1. Consider the simple regression model: V/i = Po+ Piri +ui, for i = 1, ..., n, with E(ur,) 7 0 and let z be a dummy instrumental variable for a, such that we can write: with E(uilz;) = 0 and E(vilzi) = 0. (c) Denote by no, the number of observations for which z =0 and by n, the number of observations for which z, = 1. Show that: (a - 2) = =(n-m). 1=1 and that: [(8 -=)(3: - 9) = 7 "(n - ni) (31 - 90) . 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 81. From the standard formula of the slope estimator for an OLS regression and using the result in (c), show that: By1 - 90 I1 - To This estimator is called the Wald estimator.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-12. Consider the general linear model Y = XA + c, with : ~ N,,(0,o'I,, ), 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 and Var (8) = 03(XTX)-1. (b) Let H = X(X X) 'X be the hat matrix, Y = HY be the fitted values and e = (I - H)Y be residuals. Using these matrices and properties derived in class, show that E[( x - P) ( 1 - P)] = 0. im This fact is used to provide the ANOVA decomposition SSTO = SSE + SSR for multiple linear regression.\f2. Again, consider the general linear model Y = XB + , with & ~ Nn(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) = B and Var(B) = 03(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 n Ex = 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.)1. 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)?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.
