Amherst College epartment of Economics Economics 3 Fall 2015 Monday, ecember 7 roblem et olutions 1. Consider the following linear model: tate residential Election ata: Cross section data for the fifty states and the istrict of Columbia from 19 to 2012. IncumbRunning t 1 if incumbent resident running, 0 otherwise in year t UnemRate i t Unemployment rate (percent) in state i during year t Unemployment rate trend (percent) in state i during year; that is, the annual change in the unemployment rate VoteresartyTwo i t ercent of popular vote received by the resident s party based on the two major parties in state i during year VoteresartyTwo i t = Const + Trend UnemTrendi t + Mood + ei t where = National mood, positive or negative, in year t a. evelop a theory regarding how the explanatory variables influence the dependent variable. What does your theory imply about the sign of the coefficients? As we discussed before a rising unemployment rate should decrease the vote of the incumbent resident s party. Trend < 0. Conventional wisdom suggests that a positive nation mood increases the vote for the incumbent s residents party. Mood > 0. b. ince no measures of the national mood included in the data, what problems, if any, might concern you when using the ordinary least squares (OL) estimation procedure to calculate the estimate of the unemployment trend? Explain. VoteresartyTwo i t = Const + Trend + Mood = Const + Trend + t where t = Mood Included variable up and negatively correlated Typically, omitted variable down t = Mood é t down Mood > 0 ã and t negatively correlated Ordinary least squares (OL) estimation procedure for the value of the UnemTrend coefficient will be biased downward.
2 c. Using the ordinary least squares (OL) estimation procedure to estimate the value of the unemployment trend coefficient from the data included in the workfile. Interpret the coefficient estimate. EViews residential Elections Results by tate 19 to 2012 ependent Variable: VOTEREARTYTWO Method: anel Least quares ample (adjusted): 19 2012 eriods included: 9 Cross-sections included: 51 Total panel (balanced) observations: 459 Variable Coefficient td. Error t-tatistic rob. UNEMTREN -3.172853 0.449397-7.0249 0.0000 C 49.78256 0.489894 101.6189 0.0000 Estimated Equation: VoteresartyTwo = 49.8 3.2UnemTrend Interpret the estimate: A 1 percent rise in the unemployment rate decreases the vote for the resident s party by 3.2 percent d. Introduce period fixed effects to estimate the the value of the unemployment rend coefficient. ependent Variable: VOTEREARTYTWO Method: anel Least quares ample (adjusted): 19 2012 eriods included: 9 Cross-sections included: 51 Total panel (balanced) observations: 459 Variable Coefficient td. Error t-tatistic rob. UNEMTREN -0.3390 0.796565-0.477538 0.6332 C 50.21956 0.4850 104.4392 0.0000 eriod fixed (dummy variables) Effects pecification Estimated Equation: VoteresartyTwo = 50.2.4UnemTrend Interpret the estimate: A 1 percent rise in the unemployment rate decreases the vote for the resident s party by.4 percent e. How do your answers to this problem tie together? art b: When we omit the national mood, as we must, we fear that the ordinary least squares (OL) estimation procedure for the unemployment trend coefficient estimate will be biased downward. art c: The ordinary least squares (OL) estimation procedure estimates the value of the unemployment trend coefficient estimate to be 3.2. art d: We include period fixed effects to account for the national mood affecting all states during the election year. Our estimate of the for the coefficient of unemployment trend increases to.4. This is consistent with our logic.
3 2. (rep roblem) In words answer the following questions: a. What is the goal of multiple regression analysis? Multiple regression analysis attempts to sort out the individual effect that each explanatory variable has on the dependent variable. b. What is the interpretation of each coefficient in the regression model? Each explanatory variable s coefficient reveals the individual impact that the explanatory variable has on the dependent variable; that is, each explanatory variable s coefficient tells us how changes in that explanatory variable affect the dependent variable while all other explanatory variables remain constant. Consider the following multiple regression model: General Regression Model: y = Const + x1 x1 + x2 x2 + e ince the actual parameters of the model, the s, are unobservable, we estimate them. The estimated parameters are denoted by italicized Roman b s: Esty = b Const + b x1 x1 + b x2 x2 In terms of the estimated coefficients, b x1 and/or b x2, what is the expression for the estimated change in y? c. If x1 changes by x1 while x2 constant: y = b x1 x1 d. If x2 changes by x2 while x1 constant: y = b x2 x2 e. utting parts c and d together, if both x1 and x2 change: y = b x1 x1 + b x2 x2
4 3. (rep roblem) Consider the following model of the U.. beef market: emand Model: = 100,000 10,000 + 150Inc upply Model: = 190,000 + 5,000 6,000Feed Equilibrium: = = where uantity of beef (millions of pounds) Real price of beef (cents per pound) Inc Real disposable income (thousands of dollars) Feed Real price of cattle feed (cents per pounds of corn cobs) a. Use algebra to solve for the equilibrium price and quantity. That is, 1) Express the equilibrium price,, in terms of Feed and Inc. = 100,000 10,000 + 150Inc = 190,000 + 5,000 6,000Feed 90,000 + 150Inc + 6,000Feed = 15,000 6 +.01Inc +.4Feed = 2) Express the equilibrium quantity,, in terms of Feed and Inc. = 190,000 + 5,000 6,000Feed = 190,000 + 5,000(6 +.01Inc +.4Feed) 6,000Feed = 190,000 30,000 + 50Inc + 2,000Feed 6,000Feed = 1,000 + 50Inc 4,000Feed These two equations are called the reduced form equations. b. b. uppose that Feed equals and Inc equals 4,000. 1) What are the numerical values of the equilibrium price and quantity? = 6 +.01Inc +.4Feed = 6 +.014,000 +.4 = 6 + + 16 = 50 = 1,000 + 500Inc 4,000Feed =1,000 + 504,000 4,000 =1,000 + 200,000 1,000 =200,000 2) On a sheet of graph paper, plot the demand and supply curves to illustrate the equilibrium. 20
5 c. Assume that you did not know the equations for the demand and supply models. On the other hand, you do know the reduced form equations that you derived in part a. 1) uppose that Inc were to rise from 4,000 to 6,000 while Feed remains constant at. Using the reduced form equations calculate the new equilibrium price and quantity. i. Will the demand curve shift? Explain. Yes. Higher income causes the demand curve to shift to the right. ii. Will the supply curve shift? Explain. No. The supply curve does not shift because feed prices are constant. iii. On a sheet of graph paper, plot the demand curve(s), the supply curve(s), and the two equilibria. iv. Based on the numerical values of the two equilibria can you calculate the slope of the supply curve? Explain. Yes. ince the supply curve has not shifted the two equilibria lie on the same supply curve. Accordingly, we can calculate its slope. v. Based on the numerical values of the two coefficient of the demand model? Explain. No. The demand curve has shifted. Hence, the two equilibria lie on different demand curves. 2) Instead, suppose that Feed were to rise from to while Inc remains constant at 4,000. Using the reduced form equations calculate the new equilibrium price and quantity. i. Will the demand curve shift? Explain. No. The demand curve does not shift because income is constant. ii. Will the supply curve shift? Explain. Yes. Higher feed prices cause the supply curve to shift to the left. iii. On a sheet of graph paper, plot the demand curve(s), the supply curve(s), and the two equilibria. iv. Based on the numerical values of the two coefficient of the demand model? Explain. Yes. ince the demand curve has not shifted the two equilibria lie on the same demand curve. Accordingly, we can calculate its slope. v. Based on the numerical values of the two coefficient of the supply model? Explain. No. The supply curve has shifted. Hence, the two equilibria lie on different supply curves. 20 20 Δ Δ Δ Δ