Question: Suppose that you are interested in estimating the ceteris paribus relationship between \(y\) and \(x_{1} .\) For this purpose, you can collect data on two control variables, \(x_{2}\) and \(x_{3}\). (For concreteness, you might think of \(y\) as final exam score, \(x_{1}\) as class attendance, \(x_{2}\) as GPA up through the previous semester, and \(x_{3}\) as SAT or ACT score.) Let \(\tilde{\beta}_{1}\) be the simple regression estimate from \(y\) on \(x_{1}\) and let \(\hat{\beta}_{1}\) be the multiple regression estimate from \(y\) on \(x_{1}, x_{2}, x_{3}\).

1. If \(x_{1}\) is highly correlated with \(x_{2}\) and \(x_{3}\) in the sample, and \(x_{2}\) and \(x_{3}\) have large partial effects on y, would you expect \(\tilde{\beta}_{1}\) and \(\hat{\beta}_{1}\) to be similar or very different? Explain.
2. If \(x_{1}\) is almost uncorrelated with \(x_{2}\) and \(x_{3}\), but \(x_{2}\) and \(x_{3}\) are highly correlated, will \(\tilde{\beta}_{1}\) and \(\hat{\beta}_{1}\) tend to be similar or very different? Explain.
3. If \(x_{1}\) is highly correlated with \(x_{2}\) and \(x_{3}\), and \(x_{2}\) and \(x_{3}\) have small partial effects on \(\mathrm{y}\), would you expect \(\operatorname{se}\left(\tilde{\beta}_{1}\right)\) or \(\operatorname{se}\left(\hat{\beta}_{1}\right)\) to be smaller? Explain.
4. If \(x_{1}\) is almost uncorrelated with \(x_{2}\) and \(x_{3}, x_{2}\) and \(x_{3}\) have large partial effects on \(\mathrm{y}\), and \(x_{2}\) and \(x_{3}\) are highly correlated, would you expect \(\operatorname{se}\left(\tilde{\beta}_{1}\right)\) or \(\operatorname{se}\left(\hat{\beta}_{1}\right)\) to be smaller? Explain.

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