Trying to understand log-likelihood estimation for exponential smoothing models in R forecast function ets()

Question

I am doing work on AIC comparisons. For this purpose, I am trying to understand how log-likelihood is calculated for exponential smoothing models (ETS models) in different R packages.

In particular, ets in R forecast and es in R smooth appear to give very different log-likelihood values for the same model fitted to the same data set, as follows.

# data simulation y as ETS(A,N,N)
set.seed(546)
n = 300
y <- smooth::sim.es("ANN", obs = n)

# fit E(A,N,N) and ARIMA(0,1,1) models to simulated data with R packages smooth and forecast
es_smooth_model <- smooth::es(y$data, model = "ANN", initial ="optimal")
es_forecast_model <- forecast::ets(y$data, model = "ANN", lambda = NULL)

Not surprisingly, both models give near identical fitted values, residuals, forecasts, coefficients, error variances, etc.

However, their log-likelihood estimates give very different values

# comparing loglikelihoods
es_smooth_model$logLik    # -1215.57
es_forecast_model$loglik  # -1645.456

Manual calculation for both models (assuming Gaussian error distributions) gives:

#manual loglikelihoods:

# es_smooth_model :         -1215.578
- 0.5 * (n * (log(2) + log(pi) + log(es_smooth_model$s2)) +
  1/ es_smooth_model$s2 * sum(es_smooth_model$residuals^2))

# es_smooth_model :         -1215.574
- 0.5 * (n * (log(2) + log(pi) + log(es_forecast_model$sigma2)) +
           1/ es_forecast_model$sigma2 * sum(es_forecast_model$residuals^2))

Can somebody help me to understand why forecast::ets gives a different log-likelihood value? And which of the two log-likelihoods is the correct value?

Answer 1

For this model, the log likelihood is given by $$-\frac{n}{2}\log(2\pi\sigma^2) - \frac12 \sum_{t=1}^n e_t^2 / \sigma^2.$$ Taking the partial derivative with respect to $\sigma^2$ and setting it to zero gives the MLE of $\sigma^2$ to be $$\hat{\sigma}^2 = \frac1n\sum_{t=1}^n e_t^2$$ Substituting $\hat\sigma^2$ for $\sigma^2$ in the log likelihood gives the concentrated log likelihood: $$-\frac{n}{2}\log\left(\frac{2\pi e}n\right) -\frac{n}{2} \log\left( \sum_{t=1}^n e_t^2\right) = c_n -\frac{n}{2} \log\left( \sum_{t=1}^n e_t^2\right), $$ where $c_n = -\frac{n}{2}\log(2\pi e/n)$, which does not depend on the data. The forecast package drops $c_n$ as it makes no difference to the parameter estimates, or the calculation of the AIC, and saves a small amount of computation.

Reference: Hyndman et al (2008, p69).