How to simulate a MA(1) model in r

Question

I'm trying to simulate an MA(1) model in R from scratch in order to obtain more insight into the theoretical underpinnings of the model.

The moving average is defined as follows:

$X_t = \mu + \varepsilon_t + \theta_1 \varepsilon_{t-1} + \cdots + \theta_q \varepsilon_{t-q}$

Where $\epsilon_{t}$ is defined as white noise terms.

This MA(1) model has expected value and autocovariances as written down below:

$E(X_t) = 0$

$Cov(Y_{t} , Y_{t - 1}) = \gamma_{1} = -\theta \sigma_{e}^{2}$

$Cov(Y_{t} , Y_{t - k}) = \gamma_{k} = 0$

I will post my R code below.

ma.sim = function( ma_parameter = c(0.4 , 0.5) , number){


ma_mod_vals = rep(0 , times = number)


# Obtain the sequence of coefficients.

for(i in 1:number){

# create two white noise terms:


white_noise = arima.sim(model = list() , n = 2)


ma_mod_vals[i] = white_noise[1] + ma_parameter[1] * white_noise[2]


}

lister = list(output = ma_mod_vals)


}

I used the code below to generate the 2 white noise terms present in the MA(1) model.

white_noise = arima.sim(model = list() , n = 2)

What I don't understand is why I don't obtain a similar acf plot to the arima.sim function which is used to generate arima models.

For example values simulated using my function yields the following acf plot:

It is clear according to the theory that the acf plot of a MA(1) model should be significant at lag 1. But we see in this plot that the acf value at lag 1 is not significant at all. I suspect that the white noise terms that I am generating are not correct. I think that they should be correlated in some way, but I do not know how to do this.

Answer 1

You're right that you the issue is with correlation. The problem is that your code is generating two new white noise points each time it calculates an $x_t$ value, when it should be reusing the previous white noise points.

For example, your code will generate white noise points $w_1$ and $w_0$ and calculate $x_1 = w_1 + \theta w_0$. Then it generates two new points $w_2$ and $w_3$ to calculate $x_2 = w_3 + \theta w_2$. Your $x_t$ won't be correlated because they're not using the same $w_t$. Instead, for $x_2$ you need $x_2 = w_2 + \theta w_1$

To amend the code I've done:

ma.sim = function( ma_parameter = c(0.4 , 0.5) , number){
  
  
  ma_mod_vals = rep(0 , times = number)
  
  
  # Generate the white noise sequence.
  
  white_noise = arima.sim(model = list() , n = number)
  
  for(i in 2:number){
    
    # Calculate x values
    
    ma_mod_vals[i] = white_noise[i] + ma_parameter[1] * white_noise[i-1]
    
    
  }
  
  lister = list(output = ma_mod_vals)
  
  return(lister)
  
}

You'll notice that the for loop starts at 2 instead of 1. The code will through an error otherwise. Running the above should get your desired ACF.