arch.univariate.Normal.loglikelihood

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

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

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