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Lecture 22 Preview: Simultaneous Equation Models – Introduction
Review: Explanatory Variable/Error Term Correlation Simultaneous Equation Models – Demand and Supply Endogenous versus Exogenous Variables Single Equation versus Simultaneous Equation Models Demand Model Ordinary Least Squares (OLS) Estimation Procedure: Our Suspicions Confirming Our Suspicions Supply Model Ordinary Least Squares (OLS) Estimation Procedure: Our Suspicions Confirming Our Suspicions An Example: The Market for Beef Demand and Supply Models Ordinary Least Squares (OLS) Estimation Procedure Reduced Form (RF) Estimation Procedure Comparing Ordinary Least Squares (OLS) and Reduced Form Estimates Justifying the Reduced Form (RF) Estimation Procedure Two Paradoxes Resolving the Paradoxes: The Coefficient Interpretation Approach The Coefficient Interpretation Approach: Intuition and a Bonus
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Explanatory Variable and Error Term Are Positively Correlated
Review: The Ordinary Least Squares (OLS) Estimation Procedure, Explanatory Variable/Error Term Correlation, and Bias Esty = bConst + bxx Esty = bConst + bxx Explanatory Variable and Error Term Are Positively Correlated Explanatory Variable and Error Term Are Negatively Correlated Estimated Equation More Steeply Sloped Than Actual Equation Estimated Equation Less Steeply Sloped Than Actual Equation OLS Estimation Procedure for Coefficient Value Is Biased Up OLS Estimation Procedure for Coefficient Value Is Biased Down
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Qt = Equilibrium Quantity Pt = Equilibrium Price
Simultaneous Equation Models: Demand and Supply Demand and Supply Models Price Equilibrium: Notation. NB: Both Qt and Pt are determined simultaneously. S Endogenous versus Exogenous Variables Pt Endogenous variables – Variables we are trying to explain, variables determined “within” the model: Qt and Pt D Exogenous variables – Variables we take as given, variables determined “outside” the model: Other Demand Factors and Other Supply Factors Quantity Qt Qt = Equilibrium Quantity Pt = Equilibrium Price Single Equation Models versus Simultaneous Equation Models In both types of models the dependent variable is endogenous. In a single equation model, all explanatory variables are exogenous. In a simultaneous equation model, an explanatory variable can be either an endogenous or an exogenous variable. In the demand and supply models Pt, the price, is an endogenous explanatory variable.
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Pt and eD are positively correlated
Ordinary Least Squares (OLS) Estimation Procedure and Endogenous Explanatory Variables Price Claim: The ordinary least squares (OLS) estimation procedure for the value of an endogenous explanatory variable’s coefficient will be biased. S D (eD up) Strategy: To justify this claim first focus on the demand model. Pt Pt = Equilibrium Price Question: Is the ordinary least squares (OLS) estimation procedure for the value of the price coefficient unbiased? D D (eD down) Quantity eD down eD up Pt down Pt up Pt and eD are positively correlated The price is an endogenous explanatory variable. OLS estimation procedure for the value of the price coefficient biased upward
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Confirming Our Logic: Endogenous Explanatory Variables
Is the estimation procedure for the price coefficient’s value unbiased?
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Demand Price Coefficient
Ordinary Least Squares (OLS) Estimation Procedure and Endogenous Explanatory Variables Price Claim: The ordinary least squares (OLS) estimation procedure for the value of an endogenous explanatory variable’s coefficient will be biased. S D (eD up) Question: In the demand model, is the ordinary least squares (OLS) estimation procedure for the value of the price coefficient unbiased? Pt Pt = Equilibrium Price D D (eD down) Quantity The price is an endogenous explanatory variable. eD down eD up Pt down Pt up Question: Is the OLS estimation procedure for the coefficient value of an endogenous explanatory variable: Pt and eD are positively correlated OLS estimation procedure for the value of the price coefficient biased upward Unbiased? No. Consistent? No. Lab 22.1 Actual Mean (Average) Estimation Sample Coef of Estimated Magnitude Procedure Size Value Coef Values of Bias OLS 4.0 OLS 4.0 OLS 4.0 1.4 Demand Price Coefficient
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Supply Price Coefficient
Claim: The ordinary least squares (OLS) estimation procedure for the value of an endogenous explanatory variable’s coefficient will be biased. Price S (eS down) S Pt = Equilibrium Price Question: In the supply model, is the ordinary least squares (OLS) estimation procedure for the value of the price coefficient unbiased? S (eS up) Pt D The price is an endogenous explanatory variable. Quantity eS down eS up Question: Is the OLS estimation procedure for the coefficient value of an endogenous explanatory variable: Pt up Pt down Pt and eS are negatively correlated Unbiased? No. OLS estimation procedure for the value of the price coefficient biased downward Even worse, the bias can be so severe that the mean of the coefficient estimate’s probability distribution could have the wrong sign. Actual Mean (Average) Estimation Sample Coef of Estimated Magnitude Procedure Size Value Coef Values of Bias Consistent? No. Lab 22.2 OLS OLS OLS 1.4 Supply Price Coefficient
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An Example: The Beef Market
Beef Market Data: Monthly time series data relating to the market for beef from 1977 to 1986. Qt Quantity of beef in month t (millions of pounds) Pt Real price of beef in month t ( cents per pound) FeedPt Real price of cattle feed in month t ( cents per pounds of corn cobs) Inct Real disposable income in month t (thousands of chained 2005 dollars) ChickPt Real price of whole chickens in month t ( cents per pound) Yeart Year Endogenous Variables: Qt and Pt Exogenous Variables: FeedPt and Inct We are trying to explain how Qt and Pt are determined We are not trying to explain how FeedPt and Inct are determined Qt and Pt are determined simultaneously within the model. FeedPt and Inct are determined outside the model. We take FeedPt and Inct as given.
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Ordinary Least Squares (OLS) Ordinary Least Squares (OLS)
Equilibrium: Endogenous Variables: Qt and Pt Exogenous Variables: FeedPt and Inct Beef Market Demand Model: Dependent Variable: Qt Explanatory Variables: Pt and Inct Ordinary Least Squares (OLS) Dependent Variable: Q Explanatory Variable(s): Estimate SE t-Statistic Prob P 0.0000 Inc Const Number of Observations 120 EViews Beef Market Supply Model: Dependent Variable: Qt Explanatory Variables: Pt and FeedPt Ordinary Least Squares (OLS) Dependent Variable: Q Explanatory Variable(s): Estimate SE t-Statistic Prob P 0.0000 FeedP Const Number of Observations 120 Is there a troubling result? If so, how might it be explained?
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Reduced Form (RF) Estimation Procedure
The Simultaneous Equation Model Equilibrium: Endogenous Variables: Qt and Pt Exogenous Variables: FeedPt and Inct Goal: Estimate the price coefficients of the demand and supply models. Step 1: Derive the Reduced Form (RF) Equations from the Original Models. The reduced form equations express each endogenous variable in terms of the exogenous variables only. Express the original simultaneous equation model’s parameters in terms of the reduced form parameters. Step 2: Use Ordinary Least Squares (OLS) Estimation Procedure to Estimate the Parameters of the Reduced Form (RF) Equations. Step 3: Calculate Coefficient Estimates for the Original Model Using the Estimates of the Reduced Form (RF) Equations.
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Step 1: Derive the Reduced Form (RF) Equations from the Original Models.
Endogenous Variables: Qt and Pt Exogenous Variables: FeedPt and Inct The reduced form (RF) equations express each endogenous variables in terms of the exogenous variables only: Notation: Express the original simultaneous equation model’s parameters in terms of the reduced form parameters. After elementary, yet laborious algebra: Express the reduce form parameters in terms of the original model’s parameters. Since the actual parameters are unobservable, replace them with their estimates. Divide Divide
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Ordinary Least Squares (OLS) Ordinary Least Squares (OLS)
Step 2: Use Ordinary Least Squares (OLS) Estimation Procedure to Estimate the Parameters of the Reduced Form (RF) Equations. Quantity Reduced Form Equation: Dependent Variable: Q Explanatory Variables: FeedP and Inc Ordinary Least Squares (OLS) Dependent Variable: Q Explanatory Variable(s): Estimate SE t-Statistic Prob FeedP 0.0073 Inc 0.0000 Const Number of Observations 120 EViews Price Reduced Form Equation: Dependent Variable: P Explanatory Variables: FeedP and Inc Ordinary Least Squares (OLS) Dependent Variable: P Explanatory Variable(s): Estimate SE t-Statistic Prob FeedP 0.0003 Inc Const 0.2895 Number of Observations 120 Step 3: Calculate Coefficient Estimates for the Original Model Using the Estimates of the Reduced Form Equations 332.00 17.347 = 314.3 = 921.5 1.0562
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Comparing Ordinary Least Squares (OLS) and Reduced Form (RF) Estimates
Price S Estimate “Slope” of Demand Curve Estimate “Slope” of Supply Curve Pt Qt = Equilibrium Quantity Pt = Equilibrium Price Ordinary Least Squares (OLS) 364.4 231.5 D Reduced Form (RF) 314.4 921.5 Quantity Qt Justifying the Reduced Form (RF) Estimation Procedure Demand Price Coefficient Supply Price Coefficient Actual Mean of Variance of Actual Mean of Variance of Estimation Sample Coef Estimated Estimated Coef Estimated Estimated Procedure Size Value Coef Values Coef Values Value Coef Values Coef Values Lab 22.3 5.4 RF 5.4 1.2 RF 1.2 .6 RF .6 Bad news: The reduced form (RF) estimation procedure for the coefficient value is biased. Good news: The reduced form (RF) estimation procedure for the coefficient value is consistent. As the sample size increases: The magnitude of the bias decreases. The variance of the coefficient estimates decreases.
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Two Paradoxes Endogenous Variables: Qt and Pt Exogenous Variables: FeedPt and Inct 332.00 17.347 = 314.3 = 921.5 1.0562 The estimate of the demand model’s price coefficient, , depends on the reduced form coefficients of feed price, and But feed price affects supply, not demand. Similarly, the estimate of the supply model’s price coefficient, , depends on the reduced form coefficients of income, and But income affects demand, not supply. We shall resolve these paradoxes by: Reviewing the goal of multiple regression analysis and the interpretation of the coefficient estimates. Applying the interpretation of the coefficient estimates to the original simultaneous equation demand and supply models. reduced form equations.
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Regression Model: yt = Const + Coef1x1t + Coef2x2t + et
Review: Goal of Multiple Regression Analysis and the Interpretation of the Coefficients Goal of Multiple Regression Analysis: Multiple regression analysis attempts to sort out the individual effect that each explanatory variable has on the dependent variable. Interpretation of Coefficients: Each explanatory variable’s coefficient reflects the individual impact which that 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. Regression Model: yt = Const Coef1x1t Coef2x2t + et Estimated Equation: Esty = bConst bCoef1x bCoef2x2 Individual Effect of x1: Individual Effect of x2: while x2 remains constant while x1 remains constant y bCoef1 x1 y bCoef2 x2 bCoef1 allows us to estimate the change in the dependent variable, y, when explanatory variable 1, x1, changes while all other explanatory variables remain constant. bCoef2 allows us to estimate the change in the dependent variable, y, when explanatory variable 2 , x2, changes while all other explanatory variables remain constant. Question: What happens when both explanatory variables change simultaneously? Change in Explanatory Variable 1 Change in Explanatory Variable 2 y bCoef1 x bCoef2 x2
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estimates the “slope” of the demand curve
Interpreting the Price Coefficient Estimates of the Original Simultaneous Equation Models Price Price Inc constant FeedP constant S estimates the “slope” of the demand curve P P estimates the “slope” of the supply curve QS QD D Quantity Quantity
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Price Reduced Form Equation: EstP = 33.027 + 1.0562FeedP + .018825Inc
Interpreting the Price Coefficient Estimates of the Reduced Form Equations Quantity Reduced Form Equation: EstQ = 38,726 FeedP Inc Price Reduced Form Equation: EstP = FeedP Inc Suppose that FeedP increases while Inc remains constant: Suppose that Inc increases while FeedP remains constant: Does the demand curve shift? No Does the demand curve shift? Yes Does the supply curve shift? Yes Does the supply curve shift? No What happens to Q and P? What happens to Q and P? Q 332.00FeedP Q Inc P FeedP P Inc Price Price S’ S Inc constant FeedP increases FeedP constant Inc increases D’ P FeedP P Inc S Q 332.00FeedP Q Inc D D Quantity Quantity Q 332.00FeedP 332.00 Q 17.347Inc 17.347 = = 314.3 = = 921.5 P 1.0562FeedP 1.0562 P Inc QD QS = 314.3 = 921.5 P P
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The Coefficient Interpretation Approach: Intuition and a Bonus
The coefficient interpretation approach provides us with an intuitive way to derive the relationships between the estimated “slopes” of the demand and supply curves and the reduced form estimates To estimate the “slope” of the demand curve, the demand curve must remain stationary while the supply curve shifts. That is why the estimate of the “slope” of the demand curve depends on the reduced form feed price coefficients. To estimate the “slope” of the supply curve, the supply curve must remain stationary while the demand curve shifts. That is why the estimate of the “slope” of the supply curve depends on the reduced form income coefficients. Furthermore, this allows us to avoid slogging through cumbersome algebra involved in manipulating the original model to express each endogenous variable in terms of the exogenous variables only.
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