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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.Sometimes it is known in advance that the least-squares regression line must go through the origin, i.e., the regression model is of the form Y=BX;+;. i=1.2. ... /. where &;'s are i.id. N(0, 6- ). and the equation of the regression line is y = B .x. In this case, finding the least-squares line reduces to finding the value B that minimizes the expression f(B)= [[y-B.x;]2. i-l Use the derivative of f with respect to B to derive the formula for the slope of the least-squares regression line in this case.Prob 2 - Mixed Integer Program (MIP) Formulation (33%) Review the problem formulation Sections 1 1.2 through 1 1.4 of Hillier and Lieberman (2010) again. Then consider the following mathematical model Minimize = =f,(x,) +f(x2) subject to the restrictions one at a time (1) Either x, 23 or x, 23 (2) At least one of the following inequalities holds: 2x1 + 12 27 (3) |x, - X21 = 0, or 3, or 6. *+21, 27 (4) x, 20, X2 20; where fi(x , ) = ] 17+5x, if x, >0 0 if x, =0. 12( x , ) = 3 15+6X2 if x,>0 0 if x, =0. Formulate each problem as an MIP problem.The point of this exercise is to show that tests for functional form cannot be relied on as a general test for omitted variables. Suppose that, conditional on the explanatory variables x, and x2, a linear model relating y to x, and x2 satisfies the Gauss-Markov assumptions: y = Bo + Biki + Betz + u E(w/x1, x2) = 0 Var(w/x1, x2) = 0. To make the question interesting, assume B * 0. Suppose further that x, has a simple linear relationship with x1: * = bo + 6 1 + r E(rx,) = 0 Var(rx ) = 1. (1) Show that E(vx, ) = (Bo + B250) + (B, + B25,)x. Under random sampling, what is the probability limit of the OLS estimator from the simple regression of y on x,? Is the simple regression estimator generally consistent for B,? (ii) If you run the regression of y on x1, xi, what will be the probability limit of the OLS estimator of the coefficient on x7? Explain. (iii) Using substitution, show that we can write y = (Bo + Bzoo) + ( B, + B25, )x, + u + Byr. It can be shown that, if we define v = a + By/ then E(vix, ) = 0, Var(vix, ) = o' + Bir?. What consequences does this have for the / statistic on x, from the regression in part (ii)?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,$1), find the least-squares estimators of 80 and B1 for this model. Hint: Do not expand the term (xi-x). That is, do not expand ni= 1 yi*(xi-x) as nZi=1 yixi- nZi=1 yi x for easier computations. Also remember what nEi=1(xi-x) is. 3. [10 pts] Consider the simple linear regression model yi = Bo + BI(z; - 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 Hint: Do not expand the term (r; - I). That is, do not expand ) yi(x; - I) as ) Vidi - 1=1 n it for easier computations. Also remember what E(x - I) is. 1-1 i-12. 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)