To analyze the impact of a change of one exogenous variable, should we consider possible change of other exogenous variable at the time under “ceteris parabis” assumption
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This assumption means "all other things being equal" or "holding other things constant." It allows us to isolate the effect of one variable by assuming that other relevant factors remain unchanged. Show more…
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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 $x_{1}, x_{2}, x_{3}$ i. 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 $\bar{\beta}_{1}$ and $\hat{\beta}_{1}$ to be similar or very different? Explain. ii. 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. iii. If $x_{1}$ is highly correlated with $x_{2}$ and $x_{3}$, and $x_{2}$ and $x_{3}$ have small partial effects on $y$, would you expect $\operatorname{se}\left(\tilde{\beta}_{1}\right)$ or $\operatorname{se}\left(\hat{\beta}_{1}\right)$ to be smaller? Explain. iv. If $x_{1}$ is almost uncorrelated with $x_{2}$ and $x_{3}, x_{2}$ and $x_{3}$ have large partial effects on $y,$ 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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