A LAG OF A KNOWN X VARIABLE IS NEEDED

• Consider a case where the true but unknown model is: Y(t) = v0*X(t) + v1*X(t-1) + A(t) where A is an i.i.d. ( gaussian ) error distribution and where the current tentative , albeit incorrect model is Y(t) = v0*X(t) + e(t) thus we have a case where e(t) = v1*X(t-1) + A(t) It is clear that if we correlate e(t) and X(t-1) we will identify a statistically significant relationship which will then lead to the required augmentation strategy.
  
  
  The strategy is then clear , for all possible or candidate "new variables" such as the X(t-1), X(t-2),....X(t-k)....etc. compute partial correlations which will measure the expected impact of the trial candidate. Select that one which has maximum correlation.
  
  These approaches to developing new variables from existing series have had outstanding success with time-series data.
  
  A change in the value of an X variable may have a "ripple effect" for a number of future time points.