An Adjusted profile likelihood for non-stationary panel data models with fixel effects

We calculate the bias of the profile score for the autoregressive parameters p and covariate slopes in the linear model for N x T panel data with p lags of the dependent variable, exogenous covariates, fixed effects, and unrestricted initial observations. The bias is a vector of multivariate polynomials in p with coefficients that depend only on T. We center the profile score and, on integration, obtain an adjusted profile likelihood. When p = 1, the adjusted profile likelihood coincides with Lancaster's (2002) marginal posterior. More generally, it is an integrated likelihood, in the sense of Arellano and Bonhomme (2009), with fixed effects integrated out using a new data-independent prior. It appears that p and B are identified as the unique point where the large N adjusted profile likelihood reaches a local maximum (or a at inection point, as a limiting case) inside or on an ellipsoid centered at the maximum likelihood estimator. We prove this when p = 1 and report numerical calculations that support it when p > 1. The global maximum of the adjusted profile likelihood lies at infinity for any N.