Simultaneous Equation Models Learning Objectives Introduction Introduction (2) Introduction (3) Solving the Model structural equations

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1 Simultaneous Equation Models. Introduction: basic definitions 2. Consequences of ignoring simultaneity 3. The identification problem 4. Estimation of simultaneous equation models 5. Example: IS LM model Learning Obectives. Understand the problem of simultaneity and its consequences 2. Understand the identification problem through macroeconomic examples 3. Understand and use the two-stage least squares method of estimation 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. Introduction (2) Consider the well-known demand function: 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: (a) demand function, (b) supply function and (c) condition for equilibrium Introduction (3) Take the following three equations: Solving the Model Using the equilibrium condition and solving for P t we get These are called structural equations of the simultaneous equations model, and the coefficients β and γ are called structural parameters. or

2 Solving the Model (2) Substituting the expression for Pt in the supply function we get: Solving the Model (3) The two new equations specify each of the endogenous variables 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. Consequences of Ignoring Simultaneity One of the assumptions of CLRM states that the error term of an equation should be uncorrelated with each of the explanatory variables in the equation. If such a correlation exists, the OLS regression equation is biased. It should be evident from the reduced form equations that, in cases of simultaneous equation models, such a bias exists. Consequences of Ignoring Simultaneity (2) Consider the model: (.8) (.9) And think about an increase in e t assuming everything else stays constant. Consequences of Ignoring Simultaneity (3) (a) if e t increases, this causes Y t to increase because of Equation (.8); then (b) if Y t increases (assuming that β 2 is positive) Y 2t will also increase because of the relationship in Equation (.9); but (c) if Y 2t increases in Equation (.9) it also increases in Equation (.8) where it is an explanatory variable. Consequences of Ignoring Simultaneity (4) 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. 2

3 Estimation of Simultaneous Equations Estimation of exactly identified equation: ILS method To be used only when equations in simultaneous equation model are found to be exactly identified. Step Find reduced form equations Step 2 Estimate the reduced form parameters by applying simple OLS to the reduced form equations Step 3 Obtain unique estimates of the structural parameters from the estimates of the parameters of the reduced form equation in step 2 Estimation of Simultaneous Equations (2) Estimation of over-identified equation: TSLS method Basic idea behind TSLS method is to replace stochastic endogenous regressor (which is correlated with error term and causes bias) with one that is nonstochastic and, consequently, independent of the error term. This involves the following two stages (hence twostage least squares): Estimation of Simultaneous Equations (3) Stage 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 as proxies or instruments for the endogenous regressors in the original (structural form) equations VAR Models and Causality. Vector autoregressive (VAR) models 2. Causality tests Learning Obectives. Differentiate between univariate and multivariate time series models 2. Understand VAR models and discuss advantages 3. Understand the concept of causality and its importance in economic applications 4. Use Granger causality test procedure 5. Use Sims causality test procedure 6. Estimate VAR models and test for Granger and Sims causality through the use of econometric software VAR Models Quite common in economics to have models where some variables are not only explanatory variables for a given dependent variable; but also explained by variables that they are used to determine. In those cases we have models of simultaneous equations, in which it is necessary to clearly identify which are endogenous and which are exogenous or predetermined variables. 3

4 VAR Models (2) Sims (980) suggests: if there is simultaneity among a number of variables, then all these variables should be treated in the same way. So, all variables are treated as endogenous. This means that in its general reduced form each equation has the same set of regressors. VAR Models (3) For example, time series y t that is affected by current and past values of x t and, simultaneously, the time series x t to be a series that is affected by current and past values of the y t series. In this case, simple bivariate model is given by: y t = β 0 β 2 x t + γ y t + γ 2 x t + u yt x t = β 20 β 2 y t + γ 2 y t + γ 22 x t + u xt This is a first-order VAR model, because the longest lag length is unity. Rewriting the system with matrix algebra: 2 VAR Models (4) 2 yt 0 xt yt 2xt u u yt xt Or Where B 2 2 VAR Models (5) Bz t = Γ 0 + Γ z t + u t 2 yt 0, zt, 0 xt 20 2 u yt and ut 2 uxt VAR Models (6) Advantages of VAR models: (a) Very simple (no worry about which variables are endogenous or exogenous) (b) Estimation is very simple (usual OLS) (c) Forecasts from VAR models are better than those obtained from far more complex simultaneous equation models VAR Models (7) Disadvantages of VAR models: (a) A-theoretic as they are not based on any economic theory everything causes everything (resolved by statistical inference and causality tests) (b) Loss of degrees of freedom (c) Obtained coefficients of VAR models are difficult to interpret as they lack theoretical background (overcome by impulse response functions) 4

5 Causality Tests Suppose two variables, say y t and x t, affect each other with distributed lags. The relationship between those variables can be captured by a VAR model. In this case it is possible to have that: (a) y t causes x t (b) x t causes y t (c) there is bi-directional feedback (causality among variables) (d) the two variables are independent Causality Tests (2) Granger (969) developed a relatively simple test that defined causality as follows: A variable y t is said to Granger-cause x t, if x t can be predicted with greater accuracy by using past values of the y t variable rather than not using such past values, all other terms remaining unchanged. Causality Tests (3) First step is to estimate following VAR model: y a t t x a 2 n i n i x i ti x i ti m m y y t t e e t 2t Causality Tests (4) Case Lagged x terms in () may be statistically different from zero as a group, and lagged y terms in (2) not statistically different from zero. In this case we have that x t causes y t. Case 2 Lagged y terms in () may be statistically different from zero as a group, and lagged x terms in () not statistically different from zero. In this case we have that y t causes x t. Case 3 Both sets of x and y terms are statistically different from zero in () and (2), so that have bi-directional causality. Case 4 Both sets of x and y terms are not statistically different from zero in () and (2), so that x t is independent of y t. Causality Tests (5) Step Regress y t on lagged y terms and obtain RSS of this regression (which is the restricted one) and label it as RSS R Step 2 Regress y t on lagged y terms plus lagged x and obtain RSS of this regression (which now is the unrestricted one) and label it as RSS U Step 3 Set the null and the alternative hypotheses: H o : coefficients of the lagged terms of x are equal to zero, or x / y H a : coefficients of the lagged terms of x are not equal to zero, or x y Causality Tests (6) Step 4 Calculate F statistic for normal Wald test on coefficient restrictions Step 5 If computed F value exceeds F-critical value, reect the null hypothesis and conclude that x t causes y t 5

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