Statistics and Econometrics: Estimation and Inference
Statistics and Econometrics tests estimation, inference, and the interpretation of regression output. You are expected to compute a test statistic or interval and then say precisely what it does and does not establish.
Core rules
- Estimators: unbiasedness is a repeated-sampling property, not a property of any single estimate; efficiency means smaller variance among unbiased estimators.
- CLT: sample means are approximately normal for large n regardless of the population shape; the standard error of the mean is σ/√n.
- Hypothesis tests: t = (estimate − hypothesized value)/SE. A 95% interval is estimate ± 1.96 SE; the interval and the two-sided 5% test agree.
- OLS: under Gauss–Markov assumptions OLS is BLUE. Omitted variable bias equals the effect of the omitted variable on y times its coefficient in the auxiliary regression on the included regressors.
- Interpretation: in log-level regressions a unit change in x raises y by about 100β percent; in log-log regressions β is an elasticity.
Worked example
A wage regression gives ln(wage) = 0.5 + 0.08 educ with SE(β) = 0.02. The return to one year of education is approximately 8%. Testing H₀: β = 0 gives t = 0.08/0.02 = 4, above 1.96, so reject at 5%. The 95% confidence interval is 0.08 ± 1.96(0.02) = [0.041, 0.119]. The interval excludes zero, consistent with the test; it does not prove that education causes higher wages.
Traps
- Reading a p-value as the probability that the null hypothesis is true.
- Claiming high R² establishes a causal model; it does not remove omitted-variable bias.
- Thinking heteroskedasticity biases OLS coefficients; it affects standard errors, so use robust ones.
- Confusing statistical significance with economic significance.