Simultaneous equation models Prepared by Nir Kamal Dahal(Statistics)

Introduction All econometric models covered have dealt with a single dependent variable and estimations of single equations. However, in modern world economics, interdependence is very common. Several dependent variables are determined simultaneously, therefore appearing both as dependent and explanatory variables in a set of different equations.

Simultaneous equations model The statistics framework for the simultaneous equations model (SEM) is the multivariate regression. An obvious reason for the endogeneity of explanatory variables in a regression model is simultaneity: that is, one or more of the “explanatory” variables are jointly determined with the “dependent” variable. Models of this sort are known as simultaneous equations models (SEMs), and they are widely utilized in both applied microeconomics and macroeconomics.

Simultaneous equations model Consider the well-known demand function: Above model represents Single equation model. Economic analysis suggests that price and quantity typically are determined simultaneously by the market processes, and therefore a full market consists of a set of three different equations: demand function, supply function and condition for equilibrium

Simultaneous equations model Take the following three equations: These are called structural equations of the simultaneous equations model, and the coefficients β’s and γ’s are called structural parameters.

Endogenous vs. Exogenous variables Endogenous variables (Jointly determined variables) The variables which are explained by the functioning of system and values of which are determined by the simultaneous interaction of the relations in the model are endogenous variables or jointly determined variables. Exogenous variables (Predetermined variables) The variables that contribute to provide explanations for the endogenous variables and values of which are determined from outside the model are exogenous variables or predetermined variables. E.g.

Reduced-form of structural equations The reduced-form of the structural equation is an equation in which each of the endogenous variables is expressed only in terms of the exogenous variables, the parameters of the model and the stochastic error terms. These two equations are known as reduced form equations and the πs are known as reduced-form parameters. The system of equations showing reduced-form equations is known as reduced-form model.

Consequences of Ignoring Simultaneity Consider the model: (1) (2) Assuming everything else constant, (a) if e1t increases, this causes Y1t to increase then (b) if Y1t increases (assuming that β2 is positive), Y2t will also increase but (c) if Y2t increases in Equation (2) it also increases in Equation (1) where it is an explanatory variable. Increase in the error term of an equation causes increase in explanatory variable in the same equation. Assumption of no correlation among the error term and the explanatory variables is violated, leading to biased estimates.

Simultaneity bias For OLS estimate of structural parameter, the difference between , the estimate of B (the true value of the coefficient on the stochastic variable) is equal to: If the covariance between the regressor and the error term is positive, will overestimate the true B, a positive bias. If the covariance term is negative, it will underestimate B, a negative bias. 𝛽 𝛽

Identification Problem Identification: A equation belonging to simultaneous equations model is said to be identified if we are able to obtain unique estimates of the structural parameters based on reduced-form equation. Exactly or fully identified: If we can obtain unique estimates of the structural coefficients. Over-identified: If we may obtain more than one estimate of one or more of the regressors. Unidentified or under-identified: If we cannot obtain unique values of the regression coefficients.

Conditions for Identification 1. Order Condition It compares the number of endogeneous variables (G) in the system, and number of variables excluded from the equation under consideration) M If M<G-1, the equation is under-identified If M=G-1, the equation is exactly-identified If M>G-1, the equation is over-identified The order condition is necessary but not sufficient 2. Rank Condition If we can obtain one non-zero determinant of order (G-1)(G-1) of the coefficients of the variables which are excluded in the equation under consideration but may be included in other equations, the equation is said to be identified.

Example 1: Market Demand-Supply model Here, we have three endogenous variables (Qd, Qs and P), so G = 3 and G-1 = 2. ORDER CONDITION Demand function: The number of excluded variables is 1, so M = 1, and because M < G- 1 then the demand function is not identified. Supply function: M = 1 and because M = G – 1 then the supply function is exactly identified . RANK CONDITION We need to check only for the supply function (because we saw that the demand is not identified). The resulting array if G-1=2 rows and columns matrix. Thus, rank condition is satisfied. The supply function is identified.

Estimation of Simultaneous equations model Single equation methods: These methods estimate each equation in the model individually. 1) Indirect Least Squares (ILS), for exactly identified equations 2) Two-stage Least Squares (2SLS), for over identified equations 3) Limited information maximum likelihood (LIML) method System methods of equations: These methods estimate all the equations of the model simultaneously 4) Three-stage least squares (3SLS) method 5) Full information maximum likelihood (FIML) method The problem of identification discussed above is related to the problem of estimating the structural parameters. An application of OLS to the structural equations of a simultaneous equation system will lead to simultaneity bias and inconsistent estimators. This is because the endogenous variables are not independent of the structural error terms.

Indirect Least Squares method To be used only when equations in simultaneous equation model are found to be exactly identified. Step 1 Obtain reduced form of structural model Step 2 Estimate the reduced form parameters (πs) by applying simple OLS to the reduced form equations Step 3 Obtain unique estimates of the structural parameters from the estimates of the reduced form parameters obtained in step 2

Estimation of over-identified equation: TSLS method Basic idea behind TSLS method is to replace stochastic endogenous regressors (which is correlated with error term and causes bias) with one that is non-stochastic regressors (known as proxy or instrument variables) and, consequently, independent of the error term. This method is widely used as a single equation method for the estimation of the overidentified equation which forms the part of SEM. It involves the application of least squares at two stages thus known as 2SLS method. This involves the following two stages (hence two-stage least squares): Stage 1 Regress each endogenous variable that is also a regressor on all the endogenous and lagged endogenous variables in the entire system by using simple OLS (equivalent to estimating the reduced form equations) and obtain fitted values of the endogenous variables of these regressions Stage 2 Use the fitted values from stage 1 as proxies or instruments for the endogenous regressors in the original (structural form) equations

Assumptions of 2SLS 1. The equation should be over identified 2. The model should be correctly specified 3. All the OLS assumptions related to the disturbance term of reduced form model must be satisfied i.e., E(w1)=0, E(w2)=0 4. The error term should be uncorrelated with the predicted value of endogeneous regressor Under the above assumptions, 2SLS estimators are unbiased, consistent and efficient estimators.

Properties of 2SLS estimators The 2SLS estimator is a biased estimator, but it is consistent. In large samples, the 2SLS estimator is approximately normally distributed. The variances and covariances of the 2SLS estimator are unknown in small samples, but for large samples we have expressions for them that we can use as approximations. These formulas are built into econometric software packages, which report standard errors and t-values, just like an ordinary least squares regression program. If you obtain 2SLS estimates by applying two least squares regressions using ordinary least squares regression software, the standard errors and t-values reported in the second regression are not correct for the 2SLS estimator. Always use specialized 2SLS or instrumental variables software when obtaining estimates of structural equations.

3SLS: Three-stage least squares method Step 1: The same as step 1 of 2SLS method. Regress all the endogenous variables on all the exogenous variables of the model and obtain the predicted values of the endogenous variables. Step 2: The same as step 2 of 2SLS method. Substitute the prediction of the endogenous variables found in step 1 in place of the corresponding endogenous variables on the right-hand side of each equation and apply OLS. Step 3: Using the results of step 1 and step 2, apply the GLS estimator, i.e., compute the 3SLS estimators.

3SLS estimation 1. 3SLS estimators are considered as best estimators. 2. 3SLS estimates are better than 2SLS estimates since it uses more information in the calculation of 3SLS estimates as compared to 2SLS estimates. 3. 2SLS estimators correspond to single equation estimation, while 3SLS can be applied to system of equations simultaneously. 4. No doubt 3SLS estimates are also biased but it provides consistent and more efficient estimates than 2SLS estimates.

Limited and Full information maximum likelihood methods LIML is used far less frequently than 2SLS even though Anderson and Sawa (1979) have shown that LIML is superior to 2SLS in terms of its approach to asymptotic normality and moreover the likelihood is a very useful basis for inference. The single equation LIML estimator and the full information maximum likelihood (FIML) estimator are polar cases, the latter using all of the information in the complete system, and the former using only the restrictions in the structural equation being estimated with the restrictions in the remaining structural equations ignored.

LIML Limited Information Maximum Likelihood (LIML) is a form of instrumental variable estimation that is quite similar to TSLS. As with TSLS, LIML uses instruments to rectify the problem where one or more of the right hand side variables in the regression are correlated with residuals. LIML was first introduced by Anderson and Rubin (1949), prior to the introduction of two- stage least squares. However, traditionally TSLS has been favored by researchers over LIML as a method of instrumental variable estimation. If the equation is exactly identified, LIML and TSLS will be numerically identical. Recent studies (for example, Hahn and Inoue 2002) have, however, found that LIML performs better than TSLS in situations where there are many “weak” instruments.

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