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Stationarity: the assumption every test makes and every market breaks

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Start with the idea

Stationarity is a property of a stochastic process across time, not a statement that a chart looks flat. A weakly stationary process has stable first and second moments; one realised sample will still fluctuate.

Symbols, units & horizon

xₜ: process value at observation time t
μ: constant mean
σ²: constant unconditional variance
γ_k: covariance at lag k
∀t: for every time t
E, Var, Cov: expectation, variance, covariance
c: constant AR intercept, so μ=c/(1−φ)
φ: AR coefficient, absolute value below 1
εₜ: independent mean-zero innovation
σ_ε²: innovation variance
k: nonnegative lag in observation periods

When and why to use this

Use stationarity assumptions when fitting autoregressions and estimating dependence from repeated observations. Recheck them when a market changes its trading hours, policy regime or liquidity.

A series is (weakly) stationary if its mean, variance and autocovariances do not change over time. Almost every statistical tool — regression, correlation, t-tests, the CLT — quietly assumes this. Prices are not stationary: they wander. Returns are closer, but their variance shifts with regime.

𝔼 [x_{t}] = μ, Var (x_{t}) = σ^{2}, Cov (x_{t}, x_{t - k}) = γ_{k} \forall t

Model-implied moment algebra

Verify moments in a stable AR model

For $x_{t} = c + ϕ x_{t - 1} + ε_{t}$ , take expectations under stationarity: $μ = c + ϕ μ$ , so $μ = \frac{c}{(1 - ϕ)}$ .
With independent innovation variance σ²ε, $v = ϕ^{2} v + σ_{ε}^{2}$ , hence $v = \frac{σ_{ε}^{2}}{(1 - ϕ^{2})}$ for |φ|<1. Iterating covariance gives $γ_{k} = ϕ^{k} v$ for nonnegative k.
The moments depend on lag k, not calendar time t. These calculations verify the definition under the model assumptions; they are not a test that a market obeys it.

Work it by hand

c=1, φ=.5, innovation variance 3 gives mean 2, variance 4, lag-one covariance 2.

This is why strategies decay. You estimated $P (win | setup)$ on a sample from one regime. The regime changed — rates rose, volatility doubled, the market-maker population turned over — and the conditional probability you measured no longer describes the process generating today's prices. The math did not fail; its precondition did.

You regress the S&P 500 price level on the number of Netflix subscribers, 2012–2021. The

R^{2}

will be…

Both series trend upward over the decade, so a straight line through them fits extremely well. Neither causes the other. Regress the changes and the relationship vanishes. Trending (non-stationary) series make everything look related.

The standard check is the augmented Dickey–Fuller test: it asks whether the series has a unit root (a random-walk component). A small p-value rejects the unit-root null under the chosen lag and deterministic-term specification; it does not prove stable dynamics or a tradable edge. Estimated cointegration residuals require cointegration-specific critical values, not an ordinary ADF p-value.

Python implementation

Self-contained teaching example. Python 3.10+; dependencies and input conventions are shown in the code and notation. Run in your own Python environment.

def ar1_stationary_moments(mean_level, phi, innovation_variance, lag=0):
    """Stable AR(1): x_t-mu = phi*(x_prev-mu)+epsilon_t."""
    if abs(phi) >= 1 or innovation_variance < 0 or lag < 0:
        raise ValueError("Require |phi| < 1, variance >= 0, lag >= 0")
    variance = innovation_variance/(1-phi**2)
    return mean_level, variance, phi**lag*variance

print(ar1_stationary_moments(0, .8, 1, 1))

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