Introduction to Bayesian Econometrics

THE ANALYSIS OF time series data has generated a vast literature from both frequentist and Bayesian viewpoints. This chapter considers a few standard models to illustrate how they can be analyzed with MCMC methods. Section 11.6 provides references to more detailed explanations and additional models.

Autoregressive Models

This section is concerned with models of the general form

where and t = 1, …, T. The disturbance ∊t is said to be autoregressive of order p, denoted by ∊t ∼ AR(p). This model is a way to capture the possibility that disturbances in a particular time period continue to affect y in later time periods, a property that characterizes many time series in economics and other areas.

Assume that the stochastic process defining ∊t is second-order stationary, which means that the means E(∊t) and all covariances E(∈s∊t) of the process are finite and independent of t and s, although the covariances may depend on ∣t – s∣. Because the variance is the special case of the covariance when t = s, it is finite and independent of time.

The stationarity property imposes restrictions on the ϕs. To state these, I define the lag operator L. It operates on time-subscripted variables as Lzt = zt−1, which implies that for integer values of r. The polynomial in the lag operator

allows you to relate ∊t and ut as Φp(L)∊t = ut.