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 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: t = Const + t + I Inc t + e t Equilibrium: upply Model: t = Const + t + F Feed t + e t t = t = t Variables: t and t Exogenous Variables: Feed t and Inc t roject: Estimate the price coefficients of the demand and supply models, and. Review: Ordinary Least quares (OL) Estimation rocedure and imultaneous Equation Models uestion: When endogenous explanatory variable is present, is the ordinary least squares (OL) estimation procedure for its coefficient s value: Unbiased? Consistent?
2 Review: Reduced Form (RF) Estimation rocedure Reduced Form Equation: t = Const F Feed t + I Inc t + t rice Reduced Form Equation: t = Const Reduced Form Estimates: a Const = 138,726 a Const = 33.027 F Feed t + I Inc t + t a F = 332.00 a I = 17.347 a F = 1.0562 a I =.018825 rice Coefficient Estimates: emand Model The ratio of the reduced form feed price coefficient estimates: af 322.00 b 314.3 af 1.0562 upply Model The ratio of the reduced form income estimates: ai 17.347 b 921.5 ai.018825 uestion: 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) Estimation rocedure: An Instrumental Variable Two tep Approach Consider the model for the beef market: emand Model: t = Const + t + I Inc t + e t Equilibirum: upply Model: t = Const + t + F Feed t + e t t = t = t Variables: t and t Exogenous Variables: Feed t and Inc t 1st tage: Estimate the variable that is creating the problem, the explanatory endogenous variable ependent variable: roblem explanatory variable. The endogenous explanatory variable in the original simultaneous equation model. The variable that creates the bias problem. Explanatory variables: All exogenous variables. 2nd 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 1st stage and any relevant exogenous explanatory variables.
3 1st tage: Estimate the variable that is creating the problem, the explanatory endogenous variable: ependent variable: roblem 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,, is the problem explanatory variable. Explanatory variables: All exogenous variables. In this case, the exogenous variables are Feed and Inc. Ordinary Least quares (OL) ependent Variable: Explanatory Variable(s): Estimate E t-tatistic rob Feed 1.056242 0.286474 3.687044 0.0003 Inc 0.018825 0.005019 3.750636 0.0003 Const 33.02715 31.04243 1.063936 0.2895 Est = 33.037 + 1.0562Feed +.018825Inc 2nd 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,. Explanatory variables: Estimate of the problem explanatory variable, the endogenous explanatory variable, based on the 1st stage and any relevant exogenous explanatory variables. 2 nd tage Beef Market emand Model Ordinary Least quares (OL) ependent Variable: Explanatory Variable(s): Estimate E t-tatistic rob Est 314.3312 115.2117-2.728293 0.0073 Inc 23.26411 2.161914 10.76089 0.0000 Const 149106.9 16280.07 9.158860 0.0000 b = 314.3 Est = 149,107 314.3Est + 23.26Inc b = 23.26 We estimate the slope of the demand curve to be 314.3. I
4 2nd tage Beef Market upply Model Ordinary Least quares (OL) ependent Variable: Explanatory Variable(s): Estimate E t-tatistic rob Est 921.4783 113.2551 8.136309 0.0000 Feed 1305.262 121.2969-10.76089 0.0000 Const 108291.8 16739.33 6.469303 0.0000 b = 921.5 Est = 108,292 + 921.5Est 1,305.3 Feed b = 1,305.3 F We estimate the slope of the slope curve to be 921.5. Two tage Least quares (TL) the Easy Way: Let statistical software do the work: Getting tarted in EViews Highlight all relevant variables: Inc Feed ouble Click. In the Equation settings window, click the Method drop down list and select TL Two tage Least quares (TNL and ARIMA). Instrument List: The exogenous variables Inc Feed Equation pecification: The dependent variable followed by the explanatory variables o emand Model: Inc o upply Model: Feed _ Reduced Form (RF) Approach and Two tage Least quares (TL) Estimates: A Comparison Estimate Reduced Form (RF) Two tage Least quares (TL) b 314.3 314.3 921.5 921.5 b The estimates are. Identification of imultaneous Equation Models: Order Condition uestion: 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.
5 Review: Reduced Form Coefficient Interpretation Approach imultaneous Equation emand and upply Models: emand Model: t = Const + t + I Inc t + e t Equilibirum: upply Model: t = Const + t + F Feed t + e t t = t = t Variables: t and t Exogenous Variables: Feed t and Inc t Reduced Form Estimates Reduced Form Estimates: Est = 138,726 332.00Feed + 17.347Inc rice Reduced Form Estimates: Est = 33.027 + 1.0562Feed +.018825Inc uppose that Feed increases while uppose that Inc increases while Inc remains constant: Feed 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 and? What happens to and? = Feed = Inc = Feed = Inc rice Inc constant Feed increases = rice Feed constant Inc increases = = = = Feed Feed = = = Inc Inc = = b = = b = = t = Const + t + I Inc t + e t t = Const + t + F Feed t + e t Changes in allows us demand model s Changes in allows us supply model s Feed to estimate coefficient Inc to estimate 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 Observation: The absent exogenous explanatory variable allows us to estimate the endogenous variable s coefficient.
6 review: 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: t = Const + t + I Inc t + e t Equilibirum: upply Model: t = Const + t + F Feed t + e t t = t = t Variables: t and t Exogenous Variables: Feed t and Inc t Reduced Form Equation: Ordinary Least quares (OL) ependent Variable: Explanatory Variable(s): Estimate E t-tatistic rob Feed 821.8494 131.7644-6.237266 0.0000 Const 239158.3 5777.771 41.39283 0.0000 Est = 239,158 821.85Feed a F = 821.85 rice Reduced Form Equation Ordinary Least quares (OL) ependent Variable: Explanatory Variable(s): Estimate E t-tatistic rob Feed 0.524641 0.262377 1.999571 0.0478 Const 142.0193 11.50503 12.34411 0.0000 Number of Observations Est = 142.0.52464Feed a F =.52464 Reduced Form Estimates: Est = 239,158 821.85Feed rice Reduced Form Estimates: Est = 142.02 +.52464Feed uppose that Feed increases while Inc uppose that Inc increases while Feed 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 and? What happens to and? = Feed = Inc = Feed = Inc 120
7 Reduced Form Estimates: rice Reduced Form Estimates: rice Inc constant Feed increases = Est = 239,158 821.85Feed Est = 142.02 +.52464Feed rice Feed constant Inc increases = = = = Feed Feed = = = Inc Inc = = b = = b = = t = Const + t + e t t = Const + t + F Feed t + e t Changes in allows us demand model s Changes in allows us supply model s Feed to estimate coefficient Inc to estimate 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 Observation: The absent exogenous explanatory variable allows us to estimate the endogenous variable s coefficient. Two tage Least quares (TL) Estimation rocedure Beef Market emand Model Two tage Least quares (TL) ependent Variable: Instrument(s): Feed Explanatory Variable(s): Estimate E t-tatistic rob 1566.499 703.8335-2.225667 0.0279 Const 461631.4 115943.8 3.981510 0.0001 b = 1,566.5 Est = 461,631 1,566.5 Beef Market upply Model: ependent variable: Explanatory variable: Instrument List: Feed _ uestion: How do the reduced form and two stage least squares estimates compare?
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: upply Model: Variables: t and t Reduced Form Equations Reduced Form Equation: rice Reduced Form Equation: t = Const + t + I Inc t + C Chick t + e t Equilibrium: t = Const + t + F Feed t + e t t = t = t Exogenous Variables: Feed t, Inc t, and Chick t t = Const + F Feed t + I Inc t + C Chick t + t t = Const + F Feed t + I Inc t + C Chick t + t Reduced Form Equation Ordinary Least quares (OL) ependent Variable: Explanatory Variable(s): Estimate E t-tatistic rob Feed 349.5411 135.3993-2.581558 0.0111 Inc 16.86458 2.675264 6.303894 0.0000 Chick 47.59963 158.4147 0.300475 0.7644 Const 138194.2 13355.13 10.34765 0.0000 Est = 138,194 349.54Feed + 16.865Inc + 47.600Chick a F = 349.54 a I = 16.865 a C = 47.600 rice Reduced Form Equation Ordinary Least quares (OL) ependent Variable: Explanatory Variable(s): Estimate E t-tatistic rob Feed 0.955012 0.318135 3.001912 0.0033 Inc 0.016043 0.006286 2.552210 0.0120 Chick 0.274644 0.372212 0.737870 0.4621 Const 29.96187 31.37924 0.954831 0.3416 Est = 29.96 +.95501Feed +.016043Inc +.27464Chick a F =.95501 a I =.95501 a C = 27464 Reduced Form Estimates: rice Reduced Form Estimates: Est = 138,194 349.54Feed + 16.865Inc + 47.600Chick Est = 29.962 +.95501Feed +.016043Inc +.27464Chick uppose that Feed increases while Inc and Chick remain constant: oes the demand curve shift? oes the supply curve shift? What happens to and? = Feed = Feed
9 Reduced Form Estimates: rice Reduced Form Estimates: Est = 138,194 349.54Feed + 16.865Inc + 47.600Chick Est = 29.962 +.95501Feed +.016043Inc +.27464Chick rice Inc constant Chick constant Feed increases = = = Feed Feed = = b = = t = Const + t + I Inc t + C Chick t + e t Changes in allows us demand model s Feed to estimate coefficient Exogenous Variable(s):. A total of exogenous explanatory variables. emand Model Exogenous explanatory explanatory variables variables variables included absent included Critical Observation: The absent exogenous explanatory variable allows us to estimate the endogenous variable s coefficient. Two-tage Least quares (TL) Estimation rocedure Beef Market emand Model Two tage Least quares (TL) ependent Variable: Instrument(s): Feed, Inc, and Chick Explanatory Variable(s): Estimate E t-tatistic rob 366.0071 68.47718-5.344950 0.0000 Inc 22.73632 1.062099 21.40697 0.0000 Chick 148.1212 86.30740 1.716205 0.0888 Const 149160.5 7899.140 18.88313 0.0000 Est = 149,160 366.0Est + 22.74Inc b = 366.0 b = 22.74 I b = 22.74 uestion: How do the reduced form and two stage least squares estimates compare? C
10 Reduced Form Estimates: rice Reduced Form Estimates: Est = 138,194 349.54Feed + 16.865Inc + 47.600Chick Est = 29.962 +.95501Feed +.016043Inc +.27464Chick uppose that Inc increases while uppose that Chick increases while Feed and Chick remain constant: Feed 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 and? What happens to and? = Inc = Chick = Inc = Chick rice Feed constant Chick constant Inc increases rice Feed constant Inc constant Chick increases = = = = = Inc Inc = = = Chick Chick = = b = = b = = t = Const + t + F Feed t + e t Changes in allows us demand model s Changes in allows us supply model s Inc to estimate coefficient Chick to estimate coefficient Exogenous Variable(s):. A total of exogenous explanatory variables. upply Model Exogenous explanatory explanatory variables variables variables included absent included Critical Observation: The absent exogenous explanatory variable allows us to estimate the endogenous variable s coefficient.
11 Two-tage Least quares (TL) Estimation rocedure Beef Market upply Model Two tage Least quares (TL) ependent Variable: Instrument(s): Feed, Inc, and Chick Explanatory Variable(s): Estimate E t-tatistic rob 893.4857 335.0311 2.666874 0.0087 Feed 1290.609 364.0891-3.544761 0.0006 Const 112266.0 49592.54 2.263769 0.0254 b = 893.5 ummary of Overidentification Est = 112,266 + 893.5Est 1,290.6Feed b = 1,290.6 F rice Coefficient Estimates: Estimated lope of emand Curve ( b ) upply Curve ( b ) Reduced Form (RF) 366.0 Based on Income Coefficients 1,051.2 Based on Chicken rice Coefficients 173.3 Two tage Least quares (TL) 366.0 893.5 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 to RF to RF