arch.univariate.GeneralizedError.loglikelihood¶
-
GeneralizedError.loglikelihood(parameters: Sequence[float] | ndarray | Series, resids: ndarray | DataFrame | Series, sigma2: ndarray | DataFrame | Series, individual: bool =
False) ndarray[source]¶ Computes the log-likelihood of assuming residuals are have a Generalized Error Distribution, conditional on the variance.
- Parameters:¶
- parameters: Sequence[float] | ndarray | Series¶
Shape parameter of the GED distribution
- resids: ndarray | DataFrame | Series¶
The residuals to use in the log-likelihood calculation
- sigma2: ndarray | DataFrame | Series¶
Conditional variances of resids
- individual: bool =
False¶ Flag indicating whether to return the vector of individual log likelihoods (True) or the sum (False)
- Returns:¶
ll – The log-likelihood
- Return type:¶
Notes
The log-likelihood of a single data point x is
\[\ln\nu-\ln c-\ln\Gamma(\frac{1}{\nu})-(1+\frac{1}{\nu})\ln2 -\frac{1}{2}\ln\sigma^{2} -\frac{1}{2}\left|\frac{x}{c\sigma}\right|^{\nu}\]where \(\Gamma\) is the gamma function and \(\ln c\) is
\[\ln c=\frac{1}{2}\left(\frac{-2}{\nu}\ln2+\ln\Gamma(\frac{1}{\nu}) -\ln\Gamma(\frac{3}{\nu})\right).\]