Statistical Inference

Linear Regression (OLS)

Please select the aspect of linear regression you would like practice calculating:

In linear regression, we are modeling the dependent variable using this model:

Y = β0 + β1X + ε

Here, Y is the dependent variable, X is the independent variable, β0 is the expected value of Y when X = 0 in the population, β1 is the effect of X on Y in the population, and ε is random variation unexplained by the model.

To perform statistical inference, we make the usual assumption that

ε ~ Normal(0, σ²)

Within that single assumption, there are several assumptions:

1. The relationship between the dependent (response, y-) variable and the independent (explanatory, x-) variable is linear
2. The residuals are from a Normal distribution.
3. The residuals have constant variance.

These assumptions are in addition to the usual two assumptions that the sample is representative of the target population and that the sample values are independent.

If any of these assumptions are not met, then this is not the appropriate procedure to use. Note, however, that ordinary least squares (OLS) is rather robust to violations of Normality and that adjustments can be made when the variances are not equal.