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1 Amherst College Department of Economics Economics 360 Fall 2012 Solutions: Wednesday, December 12 Beef Market Data: Monthly time series data relating to the market for beef from 1977 to 1986 Q t P t FeedP t Inc t ChickP t Year t Quantity of beef in month t (millions of pounds) Real price of beef in month t ( cents per pound) Real price of cattle feed in month t ( cents per pounds of corn cobs) Real disposable income in month t (thousands of chained 2005 dollars) Real rice of whole chickens in month t ( cents per pound) Year [Link to MIT-BeefMarket wf1 goes here] Consider the linear model of the beef market we used in class: Equilibrium: Q D t = QS t = Q t and the estimates of the reduced form equation: EstQ = 138, Inc 33200FeedP EstP = Inc FeedP 1 Focus on the linear demand model a In words, interpret the Inc coefficient estimate, b D I Express this more mathematically by filling in the following blanks: b D I estimates how the quantity of beef demanded changes when income changes while the price of beef remains constant b D I = ΔQD ΔInc while P remains constant
2 2 b Consider the price reduced form (RF) estimates: EstP = FeedP Inc 1) What equation estimates the change in the price, ΔP, when both income ΔP = 10562ΔFeedP ΔInc 2) When the while condition cited in part a is satisfied, how must the change in income, ΔInc, and the change in feed prices, ΔFeedP, be related? Solve the equation for ΔFeedP? While condition: P remains constant means that ΔP equals 0: 0 = 10562ΔFeedP ΔInc Solving for ΔFeedP 10562ΔFeedP = ΔInc ΔFeedP = ΔInc = ΔInc c Consider the quantity reduced form (RF) estimates: EstQ = 138, FeedP Inc 1) What equation estimates change in the quantity, ΔQ, when both income ΔQ = 33200ΔFeedP ΔInc 2) Substitute in your answer to part b2 Then, recall your answer to part a to calculate the numerical value of b D I ΔQ = 33200ΔFeedP ΔInc From b2: ΔFeedP = ΔInc = ( ΔInc) ΔInc = 5917ΔInc ΔInc = 23264ΔInc ΔQ ΔInc = b D I = 23264
3 3 2 Focus on the original supply model a In words, interpret the FeedP coefficient estimate, b S Express this more mathematically by filling in the following blanks: b S estimates how the quantity of beef supplied changes when the feed price changes while the price of beef remains constant b S = ΔQ S ΔFeedP while P remains constant b Consider the price reduced form (RF) estimates: EstP = FeedP Inc 1) What equation estimates the change in the price, ΔP, when both income ΔP = 10562ΔFeedP ΔInc 2) When the while condition cited in part a is satisfied, how must the change in income, ΔInc, and the change in feed prices, ΔFeedP, be related? Solve the equation for ΔInc While condition: P remains constant means that ΔP equals 0: 0 = 10562ΔFeedP ΔInc Solving for ΔInc ΔInc = 10562ΔFeedP ΔInc = ΔFeedP = 56106ΔFeedP c Consider the quantity reduced form (RF) estimates: EstQ = 138, FeedP Inc 1) What equation estimates change in the quantity, ΔQ, when both income ΔQ = 33200ΔFeedP ΔInc 2) Substitute in your answer to part b2 Then, recall your answer to part a to calculate the numerical value of b D I ΔQ = 33200ΔFeedP ΔInc From b2: ΔInc = 56106ΔFeedP = 33200ΔFeedP ( 56106ΔFeedP) = 33200ΔFeedP 97327ΔFeedP = 1,305ΔFeedP ΔQ ΔFeedP = 1,305 b S = 1,305
4 4 Beef Market Data: Monthly time series data relating to the market for beef from 1977 to 1986 Q t P t FeedP t Inc t ChickP t Year t Quantity of beef in month t (millions of pounds) Real price of beef in month t ( cents per pound) Real price of cattle feed in month t ( cents per pounds of corn cobs) Real disposable income in month t (thousands of chained 2005 dollars) Real rice of whole chickens in month t ( cents per pound) Year Consider the model for the beef market that we used in the last chapter: Equilibrium: Q D t = QS t = Q t Endogenous Variables: Q t and P t Exogenous Variables: FeedP t and Inc t 3 We shall now introduce another estimation procedure for simultaneous equation models, the two stage least squares (TSLS) estimation procedure: 1 st Stage: Estimate the variable that is creating the problem, the explanatory endogenous variable: Dependent variable: Original endogenous explanatory variable that creates the bias problem Explanatory variables: All exogenous variables 2 nd Stage: Estimate the original models using the estimate of the problem explanatory endogenous variable Dependent variable: Original dependent variable Explanatory variables: 1st stage estimate of the problem explanatory endogenous variable and any relevant exogenous explanatory variable Naturally, begin by focusing on the first stage of the two stage least squares estimation procedure 1 st Stage: Estimate the variable that is creating the problem, the explanatory endogenous variable: Dependent variable: Original endogenous explanatory variable that creates the bias problem In this case, the price of beef, P t, is the problem explanatory variable Explanatory variables: All exogenous variables In this case, the exogenous variables are FeedP t and Inc t Using the ordinary least squares (OLS) estimation procedure, what equation estimates the problem explanatory variable, the price of beef? [Link to MIT-BeefMarket wf1 goes here]
5 5 Dependent Variable: P Explanatory Variables: FeedP and Inc Dependent Variable: P Method: Least Squares Sample: 1977M M12 Included observations: 120 Coefficient Std Error t-statistic Prob FEEDP INC C EstP = FeedP Inc Generate a new variable, EstP, that estimates the price of beef based on the 1 st stage 4 Next, we focus on the 2 nd stage and consider the demand model: 2 nd Stage: Estimate the original models using the estimate of the problem explanatory endogenous variable Dependent variable: Original dependent variable In this case, the original explanatory variable is the quantity of beef, Q t Explanatory variables: 1st stage estimate of the problem explanatory endogenous variable and any relevant exogenous explanatory variable In this case, the estimate of the price of beef and income, EstP t and Inc t Demand Model: Explanatory Variables: EstPrice and Inc a Using the ordinary least squares (OLS) estimation procedure, estimate the parameters of the demand model Explanatory Variables: EstPrice and Inc Method: Least Squares Sample: 1977M M12 Included observations: 120 Coefficient Std Error t-statistic Prob ESTP INC C Estimates: b Const = 149,106 b EstP = 3143 b Inc = 23264Inc b Compare these two stage least squares coefficient estimates for the demand model with the estimates computed using the reduced form estimation procedure in the previous chapter They are identical
6 6 5 Now, consider the supply model: and the second stage of the two stage least squares estimation procedure 2 nd Stage: Estimate the original models using the estimate of the problem explanatory endogenous variable Dependent variable: Original dependent variable In this case, the original explanatory variable is the quantity of beef, Q t Explanatory variables: 1st stage estimate of the problem explanatory endogenous variable and any relevant exogenous explanatory variable In this case, the estimate of the price of beef and income, EstP t and PFeed t Supply Model: Explanatory Variables: EstPrice and FeedP a Using the ordinary least squares (OLS) estimation procedure, estimate the parameters of the supply model Explanatory Variables: EstPrice and FeedP Method: Least Squares Sample: 1977M M12 Included observations: 120 Coefficient Std Error t-statistic Prob ESTP FEEDP C Estimates: b Const = 108,292 b EstP = 9215 b = 1,305 b Compare these two stage least squares coefficient estimates for the demand model with the estimates computed using the reduced form estimation procedure in the previous chapter They are identical
7 7 6 Reconsider the following simultaneous equation model of the beef market and the reduced form estimates: Demand and Supply Models: Equilibrium: Q D t = QS t = Q t Endogenous Variables: Q t and P t Exogenous Variables: Inc t and FeedP t Reduced Form (RF) Estimates Quantity Reduced Form Estimates: EstQ = a Q Const + aq FeedP t + Price Reduced Form Estimates: EstP = a P Const + ap FeedP t + aq I Inc t ap I Inc t a Focus on the reduced form estimates for the income coefficients: 1) The reduced form income coefficient estimates, a Q I and a P I, allowed us to estimate the slope of which curve? Demand X Supply 2) If the reduced form income coefficient estimates were not available, would we be able to estimate the slope of this curve? No b Focus on the reduced form estimates for the feed price coefficients: 1) The reduced form feed price coefficient estimates of these coefficients, a Q and a P, allowed us to estimate the slope of which curve? X Demand Supply 2) If the reduced form feed price coefficient estimates were not available, would we be able to estimate the slope of this curve? No
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