2 marksEconometricsLinear regression and OLSt-tests, F-tests and testing restrictionsHardabout 3 min

IIT JAM 2022 Question 55: Linear regression and OLS

For the model $Y_{n\times1} = X_{n\times3}\,\beta_{3\times1} + U_{n\times1}$ with $\beta = [\beta_1\ \beta_2\ \beta_3]^T$ and $n = 23$ , a researcher obtains $(X^TX)^{-1} = \begin{bmatrix} 0.3 & 0.5 & 0.8 \\ 0.4 & -0.6 & 0.2 \\ 0.4 & 0.5 & 0.3 \end{bmatrix}$ , $X^TY = [0.3\ 0.2\ 0.1]^T$ and $e^Te = 0.7$ . The $t$ -statistic to test $\beta_3 = 0$ is _______ (round off to 2 decimal places).

Answer and worked solution

Answer: MTA

In the exam this question was dropped and every candidate received the marks.

The reason is that $(X^TX)^{-1}$ must be symmetric, because $X^TX$ is symmetric, yet the matrix printed is not symmetric (for example the $(1,2)$ entry $0.5$ differs from the $(2,1)$ entry $0.4$ ). The data are therefore internally inconsistent and the $t$ -statistic is not well defined, which is why the question was dropped.

Plain-text version

For the model Y_(n×1) = X_(n×3) β_(3×1) + U_(n×1) with β = [β₁ β₂ β₃]ᵀ and n = 23, a researcher obtains (XᵀX)⁻¹ = [0.3 0.5 0.8; 0.4 -0.6 0.2; 0.4 0.5 0.3], XᵀY = [0.3 0.2 0.1]ᵀ and eᵀe = 0.7. The t -statistic to test β₃ = 0 is _ (round off to 2 decimal places).

Also written as: For the model Y_(n*1) = X_(n*3) beta_(3*1) + U_(n*1) with beta = [beta_1 beta_2 beta_3]^T and n = 23, a researcher obtains (X^TX)^(-1) = [0.3 0.5 0.8; 0.4 -0.6 0.2; 0.4 0.5 0.3], X^TY = [0.3 0.2 0.1]^T and e^Te = 0.7. The t -statistic to test beta_3 = 0 is _ (round off to 2 decimal places).

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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