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 = β0 + β1xi + β2ai + εi. Suppose we run an OLS regression of y on x (without a ) and recover βˆ OLS
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• Statement (a) : βˆOLS provides a biased estimate of β .
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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 λ using 10-fold cross-validation. Specifically we choose the λ 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 λ.
(d) Consider random variables yi and xi.
• Statement (d): The difference yi − E[yi|xi] is uncorrelated with log(xi).
Author: CJH
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