# Temporal Order of Control Variables in 2SLS

I am analyzing pooled cross-sectional data with the instrumental variable being educational policy, the independent variable as high school graduation, and the dependent variable as subjective class consciousness. I am considering whether 'marriage' can be used as a control variable in this context.

Regarding the temporal sequence, high school graduation and marriage typically precede the measurement of subjective class consciousness. Hence, in stage two of the analysis, there seems to be no issue with the causal order. The model for subjective class consciousness is: b1 + b2 * high school graduation hat + b3 * marriage + error.

However, the problem arises in stage one. Commonly, if marriage is included as a control variable in stage two, it is also included in stage one. Therefore, it is used to predict high school graduation. This presents a challenge as marriage usually occurs after high school graduation. It implies that past high school graduation is being predicted based on future marriages, creating a temporal causality issue. The stage one model is: high school graduation = a1 + a2 * educational policy + a3 * marriage + error.

My question is, to avoid this problem in instrumental variable two-stage least squares (2SLS) analysis, do control variables need to temporally precede not only the dependent variables but also the independent variables? Additionally, if the time order is problematic and a variable like 'marriage' must be excluded from the control variable in stage one, can it be excluded from the stage one model but included in the stage two model?

Because marriage is an outcome (potentially) of your independent variable, it is a mediator, or in the words of Angrist & Pischke (2009, 2014), a bad control. Here is a simulation of the issue with adjusting for a mediator ($$M$$) for the effect of $$X$$ on $$Y$$ and why it would be bad. Assume the following equations:

$$X = \rho Z + \lambda U + \varepsilon_X$$ $$M = \phi X + \theta U + \varepsilon_M$$ $$Y = \beta X + \gamma M + \delta U + \varepsilon_Y$$

where $$U$$ is an unmeasured confounder and $$Z$$, $$U$$, $$\varepsilon_X$$, $$\varepsilon_M$$, and $$\varepsilon_Y$$ are independent random variables. We can use $$Z$$ as an instrument for $$X$$ to estimate the total effect of $$X$$ on $$Y$$ (this is $$\beta + \gamma\phi$$) but not its partial effect ($$\beta$$) by adjusting for $$M$$, because this would introduce bias from $$U$$.

rho <- 1
lambda <- 1
beta <- 0.25
gamma <- 0.25
delta <- 1
phi <- 1
theta <- 1


Based on these numbers the partial effect is 0.25 and the total effect is 0.5.

library(AER)
n <- 100
nsim <- 10000
res <- list()
set.seed(123)

for (i in 1:nsim) {

if(i%%1000==0) {message(paste(i, "simulations done"))}
z <- rnorm(n)
u <- rnorm(n)
e_x <- rnorm(n)
e_y <- rnorm(n)
e_m <- rnorm(n)

x <- rho*z + lambda*u + e_x
m <- phi*x + theta*u + e_m
y <- beta*x + gamma*m + delta*u + e_y

fit0 <- ivreg(y ~ x | z)
fit1 <- ivreg(y ~ x + m | z + m)

res[[i]] <- c(coef(fit0)["x"], coef(fit1)["x"])

}

res <- do.call(rbind, res)

> round(apply(res, 2, mean),3)
x      x
0.488 -0.433


Without the mediator, we get something very close to 0.5 (the difference seems slightly too large to be a coincidence; my guess is that it is due to the fact that 2SLS is not unbiased, only consistent, so there is a small bias that should shrink with the size of $$\rho$$). However, adjusting for $$M$$, we are way off from 0.25.

So, to answer your questions more directly. Yes, the control variables have to be determined before the independent variable. This applies also when your not doing 2SLS (if $$\lambda=0$$ you could have esimated the total effect by OLS, but not the parital effect unless also $$\theta=0$$). And No. That would be a form of forbidden regression. I am not sure if this question/answer covers it perfectly, but it should get you started.

Consider why you would want to adjust for marriage to begin with? The point of a control variable in a case like this would typically be to adjust for the fact that the policy variable was not randomly assigned, so there could be some unobserved non-random differences between regions (?) with and without the policy.

REFERENCES:

• Angrist, J. D., & Pischke, J. S. (2009). Mostly harmless econometrics: An empiricist's companion. Princeton University Press.
• Angrist, J. D., & Pischke, J. S. (2014). Mastering'metrics: The path from cause to effect. Princeton University Press.