Help on method selection

#1
Hi all,

I wonder whether some of you have some valuable input on which method I should use in my research.

The main idea is that I decompose a explanatory variable from a standard OLS regression (x_t) into two components, predictable and unpredictable. I would like to test whether the effect of the unpredictable component is different from the predictable component on the variable y_t.

The standard equation would be y_t = b1 * x_t + e_t.

I decompose y_t into a predictable and unpredictable component with an AR(1) model.

x_t = c + b1 * x_t-1 + e_t

where predictable component equals `x_t^e = c + b1 * ´x_t-1, and the unpredictable component would be `x_t^u = `e_t.

Subsequently I substitute x_t in the previous regression with `x_t^e and `x_t^u, which leads to

y_t = b1 * (`x_t^e + `x_t^u) + e_t => y_t = b1 * `x_t^e + b2 * `x_t^u + e_t

I guess that because `x_t^e and `x_t^u are estimated, this leads to measurement errors in the explanatory variables?

What would be an appropriate way to asses this problem? What model and method would you guys suggest?
 
Last edited: