Question

Suppose that the regression model is $Y_i =\beta_0 + \beta_1 X_{1i} + \beta_2 X_{2i} + \mu_i, i = 1, 2,..., n$. Which of the following null hypotheses could be tested using the F-test?

• A. $\beta_1 /\beta_2 =0$
• B. $\beta_0 =0$
• C. $\beta_1\beta_2 =0$
• D. $\beta_1 =\beta_2 =0$

Answer 1

The Correct Option is B, D

The given regression model is stated as:

\(Y_i = \beta_0 + \beta_1 X_{1i} + \beta_2 X_{2i} + \mu_i, \, i = 1, 2, \ldots, n\)

An F-test in regression analysis is commonly used to test multiple hypotheses simultaneously. Specifically, it assesses the overall significance of a regression model. In the context of this question, we need to know which null hypotheses could be tested using this F-test.

Let's evaluate the given options:

• \(\frac{\beta_1}{\beta_2} = 0\): This involves testing a ratio, which is typically not tested using an F-test. The F-test is used for evaluating the significance of a set of coefficients, not their ratios.
• \(\beta_0 = 0\): Testing whether the intercept is significantly different from zero (that is, testing \(\beta_0 = 0\)) can typically be done using a t-test, but an F-test can be employed if testing in combination with other parameters or constraints. Thus, it is a theoretically valid possibility for an F-test in certain scenarios.
• \(\beta_1 \beta_2 = 0\): This is a product test. We rarely use F-tests to examine products of coefficients directly. This scenario does not align with the typical uses of an F-test.
• \(\beta_1 = \beta_2 = 0\): This is a classical use of the F-test, where we assess the null hypothesis that the coefficients \(\beta_1\) and \(\beta_2\) are simultaneously equal to zero. It tests the joint significance of these predictors in the model.

Thus, the null hypotheses that could be tested using the F-test are:

• \(\beta_0 = 0\)
• \(\beta_1 = \beta_2 = 0\)

This supports the correct options given in the question.