DISPERSION OF THE ERRORS IS NON-CONSTANT AND NOT RELATED TO THE MEAN (TIME SERIES DATA)

• Consider a regression : Y(t) = v0*X(t) + e(t) where e has variance W(t)**2 * VAR and a mean of zero and where W is a vector of weights t in length.
  
  
  This is referred to in the statistical literature as weighted regression . It is standard practice for the weights to be known before-hand rather than boot-strapped from estimated errors (residuals).
  
  The residuals from a model may not have constant variance and consequently estimated parameters may be deficient. One form of non-constant variance is treated by the Box-Cox or lambda transformations. However the best set of weights may not be a function of anything and just simply a vector of values. There are cases in which the user has an "a priori" knowledge of the weights. For example consider, the actuarial economist who has data where each reading is based upon "n" samples. Thus a reading with a large "n" is more credible than one with small "n".
  
  A far more common occurrence is when there is no prior knowledge existing and an empirical approach is required. Consider the case where an upward trending series has a residual variance of say 10 for the first half and a variance of the residuals of 20 for the second half. It would be totally incorrect to either ignore the change in variance or to use the power transform procedures of Box-Cox. The suggested procedure is to simply identify a model and compute a vector of residuals.
  
  By breaking the residuals into consecutive but non-overlapping sections one can perform the standard F test for variance change. The time period with the greatest F value is then a potential point of variance change. This is referred to as a structural change in the variance . The vector of weights would then be the inverse of the estimated local variances.