Financial Risk Manager Part 1

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For a sample of 40 years, the relationship between GDP growth (Y_i), inflation (X_1) and interest rates (X_2) is modeled as follows:

y_t = \beta_1 + \beta_2 x_{1t} + \beta_3 x_{2t} + u_t

An economist wishes to test the joint hypothesis that $\beta_1 = 0$ , $\beta_2 = 0$ , and $\beta_3 = 0$ at the 90% confidence level.

The p-value for the t-statistic for $\beta_1$ is 0.11, and the p-value for the t-statistic for $\beta_2$ is 0.12. The p-value for the F-statistic for the regression is 0.09. Which of the following statements is correct?

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NNikitesh

Last updated: February 2, 2026 at 10:22

We cannot reject the null hypothesis because none of the $\beta_s$ is different from zero at 90% confidence

We can reject the null hypothesis because each of the $\beta_s$ is different from zero at 90% confidence

We cannot reject the null hypothesis because the F-statistic is not significant at 90% confidence

We can reject the null hypothesis because the F-statistic is significant at 90% confidence

Explanation:

Explanation

To test the joint hypothesis that all coefficients are simultaneously equal to zero ( $\beta_1 = 0$ , $\beta_2 = 0$ , and $\beta_3 = 0$ ), we need to use the F-test, not individual t-tests.

Key Points:

Individual t-tests vs. Joint F-test:
- Individual t-tests examine whether each coefficient is individually different from zero
- The F-test examines whether at least one coefficient is different from zero, considering the joint effects and interactions among variables
P-value interpretation:
- For $\beta_1$ : p-value = 0.11 > 0.10 → cannot reject $\beta_1 = 0$ at 90% confidence
- For $\beta_2$ : p-value = 0.12 > 0.10 → cannot reject $\beta_2 = 0$ at 90% confidence
- For F-statistic: p-value = 0.09 < 0.10 → can reject the joint null hypothesis at 90% confidence
Decision rule:
- Reject the null hypothesis when p-value < significance level (α = 0.10 for 90% confidence)
- Since F-test p-value (0.09) < 0.10, we reject the joint null hypothesis
Correct interpretation:
- The F-test result means that at least one of the coefficients is statistically different from zero
- This doesn't mean all coefficients are significant individually
- The F-test accounts for multicollinearity and joint effects that individual t-tests don't capture

Why other options are incorrect:

A: Incorrect because the F-test allows us to reject the joint null hypothesis
B: Incorrect because individual t-tests don't show each β is different from zero (p-values > 0.10)
C: Incorrect because the F-statistic IS significant at 90% confidence (p-value = 0.09 < 0.10)

Answer D is correct: We can reject the null hypothesis because the F-statistic is significant at 90% confidence level.

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