# AIC only applicable to maximum likelihood fit (not least squares)?

When I read about AIC I see that it is calculated for maximum likelihood model estimation. For example, R function arima0 estimated by method=ML will give AIC value in model summary; but if I estimate the model by method=CSS the summary will not give me an AIC value. So my questions are:

1. Is AIC computable only for maximum likelihood parameters estimation?
2. If the answer for first question is "No", then how can I calculate AIC for CSS case? arima0 gives "partial log likelihood". Can I pass this to standard AIC formula to calculate AIC "by hand"?

Please do not advice to use ML estimation because I can not do this (have my reasons). I need strict academic explanation for my questions.

## 1 Answer

1. Strictly speaking, the AIC only works with MLEs, by construction (Akaike, 1974).

2. Since both MLEs and LSEs (i.e. estimates by OLS) are consistent in your specific model, you can use LSEs as an approximation of MLEs for large samples. Since both estimators are consistent, $L(\hat{\theta}_{MLE})\approx L(\hat{\theta}_{LSE})$ for large samples. Moreover,$L(\hat{\theta}_{MLE})$ and $L(\hat{\theta}_{LSE})$ have the same limit (continuous mapping theorem). So, you can use it as an approximation, but they are not the same. Some authors have noticed that the two estimators tend to be similar.

References:

• Akaike, Hirotugu. "A new look at the statistical model identification." IEEE transactions on automatic control 19.6 (1974): 716-723.