linearmodels.iv.model.IVGMMCUE.j¶

IVGMMCUE.j(params: ndarray, x: ndarray, y: ndarray, z: ndarray) → float[source]¶

Optimization target

Parameters:¶

params: ndarray¶

Parameter vector (nvar)

x: ndarray¶

Regressor matrix (nobs by nvar)

y: ndarray¶

Regressand matrix (nobs by 1)

z: ndarray¶

Instrument matrix (nobs by ninstr)

Returns:¶

GMM objective function, also known as the J statistic

Return type:¶

float

Notes

The GMM objective function is defined as

\[J(\beta) = \bar{g}(\beta)'W(\beta)^{-1}\bar{g}(\beta)\]

where \(\bar{g}(\beta)\) is the average of the moment conditions, \(z_i \hat{\epsilon}_i\), where \(\hat{\epsilon}_i = y_i - x_i\beta\). The weighting matrix is some estimator of the long-run variance of the moment conditions.

Unlike tradition GMM, the weighting matrix is simultaneously computed with the moment conditions, and so has explicit dependence on \(\beta\).