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Construct, apply and interpret confidence intervals and hypothesis tests that concern a single regression coefficient.

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

The idea

A hypothesis test on a single coefficient compares the estimate with the hypothesised value in units of standard error. t = (estimate − hypothesised value) ÷ standard error. In large samples the statistic is compared with normal critical values, 1.96 for a two-sided 5% test and 2.576 for 1%. In small samples with one regressor, use the t distribution with n − 2 degrees of freedom. A 95% confidence interval is the estimate plus or minus 1.96 × standard error, or the t critical value in small samples. A one-sided alternative uses a single tail, so the critical value is smaller. The usual mistake is to use the default regression output, which tests the coefficient against zero, when the hypothesis is some other value such as a beta of 1. For the exam, recompute t yourself.

Formulas

t-statistic for a single regression coefficient

t = (b̂ − b₀) ÷ SE(b̂)

b̂
the estimated coefficient
b₀
the value under the null hypothesis
SE(b̂)
the standard error of the estimate

Compare with normal critical values in large samples, or t with n − 2 degrees of freedom with one regressor in small ones. The trap is using the default output, which sets b₀ to zero.

A practice question

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

In a large sample, the 95% confidence interval for a regression slope is:

  1. the slope estimate plus or minus 1.645 times its standard errorWhy not: 1.645 is the one-sided 5% (two-sided 10%) critical value, giving a 90% interval.
  2. the slope estimate plus or minus 1.96 times its standard error - correct
  3. the slope estimate plus or minus 1.96 times the standard deviation of YWhy not: Uses the dispersion of the dependent variable, not the standard error of the estimator.
  4. the slope estimate plus or minus 1.96 times the variance of the estimateWhy not: Uses the variance rather than the standard error, which is the square root of the variance.

A 95% interval covers the middle 95% of the approximately normal sampling distribution: estimate +/- 1.96 x SE.

A harder one

A regression of a fund's excess return on market excess return gives a beta estimate of 0.29 with a standard error of 0.12. Testing H0: beta = 0 against a two-sided alternative at the 5% level in a large sample, the t-statistic and conclusion are:

  1. t is about 0.41; fail to reject H0Why not: Divides the standard error by the estimate (0.12 / 0.29), which inverts the statistic.
  2. t is about 2.42; fail to reject H0Why not: Computes t correctly but compares it with 2.576, the 1% critical value, not the 5% level of 1.96.
  3. t is about 0.29; fail to reject H0Why not: Uses the estimate alone as the test statistic and forgets to divide by the standard error.
  4. t is about 2.42; reject H0 - correct

t = (0.29 - 0) / 0.12 = 2.42, which exceeds 1.96 in absolute value, so H0 is rejected at the 5% level. It would not be rejected at 1%, where the critical value is 2.576.

The hardest tier

A one-variable regression uses 27 observations (25 degrees of freedom). The slope estimate is -0.28 with a standard error of 0.15. Test H0: slope = 0 against H1: slope < 0 at the 5% level. Critical values for 25 degrees of freedom: one-tailed 5% = 1.708, one-tailed 1% = 2.485, two-tailed 5% = 2.060. Which conclusion is correct?

  1. Fail to reject H0 at 5%, since |t| = 1.87 does not exceed 2.060Why not: Applies the two-tailed critical value to a one-sided alternative.
  2. Reject H0 at 5% but not at 1%, since t = -1.87 lies beyond -1.708 but not beyond -2.485 - correct
  3. Reject H0 at both 5% and 1%, since t = -1.87 is beyond -1.708Why not: Uses the 5% critical value for the 1% test; the 1% one-tailed value is 2.485.
  4. Fail to reject H0 at 5%, since t = -1.87 is negative and the test needs a positive statisticWhy not: Treats the sign as a failure; for H1: slope < 0 a negative t is the evidence against H0.

t = (-0.28 - 0) / 0.15 = -1.87. For a one-sided lower-tail test, reject if t < -1.708 (5%): yes. At 1% the cut-off is -2.485: no. The two-tailed value 2.060 does not apply.

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

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