statsmodels.tsa.exponential_smoothing.ets.ETSModel.loglike¶

ETSModel.loglike(params, **kwargs)[source]¶

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]:

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

with

$$s^2=\frac{1}{n}\sum _{t=1}^n\frac{({\hat{y}}_t-y_t{)}^2}{k_t}$$

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