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

GATE 2021 Question 34: Linear regression and OLS

Consider two regression models estimated on a sample of 350 observations.

$y = \beta_0+\beta_1x_1+\beta_2x_2+\beta_3x_3+\beta_4x_4+\beta_5x_5+u \quad (1)$ $y = \alpha_0+\alpha_1x_1+\alpha_2x_2+v \quad (2)$

The $R^2$ in model (1) is $R_1^2=0.3521$ and in model (2) is $R_2^2=0.2314$ . The value of the test statistic to test $H_0: \beta_3=\beta_4=\beta_5=0$ is ______ (round off to three decimal places).

Answer and worked solution

Answer: 21.362

This is a standard F-test for joint significance of a subset of coefficients, using the R-squared form of the test statistic:

$F = \frac{(R^2_{UR}-R^2_R)/q}{(1-R^2_{UR})/(n-k-1)}$

where $R^2_{UR}=0.3521$ (unrestricted model 1), $R^2_R=0.2314$ (restricted model 2), $q=3$ (number of restrictions: $\beta_3=\beta_4=\beta_5=0$ ), $n=350$ , and $k=5$ (number of regressors in the unrestricted model).

$F = \frac{(0.3521-0.2314)/3}{(1-0.3521)/(350-5-1)} = \frac{0.1207/3}{0.6479/344} = \frac{0.040233}{0.001883} \approx 21.362$

Therefore the F-statistic is approximately 21.362.

The trap: 18.573

A common error is using the wrong degrees of freedom in the denominator — using $n-k$ instead of $n-k-1$ (forgetting to subtract 1 for the intercept), or miscounting $k$ as the total number of parameters instead of the number of regressors (excluding the intercept), producing a slightly different, incorrect F-statistic.

Plain-text version

Consider two regression models estimated on a sample of 350 observations. y = β₀+β₁x₁+β₂x₂+β₃x₃+β₄x₄+β₅x₅+u (1) y = α₀+α₁x₁+α₂x₂+v (2) The R² in model (1) is R₁²=0.3521 and in model (2) is R₂²=0.2314. The value of the test statistic to test H₀: β₃=β₄=β₅=0 is (round off to three decimal places).

Also written as: Consider two regression models estimated on a sample of 350 observations. y = beta_0+beta_1x_1+beta_2x_2+beta_3x_3+beta_4x_4+beta_5x_5+u (1) y = alpha_0+alpha_1x_1+alpha_2x_2+v (2) The R^2 in model (1) is R_1^2=0.3521 and in model (2) is R_2^2=0.2314. The value of the test statistic to test H_0: beta_3=beta_4=beta_5=0 is (round off to three 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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