# Biased OLS coefficient of endogenous interaction term

Assume we have the following system of equations

Eq. 1: $$z = x\gamma + \epsilon$$

Eq. 2: $$y = z\theta + x\beta + \eta$$

where the error terms fulfill standard assumptions. Simulating data from Eq. 1 and Eq. 2 and estimating Eq. 2 using OLS gives an unbiased estimate of $$\theta$$ because we condition on $$x$$. Similarly, we can use a different parametrization of the problem and write

Eq. 3: $$y = (z - x\gamma)\theta + x\rho + \eta$$

where we made explicit that $$\theta$$ is the effect of the exogenous variation in $$z$$ on $$y$$. Using Eq. 3 to simulate the data and Eq. 2 to estimate the parameters results in the same estimate of $$\theta$$ as before. So far, so good.

I am interested in including an interaction term between $$x$$ and $$z$$ in the model like follows

Eq. 4: $$y = z\theta + zx\delta + x\tau + \kappa.$$

Simulating data from Eq. 4 and estimating the parameters using OLS gives unbiased estimates of $$\delta$$ and $$\theta$$. Based on these results, I was expecting that simulating from

Eq. 5: $$y = (z - x\gamma)\theta + (z - x\gamma)x\delta + x\nu + \zeta$$

and estimating a specification like in Eq. 4 using OLS will also give unbiased results for $$\theta$$ and $$\delta$$. However, while the estimate of $$\theta$$ is unbiased, the estimate of $$\delta$$ is now heavily biased. I do not really have an explanation for this, but I assume that it has to do with the fact that $$z$$ is exogenous conditional on $$x$$, while $$z*x$$ is not exogenous conditional on $$x$$. Any thoughts are appreciated. Here is my simulation code in R.

• Update: Estimating Eq. 4 using data generated from Eq. 5 gives unbiased estimates of $\theta$ and $\delta$ when additionally controling for $x^2$. However, I estimated this specification due to intuition and not based on a formal derivation. When controlling for $x$ orthogonalizes $z$, we'd need to control for $x*x$ to orthogonalize $z*x$. While this seems to work, I'd still be happy to receive a pointer to a formal explanation of this. Mar 31 at 11:04

When you $$z-\gamma x$$, you recover the error $$\eta$$ and, since it is an additive linear model, the information in $$z$$ about $$\gamma x$$ is given to $$\beta$$. This do not apply for $$xz$$ or $$x(\gamma x + \eta)$$, which we can easily see is not equal to $$(z - x\gamma)x$$ or $$\eta x$$. To get a correct eq 5, it should be :

y = beta * x + (theta + delta * x) * z + err


If you want to remove $$z$$, then

x = rnorm(n)
eta = rnorm(n)
z = x * gamma + eta
y = beta * x + (theta + delta * x) * (gamma*x + eta) + err


Regarding your comment, Eq 4 is correct, and you see where $$x^2$$ comes from.

Hope, it helps.

• Thanks, that did help. Now it's clear why my "reparametrization" actually is not a reparametrization but a different model. Mar 31 at 13:59
• It is a nice illustration on why measurement error ($\eta$) is so problematic in moderation analysis.
– POC
Apr 7 at 12:30