Q:

Why is my probit analysis resulting in numbers outside 0 and 1?

I am doing a probit on a binary dependent variable.  The binary dependent variable has observed outcomes of 227 successes and 1704 failures.  If I run predict after the probit I get probabilities that make sense for each set of observations, but I want to be able to make sense of the beta coefficients that I get from a straight probit analysis. The same thing happens with logit.

A:

I think Dason's reply is appropriate: your title does not fit your question. If you estimate a probit/logit model, $P(Y=1|X,\beta)$, by the plugg-in solution, $P(Y=1|X,\hat\beta)$ your probability is always between 0 and 1, no matter what the values of the covariates $X$ are. If you find a probability outside $(0,1)$, you are not using a probit/logit model. Now, if you ask about the meaning of $\beta$, this is indeed a real number. For instance, in the logit model, since
$$
\frac{P(Y=1|X,\beta)}{P(Y=0|X,\beta)} = \exp\{ \beta^t X \}
$$
the coefficients of $\beta$ can be explained in terms of log-odds-ratios:
$$
\log\left( \frac{P(Y=1|X,\beta)}{P(Y=0|X,\beta)} \right) = \beta^t X
$$
so, when considering covariate $x_1$ for instance,
$$
\log\left( \frac{P(Y=1|x_1=2,X_{-1},\beta)}{P(Y=0|x_1=2,X_{-1},\beta)} \right) 
- \log\left( \frac{P(Y=1|x_1=1,X_{-1},\beta)}{P(Y=0|x_1=1,X_{-1},\beta)} \right)
= \beta_1 
$$
which means that the coefficient $\beta_1$ of $x_1$ is the amount the log-odds-ratio changes when the covariate $x_1$ is changed by one unit.

