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I am looking into properties of two-stage estimators (2SLS).

My setting is as follows:

1) $y_1 = y_2\beta+\epsilon$

2) $y_2 = z \pi_2 + \eta$.

Equation 1 represents the second-stage estimation, equation 2 the first-stage estimation.

The error terms, $\epsilon, \eta$ are assumed homoscedastic, i.e. $cov(\epsilon,\epsilon) = cov(\eta, \eta) = 1$.

Now, I am interested in the covariance between these error terms, $cov(\epsilon, \eta)$. What can be said about this covariance when the coefficients in equations 1) and 2) are being shifted?

In particular, if I reduce $\beta$ and $\pi$ by x%, is there a way to bound the change of the $cov(\epsilon, \eta)$ term? Especially if the other two covariance terms are known to be 1?

My ideas: Approximate $cov(\epsilon, \eta)$ by $cov(\epsilon, \epsilon)$ and $cov(\eta, \eta)$, use them to find a bound

Any help (including just pointing me in the right direction) highly appreciated. Thanks.

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