Consider the following simple regression model: y = β0 + β1x1 + u (1) and the following multiple regression model: y = β0 + β1x1 + β2x2 + u (2), where x1 is the variable of primary interest to explain y. Which of the following statements is correct?

Question

Consider the following simple regression model: y = β0 + β1x1 + u (1) and the following multiple regression model: y = β0 + β1x1 + β2x2 + u (2), where x1 is the variable of primary interest to explain y.  Which of the following statements is correct?

BlackTom AI · Accepted Answer

Let’s unpack the two regression setups and the implications for the individual statements one by one.
Option 1: 'If x2, contained in u, is correlated with x1, the OLS estimator of β1 in model (1) will be biased.' In model (1), x2 is omitted from the regression and is part of the error term u when x2 is correlated with x1. According to the omitted variable bias/omitted regressor intuition, correlation between a regressor of interest (x1) and an omitted explanatory variable that is a determinant of y (x2) induces a correlation between x1 and the error term, biasing the estimated β1. Therefore this statement is correct under the usual OLS assumptions for model (1).
Option 2: 'When drawing ceteris paribus conclusions about how x1 affects y, with model (1), we must assume that x2, contained in u, is uncorrelated with x1.' This is consistent with the standard identification requirement for unbiased causal inference in a simple regression: to attribute changes in y to x1 holding other factors constant, the included regressor must be exogenous relative to the error term. Since x2 is part of u in model (1), the assumption that x2 is uncorrelated with x1 is exactly the exogeneity condition needed for the β1 estimate to reflect a causal effect of x1, free of omitted variable bias. Hence this statement is correct.
Option 3: 'All of the above.' If both Statement 1 and Statement 2 are correct, then 'All of the above' would also be correct, provided there are no contradictions between the statements. Given the standard interpretation, there is no contradiction: they are consistent with the consequences of including or omitting x2 in the regression and the exogeneity conditions required for causal interpretation in model (1).
Option 4: 'When drawing ceteris paribus conclusions about how x1 affects y, with model (2), because x2 is explicitly in the model equation, we are able to measure the effect of x1 on y, holding x2 fixed—assuming all other factors contained in u, are uncorrelated with x1 and x2.' Model (2) includes x2, so the coefficient β1 is intended to reflect the effect of x1 on y controlling for x2, i.e., holding x2 fixed in the linear specification. This relies on the standard exogeneity condition: the error term u (which includes any omitted factors) should be uncorrelated with x1 and x2. If that assumption holds, the partial effect of x1 on y is identified in the presence of x2. Thus this statement is also correct under the stated assumptions.
In summary, each option analyzes a distinct but compatible aspect of the relationships between x1, x2, and the error term, and together they portray the standard identification logic for simple versus multiple regression models. The combination of these insights aligns with the idea that all of the above statements are correct, given the typical econometric assumptions and interpretations.

Consider the following simple regression model: y = β0 + β1x1 + u (1) and the following multiple regression model: y = β0 + β1x1 + β2x2 + u (2), where x1 is the variable of primary interest to explain y. Which of the following statements is correct?

View Explanation

Log in for full answers

Similar Questions

The bias caused by leaving a variable out of an equation is called

Consider the following specification "situation:" In an equation explaining student GPAs at HUTB, the bias imparted to the coefficient of hours devoted to study (per course) of omitting the number of campus social organizations is

Consider the following simple regression model: y = β0 + β1x1 + u (1) and the following multiple regression model: y = β0 + β1x1 + β2x2 + u (2), where x1 is the variable of primary interest to explain y. Which of the following statements is correct?

The correlation between the regressor educ and the error term u is most likely caused by

More Practical Tools for Students Powered by AI Study Helper

Homework AI Solver

Stylized AI Paper Writer

Plagiarism Checker Assistant

Citation AI Academic Writing Tool

In-Class Translation Assistant

AI Note Generator

AI Quiz Answers

Past Exam Questions from University Test Bank

Smart Practice Assistant

Adaptive Practice

Making Your Study Simpler