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I what to know if the error term referred to in moving average model of time series the same as white noise? which is usually define in r as

e <- rnorm(n=20, mean=0, sd=1)

If y[t] = e[t] + theta*e[t] where theta is a parameter provided makin the ma(1) model to be y[t] = e[t] + 0.8*e[t] if `t = 1,2,...,100. Can I figure it out in ar as:

n=100
theta <- 0.8
wn <- ts(rnorm(n, mean=0, sd=1))
ma1 <- wn[1]
for(i in 2:n){
  ma1[i] <- wn[i - 1] * theta + wn[i]
}
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2 Answers 2

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One of the nice things about MA models (and ARMA models more generally) is that we can based them on an underlying "error" sequence that is of whatever form we want it to be. Usually we want the underlying error sequence to be a white noise series, but there is no necessity in this. When you encounter an ARMA model, the underlying error sequence should be stipulated in the model, but if it is not, it would be reasonable to presume that Gaussian white noise is intended.

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  • $\begingroup$ You mean what I did is correct? $\endgroup$ Commented Apr 22, 2020 at 0:47
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    $\begingroup$ Unfortunately, no. There are other problems with what you did. When you generate an MA(1) process recursively, the first value in the generated vector needs to be set to have variance equal to the stationary variance of the series. Your present code does not do this. $\endgroup$
    – Ben
    Commented Apr 22, 2020 at 1:40
  • $\begingroup$ can you please give me what the starting point of this my MA(1) should be so I can learn from it $\endgroup$ Commented Apr 22, 2020 at 13:02
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    $\begingroup$ For an MA(1) you have $\mathbb{V}(Y_t) = (1+\theta^2) \sigma^2$ so your starting values should have this variance. Your code should have ma1 <- sqrt(1+0.8^2)*wn[1] instead of ma1 <- wn[1]. $\endgroup$
    – Ben
    Commented Apr 22, 2020 at 13:30
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    $\begingroup$ Try writing out the recursive equation for the model and then take the variance of both sides --- see what you get. $\endgroup$
    – Ben
    Commented Apr 24, 2020 at 4:51
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Yes, the error term in the formula is white noise. Couple of caveats,

  • There is a difference between a theoretical white noise process, and simulated samples from it, which you can get with the rnorm formula. The formula is more like the idealized version, a hypothesis for how the real world works, while with rnorm, you get real numbers that are an example of how it works out
  • White noise with discrete time steps is quite simple to construct and think about, but if you want to define or sample from white noise in continuous time, it is a lot more involved and this difference may be confusing when looking at books or articles on the topic

Edit

See the answer by Ben below, he is absolutely right by adding that the error terms in a MA model can actually also come from other distributions. So to be more precise, the answer is No, it doesn't have to be.

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