I am estimating the model $$E(Y|X) = Pr(Y=1|X) = \alpha_0 + (1 - \alpha_0 - \alpha_1)\phi(X'\beta),$$ where $\alpha_0$ and $\alpha_1$ are parameters, $\beta$ is a $p$-length vector of parameters, $X$ is a $p \times n$ matrix of data, $Y$ the dependent variable is a binary, and $\phi()$ is a probit model, so the cumulative distribution function of the standard normal distribution. To derive the expectation the assumption was made that the errors are normal and mean zero.
The source for the model is here (see equations 6 and 7), and per the paper I can estimate the model either via nonlinear least squares or maximum likelihood. I tried both approaches in R, using the
nls() function for nonlinear least squares and the
nlm() function for maximum likelihood. Experimentation suggests the results are very similar for my application, but
nls() is faster. Is there a reason to favor one approach over the other? How should I think about picking a method, e.g. do similar assumptions underly both approaches?
Any suggestions for thinking through the differences between these two approaches, or suggestions for relevant literature to consult would be greatly appreciated.