1 mark, −0.333EconometricsMisspecification and measurement errorOmitted variable biasHardabout 3 min

GATE 2021 Question 5: Misspecification and measurement error

To determine the relationship between $y$ and $x_1$ , Rohit estimated two different OLS models. In the first model, Rohit regressed $y$ on $x_1$ and $x_2$ as given below

$y = \beta_0+\beta_1x_1+\beta_2x_2+u \quad (1)$

In the second model, Rohit regressed $y$ only on $x_1$ as given below

$y = \delta_0+\delta_1x_1+v \quad (2)$

The estimated coefficients of $x_1$ in the above two models are exactly same. From this observation we can state conclusively that

(i) $Cov(x_1,y)=0$

(ii) $\hat\beta_2=0$

(iii) $Cov(x_2,x_1)=0$

where $\hat\beta_2$ is the estimated coefficient of $x_2$ in equation (1)

Tap an option to check your answer.

Answer and worked solution

Answer: (C)

This is the classic omitted variable bias relationship between a 'short' regression (2) and a 'long' regression (1). The exact algebraic relationship is:

$\hat\delta_1 = \hat\beta_1 + \hat\beta_2 \cdot \hat\pi_1$

where $\hat\pi_1$ is the coefficient from regressing $x_2$ on $x_1$ (which is proportional to $Cov(x_1,x_2)$ ).

For $\hat\delta_1 = \hat\beta_1$ exactly (as observed), we need the extra term $\hat\beta_2\cdot\hat\pi_1 = 0$ . This product is zero whenever either $\hat\beta_2=0$ or $\hat\pi_1=0$ (equivalently $Cov(x_1,x_2)=0$ in-sample) — or both. Neither condition alone is necessary; either one alone is sufficient.

So we cannot conclude specifically that only (ii) holds, or specifically that only (iii) holds — what we CAN conclude is that at least one of them (possibly both) must hold. Statement (i), about $Cov(x_1,y)$ , isn't directly implicated by this specific algebraic relationship at all.

Therefore the answer is (C): Either (ii) or (iii) or both are true.

The trap: (B) Only (ii) is true

Students who recall 'omitted variable bias is zero when the omitted variable's coefficient is zero' may focus only on condition (ii) and miss that condition (iii) — the omitted variable being uncorrelated with the included regressor — is an equally valid, independent sufficient condition for the same observed outcome, since either one alone (not necessarily both, and not necessarily specifically one) can produce identical coefficients.

Plain-text version

To determine the relationship between y and x₁, Rohit estimated two different OLS models. In the first model, Rohit regressed y on x₁ and x₂ as given below y = β₀+β₁x₁+β₂x₂+u (1) In the second model, Rohit regressed y only on x₁ as given below y = δ₀+δ₁x₁+v (2) The estimated coefficients of x₁ in the above two models are exactly same. From this observation we can state conclusively that (i) Cov(x₁,y)=0 (ii) β̂₂=0 (iii) Cov(x₂,x₁)=0 where β̂₂ is the estimated coefficient of x₂ in equation (1) (A) Only (i) is true (B) Only (ii) is true (C) Either (ii) or (iii) or both are true (D) Neither (ii) nor (iii) is true

Also written as: To determine the relationship between y and x_1, Rohit estimated two different OLS models. In the first model, Rohit regressed y on x_1 and x_2 as given below y = beta_0+beta_1x_1+beta_2x_2+u (1) In the second model, Rohit regressed y only on x_1 as given below y = delta_0+delta_1x_1+v (2) The estimated coefficients of x_1 in the above two models are exactly same. From this observation we can state conclusively that (i) Cov(x_1,y)=0 (ii) beta_hat_2=0 (iii) Cov(x_2,x_1)=0 where beta_hat_2 is the estimated coefficient of x_2 in equation (1)

Misspecification and measurement error has come up 11 times in 4 exams since 2011.

What this question tests

This question is on Omitted variable bias, part of Misspecification and measurement error in Econometrics.

How leaving out a relevant variable biases the estimates, and in which direction.

TestStop has 4 past-paper questions on this subtopic.

Take this paper as a timed mock More on Omitted variable bias

Question from the GATE 2021 paper. Solutions are TestStop's own. Not affiliated with or endorsed by any examination authority.