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Assume a triangular system such as

\begin{eqnarray*} Y = X'\beta_1 + D\gamma_1 + \varepsilon_1 \\ D = X'\beta_2 + \varepsilon_2 \end{eqnarray*}

with $Y$ and $D$ as observed endogenous variables, $X$ a vector of observed exogenous variables, and $\varepsilon_j$ as the unobserved error terms.

What is the meaning of this assumption $Cov(X, \varepsilon_1\varepsilon_2)=0$? When is it violated?

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  • $\begingroup$ What is the assumed relation between the two error terms? $\endgroup$ Dec 4, 2014 at 21:03
  • $\begingroup$ ε1 and ε2 may be correlated and there is some heteroscedasticity in the form of $cov(X, e_2^2) != 0$ but I can't get my head around when the assumption is satisfied or violated. $\endgroup$
    – greg
    Dec 4, 2014 at 22:21

1 Answer 1

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This must be a strict exogeneity from classical OLS model. There's also a weak exogeneity, which is more realistic.

It's violated when your errors are correlated with regressors. It's a very common issue in econometrics, see the discussion here.

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  • $\begingroup$ Are you sure about that? The context also states that the exogeneity assumption E(εX)=0 holds and the assumption refers to the term ε1ε2 not just εj. E(εj|X)=0 would be the standard exogeneity assumption. The text also states that the assumption is automatically satisfied if the "mean zero errors are conditionally independent" ε1 ⊥ ε2|Z = 0 $\endgroup$
    – greg
    Dec 4, 2014 at 22:15
  • $\begingroup$ Actually, this is not OLS, you're right. It's a [linear simultaneous equations model] (hec.unil.ch/docs/files/23/100/…). The condition though is similar to strict exogeneity. $\endgroup$
    – Aksakal
    Dec 4, 2014 at 22:33

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