Introduction to Non-stationary Models and Differencing

Chapter 7: Lesson 1

Learning Outcomes

Explain the concept of non-stationarity in time series

Define non-stationarity and its implications for time series analysis
Identify non-stationary behavior in time series plots

Apply differencing to remove non-stationarity

Explain the concept of differencing and its role in removing non-stationarity
Use differencing to transform non-stationary time series into stationary ones
Interpret the results of differencing on time series plots and ACF/PACF

Identify integrated models and ARIMA notation

Define integrated models and their relationship to differencing
Understand the ARIMA notation and its components (p, d, q)
Recognize the role of the ‘d’ parameter in ARIMA models

Preparation

Read Sections 7.1-7.2
Read Prof. Frenzel’s Blog Post

Learning Journal Exchange (10 min)

Review another student’s journal
What would you add to your learning journal after reading another student’s?
What would you recommend the other student add to their learning journal?
Sign the Learning Journal review sheet for your peer

Class Activity: Non-seasonal ARIMA Models (15 min)

Effect of Differencing

In Chapter 4 Lesson 2, we found that if we compute the first difference of the price of McDonald’s stock from July 2020 through December 2023, the differences can be modeled as white noise. Sometimes differencing can remove trends.

Consider the case of a random walk and a linear trend with white noise errors

Random Walk

Check Your Understanding

Consider the random walk

\[ x_t = x_{t-1} + w_t \]

where $\{w_t\}$ is a white noise process.

What is the model for the first differences of this time series?

Linear Trend with White Noise Errors

Check Your Understanding

Consider a time series with a linear trend and white noise errors.

\[ x_t = a + bt + w_t \]

where $\{w_t\}$ is a white noise process.

What is the model for the first differences of this time series?
What is the model obtained by subtracting $a+bt$ from this series?
What are some potential concerns of using differencing to eliminate a deterministic trend?

Fitting an ARIMA Model when the Difference Model has a Non-Zero Mean

(See the last sentence in the first paragraph on page 138.)

Differencing a Time Series or the Logarithm of a Time Series

If the difference of a time series demonstrates an increasing trend, taking the logarithm before differencing can eliminate the increasing variation in the differences. As an example, consider the Australian electricity production series given in the book.

Show the code

pacman::p_load("tsibble", "fable", "feasts",
    "tsibbledata", "fable.prophet",
    "tidyverse", "patchwork")
cbe <- read_table("https://byuistats.github.io/timeseries/data/cbe.dat") |>
  select(elec) |>
  mutate(
    date = seq(
      ymd("1958-01-01"),
      by = "1 months",
      length.out = n()),
      year_month = tsibble::yearmonth(date)) |>
  as_tsibble(index = year_month)

cbe |>
  mutate(
    `Diff series` = elec - lag(elec),
    `Diff log-series` = log(elec) - lag(log(elec))) |>
  pivot_longer(
    cols = all_of(c("elec", "Diff series", "Diff log-series"))) |>
  mutate(name = factor(name, levels =c("elec","Diff series", "Diff log-series"))) |>
  ggplot(aes(x = date, y = value)) +
  geom_line() +
  facet_wrap(~name, ncol = 1, scales = "free", strip.position = "left") +
  labs(x = "Time", y = "") +
  scale_x_date(breaks = "5 years", date_labels = "%Y") +
  theme_bw()

Figure 1: Plot of Australian electricity production, first differences, and first differences of the logarithm of the series

Integrated Time Series of Order d

Definition of an Integrated Series of Order $d$, $I(d)$

We say that a time series is integrated of order d if the $d^{th}$ difference of $\{x_t\}$ is a white noise process $\{w_t\}$. Expressed differently, we write this as ${\nabla^d x_t = w_t}$. We denote an integrated time series of order $d$ as $I(d)$.

Recall that $\nabla^d \equiv \left( 1 - \mathbf{B} \right)^d$. So, either of the following can be used to indicate an integrated time series of order $d$:

\[\begin{align*} \nabla^d x_t &= w_t \\ ~\\ \left( 1 - \mathbf{B} \right)^d x_t &= w_t \end{align*}\]

Special Case: $I(1)$

Check Your Understanding

What model is given by the special case $I(1)$?

Second-Order Differencing and Lagged Differences

A linear trend can be removed by first-order differencing. A curved trend can sometimes be eliminated by second order differencing.

In some cases, a lagged difference is more appropriate. For example, if you have monthly data and need to remove additive seasonal effects, you may want to take a difference with a lag of 12. This subtracts sequential January observations from each other. This models the year-over-year growth.

Notice that taking a lag 12 difference

\[ \left( 1 - \mathbf{B}^{12} \right) x_t = x_t - x_{t-12} \]

is very different from taking the twelfth differences

\[\begin{align*} \nabla^{12} x_t &= \left( 1 - \mathbf{B} \right)^{12} x_t \\ &= x_t - 12 x_{t-1} + 66 x_{t-2} - 220 x_{t-3} + 495 x_{t-4} - 792 x_{t-5} + 924 x_{t-6} \\ & ~~~~~~~~~~~~~~~~~~~ - 792 x_{t-7} + 495 x_{t-8} - 220 x_{t-9} + 66 x_{t-10} - 12 x_{t-11} + x_{t-12} \end{align*}\]

ARIMA Process

ARIMA

Definition of an ARIMA Process

A time series is said to follow an $ARIMA(p,d,q)$ process if the $d^{th}$ differences of the time series follow an $ARMA(p,q)$ process.

Suppose we let $y_t = \left( 1 - \mathbf{B} \right)^d x_t$. The series $\{y_t\}$ follows an $ARMA(p,q)$ process if $\theta_p \left(\mathbf{B} \right) y_t = \phi_q \left(\mathbf{B} \right)w_t$.

Substituting, we find that $\{x_t\}$ follows an $ARIMA(p,d,q)$ process if

\[ \theta_p \left(\mathbf{B} \right) \left( 1 - \mathbf{B} \right)^d x_t = \phi_q \left(\mathbf{B} \right) w_t \]

where $\theta_p \left(\mathbf{B} \right)$ and $\phi_q \left(\mathbf{B} \right)$ are polynomials of orders $p$ and $q$, respectively.

Special Case: $IMA(d,q)$ Process

Definition of an IMA Process

A time series $\{x_t\}$ follows an $IMA(d,q)$ process if it can be expressed as:

\[ \left( 1 - \mathbf{B} \right)^d x_t = \phi_q \left(\mathbf{B} \right) w_t \]

Note that $IMA(d,q) \equiv ARIMA(0,d,q)$.

Check Your Understanding

Solve for $x_t$ in an $IMA(1,1)$ process.

Special Case: $ARI(p,d)$ Process

Definition of an ARI Process

A time series $\{x_t\}$ follows an $ARI(p,d)$ process if it can be expressed as:

\[ \theta_p \left(\mathbf{B} \right) \left( 1 - \mathbf{B} \right)^d x_t = w_t \]

Note that $ARI(p,d) \equiv ARIMA(p,d,0)$.

Check Your Understanding

Solve for $x_t$ in an $ARI(1,1)$ process.

Simulating an ARIMA Process

We can simulate data from the ARIMA process

\[ x_t = 0.5 x_{t-1} + x_{t-1} - 0.5 x_{t-2} + w_t + 0.3 w_{t-1} \]

using the following R code.

set.seed(1)
n <- 10000
x <- rnorm(n)
w <- rnorm(n)
for (i in 3:n) {
  x[i] <- 0.5 * x[i - 1] + x[i - 1] - 0.5 * x[i - 2] + w[i] + 0.3 * w[i - 1]
}
arima(x, order = c(1, 1, 1))


Call:
arima(x = x, order = c(1, 1, 1))

Coefficients:
         ar1     ma1
      0.4811  0.3170
s.e.  0.0125  0.0134

sigma^2 estimated as 0.9814:  log likelihood = -14094.25,  aic = 28194.49

This is an ARIMA(1,1,1) process with parameters $\alpha = 0.5$ and $\beta = 0.3$.

Check Your Understanding

Modify the code above to simulate from an $ARIMA(2,1,2)$ process with parameters $\alpha_1 = 0.5$, $\alpha_2 = 0.2$, $\beta_1 = 0.4$, and $\beta_2 = 0.1$.

Class Activity: Fitting an ARIMA Process - Exchange Rates (10 min)

The data file exchange_rates.parquet gives the exchange rates for foreign currencies. The daily-observed values in the time series are the amount in the foreign currency equivalent to one U. S. dollar. We will consider the exchange rates to convert one dollar into Euros.

Show the code

exchange_ts <- rio::import("data/exchange_rates.parquet") |>
  filter(currency == "USD.EUR") |>
  as_tsibble(index = date) |>
  na.omit()

exchange_ts |>
  display_partial_table(6,3)

Table 1: Select values of the time series representing the exchange rate to convert US$1 into Euros

date	currency	rate
2023-08-24	USD.EUR	0.922
2023-08-25	USD.EUR	0.926
2023-08-26	USD.EUR	0.926
2023-08-27	USD.EUR	0.926
2023-08-28	USD.EUR	0.925
2023-08-29	USD.EUR	0.923
⋮	⋮	⋮
2024-06-30	USD.EUR	0.933
2024-07-01	USD.EUR	0.931
2024-07-02	USD.EUR	0.932

Show the code

exchange_ts |>
  autoplot(.vars = rate) + labs(title = exchange_ts$currency[1])

Figure 2: Time plot of the exchange rate to convert US$1 into Euros

Show the code

acf_plot <- exchange_ts |> select(rate) |> ACF() |> autoplot(var = .resid)

pacf_plot <- exchange_ts |> select(rate) |> PACF() |> autoplot(var = .resid)

acf_plot | pacf_plot

Figure 3: Correlogram and partial correlogram for the time series representing the exchange rate to convert US$1 into Euros

# Fit the ARIMA Model

exchange_model <- exchange_ts |>
  model(
    auto = ARIMA(rate ~ 1 + pdq(0:2,0:1,0:2) + PDQ(0, 0, 0)),
    
    a000 = ARIMA(rate ~ 1 + pdq(0,0,0) + PDQ(0, 0, 0)),
    a001 = ARIMA(rate ~ 1 + pdq(0,0,1) + PDQ(0, 0, 0)),
    a002 = ARIMA(rate ~ 1 + pdq(0,0,2) + PDQ(0, 0, 0)),
    a100 = ARIMA(rate ~ 1 + pdq(1,0,0) + PDQ(0, 0, 0)),
    a101 = ARIMA(rate ~ 1 + pdq(1,0,1) + PDQ(0, 0, 0)),
    a102 = ARIMA(rate ~ 1 + pdq(1,0,2) + PDQ(0, 0, 0)),
    a200 = ARIMA(rate ~ 1 + pdq(2,0,0) + PDQ(0, 0, 0)),
    a201 = ARIMA(rate ~ 1 + pdq(2,0,1) + PDQ(0, 0, 0)),
    a202 = ARIMA(rate ~ 1 + pdq(2,0,2) + PDQ(0, 0, 0)),
    
    a011 = ARIMA(rate ~ 1 + pdq(0,1,1) + PDQ(0, 0, 0)),
    a012 = ARIMA(rate ~ 1 + pdq(0,1,2) + PDQ(0, 0, 0)),
    a110 = ARIMA(rate ~ 1 + pdq(1,1,0) + PDQ(0, 0, 0)),
    a111 = ARIMA(rate ~ 1 + pdq(1,1,1) + PDQ(0, 0, 0)),
    a112 = ARIMA(rate ~ 1 + pdq(1,1,2) + PDQ(0, 0, 0)),
    a210 = ARIMA(rate ~ 1 + pdq(2,1,0) + PDQ(0, 0, 0)),
    a211 = ARIMA(rate ~ 1 + pdq(2,1,1) + PDQ(0, 0, 0)),
    a212 = ARIMA(rate ~ 1 + pdq(2,1,2) + PDQ(0, 0, 0))
    )

Here is one way to determine which model is selected by the “auto” process.

Show the code

exchange_model |>
  select(auto)

# A mable: 1 x 1
                     auto
                  <model>
1 <ARIMA(1,1,1) w/ drift>

We now examine all the fitted models to determine the value of the residual mean squared error (sigma2), log-likelihood, AIC, AICc, and BIC. For the log-likelihood, larger values are preferable. For all other measures, smaller values are preferred.

Show the code

exchange_model |>
  glance()

Table 2: Values used in the model selection process for the time series representing the exchange rate to convert US$1 into Euros

Model	sigma2	log_lik	AIC	AICc	BIC
auto	0	1485.2	-2962.5	-2962.3	-2947.5
a000	0	979.2	-1954.4	-1954.3	-1946.9
a001	0	1176.6	-2347.1	-2347.1	-2335.9
a002	0	1296.1	-2584.2	-2584.1	-2569.2
a100	0	1469.5	-2932.9	-2932.9	-2921.7
a101	0	1490.6	-2973.2	-2973.1	-2958.2
a102	0	1491.5	-2973	-2972.8	-2954.2
a200	0	1484.2	-2960.4	-2960.3	-2945.4
a201	0	1491.3	-2972.7	-2972.5	-2953.9
a202	0	1491.6	-2971.2	-2970.9	-2948.7
a011	0	1484.2	-2962.3	-2962.2	-2951.1
a012	0	1485.6	-2963.1	-2963	-2948.1
a110	0	1477.6	-2949.1	-2949	-2937.9
a111	0	1485.2	-2962.5	-2962.3	-2947.5
a112	0	1485.9	-2961.7	-2961.5	-2943
a210	0	1485.1	-2962.2	-2962	-2947.2
a211	0	1485.9	-2961.8	-2961.6	-2943.1
a212	0	1486	-2959.9	-2959.6	-2937.4

Suppose we choose to apply the “auto” model, which is $ARIMA(1,1,1)$. The model parameters are summarized here:

Show the code

exchange_model |>
  select(auto) |>
  coefficients()

.model	term	estimate	std.error	statistic	p.value
auto	ar1	-0.1898907	0.1302066	-1.4583798	0.1457386
auto	ma1	0.5648437	0.1095075	5.1580342	0.0000004
auto	constant	0.0000450	0.0001977	0.2275258	0.8201635

The following plots give the acf and pacf of the residuals from this model.

Show the code

model_resid <- exchange_model |>
  select(auto) |>
  residuals()

acf_plot <- model_resid |> ACF() |> autoplot(var = .resid)

pacf_plot <- model_resid |> PACF() |> autoplot(var = .resid)

acf_plot | pacf_plot

Figure 4: Correlogram and Partial Correlogram for the residuals from the ARIMA(1,1,1) model for the daily exchange rates to convert US$1 into Euros

Here is a histogram of the residuals from our model.

Show the code

model_resid |>
  mutate(density = dnorm(.resid, mean(model_resid$.resid), sd(model_resid$.resid))) |>
  ggplot(aes(x = .resid)) +
    geom_histogram(aes(y = after_stat(density)),
        color = "white", fill = "#56B4E9", binwidth = 0.001) +
    geom_line(aes(x = .resid, y = density)) +
    theme_bw() +
    labs(
      x = "Values",
      y = "Frequency",
      title = "Histogram of Residuals"
    ) +
    theme(
      plot.title = element_text(hjust = 0.5)
    )

Figure 5: Histogram of the residuals from the ARIMA(1,1,1) model for the daily exchange rates to convert US$1 into Euros

Check Your Understanding

Write the fitted model: \[ x_t = ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \]
Does the model provide an appropriate fit for the data?

Here is a forecast for the next 7 days based on our model.

final_model <- exchange_model |>
  select(auto)

temps_forecast <- final_model |>
  select(auto) |>
  forecast(h = "7 days")

temps_forecast |>
  autoplot(exchange_ts, level = 95) +
  geom_line(aes(y = .fitted, color = "Fitted"),
    data = augment(final_model)) +
  scale_color_discrete(name = "") +
  labs(
    x = paste0(
                "Date (",
                format(ymd(min(exchange_ts$date)), "%d %b %Y"),
                  " - ",
                format(ymd(max(exchange_ts$date)), "%d %b %Y"),
                ")"
              ),
    y = "Exchange Rate",
    title = "Exchange Rate for Converting US$1 to Euros",
    subtitle = "7-Day Forecast Based on our AR(1,1,1) Model"
  ) +
  theme_minimal() +
  theme(
    plot.title = element_text(hjust = 0.5),
    plot.subtitle = element_text(hjust = 0.5)
  )

Small-Group Activity: Fitting an ARIMA Process - Microsoft Stock Prices (20 min)

A time series given the daily closing price for Microsoft (MSFT) stock is given below. To handle the gaps in the data, we define a new variable, t, which gives the observation number.

Show the code

# Set symbol and date range
symbol <- "MSFT"                # Abercrombie & Fitch stock trading symbol
date_start <- "2020-01-01"
date_end <- "2024-03-28"

# Fetch stock prices
df_stock <- tq_get(symbol, from = date_start, to = date_end, get = "stock.prices")

# Transform data into tsibble
stock_ts <- df_stock |>
  mutate(
    dates = date, 
    value = close
  ) |>
  dplyr::select(dates, value) |>
  as_tibble() |> 
  arrange(dates) |>
  mutate(t = 1:n()) |>
  as_tsibble(index = t, key = NULL)

Check Your Understanding

Using the daily closing prices of Microsoft stock, do the following.

Make a time plot of the data.
Create a correlogram and partial correlogram of the stock prices.
Fit candidate $ARIMA(p,d,q)$ models to the data.
Choose the “best” model, and justify your selection.
Generate a correlagram and partial correlogram of the residuals from your chosen model.
Make a histogram of the residuals from your model.
Did your your model account for the the time series?
Predict the value 60 trading days in the future.

Note: The “time” index is just an integer sequence in the stock_ts tsibble. So, apply the forecast() function as forecast(h = 60), rather than forecast(h = "60 days").

Homework Preview (5 min)

Review upcoming homework assignment
Clarify questions

Download Homework

homework_7_1.qmd

Class Activity

Introduction to Non-stationary Models and Differencing

Learning Outcomes

Preparation

Learning Journal Exchange (10 min)

Class Activity: Non-seasonal ARIMA Models (15 min)

Effect of Differencing

Random Walk

Linear Trend with White Noise Errors

Fitting an ARIMA Model when the Difference Model has a Non-Zero Mean

Differencing a Time Series or the Logarithm of a Time Series

Integrated Time Series of Order d

Special Case: \(I(1)\)

Second-Order Differencing and Lagged Differences

ARIMA Process

ARIMA

Special Case: \(IMA(d,q)\) Process

Special Case: \(ARI(p,d)\) Process

Simulating an ARIMA Process

Class Activity: Fitting an ARIMA Process - Exchange Rates (10 min)

Small-Group Activity: Fitting an ARIMA Process - Microsoft Stock Prices (20 min)

Homework Preview (5 min)