Exercise

# ARIMA - plug and play

As you saw in the video, a time series is called ARIMA(\(p,d,q\)) if the differenced series (of order \(d\)) is ARMA(\(p,q\)).

To get a sense of how the model works, you will analyze simulated data from the integrated model $$ Y_t = .9 Y_{t-1} + W_t\, $$ where \(Y_t = \nabla X_t = X_t - X_{t-1}\). In this case, the model is an ARIMA(1,1,0) because the differenced data are an autoregression of order one.

The simulated time series is in `x`

and it was generated in R as

`x <- arima.sim(model = list(order = c(1, 1, 0), ar = .9), n = 200)`

.

You will plot the generated data and the sample ACF and PACF of the generated data to see how integrated data behave. Then, you will difference the data to make it stationary. You will plot the differenced data and the corresponding sample ACF and PACF to see how differencing makes a difference.

As before, the astsa package is preloaded in your workspace. Data from an ARIMA(1,1,0) with AR parameter .9 is saved in object `x`

.

Instructions

**100 XP**

- Plot the generated data.
- Use
`acf2()`

from`astsa`

to plot the sample P/ACF pair for the generated data. - Plot the differenced data.
- Use another call to
`acf2()`

to view the sample P/ACF pair for the differenced data. Note how they imply an AR(1) model for the differenced data.