A Bracketing Rule for the Estimation of Simple Distributed Lag Models - Oi, Walter Y. (The Review of Economics and Statistics, 1969)
T HE geometric distributed lag model developed by Koyck [8] and Nerlove [11] has been widely used in many empirical studies. The appropriate estimation method for parameters of these models is determined by the true probability distribution of random errors. If the random errors follow a first-order Markov process, Klein [7] has shown that the method of weighted regressions yields maximum likelihood estimators.' However, Klein's method requires prior knowledge of the true serial correlation of random errors. In the absence of such prior knowledge, one can resort to several alternative estimation methods.2 In this paper, I establish a bracketing rule applicable to a subset of the admissible probability distributions of random errors. Consider two special cases of Klein's method in which the true serial correlation p is (i) equal to zero and (ii) equal to the coefficient of the lagged dependent variable (1 X). The former case implies an orthogonal regression while the latter is equivalent to ordinary least squares (OLS). The orthogonal and OLS regressions yield two sets of parameter estimates lying on either side of the maximum likelihood parameter estimates provided that the true serial correlation lies in the interval 0 < p < 1 L, the OLS estimate of the population parameter 1 X. This basic bracketing theorem can be shown to hold even for fixed sample size. Moreover, the width of the interval bracketing the maximum likelihood parameter estimates is narrower, the larger is the partial correlation with the lagged dependent variable. This last result leads to an important implication. If a geometric distributed lag constitutes the correct specification of economic behavior, the dependent variable should be highly correlated with the lagged dependent variable. In this event, the discrepancy between OLS and orthogonal parameter estimates (which bracket the maximum likelihood estimates) will be small. Thus, the bias due to least squares is negligibly small provided that the true serial correlation lies in the interval 0 < p < 1 X. A simple geometric distributed lag model is described by a system of two structural equations. Yt a + 8Zt* + Ut + (1)