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)

 A: 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.
