Financial Risk Manager Part 1
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A linear regression model gave the following results: S<sub>yy</sub> = 10.6; S<sub>xx</sub> = 12.0; S<sub>xy</sub> = 8.0; n = 18
Test (at 1% significance) whether β is significantly different from zero, given that its standard error = 0.16 and give the value of the test statistic and the conclusion.
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NNikitesh
A
0.667, β is not significantly different from zero
B
0.4169, β is not significantly different from zero
C
0.667, β is significantly different from zero
D
4.169, β is significantly different from zero
Explanation:
Explanation
We are testing the hypothesis:
- H₀: β = 0 (null hypothesis)
- H₁: β ≠ 0 (alternative hypothesis)
Step 1: Calculate the slope coefficient β β = S<sub>xy</sub>/S<sub>xx</sub> = 8.0/12.0 = 0.667
Step 2: Calculate the test statistic The test statistic follows a t-distribution with n-2 degrees of freedom: t = (β - β₀)/se(β) = (0.667 - 0)/0.16 = 4.169
Step 3: Determine the critical value For a two-tailed test at 1% significance level with n-2 = 16 degrees of freedom:
- α = 0.01
- α/2 = 0.005 for each tail
- t<sub>0.005,16</sub> = 2.921 (from t-distribution table)
Step 4: Make the decision Since |4.169| > 2.921, we reject the null hypothesis H₀.
Conclusion: β is significantly different from zero at the 1% significance level.
Why the other options are incorrect:
- A (0.667, β is not significantly different from zero): 0.667 is the slope coefficient, not the test statistic, and the conclusion is wrong.
- B (0.4169, β is not significantly different from zero): 0.4169 is incorrect calculation of the test statistic, and the conclusion is wrong.
- C (0.667, β is significantly different from zero): While the conclusion is correct, 0.667 is the slope coefficient, not the test statistic.
Only option D provides both the correct test statistic (4.169) and the correct conclusion.
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