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Consider the data on the variables X, Y, and Z in multireg_eg.csv. The variables X and Y are continuous variables, whereas Z takes values 1, 2, 3, 4, and 5. The data are displayed in Session 3 Slide 60. The data can be considered to be a random sample from a population where Y = β₀ + β₁X + β₂Z + ε, E(ε | X, Z) = 0, β₁ > 0, β₂ > 0, Cov(X, Z) < 0. Suppose your interest is in measuring the effect of X on Y, controlling for Z. As illustrated in Session 3 Slide 60, and in line with our assumptions about the population, for any fixed Z, the effect of X on Y is positive. However, because Z is positively correlated with Y and negatively correlated with X, a simple linear regression of Y on X produces a negative coefficient estimate on X. We argued in class that we cannot hold Z fixed and argued that the solution was to use multiple linear regression, regressing Y on X and Z together. For estimating the effect of X on Y, this has the effect of stripping out the effect of Z on both Y and X, thereby 'controlling' for Z.

Explore a detailed analysis of how to measure the effect of X on Y in a regression model with the control variable Z. This problem covers multiple regression, linear regression concepts, and the calculation of unbiased estimators. Suitable for undergraduate students in statistics or econometrics.

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