A Further Look at Modified ML Estimation of the Panel AR(1) Model with Fixed Effects and Arbitrary Initial Conditions

In this paper we reconsider Modified ML estimation of the panel AR(1) model with fixed effects and arbitrary initial conditions, viz. y_{i,t}=ρy_{i,t-1} x_{i,t}′β(1-ρ) μ_{i}(1-ρ) ε_{i,t} with ε_{i,t}|(1(ρ≠1)((y_{i,0}-μ_{i}-T⁻¹ι′X_{i}β), QX_{i}))∼i.i.d.N(0,σ²), i=1,...,N and t=1,...,T, when T is fixed. Lancaster (Review of Economic Studies, 2002) introduced a Bayesian estimator for this model, which can be re-interpreted as a Modified ML estimator (MMLE). We propose two new MMLEs that extend the definition of Lancaster's estimator so that our estimators also exist for samples for which his estimator fails to exist, and exist w.p.a.1 for any ρ≥-1. Furthermore, unlike Lancaster, we show that our MMLEs are uniquely defined w.p.1. and consistent for any ρ≥-1. We also derive the limiting distributions of our two MMLEs. They are generally asymmetric when ρ=1. In both cases, when ρ=1, ρ is first-order unidentified but second-order identified and the rate of convergence is N^{1/4}. One of the MMLEs depends on a weight matrix W_{N} and we show that a suitable choice of W_{N} yields an asymptotically unbiased MMLE. Although the MMLEs for ρ are asymptotically inefficient, our Monte Carlo results show that even in panels as large as T=9 and N=500, they can have a significantly smaller RMSE than the asymptotically more precise REMLE. Finally, we show how the model can still be consistently estimated by MMLEs when the covariates are predetermined or endogenous and ρ∈ℝ