Simultaneous equation system

• In first part we study the estimation of single equation regression model. In such a models dependent variable are expresses as linear function of one or more explanatory variables. The cause and effect relationship in such models is unidirectional i.e. the explanatory variables are cause and dependent variables are effect. There are some situations where two way flow of influence between two types of variables, i.e. explanatory and response. In such a situation we need to consider more than one regression equation, one for each dependent variable to understand the relation among the variables.

• Where y is dependent , x is independent variable and epsilon is the disturbance term.

iii xy

iii

iii

Icyyc

• Where C is consumption, y is income and I is investment.

• In this system the explanatory variable y is not only independent at the same time y is dependent on consumption and investment, and also C consumption in dependent in equation , but at same time independent in second equation.

• In this system C and y are dependent variables and we need two equations for the solution of system.

• When the dependent variable in one equation, also independent variable in some other equation. We call it the simultaneous equation system.

• The dependent variable is called endogenous variable (these variables are determined by interplay of some factor inside the system)

• The independent variable is called the exogenous variable

• Problems of simultaneous equation system1. Mathematical completeness of the model2. Identification problem3. Statistical estimation of each equation of the

model

• Any model is said to be mathematically complete when it poses as many independents equations –as number of endogenous variables. In other words if we know values of disturbance term, exogenous variables and structural parameters then all the endogenous variables are uniquely determined. Suppose

• We can easily determined yt and xt if other term are known, also error term should known.

• Problem of identification refers to model formulation. It does not concern with the estimation of the model. A model is said to be identified only when it is in unique statistical form which enable us to get unique estimates of its parameters from sample data. In other words a model is said to be identified if we obtain the numerical estimates of the parameters of the structural model from the estimated reduce form coefficients. Otherwise the model under consideration is unidentified.

• Since in simultaneous equation we see that some of the explanatory variables are included disturbance terms, i.e. they are not independent of error term. So in this case we cannot get unbiased estimators. It means that application of OLS yield biased and inconsistent estimators. Therefore different statistical techniques are developed to estimate the parameters which are unbiased and consistent. These methods are

1. Reduced form method or Indirect least square

2. Method of instrumental variables3. 2 stage least squares4. Limited information maximum likelihood 5. Mixed estimation method6. 3 Stage least squares

• Simultaneous equation • Bias and inconsistency of OLS estimators• In SE system some of explanatory variables are

correlated with error terms, so by applying the OLS method to obtain estimators of parameters is not suitable because estimator are bias and inconsistent.

• Structural model:• A Structural model describes the complete

structure of the relationship among the economic variables. The SE maybe expressed in terms of endogenous variables, exogenous variables and disturbance term. Structural eq are the actual equations in which direct relationship can be observed.

• The parameters of structural equations are called structural parameters which express direct effect of each explanatory variable on dependent variables.

• Reduced form Model:• These are the transformed model obtained by

solving the structural equation, such that to express each endogenous variable is a function of only exogenous variables and the disturbance term. In reduced form model the exogenous variables are independent of disturbance terms.

• The OLS technique is applied on reduced form equation to get the best estimators of parameters. The parameters in reduced form equation are called the reduced form parameters and denoted by π.

• Simultaneous equation methods• 1. Ordinary least square method• 2. Indirect least square method• 3. Two stage least square method

• OLS• In the SE system the endogenous variable is

function of endogenous and exogenous variable. Therefore all the explanatory variable of SE is not independent of error term. So least square estimator of the SE parameters are bias and inconsistent,

• 2. ILS• As we discuss in OLS that explanatory variable

are uncorrelated from disturbance term , so this give bias and inconsistent estimates. So we look for alternative methods of estimation which give unbiased and consistent estimates.

• Procedure• Step 1: In 1st step we obtained the reduced

form equation. The reduced form equations are obtained from the SE in such a way that the dependent variable in each equation is only function of independent variable and disturbance term.

• Step 2: In this step we apply OLS method to reduced form equation individually because the assumption about error term of the reduced form equation are satisfied. We obtained the estimates of the reduced form coefficient. They are denoted by Phi and mixture of the structural parameters.

• Step 3: we obtain estimates of the original structural coefficients from estimated reduced form coefficients.

• Assumption of ILS:• The SE must be exactly identified. Because if the

SE model is not identified it is impossible to obtain the estimates for all its parameters.

• The error or the disturbance term of reduced form equation must satisfy all the assumptions of OLS. The error in reduced form equation, which is linear function of error term and structural parameters, must satisfy the following.

• V is random• E(Vi)=0

• E(V2i)=σ2

• E(ViVj)=0• Vi is normally distributed.• Vi is uncorrelated from exogenous variable.

• So the ILS is based on the assumption of OLS .But herein ILS we have an additional assumption that model will be exactly identified.

• The ILS fulfill all the assumption of OLS, the estimates of the reduced form equation are BLUE, however the estimates of Structural parameters obtained from the (Phi) are biased from small sample and asymptotically unbiased and consistent in both cases.

• Two Stage Least Square method• The 2SLS is method of estimating consistent

structural parameters values for exactly identified or over identified equation of simultaneous equations. 2 SLS equation involves the application of OLS in two stages.

• In stage 1 endogenous variable is expressed on all the exogenous variable of the system .In first stage we apply OLs to get the estimated values of endogenous variable.

• In second stage the predicted values instead of actual values of endogenous variable are used to estimate the structural parameters of the model. The predicted values of endogenous variable are uncorrelated with error term. So the parameters estimated form 2 SLS are consistent.