# Simple Linear Regression

Ordinary Least Squares (OLS) is a statistical method that produces a best-fit line between some outcome variable \(Y\) and any number of predictor variables \(X_1, X_2, X_3, ...\). These predictor variables may also be called independent variables or right-hand-side variables.

For more information about OLS, see Wikipedia: Ordinary Least Squares.

## Keep in Mind

- OLS assumes that you have specified a true linear relationship.
- OLS results are not guaranteed to have a causal interpretation. Just because OLS estimates a positive relationship between \(X_1\) and \(Y\) does not necessarily mean that an increase in \(X_1\) will cause \(Y\) to increase.
- OLS does
*not*require that your variables follow a normal distribution.

## Also Consider

- OLS standard errors assume that the modelâ€™s error term is IID, which may not be true. Consider whether your analysis should use heteroskedasticity-robust standard errors or cluster-robust standard errors.
- If your outcome variable is discrete or bounded, then OLS is by nature incorrectly specified. You may want to use probit or logit instead for a binary outcome variable, or ordered probit or ordered logit for an ordinal outcome variable.
- If the goal of your analysis is predicting the outcome variable and you have a very long list of predictor variables, you may want to consider using a method that will select a subset of your predictors. A common way to do this is a penalized regression method like LASSO.
- In many contexts, you may want to include interaction terms or polynomials in your regression equation.

# Implementations

## Gretl

```
# Load auto data
open https://github.com/LOST-STATS/lost-stats.github.io/blob/master/Data/auto.gdt
# Run OLS using the auto data, with mpg as the outcome variable
# and headroom, trunk, and weight as predictors
ols mpg const headroom trunk weight
```

## Matlab

```
% Load auto data
load('https://github.com/LOST-STATS/lost-stats.github.io/blob/master/Data/auto.mat')
% Run OLS using the auto data, with mpg as the outcome variable
% and headroom, trunk, and weight as predictors
intercept = ones(length(headroom),1);
X = [intercept headroom trunk weight];
[b,bint,r,rint,stats] = regress(mpg,X);
```

## Python

```
# Use 'pip install statsmodels' or 'conda install statsmodels'
# on the command line to install the statsmodels package.
# Import the relevant parts of the package:
import statsmodels.api as sm
import statsmodels.formula.api as smf
# Get the mtcars example dataset
mtcars = sm.datasets.get_rdataset("mtcars").data
# Fit OLS regression model to mtcars
ols = smf.ols(formula='mpg ~ cyl + hp + wt', data=mtcars).fit()
# Look at the OLS results
print(ols.summary())
```

## R

```
# Load Data
# data(mtcars) ## Optional: automatically loaded anyway
# Run OLS using the mtcars data, with mpg as the outcome variable
# and cyl, hp, and wt as predictors
olsmodel <- lm(mpg ~ cyl + hp + wt, data = mtcars)
# Look at the results
summary(olsmodel)
```

## SAS

```
/* Load Data */
proc import datafile="C:mtcars.dbf"
out=fromr dbms=dbf;
run;
/* OLS regression */
proc reg;
model mpg = cyl hp wt;
run;
```

## Stata

```
* Load auto data
sysuse https://github.com/LOST-STATS/lost-stats.github.io/blob/master/Data/auto.dta
* Run OLS using the auto data, with mpg as the outcome variable
* and headroom, trunk, and weight as predictors
regress mpg headroom trunk weight
```