Selecting the Optimal Order of Polynomial in the Almon Distributed Lag - Amemiya, Takeshi; Morimune, Kimio (The Review of Economics and Statistics, 1974)
HE method proposed by Almon (1965) has been extensively used in the estimation of distributed lag models. It may be regarded as the least squares method under the linear constraint that the regression coefficients lie on a polynomial of a chosen order. Therefore, the loss or inefficiency of Almon's method (defined as some reasonable function of the mean square error matrix) could be smaller than that of the unconstrained least squares method. Then, an interesting question arises: for what order of polynomial is the loss minimized in a given distributed lag model? The answer depends upon several variables: the true values of the regression coefficients, the number of lags assumed in the model, the sample size, the ratio of the variance of the dependent variable to that of the error term, and the degree of the autocorrelation of the independent variable. In this paper we will evaluate numerically how the optimal order of polynomial is determined by these variables. Because the answer depends on so many variables, it is extremely important to design the study to produce meaningful conclusions. For this purpose we adopt one important simplifying assumption -that the independent variable follows a first-order autoregressive process with a varying correlation coefficient. Such a process is a good approximation of the processes of many economic variables. Given this simplification, we obtain definitive conclusions by judiciously defining the loss function so it depends simply and nicely on the parameters that we allow to change. As a result we can calculate the optimal order of polynomial for a given distributed lag model at a minimal computational cost. The essential part of our definition of the loss function is the trace of the product of the mean square error matrix and the autocovariance matrix of the independent variable. In section II we will offer rationales for this definition, as we believe that this definition has intrinsic merit as well as the advantage of simplifying our computation. Section II defines the model, defines the loss function for Almon's method, and discusses the rationale for and mathematical properties of the loss function. Section III presents and'analyzes the results of the numerical evaluation of the loss function for twelve models. Conclusions are presented in section IV.