statsmodels.tsa.ar_model.AR.loglike

AR.loglike(params)

The loglikelihood of an AR(p) process.

Parameters

paramsndarray

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.