2 marks, −0.667EconometricsLinear regression and OLSt-tests, F-tests and testing restrictionsModerateabout 2 min

GATE 2021 Question 7: Linear regression and OLS

In a demand function estimation of a good $X$ , a researcher collected data on various households' consumption of good $X$ ( $Q_x$ ) for various price levels. The researcher also collected data on household income ( $M$ ) and household size ( $S$ ). The estimated regression result is

$\log Q_x = -0.345(0.111) - 1.543(2.345)\log P_x + 1.123(0.012)\log M + 0.234(0.123)\log S$

where $P_x$ is price per unit of $X$ . The values in the parentheses are the standard errors of the estimated coefficients. From the estimation one can conclude that

Tap an option to check your answer.

Answer and worked solution

Answer: (C)

In a log-log regression, the coefficient on $\log P_x$ directly IS the price elasticity of demand: here, $\hat\beta = -1.543$ (NOT the standard error 2.345, which rules out option (D) immediately — the standard error is not the elasticity itself).

Testing significance: $t = \dfrac{\hat\beta}{SE(\hat\beta)} = \dfrac{-1.543}{2.345} \approx -0.658$ .

Since $|t|\approx0.658$ is far below the conventional critical value (~1.96 for a 5% two-tailed test, or even ~1 as a very loose rule of thumb), the price elasticity coefficient is not statistically significant — we cannot reject the null hypothesis that the true price elasticity is zero, given this data.

(A) is unsupported since we can't even confidently say the coefficient is non-zero, let alone characterize its magnitude as 'highly elastic.' (B) is about the income coefficient's sign (which is positive, $1.123>0$ , indicating a NORMAL, not inferior, good) — the reverse of what's stated.

Therefore the answer is (C).

The trap: (A) the demand for good X is highly elastic

The coefficient magnitude ( $-1.543$ ) does look 'elastic' at face value (greater than 1 in absolute value), tempting students to pick (A) without checking whether that estimate is even statistically distinguishable from zero given its very large standard error — a large point estimate with an even larger standard error is not reliable evidence of a real effect.

Plain-text version

In a demand function estimation of a good X, a researcher collected data on various households' consumption of good X (Qₓ) for various price levels. The researcher also collected data on household income (M) and household size (S). The estimated regression result is log Qₓ = -0.345(0.111) - 1.543(2.345)log Pₓ + 1.123(0.012)log M + 0.234(0.123)log S where Pₓ is price per unit of X. The values in the parentheses are the standard errors of the estimated coefficients. From the estimation one can conclude that (A) the demand for good X is highly elastic (B) X is an inferior good (C) the estimated price elasticity of demand is not statistically significant (D) the estimated price elasticity of good X is 2.345

Also written as: In a demand function estimation of a good X, a researcher collected data on various households' consumption of good X (Q_x) for various price levels. The researcher also collected data on household income (M) and household size (S). The estimated regression result is log Q_x = -0.345(0.111) - 1.543(2.345)log P_x + 1.123(0.012)log M + 0.234(0.123)log S where P_x is price per unit of X. The values in the parentheses are the standard errors of the estimated coefficients. From the estimation one can conclude that

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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