Help on method selection

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?
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