# Unit root, or not ? is it a big deal ?

Consider a time series, generated using

```set.seed(1)
E=rnorm(240)
X=rep(NA,240)
rho=0.8
X[1]=0
for(t in 2:240){X[t]=rho*X[t-1]+E[t]}```

The idea is to assume that an autoregressive model can be considered, but we don’t know the value of the parameter. More precisely, we can’t choose if the parameter is either one (and the series is integrated), or some value strictly smaller than 1 (and the series is stationary). Based on past observations, the higher the autocorrelation, the lower the variance of the noise.

```rhoest=0.9; H=260
u=241:(240+H)
P=X[240]*rhoest^(1:H)
s=sqrt(1/(sum((rhoest^(2*(1:300))))))*sd(X)```

Now that we have a model, consider the following forecast, including a confidence interval,

```
plot(1:240,X,xlab="",xlim=c(0,240+H),
ylim=c(-9.25,9),ylab="",type="l")
V=cumsum(rhoest^(2*(1:H)))*s
polygon(c(u,rev(u)),c(P+1.96*sqrt(V),
rev(P-1.96*sqrt(V))),col="yellow",border=NA)
polygon(c(u,rev(u)),c(P+1.64*sqrt(V),
rev(P-1.64*sqrt(V))),col="orange",border=NA)
lines(u,P,col="red")```
Here, forecasts can be derived, with any kind of possible autoregressive coefficient, from 0.7 to 1. I.e. we can chose to model the time series either with a stationary, or an integrated series,

As we can see above, assuming either that the series is stationary (parameter lower – strictly – than 1) or integrated (parameter equal to 1), the shape of the prediction can be quite different. So yes, assuming an integrated model is a big deal, since it has a strong impact on predictions.

Cite this blog post
Arthur Charpentier (2012, September 10). Unit root, or not ? is it a big deal ? Freakonometrics. Retrieved June 25, 2024, from https://doi.org/10.58079/oume

## One thought on “Unit root, or not ? is it a big deal ?”

1. Hi,
I think the issue is more conceptual and is not perfectly related to interval forecasts. ar.coeff = .995 still means that you are not trying to forecast a random walk and that your process will not drift to (+-) infinity (a.s). So regardless of the CI you produce, it is still ok to work with it. However, with unit root there is no information to be extracted.

Thanks
Eran

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