Financial Risk Manager Part 1

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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:

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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TTanishq

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 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._

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