linearmodels.system.covariance.KernelCovariance¶

class linearmodels.system.covariance.KernelCovariance(x: list[ndarray], eps: ndarray[tuple[int, ...], dtype[float64]], sigma: ndarray[tuple[int, ...], dtype[float64]], full_sigma: ndarray[tuple[int, ...], dtype[float64]], *, gls: bool = False, debiased: bool = False, constraints: LinearConstraint | None = None, kernel: str = 'bartlett', bandwidth: float | None = None)[source]¶

Kernel (HAC) covariance estimation for system regression.

Parameters:¶

x: list[ndarray]¶

Regressors of each equation, with shape (nobs, k_i) for equation i.

eps: ndarray[tuple[int, ...], dtype[float64]]¶

Model residuals, with shape (nobs, neq).

sigma: ndarray[tuple[int, ...], dtype[float64]]¶

Covariance of the residuals of the equations that is used to weight the equations if gls is True, with shape (neq, neq).

full_sigma: ndarray[tuple[int, ...], dtype[float64]]¶

Covariance of the residuals of the equations at the final estimates, with shape (neq, neq). It differs from sigma if the weights are not the covariance of the final residuals, e.g., if the model was given sigma or did not iterate. It is used to estimate the covariance of the scores if gls is True.

gls: bool = False¶

Flag indicating to compute the GLS covariance estimator. If False, assume OLS was used. The default is False.

debiased: bool = False¶

Flag indicating to apply a small sample adjustment. The default is False.

constraints: LinearConstraint | None = None¶

Constraints used in estimation. The default is no constraints.

kernel: str = 'bartlett'¶

Name of the kernel. The default is “bartlett”. The names are not case sensitive. Supported kernels are:

• ”bartlett”, “newey-west” - Bartlett’s kernel

• ”parzen”, “gallant” - Parzen’s kernel

• ”qs”, “quadratic-spectral”, “andrews” - Quadratic spectral kernel

bandwidth: float | None = None¶

Non-negative bandwidth to use for the kernel. If not provided, the optimal bandwidth will be estimated. It needs at least 4 observations with the Bartlett kernel, 5 with the Parzen kernel and 6 with the quadratic spectral kernel, and a ValueError is raised for fewer when the covariance is first used. See kernel_optimal_bandwidth().

See also

linearmodels.iv.covariance.kernel_weight_bartlett

Weights of the Bartlett kernel.

linearmodels.iv.covariance.kernel_weight_parzen

Weights of the Parzen kernel.

linearmodels.iv.covariance.kernel_weight_quadratic_spectral

Weights of the quadratic spectral kernel.

Notes

If GLS is used, the covariance is estimated by

\[(X'\Omega^{-1}X)^{-1}\tilde{S}(X'\Omega^{-1}X)^{-1}\]

where X is a block diagonal matrix of exogenous variables and where \(\tilde{S}\) is a estimator of the covariance of the model scores based on the model residuals and the weighted X matrix \(\Omega^{-1/2}X\).

When GLS is not used, the covariance is estimated by

\[(X'X)^{-1}\hat{S}(X'X)^{-1}\]

where \(\hat{S}\) is a estimator of the covariance of the model scores.

Changed in version 7.1: The optimal bandwidth raises a ValueError if there are fewer observations than the kernel needs, and if it is not finite. Earlier versions raised “cannot convert float infinity to integer” (an OverflowError) or “cannot convert float NaN to integer” for some data, and used a bandwidth of nobs - 1, or another bandwidth, for other data.

Methods

Properties

bandwidth	Bandwidth used in estimation.
cov	Covariance of the parameter estimates.
cov_config	Configuration of the covariance estimator.
kernel	Kernel used in estimation.
sigma	Covariance of the residuals of the equations.