arch.univariate.StudentsT.loglikelihood¶
-
StudentsT.
loglikelihood
(parameters, resids, sigma2, individual=False)[source]¶ Computes the log-likelihood of assuming residuals are have a standardized (to have unit variance) Student's t distribution, conditional on the variance.
- Parameters
parameters (ndarray) -- Shape parameter of the 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
Notes
The log-likelihood of a single data point x is
\[\ln\Gamma\left(\frac{\nu+1}{2}\right) -\ln\Gamma\left(\frac{\nu}{2}\right) -\frac{1}{2}\ln(\pi\left(\nu-2\right)\sigma^{2}) -\frac{\nu+1}{2}\ln(1+x^{2}/(\sigma^{2}(\nu-2)))\]where \(\Gamma\) is the gamma function.