Lecture 23 Preview: Simultaneous Equations – Identification Review 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 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 332.00 Supply Model 17.347 = 314.3 = 921.5 1.0562 .018825 Question: When an endogenous explanatory variables is present, is the reduced form (RF) estimation procedure for its coefficient value Unbiased? No. Consistent? Yes. 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 Ordinary Least Squares (OLS) Dependent Variable: P Explanatory Variable(s): Estimate SE FeedP 1.056242 0.286474 Inc 0.018825 0.005019 Const 33.02715 31.04243 Number of Observations 120 t-Statistic Prob 3.687044 0.0003 3.750636 0.0003 1.063936 0.2895 Estimated Equation: EstP = 33.027 + 1.0562FeedP + .018825Inc EViews 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 EstP 314.3312 115.2117 Inc 23.26411 2.161914 Const 149106.9 16280.07 Number of Observations 120 EViews t-Statistic Prob -2.728293 0.0073 10.76089 0.0000 9.158860 0.0000 Estimated Equation: EstQD = 149,107 314.3EstP + 23.26Inc 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 EstP 921.4783 113.2551 FeedP 1305.262 121.2969 Const 108291.8 16739.33 Number of Observations 120 t-Statistic Prob 8.136309 0.0000 -10.76089 0.0000 6.469303 0.0000 Estimated Equation: EstQS = 108,292 + 921.5EstP 1,305.2 FeedP 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) 314.3 314.3 921.5 921.5 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. Review: Reduced Form Coefficient Interpretation Approach Quantity Reduced Form Equation: EstQ = 38,726 332.00FeedP + 17.347Inc Price Reduced Form Equation: EstP = 33.027 + 1.0562FeedP + .018825Inc Suppose that FeedP increases while Suppose that Inc increases while Inc remains constant: FeedP remains constant: Does the demand curve shift? No Does the demand curve shift? Does the supply curve shift? Yes Does the supply curve shift? What happens to Q and P? What happens to Q and P? Q 332.00FeedP Q 17.347Inc P 1.0562FeedP P .018825Inc Price Price S’ Inc constant FeedP increases P = 1.0562FeedP Q = 332.00FeedP S FeedP constant Inc increases D’ P = .018825Inc S D Yes No Q = 17.347Inc D Quantity Q 332.00FeedP 332.00 = = 314.3 P 1.0562FeedP 1.0562 QD = 314.3 P Quantity Q 17.347Inc 17.347 = = 921.5 P .018825Inc .018825 QS = 921.5 P Exogenous variables: FeedP and Inc. A total of 2 exogenous explanatory variables. Price Price S’ P = 1.0562FeedP S Q = 332.00FeedP QD P D Critical role played by the absent exogenous variables. Demand Model Changes in allows us FeedP to estimate demand model’s P coefficient Demand Model Explanatory Variables Endogenous Exogenous explanatory explanatory variables variables variables included absent included 1 1 21=1 1 equals 1 D’ P = .018825Inc Q = 17.347Inc D Quantity = 314.3 S FeedP constant Inc increases Inc constant FeedP increases QS P Quantity = 921.5 Supply Model Changes in allows us Inc to estimate supply model’s P coefficient Supply Model Explanatory Variables Endogenous Exogenous explanatory explanatory variables variables variables included absent included 1 1 21=1 1 equals 1 Identification of a Simultaneous Equation Model – Order Condition Number of exogenous explanatory variables absent from the model Model Underidentified No RF Estimate Less Than Equal To Greater Than Model Identified Unique RF Estimates Number of endogenous explanatory variables included in the model Model Overidentified Multiple RF Estimates Underidentified Suppose that no income data were available? Simultaneous Equation Demand and Supply Models Endogenous Variables: Qt and Pt Quantity Reduced Form Equation: Dependent Variable: Q Explanatory Variables: FeedP Ordinary Least Squares (OLS) Dependent Variable: Q Explanatory Variable(s): Estimate SE FeedP 821.8494 131.7644 Const 239158.3 5777.771 Number of Observations 120 Price Reduced Form Equation: Exogenous Variables: FeedPt and Inct t-Statistic Prob -6.237266 0.0000 41.39283 0.0000 Dependent Variable: P Explanatory Variables: FeedP Ordinary Least Squares (OLS) Dependent Variable: P Explanatory Variable(s): Estimate SE FeedP 0.524641 0.262377 Const 142.0193 11.50503 Number of Observations 120 t-Statistic Prob 1.999571 0.0478 12.34411 0.0000 EViews Quantity Reduced Form Equation: Price Reduced Form Equation: Suppose that FeedP increases while Inc remains constant: Does the demand curve shift? Does the supply curve shift? What happens to Q and P? Q 821.85 FeedP P .52464FeedP Price EstQ = 239,158 821.85FeedP EstP = 142.02 + .52464FeedP Suppose that Inc increases while FeedP remains constant: No Does the demand curve shift? Yes Does the supply curve shift? What happens to Q and P? Q ??????Inc P ??????Inc Price S’ Inc constant FeedP increases P = .52464FeedP S FeedP constant Inc increases D’ P = ??????Inc S Q = 821.85FeedP D Yes No Q = ??????Inc D Quantity Quantity Q 821.85 FeedP 821.85 = = 1,566.5 P .52464FeedP .52464 QD = 1,566.5 P Q ?????? Inc P ?????? Inc = ?????? ?????? QS = ?????? P = ????? Exogenous variable: FeedP A total of 1 exogenous explanatory variables. Price Price S’ S FeedP constant Inc increases Inc constant FeedP increases P = .52464FeedP S Q = 821.85FeedP QD P D Critical role played by the absent exogenous variables. D Quantity Quantity = 1,566.5 Supply Model Demand Model Changes in allows us demand model’s FeedP to estimate P coefficient Demand Model Explanatory Variables Endogenous Exogenous explanatory explanatory variables variables variables included absent included 0 D’ 10=1 1 1 equals 1 Changes in allows us supply model’s P Inc to estimate coefficient Supply Model Explanatory Variables Endogenous Exogenous explanatory explanatory variables variables variables included absent included 1 11=0 1 0 less than 1 Two Stage Least Squares (TSLS) Estimation Procedure Simultaneous Equation Demand and Supply Models Endogenous Variables: Qt and Pt Beef Market Demand Model: Exogenous Variables: FeedPt and Inct 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 1566.499 703.8335 -2.225667 0.0279 Number of Observations 120 Beef Market Supply Model: Error Message: Order condition violated. Comparison of Estimates Reduced Form (RF) 1,566.5 None EViews = 1,566.5 Dependent variable: Q Explanatory Variables: P and FeedP Instrument List: FeedP Two Stage Least Squares (TSLS) 1,566.5 None The reduced for (RF) and two stage least squares estimates (TSLS) are identical. Overidentified Suppose that the price of chicken is also available. Simultaneous Equation Demand and Supply Models Endogenous Variables: Qt and Pt Quantity Reduced Form Equation: Exogenous Variables: FeedPt, Inct, and ChickPt Dependent Variable: Q Explanatory Variables: FeedP, Inc, and ChickP Ordinary Least Squares (OLS) Dependent Variable: Q Explanatory Variable(s): FeedP Inc ChickP Const Number of Observations Estimate 349.5411 16.86458 47.59963 138194.2 120 Price Reduced Form Equation: SE 135.3993 2.675264 158.4147 13355.13 t-Statistic -2.581558 6.303894 0.300475 10.34765 Prob 0.0111 0.0000 0.7644 0.0000 Dependent Variable: P Explanatory Variables: FeedP, Inc, and ChickP Ordinary Least Squares (OLS) Dependent Variable: P Explanatory Variable(s): FeedP Inc ChickP Const Number of Observations EViews Estimate 0.955012 0.016043 0.274644 29.96187 120 SE 0.318135 0.006286 0.372212 31.37924 t-Statistic 3.001912 2.552210 0.737870 0.954831 Prob 0.0033 0.0120 0.4621 0.3416 First, we will estimate the price coefficient in the demand model. Quantity RF Equation: EstQ = 138,194 349.54FeedP + 16.865Inc + 47.600ChickP Price RF Equation: EstP = 29.962 + .95501FeedP + .016043Inc + .27464ChickP Suppose that FeedP increases while Exogenous variables: FeedP, Inc, and ChickP Inc and ChickP remains constant: 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? Q 349.54FeedP P .95501FeedP Demand Model Price S’ Inc constant ChickP constant FeedP increases P = .95501FeedP Q = 349.54FeedP S D Quantity Q 349.54FeedP 349.54 = = 366.0 P .95501FeedP .95501 QD = 366.0 P Changes in allows us FeedP to estimate demand model’s P coefficient Demand Model Explanatory Variables Endogenous explanatory Exogenous explanatory variables variables variables included included absent 2 32=1 1 1 equals 1 Critical role played by the absent exogenous variables. 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 366.0071 68.47718 -5.344950 0.0000 Inc 22.73632 1.062099 21.40697 0.0000 ChickP 148.1212 86.30740 1.716205 0.0888 Const 149160.5 7899.140 18.88313 0.0000 Number of Observations 120 Comparison of Estimates Reduced Form (RF) Two Stage Least Squares (TSLS) 366.0 366.0 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. Quantity RF Equation: EstQ = 138,194 349.54FeedP + 16.865Inc + 47.600ChickP Price RF Equation: EstP = 29.962 + .95501FeedP + .016043Inc + .27464ChickP Suppose that Inc increases while FeedP and ChickP remain constant: Does the demand curve shift? Yes Does the supply curve shift? No Suppose that ChickP increases while FeedP and Inc remain constant: Does the demand curve shift? Yes Does the supply curve shift? No Q 16.865Inc P .016043Inc Q 47.600ChickP P .27464ChickP Price Price FeedP constant ChickP constant Inc increases S D’ P = .016043Inc D Quantity = 16.865 = 1,051.2 .016043 QS = 1,051.2 P S D’ P = .27464ChickP Q = 47.600ChickP Q = 16.865Inc Q 16.865Inc P .016043Inc FeedP constant Inc constant ChickP increases D Q 47.600ChickP 47.600 = P .27464ChickP .27464 QS = 173.3 P Quantity = 173.3 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 D’ P = .016043Inc S D’ P = .27464Inc Q = 16.865Inc Q = 47.600Inc D D QS P Quantity = 1,051.2 = Supply Model QS P Quantity = 173.3 supply model’s P Changes allows us supply model’s P coefficient in ChickP to estimate coefficient Supply Model Explanatory Variables Endogenous Critical role explanatory Exogenous explanatory played by the variables variables variables absent exogenous included included absent variables. 1 1 31=2 Changes allows us in Inc to estimate 2 greater than 1 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 EViews Two Stage Least Squares (TSLS) Dependent Variable: Q Instrument(s): FeedP, Inc, and ChickP Explanatory Variable(s): Estimate SE t-Statistic Prob P 893.4857 335.0311 2.666874 0.0087 FeedP 1290.609 364.0891 -3.544761 0.0006 Const 112266.0 49592.54 2.263769 0.0254 Number of Observations 120 Summary of Reduced Form (RF) and Two Stage Least Squares (TSLS) Reduced Form (RF) Based on Income Coefficients Based on Chicken Price Coefficients Two Stage Least Squares (TSLS) Price Coefficient Estimates: Estimated “Slope” of Demand Curve Supply Curve 366.0 1,051.2 173.3 366.0 893.5 A difference emerges when the model is overidentified. There are two reduced form estimates and only one two stage least squares estimate. Identification Summary Number of exogenous explanatory variables absent from the model Less Than Equal To Greater Than Number of endogenous explanatory variables included in the model Model Underidentified No RF Estimate Model Identified Unique RF Estimate Model Overidentified 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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