[SOLVED] CS Problem 1: 50 Marks

Problem 1: 50 Marks
Evaluate whether each of statements (a)-(d) below is True, False or Uncertain. Explain the rea- soning behind your answers. Your mark will depend on the quality and clarity of your explanation. Each is worth 12.5 marks.
(a) Consider the following model: yi = 0 + 1xi + 2ai + i. Suppose we run an OLS regression of y on x (without a ) and recover OLS
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Statement (a) : OLS provides a biased estimate of .
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(b) Suppose we have a dataset with variables yi and xi, and are interested in estimating E[yi|xi] with non-parametric methods.
Statement (b): Nearest neighbors regression is equivalent to Nadaraya-Watson with a uniform kernel.
(c) Suppose we are interested in building a model for predicting outcome yi out of sample using variables x1i, x2i, , x50i.. We decide to use a LASSO approach and select the hyper-parameter using 10-fold cross-validation. Specifically we choose the the provides the minimum cross validated error (that is, the minimum mean squared error on average across our 10 cross- validation samples).
Statement (c): The model described above will provide a lower out-of-sample mean squared error when compared to a model built using LASSO and a different value of .
(d) Consider random variables yi and xi.
Statement (d): The difference yi E[yi|xi] is uncorrelated with log(xi).
Author: CJH
Imperial College London 2021