Download Statistical implications of distributed lag models

Statistical implications of distributed lag models
Suppose we have:
Yt   Yt 1  ut
 Yt  ut   ut 1   2ut  2   3ut 3  ...

   i ut i
i 0
This is the autoregressive model in moving average form.
Is OLS consistent?
Yt 1Yt  Yt 1   Yt 1  ut 
Yt 1ut


ˆ



2
2
2
Y
Y
Y
 t 1
 t 1
 t 1
plim ˆ  
if
1
plim  Yt 1ut  0
T
1
plim  Yt 21   Y2  0
T
These conditions are not obviously satisfied. If u is itself
serially correlated then the first will fail. The second will fail
if β ≥1 as the plim will not converge.
Example:
Yt   Yt 1  ut
1    1
ut   ut 1   t
1    1
We have
Yt  ut   ut 1   2ut 2   3ut 3  ....
utYt 1    ut 1   t   ut 1   ut 2   2ut 3   3ut 4  ....
and taking expectations yields
2

u
E  utYt 1    u2   2  u2   3  2 u2  .... 
1  
Therefore it follows that if ρ>0 then the OLS estimator of
β will be biased upwards.
It is also possible to show that the estimate of ρ based on the
OLS residuals will be biased downwards when ρ>0 .
Finally, we can show that the Durbin-Watson statistic will
be biased towards 2 when ρ>0.
Given these results we should not rely on the Durbin-Watson
test when the regression equation contains lagged endogenous
variables. We should use an alternative test such as Durbin’s
h test or the Breusch-Godfrey test.
We have seen that the AR(1) model can be written as an infinite
moving average. If the u’s have the standard Gaussian properties
then:

Yt    i ut i
i 0
2

u
E Yt 2     2i u2 
1  2
i 0

If the variance of Y is to be positive and finite we therefore require
 1
If this condition is not satisfied then we say that the series
contains a unit root or is non-stationary.
If a series is non-stationary then:
1 T 2
Yt 1   as

T t 1
T 
Therefore the probability limit of the OLS estimator may
equal the true value even if cov(Yt-1, ut) ≠ 0.
In these circumstances we say that the OLS estimator is
superconsistent.
However, non-stationarity creates many problems for statistical
analysis in that the distributions we have derived for all the
standard test statistics assume that the series are stationary.
The Lag Operator
When we have models with lags then it is often convenient to
write them using the Lag Operator.
LYt  Yt 1
or
Lk Yt  Yt k
Therefore if we have a model of the form:
Yt  1 X t  2 X t 1  3Yt 1  ut
This can be written as:
Yt   1  2 L  X t  3 LYt  ut
The lag operator makes it easier to manipulate equations of
this type because the usual rules of algebra apply.
For example:
Yt 1  3 L    1   2 L  X t  ut
1   2 L 

1
 Yt 
Xt 
ut
1  3 L 
1  3 L 
The lag polynomial for X determines the pattern of the
distributed lag effect of a change in X on Y.
A general result is that almost any shape of distributed lag
response can be modelled as the ratio of two low order lag
polynomials.