2 marksEconometricsLinear regression and OLSt-tests, F-tests and testing restrictionsEasyabout 1 min

GATE 2023 Question 61: Linear regression and OLS

Using the Ordinary Least Squares (OLS) method, a researcher estimated the relationship between initial salary ( $S$ ) of MBA graduates and their cumulative grade point average ( $CGPA$ ) as

$\hat S_i = \hat\beta_0 + \hat\beta_1 CGPA_i \quad ; \quad i=1,2,\ldots,100$

where $\hat\beta_0=4543$ and $\hat\beta_1=645.08$ . The standard errors of $\hat\beta_0$ and $\hat\beta_1$ are 921.79 and 70.01, respectively.

The $t$ -statistic for testing the null hypothesis $\beta_1=0$ is ______ (round off to two decimal places).

Answer and worked solution

Answer: 9.20 to 9.24

The t-statistic for testing $H_0: \beta_1=0$ is simply the estimated coefficient divided by its standard error:

$t = \frac{\hat\beta_1}{SE(\hat\beta_1)} = \frac{645.08}{70.01} \approx 9.214$

Therefore the t-statistic is approximately 9.21, within the key's accepted range of 9.20 to 9.24.

The trap: 4.93

Students sometimes accidentally use the standard error of the intercept ( $\hat\beta_0$ 's SE of 921.79) or divide the wrong pair of numbers (e.g. $4543/921.79\approx4.93$ , the t-statistic for testing the intercept, not the slope) when the question specifically asks about $\beta_1$ , the CGPA coefficient.

Plain-text version

Using the Ordinary Least Squares (OLS) method, a researcher estimated the relationship between initial salary (S) of MBA graduates and their cumulative grade point average (CGPA) as Ŝᵢ = β̂₀ + β̂₁ CGPAᵢ; i=1,2,…,100 where β̂₀=4543 and β̂₁=645.08. The standard errors of β̂₀ and β̂₁ are 921.79 and 70.01, respectively. The t -statistic for testing the null hypothesis β₁=0 is (round off to two decimal places).

Also written as: Using the Ordinary Least Squares (OLS) method, a researcher estimated the relationship between initial salary (S) of MBA graduates and their cumulative grade point average (CGPA) as S_hat_i = beta_hat_0 + beta_hat_1 CGPA_i; i=1,2,...,100 where beta_hat_0=4543 and beta_hat_1=645.08. The standard errors of beta_hat_0 and beta_hat_1 are 921.79 and 70.01, respectively. The t -statistic for testing the null hypothesis beta_1=0 is (round off to two 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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