Differences between formulas for AIC and BIC

Question

I have a question regarding the information criteria AIC and BIC:

I found different formulas for the AIC/BIC, the common ones including the likelihood $\mathcal{L}$ are $$AIC = 2K - 2 ln(\mathcal{L})\quad\text{and}\quad BIC =K\;ln(n)- 2 ln(\mathcal{L}).$$ In Diebold's "Elements of Forecasting" and in Greene's "Econometric Analysis" I found some very similar formulations with MSE (or RSS), $$AIC = ln(\frac{RSS}{n}) + \frac{2K}{n} \quad\text{and}\quad BIC = ln(\frac{RSS}{n}) + \frac{K \;ln(n)}{n}.$$ Aside from the fact that values obtained by one of the former formulas can't be compared to those of the latter ones: In which way do they differ or are they all equivalent? Do they all assume i.i.d. normal distribution, or are there different assumptions underlying these formulas?

Answer 1

If you assume normally distributed errors, then minimizing the MSE is equivalent to maximizing the likelihood function. Your later expression for AIC and BIC are therefore spesial cases of the general formula (up to a propotional constant):

$$\text{AIC} = 2K - 2 \ln(\mathcal{L})\quad\text{and}\quad \text{BIC} =K\;\ln(n)- 2 \ln(\mathcal{L}).$$

If you assume a different distribution for your data, then the MSE estimates will no longer be the same as the maximum likelihood estimates, and you can no longer use the MSE in place of $\mathcal{L}$, since it is not the likelihood of your model. See this post for more information about the use of AIC.