Exercise solutions: Section 9.11

fpp3 9.11, Ex2

A classic example of a non-stationary series are stock prices. Plot the daily closing prices for Amazon stock (contained in gafa_stock), along with the ACF and PACF. Explain how each plot shows that the series is non-stationary and should be differenced.

gafa_stock |>
  filter(Symbol == "AMZN") |>
  mutate(t = row_number()) |>
  update_tsibble(index = t) |>
  gg_tsdisplay(Close, plot_type = "partial")

• The time plot shows the series “wandering around”, which is a typical indication of non-stationarity. Differencing the series should remove this feature.

• ACF does not drop quickly to zero, moreover the value \(r_1\) is large and positive (almost 1 in this case). All these are signs of a non-stationary time series. Therefore it should be differenced to obtain a stationary series.

• PACF value \(r_1\) is almost 1. All other values \(r_i, i>1\) are small. This is a sign of a non-stationary process that should be differenced in order to obtain a stationary series.

fpp3 9.11, Ex3

For the following series, find an appropriate Box-Cox transformation and order of differencing in order to obtain stationary data.

1. Turkish GDP from global_economy.

turkey <- global_economy |> filter(Country == "Turkey")
turkey |> autoplot(GDP)

turkey |> autoplot(log(GDP))

turkey |> autoplot(log(GDP) |> difference())

turkey |> features(GDP, guerrero)

# A tibble: 1 × 2
  Country lambda_guerrero
  <fct>             <dbl>
1 Turkey            0.157

• Logs and differences make the data appear stationary.
• Using a Box-Cox transformation with \(\lambda\) between 0 and 0.2 would also have worked well.

2. Accommodation takings in the state of Tasmania from aus_accommodation.

tas <- aus_accommodation |> filter(State == "Tasmania")
tas |> autoplot(Takings)

tas |> autoplot(log(Takings))
tas |> autoplot(log(Takings) |> difference(lag = 4))
tas |> autoplot(log(Takings) |> difference(lag = 4) |> difference())
tas |> features(Takings, guerrero)

• Logs followed by seasonal and first differences make the data appear stationary.
• The automatically selected Box-Cox \(\lambda\) value is very close to zero, confirming the choice of using logs.

3. Monthly sales from souvenirs.

souvenirs |> autoplot(Sales)

souvenirs |> autoplot(log(Sales))

souvenirs |> autoplot(log(Sales) |> difference(lag=12))

souvenirs |> autoplot(log(Sales) |> difference(lag=12) |> difference())

souvenirs |> features(Sales, guerrero)

# A tibble: 1 × 1
  lambda_guerrero
            <dbl>
1         0.00212

• Logs followed by seasonal and first differences make the data appear stationary.
• The automatically selected Box-Cox \(\lambda\) value is very close to zero, confirming the choice of using logs.

fpp3 9.11, Ex4

For the souvenirs data, write down the differences you chose above using backshift operator notation.

Let \(y_t =\) log(Sales). Then \((1-B^{12})(1-B)y_t\) gives the differences used above.

fpp3 9.11, Ex5

For your retail data (from Exercise 8 in Section 2.10), find the appropriate order of differencing (after transformation if necessary) to obtain stationary data.

set.seed(12345678)
myseries <- aus_retail |>
  filter(
    `Series ID` == sample(aus_retail$`Series ID`, 1),
    Month < yearmonth("2018 Jan")
  )
myseries |> autoplot(Turnover)

Data requires a transformation as the variation is proportional to the level of the series. A log transformation will be okay, although it does seem to be too strong.

myseries |> autoplot(log(Turnover))

Data contains seasonality, so a seasonal difference is required.

myseries |> autoplot(log(Turnover) |> difference(lag = 12))

Data still appears to have extended periods of high values and low values. A first order differencing may be useful.

myseries |> autoplot(log(Turnover) |> difference(lag = 12) |> difference())

The dataset is now clearly stationary. Either just a seasonal difference, or both a seasonal and first difference, could be used here.