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I have to simulate a sample of size $n=1000$ for the model $X_t=0.9 X_{t-1}+ Z_t$ with $(Z_t)$ iid student t-distributed with 10 degrees of freedom. I have to plot the sample ACF for the first 20 lags with the program acf in R. I'm not the best to R but I thought that I have made a plot:

set.seed(123)

# Simulate 250 observations from the described MA(1) model
ma1_sim <- arima.sim(model = list(ma = 0.9), n = 1000, mean = 0, 
    sd = 0.1)

# Generate the theoretical ACF with upto lag 20
acf_ma1_model <- 
  
# Split plotting window in three rows
par(mfrow = c(3, 1))

# First plot: The simulated observations
plot(ma1_sim, type = "l", main = "MA(1) Process: theta = 0.9", 
    xlab = "time", ylab = "y(t)")
abline(h = 0)

# Second plot: Theoretical ACF
plot(1:20, acf_ma1_model[2:21], type = "h", col = "blue",  
    ylab = "ACF", main = "theoretical ACF")

# Third plot: Sample ACF

tmp <- 

  par(mfrow = c(1, 1))

Then I get the plot:

enter image description here

But I'm not sure that this is correct and I can't see anywhere in my problem the value for sd so I have just set it to 0.1. Can anyone help me with the problem and maybe correct some of my code?

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You're describing an AR (autoregressive) model, not an MA (moving average) model.

Find below some simpler code that does what you need, I think. You didn't specify what your time series $X$ looks like so here I'm arbitrarily assuming $X\sim N(100,10)$ - but feel free to change this.

# set random seed
set.seed(123)
# 1000 observations
n <- 1000
# simulate the time series (X)
x <- rnorm(n, 100, 10)
# create X_{t-1} - note that the length of this series will be 999...
# ...because I'm removing the first NA term
x_1 <- lag(x)[2:length(lag(x))]
# simulate the data coming from the AR(1) model with errors distributed t(10)
ar1 <- 0.9*x_1 + rt(n-1, 10)
# plot the ACF for the first 20 lags
acf(ar1, lag.max = 20)

ACF

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