arch.univariate.SkewStudent.loglikelihood

SkewStudent.loglikelihood(parameters, resids, sigma2, individual=False)[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 (ndarray) – Shape parameter of the skew-t distribution

  • resids (ndarray) – The residuals to use in the log-likelihood calculation

  • sigma2 (ndarray) – Conditional variances of resids

  • individual (bool, optional) – Flag indicating whether to return the vector of individual log likelihoods (True) or the sum (False)

Returns

ll – The log-likelihood

Return type

float

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.