Specification tests under the GMM in PROC PANEL follow Arellano and Bond (1991) very generally. The first test available is a Sargan/Hansen test of over-identification. The test for a one-step estimation is constructed as

where is the stacked error term (of the differenced equation and level equation).

When the robust weighting matrix is used, the test statistic is computed as

This definition of the Sargan test is used for all iterated estimations. The Sargan test is distributed as a with degrees of freedom equal to the number of moment conditions minus the number of parameters.

In addition to the Sargan test, PROC PANEL tests for autocorrelation in the residuals. These tests are distributed as standard normal. PROC PANEL tests the hypothesis that the autocorrelation of the th lag is significant.

Define as the lag of the differenced error, with zero padding for the missing values generated. Symbolically,

You define the constant as

You next define the constant as

Note that the choice of is dependent on the stage of estimation. If the estimation is first stage, then you would use the matrix with twos along the main diagonal, and minus ones along the primary subdiagonals. In a robust estimation or multi-step estimation, this matrix would be formed from the outer product of the residuals (from the previous step).

Define the constant as

The matrix is defined as

The constant is defined as

Using the four quantities, the test for autoregressive structure in the differenced residual is

The statistic is distributed as a normal random variable with mean zero and standard deviation of one.