[Limdep Nlogit List] Inverting AR Coefficients

[William Greene]

David.
I'd love to take credit for the result that every AR(P) model can
be turned into an AR(1) model by writing it as a VAR. But, someone
else taught it to me - I think George Box. One place you can find
the result used to great advantage is in the ARCH literature in the
derivations of the equilibrium (long run) variances.
   In any event, you are correct about the equilibrium multiplier 
in the one equation. It is, indeed, Sum(b(j) / [1 - Sum(a(i))] assuming
the stationarity condition, which is as you note, with an obvious notation. 
As for coding it, you can do it trivially, and get an asymptotic standard
error for the result by using a WALD command after you fit the
regression. Since it's only a single line command, I guess you could
say it's already in the program. It does require a very small amount of
programming.
/Bill Greene



----- Original Message -----
From: "David Tufte" <tufte at suu.edu>
To: limdep at limdep.itls.usyd.edu.au
Sent: Saturday, August 22, 2009 9:07:17 PM GMT -05:00 Columbia
Subject: [Limdep Nlogit List] Inverting AR Coefficients

It's taken me a while to get back to this and to work out the math. I have appended my original message and the reply below.

I'm perplexed: 1) I thought there would be commands to do this, and I was really asking for help on those, but if it has to be coded that is OK, 2) if it has to be coded, then I'm so surprised at my own math that I'd like someone on the list to eyeball this for me because I don't quite believe it can be this simple.

First, let me thank Bill Greene for the suggestion below. It made me laugh: it was very obvious and I'd never even considered approaching my problem this way. I was persevering with a bad approach to the problem instead of being clever about how to start over. Anyway ...

Using this method, is it correct that the equilibrium multiplier from a single equation of a 2 variable VAR with p lags is always equal to:

(sum_of_the_x_coefficients)/(1-sum_of_the_y_coefficients)

if the denominator is between 0 and 1? This is clearly true when the original equation has 1 lag of y and x, but working through Bill's suggestion leads me to think this is a general proposition.

If this is true, it seems to me that it also follows that in my original ECM model that the equilibrium multiplier is always a2/(1+a1) no matter how many lagged differences are in the model?

If this is true, then the method for computing the variance is almost identical to the long-run MPC example in the section on testing non-linear restrictions that's been in Greene for many editions (I'm looking at pp. 220-1 in the 2nd edition).

Regards,

Dave Tufte

=============================================
REPLY

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 POST

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
School of Business
Southern Utah University  
=============================================

David Tufte
Associate Professor
Department of Economics and Finance
School of Business
Southern Utah University
351 W. University Blvd.
Cedar City  UT  84720
351 W. University Blvd.
Cedar City  UT  847820


351 West Center St.
Cedar City, UT 84720

Office: (435) 586-5407
Fax:     (435) 586-5493 

http://www.suu.edu/faculty/tufte
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