linearmodels.iv.covariance.KernelCovariance

class KernelCovariance(x, y, z, params, kernel='bartlett', bandwidth=None, debiased=False, kappa=1)

Kernel weighted (HAC) covariance estimation

Parameters

xndarray

Model regressors (nobs by nvar)

yndarray

Series ,modeled (nobs by 1)

zndarray

Instruments used for endogenous regressors (nobs by ninstr)

paramsndarray

Estimated model parameters (nvar by 1)

kernelstr

Kernel name. Supported kernels are:

• “bartlett”, “newey-west” - Triangular kernel

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

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

bandwidth{int, None}

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

debiasedbool

Flag indicating whether to use a small-sample adjustment

kappafloat

Value of kappa in k-class estimator

See also

linearmodels.iv.covariance.kernel_weight_bartlett

linearmodels.iv.covariance.kernel_weight_parzen

linearmodels.iv.covariance.kernel_weight_quadratic_spectral

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.

Attributes

config

cov

Covariance of estimated parameters

debiased

Flag indicating if covariance is debiased

s

HAC score covariance estimate

s2

Estimated variance of residuals.

Methods

Properties

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