CFA Level IQuantitative MethodsHard
A regression analysis produces a slope coefficient (b1) of 1.5 for a factor exposure model, with a standard error of the coefficient of 0.60, based on n = 40 observations (38 degrees of freedom). An analyst wants to test H0: b1 = 1 versus Ha: b1 ≠ 1 at the 5% significance level, where the two-tailed critical t-value is approximately ±2.024. What is the calculated t-statistic and conclusion?
- At = 0.83; reject H0
- Bt = 1.50; fail to reject H0
- Ct = 0.83; fail to reject H0
- Dt = 2.50; reject H0
Show answer & explanationAnswer & explanation
Correct answer: C. t = 0.83; fail to reject H0
t = (b1 - hypothesized value) / standard error = (1.5 - 1) / 0.60 = 0.5 / 0.60 = 0.833. Since |0.833| < 2.024, the analyst fails to reject H0; the slope is not statistically significantly different from 1.
Why the other options are wrong
- A. The t-statistic value is correct, but the conclusion is wrong since 0.83 is below the critical value.
- B. This uses the slope coefficient itself as the t-statistic, omitting the subtraction and division steps.
- D. This value does not match the correct calculation of (1.5-1)/0.60.
Hypothesis Test on a Regression Coefficient
A t-test on a regression slope coefficient tests whether the estimated coefficient differs significantly from a hypothesized value, using the coefficient's standard error.
- t = (b1 - hypothesized value) / standard error of b1
- Degrees of freedom = n - k - 1, where k is the number of independent variables
- Fail to reject H0 if |t-statistic| is less than the critical t-value
Memory trick: Subtract the hypothesis, divide by the error, compare to the fence.