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.