arch.univariate.Normal.loglikelihood

Normal.loglikelihood(parameters: Sequence[float] | ndarray | Series, resids: ndarray | DataFrame | Series, sigma2: ndarray | DataFrame | Series, individual: bool = False) float | ndarray[source]

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

Parameters:
parameters: Sequence[float] | ndarray | Series

The normal likelihood has no shape parameters. Empty since the standard normal has no shape parameters.

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:

float

Notes

The log-likelihood of a single data point x is

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