Test a coefficient and explain t-distribution

In OLS, determine whether feature j is relevant. a) State $H_0: \beta_j = 0$ against $H_1: \beta_j \neq 0$, form the t-statistic $t_j = \hat{b}_j / \mathrm{se}(\hat{b}_j)$, and give the exact formula for $\mathrm{se}(\hat{b}_j)$ along with the degrees of freedom. b) Prove or justify why $t_j$ follows a t distribution under the classical linear model (normal errors, full rank, independence): that is, a normal numerator divided by the square root of an independent scaled $\chi^2$ variance estimate. c) Provide intuition for why estimating $\sigma$ inflates uncertainty compared with a Z-test where $\sigma$ is known. d) Describe how heteroskedasticity or clustering changes the test (HC/cluster-robust standard errors) and what happens to the reference distribution.

Overview: This question assesses a candidate's ability in statistical inference for linear regression—specifically hypothesis tests for OLS coefficients, forming t-statistics, estimating standard errors, and the effect of estimated error variance—within the Statistics & Math area for Data Scientist positions.