We calculate the exact bias of the profile score for the first-order autoregressive
parameter, ρ, in a Gaussian N ×T panel data model with
arbitrary initial conditions and arbitrary heterogeneity in intercepts,
trends and error variances. The bias is a polynomial in ρ and does not
depend on the initial values or the incidental parameters. Subtracting
its integral from the profile loglikelihood leads to an adjusted profile
likelihood which, in the case without incidental trends and error variances, coincides with Lancaster’s (2002) marginal posterior density for ρ. We show, largely by simulation, that the expected adjusted profile loglikelihood (and hence the expected marginal posterior log-density), in addition to attaining a local maximum on [−1, 1] at the true value
of ρ, may attain a global maximum at 1. The latter occurs when the initial values are strong inliers relative to the stationary distribution, which leads to weakly informative data when the autoregressive parameter is moderate to large, even with very large N.