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I have a question about terminology. I've constructed a multiple linear regression model for a variable X. This model is in turn used to estimate values of X that are used as input when constructing another multiple linear regression for a variable Y. Is there a term for this sort of approach?

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It sounds a bit like Two Stage Least Squares from instrumental variable econometrics. This is done when you have simultaneity, measurement error in X, or an omitted variable that is correlated with X in your Y regression. However, there are some stringent conditions that X has to to satisfy for the approach to be appropriate.

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