# Probit Regressions

A Probit regression is a statistical method for a best-fit line between a binary [0/1] outcome variable \(Y\) and any number of independent variables. Probit regressions follow a standard normal probability distribution and the predicted values are bounded between 0 and 1.

For more information about Probit, see Wikipedia: Probit.

## Keep in Mind

- The beta coefficients from a probit model are maximum likelihood estimations. They are not the marginal effect, as you would see in an OLS estimation. So you cannot interpret the beta coefficient as a marginal effect of \(X\) on \(Y\).
- To obtain the marginal effect, you need to perform a post-estimation command to discover the marginal effect. In general, you can ‘eye-ball’ the marginal effect by dividing the probit beta coefficient by 2.5.

# Implementations

## Gretl

```
# Load auto data
open auto.gdt
# Run probit using the auto data, with mpg as the outcome variable
# and headroom, trunk, and weight as predictors
probit mpg const headroom trunk weight
```

## Python

The **statsmodels** package has methods that can perform probit regressions.

```
# Use pip or conda to install pandas and statsmodels
import pandas as pd
import statsmodels.formula.api as smf
# Read in the data
df = pd.read_csv('https://vincentarelbundock.github.io/Rdatasets/csv/datasets/mtcars.csv',
index_col=0)
# Specify the model
mod = smf.probit('vs ~ mpg + cyl', data=df)
# Fit the model
res = mod.fit()
# Look at the results
res.summary()
# Compute marginal effects
marge_effect = res.get_margeff(at='mean', method='dydx')
# Show marginal effects
marge_effect.summary()
```

## R

R can run a probit regression using the `glm()`

function. However, to get marginal effects you will need to calculate them by hand or use a package. We will use the **mfx** package, although the **margins** package is another good option, which produces tidy model output.

```
# If necessary, install the mfx package
# install.packages('mfx')
# mfx is only needed for the marginal effect, not the regression itself
library(mfx)
# Load mtcars data
data(mtcars)
# Use the glm() function to run probit
# Here we are predicting engine type using
# miles per gallon and number of cylinders as predictors
my_probit <- glm(vs ~ mpg + cyl, data = mtcars,
family = binomial(link = 'probit'))
# The family argument says we are working with binary data
# and using a probit link function (rather than, say, logit)
# The results
summary(my_probit)
# Marginal effects
probitmfx(vs ~ mpg + cyl, data = mtcars)
```

## Stata

```
* Load auto data
sysuse auto.dta
* Probi Estimation
probit foreign mpg weight headroom trunk
* Recover the Marginal Effects (Beta Coefficient in OLS)
margins, dydx(*)
```