# Logit Regressions

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

For more information about Logit, see Wikipedia: Logit.

## Keep in Mind

- The beta coefficients from a logit 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 logit beta coefficient by 4.

# Implementations

## Gretl

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

## Python

There are a number of Python packages that can perform logit regressions but the most comprehensive is probably **statsmodels**. The code below is an example of how to use it.

```
# Install pandas and statsmodels using pip or conda, if you don't already have them.
import pandas as pd
import statsmodels.formula.api as smf
df = pd.read_csv('https://vincentarelbundock.github.io/Rdatasets/csv/datasets/mtcars.csv',
index_col=0)
# Specify the model, regressing vs on mpg and cyl
mod = smf.logit('vs ~ mpg + cyl', data=df)
# Fit the model
res = mod.fit()
# Look at the results
res.summary()
# Compute marginal effects
marg_effect = res.get_margeff(at='mean', method='dydx')
# Show marginal effects
marg_effect.summary()
```

## R

R can run a logit 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 logit
# Here we are predicting engine type using
# miles per gallon and number of cylinders as predictors
my_logit <- glm(vs ~ mpg + cyl, data = mtcars,
family = binomial(link = 'logit'))
# The family argument says we are working with binary data
# and using a logit link function (rather than, say, probit)
# The results
summary(my_logit)
# Marginal effects
logitmfx(vs ~ mpg + cyl, data = mtcars)
```

## Stata

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