Discrete Approximations to Continuous Time Distributed Lags in Econometrics

A model is considered in which a covariance-stationary exogenous process is related to an endogenous process by an unrestricted, infinite, linear distributed lag. It is shown that when an underlying continuous time model is sampled at unit intervals to yield endogenous and exogenous discrete time processes, the discrete time processes are related by a discrete time equivalent of the underlying continuous model. The relationship between the underlying continuous lag distribution and its discrete time equivalent is close when the exogenous process is smooth. Even then, however, it is interesting to note that (i) a monotone continuous time distribution does not in general have a monotone discrete time equivalent and (ii) a one-sided continuous time distribution does not in general have a one-sided discrete time equivalent. The implications of the results for statistical practice are considered in the latter part of the paper.