Stationarity & Differencing

What is Stationarity?

Classical statistical time-series algorithms (ARIMA) assume statistical properties of historical data will remain constant in the future.

       Non-Stationary Series (Upward Trend + Growing Variance)
        100 ┤                                              /\  /\
         50 ┤                                /\  /\  /\   /  \/  \
          0 ┴───────────────────────────────/──\/──\/──\/────────► Time
            (Mean μ_t increases; Variance σ²_t grows! Spurious predictions)

       Stationary Series (Constant Mean, Constant Variance around Zero)
         10 ┤    /\  /\  /\  /\  /\  /\  /\  /\  /\  /\  /\  /\
          0 ┼───/──\/──\/──\/──\/──\/──\/──\/──\/──\/──\/──\/────► Time
        -10 ┤
            (Mean E[Y_t] = 0, Variance Var(Y_t) = σ² constant across time)

1. Strict Stationarity

Joint distribution of $(Y_{t_1}, \dots, Y_{t_n})$ is identical to shifted joint distribution $(Y_{t_1+k}, \dots, Y_{t_n+k})$ for all shifts $k$.

2. Weak (Covariance) Stationarity (Industry Standard)

1. Constant Mean: $\mathbb{E}[Y_t] = \mu$ for all $t$.
2. Constant Variance: $\text{Var}(Y_t) = \sigma^2 < \infty$ for all $t$.
3. Lag-dependent Autocovariance: $\text{Cov}(Y_t, Y_{t+k}) = \gamma(k)$ (Depends only on lag $k$, not timestamp $t$).

Transforming Non-Stationary Series

1. First Differencing ($\Delta Y_t$): Removes linear trend:

$$\Delta Y_t = Y_t - Y_{t-1}$$

2. Seasonal Differencing ($\Delta_s Y_t$): Removes seasonal cycles of period $s$ (e.g. 12 months):

$$\Delta_s Y_t = Y_t - Y_{t-s}$$

3. Log Transformation ($\ln Y_t$): Stabilizes exponential growth / heteroscedastic growing variance before differencing.

Testing Stationarity: ADF Test

Augmented Dickey-Fuller (ADF) Test:

• Null Hypothesis ($H_0$): Unit root exists ($\gamma = 0$ in $\Delta Y_t = \alpha + \beta t + \gamma Y_{t-1} + \dots$). Series is non-stationary.
• Alternative ($H_1$): Series is stationary.

If ADF statistic $p\text{-value} < 0.05$, reject $H_0 \implies$ Confirm series is stationary.

Say this out loud

"Stationarity requires constant mean, constant variance, and autocovariance depending only on lag distance k. Non-stationary data containing trends or growing variance causes classical forecasting algorithms like ARIMA to fail. We achieve stationarity by applying log transforms to stabilize variance, first differencing (Y_t - Y_t-1) to remove linear trends, and validating with the Augmented Dickey-Fuller (ADF) test (p < 0.05)."

Follow-ups to expect

• What is Spurious Regression? Regressing one non-stationary series on another unrelated non-stationary series (e.g. US GDP vs Global Temperature) produces falsely high R² and low p-values due to shared trends, despite zero causal relationship.
• What is Cointegration? Two non-stationary time series $X_t$ and $Y_t$ are cointegrated if a linear combination $Z_t = Y_t - \beta X_t$ is stationary. Used in algorithmic pairs trading (long/short stock pairs).