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Assume the following scenario:

  1. I have four features: $x_1$, $x_2$, $x_3$, and $x_4$
  2. There are non-negligible multi-collinearity among the features.
  3. I want to predict $y$ (response variable) with those 4 features.
  4. I use simple multiple linear regression model: $y = a_1x_1 + a_2x_2 + a_3x_3 + a_4x_4$

Let's say that I want to understand the impact that different components of a chair have on the chair's retail price. For example:

$y\,\,\,$ = chair's retail price

$x_1$ = type or color of cushion used

$x_2$ = overall design of a chair

$x_3$ = strength of a chair

$x_4$ = softness of a chair

$x_1$ is completely independent, but other features are all somewhat impacted by the other features due to multi-collinearity, and their relationships are complicated. I've heard that the analysis of regression coefficients are unreliable under severe multi-collinearity.

Assuming that the multiple regression model fits the chair price well, can I naively use each feature's regression coefficient to understand the impact of each feature on the response variable? If not, what technique should I use?

Ex 1: If I use a red cushion ($x_1$), I can increased the retail price by 3 dollars

Ex 2: If I use conference room style chair ($x_2$), I can increase the retail price by 12 dollars

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