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Answer: We can reject the null hypothesis because the F-statistic is significant at 90% confidence.
## Explanation To test the joint hypothesis that all coefficients are zero ($\beta_1 = 0$, $\beta_2 = 0$, and $\beta_3 = 0$), we need to use the F-test, not individual t-tests. ### Key Points: - **F-test purpose**: Tests whether at least one of the independent variables has a statistically significant relationship with the dependent variable - **Significance level**: 90% confidence level corresponds to a 10% significance level (α = 0.10) - **P-value interpretation**: We reject the null hypothesis when p-value < significance level ### Analysis: - **F-statistic p-value = 0.09** - **Significance level = 0.10** - Since 0.09 < 0.10, we **reject** the null hypothesis ### Why individual t-tests are insufficient: - Individual t-tests (p-values: β₁ = 0.11, β₂ = 0.12) are not significant at 10% level - However, t-tests don't account for interactions among independent variables - The F-test considers the joint effect of all variables ### Conclusion: We can reject the joint null hypothesis because the F-statistic is significant at the 90% confidence level, indicating that at least one of the β coefficients is different from zero.
Author: Tanishq Prabhu
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For a sample of 40 years, the relationship between GDP growth (Y_t), inflation (X_1) and interest rates (X_2) is modeled as follows:
An economist wishes to test the joint hypothesis that , , and at the 90% confidence level.
The p-value for the t-statistic for is 0.11, and the p-value for the t-statistic for is 0.12. The p-value for the F-statistic for the regression is 0.09. Which of the following statements is correct?
A
We cannot reject the null hypothesis because none of the is different from zero at 90% confidence.
B
We can reject the null hypothesis because each of the is different from zero at 90% confidence.
C
We cannot reject the null hypothesis because the F-statistic is not significant at 90% confidence.
D
We can reject the null hypothesis because the F-statistic is significant at 90% confidence.