Relate coefficients under linear feature transformation

Q: Relate coefficients under linear feature transformation

This question evaluates understanding of linear regression parameter relationships under linear feature transformations, model equivalence and identifiability, and detection and consequences of multicollinearity, testing competencies in linear algebra, statistical inference, and regression modeling.

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Question

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Answer both parts.

Part 1: Coefficients under feature transformation

You have original predictors $x_1, x_2$ and define transformed predictors:

$z_1 = x_1 + x_2$
$z_2 = x_1 - x_2$

Consider two linear regression specifications (with the same response $y$ ):

Model A: $y = \alpha_0 + \alpha_1 x_1 + \alpha_2 x_2 + \varepsilon$
Model B: $y = \beta_0 + \beta_1 z_1 + \beta_2 z_2 + \varepsilon$

What is the relationship between $(\alpha_1, \alpha_2)$ and $(\beta_1, \beta_2)$ ?
Are Model A and Model B equivalent (i.e., do they represent the same set of functions of $x_1, x_2$ )? State any assumptions.

Part 2: Multicollinearity in a wage/gender regression

You want to estimate/assess the relationship between gender and income using a regression model with additional covariates such as:

age
education level
years since receiving degree

Explain whether multicollinearity could be present and why.
Describe how you would diagnose it.
If it exists, explain multiple ways to address it, and discuss trade-offs (interpretability vs variance vs bias).

Relate coefficients under linear feature transformation

Quick Overview

Part 1: Coefficients under feature transformation

Part 2: Multicollinearity in a wage/gender regression

Solution

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