Calculation of Pseudo-R2 in Probit Out-of-Sample Forecasting

[Ciaran]

To whom it may concern

I have a question regarding Probit
models. For my project, my dependent is a binary variable equalling 1 if a
recession, and 0 otherwise. My explanatory variables are the yield spread,
lagged k, and the kth lag of the dependent variable.

What I am trying to do is an out-of-sample forecasting analysis. My
original regression ran the probit model for the whole sample. But to
correct for the fact that, when predicting a recession in 1980 I can't use
data from, say, 2009, I perform out-of-sample forecasting. The procedure is
running a regression from the start of my sample, q1 1971, to q41988, and
then use these coefficient estimates to get predicted value k periods ahead
of q4 1988. I record this fitted forecast value. I then run regression q1
1971 to q1989, and then use these coefficient estimates to get predicted
value k periods ahead from q1 1989, and so on...

I want to calculate the pseudo R2 for this out-of-sample forecasting.
Getting the unconstrained log likelihood is fine because I can calculate it
from the fitted probability values. However, how do you get the constrained
log likelihood? Is this the exact same procedure as unconstrained problem,
except that I just use the intercept, or do I use the constrained log
likelihood from the in-sample regression?

The reason I am confused is because i says in textbooks that, with out-of-sample forecasting, it is possible to get negative pseudo R2, which I can't see as being possible when I estimate constrained log likelihood the same way as unconstrained likelihood

Any help would be greatly appreciated

Kind Regards
Ciaran