Stationarity in Time Series

Author

AS

1 1. What is Stationarity?

A stationary time series is one whose statistical properties do not change over time.

  • Mean is constant
  • Variance is constant
  • Autocovariance depends only on lag, not on time

https://towardsdatascience.com/stationarity-in-time-series-a-comprehensive-guide-8beabe20d68/

2 2. Types of Stationarity

Strict Stationarity
- The joint distribution of \((Y_t, Y_{t+k}, …)\) is the same no matter the time index. This is a very strong condition: it means not just the mean/variance, but all higher-order moments (skewness, kurtosis, etc.) are invariant over time.
- Strong requirement, rarely used directly in applications.

Weak (Covariance) Stationarity

Only the first two moments are time-invariant and much easier to satisfy:
- \(E[Y_t] = \mu\) (constant mean)
- \(Var(Y_t) = \sigma^2\) (constant variance)
- \(Cov(Y_t, Y_{t+k})\) depends only on \(k\) (lag)

2.0.1 Key Difference

  • Strict Stationarity → requires invariance of the entire probability distribution.

  • Covariance Stationarity → only requires invariance of the mean, variance, and autocovariance function.

2.0.2 In practice:

  • Econometricians and data scientists usually assume covariance stationarity, because it’s sufficient for most linear time series methods.

  • Strict stationarity is more theoretical and rarely testable with real data.

3 3. Why It Matters

  • Many econometric models (ARMA, ARIMA, VAR, GARCH) require stationarity.
  • Without it:
    • Risk of spurious regressions (high \(R^2\) but meaningless results).
    • Standard errors and tests become unreliable.
  • Stationarity ensures sample moments converge to population moments.

4 4. Sources of Non-Stationarity

  • Trends

    • Deterministic: linear or polynomial trend.

      Code
      # --- Packages ---------------------------------------------------------------
suppressPackageStartupMessages({
  library(tseries)       # ADF, KPSS tests
  library(forecast)      # autoplot.ts, ggtsdisplay (optional)
})

set.seed(123)

# --- Deterministic trend: linear & quadratic --------------------------------
n          <- 300
time       <- 1:n
eps        <- rnorm(n, 0, 1)

y_lin      <- 0.5 + 0.02*time + eps                      # linear trend
y_poly     <- 0.5 + 0.02*time - 0.00005*time^2 + eps     # polynomial (quadratic)

# --- Stochastic trend: random walk ------------------------------------------
u          <- rnorm(n, 0, 1)
y_rw       <- cumsum(u)                                   # random walk (unit root)

# --- Quick plots -------------------------------------------------------------
par(mfrow = c(1,3))
plot.ts(y_lin,  main = "Deterministic: Linear Trend", ylab="y")
plot.ts(y_poly, main = "Deterministic: Quadratic Trend", ylab="y")
plot.ts(y_rw,   main = "Stochastic: Random Walk", ylab="y")

    • Stochastic: random walk.

  • Changing Variance

    • E.g., volatility clustering in finance (ARCH/GARCH).

      Code
      # amazon closing price clearly looks like a stochastic trend (unit-root-ish)
amzn <-
  gafa_stock |>
  filter(Symbol == "AMZN") |>
  as_tsibble(index = Date)

#autoplot(amzn, Close) +
#  labs(title = "gafa_stock (AMZN): price with stochastic trend", y = "Close")

# difference the log price to remove the stochastic trend
amzn_ret <-
  amzn |>
  mutate(ret = difference(log(Close))) |>
  drop_na()

#autoplot(amzn_ret, ret) +
#  labs(title = "AMZN: differenced log price (≈ stationary)", y = "log return")

# rolling volatility visualization
amzn_ret |>
  mutate(roll_sd = slider::slide_dbl(ret, sd, .before = 20, .complete = TRUE)) |>
  ggplot(aes(Date)) +
  geom_line(aes(y = ret)) +
  geom_line(aes(y = roll_sd)) +
  labs(title = "AMZN: returns and rolling SD (≈ volatility)",
       y = "return / rolling sd", x = NULL)

      Code
      # squared returns often show autocorrelation when volatility clusters
amzn_ret |>
  mutate(r2 = ret^2) |>
  ACF(r2) |>
  autoplot() +
  labs(title = "AMZN: ACF of squared returns → volatility persistence")

  • Structural Breaks

    • Policy change, financial crisis, regime shifts.

      Code
      # aggregate half-hourly demand to daily; visualize possible shifts
elec_daily <-
  vic_elec |>
  index_by(Date = as_date(Time)) |>
  summarise(Demand = sum(Demand))

autoplot(elec_daily, Demand) +
  labs(title = "vic_elec: daily demand (visual check for breaks)", x = NULL, y = "MWh")

      Code
      # real GDP growth for united states; look for break around the GFC
library(strucchange)

us_growth <-
  global_economy |>
  filter(Country == "United States") |>
  mutate(g = difference(log(GDP))) |>
  drop_na()

autoplot(us_growth, g) +
  labs(title = "United States: GDP growth (log diff of GDP)", y = "growth")

      Code
      # estimate possible break(s) in the mean of growth
bp <- breakpoints(g ~ 1, data = as.data.frame(us_growth))
bp
      
     Optimal 4-segment partition: 

Call:
breakpoints.formula(formula = g ~ 1, data = as.data.frame(us_growth))

Breakpoints at observation number:
11 21 47 

Corresponding to breakdates:
0.1964286 0.375 0.8392857
      Code
      autoplot(us_growth, g) +
  geom_vline(xintercept = year(breakdates(bp, breaks = 1)), linetype = 2, color = "red") +
  labs(title = "Break in mean growth (Bai–Perron 1-break)")

5 5. How to Achieve Stationarity

  • Detrending: Remove deterministic trend.
  • Differencing: \(\Delta Y_t = Y_t - Y_{t-1}\).
  • Log Transformations: Stabilize variance.
  • Seasonal Differencing: Handle seasonal effects.

6 6. Testing for Stationarity

  • Visual Inspection: Time plot.
  • Correlogram (ACF/PACF): Slowly decaying ACF \(\Rightarrow\) likely non-stationary.
  • Formal Tests:
    • Augmented Dickey-Fuller (ADF)
    • Phillips-Perron (PP)
    • KPSS (tests stationarity as null)
Code
library(forecast) 
library(tseries)

set.seed(123) 
y1 <- arima.sim(model = list(ar = 0.6),
                n = 200) # stationary AR(1) 
y2 <- cumsum(rnorm(200)) # non-stationary random walk

#par(mfrow = c(2,1)) 
plot.ts(y1, main = "Stationary AR(1) Process")

Code
plot.ts(y2, main = "Non-Stationary Random Walk")