Infinite distributed lags
exist different specifications for infinite distributed lags. Some
of them are based on economic theory, others are of a more inductive
Koyck lags are frequently
used in econometric practice. Formally this means that
that the Koyck lag is sometimes also called the geometric
distributed lag due to the fact that the regression parameters
are exponentially decreasing.
statistical model is, when using Koyck lags, assumed to be of the
that (III.VI.2-2) can be substituted into (III.VI.2-3) yielding
it is suggested in econometric literature that Koyck lags can be
estimated by OLS, using a mathematical trick.
can be used in OLS estimation.
we rewrite (III.VI.2-4) as
if et is normally distributed with zero mean and constant
variance then MLE can be applied to
it is remarked that the Koyck distributed lags do not have to start
from lag zero. It is possible to set the starting point to some
specified lag, and capture the early lags by a finite distributed
lag method such as the Almon lag. This way great flexibility can be
obtained by combining finite distributed lags with postponed
infinite distributed lags.
way of combining, say, Almon lags and Koyck lags is the following
that the model becomes
method of distributing parameters over time is the Pascal
distributed lag or formally
are the weights of time. Hence the complete model is
for instance r = 3 then the Pascal distributed lag becomes
which it can be seen that this is a special case of (III.VI.2-12).
distributed lag as the ratio of two polynomials
may be illustrated by a simple example
same remarks as with the Koyck lags hold for estimation of Jorgenson
we define the (adapted) gamma distributed lags by
complete model is written as
which it can be shown that the deletion of the truncation remainder
does not affect the asymptotic properties.
furthermore remark that the omission of important variables in the
regression equation can have devastating effects on the estimation
of distributed lag parameters (c.q. the UVB).
of the most important inductive
distributed lag models will be considered later (see Box-Jenkins
Transfer Function Analysis).