linearmodels.iv.covariance.KernelCovariance¶

class linearmodels.iv.covariance.KernelCovariance(x: ndarray, y: ndarray, z: ndarray, params: ndarray, kernel: str = 'bartlett', bandwidth: int | float | None = None, debiased: bool = False, kappa: int | float = 1)[source]¶

Kernel weighted (HAC) covariance estimation

Parameters:¶

x: ndarray¶

Model regressors (nobs by nvar)

y: ndarray¶

Series ,modeled (nobs by 1)

z: ndarray¶

Instruments used for endogenous regressors (nobs by ninstr)

params: ndarray¶

Estimated model parameters (nvar by 1)

kernel: str = 'bartlett'¶

Kernel name. Supported kernels are:

”bartlett”, “newey-west” - Triangular kernel
”qs”, “quadratic-spectral”, “andrews” - Quadratic spectral kernel
”parzen”, “gallant” - Parzen’s kernel;

bandwidth: int | float | None = None¶

Non-negative bandwidth to use with kernel. If None, automatic bandwidth selection is used.

debiased: bool = False¶

Flag indicating whether to use a small-sample adjustment

kappa: int | float = 1¶

Value of kappa in k-class estimator

Notes

Covariance is estimated using

\[n^{-1} V^{-1} \hat{S} V^{-1}\]

where

\[\begin{split}\hat{S}_0 & = n^{-1} \sum_{i=1}^{n} \hat{\epsilon}^2_i \hat{x}_i^{\prime} \hat{x}_{i} \\ \hat{S}_j & = n^{-1} \sum_{i=1}^{n-j} \hat{\epsilon}_i\hat{\epsilon}_{i+j} (\hat{x}_i^{\prime} \hat{x}_{i+j} + \hat{x}_{i+j}^{\prime} \hat{x}_{i}) \\ \hat{S} & = \sum_{i=0}^{bw} K(i, bw) \hat{S}_i\end{split}\]

where \(\hat{\gamma}=(Z'Z)^{-1}(Z'X)\), \(\hat{x}_i = z_i\hat{\gamma}\) and \(K(i,bw)\) is a weight that depends on the kernel. If debiased is true, then \(S\) is scaled by n / (n-k).

\[V = n^{-1} X'Z(Z'Z)^{-1}Z'X\]

where \(X\) is the matrix of variables included in the model and \(Z\) is the matrix of instruments, including exogenous regressors.

`config`
`cov`	Covariance of estimated parameters
`debiased`	Flag indicating if covariance is debiased
`s`	HAC score covariance estimate
`s2`	Estimated variance of residuals.