I have a time series data y (response variable) which I want to predict using 3 independent time series data x1, x2 and x3. My goal is to build a multivariate regression model for which I want to identify useful lags of x1, x2 and x3 (also possibly lags of y) that might be useful predictors for y. Snapshot of y, x1, x2 and x3 respectively are shown below My questions are-
If I determine cross correlation between y and x1 (without taking into account x2 and x3), y and x2 (without taking into account x1 and x3), y and x3 (without taking into account x1 and x2) separately, would that give me useful insight about how many lags of x1, x2 and x3 I should use in my regression model? What would be a better approach? Is there any way to take into account other independent variables while performing cross correation between two time series?
Since there is more than one independent variable, would prewhitening the independent variables before determining cross-correlation give me correct result? Or do I have to use spectral analysis?
independent variable x1 and x2 are not stationary. Because of that I have to make the data stationary before determining the ARMA model. Since for determining CCF, the same filter has be applied on y, should I apply the same steps that I took to make x1 stationary on y too? Or do I have to make y stationary by other separate steps? Or do I just apply the ARMA model filter?
Is it advisable to use Granger causality here do determine if y has causal relationship with x1, x2 and x3?
Another question is could you give me insight from looking at the data if y, x1, x2 and x3 have trend? Is there any test to find out trend? if my time series have trend, do I have to detrend them before determining the regression model?