# Why is the difference between 2 time series drawn from the same process not White Noise?

I take the difference between 2 time series (each with 200,000 observations) drawn from the same ARMA(2,1) process and find that (at least the first 1000 observations of) this difference looks like White Noise (visually; for what it is worth; see top plot) but the ACF plot and Breusch-Godfrey test suggest it is not White Noise. Shouldn't I expect the difference to be White Noise? If so, where have I made the mistake(s)?

library(forecast)
library(lmtest)
par(mfrow=c(2,1))
set.seed(1000)
simA1<-arima.sim(list(ar=c(0.88,-0.4),ma=c(-0.22)),sd=0.1,n=200000)
set.seed(2000)
simA2<-arima.sim(list(ar=c(0.88,-0.4),ma=c(-0.22)),sd=0.1,n=200000)
dif<-simA1-simA2
plot(dif[1:1000],type="l")
Acf(dif,lag.max=100,ylim=c(-0.02,0.02))
lmtest::bgtest(dif~1,order=100)

• An ARMA process is essentially the result of applying an infinite-duration impulse response filter to white noise. Why do you expect the difference of two such time series to have the same properties as white noise? At best, the difference process is essentially the result of applying an infinite-duration impulse response filter to white noise with twice the variance, and so the difference you are looking at is still an ARMA process, and not white noise. Things are seldom what they seem; ARMA masquerades as cream-(of-white noise)..... Feb 1 at 14:56
• @DilipSarwate: Thank you for that point. Feb 1 at 15:07

One formulation of this model in terms of the back-shift operator $$L$$ is that

$$\phi(L) X = \theta(L)\varepsilon$$

where $$X$$ is the time series process, $$\varepsilon$$ is the "white noise" process of innovations, and $$\phi$$ and $$\theta$$ are polynomials.

When $$X$$ and $$Y$$ are two such processes, let $$\delta$$ be the innovations for $$Y.$$ Then because these polynomial functions of $$L$$ are linear operators,

$$\phi(L)(X-Y) = \phi(L)X - \phi(L)Y = \theta(L)\varepsilon - \theta(L)\delta = \theta(L)(\varepsilon-\delta),$$

thereby exhibiting the difference $$\varepsilon - \delta$$ as the innovations for $$X-Y$$ and explicitly showing $$X-Y$$ is an AR process with the same parameters as both $$X$$ and $$Y.$$

Notice that since $$\operatorname{Var}(\varepsilon_t - \delta_t) = \operatorname{Var}(\varepsilon_t) + \operatorname{Var}(\delta_t)$$ at any time $$t,$$ the scale of $$X-Y$$ is greater than the scales of either $$X$$ or $$Y.$$ Of course this distinction will not be revealed in the autocorrelation function (ACF) plot, but it will show up in plots of autocovariance.

• Brilliant! Thank you. Feb 1 at 15:06