Chapter 15 Simultaneous Equation Models
Chapter 15 Simultaneous Equation Models
Single equation regression models - The dependent variable, Y, is expressed as a linear function of one or more explanatory variables, the Xs Assumption the cause-and-effect relationship, if any, between Y and the Xs is unidirectional: explanatory variables are the cause; the dependent variable is the effect
• Single equation regression models: ——The dependent variable, Y, is expressed as a linear function of one or more explanatory variables, the Xs. Assumption the cause-and-effect relationship, if any, between Y and the Xs is unidirectional: ·explanatory variables are the cause； ·the dependent variable is the effect
Simultaneous equation regression models: regression models in which there is more than one equation in which there are feedback relationships among variables
• Simultaneous equation regression models: ——Regression models in which there is more than one equation in which there are feedback relationships among variables
15.1 The Nature of Simultaneous Equation Models C=B+B. Io YC+ Endogenous variable Variable that is an inherent part of the system being studied and that is determined within the system Variable that is caused by other variables in a causal system Exogenous variable/predetermined variable Variable entering from and determined from outside the system being studied C If there are more endogenous variables, there will be more equations
15.1 The Nature of Simultaneous Equation Models Ct=B1+B2Yt+ut Yt=Ct+It Endogenous variable: Variable that is an inherent part of the system being studied and that is determined within the system. Variable that is caused by other variables in a causal system Exogenous variable/predetermined variable: Variable entering from and determined from outside the system being studied. ◆ If there are more endogenous variables, there will be more equations
15.2 The Simultaneous Equation Bias Inconsistency of ols Estimators C+ =(Bo+B1Y+u)+1 =B+B,Y+u+ B 1-B,1-B 1-B, The explanatory variable in a regression equation is correlated with the error term, this explanatory variable becomes a random, or stochastic variable
15.2 The Simultaneous Equation Bias: Inconsistency of OLS Estimators Yt=Ct+It =(B0+B1Yt+ut )+It =B0+B1Yt+ut+It • The explanatory variable in a regression equation is correlated with the error term, this explanatory variable becomes a random, or stochastic variable. t t ut B I B B B Y 1 1 1 0 1 1 1 1 1 − + − + − =
In the presence of simultaneous problem the Ols estimators are generally not BLUE They are biased in small sample) and inconsistent (in large sample) Inconsistent estimator is the estimator which does not approach the true parameter value even if the sample size increases definitely
In the presence of simultaneous problem, the OLS estimators are generally not BLUE. They are biased （ in small sample ） and inconsistent（in large sample） Inconsistent estimator is the estimator which does not approach the true parameter value even if the sample size increases indefinitely
153. The method of Indirect Least Squares (ILS) 1. Simplify the original model, and get the reduced form regression model C-B,+BY+ox B B 1-B21-B21-B2 C=A+A2 2+Vt A1=B/(1B2)A2=B2/(1-B2),andu=u/(1-B2
15.3. The Method of Indirect Least Squares（ILS） 1. Simplify the original model, and get the reduced form regression model Ct=B1+B2Yt+ut Ct=A1+A2 I 2+vt A1=B1 /(1-B2 ),A2=B2 /(1-B2 ),andυt=ut /(1-B2 ). t t ut B I B B B B C 2 2 2 2 1 1 1 1 1 − + − + − =
2. Applying OLS to the reduced form of the model, get the OLS estimators of the reduced form model 3. According to the relationship between the parameters of the reduced form model and the parameters of the original model, obtain the estimators of the original parameters these estimators are the indirect least squares estimators A B 11+2 1+A
2. Applying OLS to the reduced form of the model, get the OLS estimators of the reduced form model. 3.According to the relationship between the parameters of the reduced form model and the parameters of the original model, obtain the estimators of the original parameters, these estimators are the indirect least squares estimators. 2 1 1 1 A A B + = 2 2 2 1 A A B + =
The Ils estimators are consistent estimators, as the sample size increases indefinitely, there estimators converge to their true population values. In small samples, the ILS estimators may be biased. In contrast, the Ols estimators are biased as well as inconsistent
The ILS estimators are consistent estimators,as the sample size increases indefinitely, there estimators converge to their true population values. In small samples, the ILS estimators may be biased. In contrast, the OLS estimators are biased as well as inconsistent
Whether we can use the method of indirect least squares to estimate the parameters of Simultaneous equation models, depends on whether we can retrieve the original structural parameters from the reduced form estimates: the answer depends on the so-called identification problem
Whether we can use the method of indirect least squares to estimate the parameters of simultaneous equation models, depends on whether we can retrieve the original structural parameters from the reduced form estimates: the answer depends on the so-called identification problem