Regression with a constructed dependent variable

Question

I'm attempting to construct a regression model from two different dependent variables. The issue I'm having is that I would like to condense these down to one dependent variable and use time as my dependent variable.

To be more in depth, my research is on economic impact and the factors I am looking at are GDP per Capita and unemployment rates. I would like to condense GDP per capita and unemployment rates into a single dependent variable "economic impact".

Anyone have any ideas if this is possible?

Answer 1

Assuming you two variables $y_1$ and $y_2$ span a 2D space Principal Component Analysis will return the two principal modes of variation $\phi_1$ and $\phi_2$ ( $\phi_1 \perp \phi_2$). Get the projections scores $\xi_1 = \langle Y - \mu_Y, \phi_1 \rangle$ along the first mode of variation $\phi_1$ and use $\xi_1$ as a surrogate variable. By definition $\xi_1$ will encapsulate the most variation in the sample $Y$ in terms of fraction of variance explained when using a single number. This is practically Principal Component Regression but instead of reducing the dimensions of your independent data $X$ you do that on the dependent data $Y$; here $Y = [ y_1, y_2]$.

Answer 2

It is certainly possible.

Adding the two variables together would implicitly give more weight to the variable with greater variance. Alternatively, you could standardize the variables before summing them.

A different approach would be to let your theory (and prior work in the field) guide the weights you assign to each variable. For example, if the consensus in your field is that GDP per capita is twice as important to economic impact as unemployment rates, you could give GDP per capita a weight of two, and unemployment rates a weight of one. Whatever your theory/research suggests for weights can also work.

Of course, the statistical approach recommended by @usεr11852 works as well.