Re: st: using xtabond and xtabond2

["Rodrigo A. Alfaro" <raalfaroa@gmail.com>]

Hi,

The 'answer' to Natalie's question is in Alvarez-Arellano paper (AA, 
Econometrica 2003), which is probably the original source of Baltagli's 
textbook mentioned by Branko. They derived the asymptotic properties of 
Arellano-Bond estimator using GMM (AB/GMM, the standard procedure) and LIML 
(AB/LIML, a feasible estimator under homoscedastic errors), and the Within 
Group (WG, fixed-effects) estimator under the alternative sequences where 
both n (cross sectional units) and T (time periods) are large for the case 
of the AR(1) model. You should also consider Hahn-Kuersteiner (HK, 
Econometrica 2002) for a complete understanding of the material.

There are conditions on T and n in AA's paper that you should check for the 
validity of the results. They lazy-conclusion is AB/GMM and WG are 
asymptotically biased in the following magnitud (1+theta)/n and (1+theta)/T 
where theta is the autoregressive coefficient. Usually, T<n then AB/GMM is 
less biased than WG. It should be noted that the theoretical results as well 
the numerical simulations are based in a balanced panel. Also, the number of 
moment conditions for AB/GMM increases with T at the rate of T^2, which 
produces a similar finite-sample bias as we have in the case of 2SLS 
estimator.

HK's paper presents the same result for WG, but the motivation is to 
construct a bias-corrected estimator. Basically, the bias-corrected 
estimator (obtained in that paper) is b1 = (T+1)*b0 + 1/T, where b0 is the 
original WG estimator. Something that is interesting about HK's paper is the 
fact that their correction is a feasible solution of Kiviet's 
bias-correction (Journal of Econometrics 1995).

Well, I tried both topics in one chapter of my dissertation (now editing) 
and I found some interesting results:

(a) Unbalancedness: I found that numerical observation of Bruno's WG 
bias-corrected (Economics Letters 2005), which is based on a generalization 
of Kiviet's bias-correction for the case of unbalanced panels, meets the 
HK's bias-correction for the case of unbalanced panels. This is valid for 
small positive theta. This means that the bias for WG in the case of 
unbalanced panels is similar to the case of balanced panels. Following Bruno 
(2005), I used Ta and Th as the arithmetic and harmonic average of time 
periods. It seems more accurate to use Th than Ta in HK's formula.

The bias for AB/GMM is very complicated and it is not clear that meets the 
theoretical result presented in AA's paper for the case of balanced panels. 
However, if the degree of unbalancedness is mild (Th/Ta close to one) it 
should be safe to use Ta or Th as the "total number of time-periods".

(b) Truncated-AB/GMM estimator (TAB): I define TAB as the same AB/GMM, but 
restricting the number of lags -maxldep- option in -xtabond-. TAB estimator 
seems to be very common in empirical applications. Based on the assumptions 
of AA's paper (homoscedastic errors and balanced panel) I proved that TAB is 
asymptotically unbiased. I checked my proof with Monte Carlo simulation, 
however, the improvement in bias pays a price in standard errors (there are 
less instruments, then TAB is more inefficient than AB/GMM).