A Simple Unimodal Lag Distribution

Abstract

Distributed lags are of considerable importance in the analysis of the relationships between two or more economic variables. They will presumably become of greater importance in the future because of the increased number of quarterly and monthly time series which many statistical bureaus are making available. These data, formerly computed on only an annual basis, will contribute significantly to the possibility of estimating lagged relationships. In spite of this favourable development, however, a serious difficulty exists in the estimation of lag distributions because of the intercorrelation of successive values of explanatory variables. To see this, consider the case in which a dependent variable (y) depends linearly on one explanatory variable (x), apart from certain random disturbances (u):

$${y_t} = {\rm{\alpha }} + {{\rm{\beta }}_{\rm{0}}}{x_t} + {{\rm{\beta }}_{\rm{1}}}{x_{t - 1}} + \ldots + {u_t},$$

where t = 1, ..., T refers to successive time periods. The problem is to estimate, on the basis of a sample of T observations, the constant term α and the parameters β ₀ , β ₁ , ... of the lag distribution.¹ However, when the successive values x _t, x _t-1 , x _t-2 , ... are highly correlated (and they usually are, in particular when the time unit is small), the estimation problem is characterized by a considerable degree of multicollinearity, which leads to large standard errors of the β-estimates and hence to unreliable results. This problem is aggravated further when the explanatory variable is subject to observational errors, for it is well-known that the classical least-squares standard errors give a too optimistic picture of the reliability of the estimates when there are such errors.²

This article first appeared in Metroeconomica, 12 (1960), 112-119. Reprinted here with the permission of Metroeconomica.