Linear Regression : Can I use both levels and changes in the same model? I have a linear model with 1 predictor variable in the form of:
$Y = a + b_{1}*X$
Both $X$ and $Y$ are stationary variables and the fit of the model is good.
I have also created 2 other models based on the 3-month and 6-month % changes of the above variables, so I have another 2 models in the form of:
$ΔY_{3m} = a + b_{2} * ΔX_{3m}$ and
$ΔY_{6m} = a + b_{3} * ΔX_{6m}$
My question is: Does it make sense to have another model (as an improvement to my first one) in the form of:
$Y = a + b_{1}*X + b_{2} * ΔX_{3m} + b_{3} * ΔX_{6m}$
to improve forecasts? But if yes, the question becomes:


*

*What kind of model should I use (just a simple one like the above)?

*How many lags should I use?

*What are the potential econometric issues I may have with such a model?

 A: If I am reading your notation correctly it seems like you have a variable $ x $ which is measured at $t$ times.  You seem to be using $ \Delta$ to denote the difference between these times on the $x$ variable. By adding multiple predictors which are difference scores of the same variable, your model would suffer from having highly correlated predictors, this is called high multicollinearity. It is assumed that your predictors are not highly correlated, therefore you may suffer other violations regarding your residuals, like normality and constant variance assumptions. 


*

*I would recommend a model which accounts for repeated measures, I would reshape your data to "long" format, adding a categorical time variable (T1, T2, T3). Then specify a model such that the error term accounts for the repeated measurements. Something like this:


$$ Y_i = a + b_1x + b_2t + b_3(x*t) + \epsilon_i$$ 
Where $x$ is your predictor variable, and $t$ is your time variable which is a categorical index of when the measurement occurred. $\epsilon_i$ represents the error term which accounts for the fact that cases were measures multiple times, the $i$ subscript denotes this. 


*Using this model no lags are necessary as the variable is expressed as an interaction with time.

*This model will not suffer the issues of multi-colinearity, it will have more power, and a lower error rate.
I dont know what language you use, but in R I would specify this model with the lme4 library like so:
library(lme4)
library(lmerTest)

lmer(Y ~ x + t + x:t + (1|ID))

Where ID is the variable which denotes which observation is which.
