# Hypothesis testing for linear regression We are typically interested in whether or not a predictor has a statistically significant association with an outcome. $ H_0: \beta_j = 0 \ \text{ vs } \ H_a: \beta_j \neq 0 $ [[Statistical properties of the OLS estimators]] tell us that they are normally distributed under the null hypothesis. They also give us an expression for the covariance of the OLS estimates: $ \text{Cov}(\beta_j \mid \mathbf{X}) = \sigma^2(X'X)^{-1} $ From this matrix, we can single out the standard error for $\hat{\beta}_j$ — $\text{se}(\hat{\beta}_j)$ — and use this to form a Wald-type statistic: $ T = \frac{\hat{\beta}_j - \beta_j}{\text{se}(\hat{\beta}_j)} = \frac{\hat{\beta}_j}{\text{se}(\hat{\beta}_j)} $ Since we need to estimate the standard error of the parameter, $T$ will have a $t$-distribution with $n-p$ degrees of freedom. ## Note - For [[Multiple linear regression|multiple linear regression]], you should acknowledge that this test is implicitly conditioned on the other covariates used in the model. A hypothesis test on for the same coefficient in another model using different covariates is technically different. --- # References - [[Applied Linear Regression#3. Multiple Linear Regression]]