Error in the standard deviation when simulating an ARMA(1,1) using arima.sim

Question

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

set.seed(100)

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.

mean(fit) 
#922.749

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.

sd(fit)
#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?

Answer 1

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

set.seed(100)
b <- 0.5
s <- 0.9
fit = arima.sim(list(order = c(1,0,0), ar = b), n = 100000, 
  rand.gen= rnorm, sd = s) 

sd(fit)
sqrt(s^2/(1-b^2))

returns the output

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

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