# Sufficiency of CIA for identification in constant effects linear model

Everything is from page 58-59 of Mostly Harmless Econometrics (Angrist & Pischke, 2009)

Assume the following causal constant effects model:

Potential outcomes of person $$i$$ as a function of $$s$$ are given by

$$f_i(s)=\alpha+\rho s + \eta_i$$

assume (w.l.o.g, due to the presence of $$\alpha$$) $$E[\eta_i]=0$$

Hence we observe outcome:

$$y_i=\alpha+\rho s_i +\eta_i$$

The conditional independence assumption is assumed to hold. So:

$$s_i \perp f(s_i)|X_i$$

$$X_i$$ is a vector of observables.

Then the authors make the following argument:

We can write $$\eta_i=X_i'\gamma+\nu_i$$ Where $$\gamma$$ are regression coefficients so $$cov(X_i, \nu_i)=0$$ by construction.

Then they assume $$E[\eta_i|X_i]=X_i' \gamma$$ or in other words $$E[\nu_i|X_i]=0$$. This seems like a very strong parametric assumption on the form of $$\eta_i$$. Using this assumption they show that in this case the model $$y_i=\alpha+\rho s_i + X_i' \gamma +\nu_i$$ is a regression and in particular $$\rho$$ is identified and can be estimated by OLS, including these covariates. I understand all this.

Commonly however, it is claimed by people that if CIA holds, then $$\rho$$ is identified and can be estimated by OLS, including these covariates. A weaker claim, since it is not assumed that $$E[\nu_i | X_i]=0$$. Can this be proved?

My own workings

Iff $$\rho$$ is regression coefficient in model of form

$$y_i=\alpha+\rho s_i +\beta' X_i+e_i$$ (1.1)

(I assume full rank assumption holds, sucht that there exist unique set of $$\alpha$$ and coefficients on $$s_i$$ and $$X_i$$ that make it regression). Then it must hold that there exist $$\alpha, \beta$$ such that

(i) $$cov(X_i, e_i)=0$$ ($$e_i$$ is a function of coefficients)

(ii) $$cov(s_i, e_i)=0$$

We assume instead of the linear specification of the CEF (condtional expectation function) of $$\eta_i$$ the following:

$$\eta_i=f(X_i)+u_i$$ with $$E[u_i | X_i]=0$$.

Then we can rewrite equation (1.1) as

$$y_i=\alpha+\rho s_i +\beta' X_i+[u_i+f(X_i)-X_i'\beta]$$ (1.1)

Then it is straightforward to see that for identification we must have

1. $$cov(s_i, f(X_i)-X_i' \beta)=0$$

2. $$cov(X_i'\beta, f(x_i))=Var(X_i' \beta)$$

I don't see how the CIA is sufficient here.

Simulation

I have ran a simulation in R with $$f$$ linear, quadratic, sine and logarithm for the case $$X_i \in \mathbb{R}$$. This suggests that indeed the CIA is sufficient for identification of $$\rho$$ via OLS:

library(tidyverse)
##params
alpha=1
rho=1
###initialize data
nspecific=4
nsims=10^4
n=10^3
#matrix for results
results=matrix(data=NA, nrow=nsims, ncol=nspecific)
#grid for graphs
par(mfrow=c(1,nspecific))
#specs in string
specifications=c("X^2", "log(|X|)", "aX", "sin(X)")
for (spec in c(1:nspecific)){
for (i in c(1:nsims)){
#s is influenced by X in the same direction
X=rnorm(n, 10, 4)
s=rnorm(n, X)
Z=as.data.frame(cbind(X,s))
#X also influences earnings. NOTE: in absence of this, the endogeneity of s
#would not be a problem.
#X influences earnings in some nonlinear way.
nu=rnorm(n)
if (spec==1){
eta=X^2+nu
}
if (spec==2){
eta=log(abs(X))+nu
}
if (spec==3){
eta=X+nu
}
if (spec==4){
eta=sin(X)+nu
}
y=alpha+rho*s+eta
#OLS identifies rho?
reg=lm(y~Z$X+Z$s)
results[i, spec]=reg$coefficients } hist(results[, spec], col = "blue",main = specifications[spec]) abline(v = rho, col="red") }  This produces histograms of sampling distribution for OLS estimate for $$\rho$$: I will check back later, since I am writing in a hurry. I am very curious though. Thank you. Addendum: CIA insufficient? I run the same simulation as above, but with different specifications. To be precise: $$y_i=\alpha+s_i + \eta_i$$ with $$\eta_i=X_i^2+\nu_i, \qquad \nu_i \sim N(0,1), X_i \sim Poisson(10)$$ Furthermore $$s_i \sim Bin(X_i^2, 0.5)$$ ##investigate the failure of CEF X^2 library(tidyverse) ##params alpha=1 rho=1 ###initialize data nspecific=1 nsims=10^4 n=10^3 #matrix for results results=matrix(data=NA, nrow=nsims, ncol=nspecific) regressions=list() #grid for graphs par(mfrow=c(1,nspecific)) #specs in string specifications=c("X^2") for (spec in c(1:nspecific)){ for (i in c(1:nsims)){ #s is influenced by X in the same direction X=rpois(n, 10) #s=rnorm(n, sin(X), X^2) s=rbinom(n, X^2, 0.5) Z=as.data.frame(cbind(X,s)) #X also influences earnings. NOTE: in absence of this, the endogeneity of s #would not be a problem. #X influences earnings in some nonlinear way. nu=rnorm(n) if (spec==1){ eta=X^2+nu } if (spec==2){ eta=log(abs(X+1))+nu } if (spec==3){ eta=X+nu } if (spec==4){ eta=sin(X)+nu } y=alpha+rho*s+eta #OLS identifies rho? reg=lm(y~Z$$X+Z$$s) results[i, spec]=reg$coefficients
regressions[[i]]=reg
if (i%%100==0){
print(summary(reg))
}

}
hist(results[, spec], col = "blue",main = specifications[spec], xlim = c(0.9, 2.6))
abline(v = rho, col="red")
} Now OLS seems (very) inconsistent. Even though CIA is satisfied, it seems to me.

As an informal test of conditional independence I fix a range of $$X$$ values and generate $$s_i$$ and $$\eta_i$$ conditional on $$X_i=x$$. PLotting seems to indicate CIA is satisfied.

fix_X=c(1:max(X))
#fix_X=sample(fix_X, 20, replace = FALSE)
trials=10^3

number_cols=7
par(mfrow=c(4, number_cols), mar=c(2,2,2,2))

for (x in fix_X){
for (i in trials){
nu=rnorm(trials)
eta=x^2+nu
s=s=rbinom(trials, x^2, 0.5)

}
#Chi squared test
plot(eta,s, xlim = c(min(eta), max(eta)), main = paste("X=",x))

} $\eta$ on the horizontal axis, $$s$$ on the vertical axis" />

I am curious what the right conclusion is here. In particular

(i) Maybe my approach is faulty and CIA is in fact sufficient to identify linear constant causal effects. In this case I am happy to be corrected.

(ii) Maybe I stumbled upon an unnatural specification in which CIA is insufficient. But in most applications CIA is sufficient. In this case I am curious for sources that show this robustness.

• What do you mean if it can be supported? If I am not mistaken they actually have proof in the book Nov 3, 2022 at 16:09
• I mean proved @csilvia. I cannot find the proof Nov 4, 2022 at 8:19