ECONOMETRICS代写| ECON 120B, ECONOMETRICS代写| midterm exam代写

(a) Let $X_{i} i=1, \ldots, n$ denote a sequence of random variables which are independent and identically distributed. Write down a formula for the sample mean.
(b) Consider the regression model $Y_{i}=\beta_{0}+\beta_{1} X_{i}+u_{i},$ and assume the first three least squares assumptions hold. State the Gauss Markov Theorem, including any additional assumptions needed for the theorem to hold (beyond the first three least squares assumptions).
(c) Define the average treatment effect (ATE) in terms of potential outcomes.

Consider the regression equation $Y_{i}=\beta_{0}+\beta_{1} X_{i}+u_{i} .$ Show that $\hat{\beta}_{1}=r_{X Y}\left(\frac{s_{Y}}{s_{X}}\right),$ where $r_{x y}$ is the sample correlation between $X$ and $Y,$ and $s_{x}$ and $s_{Y}$ are the sample standard deviations.

Females, on average, are shorter and weigh less than males. One of your friends tells you that in addition, females will weigh less for a given height. To test this hypothesis, you use a simple random sample to collect the height and weight of 59 female and 54 male students at UCSD. A regression of the weight on a constant, height, and a binary variable which takes a value of one for females and is zero otherwise, yields the following result:
weight $_{i}=-229.21-6.36 \times$ female $_{i}+5.58 \times$ height $_{i} \quad \mathrm{R}^{2}=0.50, \mathrm{RMSE}=20.99$
Where weight is weight measured in pounds, female takes a value of 1 for females and is 0 otherwise, and height is measured in inches.
(a) What is the literal interpretation of the intercept term? Does a negative intercept term imply a problem with the regression you ran? Explain.
(b) Is the estimated coefficient for height significantly different from zero at the $5 \%$ confidence level? Show your work.

(c) You decide to rescale the height variable by subtracting 5 feet (or 60 inches) from the original variable height. Will this change the estimated intercept term if you re-run the regression with this rescaled variable instead? Explain.
(d) Suppose you collected this data by having students fill out a survey. If men are not good at remembering their height, so that you have a noisy measure of their height, what problems (if any) does this create?
(e) Come up with one omitted variable which you think would create a bias for the estimated height coefficient. Which direction would this omitted variable bias the coefficient? Explain your reasoning

A subsample from the Current Population Survey (a survey the U.S. government uses to calculate official statistics) is taken on weekly earnings of individuals, their age, and their gender. You have read in the news that women make 70 cents for every $\$ 1$ that men earn. To examine this hypothesis, you first regress weekly earnings on a constant and a binary variable female, which takes on a value of 1 for females and 0 otherwise. The regression results are:
earn $_{i}=570.70-170.72 \times$ female $_{i} \quad \mathrm{R}^{2}=0.084, \mathrm{RMSE}=282.12, \mathrm{~N}=1740$
(a) What are the mean earnings of males and females in this sample?
(b) A classmate argues that you should control for education. He breaks education into 3 mutually exclusive and exhaustive categories: high school dropout or less, high school graduate, and at least one year of college. He says you should add three dummy variables into the model, one for each category. Is this good advice? Explain.

You decide to control for age (in years) in your regression results because you hypothesize that older people earn more on average than younger people, but at a decreasing rate as they age. This regression output is as follows:
$$
\begin{array}{l}
\text { earn }_{i}=323.70-169.78 \times \text { female }_{i}+15.15 \times \text { age }_{i}-.021 \times \text { age }^{2} \text { i } \\
\mathrm{R}^{2}=0.135, \mathrm{RMSE}=274.45, \mathrm{~N}=1,740
\end{array}
$$
(c) What words do the initials RMSE stand for? Interpret the RMSE.
(d) How much more, on average, does a 30-year-old male make per year in your sample compared to a 25 -year-old female? (note: you do not need to simplify your answer)
(e) Assuming homoskedasticity, test whether age and $a g e^{2}$ are jointly significant. Show your work. To help get you started, the formula for the homoskedasticity only F statistic is:
$$
\frac{\left(R_{\text {unrestricted }}^{2}-R_{\text {restricted }}^{2}\right) / q}{\left(1-R_{\text {unrestricted }}^{2}\right) /(n-k-1)}
$$