23
Oct 17

## Canonical form for time series

We start with a doubly-infinite time series $\{ Y_t:t=0,\pm 1,\pm 2,...\} .$ At each point in time $t$, in addition to $Y_t$, we are given an information set $I_t.$ It is natural to assume that with time we know more and more: $I_t\subset I_{t+1}$ for all $t$. We want to apply the idea used in two simpler situations before:

1) Mean-plus-deviation-from-mean representation: $Y=\mu+\varepsilon$, where $\mu=EY$, $\varepsilon=Y-EY$, $E\varepsilon=0.$

2) Conditional-mean-plus-remainder representation: having some information set $I$, we can write $Y=E_IY+\varepsilon$, where $E_IY=E(Y|I)$, $\varepsilon=Y-E_IY$, $E_I\varepsilon=0.$

Notation: for any random variable $X$, the conditional mean $E(X|I_t)$ will be denoted $E_{t}X$.

Following the above idea, we can write $Y_{t+1}=Y_{t+1}-E_tY_{t+1}+E_tY_{t+1}$. Hence, denoting

$\mu_{t+1}=E_tY_{t+1}$, $\varepsilon_{t+1}=Y_{t+1}-E_tY_{t+1}$, we get the canonical form

(1) $Y_{t+1}=\mu_{t+1}+\varepsilon_{t+1}.$

## Properties

a) Conditional mean of the remainder: $E_t\varepsilon_{t+1}=E_t(Y_{t+1}-E_tY_{t+1})=0$, because $E_tE_t=E_t$. This implies for the unconditional mean $E\varepsilon _{t+1}=0$ by the LIE.

b) Conditional variances of $Y_{t+1}$ and $\varepsilon _{t+1}$ are the same:

$V_t(\varepsilon_{t+1})=E_t(\varepsilon_{t+1}-E_t\varepsilon_{t+1})^2=E_t\varepsilon_{t+1}^2=E_t(Y_{t+1}-E_tY_{t+1})^2=V_t(Y_{t+1}).$

c) The two terms in (1) are conditionally uncorrelated:

$Cov_t(\mu_{t+1},\varepsilon_{t+1})=E_t[(\mu_{t+1}-E_t\mu_{t+1})\varepsilon_{t+1}]=E_t[(\mu_{t+1}-\mu_{t+1})\varepsilon_{t+1}]=0$

($\mu_{t+1}$ is known at time $t$).

They are also unconditionally uncorrelated: by the LIE

$Cov(\mu_{t+1},\varepsilon_{t+1})=E[(\mu_{t+1}-E\mu_{t+1})\varepsilon_{t+1}]=EE_t[(\mu_{t+1}-E\mu_{t+1})\varepsilon_{t+1}]=E[(\mu_{t+1}-E\mu_{t+1})E_t\varepsilon_{t+1}]=0.$

d) Full (long-term) variance of $Y_{t+1}$, in addition to $V(\varepsilon_{t+1})$, includes variance of the conditional mean $\mu _{t}$:

$V(Y_{t+1})=V(\mu_{t+1}+\varepsilon_{t+1})=V(\mu_{t+1})+2Cov(\mu_{t+1},\varepsilon_{t+1})+V(\varepsilon_{t+1})=V(\mu_{t+1})+V(\varepsilon_{t+1}).$

e) The remainders are uncorrelated. When considering $Cov(\varepsilon_t,\varepsilon_s)$ for $s\neq t$, by symmetry of covariance we can assume that $t\leq s-1$. Then, remembering that $\varepsilon_t$ is known at time $s-1$, by the LIE we have

$Cov(\varepsilon_t,\varepsilon_s)=E[E_{s-1}(\varepsilon_t\varepsilon_s)]=E[\varepsilon_tE_{s-1}\varepsilon_s] =0.$

Question: do the remainders represent white noise?