Linear regression in continuous time
E. J. Hannan · Journal of the Australian Mathematical Society · 1975
We consider a regression relation of the from whereiny(t) andx(t) are real (column) vectors ofqandpcomponents ande(t) is real and is generated by a stationary generalised vector process ofqcomponents with zero mean and covariance function (aqrowed matrix) Γ(t–s) =E{x(s)x(t)′}. (See Hannan (1970; pages 23–26, 91–94) and references therein for definitions of terms used.) We assumee(t) to be independent ofx(s) for alls,t. Thus we may regardx(t) as a fixed time function and not stochastic and we shall henceforth do that. We take Γ(t) to be continuous and to correspond to an absolutely continuous spectral function with spectral density which is uniformly bounded and continuous. Then we have We do not exclude the possibility that for theyth diagonal element,fjj, offwe have