🤖 AI Summary
This paper addresses the challenge of identifying multiple long-run equilibrium relationships in large-dimensional unbalanced panel data (where $n gg T$). Existing methods typically assume a single long-run relationship and struggle to simultaneously eliminate short-run dynamic disturbances while ensuring stable identification of long-run structures. We propose the “Pooled Minimum Eigenvalue” (PME) method—a novel, fully data-driven approach that requires no pre-specified causal ordering and applies to generalized linear processes with interactive time-varying effects. PME constructs bias-corrected sequences via time-averages over non-overlapping subsamples and jointly estimates eigenvalues and eigenvectors of the long-run covariance matrix to consistently determine both the number and coefficients of long-run relationships. We establish its consistency and asymptotic normality under $n,T o infty$ with $T approx n^d$, $d > 1/2$. Monte Carlo experiments confirm its high accuracy in rank selection, low coefficient bias, small RMSE, and strong test power.
📝 Abstract
This paper provides a new methodology for the analysis of multiple long run relations in panel data models where the cross section dimension, $n$, is large relative to the time series dimension, $T$. For panel data models with large $n$ researchers have focused on panels with a single long run relationship. The main difficulty has been to eliminate short run dynamics without generating significant uncertainty for identification of the long run. We overcome this problem by using non-overlapping sub-sample time averages as deviations from their full-sample counterpart and estimating the number of long run relations and their coefficients using eigenvalues and eigenvectors of the pooled covariance matrix of these sub-sample deviations. We refer to this procedure as pooled minimum eigenvalue (PME) and show that it applies to unbalanced panels generated from general linear processes with interactive stationary time effects and does not require knowing long run causal linkages. To our knowledge, no other estimation procedure exists for this setting. We show the PME estimator is consistent and asymptotically normal as $n$ and $T
ightarrow infty$ jointly, such that $Tapprox n^{d}$, with $d>0$ for consistency and $d>1/2$ for asymptotic normality. Extensive Monte Carlo studies show that the number of long run relations can be estimated with high precision and the PME estimates of the long run coefficients show small bias and RMSE and have good size and power properties. The utility of our approach is illustrated with an application to key financial variables using an unbalanced panel of US firms from merged CRSP-Compustat data set covering 2,000 plus firms over the period 1950-2021.