Time-Series Models for Financial Data › Stationarity: The Foundation of Time-Series Analysis

Before building any time-series model, we must understand stationarity, a property that determines which modeling techniques apply and whether our statistical estimates are meaningful. The concept of stationarity addresses a fundamental challenge in time-series analysis: how can we learn about a process when we observe only a single realization of it? If the statistical properties of a series change arbitrarily over time, then observations from the past tell us nothing reliable about the future. Stationarity provides the stability assumption that makes statistical inference possible.

Think of stationarity as a form of temporal homogeneity. Just as we assume in cross-sectional statistics that observations come from the same underlying distribution, stationarity assumes that the time series comes from a process whose statistical character remains consistent across time. This does not mean the series is constant or predictable. Rather, it means the rules governing its behavior do not change.

A time series $\{Y_t\}$ is strictly stationary if its joint probability distribution is invariant to time shifts. For any set of time indices $t_1, t_2, \ldots, t_k$ and any shift $h$, the joint distribution of $(Y_{t_1}, Y_{t_2}, \ldots, Y_{t_k})$ equals that of $(Y_{t_1+h}, Y_{t_2+h}, \ldots, Y_{t_k+h})$. This is a strong requirement rarely verified in practice. Strict stationarity essentially demands that if you were to examine any collection of observations from the series, you could not tell what time period they came from based solely on their statistical properties. The distribution of Monday's return and the following Wednesday's return must be identical to the distribution of any other Monday-Wednesday pair, regardless of when those observations occur.

More useful is weak stationarity (or covariance stationarity), which requires only:

1. Constant mean: $E[Y_t] = \mu$ for all $t$
2. Constant variance: $\text{Var}(Y_t) = \sigma^2$ for all $t$
3. Autocovariance depends only on lag: $\text{Cov}(Y_t, Y_{t-k}) = \gamma_k$ for all $t$

Weak stationarity relaxes the strict requirement by focusing only on the first two moments of the distribution. We do not require the entire distribution to be time-invariant, only the mean, variance, and the covariance structure. This makes weak stationarity much easier to test and work with in practice. The third condition is particularly important: it states that the covariance between observations depends only on how far apart they are in time (the lag $k$), not on their absolute position in the series. The covariance between observations 5 periods apart must be the same whether we are looking at periods 1 and 6 or periods 100 and 105.

The autocovariance function $\gamma_k$ measures how observations separated by $k$ periods covary. This function captures the memory of the process, telling us how much information about the current value is contained in values from $k$ periods ago. The autocorrelation function (ACF) normalizes this to a scale that is easier to interpret:

$$\rho_k = \frac{\gamma_k}{\gamma_0} = \frac{\text{Cov}(Y_t, Y_{t-k})}{\text{Var}(Y_t)}$$

where:

• $Y_t$: value of the series at time $t$
• $\rho_k$: autocorrelation at lag $k$
• $\gamma_k$: autocovariance at lag $k$
• $\gamma_0$: variance of the series ($\text{Var}(Y_t)$)
• $\text{Cov}(Y_t, Y_{t-k})$: covariance between observations $k$ periods apart

The normalization by $\gamma_0$ (the variance) ensures that autocorrelations are bounded between -1 and 1, just like ordinary correlation coefficients. For a stationary series, $\rho_0 = 1$ always holds because any observation is perfectly correlated with itself. The constraint $|\rho_k| \leq 1$ for all lags follows from the Cauchy-Schwarz inequality. The ACF gives a complete summary of the linear dependence structure of a stationary time series and is our primary diagnostic tool for model identification.

Asset prices are almost never stationary: their levels trend and their variance changes over time. The S&P 500 today is systematically higher than it was 50 years ago, violating the constant mean requirement. This non-stationarity reflects economic growth rather than a modeling failure. Over long horizons, stock prices tend to rise because companies generate earnings and reinvest profits, creating compound growth in value. A statistical model that assumed stock prices fluctuate around a fixed mean would fundamentally mischaracterize how equity markets work.

However, returns are often approximately stationary:

$$\begin{aligned} r_t &= \frac{P_t - P_{t-1}}{P_{t-1}} && \text{(simple return)} \\ r_t &= \log(P_t) - \log(P_{t-1}) && \text{(log return)} \end{aligned}$$

where:

• $r_t$: return at time $t$
• $P_t$: price at time $t$
• $P_{t-1}$: price at time $t-1$

The transformation from prices to returns removes the trending behavior. Log returns fluctuate around a relatively stable mean (often close to zero for daily data) with variance that, while exhibiting clustering, maintains a long-run average. Intuitively, while the price level can wander arbitrarily far from any fixed reference point, the percentage changes in price tend to stay within a bounded range. This is why most time-series modeling in finance focuses on returns rather than prices. The transformation also has an economic interpretation: investors care about percentage gains and losses rather than absolute price levels.