statsmodels.tsa.ar_model.AR.loglike

AR.loglike(params)

The loglikelihood of an AR(p) process

Parameters:params (array) – The fitted parameters of the AR model Returns:llf – The loglikelihood evaluated at params Return type:float

Notes

Contains constant term. If the model is fit by OLS then this returns the conditonal 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.