A topic in the Open Knowledge Graph — a free, open map of 15,495 topics and the order to learn them in.
A simultaneous equations model (SEM) has several structural equations whose dependent variables are determined jointly, as price and quantity are set by supply and demand together. Each endogenous variable is correlated with the errors of the equations it appears in, so OLS on one equation is biased and inconsistent. An equation can be estimated only if it is identified: some exogenous variable that is excluded from it shifts the other equation. The order condition requires at least as many excluded exogenous variables as included endogenous explanatory variables; the rank condition requires that those excluded variables actually appear, with nonzero coefficients, elsewhere in the system. Identified equations are estimated by two-stage least squares.
Draw a supply-and-demand diagram where both curves shift randomly, and show that the observed price-quantity points trace out neither curve. Then add a variable that shifts only supply, such as weather in a crop market, and show that its movements trace out the demand curve. Writing the two structural equations and counting excluded variables for each turns this picture into the order condition.
From your study of endogeneity you know that simultaneity is one way a regressor becomes correlated with the error. Simultaneous equations models make that idea systematic. The classic case is a market. Supply and demand are two structural equations, each describing the behavior of one side, and price and quantity are both determined by where they meet.
Write the model as supply, q = α₁p + β₁z₁ + u₁, and demand, q = α₂p + β₂z₂ + u₂. Here p and q are endogenous, determined inside the system. The variables z₁ and z₂ are exogenous, determined outside it, such as rainfall for a crop and consumer income. Solve the two equations together and you get the reduced form: p and q each written as functions of z₁, z₂, and both errors. Because p depends on u₂, OLS on the demand equation is inconsistent. The same holds for supply.
This problem was recognized early. Elmer Working's 1927 article "What Do Statistical 'Demand Curves' Show?" explained that price-quantity data trace out a cloud of intersections. Trygve Haavelmo's 1943 work on simultaneous equations made simultaneity bias a central topic in econometrics.
The solution is identification. To estimate the demand curve, you need movements in supply that leave demand unchanged. Rainfall does that: it shifts supply, is excluded from demand, and so traces out points along the demand curve. Rainfall serves as the instrument for price in the demand equation. Income, excluded from supply, plays the same role for the supply equation.
Two conditions formalize this. The order condition says an equation can be identified only if the number of exogenous variables excluded from it is at least the number of endogenous explanatory variables it includes. It is a counting rule and is necessary, but not enough. The rank condition adds that the excluded variables must really matter in the other equation. If rainfall's true coefficient in supply were zero, it would pass the count but fail to identify demand. An equation with exactly enough instruments is just identified; one with more is overidentified, and the extra restrictions can be tested.
Once identified, each equation is estimated by two-stage least squares, using all exogenous variables in the system as instruments. Applications go beyond markets: labor supply and wage offers, crime and police spending, and money demand and interest rates have all been modeled this way.
No topics depend on this one yet.