Distributed Lags Analysis - Almon Distributed Lag
A common problem that often arises when computing the weights for the multiple linear regression model shown above is that the values of adjacent (in time) values in the x variable are highly correlated. In extreme cases, their independent contributions to the prediction of y may become so redundant that the correlation matrix of measures can no longer be inverted, and thus, the Beta weights cannot be computed. In less extreme cases, the computation of the Beta weights and their standard errors can become very imprecise, due to round-off error. In the context of Multiple Regression this general computational problem is discussed as the multicollinearity or matrix ill-conditioning issue.
Almon (1965) proposed a procedure that will reduce the multicollinearity in this case. Specifically, suppose we express each weight in the linear regression equation in the following manner:
βi = α0 + α1*i + ... + αq*iq
Almon could show that in many cases it is easier (i.e., it avoids the multicollinearity problem) to estimate the Alpha values than the Beta weights directly. Note that with this method, the precision of the Beta weight estimates is dependent on the degree or order of the polynomial approximation.
Misspecifications
A general problem with this technique is that, of course, the lag length and correct polynomial degree are not known a priori. The effects of misspecifications of these parameters are potentially serious (in terms of biased estimation). This issue is discussed in greater detail in Frost (1975), Schmidt and Waud (1973), Schmidt and Sickles (1975), and Trivedi and Pagan (1979).