Two Results Useful for Implementing Litterman's Procedure for Interpolating a Time Series

Abstract

Litterman (1983) recently suggested a modification to the rameter to be accurate. Litterman's notation is followed Chow and Lin (1971) and Fernandez (1981) procedures for throughout. interpolating monthly data from an observed quarterly time series. ~riefl~ stated, the Litterman proceduri involves obtaining generalized least squares (GLS) forecasts of the monthly 2. THE LITTERMAN PROCEDURE IN BRIEF values of interest-the GLS estimates determined from quar- Letting Y* denote a vector of 3N monthly observations terly observations-allowing for both a random drift and a on the dependent variable and Y = BY* denote the associated first-order Markov process in the residuals of the monthly N observed quarterly values, Litterman shows that model. Litterman provides evidence that his technique may offer significant improvement over the Chow-Lin and Fernandez methods. Unfortunately for many applied researchers, in order to implement the Litterman procedure one must either solve for the roots of a seventh order polynomial or, when sample size is small, iterate on an N X N matrix to solve for the parameter of the Markov process. Hence, those without rather advanced computer programming skills or those without ready access to quality programming assistance may tend to avoid using the Litterman procedure. The purpose of this note is to provide values of the Markov parameter in the Litterman procedure. Since this parameter is shown to be a function of the first-order autocorrelation coefficient of the first difference of quarterly residuals (see Litterman 1983, p. 172), the results presented here are obtained assuming this latter quantity is known. I also adwhere X* are monthly observations on K indicator variables, 6 = {Xt[B(D'H'HD)-'Bt]-'XI-'X'