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In regression, each coefficient β̂ⱼ has an associated standard error se(β̂ⱼ), and the t-statistic t = (β̂ⱼ − β₀)/se(β̂ⱼ) tests whether βⱼ equals some hypothesized value (usually zero) in the population. Under the null, this t-statistic follows a t-distribution with n−k−1 degrees of freedom; for large samples it approaches the standard normal. Statistical significance at the 5% level means the p-value is below 0.05, but economic significance — whether the effect size matters practically — is a separate judgment. Confidence intervals for coefficients convey both magnitude and precision.
Interpret regression tables from published papers, explaining each coefficient's sign, magnitude, standard error, and significance level. Practice constructing confidence intervals manually from reported standard errors.
When you learned t-tests for comparing means, you computed a test statistic by dividing an estimate by its standard error: t = (x̄ − μ₀)/se(x̄). Hypothesis testing in regression is exactly the same idea applied to a regression coefficient. The OLS estimator β̂ⱼ is a random variable with a sampling distribution — it varies across hypothetical repeated samples. The standard error se(β̂ⱼ) measures how much β̂ⱼ varies across those samples. The t-statistic t = (β̂ⱼ − β₀)/se(β̂ⱼ) measures how many standard errors the estimate lies from the hypothesized value β₀ (usually zero). Under the null, this follows a t-distribution with n − k − 1 degrees of freedom; in large samples it is approximately standard normal.
Reading a regression table fluently means interpreting four things for each coefficient: its sign (direction of effect), its magnitude (how large is the effect), its standard error (how precisely estimated), and its p-value or significance stars (whether you would see this estimate by chance under H₀). The stars tell you whether the effect is statistically distinguishable from zero; they do not tell you whether the effect is large enough to matter. Every published table reports these together, and conflating them is one of the most common errors in applied work.
The statistical vs. economic significance distinction is the central lesson of this topic, and it is non-obvious. With a large enough sample, even a microscopic true effect will produce a tiny standard error, generating a significant p-value. An economist studying wages with n = 2,000,000 records might find that having a window seat at work raises wages by $0.12 per year with p < 0.001. That effect is real — it is not due to noise — but it is economically meaningless. Conversely, with n = 50 observations, a genuinely large effect may fail to reach significance simply because the sample is too small to detect it. Always report effect sizes alongside significance, and ask whether the magnitude of β̂ is large enough to matter for any real decision.
What does it actually mean when a result is significant at the 5% level? It means that if the null hypothesis were true, you would observe a t-statistic at least this large only 5% of the time by chance. It does not mean there is a 95% chance the null is false, and it does not mean the coefficient is 'probably' the right sign. The p-value is a statement about what you would observe in hypothetical repeated sampling under H₀ — not a probability assigned to the null hypothesis being true or false.
Confidence intervals are often more informative than p-values. A 95% confidence interval for β̂ⱼ is approximately β̂ⱼ ± 1.96 × se(β̂ⱼ). This interval gives you the range of effect sizes consistent with the data, rather than forcing a binary significant/not-significant verdict. A very wide confidence interval that just excludes zero is technically significant but tells you almost nothing about the true effect size. A narrow interval centered on a small value tells you the effect is real and small. Reporting and interpreting confidence intervals is the best habit to develop for honest empirical work.