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Re: st: using xtabond and xtabond2


From   "Rodrigo A. Alfaro" <raalfaroa@gmail.com>
To   <statalist@hsphsun2.harvard.edu>
Subject   Re: st: using xtabond and xtabond2
Date   Sat, 28 Jul 2007 22:09:14 -0400

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).





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