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1. The simple linear regression model Y, = Bo + Bir; +;, &; " N(0, 0'), i = 1, ..., n, can also be written as a general linear model Y = XB+E, where Y = (Y1, . . ., Yn), E = (61, . . . .E.)? ~ Na. (0, 621), B = (Bo, Bi) and T2 X = In In this problem, we will show that the least squares estimate B = (XTX)-1XTY derived from multiple linear regression coincides with the least squares estimates from simple linear regression. (a) Using straightforward matrix multiplication, show that XX= (ni _mix ) =n( x Smo+2) XTY = _ _X ) =(SxY + niY (b) Using the identity ( " * ) d ad - bc -c a for a 2 x 2 matrix, show that Sxx + 12 (x X)= -I SXX -T 12. Again, consider the general linear model Y = X8 + , with & ~ N,(0, o'/), where the first column of X consists of all ones. (a) Using facts about the mean and variance/covariance of random vectors given in lecture, show that the least squares estimate from multiple linear regression satisfies E(B) = 8 and Var(B) = ?(XTX)-1. (b) Let H = X(X X)-1XT be the hat matrix, Y = HY be the fitted values and e= (I - H)Y be residuals. Using properties derived in class, show that Can = 0. i=1 This fact is used to provide the ANOVA decomposition SSTO = SSE + SSR for multiple linear regression. (Hint: The sum above can be written as e Y. Apply properites of H.)3. [10 pts] Consider the simple linear regression model yi = 60+81(xi-x) +ci(i= 1,2,..,n), where x= ni=1 xi. From the least-squares criterion S(80,81), find the least-squares estimators of 80 and 1 for this model. Hint: Do not expand the term (xi-x). That is, do not expand nti= 1 yi*(xi-x) as nZi=1 yixi- nZi=1 yi x for easier computations. Also remember what ni=1(xi-x) is. 3. [10 pts] Consider the simple linear regression model yi = Po + Bi(x; - I) + &; (i=1,2,...,n), where I = _, I;. From the least-squares criterion S(Bo, 81), find the least-squares estimators of Bo and B, for this model. n n Hint: Do not expand the term (r; - I). That is, do not expand ) yi(x, - F) as ur - 1= 1 1=1 n it for easier computations. Also remember what E(x - 1) is. i-1 i-11. Some (More) Math Review a) Let N = 3. Expand out all the terms in this expression: Cov Xi b) Now write out all the terms using the formula from last class for variance of a sum: Var( X:) = _Var(X) + > > Cov(X, X;) i-1 1=1 i-lj=1ifi Verify that (a) and (b) are giving you the same thing. Hint: Cov(X, X) = Var(X). c) Suppose that D is a roulette wheel that takes on the values {1, 2, 3} all with probability 1/3. What is the expected value of this random variable? What about the variance? d) Now suppose that M is another roulette wheel that takes on the values {1, 2,3} all with probability 1/3. Solve for the expected value of 1/M. e) Finally, suppose that D and M are independent. Solve for: E Hint: You do not need to do any new calculations here. Just remember that for independent RVs, E(XY) = E(X)E(Y). f) Does E(D/M) = E(D)/ E(M)?A researcher has data on class size (CS) and average test scores from 100 econometrics classes (TS). He estimates the OLS regression and obtains the following regression line: TS = 520.4-5.82 x CS reg (20.4) (2.21) where the R2 = 0.08, SER = 11.5, and the standard errors of each coefficient is in parentheses underneath. inf a) What are Bo and B1, and what are their standard errors? 51 b) Interpret the R for this regression. c) A classroom has 22 students. What is the regression's prediction for that classroom's average test score? d) Last year a classroom had 19 students and this year it has 23 students. What is the regression's prediction for the change in the classroom average test score? (Hint: A change of 1 unit of X induces a change of B 1 units in Y, so a change in 4 units of X will induce what change in Y?) e) The sample average class size across the 100 classrooms is 21.4. What is the sample average of test scores across the 100 classrooms? (Hint: The estimated regression line (above) gives you the average relationship between Y and X!) f) Is Bi significant at the 5% significance level (i.e. can you reject the null that B 1 = 0)? (Hint: Same procedure as 4b where the critical value is now for 98 degrees of freedom at the 5% level-see Table 2) . g) Construct a 95% confidence interval for 1, the regression slope coefficient.3. An economics department at a large state university keeps track of its majors' starting salaries. Does taking econometrics affect starting salary? Let SAL = salary in dollars, GPA = grade point average on a 4.0 scale, METRICS = 1 if student took econometrics, and METRICS = 0 otherwise. Using the data file metrics.dat, which contains information on 50 recent graduates, we obtain the estimated regression SAL = 24200 + 1643GPA + 5033METRICS R= = 0.74 (se ) (1078) (352) (456) (a) Interpret the estimated equation. (b) How would you modify the equation to see whether women had lower starting salaries than men? (Hint: Define an indicator variable FEMALE = 1, if female; zero otherwise.) (c) How would you modify the equation to see if the value of econometrics was the same for men and women?1. Consider the simple regression model: Vi = Bo+ Biri + Hi, for i = 1, ... . n, with E(uilz,) - 0 and let a be a dummy instrumental variable for I, such that we can write: Ci =not matu with E(uilz;) = 0 and E(viz) =0. (c) Denote by no, the number of observations for which = = 0 and by n, the number of observations for which a, = 1. Show that: (a - 2) = =(n-m). 1=1 and that: [( - =)(y: - 9) = -(n - n) (31 - 30) . where to and g are the sample means of y for z equal to 0 and 1 respectively. ( Hint: Use the fact that n = nj + no, and that = = m). (d) Now we regress y on i to obtain an estimator of &. From the standard formula of the slope estimator for an OLS regression and using the result in (c), show that: B - 91 -90 $1 - To This estimator is called the Wald estimator.