A further look at Modified ML estimation of the panel AR(1) model with fixed effects and arbitrary initial conditions

In this paper we consider two kinds of generalizations of Lancaster's (Review of Economic Studies, 2002) Modified ML estimator (MMLE) for the panel AR(1) model with fixed effects and arbitrary initial conditions and possibly covariates when the time dimension, T, is fixed. When the autoregressive parameter ρ=1, the limiting modified profile log-likelihood function for this model has a stationary point of inflection and ρ is first-order underidentified but second-order identified. We show that the generalized MMLEs exist w.p.a.1 and are uniquely defined w.p.1. and consistent for any value of ρ≥-1. When ρ=1, the rate of convergence of the MMLEs is N^{1/4}, where N is the cross-sectional dimension of the panel. We then develop an asymptotic theory for GMM estimators when one of the parameters is only second-order identified and use this to derive the limiting distributions of the MMLEs. They are generally asymmetric when ρ=1. One kind of generalized MMLE depends on a weight matrix W_{N} and we show that a suitable choice of W_{N} yields an asymptotically unbiased MMLE. We also show that Quasi LM tests that are based on the modified profile log-likelihood and use its expected rather than observed Hessian, with an additional modification for ρ=1, and confidence regions that are based on inverting these tests have correct asymptotic size in a uniform sense when |ρ|≤1. Finally, we investigate the finite sample properties of the MMLEs and the QLM test in a Monte Carlo study.