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This is called an opportunity for restructuring your regression equation. For example, why not do this: $$ GDP/Pop = \beta_0 + \beta_2 X_2 + \beta_3 X_3 $$ Then you only have to multiply to the population to get GDP estimates. OR $$ log(GDP / Pop) = \beta_0 + \beta_2 X_2 + \beta_3 X_3 $$ which is effectively: $$ log(GDP) = log(Pop) + \beta_0 + \beta_2 X_2 + ...


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Coefficient change Let some there be some data distributed according to a quadratic curve: $$y \sim \mathcal{N}(\mu = a+bx+cx^2, \sigma^2 = 10^{-3})$$ For instance with $x \sim \mathcal{U}(0,1)$ and $a=0.2$, $b=0$ and $c=1$. Then a linear curve and a polynomial curve will have very different coefficients for the linear term. set.seed(1) x <- runif(100, 0,...


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The ordinary least squares solution is simply given by: $$\beta = (X'X)^{-1}X'y$$ Let's imagine we augment $X_{n\times p}$ with one or more variables $\tilde X_{n\times \tilde p}$, appending its corresponding values as columns, and call the resulting matrix ${X^*}_{n\times p^*}$, $p^* = p + \tilde p$. Now, given enough degrees of freedom, coefficients will ...


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Since the Extraversion score is just the average of the Assertiveness and Enthusiasm scores, each of these variables is a linear function of the other two. Thus, once you already have two of the variables in the model, adding the third gives you non-identifiable effect terms. I recommend you include only the latter variables in your regression model and ...


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