# Error term covariance of two-stage estimators

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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