Canonical form for time series

Oct 17

Canonical form for time series

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