Differences between formulas for AIC and BIC

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?

• Is there a typo in your formula? Should not RRS in fact be RSS? Commented Aug 22, 2020 at 10:36
• Fixed that, thank you! Commented Aug 22, 2020 at 10:41

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