2 marksEconometricsLinear regression and OLSt-tests, F-tests and testing restrictionsModerateabout 2 min

IIT JAM 2022 Question 33: Linear regression and OLS

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

Select all that apply.

Answer and worked solution

Answer: (B) and (D)

The $F$ -test evaluates linear restrictions on the coefficients. $\beta_0 = 0$ is a single linear restriction and $\beta_1 = \beta_2 = 0$ is a joint linear restriction, both testable with an $F$ -test (the single-restriction case coincides with the square of a $t$ -statistic).

The ratio $\beta_1/\beta_2 = 0$ and the product $\beta_1\beta_2 = 0$ are nonlinear in the parameters and cannot be cast as the linear restrictions the standard $F$ -test requires. The testable hypotheses are (B) and (D).

The trap: (A)

\beta_1/\beta_2 = 0

$\beta_1/\beta_2 = 0$ looks like it reduces to $\beta_1 = 0$ , but as written it is a nonlinear (ratio) restriction, outside the linear framework of the standard $F$ -test.

Plain-text version

Suppose the regression model is Yᵢ = β₀ + β₁ X₁ᵢ + β₂ X₂ᵢ + μᵢ, i = 1, 2, …, n. Which of the following null hypotheses could be tested using the F -test? (A) β₁/β₂ = 0 (B) β₀ = 0 (C) β₁β₂ = 0 (D) β₁ = β₂ = 0

Also written as: Suppose 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?

Linear regression and OLS has come up 74 times in 7 exams since 2008.

What this question tests

This question is on t-tests, F-tests and testing restrictions, part of Linear regression and OLS in Econometrics.

Testing a single coefficient, several coefficients, or a linear restriction.

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