Automatic Inference for Infinite Order Vector Autoregressions
Infinite order vector autoregressive (VAR) models have been used in a number of applications ranging from spectral density estimation, impulse response analysis, tests for cointegration and unit roots, to forecasting. For estimation of such models it is necessary to approximate the infinite order lag structure by finite order VAR’s. In practice, the order of approximation is often selected by information criteria or by general-to-specific specification tests. Unlike in the finite order VAR case these selection rules are not consistent in the usual sense and the asymptotic properties of parameter estimates of the infinite order VAR do not follow as easily as in the finite order case. In this paper it is shown that the parameter estimates of the infinite order VAR are asymptotically normal with zero mean when the model is approximated by a finite order VAR with a datadependent lag length. The requirement for the result to hold is that the selected lag length satisfies certain rate conditions with probability tending to one. Two examples of selection rules satisfying these requirements are discussed. Uniform rates of convergence for the parameters of the infinite order VAR are also established.