Applying the Bayesian Information Criterion for Stepwise Selection Algorithms on Time Series The title sounds rather complicated for fairly simple statistics issue.  I've created a factor model that tests adding additional factors by checking if the improvement in the mean squared error between the factor returns and the returns has improved by more than a given percent.
However, I understand from Monarcha (2009) and other statistics sources that this does not penalize enough for the risk of over-specification.  After some reading, I've been looking to apply the Schwarz Criterion which is unfortunately has a relatively poor wiki page.
My question is fairly simple and shows my lack of understanding, what is $\hat L$ in  $\mathrm{BIC} = {-2 \cdot \ln{\hat L} + k \cdot \ln(n)} $ for my problem?  Is it just the mean squared error?
 A: $\hat{L}$ is the likelihood: it's a number that comes from a Maximum Likelihood Estimation (MLE), which estimates the value of your model's parameters.
The Likelihood $L(\theta | X)$ says how likely the parameter values in vector $\theta$ fit the data in $X$. It is derived from the Bayes theorem:
$$L(\theta | X) P(X) = P(\theta) P(X | \theta) \quad \Rightarrow \quad L(\theta | X) = \frac{1}{P(X)} P(\theta) P(X | \theta)$$
$L(\theta | X)$ is the likelihood, $P(\theta)$ is the prior probability of the parameters and $P(X | \theta)$ is the posterior probability, $P(X)$ is the probability distribution you assume from which your data is drawn.
Considering $X$ as the known variable and $\theta$ as the unknown, the MLE method finds $\theta$ that maximizes the right hand side; to this purpose, $P(X)$ is ignored since it is independent of $\theta$ and it does not affect the maximization. Often, the distribution of $\theta$ is assumed uniform and the maximization is operated only on the posterior probability $P(X | \theta)$ since $P(\theta)$ becomes irrelevant for the maximization.
$P(X | \theta)$ is therefore the joint probability distribution of your data that you assume. Usually, it is assumed that the data is i.i.d., therefore the joint probability is just a product of probability densities and the MLE method finds $\theta$ that maximizes this product.
Coming back to your problem, you can compute the likelihood: your estimation method gives you the parameters of the residuals, therefore you just have to use the correct R function to compute it given the data and the parameters. Here is a very simple example.
To add more about your selection process, AIC (Akaike Information Criterion) is an alternative to BIC. As far as I know, there is no rule about when one criterion is better than the other. Wikipedia summarizes a comparison between the two, coming from some papers about them.
