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I have a set of LHS variables ($y_{1i},y_{2i},...y_{ki})$ where $i=1..N$ represents observations, and $j=1...k$ represent different possible $y$ variables. These variables are highly correlated- think of them as different measures of economic activity, such as GDP, GNP, Gross National Income, Domestic Absorption. etc. While each of them is interesting in its own right, theory tells me to use some measure of economic activity, in a regression of the form: $$ y=\boldsymbol{x}\beta+\epsilon $$ where $\boldsymbol{x}$ represents the same set of explanatory variables. It is therefore not clear which of these measures to use in my regression. I could run individual regressions for each of the different $y$ and compare results, but that would essentially be a form of p-hacking, where I could choose which measure produces most significant coefficients, for instance. I could take an average of each, such that for any observation $i,$ I would construct:~ $$ \bar{y}_{i}=\frac{1}{K}\sum y_{ik} $$ The problem with this is that I would be weighing each of these measures equally, and there is no reason to think why that would be correct. Another thought I had was to use Principal components, and construct for instance, the first principal component of each of these $y$ variables. This would be a variance-maximizing weighted average of the different $y,$ which is less ad-hoc than a simple average. My question then is this: is it OK to use PCA to construct a dependent variable when several similar dependent variables are available?

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You could do that, or you could use a method that weights the composites (on both sides) based on the relationships between predictor and outcome variables.Back in the day, Jacob Cohen (1982) described what he called "set correlation," a generalization of regression that allowed the LHS to include a set of variables. However, there are a range of techniques for modeling relations between composites, where the composite weights are parameters of the same model. These include partial least squares (PLS) path anaysis and generalized structured component analysis (GSCA). These methods, unlike Cohen's innovation, are lively areas with much recent work and ready-made software options.

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