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.

  • $\begingroup$ 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. $\endgroup$
    – mrz1702
    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.

  • $\begingroup$ Thanks, that did help. Now it's clear why my "reparametrization" actually is not a reparametrization but a different model. $\endgroup$
    – mrz1702
    Mar 31 at 13:59
  • $\begingroup$ It is a nice illustration on why measurement error ($\eta$) is so problematic in moderation analysis. $\endgroup$
    – POC
    Apr 7 at 12:30

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