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1. Applied microeconomists have often estimated systems of demand equations in which expenditures on each category of goods depends on total expenditure. Such a demand system would be consistent with a two-stage decision-making process in which house- holds first decide how much to spend and how much to save, and then decide how to divide their total spending across different categories of goods. Consider a very simple system in which households must divide their total expenditure Y, between food (F;). clothing (C;), and entertainment (E;). Household i's system of demand equations can be written as Fi = (1 + 02PF; + 03PC; + aAPEi + asYi + of Ci = Bi + B2PF; + B&PC. + BAPE, + BsVi+ , E = 1 + 72PFi + 13PC; + yPE + 7site, where PF, is the price of food; PC, is the price of clothing; PE, is the price of enter- tainment; 67 , e, , and er are random error terms; and Y, = Fi+ C;+ E; for all i. Show that if each equation is estimated using OLS, then as + bs + cs = 1 and ax + by + Ck = 0, k = 1, ...,4, where a, is the OLS estimator of ox, b, is the OLS estimator of By, and c is the OLS estimator of 7%, k = 1, . ..,5. 2. Consider the following two linear regression models: y = XIB + X2B2 + E (1) and y =XIBite. (2) Suppose that equation (1) is the true specification of the model, but due to a lack of data on the variables in X2, you have estimated equation (2) instead. Let b, and b2 be the OLS estimators of , and B2 when equation (1) is estimated, and let ba be the OLS estimator of B, when equation (2) is estimated. (a) Write down a formula for ba. (b) Derive E(b. | X1,X2). (c) Is be an unbiased estimator of 31, conditional on X, and X2? (d) Derive var(ba | X1, X2)