# Why is simulation used for probit choice probabilities?

As explained in this answer, the probability of a certain outcome in a Logit model can be written as $$P=\int_{\varepsilon=-\beta'x}^{\infty} f(\varepsilon)d\varepsilon\\ = 1- F(-\beta'x) = 1-\dfrac{1}{1+\exp(\beta'x)} = \dfrac{\exp(\beta'x)}{1+\exp(\beta'x)}$$ I think I understand the reasoning behind this; the logit model assumes that the error term is distributed logistically, so the CDF of the logistic distribution is used. The probit model assumes that the error term follows a normal distribution giving

$$\Pr(Y=1 \mid X) = \Phi(X'\beta) =\int_{\varepsilon=-\beta'x}^{\infty} f(\varepsilon)d\varepsilon = \Phi(x_{i}'\beta) = \int_{-\infty}^{x_{i}'\beta} \dfrac{1}{\sqrt{2\pi}}e^{-\dfrac{1}{2}t^{2}}dt$$ where $\Phi(x)$ is the CDF of the standard normal distribution. Now the referred answer, along with other sources, say that because these probabilities aren't closed-form, unlike those of the logit model, probit models require simulation.

But why use something complicated as simulation when there are a variety of methods that very accurately approximate $\Phi(x)$? It would seem to me that this would both be less complicated and more accurate.

• This is ancillary to your question, but neither the logit nor the probit can be properly said to have an error term. Mar 26, 2015 at 1:48
• @gung I agree, but there are people (economists) who think of logit/probit as a linear model for a latent propensity for success that includes an Gaussian/logisticly distributed error term, and we observe a success when the latent propensity is larger than 0. I suspect that that is what Max meant. This way of thinking is not to my taste: solid empirical research is about stuff we can see not about stuff we imagine. Nevertheless, it is in some disciplines a common way of thinking about these models. Mar 26, 2015 at 10:57
• @Maarten You're completely right in that suspicion - I was approaching it from a latent variable perspective.
– JH-
Mar 26, 2015 at 15:40

In both cases the model is $$Y\sim Be(p)$$ in which Y is the dependent variable. Here we model the effect of independent variables on Y through a link function. Logit and probit functions are used as the link function for logisic and probit regression models respectively $$logit(p)=X'\beta~~~~or~~~~~\Phi(p)=X'\beta$$ So it's easy to see if you want to simulate Y based on an observation of X, you have to compute p based on the $$X'\beta$$ In logistic regression it is straight forward as you have mentioned. But in probit regression model you have to compute $$p=\Phi^{-1}(X'\beta)$$ to be able to simulate Y. So it's up to you which method you like to use to compute this quantity. You can use numerical methods or interpolating from a table or any other method you know.