Off-diagonal elements of projection matrices and dimension asymptotics. Economics Letters 239 (2024): 111761. (with Stanislav Anatolyev) [link]

We provide insights on asymptotic behavior of the off-diagonal elements of projection matrices in settings, where the dimension of underlying vectors grows with the sample size. Under designs favorable to application of the random matrix theory, the off-diagonal elements are asymptotically normal with a simple variance expression. We also discuss the robustness of the result to deviations of the design from the ideal setup.

Asymptotics of large-dimensional projection matrices (with Stanislav Anatolyev)

We characterize the joint asymptotic behavior of diagonal and off-diagonal elements of projection matrices, say $P = Z(Z'Z)^{-1}Z'$, whose underlying dimension $\ell$ is asymptotically proportional to sample size $n$, that is, $\ell/n \xrightarrow[]{} \alpha$ as $n \xrightarrow[]{} \infty$, with the aspect ratio $\alpha \in (0,1)$, under the rotated i.i.d. assumption for elements of $Z$. The rate of convergence is $\sqrt{n}$, and the limiting distribution is multivariate centered Gaussian. The formulas for the asymptotic variances and covariances are expressed as functions of $\alpha$ and moments of elements of $Z$. The instrumental tools in deriving the asymptotic results are the Woodbury matrix identity, CLT for quadratic and bilinear forms, and elements of the random matrix theory such as the Marchenko-Pastur law.

Many instruments estimation and inference under clustered dependence (with Stanislav Anatolyev) [slides]

We study many weak instruments in a heteroskedastic environment under clustering. We show that clustering either deems the jackknife instrumental variables estimation inconsistent, or makes its inferences hugely distorted. We suggest, instead of following the "save the Jackknife" approach, an alternative approach, which is computationally attractive and allows general structures of intra-cluster correlations. We use the natural extension of jackknifing, the leave-cluster-out methodology, applied to the instrument projection matrix, which allows one to dispose of the cross-cluster dependencies in the influence function of the structural parameter estimator. We set out a formal asymptotic framework to analyze the proposed cluster-jackknife instrumental variables (CJIV) estimator, with an increasing number of clusters, possibly increasing heterogeneous cluster sizes, and possible presence of many weak instruments. We prove a central limit theorem for the influence function embedded in the CJIV estimator, and show consistency of the associated CJIV variance estimator. We study the importance of instrument design on the properties of CJIV, run a simulation study revealing its finite sample properties, and compare with other estimators in relevant empirical contexts.

Heterogeneity analysis in judge designs