Test a coefficient and explain t-distribution

In OLS, test whether feature j is relevant. a) State H0: β_j = 0 versus H1: β_j ≠ 0 and construct the t‑statistic t_j = b̂_j / se(b̂_j), giving the exact formula for se(b̂_j) and 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): i.e., a normal numerator divided by the square root of an independent scaled χ² variance estimate. c) Provide intuition for why estimating σ inflates uncertainty compared with a Z‑test with known σ. d) Describe how heteroskedasticity or clustering changes the test (HC/cluster‑robust SEs) and what happens to the reference distribution.