Harmonizing Sim Results and Theory

[ijyliu]

Basically on the theory side we have IV. But on the simulation side we have taking the mean of the standardized mismeasured covariates.

@nicomarto @marionoro which should we include- or should we include both?

In terms of what we can do for empirics, IV will be probably a bit harder- unless we take the approach of instrumenting for one mismeasured variable with another. Taking the mean of standardized variables seems not too hard.

[paul-opheim]

I think it's good that Nico included IV in the theory, but think that we shouldn't include it in the simulation or empirical sections because one of the main ideas of the paper is that the conditions for IV are not always met. Since the downsides of IV are not something that can be easily modeled in e.g the simulation, it's better that we don't include it in either of the latter sections.

[paul-opheim]

I'm not sure if we need to have theoretical arguments for the mean of the mismeasured covariates, or if we can just propose that in later sections as a simple idea that might serve a similar purpose as using PCA. Either way I'm inclined to not include the mean option in the theory section for now, so that we can see what the paper looks like without it.

[ijyliu]

I think if we do the mean we should have some simple theory behind it, should not take up a ton of space, basically just the law of large numbers I guess, over the mismeasured items centered at the true mean.

[paul-opheim]

Makes sense to me.

[ijyliu]

Seems to be resolved: IV has been de-emphasized in favor of the mean approach and theory has been added.