Activities: Week 9

The effect of c and d on ARIMA forecasts

In an ARIMA model, c is the intercept and d is the degree of differencing.

For the Australian imports series from global_economy, try fitting ARIMA models with the following combinations of c and d:

c	d	Effect on mean	Effect on variance
0	0
\ne0	0
0	1
\ne0	1
0	2
\ne0	2

e.g., ARIMA(Imports ~ 1 + pdq(d = 0)) specifies c\ne 0 and d=0

and ARIMA(Imports ~ 0 + pdq(d = 2)) specifies c = 0 and d=2

In each case, forecast 50 years ahead. Here is some starting code with c\ne0 and d=0:

global_economy |>
  filter(Country == "Australia") |>
  model(ARIMA(Imports ~ 1 + pdq(d = 0))) |>
  forecast(h = 50) |>
  autoplot(global_economy)

Fill in the table above with what you see in the forecast plot. What can you conclude about the effect of c and d on the long-term forecast mean and variance?

Simulate data from an AR(1) model

Use the following R code to generate data from an AR(1) model with \phi_{1} = 0.6 and \sigma^2=1. The process starts with y_1=0.

y <- numeric(100)
e <- rnorm(100)
for(i in 2:100)
  y[i] <- 0.6*y[i-1] + e[i]
sim <- tsibble(idx = seq_len(100), y = y, index = idx)

Produce a time plot for the series. How does the plot change as you change \phi_1?

Simulate data from an MA(1) model

Write your own code to generate data from an MA(1) model with \theta_{1} = 0.6 and \sigma^2=1.

Produce a time plot for the series. How does the plot change as you change \theta_1?

Simulate data from an AR(2) model

Generate data from an AR(2) model with \phi_{1} = 1.35, \phi_{2} = -0.75 and \sigma^2=1.

What happens if you change \phi_2 to -0.45? Why?

What happens if you change \phi_1 to -0.25? Why?