What is the relationship between R² and Log Likelihood?

I estimated a non-linear model using the MATLAB function @fmincon which returns me a Log-likelihood value.

I also estimate a linear model (OLS) from which I can compute the R².

Here I need to compare the goodness of fit of the two models and see whether the non-linear model is statistically different from the linear model. I am thus wondering if there is a formula to express R² as a function of Log-Likelihood or the other way around.

• There is a simple relationship for certain kinds of nonlinear models: they are the ones fit using ordinary least squares. With other fitting methods, such as maximum likelihood, if there is a relationship it usually is only an approximate asymptotic one. What exactly is your model? – whuber Feb 26 '13 at 15:13
• I estimate a Markov-Switching model on fiscal policy in the United States and thus allow for structural breaks in my series (I break my regression in two parts : the one pertaining to, say, regime 1 and then the one pertaining to regime 2.) I fit my series with the coefficients of regime 1 when the model identifies regime 1 and so forth. I end up with a value of Log-Likelihood which I would like to compare to a simple R² for the linear case (when I do not allow for switches) and see whether the difference is large or not. I doubt that comparing corr(y,y_hat)^2 for the two models is enough. – Olivier Hubert Feb 27 '13 at 16:50