# Degrees of Freedom in Simple Linear Regression

Why is the degrees of freedom for Simple Linear Regression $n-2$?

Specifically looking at a hypothesis $t$-test to determine if there's a relationship between the one independent variable, $x$, and the dependent variable $y$.

I would assume it would be $n-k$ where $k = 1$ in the case where we're using the slope in the simple linear regression equation as the parameter.

• Don't forget the intercept.
– whuber
Oct 31, 2017 at 22:02
• Because you have two parameters. Oct 31, 2017 at 22:03

You are correct that the degrees of freedom are $n-k$, however, in simple linear regression you estimate both a y-intercept and a slope, so $k=2$. Even though we generally don't worry about testing the intercept, it still uses up a degree of freedom, the slope would be very different and have a very different interpretation if we did not estimate an intercept along with the slope.
If our model is $y_{i} = \beta_{0} + \beta_{x}x_{i} + \varepsilon_{i}$, and we want to test, for example, $H_{0}: \beta_{x} = 0$, against the alternative $\beta_{x} \ne 0$... we require an estimate: $\hat{\beta}$.
$$\hat{\beta} = \frac{\sum_{i=1}^{n}{\left(x_{i} - \bar{x}\right)\times \left(y_{i} - \bar{y}\right)}}{\sum_{i=1}^{n}{\left(x_{i} - \bar{x}\right)^{2}}}$$
Notice that to obtain $\hat{\beta}$ we require an estimate of two quantities: $\mu_{x}$, and $\mu_{y}$, which we have in $\bar{x}$ and $\bar{y}$. We lose a degree of freedom for each of these estimates.
• @whuber I appreciate this, and Greg Snow's answer also... I think the "not very convincing" in my answer is from its coming "long way around" to the intercept which also depends on the estimates of $\bar{x}$ and $\bar{y}$. Oct 31, 2017 at 22:38