)

Limit distributions of likelihood ratio statistics are well-known to be discontinuous in the presence of nuisance parameters at the boundary of the parameter space, which lead to size distortions when standard critical values are used for testing. In this paper, we propose a new and simple way of constructing critical values that yields uniformly correct asymptotic size, regardless of whether nuisance parameters are at, near or far from the boundary of the parameter space. Importantly, the proposed critical values are trivial to compute and at the same time provide powerful tests in most settings. In comparison to existing size-correction methods, the new approach exploits the monotonicity of the two components of the limiting distribution of the likelihood ratio statistic, in conjunction with rectangular confidence sets for the nuisance parameters, to gain computational tractability. Uniform validity is established for likelihood ratio tests based on the new critical values, and we provide illustrations of their construction in two key examples: (i) testing a coefficient of interest in the classical linear regression model with non-negativity constraints on control coefficients, and, (ii) testing for the presence of exogenous variables in autoregressive conditional heteroskedastic models (ARCH) with exogenous regressors. Simulations confirm that the tests have desirable size and power properties. A brief empirical illustration demonstrates the usefulness of our proposed test in relation to testing for spill-overs and ARCH(-X) effects.

Identification and Estimation of Causal Effects in High-Frequency Event Studies (with Alessandro Casini)

Replication Code

We consider identification, estimation and inference in high-frequency event study regressions, which have been used widely in the recent macroeconomics, financial economics and political economy literatures. The high-frequency event study method regresses changes in an outcome variable on a measure of unexpected changes in a policy variable in a narrow time window around an event or a policy announcement (e.g., a 30-minute window around an FOMC announcement). We show that, contrary to popular belief, the narrow size of the window is not sufficient for identification. Rather, the population regression coefficient identifies a causal estimand when (i) the effect of the policy shock on the outcome does not depend on the other variables (separability) and (ii) the surprise component of the news or event dominates all other variables that are present in the event window (relative exogeneity). Technically, the latter condition requires the ratio between the variance of the policy shock and that of the other variables to be infinite in the event window. Under these conditions, we establish the causal meaning of the event study estimand corresponding to the regression coefficient and super-consistency of the event study estimator with rate of convergence faster than the parametric rate. We show the asymptotic normality of the estimator and propose bias-corrected inference. We also provide bounds on the worst-case bias and use them to quantify its impact on the worst-case coverage properties of confidence intervals, as well as to construct a bias-aware critical value. Notably, this standard linear regression estimator is robust to general forms of nonlinearity. We apply our results to Nakamura and Steinsson’s (2018a) analysis of the real economic effects of monetary policy, providing a simple empirical procedure to analyze the extent to which the standard event study estimator adequately estimates causal effects of interest. 

Inference for Interval-Identified Parameters Selected from an Estimated Set (with Sukjin Han)

Interval identification of parameters such as average treatment effects, average partial effects and welfare is particularly common when using observational data and experimental data with imperfect compliance due to the endogeneity of individuals' treatment uptake. In this setting, a treatment or policy will typically become an object of interest to the researcher when it is either selected from the estimated set of best-performers or arises from a data-dependent selection rule.  In this paper, we develop new inference tools for interval-identified parameters chosen via these forms of selection.  We develop three types of confidence intervals for data-dependent and interval-identified parameters, discuss how they apply to several examples of interest and prove their uniform asymptotic validity under weak assumptions.