linearmodels.iv.covariance.KernelCovariance

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

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

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.