arch.univariate.Normal.loglikelihood

Normal.loglikelihood(parameters, resids, sigma2, individual=False)[source]

Computes the log-likelihood of assuming residuals are normally distributed, conditional on the variance

Parameters
  • parameters (ndarray) – The normal likelihood has no shape parameters. Empty since the standard normal has no shape parameters.

  • 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

\[\ln f\left(x\right)=-\frac{1}{2}\left(\ln2\pi+\ln\sigma^{2} +\frac{x^{2}}{\sigma^{2}}\right)\]