# Types of noise processes and the one assumed in arima() estimation in R

Here is a time series class defining white noise incorrectly as an independent sequence of random variables.

Aside from the widespread mix-up of White noise and iid noise, a further complicating factor in understanding the noise process is that we almost never seem to be given all the information about it. In ARMA models it is most often assumed to be:

1. Gaussian White noise (uncorrelated and Normally distributed)
2. Gaussian iid noise (independent and Normally distributed)
3. White noise (uncorrelated and distribution unspecified)
4. iid noise (independent and distribution unspecified).

Some estimation methods, like Maximum Likelihood or Kalman filter, require the specification of a distribution for the noise, while other, such as OLS for a pure AR model, do not require an assumption about the distribution of the noise process.

Let's take the arima() function in R. The default estimation method in arima() is described as follows: "The default (unless there are missing values) is to use conditional-sum-of-squares to find starting values, then maximum likelihood." Since it is Maximum Likelihood estimation, it is reasonable to assume that the noise is taken to be Gaussian, but there is however no mention of whether the noise is assumed to be Gaussian White noise or Gaussian iid noise. Can someone point me to where this distinction is made or explain why the type of noise assumed must be self-evident? While uncorrelatedness and joint normality imply independence, I don't believe we assume joint normality, just marginal normality - so there should still be a distinction between Gaussian White noise and Gaussian iid noise. See this Wikipedia article for more on this distinction.

Or the same question with a different flavor: draw 2 observations from the particular AR(2) process below:

arima.sim(n=2,list(ar=c(1.2,-0.8)),sd=0.5)


What have I just implicitly assumed about the noise process? Normally distributed, I presume. But iid noise or just white noise?

• IID is white noise, you can weaken this and say serially uncorrelated instead of independent. I think in discrete time signal processing white noise is IID, also Jan 5, 2022 at 15:29
• I don't think that in discrete time signal processing - white noise is IID, without some further qualifying statement. Let $\epsilon_t$ ~ ARCH(p) then $\epsilon_t$ is serially uncorrelated (white noise) but not serially independent (not iid white noise). Jan 5, 2022 at 15:58
• @Aksakal: finding this wikipedia article over here made my day en.wikipedia.org/wiki/… take a look; it addresses your point Jan 7, 2022 at 23:16
• The qualifying statement we were looking for was Gaussian; if you rephase your statement as "in discrete time signal processing Gaussian white noise is IID" then it's all good. Thanks for the comment. Jan 8, 2022 at 0:27