linearmodels.iv.model.IVGMMCUE.j¶
- IVGMMCUE.j(params: ndarray[tuple[int, ...], dtype[float64]], x: ndarray[tuple[int, ...], dtype[float64]], y: ndarray[tuple[int, ...], dtype[float64]], z: ndarray[tuple[int, ...], dtype[float64]]) float[source]¶
Optimization target
- Parameters:¶
- params: ndarray[tuple[int, ...], dtype[float64]]¶
Parameter vector (nvar)
- x: ndarray[tuple[int, ...], dtype[float64]]¶
Regressor matrix (nobs by nvar)
- y: ndarray[tuple[int, ...], dtype[float64]]¶
Regressand matrix (nobs by 1)
- z: ndarray[tuple[int, ...], dtype[float64]]¶
Instrument matrix (nobs by ninstr)
- Returns:¶
GMM objective function, also known as the J statistic
- Return type:¶
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 \(\beta\).