Comments on the Simulation Section

[ijyliu]

• I think we should pick one or two of the tables that best make our point and move them from the appendix to the main text (I think most people find tables in the appendix for important things annoying to flip to). Also, we should add titles and labels/numbering to them (need to do this in the python code that generates the tables). Instead of numbering appendix 1, appendix 2, etc. we can then list the numbers of the individual tables and latex will update them automatically.
• Also open to debate as to if we should produce histograms of coefficients for any of the results, maybe the most important ones. We could make seaborn facetGrids
• Should probably remove the first paragraph since it's basically reexplaining @nicomarto 's bit. Up to you guys whose notation you want to use
• Need to explain why there is no improvment for covariance 0, -1, 1. I feel like 0 is a very important/possible case of orthogonal regressors. Also, what's performance like near, but not at these points? Is there a continuous improvement or are these special points discontinuities?
• I think we will definitely want to think about or look to the literature for cases when PCA might do better than just taking the average value of the covariate among mismeasurements or including all the measurements as separate covariates. I think maybe if we bump up p enough the multicollinearity might screw up at least the tactic of including all the measurments as separate covariates... I know you bumped it up to 50 but we will probably want to go higher (this seems like it will work because the standard errors are getting larger). I really don't know what will screw up the averaging. Maybe we can look at or at least discuss a scenario where the measurements are in different units or something? (good to discuss with Prof or Nadav)
• I really think we should add IV also since it's discussed in the previous section
• We should do more observations. The empirical setting has nearly 2000, 100 seems low for a good comparison. I think this is more important than doing a variety of p values.

[ijyliu]

We are rescaling already so can't beat averaging just in cases where measurements have different scale.

Maybe try non classical measurement error