$R^2$ and Model Fit

Once we have estimated a regression model, a natural question is: how well does it fit the data?

We begin by recapping fitted values and using them to describe the different sources of variation in the dependent variable. In particular, we can split the overall variation in the data into a part explained by our model and a residual part left unexplained.

This leads to one of the most common measures in linear regression: $R^2$, or the coefficient of determination. Roughly speaking, $R^2$ tells us what proportion of the variation in the dependent variable has been explained by our fitted model.

A higher $R^2$ therefore indicates a closer fit to the observed data. However, this does not by itself tell us that our model is a good one in every sense – or that we have correctly estimated the relationship we are actually interested in.