ECON2515 · Intermediate Applied Econometrics Ii
Inference: Testing a Single Population Parameter
This is ECON 2515's first full inference topic: once OLS has produced an estimate β̂ₖ, how do you decide whether the underlying population coefficient βₖ is really different from zero — or from any other value? You learn that β̂ₖ is one draw from a sampling distribution, standardise the gap into a t-statistic on n − k − 1 degrees of freedom, and run one- or two-sided tests using the correct critical value, backing them up with a confidence interval. The examiner rewards a visible six-step write-up and a plain-English conclusion, so the marks are won on getting the degrees of freedom, the side of the test and the null value right — not on algebra.
What this chapter covers
- 011. Sampling distribution of OLS — β̂ₖ ~ N(βₖ, Var(β̂ₖ)) exactly under MLR.6, approximately by the CLT in large samples
- 022. Standard errors — se(β̂ₖ) = √Var̂(β̂ₖ) with σ̂² = Σûᵢ²/(n − k − 1); what makes an estimate precise
- 033. The t-statistic — t = (β̂ₖ − c)/se(β̂ₖ) distributed t(n − k − 1); the c = 0 significance test
- 044. Degrees of freedom — always n − k − 1, where k counts the slopes and 1 is the intercept
- 055. One- vs two-sided tests — the alternative H₁ sets the direction, tail and critical column (α vs α/2)
- 066. Critical values — reading the t-table; two-sided uses the α/2 column, one-sided the α column
- 077. The six-step write-up — H₀, H₁, α and critical value, test statistic, decision rule, conclusion in words
- 088. Confidence intervals — β̂ₖ ± t·se(β̂ₖ); if c lies outside, reject H₀; a CI tests only the two-sided hypothesis
One-sided t-test of a coefficient against a non-zero value
- +2(a) The claim ('more than 3') is the ALTERNATIVE, and the null value is 3, not 0. Write H₀: β₂ ≤ 3 vs H₁: β₂ > 3 — a right-sided test with c = 3. The alternative always carries the direction the claim asserts.
- +3(b) The R/STATA 't' column tests β₂ = 0, so it is useless here — recompute by hand against c = 3: t = (β̂₂ − c)/se = (3.8 − 3)/0.55 = 0.8/0.55 = 1.45.
- +3(c) Degrees of freedom = n − k − 1 = 210 − 3 − 1 = 206 (k = 3 slopes). One-sided at 5% uses the 0.05 column: t(0.05, 206) ≈ 1.65. Since 1.45 < 1.65, the statistic is not in the upper tail, so fail to reject H₀.
Key terms
- Sampling distribution
- The distribution of an estimator β̂ₖ across all possible samples of a given size. Under Gauss-Markov plus normal errors it is exactly N(βₖ, Var(β̂ₖ)); by the Central Limit Theorem it is approximately normal in large samples. All inference rests on it.
- Standard error
- se(β̂ₖ) = √Var̂(β̂ₖ), the estimated standard deviation of the OLS estimator; it measures how precisely β̂ₖ is pinned down and shrinks with a larger sample, less error noise and more spread in xₖ.
- t-statistic
- t = (β̂ₖ − c)/se(β̂ₖ), the number of standard errors between the estimate and the null value c. It follows a t distribution with n − k − 1 degrees of freedom; the case c = 0 tests whether xₖ is statistically significant.
- Degrees of freedom
- n − k − 1, where n is the sample size, k the number of slope coefficients and the extra 1 the intercept. It selects the row of the t-table; using n or n − 1 by mistake is a classic error.
- One-sided vs two-sided test
- The form of H₁. Two-sided (βₖ ≠ c) splits α across both tails and uses the α/2 critical value; one-sided (βₖ > c or βₖ < c) puts all of α in one tail and uses the α critical value.
- Critical value
- The cut-off tₐ from the t-table that bounds the rejection region at a chosen significance level α. Reject H₀ when the test statistic falls beyond it (|t| > tₐ/₂ two-sided, or t past tₐ one-sided).
- Confidence interval
- β̂ₖ ± t(α/2, n−k−1)·se(β̂ₖ): a range that, over repeated samples, contains the true βₖ with probability 1 − α. If the null value c lies outside it, reject H₀; a CI tests only the two-sided hypothesis.
- Six-step write-up
- The examined template: state H₀, state H₁, choose α and the critical value, compute the test statistic, apply the decision rule, and conclude in words. Showing the steps earns method marks even if arithmetic slips.
Inference: Testing a Single Population Parameter FAQ
How do I get the degrees of freedom right?
Use df = n − k − 1, where k is the number of slope coefficients (regressors) and the extra 1 is the intercept. For example, n = 16 observations with one regressor gives df = 16 − 1 − 1 = 14, not 16. Adding another regressor lowers df again. Compute it explicitly before looking anything up, because the wrong t-table row flips borderline decisions.
When do I use a one-sided versus a two-sided test?
Let the wording of the claim decide. 'Is xₖ significant / does it have any effect?' is two-sided (H₁: βₖ ≠ c) and reads the α/2 column. 'Is the effect greater than / less than some value?' is one-sided (H₁: βₖ > c or < c) and reads the α column, giving a smaller, easier-to-beat critical value. Never use the two-sided critical value for a one-sided claim or vice versa.
The question tests whether a coefficient exceeds a number, not zero — what changes?
Set the null value c to that number, not 0. Then the software 't' column (which always tests βₖ = 0) is useless: recompute t = (β̂ₖ − c)/se(β̂ₖ) by hand. The alternative carries the direction of the claim, so 'more than c' gives H₁: βₖ > c. Testing against 0 instead is the wrong test for the wrong claim.
How does a confidence interval relate to the test?
A (1 − α) confidence interval and the two-sided test at level α give the same decision: if the null value c lies outside the interval, reject H₀; if it lies inside, fail to reject. A 95% CI that excludes 0 is exactly a two-sided 5% rejection of βₖ = 0. But a two-sided interval cannot settle a one-sided claim — for those, go back to the one-tailed t-test.
What must my written conclusion include to get full marks?
Two things. First the statistical verdict — 'reject / fail to reject H₀ at the α level.' Second an economic sentence in the variable's real units, e.g. 'one extra year of education is associated with about 7.4% higher wages, ceteris paribus.' Reporting only the t-value, with no decision and no interpretation, leaves easy marks behind.
Does statistically significant mean the effect is large or important?
No. A tiny coefficient can be significant in a very large sample, and a genuinely large effect can be insignificant in a small one. Significance is about whether the effect is distinguishable from zero given the noise; importance is about the magnitude in real units. Always comment on both — this distinction carries into the p-value and F-test topics.
Exam move
Drill this topic as a fixed routine rather than a set of formulas. For every practice coefficient, write the six steps in full — H₀, H₁, α and critical value, the statistic, the decision rule, and a worded conclusion — because the exam awards method marks for the visible steps, and the group report expects the same discipline. Before touching the t-table, always compute df = n − k − 1 out loud and decide the side of the test from the claim's wording; those two choices, not the arithmetic, are where most marks are lost. Practise both the c = 0 significance case and the harder 'more than / less than a non-zero value' case where you must rebuild t = (β̂ − c)/se by hand and ignore the printout's t-column. Finish every answer with a confidence interval to cross-check the two-sided verdict, and a one-line economic interpretation in the variable's units so you separate statistical significance from real-world magnitude.
