Time Series Differencing

Random Walk ARIMA(0,d=1,0)

Note the simulated time series below with differencing \(d=1\):

• model equation is \(y_t=y_{t-1}+\epsilon_{t}\)
• note the time plot displays a trend.
• note the ACF coefficients remain very high.
• the PACF(1) is the only non zero coefficient on the PACF plot indicating \(y_{t-1}\) is the only explanatory variable needed for explaining this time series.

See Random Walk model https://otexts.com/fpp2/stationarity.html#random-walk-model

require(fma)
tsdisplay(arima.sim(list(order = c(0,1,0)), n = 10000))

Comparison differencing ARIMA(0,d=1,0) with AutoRegressive model ARIMA(p=1,0,0)

Note the simulated AR(1) time series by comparison :

• model equation is \(y_t=\phi_1\ y_{t-1}+\epsilon_{t}\) with \(|\phi_1|<1\)
• the time plot does not display trend
• the ACF plot shows an exponential decrease
• the PACF(1) is the only non zero coefficient on the PACF plot indicating \(y_{t-1}\) is the only explanatory variable needed for explaining this time series.

require(fma)
tsdisplay(arima.sim(list(order = c(1,0,0), ar = 0.7), n = 10000))

Higher order differencing

Example ARIMA(0,d=2,0)

require(fma)
tsdisplay(arima.sim(list(order = c(0,2,0)), n = 10000))

Example ARIMA(0,d=3,0)

require(fma)
tsdisplay(arima.sim(list(order = c(0,3,0)), n = 10000))

Remark: White noise ARIMA(0,0,0)

See White noise https://otexts.com/fpp2/wn.html

require(fma)
tsdisplay(arima.sim(list(order = c(0,0,0)), n = 10000))