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Go through the stages of conducting a hypothesis test in a linear regression.

FRM® Part I · Quantitative Analysis · Linear Regression · FRM1.QA.7.f

The idea

A hypothesis test on a regression coefficient follows a fixed sequence. First state the null and the alternative. The claim you want evidence for goes in the alternative, for example H0: β = 1 against H1: β > 1. Second, choose the significance level, the probability of rejecting a true null, before looking at the result. Third, compute the test statistic from the estimate and its standard error. t = (estimate − hypothesised value) ÷ standard error. Fourth, find the critical value for the chosen level, tail and degrees of freedom, or compute the p-value. Fifth, compare and decide: reject if the statistic falls beyond the critical value. Finally, state the conclusion in words. Failing to reject does not prove the null true. A 5% level means about one in twenty true nulls is rejected by chance.

A practice question

From the question bank, with the reason each wrong option is wrong.

What is the first stage in conducting a hypothesis test on a regression coefficient?

  1. Computing the t-statistic from the regression outputWhy not: The statistic is computed only after hypotheses and the significance level are set.
  2. Looking up the critical value for the sample's degrees of freedomWhy not: The critical value depends on the alternative and significance level, which come first.
  3. Deciding whether to reject the null hypothesisWhy not: The decision is the last step, made by comparing the statistic with the decision rule.
  4. Stating the null and alternative hypotheses about the coefficient - correct

The sequence is: state hypotheses, choose significance level, compute the statistic, find the critical value or p-value, decide, and interpret.

A harder one

A risk manager wants statistical evidence that a fund's market beta exceeds 1. Which formulation of the hypotheses is appropriate?

  1. H0: beta > 1 against H1: beta = 1Why not: Puts the claim to be proved in the null; the null should be the status quo, with the claim in the alternative.
  2. H0: beta = 1 against H1: beta > 1 - correct
  3. H0: beta = 1 against H1: beta is not equal to 1Why not: A two-sided alternative tests for any difference, not specifically for beta above 1.
  4. H0: beta < 1 against H1: beta = 1Why not: Reverses the roles and uses an equality as the alternative, which cannot be tested with a one-sided region.

The claim the manager wants support for goes in the alternative, so a one-sided H1: beta > 1 is appropriate. Rejecting H0 then provides evidence for it.

The hardest tier

A slope estimate is 0.74 with a standard error of 0.11. Test H0: slope = 0.50 against a two-sided alternative at the 5% level in a large sample. Which statement gives the correct statistic and decision?

  1. t is about 2.18; reject H0, since it exceeds 1.96 - correct
  2. t is about 2.18; fail to reject H0, since it is below 2.576Why not: Uses the 1% critical value for a 5% test.
  3. t is about 6.73; reject H0, since it exceeds 1.96Why not: Tests against zero (0.74 / 0.11) rather than against the stated null of 0.50.
  4. t is about 0.46; fail to reject H0, since it is below 1.96Why not: Divides the standard error by the distance from the null, inverting the statistic.

t = (0.74 - 0.50) / 0.11 = 2.18. At the 5% level the two-sided critical value is 1.96, so the null is rejected; the 2.576 cut-off belongs to the 1% level.

In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.

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