## 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**

(1)

## Properties

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?

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