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Does anyone of you know what happens to the b coefficients and to R-squared when a dependent variable is expressed as a linear function of itself in a multiple regression as

$$y = b_0 + b_1x_{1}^* + b_2x_2 + \cdots + b_kx_k + u, $$

when, suppose,

$$x_{1}^* = a + cx_1$$ Where a and c are given coefficients.

What is the difference between this model and the usual model

$$y = b_0 + b_1x_{1}^* + b_2x_2 + \cdots + b_kx_k + u $$

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  • $\begingroup$ This sounds like you are describing a certain special type of what is usually called an "errors-in-variables" model. Have you read anything about those? See the wikipedia article, for instance: en.wikipedia.org/wiki/Errors-in-variables_models $\endgroup$ Apr 12 '18 at 12:24
  • $\begingroup$ I think there might be a couple typos. In the first paragraph, you say "dependent variable," but the rest of the question is about an independent variable. Also, should the x1* in the "usual model" be just x1? $\endgroup$ Apr 12 '18 at 12:30
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Linear transformations of predictors (or outcomes) in a linear regression have no meaningful effect on the model or the predictive utility of the variable.

A linear transformation of x1 will rescale the b associated with that variable and only that variable, in inverse proportion to the magnitude of c. a will also affect the intercept of y in the regression equation. It will have no effect on R-squared (or on the standardized regression coefficients).

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