Relate coefficients under linear feature transformation

Q: Relate coefficients under linear feature transformation

The question evaluates understanding of linear regression and linear-algebraic feature transformations, focusing on how coefficient vectors relate under reparameterization and how interpretability of parameters changes; Category/domain: Statistics & Math, position type: Data Scientist.

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Question

Suppose you are fitting a linear regression model and you consider two different feature parameterizations.

Original features: x1, x2. Transformed features:

z1 = x1 + x2
z2 = x1 - x2

Model A (original features):

y = β0 + β1 x1 + β2 x2 + ε

Model B (transformed features):

y = γ0 + γ1 z1 + γ2 z2 + ε

Questions:

What is the relationship between (β1, β2) and (γ1, γ2) ?
Are Model A and Model B equivalent (i.e., do they have the same expressive power / can they produce identical fitted values)? Under what assumptions could they behave differently in practice?

Relate coefficients under linear feature transformation

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