# Source for BIC with residual sum of squares?

Wikipedia states that under certain circumstances, $$\mathrm{BIC}$$ can be calculated as: $$\mathrm{BIC} = n \ln(\mathrm{RSS}/n) + k \ln(n),$$

where $$\mathrm{RSS}$$ is the residual sum of squares. However, I was not able to find a reliable source for it. This question gives another source, however that source is just a copy of the Wikipedia article.

Yaffee and McGee state that

$$\mathrm{BIC} = n \ln(\mathrm{MSE}) + k \ln(n),$$

which IMHO would be identical to the Wikipedia definition, if they did not specify

$$\mathrm{MSE} = \frac{1}{n-k}(RSS).$$

That's a definition of $$\mathrm{MSE}$$ that I have never seen before (due to $$-k$$) and does not make a lot of sense to me.

Any references to reliable sources for BIC with residual sum of squares or an explanation of the strange definition of $$\mathrm{MSE}$$ is greatly appreciated.

Thank you.

• The $-k$ is the standard adjustment for creating an unbiased estimator of residual variance in a regression, and serves the same purpose (well is exactly the same thing) as the $-1$ in the usual calculation of variance, where you divide by $n-1$. I'm a bit surprised you haven't run into it before :) Mar 4, 2020 at 19:08
• @jbowman Of course! Ok, I withdraw my hasty statement that I have never seen it before. Of course I have, I just did not make the connection because I'm used to only $1$ being subtracted.
– Nos
Mar 4, 2020 at 19:29

The formulas given by Yaffee and McGee and Wikipedia are indeed identical, except for Bessel's correction ($$-k$$). Whether Bessel's correction has to be applied depends on the use case.