Comments on the Theoretical Framework

[ijyliu]

I've already made some minor edits to things like wordings, section formatting, etc. But here are some additional comments.

• [x] I feel like the convention is to use x* as the true variable but x as the mismeasured value (no big deal though)
• [x] I think instead of E(x'x)=0 for example it's usually better/more intuitive to express this in covariance form (unless critical to the proof)
• [x] I'm not familiar with the notation used for OLS in (2), would be good to define what the Sumtx for example stands for. Is it the covariance? I think writing Cov(x,y) or <a href="https://www.codecogs.com/eqnedit.php?latex=\inline&space;\sigma{xy}" target="blank"><img src="https://latex.codecogs.com/gif.latex?\inline&space;\sigma{xy}" title="\sigma_{xy}" /> might be better.
• [x] IV section: we definitely need more discussion of the material in the "However, find a reliable source..." paragraph. You could mathematically show the problem with a bad exclusion restriction or irrelevance/a weak instrument.

By the way, copy-pasting the url from the bottom of this page is a great way to render LaTeX on Github: https://www.codecogs.com/latex/eqneditor.php

I feel like we are gonna go way over 8 pages but I don't have a problem with that and I'm pretty sure the Prof and Nadav won't either.

[ijyliu]

I think in light of what I found/remembered in https://github.com/ijyliu/ECMA-31330-Project/issues/13 we will possibly have to make changes to the IV section: the standard practice isn't to find an entirely new variable that has to satisfy an exclusion restriction to deal with measurement error. You just instrument with the other mismeasurements.

[ijyliu]

• [x] Is the example you give with some numeric values just valid for those values? I'm inclined to say yes. I think we should delete that section, and instead, in the Application section, make some guesses as to reasonable parameter values. Or, alternatively, we could use something from the Simulations section. Or, we could prove something about the sign of the bias.