Question: Augmenting features with the average. You are fitting a regression model \(\hat{y}=x^{T} \beta+v\) to data, computing the model coefficients \(\beta\) and \(v\) using least squares. A friend suggests adding a new feature, which is the average of the original features. (That is, he suggests using the new feature vector \(\tilde{x}=(x, \operatorname{avg}(x))\).) He explains that by adding this new feature, you might end up with a better model. (Of course, you would test the new model using validation.) Is this a good idea?

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