Stationary time series: ARMA

ARIMA Models in R

David Stoffer

Professor of Statistics at the University of Pittsburgh

Wold Decomposition

wold.png Wold proved that any stationary time series may be represented as a linear combination of white noise:

$$X_t = W_t + a_1 W_{t-1} + a_2 W_{t-2} + ...$$

For constants $ \ a_1,a_2,...$

Any ARMA model has this form, which means they are suited to modeling time series.

Note: Special case of MA(q) is already of this form, where constants are 0 after q-th term.

ARIMA Models in R

Generating ARMA using arima.sim()

  • Basic syntax:
arima.sim(model, n, ...)
  • model is a list with order of the model asc(p, d, q) and the coefficients
  • n is the length of the series
ARIMA Models in R

Generating and plotting MA(1)

ch1_3.013.png

ARIMA Models in R

Generating and plotting MA(1)

ch1_3.014.png

x <- arima.sim(list(order = c(0, 0, 1), ma = 0.9), n = 100)
plot(x)
ARIMA Models in R

Generating and plotting AR(2)

ch1_3.016.png

ARIMA Models in R

Generating and plotting AR(2)

ch1_3.017.png

x <- arima.sim(list(order = c(2, 0, 0), ar = c(0, -0.9)), n = 100)
plot(x)
ARIMA Models in R

Let's practice!

ARIMA Models in R

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