Statistics and Econometrics: Inference and the Linear Model
The Statistics and Econometrics domain tests whether you can choose the right estimator or test, compute it correctly, and interpret the result in the language of the problem rather than in symbols. Typical content: sampling distributions, the central limit theorem, confidence intervals, hypothesis tests, and simple or multiple OLS with its assumptions.
Core rules
- Estimator properties: unbiasedness, efficiency, consistency. Gauss–Markov gives BLUE under linearity in parameters, exogeneity, no perfect collinearity, homoskedasticity and no autocorrelation.
- Standard error of a mean: s/√n. The CLT justifies the normal approximation for large n (rule of thumb n ≥ 30).
- Confidence interval: x̄ ± t(critical, n−1) × s/√n.
- Test statistic for a slope: t = (b̂ − β0)/SE(b̂), with df = n − k. The F statistic tests joint significance of several coefficients.
- The p-value is the probability of a result at least as extreme as the observed one, computed under H0. It is not the probability that H0 is true.
- R² never falls when a regressor is added; the adjusted R² penalises extra parameters.
- Omitted variable bias: direction = sign of the omitted variable's true effect × sign of the correlation between the included and omitted regressors.
Worked example
A sample of n = 5 gives Σx = 15, Σy = 25, Σxy = 95, Σx² = 55, Σy² = 145. Then x̄ = 3, ȳ = 5, Sxx = 55 − 225/5 = 10 and Sxy = 95 − 15·25/5 = 20. The slope is b = Sxy/Sxx = 2 and the intercept a = 5 − 2·3 = −1. For Syy = 145 − 625/5 = 20, R² = b·Sxy/Syy = 40/20, which signals near-perfect fit; with SE(b) = 0.5 the statistic is t = 4 against a 5% two-tailed critical value of 3.182 at df = 3, so H0: β = 0 is rejected.
Traps that cost marks
Running a one-tailed test when the question is two-tailed; claiming heteroskedasticity biases the slopes (it biases the standard errors, not the coefficients); reading a CI as the probability that the parameter lies inside; treating R² as proof of causality; confusing statistical with economic significance; and forgetting that measurement error in x attenuates the slope towards zero.