AR models — the math under mean reversion

AR(1): a series with memory

The autoregressive model of order one ("autoregressive" — the series is regressed on itself):

$x_t = c + \phi\, x_{t-1} + \varepsilon_t$

In words: tomorrow's value is a constant $c$, plus a fraction $\phi$ of today's value, plus fresh white noise. Reading the symbols: $x_t$ ("x sub t") is the value at time $t$; $x_{t-1}$ is yesterday's value, one lag back; $\phi$ is the Greek letter phi, the fraction that carries over; $\varepsilon_t$ ("epsilon sub t") is the shock — fresh random noise arriving each period, as in D1-02. Concrete: with $c = 2$, $\phi = 0.5$, and today's value 10, tomorrow's expected value is $2 + 0.5 \times 10 = 7$.

The parameter $\phi$ is the memory dial — how much of today persists into tomorrow. The trading question it answers: if the spread I trade is stretched today, what do I expect it to do next? White noise says "nothing carries over" ($\phi = 0$); a random walk says "everything carries over" ($\phi = 1$); AR(1) covers the whole spectrum in between.

The phi dial

• $|\phi| < 1$: stationary and mean-reverting. Shocks decay geometrically — shrinking by the same multiplicative factor each step, like repeated halving — and the series is pulled back toward its long-run mean. This is the regime where trading a stretch makes sense.
• $\phi = 1$: random walk (D1-02) — the unit root of D1-03. Shocks never decay; there is no mean to revert to. The ADF test is literally a test of whether $\phi = 1$.
• $\phi > 1$: explosive — shocks compound; not a sustainable model of anything you can trade.
• $\phi < 0$: mean-reverting with overshoot — the series flip-flops around the mean. Rare in daily financial data, though bid-ask bounce (trades ping-ponging between the bid price and the ask price) induces something like it in high-frequency returns.

For the stationary case, where does the series settle? Call the settling point $\mu$ — the Greek letter mu, the long-run mean from A1-00. In a settled, noiseless world, tomorrow equals today: set $x_t = x_{t-1} = \mu$ in the equation with the noise removed. That gives $\mu = c + \phi\mu$, and solving for $\mu$:

$\mu = \frac{c}{1 - \phi}$

In words: the long-run mean is the constant divided by one minus phi. Concrete: $c = 2$, $\phi = 0.5$ gives $\mu = 2/0.5 = 4$. Deviations from that mean decay geometrically in expectation:

$E[x_{t+h}] - \mu = \phi^h\,(x_t - \mu)$

In words: the expected deviation $h$ steps from now equals phi to the power $h$, times today's deviation. Concrete: $\phi = 0.9$ and a deviation of 10 today means an expected deviation of 9 tomorrow, 8.1 the day after, and so on.

Half-life: turning phi into a holding period

How long until half of a deviation is expected to be gone? That waiting time is the half-life. Find the number of steps $h$ at which $\phi^h = \tfrac{1}{2}$ — phi multiplied by itself $h$ times equals one half. Taking logs of both sides and solving:

$h_{1/2} = \frac{\ln(0.5)}{\ln(\phi)} \qquad (0 < \phi < 1)$

In words: the half-life equals the natural log of one-half, divided by the natural log of phi ($\ln$ is the natural logarithm — the np.log function; it undoes "raise to a power," which is exactly why it appears here). Worked example: $\phi = 0.9$ gives $h_{1/2} = \ln(0.5)/\ln(0.9) = 0.6931/0.1054 \approx 6.58$ periods. A feel for the dial:

Why traders compute this before anything else: the half-life is roughly your expected holding period for a mean-reversion trade. That drives capital efficiency, cost drag, and how long you must survive being wrong. And notice how violently the half-life explodes as $\phi \to 1$ (read: "as phi approaches 1"). The difference between $\phi = 0.9$ and $\phi = 0.99$ is a week versus a quarter. Combine that with D1-03's warning that the ADF test can barely distinguish $\phi = 0.99$ from $\phi = 1$, and you get the central risk of the strategy: your "mean-reverting" spread may be a random walk wearing a good backtest.

Fitting AR(1) is a regression you already know

No new machinery needed. Estimating AR(1) is OLS of $x_t$ on $x_{t-1}$ — OLS being ordinary least squares, the plain line-fitting regression from A2-06. You regress the series on its own lag: the same series shifted back one step. In code, x[1:] is the response and x[:-1] is the predictor. The closed form is the familiar slope and intercept:

phi = np.cov(x[:-1], x[1:])[0, 1] / np.var(x[:-1], ddof=1)   # OLS slope
c   = np.mean(x[1:]) - phi * np.mean(x[:-1])                 # OLS intercept

Two honest caveats. First, in small samples the OLS estimate of $\phi$ is biased downward — it systematically comes out too low. This is the Kendall bias: on 100 points, a true $\phi = 0.98$ fits as roughly $0.94$, which flatters the half-life (makes reversion look faster than it is). Second, as always: a $\phi$ fitted on history describes history. Persistence itself drifts across market regimes.

The pairs-trading connection

Here is where the track's threads knot together. A pairs trade shorts one asset and buys a related one; the combined position's value is a spread — the gap between the two prices. If that spread is stationary — the cointegration exception flagged in D1-03 — then AR(1) with $\phi < 1$ is its natural model, and the trade writes itself. Enter when the spread is stretched multiple standard deviations from $\mu = c/(1-\phi)$. Exit near $\mu$. Expect to wait about one half-life. This is the simplest honest description of statistical arbitrage — trading measured statistical regularities rather than opinions about companies. The failure mode is equally clean: the economic link between the pair breaks, $\phi$ drifts to 1, and the spread stops reverting. That is a model-risk loss, not bad luck, and no stop-loss inside the model can see it coming.

Try it

The first test sequence is worth staring at: with no noise, $0, 2, 3, 3.5, 3.75, \ldots$ climbs toward the long-run mean $c/(1-\phi) = 2/0.5 = 4$, halving the remaining gap each step — a half-life of exactly one period, matching $\phi = 0.5$ in the table.

Map it

On paper, draw a concept map connecting: phi, random walk, unit root, ADF test, long-run mean, half-life, OLS slope, and pairs trading. Label each arrow with the relationship (which phi value is a unit root? what does ADF test about phi? which formula turns phi into a holding period? which regression estimates phi?).

Predict before the next lesson

Last stop: MA and ARMA models (D1-05). AR remembers its own past values. Predict: what would a series look like that instead remembers only its most recent past shock — and how would its ACF differ from the geometric decay of an AR?