arch.univariate.SkewStudent.loglikelihood¶
-
SkewStudent.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 standardized (to have unit variance) Skew Student’s t distribution, conditional on the variance.
- Parameters:¶
- parameters: Sequence[float] | ndarray | Series¶
Shape parameter of the skew-t 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
\[\ln\left[\frac{bc}{\sigma}\left(1+\frac{1}{\eta-2} \left(\frac{a+bx/\sigma} {1+sgn(x/\sigma+a/b)\lambda}\right)^{2}\right) ^{-\left(\eta+1\right)/2}\right],\]where \(2<\eta<\infty\), and \(-1<\lambda<1\). The constants \(a\), \(b\), and \(c\) are given by
\[a=4\lambda c\frac{\eta-2}{\eta-1}, \quad b^{2}=1+3\lambda^{2}-a^{2}, \quad c=\frac{\Gamma\left(\frac{\eta+1}{2}\right)} {\sqrt{\pi\left(\eta-2\right)} \Gamma\left(\frac{\eta}{2}\right)},\]and \(\Gamma\) is the gamma function.