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Econometrics 1 Lecture 15 Simultaneous Equation Models – Reduced Form and Simultaneity Bias 1 Main Feature of Simultaneous Equation System So far we only discussed regression model in which a dependent variable Y is explained by one or a set of explanatory variables with assumption that there one way causation from independent variables to the dependent variables. However, many interdependent. variables in economics are Consider a market model with demand and supply. How much price can a firm charge for a particular product from its customers depends on the quantity sold in the market and how much quantity is demanded by customers depend on the market price. Price determines quantity and quantity determines price. Same is true in national income determination model. 2 Main Feature of Simultaneous Equation System-2 Consumption is a component of income that determines income, but income is the major determinant of consumption. Both quantities and prices and income and consumption are determined simultaneously. We need to estimates a system of equations, not a single equation, in order to be able to capture this interdependency among variables. The main features of a simultaneous equation model are: 1. two or more dependent (endogenous) variables 2. A set of equations 3. Computationally cumbersome, highly non-linearity in parameters and errors in one equation transmitted through the whole system 3 Simultaneity-an Example Consider a relation between quantity and price Qt Pt ut 0 1 A priory it is impossible to say whether this a demand or supply model, both of them have same variables. If we estimate a regression model like this how can we be sure whether the parameters belong to a demand or supply model? We need extra information. Economic theory suggests that demand is related with income of individual and supply may be respond 4 Example of an Identified Model Demand Qtd Pt It u 0 1 2 1,t (1) Supply Qts Pt P u 0 1 2 t 1 2,t (2) where Qtd is quantity demanded andQts is quantity supplied, Pt is the price of commodity, P t 1 is price lagged by one period,It is income of an individual, u and u are independently and identically distributed 1,t 2,t (iid) error terms with a zero mean and a constant variance. Qt and Pt are endogenous variables and P andIt are t 1 exogenous variables, 0 ,1 , 2 , , and are six 0 1 2 parameters defining the system. 5 Example of an Identified Model In equilibrium Qtd =Qts , however note that quantities bought and sold (Q) depends upon market prices (P) and the equilibrium prices is determined by quantity supplied and demanded. The random terms of quantity and price equations above, u1,t and u2,t , are not independent of each other. There is simultaneity problem. 6 A Simple Keynesian Model of Income Determination A simple version of Keynesian income determination model of the following form: Ct Yt ut 0 1 Yt Ct It (3) (4) where Yt is income, C is consumptionI t is t investment, andu is the random error term. The t subscript t refers to time period. and are 1 0 structural parameters, and 0 1Y. 1 t endogenous variables andI t variable. andC are t is an exogenous 7 Reduced form of the Market Model a. Reduced for is obtained by expressing endogenous variable in terms of exogenous variables. In the demand and supply equation in example 1 the reduced form takes the following form: Qt P I v 10 11 t 1 12 t 1,t Pt P I v 20 21 t 1 22 t 2,t (5) (6) Where the reduced form coefficients are defined as: 0; 2 0 2 ; 10 11 12 ; 1 1 1 1 1 1 u u 1,t ; v 2,t 1,t 1 1 1 0 0 1 1 1 2 1 ; ; 21 12 2,0 ; 1 1 1 1 1 1 u u 1 1,t v 1 2,t ; 2,t 1 1 These coefficients are obtained by using equation (2) in (1) for quantity (Q) and using that information in (2) for price level. 8 Reduced form of the National Income Model In the income determination model (example 2) the reduced form is obtained by expressing C and Y endogenous variables in terms of I which is the only exogenous variable in the model. Ct It v 11 12 1,t Yt It v 21 22 2,t (7) (8) 0 Where 11 ; 12 1 1 ; 21 1 0 and 1 1 1 1 . 22 1 1 9 Simultaneity bias c y y t b1 t 2 t t C C y C y C y C y = = = y y y t t t t t t 2 t t 2 t t t => b1 t t t 0 t 2 t t C y y t t 2 t t b1 t t t 1Yt et y t y 2 t t e et covYt , et E Yt E Yt et E et E et 1 1 1 1 2 et y t n e2 1 1 t p lim b1 1 1 y2 2 yt n t 10