linearmodels.iv.model.IVGMMCUE.j

IVGMMCUE.j(params, x, y, z)[source]

Optimization target

Parameters
paramsndarray

Parameter vector (nvar)

xndarray

Regressor matrix (nobs by nvar)

yndarray

Regressand matrix (nobs by 1)

zndarray

Instrument matrix (nobs by ninstr)

Returns
float

GMM objective function, also known as the J statistic

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 \(\beta\).

Return type

float