statsmodels.tsa.exponential_smoothing.ets.ETSModel.loglike

ETSModel.loglike(params, **kwargs)

Log-likelihood of model.

Parameters

paramsnp.ndarray of np.float

Model parameters: (alpha, beta, gamma, phi, l[-1], b[-1], s[-1], …, s[-m])

Notes

The log-likelihood of a exponential smoothing model is [1]:

\[l(\theta, x_0|y) = - \frac{n}{2}(\log(2\pi s^2) + 1) - \sum\limits_{t=1}^n \log(k_t)\]

with

\[s^2 = \frac{1}{n}\sum\limits_{t=1}^n \frac{\hat{y}_t - y_t}{k_t}\]

where \(k_t = 1\) for the additive error model and \(k_t = y_t\) for the multiplicative error model.

References

1

J. K. Ord, A. B. Koehler R. D. and Snyder (1997). Estimation and Prediction for a Class of Dynamic Nonlinear Statistical Models. Journal of the American Statistical Association, 92(440), 1621-1629