Relate coefficients under linear feature transformation

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: $y = \beta_0 + \beta_1 z_1 + \beta_2 z_2 + \varepsilon$

1. What is the relationship between $(\alpha_1, \alpha_2)$ and $(\beta_1, \beta_2)$ ?
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

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