AR and MA together

ARIMA Models in R

David Stoffer

Professor of Statistics at the University of Pittsburgh

AR and MA Together: ARMA

ch2_2.003.png

AR and MA Together: ARMA

ch2_2.005.png

AR and MA Together: ARMA

ch2_2.006.png

x <- arima.sim(list(order = c(1, 0, 1), 
                   ar = .9, 
                   ma = -.4), 
                   n = 200)
plot(x,  main = "ARMA(1, 1)")

ACF and PACF of ARMA Models

	AR(p)	MA(q)	ARMA(p, q)
ACF	Tails off	Cuts off lag q	Tails off
PACF	Cuts off lag p	Tails off	Tails off

ACF and PACF of ARMA Models

	AR(p)	MA(q)	ARMA(p, q)
ACF	Tails off	Cuts off lag q	Tails off
PACF	Cuts off lag p	Tails off	Tails off

ch2_2.009.png

Estimation

$X_t = .9 X_{t-1} + W_t - .4 W_{t-1}$

x <- arima.sim(list(order = c(1, 0, 1), 
                    ar = .9, 
                    ma = -.4), 
                    n = 200)
x_fit <- sarima(x, p = 1, d = 0, q = 1)
x_fit$ttable

      Estimate     SE  t.value p.value
ar1     0.9083 0.0424  21.4036       0
ma1    -0.4458 0.0879  -5.0716       0
xmean  49.5647 0.4079 121.5026       0

Let's practice!

ARIMA Models in R