DIFFERENCE GMM METHODOLOGY

The GMM estimators are used for econometric analysis of dynamic panel data where different entities are observed over time, particularly where the number of observations(N) is large and time(T) is small. In autoregressive models, because of the endogeneity problem, simple OLS regression yields biased and inconsistent estimates. Even though fixed effects model eliminates the unobservable time-invariant country specific fixed effects, the within transformation does not completely eliminate the inconsistency.

The difference GMM, introduced by Arellano and Bond (1991), eliminates the country specific fixed effects by first differencing instead of using the within transformation. In addition to this, first differencing constructs instruments for the regressors which are uncorrelated with the error term and highly correlated with the original regressors.

In the first differences, values of y lagged 2 time periods or more can be constructed as instruments.  First differencing transformation eliminates the fixed effects and makes the error term moving average of order 1.Thus, the value in period t = 1 is a good instrument for the difference between the values in period t = 2 and period t = 3. As t increases until T (t = 3, …, T), for each extra period we obtain an additional valid instrument, so that in period t = T, we have (T – 2) instruments. The total number of instrumental variables or moment conditions(m) is given by m=(T − 1)(T − 2)/2. This set of instruments is used to define the instrument a [(T – 2) m] matrix Z.

If the number of instruments is greater than the number of regressors, an optimal GMM has to be derived. The GMM chooses the minimum value of the parameter ∝ by employing a weighting matrix, W.  The GMM estimator is the value of ∝ closest to solving the sample moment conditions and Q is the measure of closeness to zero.

If the data generating process is assumed to meet the conditions for some kind of law of large 

numbers, we may assume that the sample moments converge in probability to their expectation. The variance of the GMM estimator depends on the choice of W, so we can extract the most information from the moment conditions by choosing an appropriate weighting matrix.

The consistency of the GMM estimator depends on the validity of the instruments and the assumption of no autocorrelation. To address these issues, we use some specification tests. The first test that we use is the Sargan test of identifying restrictions, which tests the overall validity of the instruments by analysing the sample analogue of the moment conditions used in the estimation process. The second test is for autocorrelation. The Wald test is used for testing the significance of explanatory variables and categorical variables.