Causal Autoregressive Time Series

January 21, 2014
By

(This article was first published on Freakonometrics » R-english, and kindly contributed to R-bloggers)

In the MAT8181 graduate course on Time Series, we will discuss (almost) only causal models. For instance, with http://latex.codecogs.com/gif.latex?AR(1),

http://latex.codecogs.com/gif.latex?X_t=\phi%20X_{t-1}+\varepsilon_t

with some white noise http://latex.codecogs.com/gif.latex?(\varepsilon_t), those models are obtained when http://latex.codecogs.com/gif.latex?\vert%20\phi\vert%20%3C1. In that case, we’ve seen that http://latex.codecogs.com/gif.latex?(\varepsilon_t) was actually the innovation process, and we can write

http://latex.codecogs.com/gif.latex?X_t%20=%20\sum_{h=0}^{+\infty}%20\phi^h%20\varepsilon_{t-h}

which is actually a mean-square convergent series (using simple Analysis arguments on series). From that expression, we can easily see that http://latex.codecogs.com/gif.latex?(X_t) is stationary, since http://latex.codecogs.com/gif.latex?\mathbb{E}(X_t)=0 (which does not depend on http://latex.codecogs.com/gif.latex?t) and

http://latex.codecogs.com/gif.latex?\text{cov}(X_t,X_{t-h})=\frac{\phi^h}{1-\phi^2}\sigma^2(which does not depend on http://latex.codecogs.com/gif.latex?t).

Consider now the case where http://latex.codecogs.com/gif.latex?\vert%20\phi\vert%20%3E1. Clearly, we have some problem here, since

http://latex.codecogs.com/gif.latex?X_t%20=%20\sum_{h=0}^{+\infty}%20\phi^h%20\varepsilon_{t-h}

cannot be defined (the series does not converge, in http://latex.codecogs.com/gif.latex?L^2). Nevertheless, it is still possible to write

http://latex.codecogs.com/gif.latex?X_t=\frac{1}{\phi}%20X_{t{\color{Red}%20+1}}{\color{Red}%20-\frac{1}{\phi}}\varepsilon_{t{\color{Red}%20+1}}But it is possible to iterate (as in the previous case) and write

http://latex.codecogs.com/gif.latex?X_t%20=%20\sum_{h={\color{Red}%201}}^{+\infty}%20\frac{-1}{\phi^h}%20\varepsilon_{t{\color{Red}%20+h}}

which is actually well defined. And in that case, the sequence of random variables http://latex.codecogs.com/gif.latex?(X_t) obtained from this equation is the unique stationary solution of the recursive equation http://latex.codecogs.com/gif.latex?X_t=\phi%20X_{t-1}+\varepsilon_t. This might be confusing, but the thing is this solution should not be confused with the usual non-stationary solution of http://latex.codecogs.com/gif.latex?X_t=\phi%20X_{t-1}+\varepsilon_t obtained from http://latex.codecogs.com/gif.latex?X_0. As in the code writen to generate a time series, from some starting value http://latex.codecogs.com/gif.latex?X_0 in the previous post.

Now, let us spent some time with this stationary time series, considered as unatural in Brockwell and Davis (1991). One point is that, in the previous case (where http://latex.codecogs.com/gif.latex?\vert%20\phi\vert%20%3C1) http://latex.codecogs.com/gif.latex?(\varepsilon_t) was the innovation process. So variable http://latex.codecogs.com/gif.latex?X_t was not correlated with the future of the noise, http://latex.codecogs.com/gif.latex?\sigma\{\varepsilon_{t+1},\varepsilon_{t+2},\cdots\}. Which is not the case when http://latex.codecogs.com/gif.latex?\vert%20\phi\vert%20%3E1.

All that looks nice, if you’re willing to understand thing at some theoretical level. What does all that mean from a computational perspective ? Consider some white noise (this noise actually does exist whatever you want to define, based on that time series)

> n=10000
> e=rnorm(n)
> plot(e,type="l",col="red")

If we look at the simple case, to start with,

> phi=.8
> X=rep(0,n)
> for(t in 2:n) X[t]=phi*X[t-1]+e[t]

The time series – the latest 1,000 observations – looks like

Now, if we use the cumulated sum of the noise,

> Y=rep(0,n)
> for(t in 2:n) Y[t]=sum(phi^((0:(t-1)))*e[t-(0:(t-1))])

we get

Which is exactly the same process ! This should not surprise us because that’s what the theory told us. Now, consider the problematic case, where http://latex.codecogs.com/gif.latex?\vert%20\phi\vert%20%3E1

> phi=1.1
> X=rep(0,n)
> for(t in 2:n) X[t]=phi*X[t-1]+e[t]

Clearly, that series is non-stationary (just look at the first 1,000 values)

Now, if we look at the series obtained from the cumulated sum of future values of the noise

> Y=rep(0,n)
> for(t in 1:(n-1)) Y[t]=sum((1/phi)^((1:(n-t)))*e[t+(1:(n-t))])

We get something which is, actually, stationary,

So, what is this series exactly ? If you look that the autocorrelation function,

> acf(Y)

we get the autocorrelation function of a (stationary) http://latex.codecogs.com/gif.latex?AR(1) process,

> acf(Y)[1]

Autocorrelations of series ‘Y’, by lag

    1 
0.908 

> 1/phi
[1] 0.9090909

Observe that there is a white noise – call it http://latex.codecogs.com/gif.latex?(\eta_t) – such that

http://latex.codecogs.com/gif.latex?X_t=\frac{1}{\phi}X_{t-1}+\eta_t

This is what we call the canonical form of the stationary process http://latex.codecogs.com/gif.latex?(X_t).

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