Wednesday, December 12 Handout: Simultaneous Equations Identification

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1 Amherst College epartment of Economics Economics 360 Fall 2012 Wednesday, ecember 12 Handout: imultaneous Equations Identification Preview Review o emand and upply Models o Ordinary Least quares (OL) Estimation Procedure o Reduced Form (RF) Estimation Procedure One Way to Cope with imultaneous Equation Models Two tage Least quares (TL): An Instrumental Variable (IV) Two tep Approach A econd Way to Cope with imultaneous Equation Models o 1st tage: Use the exogenous explanatory variable(s) to estimate the endogenous explanatory variable(s). o 2nd tage: In the original model, replace the endogenous explanatory variable with its estimate. Comparison of Reduced Form (RF) and Two tage Least quares (TL) Estimates tatistical oftware and Two tage least quares (TL) Identification of imultaneous Equation Models: Order Condition o Taking tock o Underidentification o Overidentification o Overidentification and Two tage Least quares (TL) ummary of Identification Issues Review: imultaneous Equation emand and upply Models emand Model: Q t = β Const + β P P t + β I Inc t + e t Equilibrium: upply Model: Q t = β Const FP FeedP t + e t Q t = Q t = Q t Variables: Q t and P t Exogenous Variables: FeedP t and Inc t Goal: Estimate the price coefficients of the demand and supply models, β P and β P. Review: Ordinary Least quares (OL) Estimation Procedure and imultaneous Equation Models Question: When endogenous explanatory variable is present, is the ordinary least squares (OL) estimation procedure for its coefficient s value: Unbiased? Consistent?

2 2 Review: Reduced Form (RF) Estimation Procedure Reduced Form Equation: Q t = Reduced Form Equation: P t = α P Const Reduced Form Estimates: α Q Const α Q FP FeedP t + αq I Inc t + εq t a Q Const = 138,726 a P Const = αp FP FeedP t + α P I Inc t + εp t aq FP = aq I = ap FP = ap I = Coefficient Estimates: emand Model The ratio of the reduced form feed price coefficient estimates: b P = aq FP a P = = FP upply Model The ratio of the reduced form income estimates: b P = aq I a P = = I Question: When endogenous explanatory variable is present, is the reduced form (RF) estimation procedure for its coefficient s value: Unbiased? Consistent? Two tage Least quares (TL): An Instrumental Variable Two tep Approach 1 st tage: Estimate the variable that is creating the problem, the explanatory endogenous variable ependent variable: Problem explanatory variable. The endogenous explanatory variable in the original simultaneous equation model. The variable that creates the bias problem. Explanatory variables: All exogenous variables. 2 nd tage: Estimate the original models using the estimate of the problem explanatory endogenous variable ependent variable: Original dependent variable. Explanatory variables: Estimate of the problem explanatory variable, the endogenous explanatory variable, based on the 1 st stage and any relevant exogenous explanatory variables. Consider the model for the beef market: emand Model: Q t = β Const + β P P t + β I Inc t + e t Equilibirum: upply Model: Q t = β Const FP FeedP t + e t Q t = Q t = Q t Variables: Q t and P t Exogenous Variables: FeedP t and Inc t

3 3 1 st tage: Estimate the variable that is creating the problem, the explanatory endogenous variable: ependent variable: Problem explanatory variable. The endogenous explanatory variable in the original simultaneous equation model. The variable that creates the bias problem. In this case, the price of beef, P, is the problem explanatory variable. Explanatory variables: All exogenous variables. In this case, the exogenous variables are FeedP and Inc. 1 st tage: ependent variable: P Explanatory variables: FeedP and Inc ependent Variable: P Method: Least quares FEEP INC C Estimated Equation: EstP = FeedP Inc 2 nd tage: Estimate the original models using the estimate of the problem explanatory endogenous variable ependent variable: Original dependent variable. In this case, the original explanatory variable is the quantity of beef, Q. Explanatory variables: Estimate of the problem explanatory variable, the endogenous explanatory variable, based on the 1 st stage and any relevant exogenous explanatory variables. 2 nd tage Beef Market emand Model: ependent variable: Q Explanatory Variables: EstP and Inc Method: Least quares ETP INC C Estimated Equation: EstQ = 149, EstP Inc We estimate the slope of the demand curve to be

4 4 2 nd tage Beef Market upply Model: ependent variable: Q Explanatory Variables: EstP and FeedP Method: Least quares ETP FEEP C Estimated Equation: EstQ = 108, EstP 1,305.2 FeedP We estimate the slope of the demand curve to be Two tage Least quares (TL) the Easy Way: Let EViews do the work: Highlight all relevant variables: Q P Inc FeedP ouble Click, Click Options and then choose Two-tage Least quares Instrument List: The exogenous variables Inc FeedP Equation pecification: The dependent variable followed by the explanatory variables o emand Model: Q P Inc o upply Model: Q P FeedP Reduced Form (RF) Approach and Two tage Least quares (TL) Estimates: A Comparison Estimate Reduced Form (RF) Two tage Least quares (TL) b P b P The estimates are. Identification of imultaneous Equation Models: Order Condition Question: Can we always estimate models when an endogenous explanatory variable is present? trategy: We shall exploit the coefficient interpretation approach that we introduced in the last lecture to address this question, that is, to explore the simultaneous equation identification issue.

5 5 Review: Reduced Form Coefficient Interpretation Approach imultaneous Equation emand and upply Models: emand Model: Q t = β Const + β P P t + β I Inc t + e t Equilibirum: upply Model: Q t = β Const FP FeedP t + e t Q t = Q t = Q t Variables: Q t and P t Exogenous Variables: FeedP t and Inc t Reduced Form Estimates Reduced Form Estimates: EstQ = 138, FeedP Inc Reduced Form Estimates: EstP = FeedP Inc uppose that FeedP increases while uppose that Inc increases while Inc remains constant: FeedP remains constant: oes the demand curve shift? oes the demand curve shift? oes the supply curve shift? oes the supply curve shift? What happens to Q and P? What happens to Q and P? ΔQ = ΔFeedP ΔQ = ΔInc = ΔFeedP = ΔInc Inc constant FeedP increases = FeedP constant Inc increases = ΔQ = ΔQ = ΔQ = ΔFeedP ΔFeedP = = ΔQ = ΔInc ΔInc = = b P = ΔQ = b P = ΔQ Q t = β Const + β P P t + β I Inc t + e t Q t = β Const = FP FeedP t + e t Changes in allows us demand model s Changes in allows us supply model s FeedP to estimate P coefficient Inc to estimate P coefficient Exogenous Variable(s):. A total of exogenous explanatory variables. emand Model upply Model Exogenous explanatory explanatory Exogenous explanatory explanatory variables variables variables variables variables variables included absent included included absent included Critical role of absent exogenous variables.

6 6 Preview: Identification of a imultaneous Equation Model Order Condition Number of exogenous Less Than Number of endogenous explanatory variables Equal To explanatory variables absent from the model Greater Than included in the model Model Model Model Underidentified Identified Overidentified No RF Estimate Unique RF Estimates Multiple RF Estimates Underidentified: uppose that no income data were available. imultaneous Equation emand and upply Models: emand Model: Q t = β Const + β P P t + β I Inc t + e t Equilibirum: upply Model: Q t = β Const FP FeedP t + e t Q t = Q t = Q t Variables: Q t and P t Exogenous Variables: FeedP t and Inc t Reduced Form Equation: Explanatory Variable: FeedP Method: Least quares FEEP C Estimated Equation: EstQ = 239, FeedP Reduced Form Equation: Explanatory Variable: FeedP ependent Variable: P Method: Least quares FEEP C Estimated Equation: EstP = FeedP Reduced Form Estimates: EstQ = 239, FeedP Reduced Form Estimates: EstP = FeedP uppose that FeedP increases while Inc uppose that Inc increases while FeedP remains constant: remains constant: oes the demand curve shift? oes the demand curve shift? oes the supply curve shift? oes the supply curve shift? What happens to Q and P? What happens to Q and P? ΔQ = ΔFeedP ΔQ = ΔInc = ΔFeedP = ΔInc

7 7 Reduced Form Estimates: EstQ = 239, FeedP Reduced Form Estimates: EstP = FeedP Inc constant FeedP increases = FeedP constant Inc increases = ΔQ = ΔQ = ΔQ = ΔFeedP ΔFeedP = = ΔQ b P = ΔQ = b P = ΔQ Q t = β Const + β P P t + e t Q t = β Const = ΔInc ΔInc = = = FP FeedP t + e t Changes in allows us demand model s Changes in allows us supply model s FeedP to estimate P coefficient Inc to estimate P coefficient Exogenous Variable(s):. A total of exogenous explanatory variables. emand Model upply Model Exogenous explanatory explanatory Exogenous explanatory explanatory variables variables variables variables variables variables included absent included included absent included Critical role of absent exogenous variables. Two tage Least quares (TL) Estimation Procedure Beef Market emand Model: ependent variable: Q Explanatory variable: P Instrument List: FeedP b P = Method: Two-tage Least quares Instrument list: FEEP P C Beef Market upply Model: ependent variable: Q Explanatory variable: P Instrument List: FeedP Error Message: Order condition violated. Question: How do the reduced form and two stage least squares estimates compare?

8 8 Overidentified: uppose that in addition to the feed price and income information, the price of chicken is also available. imultaneous Equation emand and upply Models: emand Model: Q t = β Const + β P P t + β I Inc t + β CP ChickP t + e t Equilibrium: upply Model: Q t = β Const FP FeedP t + e t Q t = Q t = Q t Variables: Q t and P t Exogenous Variables: FeedP t, Inc t, and ChickP t Reduced Form Equations Reduced Form Equation: Q t = α Q Const + αq FP FeedP t + αq I Inc t + αq CP ChickP t + εq t Reduced Form Equation: P t = α P Const + αp FP FeedP t + αp I Inc t + αp CP ChickP t + εp t Reduced Form Equation: Explanatory Variables: FeedP, Inc, and ChickP Method: Least quares FEEP INC CHICKP C Estimated Equation: EstQ = 138, FeedP Inc ChickP Reduced Form Equation: ependent Variable: P Explanatory Variables: FeedP, Inc, and ChickP ependent Variable: P Method: Least quares FEEP INC CHICKP C Estimated Equation: EstP = FeedP Inc ChickP Reduced Form Estimates: EstQ = 138, FeedP Inc ChickP Reduced Form Estimates: EstP = FeedP Inc ChickP uppose that FeedP increases while Inc and ChickP remain constant: oes the demand curve shift? oes the supply curve shift? What happens to Q and P? ΔQ = ΔFeedP = ΔFeedP

9 9 Reduced Form Estimates: EstQ = 138, FeedP Inc ChickP Reduced Form Estimates: EstP = FeedP Inc ChickP Inc constant ChickP constant FeedP increases = ΔQ = ΔQ = ΔFeedP ΔFeedP = = b P = ΔQ = Q t = β Const + β P P t + β I Inc t + β CP ChickP t + e t Changes in allows us demand model s FeedP to estimate P coefficient Exogenous Variable(s):. A total of exogenous explanatory variables. emand Model Exogenous explanatory explanatory variables variables variables included absent included Critical role of absent exogenous variables. Two-tage Least quares (TL) Estimation Procedure Beef Market emand Model: ependent variable: Q Explanatory Variables: P, Inc, and ChickP Instrument List: FeedP, Inc, and ChickP b P = Method: Two-tage Least quares Instrument list: FEEP INC CHICKP P INC CHICKP C Question: How do the reduced form and two stage least squares estimates compare?

10 10 Reduced Form Estimates: EstQ = 138, FeedP Inc ChickP Reduced Form Estimates: EstP = FeedP Inc ChickP uppose that Inc increases while FeedP and ChickP remain constant: uppose that ChickP increases while FeedP and Inc remain constant: oes the demand curve shift? oes the demand curve shift? oes the supply curve shift? oes the supply curve shift? What happens to Q and P? What happens to Q and P? ΔQ = ΔInc ΔQ = ΔChickP = ΔInc = ΔChickP FeedP constant ChickP constant Inc increases FeedP constant Inc constant ChickP increases ΔQ = ΔQ = ΔQ = = ΔInc ΔInc = = ΔQ = = ΔChickP ΔChickP = = b P = ΔQ = b P = ΔQ Q t = β Const FP FeedP t + e t = Changes in allows us demand model s Changes in allows us supply model s Inc to estimate P coefficient ChickP to estimate P coefficient Exogenous Variable(s):. A total of exogenous explanatory variables. upply Model Exogenous explanatory explanatory variables variables variables included absent included Critical role of absent exogenous variables.

11 11 Beef Market upply Model: ependent variable: Q Explanatory Variables: P and FeedP Instrument List: FeedP, Inc, and ChickP Method: Two-tage Least quares Instrument list: FEEP INC CHICKP P FEEP C b P = Identification ummary Order Condition Number of exogenous Less Than Number of endogenous explanatory variables Equal To explanatory variables absent from the model Greater Than included in the model Model Model Model Underidentified Identified Overidentified RF Estimate RF Estimates RF Estimates TL Estimate TL Estimates TL Estimates

Chapter 23 Handout: Simultaneous Equations Models Identification

Chapter 23 Handout: Simultaneous Equations Models Identification Chapter 23 Handout: imultaneous Equations Models Identification review Review o emand and upply Models o Ordinary Least quares (OL) Estimation rocedure o Reduced Form (RF) Estimation rocedure One Way to