statsmodels.tsa.ar_model.AR.loglike

AR.loglike(params)[source]

The loglikelihood of an AR(p) process.

Parameters:

params : ndarray

The fitted parameters of the AR model.

Returns:

float

The loglikelihood evaluated at params.

Notes

Contains constant term. If the model is fit by OLS then this returns the conditional maximum likelihood.

\[\frac{\left(n-p\right)}{2}\left(\log\left(2\pi\right) +\log\left(\sigma^{2}\right)\right) -\frac{1}{\sigma^{2}}\sum_{i}\epsilon_{i}^{2}\]

If it is fit by MLE then the (exact) unconditional maximum likelihood is returned.

\[-\frac{n}{2}log\left(2\pi\right) -\frac{n}{2}\log\left(\sigma^{2}\right) +\frac{1}{2}\left|V_{p}^{-1}\right| -\frac{1}{2\sigma^{2}}\left(y_{p} -\mu_{p}\right)^{\prime}V_{p}^{-1}\left(y_{p}-\mu_{p}\right) -\frac{1}{2\sigma^{2}}\sum_{t=p+1}^{n}\epsilon_{i}^{2}\]

where

\(\mu_{p}\) is a (p x 1) vector with each element equal to the mean of the AR process and \(\sigma^{2}V_{p}\) is the (p x p) variance-covariance matrix of the first p observations.