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I would say that in principle there is no problem at all. You are simply making a different set of assumptions regarding the functional form. For instance, rather than modeling as: $$y=a_0+ a_1 * x + a_2 *z+e$$ you are modeling: $$y=a_0+ a_1 * x + a_2 *z + a_2 *z * 1(x>0)+e$$ So a different model that is, perhaps more flexible. The problem I would say, is ...

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The "model building" process is a misnomer. A well conducted analysis pre-specifies the variables, and their encoding, to be included in the final model based on the scientific expertise of the discipline and based on statistical power of the sample. We can't tell from statistical output alone whether a variable is a "confounder" or a &...

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It is to be expected that independent variables in a multivariable regression model will be correlated. Only in an experimental setting would you expect them to be orthogonal. A correlation coefficient of 0.58 is not especially large and I would not be worried about this.

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If your "treatment" is sex and your "outcome" is educational success (perhaps measured by grades), then conditioning on anything on the causal path between sex and success in your analysis will create bias. As a result, the usual advice is not to use something like PSM for this, as these methods are meant for dealing with pre-treatment ...

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This is a matter of model form, not of colinearity, so the correlations between the variables will not help you interpret this phenomenon. The fact that you get significant results with one model but not with another model just means that the predictors with significant coefficients are conditionally associated with the outcome while the predictors in the ...

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Yes, in general this is a concern. Maybe the best way to see it is with an extreme example: Consider the case where both models are just linear regression, with the same features. The base model fits, say, $\hat{y}=\beta_0+\mathbf{\beta} \mathbf{x}$. Then the second model has a continuum of best fits:  \alpha \hat{y} + (1-\alpha)(\beta_0+ \mathbf{\beta} \...

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