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Confidence intervals quantify uncertainty around parameter estimates using estimated standard errors and critical values from the t-distribution; hypothesis tests evaluate whether parameters equal specific values. Both rely on asymptotic normality and require valid estimates of standard errors, accounting for any heteroskedasticity or serial correlation.
You already know from asymptotic normality that the OLS estimator β̂ is approximately normally distributed in large samples: (β̂ − β) / se(β̂) → N(0,1). A confidence interval uses this fact directly — it constructs a range around β̂ that, under repeated sampling, would contain the true β in 95% of cases (for a 95% interval). The formula is β̂ ± t* · se(β̂), where t* is the critical value from the t-distribution with n − k − 1 degrees of freedom. In large samples, t* ≈ 1.96 for 95% confidence. The interval does not mean "there is a 95% chance β is inside this range" — β is a fixed (unknown) constant, not a random variable. What varies across samples is β̂ itself.
The t-statistic for hypothesis testing is the same building block rearranged. To test H₀: βⱼ = c (typically c = 0, the hypothesis that variable j has no effect), compute t = (β̂ⱼ − c) / se(β̂ⱼ). Under H₀, this statistic follows an approximate t-distribution. A large absolute value of t means the estimate is many standard errors from the hypothesized value — unlikely if H₀ were true. The p-value converts this to a probability: it is the probability of observing a t-statistic at least as extreme as yours if H₀ were true. A p-value below your chosen significance level (commonly 0.05) leads to rejection of H₀.
The confidence interval and hypothesis test are two sides of the same coin: β̂ ± t* · se(β̂) contains exactly the values c for which a two-sided test would fail to reject H₀: βⱼ = c. If zero is inside the 95% confidence interval, the t-test at 5% significance fails to reject H₀: βⱼ = 0. If zero is outside, you reject. This duality gives you two ways to communicate the same inference — the interval conveys both significance and effect magnitude, while the test gives a clean binary decision.
The critical ingredient in all of this is se(β̂) — the estimated standard error. You know from your prerequisites that if errors are heteroskedastic or serially correlated, the classical OLS standard errors are inconsistent; they typically understate the true uncertainty, producing confidence intervals that are too narrow and t-statistics that are too large. This is why valid inference requires using heteroskedasticity-robust or cluster-robust standard errors whenever those problems are present. The formula for β̂ does not change — only the standard errors you attach to it do. Getting this right is not a technical nicety; it is the difference between inference you can trust and inference that will mislead you.