ECON2515 · Intermediate Applied Econometrics Ii
Hypothesis Testing Using p-values
This topic re-runs the single-parameter test from the previous week, but decides it with the p-value — the probability of a test statistic at least as extreme as the one observed, computed assuming the null is true. The decision rule is one line for every case: reject H₀ if p ≤ α, and you read the number straight off the regression output's Pr(>|t|) column instead of a critical-value table. The two skills the exam rewards are reading that column correctly (it is two-sided — halve it for a one-sided test) and keeping statistical significance (is the coefficient distinguishable from zero?) separate from economic significance (is the effect big enough to matter?).
What this chapter covers
- 011. What a p-value is — a tail probability computed under H₀, measuring the strength of evidence against the null (small p = strong evidence)
- 022. The universal decision rule — reject H₀ when p ≤ α; fail to reject when p > α; the same rule for every alternative
- 033. p-value by direction — two-sided p = 2·P(T > |t|), right-sided p = P(T > t), left-sided p = P(T < t), all on df = n − k − 1
- 044. p-value as a tail area — visualising the null distribution and the shaded region beyond the observed t
- 055. p-value vs critical-value — two equivalent roads to the same verdict; |t| > t_c exactly when p < α
- 066. Reading Pr(>|t|) from R/STATA output — the printed p-value is two-sided for H₀: β = 0; halve it (with a sign check) for a one-sided test
- 077. Statistical vs economic significance — a tiny p means non-zero, not large; judge the magnitude of the effect separately
- 088. Language and traps — 'not distinguishable from zero' not 'has no effect'; p is P(data | H₀), not P(H₀ true)
Decide significance from p-values, halve for a one-sided claim, and read the elasticity
- +1Setup: k = 2 slopes, so df = n − k − 1 = 240 − 2 − 1 = 237. The reported p-values are two-sided, for H₀: β = 0 against H₁: β ≠ 0.
- +1(a) log(price): p < 0.001 ≤ 0.05 → reject H₀, statistically significant (check: t = −1.80/0.35 = −5.14, far past ≈ 2).
- +1(a) promo: p = 0.13 > 0.05 → fail to reject H₀, not statistically distinguishable from zero at 5% (t = 0.06/0.04 = 1.5 < 2).
- +2(b) The claim 'promotions raise sales' is a one-sided alternative H₁: β₂ > 0. The estimate β̂₂ = +0.06 has the sign the claim predicts, so halve the two-sided p: p_one-sided = 0.13/2 = 0.065.
- +1(b) Compare to α: 0.065 > 0.05 → fail to reject H₀ even one-sided. There is not enough evidence at 5% to support the manager's claim that promotions raise sales.
- +2(c) log-log model: β̂₁ is the price elasticity of sales directly. A 1% rise in price is associated with about a 1.8% fall in sales, ceteris paribus (elastic demand). Report this magnitude separately from the p-value — significance and size are different questions.
Key terms
- p-value
- The probability of a test statistic at least as extreme as the one observed, calculated assuming H₀ is true. It measures evidence against the null: small p = strong evidence to reject. It is P(data this extreme | H₀), NOT the probability that H₀ is true.
- Decision rule (p ≤ α)
- Reject H₀ if the p-value is at most the significance level α; otherwise fail to reject. One rule works for every alternative — the only thing that changes is how the p-value is computed.
- Significance level (α)
- The threshold probability chosen before testing (commonly 0.05), equal to the tolerated chance of rejecting a true H₀; 1 − α is the confidence level. A p = 0.03 rejects at 5% but not at 1%, so the α must be stated.
- Two-sided p-value
- The p-value for H₁: β ≠ c, adding both tails: 2·P(T > |t|). Regression software prints this by default in the Pr(>|t|) column for the null H₀: β = 0.
- One-sided test
- A test whose alternative points in a single direction (β > c or β < c). Its p-value is a single tail; taken from a two-sided printout it is half the reported p, but only if the estimate's sign matches the claimed direction.
- Statistical significance
- A coefficient is statistically significant when its p-value is at or below α, i.e. it is distinguishable from zero given the data. It says nothing about how large or important the effect is.
- Economic significance
- Whether the magnitude of the estimated effect is large enough to matter in the real world, judged from the size of the implied change in y (a dollar amount, a percentage, an elasticity) — a separate question from the p-value.
- Degrees of freedom (n − k − 1)
- The parameter of the t-distribution used to turn a t-statistic into a p-value, where k is the number of slope coefficients. Getting it wrong (e.g. using n − k) can flip a borderline decision.
Hypothesis Testing Using p-values FAQ
Is the p-value approach different from the critical-value method I learned last week?
No — they give the identical verdict. The critical-value method compares |t| to a table value t_c; the p-value method compares the p to α. Because |t| exceeds t_c exactly when the tail area beyond it (the p-value) drops below α, the two never disagree. Use the p-value when you have software output, since it is printed and needs no table.
The R output shows Pr(>|t|). Can I use that directly for a one-sided test?
Not directly. Pr(>|t|) is a two-sided p-value for H₀: β = 0. For a one-sided alternative, first check the sign of the estimate matches the direction you are testing, then halve the printed p-value before comparing to α. If the sign is opposite to your claim, the one-sided p is close to 1 and you cannot reject.
Does a large p-value prove the variable has no effect?
No. 'Fail to reject H₀' means insufficient evidence to distinguish the coefficient from zero at your chosen α — not proof the true effect is zero. Write 'not statistically distinguishable from zero at 5%', never 'has no effect'. Examiners deduct for the stronger wording.
What is the difference between statistical and economic significance?
Statistical significance (small p) says the effect is distinguishable from zero. Economic significance asks whether the effect is big enough to matter, judged from its magnitude in real units. A huge sample can make a trivial effect significant, and a genuinely large effect can be insignificant if imprecisely estimated — so a full answer comments on both.
How is this tested in the ECON 2515 exam?
It appears in Part A multiple-choice ('which regressors are significant at 5%?') and inside Part B worked answers, where you read the Pr(>|t|) column, apply the p ≤ α rule with the correct degrees of freedom and tail, and then interpret the coefficient's size using the right functional-form rule. The final exam is closed-book with statistical tables provided; exact weightings are subject to confirmation on your course outline.
A coefficient's p-value is 0.049 and another's is 0.051 — are they really that different?
Statistically they are almost identical evidence; the 0.05 line is a convention, not a cliff. Report the actual p-values rather than just 'significant/not', note that one clears 5% and the other just misses, and let the reader apply their own α. Never treat a p just above 0.05 as proof of no relationship.
Exam move
Drill the reading, not the recall: take past-style R or STATA printouts and, for each coefficient, write the one-line decision — 'p = ___ [≤/>] α, so [reject / fail to reject] H₀: β = 0 at the __% level' — then a second line on what the coefficient means in real units using the correct log/level/quadratic rule. Practise the one-sided move until it is automatic: check the estimate's sign against the claim, halve the two-sided p, compare to α. Keep a short note on your allowed A4 sheet with the p-value rule (reject if p ≤ α), the two-sided-to-one-sided halving with the sign caveat, df = n − k − 1, and the phrase 'not distinguishable from zero' so you never slip into 'has no effect'. Above all, train yourself to answer both questions the examiner is really asking — is the effect real (p-value) AND is it big (magnitude) — because marks are split across the two.
