Part III · Models · Chapter 3B

Mean reversion and momentum

The AR(1) model

1 · The AR(1) model

The most basic useful model is an autoregressive linear regression of order one, or AR(1) model. The series is regressed on its own past (autoregressive) at a single lag (order one). It captures the two most fundamental trading behaviors, mean reversion and momentum, by predicting the next log return as a linear function of the previous one:

Here $r_{t-1}$ is the last period's log return and $\hat{r}_t$ the prediction for the current period. The weight $w$ and the bias $b$ are the model's only two parameters. The sign of $w$ determines everything. The actual return is the prediction plus a fresh shock:

The innovation $\varepsilon_t$ is a random shock with mean zero and standard deviation $\sigma$, independent of everything before it. Chapter 1G wrote the same equation for the decay of a pricing error, using $\phi$ (phi), the usual statistical letter, in place of $w$. The first figure below uses $\phi$ too.

With $w = -0.05$ and $b = 0$, a day that rose 1% predicts $-0.05 \times 1\% = -0.05\%$ for the next. Adding a hypothetical daily drift of $b = 0.03\%$ makes the prediction $-0.05\% + 0.03\% = -0.02\%$.

2 · Three regimes

A negative weight, $w < 0$, means mean reversion. It flips the sign of the lag, so a positive return predicts a negative one and a negative return predicts a positive one: "what goes up must come down". This regime is common at second-to-minute timescales in liquid markets, where dealer mean reversion (from gamma hedging) and microstructure noise (the bid-ask bounce) dominate. It is also common at multi-week timescales for individual equities and indices, where overreaction followed by correction recurs.

A positive weight, $w > 0$, means momentum. It reinforces the sign: "what goes up keeps going up". This regime is most common at intermediate timescales of weeks to months, across asset classes. It is the empirical foundation of decades of trend following research. It also underlies factor strategies like Jegadeesh-Titman 12-1 momentum, from Jegadeesh and Titman (1993, Journal of Finance), which ranks stocks by their past 12-month return excluding the latest month (Chapter 10F). Hurst, Ooi and Pedersen (2017, Journal of Portfolio Management) have documented cross-asset trend in 140+ years of data.

A weight near zero, $w \approx 0$, means a random walk. The next return is independent of the previous one, so the model has no predictive content. This is the efficient market hypothesis in its weakest form, a good approximation at most timescales for most liquid markets. Trade fees, slippage and adverse selection eat any strategy that operates in this regime.

3 · How far the paths spread

A process is stationary when its statistical behavior does not drift over time: its variance stays bounded, its distribution holds steady and forecasts far ahead converge to the long-run mean. Take the variance of both sides of (3B.2), writing $V$ for the variance of each return. Because $\varepsilon_t$ is independent of $r_{t-1}$, their variances add, so a stationary $V$ must satisfy $V = w^2 V + \sigma^2$:

With $\sigma = 1\%$ and $w = 0.5$, $V = 1/0.75 = 1.33$ in squared percent, a standard deviation of 1.15% a period. A positive answer exists only when $|w| < 1$, which is the condition for stationarity. As $w$ approaches either +1 or −1, $V$ grows without limit.

What a position lives through, though, is the accumulated return, the price path, where the two ends behave in opposite ways. Over $n$ periods the standard deviation of the summed returns is approximately

A random walk, with $w = 0$, gives the square-root-of-time rule of Chapter 1A. As $w$ approaches +1 the denominator shrinks toward zero and the paths fan out without limit. As $w$ approaches −1 the denominator approaches 2 and the paths draw in, because each return is largely undone by the next.

Where the spread of the paths comes from

Step 1: add up the process. Sum (3B.2) over $t = 1$ to $n$, with $b = 0$ and $S_n = r_1 + \cdots + r_n$:

$$S_n = w\sum_{t=1}^{n} r_{t-1} + \sum_{t=1}^{n} \varepsilon_t$$ (3B.5)

Step 2: rewrite the lagged sum. The sum of $r_0$ to $r_{n-1}$ is $S_n - r_n + r_0$. Substituting it and collecting $S_n$ on the left gives

$$(1 - w)\,S_n = \sum_{t=1}^{n} \varepsilon_t + w\,(r_0 - r_n)$$ (3B.6)

Step 3: drop the small term. For a stationary process, $r_0$ and $r_n$ are single returns of fixed size, while the sum of $n$ shocks keeps growing. Over a long horizon, $S_n \approx \sum \varepsilon_t / (1 - w)$.

Step 4: take the standard deviation. The $n$ independent shocks sum to a standard deviation of $\sigma\sqrt{n}$. Dividing by $1 - w$ gives (3B.4).

Over a year of 250 steps with $\sigma = 1\%$, (3B.4) gives $1\% \times \sqrt{250} = 15.8\%$ for a random walk. At $w = 0.9$ it gives $15.8\%/0.1 = 158\%$, and at $w = -0.9$ it gives $15.8\%/1.9 = 8.3\%$.

The figure below simulates price paths whose log returns follow the AR(1) process, all starting at 100, under four controls:

• Weight φ, from −0.9 to 0.95, starting at 0.
• Innovation vol σ, from 0.5% to 5%, starting at 1%.
• Steps per path, from 100 to 500, starting at 250.
• Paths, 10, 50 or 200, starting at 50, with a seed starting at 42 and a button that regenerates them.

The readouts estimate the lag-1 autocorrelation and the Hurst exponent, both defined below, from the first path, and a verdict names the regime. The verdict reads random walk for φ from −0.3 to 0.3, momentum above 0.3, near-non-stationary from 0.85, and mean reversion below −0.3. Drag φ to 0.9, then to −0.9, to compare how far the paths spread.

weight φ

0.00

innovation vol σ

1.0%

steps per path

250

paths 10 50 200

seed regenerate

empirical ρ (lag 1) = −0.079

Hurst H (path 0) = 0.45

|φ| = 0.00

Random walk regime. Near-zero autocorrelation. The next move is statistically independent of the last. No exploitable signal at this horizon. Frictions (spread, slippage) will eat any attempted strategy operating here.

At the starting settings the paths span roughly 73 to 157. At φ = 0.9 they span about 6 to 7,000, while at φ = −0.9 they stay between about 84 and 129. Measured across the 50 paths, the standard deviations of the final log prices are 0.158, 1.57 and 0.085, as (3B.4) predicts.

Two readouts deserve a second look. At φ = 0 the estimated autocorrelation is −0.079 rather than zero, which is sampling noise, as the next section explains. At φ = −0.9 the Hurst estimate is −0.10, outside the range of zero to one that a true Hurst exponent occupies. The readout estimates it from the variance ratio at a lag of only 8 steps.

4 · Reading a fitted weight

The weight in a fitted AR(1) is interpretable as the first-order autocorrelation of returns, the correlation between each return and the one before it:

A fitted weight is a regression slope, the covariance of $r_t$ and $r_{t-1}$ divided by the variance of $r_{t-1}$, as Chapter 3C derives. For a stationary series the two returns have the same variance, so the slope equals the correlation.

A weight of −0.05 means returns are weakly mean-reverting: a 1% up move predicts roughly a 0.05% down move on average. A weight of +0.10 means returns are weakly momentum-driven. The magnitude tells you the predictive content, which in financial markets is small. Typical first-order autocorrelations of daily SPX returns lie between −0.05 and +0.05, with most of the absolute value coming from microstructure noise rather than fundamental signal.

Like the sample size calculation you would do before a trial, it shows why a weight fitted to a year or two of daily data is mostly noise.

Where the noise comes from: the bid-ask bounce

Much of the small negative autocorrelation in short-horizon returns is manufactured by the way prices are recorded. Trades print at the bid or at the ask, so a recorded price jumps between the two even when the value between them has not moved (Chapter 2C).

Suppose a stock's value sits at $100.00, with a bid of $99.95 and an ask of $100.05, while each trade prints at one or the other at random. The recorded price is then the value plus an error of ±$0.05, with some variance $s^2$. Each recorded change is the difference of two errors, with variance $2s^2$. Consecutive changes share one error with opposite signs, which gives them a covariance of $-s^2$. Their correlation is therefore $-s^2/2s^2 = -0.5$: strong apparent mean reversion in a price that never moved.

Real moves dilute the effect. If the value also moves with variance $\sigma^2$ a period, the autocorrelation becomes $-s^2/(\sigma^2 + 2s^2)$. With a half-spread of 0.05% and a hypothetical volatility of 0.03% a minute, that is $-0.0025/0.0059 = -0.42$. With a daily volatility of 1% it is $-0.0025/1.005 = -0.0025$, so the bounce that dominates minute returns is invisible in daily ones. Nor can you trade it, because capturing it means buying at the bid and selling at the ask, which is the market maker's business rather than the taker's.

This is regression to the mean from measurement error, the pattern of a stable tumor marker measured with independent assay error: successive visit-to-visit changes correlate at −0.5 while the true level never moves.

5 · The Hurst exponent and the variance ratio test

A more powerful way to detect mean reversion or momentum than fitting an AR(1) is to compute the Hurst exponent $H$ of the return series. It measures how variance scales with the time horizon. Over $k$ periods, where $\propto$ means "proportional to",

For a random walk (no autocorrelation) $H = 0.5$, so $2H = 1$ and variance scales linearly with time. For mean-reverting series $H < 0.5$: variance grows more slowly than linearly because reversals shrink the cumulative move. For momentum-driven (persistent) series $H > 0.5$: variance grows faster than linearly because trends compound.

The related variance ratio test of Lo and MacKinlay (1988, Review of Financial Studies) formalizes this into a statistical test of the random-walk null hypothesis. The variance ratio at lag $k$ is

Under the random-walk null, $\text{VR}(k) = 1$ for all $k$. A ratio above 1 indicates momentum and a ratio below 1 mean reversion. The ratio also says how wrong the square-root-of-time rule is, since the true $k$-period standard deviation is $\sqrt{kV}$ times $\sqrt{\text{VR}(k)}$. The two measures are linked: if variance scales as $k^{2H}$, then $\text{VR}(k) = k^{2H-1}$, which is how the figure above estimates $H$.

The two-period ratio is easy to work out. For any stationary series with variance $V$ and lag-1 autocorrelation $\rho_1$, a two-period sum has variance $V + V + 2\rho_1 V$, so dividing by $2V$ gives $\text{VR}(2) = 1 + \rho_1$. A weight of −0.05 gives 0.95 and a weight of +0.10 gives 1.10. For an AR(1) the ratio settles at long lags to $(1 + w)/(1 - w)$, which is 0.905 and 1.22 for the same two weights.

Lo and MacKinlay's 1988 paper found significant momentum in weekly US stock returns, with variance ratios above 1 at lags of 1 to 4 weeks, which challenged the random-walk hypothesis on those horizons. Tools like these are how the academic literature has built robust statistical support for mean reversion at very short and very long horizons, and for momentum at intermediate ones. The retail trader who claims "I'm a mean reversion guy" is making a horizon-specific assumption without perhaps realizing it.

The figure below returns to the one-step prediction of (3B.1). The solid line is sixty synthetic returns from an AR(1) with a true weight of −0.55 (mean reversion) or +0.55 (momentum). The dashed line is the prediction made with the weight and bias you set. You choose the true regime and set the weight $w$ from −1.5 to 1.5, the bias $b$ from −0.005 to 0.005 and a seed from 1 to 30, starting at mean reversion, −0.6, 0 and 7. The readouts give the sign of your weight, the true regime and the directional accuracy, the share of steps on which the prediction and the return have the same sign. Sweep the weight across zero, then switch the regime.

regime mean reversion momentum

weight w

−0.60

bias b

0.0000

seed

#7

weight sign = mean reversion

true regime = mean reversion

directional accuracy = 66.1%

With the bias at zero, only the sign of $w$ moves the accuracy: any negative weight scores 66.1% on the default returns, and any positive one 32.2%. Under momentum, positive weights score 62.7% and negative ones 35.6%. With $w$ at zero and $b$ nudged up to 0.001, the model predicts "up" at every step and scores 57.6%, the share of up steps: the base rate of Chapter 3A.

Directional accuracy ignores magnitude, unlike the squared error that Chapter 3C minimizes. The least-squares weight for the default returns is −0.58, close to the true −0.55, while across the 30 seeds it runs from −0.77 to −0.35. Sixty returns pin down even a large weight only roughly.

6 · Why AR(1) is the right starting point

You do not get rich on AR(1) alone. The reason to learn it is pedagogical. It is the most basic non-trivial model that captures the essential question every trading system must answer: do you expect the next move to continue the current move (momentum) or reverse it (mean reversion)? Every more sophisticated model refines this question:

• ARMA models add moving-average residual terms, which carry past shocks forward.
• ARIMA models handle non-stationarity.
• GARCH models add time-varying volatility.
• Kalman filters add inference about a hidden state.
• Hidden Markov models add switching between discrete regimes.
• LSTMs and transformers, two kinds of neural network, add nonlinear interactions across time.

All of them, in their cleanest formulation, answer some version of "given what has happened, what comes next?" The most basic answer is a weighted combination of recent values. Start with AR(1), adding complexity only when the residuals demand it.

7 · The regime problem

The most important practical issue with AR(1) and its descendants is that the sign of $w$ changes over time. SPX intraday autocorrelation in calm regimes is mildly negative. That is mean reversion, courtesy of positive-gamma dealer hedging, in which dealers sell into rallies and buy into falls (Part VI shows why). In stress regimes intraday autocorrelation flips positive: momentum, courtesy of negative-gamma dealer hedging that amplifies moves. A static AR(1) fitted to all available data averages across these regimes and produces an estimated $w$ near zero, which belongs to neither regime and predicts nothing.

The fix is regime-conditional modeling: fit one AR(1) for calm regimes and another for stress regimes, with a classifier that identifies which regime is in force. This is the essence of regime-switching models and the foundation of every serious quant trading framework. Chapter 8D returns to it with positive and negative gamma exposure (GEX) as the regime indicator, and Chapter 11G with the composite regime scorecard.

8 · Check yourself