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Lecture 23 Preview: Simultaneous Equations – Identification
Demand and Supply Models Ordinary Least Squares (OLS) Estimation Procedure Reduced Form (RF) Estimation Procedure One Way to Cope with Simultaneous Equation Models Two Stage Least Squares (TSLS): An Instrumental Variable Two Step Approach – A Second Way to Cope with Simultaneous Equation Models 1st Stage: Use the exogenous explanatory variable(s) to estimate the endogenous explanatory variable(s). 2nd Stage: In the original model, replace the endogenous explanatory variable with its estimate. Comparison of Reduced Form (RF) and Two Stage Least Squares (TSLS) Estimates Statistical Software and Two Stage least Squares (TSLS) Identification of Simultaneous Equation Models: Order Condition Taking Stock Underidentification Overidentification Overidentification and Two Stage Least Squares (TSLS) Summary of Identification Issues
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Review: Simultaneous Equation Demand and Supply Models
Endogenous Variables: Qt and Pt Exogenous Variables: FeedPt and Inct Goal: Estimate the price coefficients of the demand and supply models. Ordinary Least Squares (OLS) Estimation Procedure and Simultaneous Equation Models Question: When an endogenous explanatory variables is present, is the ordinary least squares (OLS) estimation procedure for its coefficient value Unbiased? No. Consistent? No. Review: Reduced Form (RF) Estimation Procedure Quantity Reduced Form Equation: Price Reduced Form Equation: Reduced Form Estimates: Price Coefficient Estimates Demand Model Supply Model 332.00 17.347 = 314.3 = 921.5 1.0562 Question: When an endogenous explanatory variables is present, is the reduced form (RF) estimation procedure for its coefficient value Unbiased? No. Consistent? Yes.
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Ordinary Least Squares (OLS)
Two Stage Least Squares (TSLS) Estimation Procedure Endogenous Variables: Qt and Pt Exogenous Variables: FeedPt and Inct 1st Stage: Estimate the variable that is creating the problem, the explanatory endogenous variable: Dependent 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. 1st Stage: Dependent variable: P Explanatory variables: FeedP and Inc EViews 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 Estimated Equation: EstP = FeedP Inc
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Ordinary Least Squares (OLS) Ordinary Least Squares (OLS)
2nd 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. Explanatory variables: Estimate of the “problem” explanatory variable, the endogenous explanatory variable, based on the 1st stage and any relevant exogenous explanatory variables. 2nd Stage – Beef Market Demand Model: Dependent variable: Q Explanatory Variables: EstP and Inc Ordinary Least Squares (OLS) Dependent Variable: Q Explanatory Variable(s): Estimate SE t-Statistic Prob EstP 0.0073 Inc 0.0000 Const Number of Observations 120 EViews Estimated Equation: EstQD = 149,107 314.3EstP Inc 2nd Stage – Beef Market Supply Model: Dependent variable: Q Explanatory Variables: EstP and FeedP Ordinary Least Squares (OLS) Dependent Variable: Q Explanatory Variable(s): Estimate SE t-Statistic Prob EstP 0.0000 FeedP Const Number of Observations 120 Estimated Equation: EstQS = 108, EstP 1,305.2 FeedP
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Two Stage Least Squares (TSLS) the Easy Way: Let statistical software do the work:
Highlight all relevant variables: Q P Inc FeedP Double Click. In the Equation settings window, click the Method drop down list and select TSLS – Two Stage Least Squares (TSNLS and ARIMA). Instrument List: The exogenous variables – Inc FeedP Equation Specification: The dependent variable followed by the explanatory variables Demand Model: Q P Inc EViews Supply Model: Q P FeedP Reduced Form and Two Stage Least Squares Estimates: A Comparison Comparison of Estimates Reduced Form (RF) Two Stage Least Squares (TSLS) 314.3 The reduced for (RF) and two stage least squares estimates (TSLS) are identical. Identification of Simultaneous Equation Models: Order Condition Question: Can we always estimate models when an endogenous explanatory variable is present? Strategy: We shall exploit the coefficient interpretation approach that we introduced in the last lecture to address this question.
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Review: Reduced Form Coefficient Interpretation Approach
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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Critical role played by the absent exogenous variables.
Exogenous variables: FeedP and Inc. A total of 2 exogenous explanatory variables. Price Price S’ S Inc constant FeedP increases FeedP constant Inc increases D’ Critical role played by the absent exogenous variables. P = FeedP P = Inc S Q = 332.00FeedP Q = Inc D D Quantity Quantity QD QS = 314.3 = 921.5 P P Demand Model Supply Model Changes in FeedP allows us to estimate demand model’s P coefficient Changes in Inc allows us to estimate supply model’s P coefficient Demand Model Explanatory Variables Supply Model Explanatory Variables Endogenous explanatory variables included Endogenous explanatory variables included Exogenous explanatory variables variables included absent Exogenous explanatory variables variables included absent 1 2 1 = 1 1 1 2 1 = 1 1 1 equals 1 1 equals 1
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Identification of a Simultaneous Equation Model – Order Condition
Number of exogenous explanatory variables absent from the model Less Than Number of endogenous explanatory variables included in the model Equal To Greater Than Model Underidentified Model Identified Model Overidentified No RF Estimate Unique RF Estimates Multiple RF Estimates
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Ordinary Least Squares (OLS) Ordinary Least Squares (OLS)
Underidentified Suppose that no income data were available? Simultaneous Equation Demand and Supply Models Endogenous Variables: Qt and Pt Exogenous Variables: FeedPt and Inct Quantity Reduced Form Equation: Dependent Variable: Q Explanatory Variables: FeedP EViews Ordinary Least Squares (OLS) Dependent Variable: Q Explanatory Variable(s): Estimate SE t-Statistic Prob FeedP 0.0000 Const Number of Observations 120 Price Reduced Form Equation: Dependent Variable: P Explanatory Variables: FeedP Ordinary Least Squares (OLS) Dependent Variable: P Explanatory Variable(s): Estimate SE t-Statistic Prob FeedP 0.0478 Const 0.0000 Number of Observations 120
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Quantity Reduced Form Equation: EstQ = 239,158 821.85FeedP
Price Reduced Form Equation: EstP = FeedP Suppose that Inc increases while FeedP remains constant: Suppose that FeedP increases while Inc 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 FeedP 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 = 821.85FeedP Q = ??????Inc D D Quantity Quantity Q FeedP 821.85 Q ?????? Inc ?????? = = 1,566.5 = = ????? P .52464FeedP .52464 P ?????? Inc ?????? QD QS = 1,566.5 = ?????? P P
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Critical role played by the absent exogenous variables.
Exogenous variable: FeedP A total of 1 exogenous explanatory variables. Price Price S’ S Inc constant FeedP increases FeedP constant Inc increases D’ Critical role played by the absent exogenous variables. P = FeedP S Q = FeedP D D Quantity Quantity QD = 1,566.5 P Supply Model Demand Model Changes in FeedP allows us to estimate demand model’s P coefficient Changes in Inc allows us to estimate supply model’s P coefficient Demand Model Explanatory Variables Supply Model Explanatory Variables Endogenous explanatory variables included Endogenous explanatory variables included Exogenous explanatory variables variables included absent Exogenous explanatory variables variables included absent 1 0 = 1 1 1 1 1 = 0 1 1 equals 1 0 less than 1
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Two Stage Least Squares (TSLS)
Two Stage Least Squares (TSLS) Estimation Procedure Simultaneous Equation Demand and Supply Models Endogenous Variables: Qt and Pt Exogenous Variables: FeedPt and Inct Beef Market Demand Model: Dependent variable: Q Explanatory Variables: P Instrument List: FeedP Two Stage Least Squares (TSLS) Dependent Variable: Q Instrument(s): FeedP Explanatory Variable(s): Estimate SE t-Statistic Prob P 0.0279 Number of Observations 120 EViews = 1,566.5 Beef Market Supply Model: Dependent variable: Q Explanatory Variables: P and FeedP Instrument List: FeedP Error Message: Order condition violated. Comparison of Estimates Reduced Form (RF) Two Stage Least Squares (TSLS) 1, 1,566.5 None None The reduced for (RF) and two stage least squares estimates (TSLS) are identical.
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Ordinary Least Squares (OLS) Ordinary Least Squares (OLS)
Overidentified Suppose that the price of chicken is also available. Simultaneous Equation Demand and Supply Models Endogenous Variables: Qt and Pt Exogenous Variables: FeedPt, Inct, and ChickPt Quantity Reduced Form Equation: Dependent Variable: Q Explanatory Variables: FeedP, Inc, and ChickP EViews Ordinary Least Squares (OLS) Dependent Variable: Q Explanatory Variable(s): Estimate SE t-Statistic Prob FeedP 0.0111 Inc 0.0000 ChickP 0.7644 Const Number of Observations 120 Price Reduced Form Equation: Dependent Variable: P Explanatory Variables: FeedP, Inc, and ChickP Ordinary Least Squares (OLS) Dependent Variable: P Explanatory Variable(s): Estimate SE t-Statistic Prob FeedP 0.0033 Inc 0.0120 ChickP 0.4621 Const 0.3416 Number of Observations 120 First, we will estimate the price coefficient in the demand model.
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Critical role played by the absent exogenous variables.
Quantity RF Equation: EstQ = 138,194 FeedP Inc ChickP Price RF Equation: EstP = FeedP Inc ChickP Suppose that FeedP increases while Inc and ChickP remains constant: Exogenous variables: FeedP, Inc, and ChickP Does the demand curve shift? No A total of 3 exogenous explanatory variables. Does the supply curve shift? Yes What happens to Q and P? Demand Model Q 349.54FeedP P FeedP Price S’ Changes in FeedP allows us to estimate demand model’s P coefficient Inc constant ChickP constant FeedP increases Demand Model Explanatory Variables P = FeedP Endogenous explanatory variables included S Exogenous explanatory variables variables included absent Q = 349.54FeedP D Quantity 2 3 2 = 1 1 Q 349.54FeedP 349.54 1 equals 1 = = 366.0 P .95501FeedP .95501 Critical role played by the absent exogenous variables. QD = 366.0 P
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Two Stage Least Squares (TSLS)
Two Stage Least Squared (TSLS) Estimation Procedure EViews Beef Market Demand Model: Dependent variable: Q Explanatory Variables: P, Inc, and ChickP Instrument List: FeedP, Inc, and ChickP Two Stage Least Squares (TSLS) Dependent Variable: Q Instrument(s): FeedP, Inc, and ChickP Explanatory Variable(s): Estimate SE t-Statistic Prob P 0.0000 Inc ChickP 0.0888 Const Number of Observations 120 Comparison of Estimates Reduced Form (RF) Two Stage Least Squares (TSLS) 366.0 The reduced form (RF) and the two stage least squares (TSLS) estimates are identical. Simultaneous Equation Demand and Supply Models Next, we will estimate the price coefficient in the supply model.
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Suppose that Inc increases while FeedP and ChickP remain constant:
Quantity RF Equation: EstQ = 138,194 FeedP Inc ChickP Price RF Equation: EstP = FeedP Inc ChickP Suppose that Inc increases while FeedP and ChickP remain constant: Suppose that ChickP increases while FeedP and Inc remain constant: Does the demand curve shift? Yes Does the demand curve shift? Yes Does the supply curve shift? No Does the supply curve shift? No Q Inc Q ChickP P Inc P ChickP Price Price FeedP constant ChickP constant Inc increases S FeedP constant Inc constant ChickP increases S D’ D’ P = Inc P = ChickP Q = Inc Q = ChickP D D Quantity Quantity Q 16.865Inc 16.865 Q 47.600ChickP 47.600 = = 1,051.2 = = 173.3 P Inc P .27464ChickP .27464 QS QS = 1,051.2 = 173.3 P P
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Critical role played by the absent exogenous variables.
Exogenous variables: FeedP, Inc, and ChickP A total of 3 exogenous explanatory variables. Price Price FeedP constant ChickP constant Inc increases S FeedP constant Inc constant ChickP increases S D’ D’ P = Inc P = Inc Q = Inc Q = Inc D D Quantity Quantity QS QS = = 1,051.2 = = 173.3 P P Supply Model Changes in Inc allows us to estimate supply model’s P coefficient Changes in ChickP allows us to estimate supply model’s P coefficient Supply Model Explanatory Variables Endogenous explanatory variables included Critical role played by the absent exogenous variables. Exogenous explanatory variables variables included absent 1 3 1 = 2 1 2 greater than 1
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EViews Two Stage Least Squared (TSLS) Estimation Procedure
Beef Market Supply Model: Dependent variable: Q Explanatory Variables: P and FeedP Instrument List: FeedP, Inc, and ChickP Two Stage Least Squares (TSLS) Dependent Variable: Q Instrument(s): FeedP, Inc, and ChickP Explanatory Variable(s): Estimate SE t-Statistic Prob P 0.0087 FeedP 0.0006 Const 0.0254 Number of Observations 120 Summary of Reduced Form (RF) and Two Stage Least Squares (TSLS) Price Coefficient Estimates: Estimated “Slope” of Demand Curve Supply Curve Reduced Form (RF) 366.0 Based on Income Coefficients 1,051.2 173.3 Based on Chicken Price Coefficients Two Stage Least Squares (TSLS) 366.0 893.5 A difference emerges when the model is overidentified. There are two reduced form estimates and only one two stage least squares estimate.
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Identification Summary
Number of exogenous explanatory variables absent from the model Less Than Number of endogenous explanatory variables included in the model Equal To Greater Than Model Underidentified Model Identified Model Overidentified No RF Estimate Unique RF Estimate Multiple RF Estimates No TSLS Estimate Identical to RF Unique TSLS Estimate Identical to RF Unique TSLS Estimate. Question: What about the two stage least squares (TSLS) estimation procedure?
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