Lecture 13 Flashcards
Simultaneity
When x and y are jointly determined, so each affects the other
- causes endogeneity
Simple market equilibrium example:
- S = k1p + m1
- D = k2p + m2
M1 and m2 are the exogenous variables influencing supply like interest rates, technology, etc
S = D
-> p = (m1-m2)/(k2-k1)
-> q = (k2m1 - k1m2)/(k2-k1)
As you can see price is correlated with both the demand and supply shock, so correlated with the error terms in both equations, violating OLS that regressions must be uncorrelated with the error term - p is endogenous,
Why do we care if p was endogenous?
Importance for understanding market primitives
- e.g. if we want to see effect of a policy change on market, we need an understanding of the structural parameters of the supply and demand functions
An interesting solution to the simultaneity issue
If the supply shifter contains an unobservable component u1, and an observable component z, which is exogenous with respect to u1 and v2
- cov(z,u1) = cov(z,v2) = 0
Intuitively, an exogenous supply shifter traces the demand curve, helps us estimate it.
After splitting error term up, supply function is now:
S = k1p + B1z + u
- new equilibrium price and quantity
- k2 = cov(z,q)/cov(z,p)
- therefore k2^ is the IV estimator we saw earlier, with z acting as an instrument for p
Summarise the Z stuff
- exogenous variable z which shows up in the supply function and is excluded from the demand function identifies the demand function
- acts as an instrument for endogenous price p, creates variation in p through supply side, but isn’t directly part of the demand equation
- z must be relevant and also not correlated with the residuals of the equation being estimated
Two equation SEM in generality
Y1 = a1y2 + B1z1 + u1
Y2 = a2y1 + B2z2 + u2
Where y1,y2 are endogenous variables, z1,z2 are exogenous variables, u1,u2 are structural shocks
- can show endogeneity by substituting y2 into first equation, and then seeing that the equation for y2 contains u1 and u2, which correlate with y1
How does TSLS play a factor?
Key requirement is that the exogenous variables in equation 2 must include at least one instrument that is excluded from equation 1, so Z1 cant equal z2
- known as the exclusion restriction, and is behind the condition for identification of a structural equation in SEM: the rank condition
What is the rank condition
A structural equation in SEM is identified iff the excluded exogenous variables provide enough independent variation to predict the endogenous variables, meaning the matrix of their coefficients has full column rank
Can you tell us which equation is identified
- 2 endo and 3 excluded exo
- 1 endo and 1 excluded exo
- 1 endo and no excluded exo
- overidentified, so use TSLS
- equation is just identified
- equation is unidentified
Order condition
A necessary but not sufficient requirement for identifying a structural equation in an SEM
- satisfied if the number of excluded exogenous variables from the equation is at least as large as the number of RHS endogenous variables
Go through a simultaneity problem,:
- Q = a1P + a2I + u1, demand
- Q = B1P + B2C + u2, supply
P is endogenous as it is determined by both supply and demand, so if we use OLS to estimate the demand equation, a1 will be biased
- we need an instrument which affects P but not demand, lets say, K - production cost
- K affects supply, shifting the supply curve and influencing price, K doesn’t directly affect demand, satisfying the exclusion restriction - now we can use TSLS
- regress K and I on P, so P = y1K + y2I + v
- now Qd = a1P^ + a2I + u1, since P is now uncorrelated with u1, estimation of a1 is now consistent.
- do the reverse for the supply equation
Main reasons for the 3 conditions
Exclusion - ensures an exogenous variable is a valid instrument
Order - ensures we have enough instruments to identify the equation
Rank - ensures the instruments are strong and provide independent variation
