# Regression with a constructed dependent variable

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?

• Please elaborate, why? – Repmat Feb 25 '16 at 20:29
• Of course it's possible. – StatsStudent Feb 25 '16 at 22:04
• I am comparing economic impact of a city compared to a hypothesized version of itself prior to the natural event. – Dan Offenbacker Feb 26 '16 at 3:03

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 $Y = [ y_1, y_2]$.