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I'm reading through Asymptotic Theory for Econometricians by Halbert White and am trying to figure out the intuition for situations where the errors would be correlated.

Here's the setup:

$Y_{t1} = Y_{t2}\alpha_o + \pmb W_{t1}^T \pmb \delta_o + \epsilon_{t1}, E(\pmb W_{t1}\epsilon_{t1}) = 0$

$Y_{t2} = \pmb W_{t2}^T\pmb \gamma_o + \epsilon_{t2}, E(\pmb W_{t2}\epsilon_{t2}) = 0$

There's a couple other assumptions made but the one I'm having a hard time reasoning about is $E(\epsilon_{t1}\epsilon_{t2}) \neq 0$. What would be some intuitive scenarios for when this would be expected?


Edit:

Using a univariate framework, here's how I currently understand simultaneity (could be dead wrong):

Assume the model $y_1 = xb + \epsilon$. We already know that \begin{align} \frac{cov(x,y_1)}{var(x)} = \hat b \end{align}

Substituting in the equation for $y_1$ and simplifying we get that $b = \hat b$.

Now assume that we have an issue with simultaneity. In particular, we discover the model should be $y_1 = y_2a + xb + \epsilon$ where $y_2 = x b_2 + \epsilon_2$.

Then what our erroneous $\hat b$ is actually equal to is:

\begin{align} \hat b = \frac{cov(x,y_1)}{var(x)} = \frac{cov(x,y_2a + xb + \epsilon_2)}{var(x)} = \frac{a\ cov(y_2,x)}{var(x)} + b \neq b \end{align}

which is a pretty simple bias term to me:

\begin{align} \text{bias} &= \frac{a\ cov(y_2,x)}{var(x)}\\ &= \frac{a\ cov(x b_2 + \epsilon_2,x)}{var(x)} \\& = a b_2 \end{align}

But this argument for simultaneity doesn't look like $cov(\epsilon_1, \epsilon_2) \neq 0$ to me. It really just looks like OVB haha

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Virtually every social science scenario qualifies, in fact it is much harder to find cases when it does not apply. Consider classic examples:

Education allows people to earn higher income. However, people with higher income can more afford to get educated. This results in $E[\epsilon_1,\epsilon_2] \neq 0$ since higher income causes education but education causes higher income.

Or for example consider spending on policing. Higher spending on policing should reduce crime. However, countries with higher crime need to spend more on policing. More policing causes lower crime, but higher crime causes more policing.

The two above are just classic examples, virtually every social science problem will have more or less obvious violation of $E[\epsilon_1,\epsilon_2] \neq 0$. It is extremely rare to find social science problems where this is not a case.


Mathematically the $E[\epsilon_1,\epsilon_2] \neq 0$ because of the following:

Suppose we have reverse causality so we have structural model given by:

$$y_i= \beta_1 x_i+ \gamma_1z_i+\epsilon_1 \\ z_i=\beta_2x_i+\gamma_2y_i+\epsilon_2 $$

Now in the first eq by assumption $E(z_{i}u_{i})\neq 0$. If we now substitute second equation into first one and solve for $z$ assuming that $1-\gamma_{1}\gamma_{2}\neq 0$ gets us

$$z_{i}={\frac {\beta _{2}+\gamma _{2}\beta _{1}}{1-\gamma _{1}\gamma _{2}}}x_{i}+{\frac {1}{1-\gamma _{1}\gamma _{2}}}\epsilon_1+\frac{\gamma _{2}}{1-\gamma_1\gamma_2}\epsilon_2$$

Assuming $x$ and $\gamma$ are uncorrelated then we get:

$$E[z, \epsilon_2]={\frac {\gamma _{2}}{1-\gamma _{1}\gamma _{2}}}\operatorname {E} (\epsilon_1,\epsilon_2)\neq 0 \implies E(\epsilon_1,\epsilon_2)\neq 0 $$

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  • $\begingroup$ Thinking if Y1 as education and Y2 as education, why would this imply the errors are corrected? We want E([Y2,W']eps1) = [E(Y2eps1),0], we subsitute in E((W+eps2)eps1) = E(eps2eps1) but how exactly does that lead to this term being non-zero? Which part of eps1 is correlated with which part of eps2? (For example, in OVB we have E(Xeps)=E(X(W+Eps2)) = E(XW) where XW is a nonzero vector because they're correlated) Having a hard time seeing something like this with E(eps1eps2) $\endgroup$ Oct 20, 2022 at 21:58
  • $\begingroup$ @financial_physician oh I thought you are just looking for intuition because you understand the math. Mathematically if income (y) causes education (e) and education causes income we can write $y=a +be + \epsilon_y$ and $e=c + dy + \epsilon_e$ with b and d $\neq 0 \implies E[\epsilon_y, \epsilon_e] \neq 0$. I mean how in detail you want me to go in here? This is the same thing you have in your opening question just simplified, if you want I can add to my answer the whole math but that in itself is not intuitive $\endgroup$
    – 1muflon1
    Oct 20, 2022 at 22:05
  • $\begingroup$ If it's not too much hassle, would really appreciate it. I googled a proof earlier and came across this pdf but it wasn't satisfying for me because (using the author's notation) the author writes [obvious y2 corr with u1] but if u1 is noise (random) than this doesn't make sense. It can't be noise if they are correlated haha. I've edited my question to better explain my current understanding which might be very wrong. $\endgroup$ Oct 20, 2022 at 22:50
  • $\begingroup$ if I should open a new question, I'm happy to accept yours and open a new one $\endgroup$ Oct 21, 2022 at 23:26
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    $\begingroup$ @financial_physician hi I am planning to write extension it’s just lot of math and I was very busy at the university today. If you don’t want to wait you can open it as new question $\endgroup$
    – 1muflon1
    Oct 21, 2022 at 23:27

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