Canonical form for time series
We start with a doubly-infinite time series At each point in time , in addition to , we are given an information set It is natural to assume that with time we know more and more: for all . We want to apply the idea used in two simpler situations before:
1) Mean-plus-deviation-from-mean representation: , where , ,
2) Conditional-mean-plus-remainder representation: having some information set , we can write , where , ,
Notation: for any random variable , the conditional mean will be denoted .
Following the above idea, we can write . Hence, denoting
, , we get the canonical form
a) Conditional mean of the remainder: , because . This implies for the unconditional mean by the LIE.
b) Conditional variances of and are the same:
c) The two terms in (1) are conditionally uncorrelated:
( is known at time ).
They are also unconditionally uncorrelated: by the LIE
d) Full (long-term) variance of , in addition to , includes variance of the conditional mean :
e) The remainders are uncorrelated. When considering for , by symmetry of covariance we can assume that . Then, remembering that is known at time , by the LIE we have
Question: do the remainders represent white noise?