# [Limdep Nlogit List] Inverting AR Coefficients

William Greene wgreene at stern.nyu.edu
Thu Aug 6 05:16:29 EST 2009

```David.
You write the system in its AR(1) form as follows:
Y(t) = [y(t),y(t-1)]'
X(t-1) = [x(t-1),x(t-2)]
A = [b1,b2 / 1,0]  B = [b3,b4 / 0,0].
Then,
Y(t) = AY(t-1) + BX(t-1) + E(t).
Skipping a couple lines of algebra, the MA form is then
Y(t) = BX(t-1) + ABX(t-1) + A^2BX(t-3) + ...
The symbolic representation of the coefficients in
B, AB, Z^2B, etc. will be a mess. But, numerically, it's
easy.  Also, if you are looking for the equilibrium multipliers
to compute standard errors for,
That is C = [I - A]^-1B.
Note, unfortunately, since A is not symmetric, this will not
be simple to deal with. But, it is how to handle the computations.
If you have variances and covariances for the terms in A and B,
you should be able to deduce a variance matrix for X.
Regards,
Bill Greene

----- Original Message -----
From: "David Tufte" <tufte at suu.edu>
To: limdep at limdep.itls.usyd.edu.au
Sent: Tuesday, August 4, 2009 7:25:13 PM GMT -05:00 US/Canada Eastern
Subject: [Limdep Nlogit List] Inverting AR Coefficients

Hoping you folks can help me (I am not getting the support I need
from one of those **other** software companies).

I have a single ECM equation that I've estimated:

dY = a0 + a1Y(t-1) + a2X(t-1) + a3dY(t-1) + a4dX(t-1)

I can unravel this into something like this:

Y = b0 + b1Y(t-1) + b2Y(t-2) + b3X(t-1) + b4X(t-2)

What I'd like to do is invert the entire AR(2) process and obtain:

Y = c0 + c1X(t-1) + c2X(t-2) + c3X(t-3) + ....

Where ci=f(b1,b2,b3,b4) for i>0

I'd like to produce a sum of the c's from 1 on up to some truncation
point (which I can do outside of a econometrics package), as well as
some sort of standard error for that sum (which is where LimDep comes in).

There is an example of this in either Greene's text, or the manual
(neither of which is handy right now), but the example uses only 1
lag of Y and X. How do I extend that to this 2 lag case?

Feel free to reference the text or manual (by the time you read this,
I'll be back in the same room with them).

Also, I'm aware that I'll have to input my own derivatives, but
figuring out which one goes where is bugging me.

Regards,

David Tufte
Associate Professor
Department of Economics and Finance