I'm trying to simulate an ARMA(1,1) process whose autoregressive and moving average parameters are, respectively, 0.74 and 0.47. Moreover, I want the simulated data to have mean equal 900 and standard deviation equal to 230. To accomplish this, I tried


fit = arima.sim(list(order = c(1,0,1), ar = 0.74, ma = 0.47), n = 10000, 
      rand.gen= rnorm, sd = 230) + 900

The mean of the synthetic time series is acceptable.


However, when I calculate the standard deviation, the difference between the calculated value and the one I stipulated as the standard deviation for fit is too large.

#511.3077 - almost two times higher than the value I thought I'd observe

How can I change my code to make sure the simulated series will have a standard deviation close to the one I stipulate inside the arima.sim function?


The sd(fit) is $\sqrt{Var(y_t)}$ where $y_t$ is ARIMA(1,1), however the sd you specify in the arima.sim call is the sd of the white noise in the series.

Consider the AR(1)-proces $$y_t = b y_{t-1} + u_t$$ $$u_t = \sigma \epsilon_t$$ $$\epsilon_t \sim \mathcal N(0,1)$$ here the $sd(y_t) = \sqrt{Var(y_t)}$ which can be found to be $$Var(y_t) = b^2Var( y_{t-1}) + \sigma^2Var( \epsilon_t)$$ such that

$$Var(y_t) = \frac{\sigma^2}{1-b^2}$$

and $\sigma$ is standard deviation of $u_t$.

Specifying a model in R

b <- 0.5
s <- 0.9
fit = arima.sim(list(order = c(1,0,0), ar = b), n = 100000, 
  rand.gen= rnorm, sd = s) 


returns the output

> sd(fit)
[1] 1.041033
> sqrt(s^2/(1-b^2))
[1] 1.03923

so the sd in arima.sim is $\sigma$.


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