Hi, I am wondering if we can "manually generate" more instrumental variables in TSLS estimation by taking high order terms?
For instance, when Z is a valid instrumental variable for X, then Z^2 must also satisfy the conditions for relevance and exogeneity. Can I make more insturments in this way?
My guess is that the answer dependents on the true relationship between X and Z, when X is linear in Z, then the additional quadratic term will not be significant in the first stage regression and hence is useless. But for cases when X has a quadratic relationship with Z, I think it is better to include the quadratic term in the first stage as well. However, I hardly see such a practice in literatures, why? Can you share some examples when quadratic or higher order terms IV are included? Thanks
Even in OLS estimates in econometrics one rarely uses quadratic terms in economic models. It means that the marginal effects are a linear function of the variable in question. If you are happy there is no reason not to use the quadratic term as an instrument.
You can motivate the use of quadratic terms, higher order terms, and interactions by stating that you aim to flexibly model the unobserved relationship between X and Z.