(Solved):   8. Consider the simple linear regression model in question 7 . a. (6) Minimizin ...

 

8. Consider the simple linear regression model in question 7 . a. (6) Minimizing the sum of squared residuals, \( S\left(\beta_{1}, \beta_{2}\right)=\sum_{i=1}^{T} e_{t}^{2} \), derive the normal equations and solve them for \( b_{1} \) and \( b_{2} \) by using Cramer's Rule. b. (4) Show that solutions to \( \mathrm{b}_{2} \) and \( \mathrm{b}_{1} \) in part (a) be simplified to \( b_{2}=\frac{\sum_{t=1}^{T}\left(x_{t}-\bar{x}\right)\left(y_{t}-\bar{y}\right)}{\sum_{t=1}^{T}\left(x_{t}-\bar{x}\right)^{2}} \) and \( b_{1}=\bar{y}-b_{2} \bar{x} \). c. (4) Using results in part (b) show that \( b_{1} \) and \( b_{2} \) are unbiased. Interpret your results. d. (4) Using your results in parts (b) and (c) derive the variance for \( b_{2} \). e. (4) Using the normal equations show that OLS estimator for \( \beta=\left(\beta_{1} \beta_{2}\right)^{\prime} \) can be expressed \( \hat{\beta}=\left(X^{\prime} X\right)^{-1} X^{\prime} Y \). f. (4) Using your result in (e) show that \( \operatorname{Var}(\hat{\beta})=\sigma^{2}\left(X^{\prime} X\right)^{-1} \). g. (4) Show that the variances you derived for \( b_{2} \) in parts (d) and (f) are identical. h. (6) Show that the matrix \( X^{\prime} X \) is positive definite. Explain what this result means in terms of minimizing the sum of squared errors. 9. Consider the simple linear regression model in question 7. a. (3) Explain what exactly the coefficient of determination \( R^{2} \) tells us. For example, if \( R^{2}=0.50 \), how exactly would you interpret it. b. (3) Indicate the lowest and highest possible values for \( R^{2} \) and prove your claim. c. (3) Is there any relationship between the coefficient of determination \( R^{2} \) and the simple correlation coefficient \( r \). Prove your claim. d. (3) Show that if a constant is not included in the regression model, i.e. \( y_{t}=\beta x_{t}+e_{t} \), \( R^{2}=\frac{S S R}{S S T}=1-\frac{S S E}{S S T} \) can take negative values. 10. The transit corporation of Greater Adana Municipality has estimated the following CobbDouglas production function using monthly observations for the past two years. \( \ln Q=2.303+0.4 \ln K+0.6 \ln L+0.2 \ln G, \quad S S E=1805, S S T=13581, D-W=2.20 \) \( (0.14) \quad(0.16) \quad(0.07) \) Where \( Q \) is the number of bus miles driven, \( K \) is the number of buses the firm operates, \( \mathrm{L} \) is number of bus drivers it employs each day, and \( G \) is the liters of diesel it uses. Numbers inside the parentheses are standard errors. a. (3) What is the economic meaning of the coefficients in the model? Prove your claim. b. (3) Interpret the estimated coefficients in an economic sense. c. (3) Do the signs of estimated coefficients agree with your expectations? Explain. d. (3) Estimate \( Q \) if \( K=200, L=400, G=4,000 \). e. (3) Rewrite the estimated production function in the form of a power function. f. (3) Find the marginal product of \( K, L \), and \( G \). a. (2) Find the expected grade in the class at the end of the semester. b. (3) Assuming three exams are independent, find the standard deviation of "grade" at the end of the semester. c. (6) Now suppose exam 3 is independent of exam 1 and 2, but the correlation between exam 1 and 2 is 0.7. Would this change alter your answers to parts (a) and (b)? If so, how? 4. The per capita income in Cukurova region has approximately normally distributed with 15,000 TL mean and 5,000 TL standard deviation. Also, the poverty level is determined as 8,000 TL. a. (2) What proportion of population lives in poverty? b. (4) Which level of income corresponds to \( 90 \% \) of income distribution? In other words, what is the level of income that is exceeded by only \( 10 \% \) of the population? 5. (6) Indicate the lowest and highest possible values of the simple correlation coefficient \( \boldsymbol{r} \) between the variables \( X \) and \( Y \) and prove your claim. 6. Consider sample size of \( T \) on an independently and identically distributed random variable \( Y \) with mean \( \mu_{y} \) and variance \( \sigma_{y}^{2} \). Suppose you calculate the variance for \( Y \) as \( s^{2}=\left[\frac{\sum_{t=1}^{T}\left(y_{t}-\bar{y}\right)^{2}}{T}\right] \) a. (3) Show that \( \operatorname{var}(\bar{y})=\sigma_{y}^{2} / T \). b. (4) Find \( E\left[s^{2}\right] \). c. (3) Is \( E\left[s^{2}\right]=\sigma_{y}^{2} \) ? If not, what modification would you need in the formula for \( s^{2} \) ? d. (3) Interpret your answers in (b) and (c) using the concept of "degrees of freedom." 7. Take the simple regression model, \( y_{t}=\beta_{1}+\beta_{2} x_{t}+e_{t} \), and consider that all usual assumptions, including normality are satisfied. a. (3) Find the expected value of the least squares residual \( \hat{e}_{t} \). Show your work. b. (4) Show that \( E(\bar{y})=E\left[\frac{\sum_{t=1}^{T} y_{t}}{T}\right]=\beta_{1}+\beta_{2} \bar{x} \), where \( \bar{x}=\sum_{t=1}^{T} x_{t} / T \). Clearly state any assumptions you are employing at each step of your answer. c. (3) Show that \( \operatorname{var}(\bar{y})=\sigma^{2} / T \). Clearly state any assumptions you are employing at each step of your answer. d. (5) "If \( \mathrm{I} \) am running a regression of \( \mathrm{y} \) on \( \mathrm{x} \), it is better to have all the \( \mathrm{x} \) values bunched up and close together, so that I have a better idea of where the least squares regression line goes." Is this statement true or false? Explain your answer.