Chapter 6

Suppose we want to estimate the effects of alcohol consumption (alcohol) on college grade point average (colGPA). In addition to collecting information on grade point averages and alcohol usage. we also obtain attendance information (say, percentage of lectures attended, called attend). A standardized test score (say, $SAT$ ) and high school GPA (hsGPA) are also available. i. Should we include attend along with alcohol as explanatory variables in a multiple regression model? (Think about how you would interpret ${\beta }_{\text{alcohbol. }}$ ) ii. Should $SAT$ and $hsGPA$ be included as explanatory variables? Explain.

Consider the equation. $$\begin{array}{rl}y& ={\beta }_0+{\beta }_{1}x+{\beta }_2{x}^2+u\\ \\ \mathrm{E}(u|x)& =0\end{array}$$ where the explanatory variable $x$ has a standard normal distribution in the population. In particular, $\mathrm{E}(x)=0,\mathrm{E}\left(x^2\right)=\mathrm{Var}(x)=1,$ and $\mathrm{E}\left(x^3\right)=0.$ This last condition holds because the standard normal distribution is symmetric about zero. We want to study what we can say about the OLS estimator if we omit $x^2$ and compute the simple regression estimator of the intercept and slope. i. Show that we can write $$y={\alpha }_0+{\beta }_{1}x+v$$ where $E(v)=0.$ In particular, find $v$ and the new intercept, ${\alpha }_0$ ii. Show that $\mathrm{E}(v|x)$ depends on $x$ unless ${\beta }_2=0$ iii. Show that $\mathrm{Cov}(x,v)=0$ iv. If ${\hat{\beta }}_1$ is the slope coefficient from regression $y_i$ on $x_i$, is ${\hat{\beta }}_1$ consistent for ${\beta }_1$ ? Is it unbiased? Explain. v. Argue that being able to estimate ${\beta }_1$ has some value in the following sense: ${\beta }_1$ is the partial effect of $x$ on $\mathrm{E}(y|x)$ evaluated at $x=0$ the average value of $x$ vi. Explain why being able to consistently estimate ${\beta }_1$ and ${\beta }_2$ is more valuable than just estimating ${\beta }_1$