linearmodels.panel.covariance.ACCovariance¶
-
class linearmodels.panel.covariance.ACCovariance(y: ndarray, x: ndarray, params: ndarray, entity_ids: ndarray, time_ids: ndarray, *, debiased: bool =
False, extra_df: int =0, kernel: str | None =None, bandwidth: float | None =None)[source]¶ Autocorrelation robust covariance estimation
- Parameters:¶
- y: ndarray¶
(entity x time) by 1 stacked array of dependent
- x: ndarray¶
(entity x time) by variables stacked array of exogenous
- params: ndarray¶
variables by 1 array of estimated model parameters
- entity_ids: ndarray¶
(entity x time) by 1 stacked array of entity ids
- time_ids: ndarray¶
(entity x time) by 1 stacked array of time ids
- debiased: bool =
False¶ Flag indicating whether to debias the estimator
- extra_df: int =
0¶ Additional degrees of freedom consumed by models beyond the number of columns in x, e.g., fixed effects. Covariance estimators are always adjusted for extra_df irrespective of the setting of debiased
- kernel: str | None =
None¶ Name of one of the supported kernels. If None, uses the Newey-West kernel.
- bandwidth: float | None =
None¶ Non-negative integer to use as bandwidth. If not provided a rule-of- thumb value is used.
Notes
Estimator is robust to autocorrelation but not cross-sectional correlation.
Supported kernels:
“bartlett”, “newey-west” - Bartlett’s kernel
“quadratic-spectral”, “qs”, “andrews” - Quadratic-Spectral Kernel
“parzen”, “gallant” - Parzen kernel
Bandwidth is set to the common value for the Bartlett kernel if not provided.
The estimator of the covariance is
\[n^{-1}\hat{\Sigma}_{xx}^{-1}\hat{S}\hat{\Sigma}_{xx}^{-1}\]where
\[\hat{\Sigma}_{xx} = n^{-1}X'X\]and
\[\begin{split}\xi_t & = \epsilon_{it} x_{it} \\ \hat{S} & = n / (N(n-df)) \sum_{i=1}^N S_i \\ \hat{S}_i & = \sum_{j=0}^{bw} K(j, bw) \hat{S}_{ij} \\ \hat{S}_{i0} & = \sum_{t=1}^{T} \xi'_{it} \xi_{it} \\ \hat{S}_{ij} & = \sum_{t=1}^{T-j} \xi'_{it} \xi_{it+j} + \xi'_{it+j} \xi_{it}\end{split}\]where df is
extra_dfand n-df is replace by n-df-k ifdebiasedisTrue. \(K(i, bw)\) is the kernel weighting function.Methods
Covariance calculation deferred until executed
Properties
Estimated covariance
Model residuals
Covariance estimator name
Error variance