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 generalizations of Lancaster's (Review of Economic Studies, 2002) Modified Maximum Likelihood estimator (MMLE) for the panel AR(1) model with fixed effects, arbitrary initial conditions, and strictly exogenous covariates when the time dimension of the panel, T, is fixed. When the autoregressive parameter rho=1, the limiting modified profile log-likelihood function for this model has a stationary point of inflection, and rho is first-order underidentified but second-order identified. We show that, unlike the Random Effects and Transformed MLEs for this type of model, the generalized MMLEs are uniquely defined in finite samples w.p.1. for any value of |rho|=<1. When rho=1, the rate of convergence of the MMLEs is N^{1/4}, where N is the cross-sectional dimension of the panel. We derive the limiting distributions of the MMLEs when rho=1. They are generally asymmetric. We also show that Quasi LM tests that are based on the modified profile log-likelihood function and use its expected rather than observed Hessian for hypotheses that include a restriction on rho, and confidence sets that are based on inverting these tests have correct asymptotic size in a uniform sense when |rho|=<1. Finally, we investigate the finite sample properties of the MMLEs and the QLM test in a Monte Carlo study.