Information Criterion to compare different linear models

#1
Dear Community,

In order to perform a price forecast, I built four linear models:1-3 have one independent variable, 4 has two independent variables. n = 28 in all cases
1. ad. R² = 0.91, AIC = -27, BIC = -25
2. ad. R² = 0.95, AIC = -37, BIC = -34
3. ad. R² = 0.97, AIC = -70, BIC = -68
4. ad. R² = 0.95, AIC = -35, BIC = -32

Which statements can be derived from these results?

As I understood the issue so far, the lower AIC/BIC score the better. However, both information criteria penalize the usage of more independent variables as the risk of overfitting increases.
Therefore, I would drive that model 3 is the best model as adjusted R² is high as well as AIC and BIC are the lowest of all four models. Further I would prefer 4 over 2 as ad. R² is the same and 4 as lower AIC/BIC scores even though it has two independent variables in comparison to one in 2. Model 1 would be the worst performing model.

Is it possible to make these statements?

Thank you very much for your clearification.

Best,

Anton
 
#2
Hi Community,

I just saw that my post has over 100 views. Has somebody a short comment on my post? I would be really intested whever I understood the theory, especially since I am not quite sure why the BIC values are negative as the formula is as follows:
BIC = -2 * ln(likelihood) + ln(n)*k
k = number of parameters estimated
n = observation

ln(likelihood) has to be in my oppinion negative, as the likelihood has values from 0 to 1, resulting in a negative value for ln(likelihood). -2 * ln(likelihood) therefore has to be positive; the other term ln(n)*k has to be positive as well, so currently I cannot understand why I received negative BIC scores.

I would highly appreciate your comments!

Best,

Anton