Sunday, May 13, 2012
Average Treatement Effects and Correlated Random Coefficients
* Average Treatement Effects and Correlated Random Coefficients
* Wooldridge, J. (2003). Further results on instrumental
* variables estimation of average treatment effects in
* the correlated random coefficient model. Economics
* Letters, 79(2), 185-191. doi:10.1016/S0165-1765
* This simulation follows the paper by Wooldridge which
* shows that it is possible to consistently estimate
* the population average of the correlated random
* coefficient (CRC) model with multiple treatment
* variables under the standard assumptions under
* which instrumental variables (IV) estimators are
* In order to understand CRC think of the following
* model. Equation (1):
* E(y|a,b,w)=a+wb=a + b1*w1 + b2*w2 ... + bg*wg (1)
* b can depend on unobserved heterogeniety as well
* as w. Writing the equation in error form (2):
* y = a + wb + e, E(e|a,b,w)=0
* b can vary for every individual observation i.
* Therefore we are not trying to estimate each individual
* b which is impossible, but instead we are attempting
* to estimate the E(b) or the average treatment effect (ATE).
* This in general is difficult when bj is correlated with
* unobserved heterogeniety of individual j.
* In order to tackle this problem Wooldridge introduces
* a instrumental variable which is redundant or ignorable
* in covariates x and instruments z ie.
* E(y|a,b,w,x,z)=E(y|a,b,w) (6)
* The next assumption is what seperates xs from zs.
* E(a|x,z)=E(a|x)=gamma0 + x*gamma (7)
* This says that the mean a can depend on the explanatory
* variables x but not on the instrumental variable z.
* E(bj|x,z)=E(b|x)=beta0 + (x-E(x))*deltaj
* This assumption says that the average coefficient can
* depend on the explanatory variables x but not on the
* instrumental variable z.
* We can write a = gamma0 + x*gamma + c , E(c|x,z)=0 (8)
* and b=beta0 + (x-E(x))*deltaj + vj, E(vj|x,z)=0 (9)
* Ultimately by substituting this back into (1) we get:
* y=gamma0 + xgamma + wbeta + w1(x-E(x))*delta1 +....
* wG(x-E(x))*deltaG + c + wv + e (10)
* The composite error term is c + wv + e. Under the
* assumptions thus far E(c|x,z)=E(e|x,z)=0. However
* E(wv|x,z)!=0 because b is generally not a
* deterministic linear function of x.
* Let us simulate up to this point imagining that we do
* have our bs as deterministic linear functions of xs.
set obs 10000
gen x1 = rnormal()
gen x2 = rnormal()
* We will force w to be uncorrelated with x.
gen w1 = rnormal()
gen w2 = rnormal()
* Each idividual has his/her own intercept or starting
* point c
gen a= -5 + 3*x1 - 2*x2 + c
* Let us first imagine b being a deterministic function
* of observables x. Each idividual has a unique response
gen b1 = 1 + .5*x1 + -2*x2
gen b2 = -2 + 1.75*x1 + 3*x2
* Imagine w1 as being years of education, w2 as being
* offered a job right out of college and x1 as intelligence
* and x2 as GPA. The interesting thing is having a high
* GPA might be correlated with years of experience
* but it also might help explain the effect years of education
* has on y (future income).
* Let us generate our reduced form error
* Now let's generate our y variables
gen y = a + b1*w1 + b2*w2+e
* First let us generate our w1(x-E(x)) variables:
* The average of observed x is of course not the E(x) but it is
* a consistent estimator of E(x).
gen w1_x1 = w1*(x1-r(mean))
gen w2_x1 = w2*(x1-r(mean))
gen w1_x2 = w1*(x2-r(mean))
gen w2_x2 = w2*(x2-r(mean))
* Now we can estimate all of our coefficients directly using
* equation 10.
* y=gamma0 + x gamma + w beta + w1(x-E(x))*delta1 +....
* wG(x-E(x))*deltaG + c + wv + e (10)
reg y x1 x2 w1 w2 w1_x1 w1_x2 w2_x1 w2_x2
* One can see that in the case where b is completely
* linearly dependant on x the above CRC estimator works fine.
* Stay tuned for what happens when there is some error in b!