1 Inference After Model Selection This question Mustrates conceptual material, and thus it has a lot of erposition. In dass we decoded model selection took is the context of building a high-quality prediction model, but custioned that statistical inference (testing, confidence intervals, etc) were unreliable following moded selection. Let us review why. In week 2 we showed that our uncertainty regarding by as an estimator off 1 comes from the fact that if the data were to change, so would our estimate. The standard errors we derived, and those reported by the computer, capture this uncertainty. However, when doing variable selection, an additional layer of uncertainty is introduced: the fact that is the data changes, the very model we select may change, above and beyond to the coefficient estimates within the model changing. This second layer of uncertainty is not at all reflected in the standard This question will explore inference after model selection and highlight some of the potential problems using Monte Carlo simulation, just like we did in work 2. The set up will be the simplest possible, just to make things easy and to Illustrate the issues. We have an outcome F and two predictors, X, and X2, in the standard multiple linear regression model: (1) Our goal is to do inference only, the coefficient on N, for example forming a confidence interval, after deciding if N, belongs in the model. If X, belongs in the model and we run the full model, i.c. In(Y - X_1 + I_2), we get an estimator by that is unbiased and Normally distributed with standard error Shy (the "F' is for 'Full"); that is, in math On the other hand, if No does not belong in the model, then we will get a better estimate off , by running the restricted model that only includes Ne, be. Ia(Y " I_1), and (3) where, typically a, i