#### III.VI.2
Infinite distributed lags
There
exist different specifications for infinite distributed lags. Some
of them are based on economic theory, others are of a more inductive
nature.
The
**Koyck lags** are frequently
used in econometric practice. Formally this means that
**(III.VI.2-1)**
for
which
**(III.VI.2-2)**
Note
that the Koyck lag is sometimes also called the **geometric
distributed lag** due to the fact that the regression parameters
are exponentially decreasing.
The
statistical model is, when using Koyck lags, assumed to be of the
form
**(III.VI.2-3)**
such
that (III.VI.2-2) can be substituted into (III.VI.2-3) yielding
**(III.VI.2-4)**
Sometimes
it is suggested in econometric literature that Koyck lags can be
estimated by OLS, using a mathematical trick.
**(III.VI.2-5)**
or
simply
**(III.VI.2-6)**
which
can be used in OLS estimation.
If
we rewrite (III.VI.2-4) as
**(III.VI.2-7)**
and
if e_{t} is normally distributed with zero mean and constant
variance then MLE can be applied to
**(III.VI.2-8)**
**(III.VI.2-9)**
with
**(III.VI.2-10)**
where
**(III.VI.2-11)**
Also,
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.
Another
way of combining, say, Almon lags and Koyck lags is the following
**(III.VI.2-12)**
such
that the model becomes
**(III.VI.2-13)**
or
**(III.VI.2-14)**
Another
method of distributing parameters over time is the **Pascal
distributed lag** or formally
**(III.VI.2-15)**
which
are the weights of time. Hence the complete model is
**(III.VI.2-16)**
If
for instance r = 3 then the Pascal distributed lag becomes
**(III.VI.2-17)**
from
which it can be seen that this is a special case of (III.VI.2-12).
Define
**Jorgenson's rational
distributed lag** as the ratio of two polynomials
**(III.VI.2-18)**
which
may be illustrated by a simple example
**(III.VI.2-19)**
The
same remarks as with the Koyck lags hold for estimation of Jorgenson
lags.
Finally
we define the (adapted) **gamma distributed lags** by
**(III.VI.2-20)**
**(III.VI.2-21)**
The
complete model is written as
**(III.VI.2-22)**
for
which it can be shown that the deletion of the truncation remainder
does not affect the asymptotic properties.
We
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).
One
of the most important **inductive
distributed lag models** will be considered later (see Box-Jenkins
Transfer Function Analysis). |