SIMULTANEOUS EQUATION MODEL ONE Equation Model (revisited) Characteristics: One dependent variable (Y): as a regressand One ore more independent variables (X): as regressors One way causality relationship: from X to Y no feedback
Illustration: Income can explain consumption patterns. However, consumption does not influence income. Income and education level can explain insurance consumption. But, insurance consumption does not influence education level.
Models of TWO or More Equations There are cases where causality does not have one way effect only, from X to Y. The values of Y are not only determined by X, but the values of X depend on Y as well. In this case, there are two-way relationships or simultaneous relationships between X and Y that are needed to be modeled using multiple equations.
Simple Illustration: Relationships between demand and supply Market price determines quantity demanded and quantity supplied; at the same time, the quantity supplied and quantity demanded also determine market price. Price and quantity are endogenous variables since they are solved simultaneously from the system of equations.
Three-Equation Model (1). Q ts = α 1 + α 2 P t + α 3 P t-1 + ε t (2). Q td = β 1 + β 2 P t + β 3 Y t + μ t (3). Q ts = Q t D
Terminologies Equation (1) represents supply equation that depends on current price and previous price Equation (2) represents demand equation that depends on current price and income Equation (3) represents market equilibrium In this multi equation system, do we still have independent variables and dependent variable?
Q ts, Q td and P t are called endogenous variables (determined by the system of equation) Y t and P t-1 are called exogenous variables (determined outside the system of equation) Remarks: P t-1 was an endogenous variable at time (t-1) but it is an exogenous variable at time t; (it is also called predetermined variable). It is assumed that Y t is known at time t
Graphical Illustration P S P t+1 P t D 2 (y t+1 ) D 1 (y t ) Q t Q t+1 Q
OLSE biased in a Simultaneous Equation. Observed the following model: Q t = α 1 + α 2 P t + ε ; Supply t Q t = β 1 + β 2 P t + β 3 Y t + u t ; Demand To simplify, define: q t = Q t Q aver ; y t = Y t Y aver ; p t = P t P aver Then: q t = α 2 p t + ε t q t = β 2 p t + β 3 y t + u t
This type of model is called Structural Model or Behavioral Model since the form is based on a theory that fit with market structure and/or market behavior. The structural model consists of endogenous variable that at the left side of the equation and both predetermined variable and endogenous variable at the right side of the equation. More specifically: in this model, endogenous variables are in both sides of equations since the model is generated based on a market structure only regardless where the position of endogenous variables are.
If the structural model is estimated using OLS, the estimator of α 2 denoted by a 2 is the following: a 2 = (Σp t q t ) / Σ p t 2 = {Σp t ( α 2 p t + ε t )} / Σ p t 2 = α 2 + (Σ p t ε t )} / Σ p t 2 Since, in general, {(Σp t ε t ) / (Σ p t2 )} 0, estimator a 2 tends to be biased estimator.
The structural model can be simplified into a reduced formed model that all endogenous variables are in the left side of the equations and all exogenous variables are in the right side of the equations. The following is the reduced form model: q t = π 12 y t + v 1t p t = π 22 y t + v 2t Parameters of this reduced form model sometimes can be estimated consistently. (Elaborate in class why?)
As an illustration, for the structural model from market equilibrium discussed earlier, the parameter of α 2 from supply equation can be estimated by: α 2 = π 12 / π 22 = ( y t q t ) / ( p t y t ) The obtained estimator is a consistent estimator since OLS can estimate parameters of the reduced form model consistently.
The procedure that estimate parameters of structural model through estimating parameters of its reduced form model using OLS is called Indirect Least Square. But, this procedure can not always be used in all cases. Sometimes, parameters estimated using this technique will not find unique estimators. Thus, we should find another technique to estimate parameters of a simultaneous equation model.
Identification Problem Identification: a problem of determining structural equation from estimated reduced from model The big question: After estimating the reduced form model, can we find its structural model?
Terminologies: An equation is called unidentified if there is no way in estimating all parameters in the equation of structural form from its reduced form. An equation is called identified if there is a way to obtain all parameters of the equation of structural form from its reduced form.
Terminologies: An equation is exactly identified if the parameters obtained have unique value. An equation is over identified if some of the parameters obtained have multiple values.
Remarks 1. In a system of equations, it is possible to have one equation identified but another equation unidentified. 2. Also, in an equation, it is possible to have some parameters identified but some others unidentified.
Order Condition for Identification. Not all equations can be identified. More over, to identify an equation is not easy. So, we need a rule to make an identification process easier.
Order condition: If an equation is identified, then, the number of exogenous variable(s) and predetermined variable(s) excluded from the equation must equal or greater than the number of endogenous variable that exist in the model minus one.
This condition can be stated this way: Necessary condition for an equation to be identified is that the number of exogenous variable(s) and predetermined variable(s) excluded from the equation must equal or greater than the number of endogenous variable that exist in the model minus one. This condition is only necessary condition. It means that it is possible to have a case where the condition is satisfied but the equation is unidentified.
Illustration Supply: Q t = α 1 + α 2 P t + ε t ; P: price Demand: Q t = β 1 + β 2 P t + β 3 Y t + u t ; Q: quantity
Demand equation is unidentified since there is no exogenous variable nor predetermined variable that excluded from the equation. Supply equation is identified since there are two endogenous variables in the equation and there is one endogenous variable that excluded from the equation (Y t ). So, the order condition fulfilled.
Two-stage Least Squares This procedure is a good method to estimate parameters of structural model for over-identified equations. This technique will give a unique parameter estimator.
See the following supply-demand model Structural Model: Supply: q t = α 2 p t + ε t Demand: q t = β 2 p t + β 3 y t + β 4 w t + u t ; w: wealth Reduced form Model: q t = π 12 y t + π 13 w t + v 1t p t = π 22 y t + π 23 w t + v 2t
Two Stages of Estimation: 1. Equation p t in the reduced form model is estimated using OLS. After π 22 and π 23 are estimated, then, p t also predicted. 2. Supply equation can be estimated using OLS using predicted p obtained from the first stage. Therefore, supply equation is then estimated. Analogously, demand equation can be estimated similarly.
Comments: Supply equation is over-identified since the number of exogenous variables excluded from the equation is 3 (three) while the number of endogenous variables exists in the equation is 2 (two). Therefore, if supply equation is estimated using ILS, the estimators will have multiple values.
Example: Demand for Electricity It will be calculated price elasticity of electricity demand in the US. The data collected from 48 states from 1961-1969.
Demand of Electricity (Q) depends on: (i). P: price of electricity (real) (ii). Y: income per capita / year (real) (iii). G: price of gas( (substitute) (iv). D: number of days using heater (v). J: average temperature in July (vi). R: percentage of pop living in suburb (vii). H: household size
Short term supply of electricity is assumed fixed while the price of electricity (P) depends on: (i). Q: quantity purchased (ii). L: wages (iii). T: time (iv). K: percentage of electricity produced by companies (v). F: price of oil to produce 1 kilowatt-hour of electricity (vi). I: Ratio of total sales of electricity for industry and total sales of household consumption
Simultaneous Model offered: 1. Ln Q = a 1 + a 2 Ln P + a 3 Ln Y + a 4 Ln G + a 5 Ln D + a 6 Ln J + a 7 Ln R + a 8 Ln H + e 2. Ln P = b 1 + b 2 Ln Q + b 3 Ln L + b 4 Ln K + b 5 Ln F + b 6 Ln R + b 7 Ln I + b 8 Ln T + u
Comments: (i). Equation (1) is a demand equation while equation (2) is a price equation. In this price equation, it is assumed that the price is cheaper when we purchase in a bigger quantity. (ii). Equation (1) is identified since the number of endogenous variable is two (P and Q). While the number of exogenous variables that are not appeared (excluded) in equation (1) is five (L, K, F, I, T)
(iii). Equation (2) is also identified since the number of endogenous variable is two and the number of exogenous variables that are not appeared (excluded) in the model is five 5 (Y, G, D, J, H) (iv). These two equations are estimated using 2SLS. (a). In the first stage, each endogenous variable is regressed with all exogenous variables. (b). In the second stage, each equation of structural model is estimated using instrumental variable that has been predicted in the first stage to replace an endogenous variable appeared in the right hand side of the equation.
(v). Estimation results: 1. Ln Q = - 0.21 1.15 Ln P + 0.51 Ln Y + (0.03) (0.06) 0.04 Ln G - 0.02 Ln D + 0.54 Ln J + (0.01) (0.02) (0.12) 0.21 Ln R + 0.24 Ln H (0.02) (0.12) R 2 = 0.91 2. Ln P = 0.57 0.60 Ln Q + 0.24 Ln L 0.02 Ln K + (0.03) (0.04) (0.01) 0.01 Ln F + 0.03 Ln R 0.12 Ln I + 0.004 T (0.003) (0.01) (0.01) (0.03) R 2 = 0.97
Comment: Price Elasticity of electricity demand in the US (in the long term) is 1.15.