Archives

# On the other hand the fixed effect

On the other hand, the fixed effect estimator, which corrects for the presence of heterogeneity in transversal units, generates and estimation of downward-biased β1 in panels with a small temporal dimension. Studies developed by Monte Carlo, Judson and Owen (1999) showed that this bias can reach 20%, even in panels where T=30. The second problem is due to the provable endogeneity of explanatory variables. In this case, an endogeneity on the right side of Eq. (1) must be treated to avoid a possible bias generated by a simultaneity problem.
Seeking to correct these problems, Arellano and Bond (1991) proposed the estimator for the Generalized Differentiated method of Moments (GMM). Such method consists of eliminating fixed effects through the first difference in Eq. (1), that is:where for a variable Z any, ΔZ=Z−Z. By the construction of Eq. (2), ΔP and Δε are correlated and therefore, OLS estimators for their coefficients shall also be tendentious and inconsistent. In this case, it is necessary to employ instrumental variables for ΔP. The set of micromolar to molar adopted in Eq. (1) imply that the conditions in moments E[ΔPΔε]=0, for t=3.4,…T and s≥2, are valid. Based on these moments, Arellano and Bond (1991) suggest to use P, for t=3.4,…T e s≥2, as equation instruments (2).
With regards to other explanatory variables, there are three possible situations. An explanatory Z may be qualified as (i) strictly exogenous, if not correlated to the terms of past, present and future errors; (ii) frankly exogenous, if it is only correlated to past error term values and; (iii) endogenous, if correlated with past, present and future error terms. In the second case, Z lagged values in one or more periods are valid instruments to estimate equation parameters (2). As for the last case, Z lagged values for two or more periods are valid instruments for Eq. (2).
Meanwhile, Arellano and Bover (1995) and Blundell and Bond (1998) argue that these instruments are weak when the dependant and explanatory variables have a strong persistence and/or the relative variance of fixed effects increases. This produces a non-consistent biased GMM estimator for panels with a small temporal dimension. Arellano and Bover (1995) and Blundell and Bond (1998) suggest a system that combines a set of equations in difference as a way to reduce the bias and imprecision problems (Eq. (2)) with a set of leveled Eq. (1). That is where the generalized moments system comes from. For difference equations, the set of instruments is the same described above. For the level regression, the most adequate instruments are the lagged differences of the respective variables.
For example, assuming that explanatory variable differences are not correlated to the fixed individual effects (for t=3.4,…T) and E[ΔPη]=0, for i=1,2,3,…,N, then the different explanatory variables, either exogenous or frankly exogenous, and ΔP, are valid instruments for level equations. The same happens to the ΔP explanatory variables in lagged differences for a given period if they are endogenous.
Results are introduced in the following section and variances estimators for parameters are robust with regards to the heteroscedasticity and the autocorrelation obtained through the GMM system. The estimator obtained is corrected through the Windmeijer’s method (2005) to avoid that the respective variance estimators underestimate the real variances in a finite sampling.

Results
The estimated results of the parameters obtained from Eq. (1) with the aid of Eq. (2) were obtained through the econometric techniques introduced in Section 5 and are now entered in Table A1.
In the model estimated through the GMM-S, explanatory variables considered endogenous were the dependent one-period lagged variable (P) and the per capita GDP (pib). The variables inf and gini were treated as frankly exogenous and the others were considered as being strictly exogenous.
Initially, it was verified that the value of the coefficient estimation of β1 for P through the GMM method (column [c]) was bigger than the one obtained through the EF (column [B] and smaller than the one obtained through MQO (Column [a]). As discussed in Section 5, MQO and EF estimations for β1 are upward and downward biased respectively, providing approximate superior and inferior limits to guide the β1 estimation through the GMM-S. In this sense, the estimation bias of β1 seems to have been minimized.