# Online Econometrics Textbook - Regression Extensions - Infinite distributed lags

#### 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 et 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).

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