Contrasting Bayesian and Frequentist Approaches to Autoregressions: the Role of the Initial Condition

The frequentist and the Bayesian approach to the estimation of autoregressions are often contrasted. Under standard assumptions, when the ordinary least squares (OLS) estimate is close to 1, a frequentist adjusts it upwards to counter the small sample bias, while a Bayesian who uses a at prior considers the OLS estimate to be the best point estimate. This contrast is surprising because a at prior is often interpreted as the Bayesian approach that is closest to the frequentist approach. We point out that the standard way that inference has been compared is misleading because frequentists and Bayesians tend to use different models, in particular, a different distribution of the initial condition. The contrast between the frequentist and the Bayesian at prior estimation of the autoregression disappears once we make the same assumption about the initial condition in both approaches.