# Re: st: RE: RE: Prober Instrument for GMM xtabond2

 From "Jo Gardener" <1243go@gmx.net> To statalist@hsphsun2.harvard.edu Subject Re: st: RE: RE: Prober Instrument for GMM xtabond2 Date Tue, 18 Jul 2006 20:07:34 +0200

```Hi Jean,
again, thank you for your reply on STATALIST. But I have one final question I want to ask you because you seem to have much GMM experience and I have following problem

I asked for the importance of the AB-test AR(2) number of the diff. residuals in the STATA output.
However, you are right as long we are in a AR(1) model world (as treated in the AB-paper in 1991. However, my question refers to the AR(2) ... case:

y=y(t-2!!)+...

Maybe I should reformulate my question:
For an AR(2), AR(4) or any other model: Is the AB-test of AR(2) in differenced residuals always a sign, that the estimation is INCONSISTENT because there is autocorrelation in the original levels equation (no matter which instruments I use).

Or put it differently:
1. Which instruments should I use? My answer: Look at the AR(.) number of the AB-test. If it is above 0.1 you can use this lag of the dependent variable accordingly as instrument, since it is not correlated with the error term. 2. Is the estimation CONSISTENT? And here, I fear, always the AB-Test of AR(2!!) of diff. res. is important no matter if I use an AR(2), AR(3) or AR(3) model.

For example:
AB-Bond test for AR(1) in first differences: z =  -8,81  Pr > z =  0,000
AB-Bond test for AR(2) in first differences: z =  -2,05  Pr > z =  0,040
AB-Bond test for AR(3) in first differences: z =  -0,04  Pr > z =  0,969
AB-Bond test for AR(4) in first differences: z =  -0,22  Pr > z =  0,822

Interpretation of output:
In both cases, I can use the level and the difference of l3.depvar and l4.depvar as instruments for l2.depvar.
BUT: The estimation is not consistent, since there is autocorrelation of second order in the differenced equation and thus a sign for autocorrelated residuals of the original levels equation.

I first thought , that the autocorrelation test is only important WHICH instrument I can use (I think your reply was going in this direction.) However, now it seems to me, that autocorrelation of second order is ALWAYS a SIGN, that the estimation is NOT CONSISTENT no matter if the original equation is an AR(1), AR(2) or an AR(5) model.

Is this assumption regarding AR(2) of the AB-test correct and my interpretation of the output correct?

Actually I hope, that the consistency of the estimation for an AR(2) model does depend on the AB-AR(3) test of r., and for an AR(4) model does depend on the AB-AR(5) test of r. and so on (and not always on the AB-AR(2) test). Hopefully this question is not too long. But I want to use this maybe once in a GMM-lifetime opportunity to ask as clearly as possible. As I said, I referred to most of your GMM-articles but they usually treat the AR(1)-model-case.

With best regards and thanks in advance
Jo Gardener

> Arellano-Bond orthogonality conditions are established under the assumption that
> the error term in the levels equation is not autocorrelated. The purpose of
> the Arellano-Bond autocorrelation test is to test this assumption. If the
> error term in the levels equation is not autocorrelated, then the error term
> in the first-difference equation has negative first-order autocorrelation,
> and 0 second order autocorrelation.
>
> If you reject the hypothesis that there is 0 2nd order autocorrelation in
> the residuals of the first-difference equation, then you also reject the
> hypothesis that the error term in the levels equation is not autocorrelated.
> This indicates that the AB orthogonality conditions are not valid--no
> matter which lags you use as instruments.
>
> Jean Salvati
>
> > -------- Original-Nachricht --------
> > Datum: Mon, 17 Jul 2006 13:58:54 +0530
> > Von: "M.Parameswaran" <mpeswaran@gmail.com>
> > An: statalist@hsphsun2.harvard.edu
> > Betreff: st: Re:
> >
> > > If there are second order serial correlation, then second
> > lags are not
> > > valid instruments, in this case one has to use 3rd lag onwards.
> > >
> > > Parameswaran
> > >
> > > On 17/07/06, Jo Gardener <1243go@gmx.net> wrote:
> > > > Dear all,
> > > >
> > > > using a simple dynamic model (DPD) I currently face
> > following problem:
> > > >
> > > > Model: y=a1+a2*y(t-1)+ ...
> > > > xi: xtabond2 y l.y i.year, gmm(y, lag(3!! 4) equ(both) coll)
> > > > iv(i.year,
> > > equ(both)) small rob twostep arte(3)
> > > >
> > > > I use gmm(y, lag(3 4) and not lag(2 4) because the
> > AB-Test for AR(2)
> > > > -
> > > not shown here - says that there is autocorrlation of second order.
> > > Thus I cannot use lag(2) as instrument.
> > > > However, my question: Can I use lag(3 4), because the 3rd
> > lag is not
> > > correlated with the differenced error term?
> > > >
> > > > Following output of the lag(3 4) estimation says, that the
> > > > differenced
> > > residuals show no AR(3) correlation and the Hansen J is ok:
> > > > Hansen test of overid. restr.: chi2(3) =   3,72     Prob
> > > chi2 =  0,432
> > > > Arellano-Bond test for AR(1) in first diff: z =  -5,76
> > Pr > z =  0,000
> > > > Arellano-Bond test for AR(2) in first diff: z =   2,21
> > Pr > z =  0,042
> > > > Arellano-Bond test for AR(3) in first diff: z =  -0,75  Pr > z =
> > > > 0,433
> > > >
> > > > I am glad for any response I can get on this issue Jo gardener
> > > >
> > > > --
> > > >
> > > > --
> > > >
> > > >
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> > >
> > >
> > > --
> > > ___________________________________
> > > M. Parameswaran,
> > > Research Associate,
> > > Centre for Development Studies,
> > > Prasanth Nagar Road, Ulloor.
> > > Trivandrum - 695 011,
> > > Kerala, India.
> > > Phone: +91-471-2448881 (O)
> > >             +91 - 09446506388 (mobile)
> > > e-mail: parameswaran@cds.ac.in
> > >            mpeswaran@gmail.com
> > >
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```