---
title: "Statistical & Quantitative Indicators"
description: "Quantify statistical properties, relationships, normalization, or model residuals."
canonical: https://www.getswoopr.com/indicators/statistical-and-quantitative/
source: "Swoopr Investment: https://www.getswoopr.com"
---

## Direct Answer



Quantify statistical properties, relationships, normalization, or model residuals.





      


# Statistical & Quantitative Indicators



> Statistical and quantitative indicators apply textbook statistics to price and return series: rolling location and scale, z-scores and percentile ranks, correlation, beta, regression and cointegration, persistence tests such as autocorrelation and the Hurst exponent, and entropy measures. They are the building blocks other indicators and systematic strategies are made from, and each carries the assumptions its textbook states.

## What do Statistical & Quantitative indicators have in common?

Every indicator in this category is a textbook statistic applied to a rolling window of prices or returns. None was invented for markets: the z-score, the correlation coefficient, linear regression, the Dickey-Fuller test and the Kalman filter each come from statistics or engineering, and their appearance here is as tools for stating how unusual a reading is, how two series move together, or whether a series tends to return to a level.

The category divides by the question each statistic answers. Location and scale (rolling mean, median, standard deviation, variance, interquartile range, median absolute deviation, coefficient of variation) describe the window. Normalization (the z-score family, percentile and percent rank, cross-sectional ranks) places today's value within that window or across a universe. Relationship measures (correlation in its rolling, log and distance forms, covariance, beta, alpha, r-squared, linear regression with its slope, intercept, angle and residual, principal component scores, Mahalanobis distance) describe how series move together.

Persistence and stationarity measures (autocorrelation and partial autocorrelation, the Hurst exponent, fractal dimension, the half-life and Ornstein-Uhlenbeck speed of mean reversion, the augmented Dickey-Fuller and KPSS statistics, the cointegration statistic and Johansen rank) test whether a series trends, reverts or wanders. Information measures (Shannon, approximate and sample entropy, mutual information, signal-to-noise ratio) describe how predictable it is, and the Kalman filter estimate and residual and the time series forecast are model-based estimates of the next value.

What they share is that each carries assumptions a textbook states and a chart does not: a z-score assumes the window's distribution is a fair description of the next observation, a correlation assumes a linear relationship, a regression assumes its residuals are independent, and a stationarity test has a known false-positive rate. Each also has a published definition, so unlike most chart indicators its formula does not vary between providers.

## How do you choose a Statistical & Quantitative indicator?

- For how unusual a reading is, use a robust or winsorized z-score when the window contains outliers, which price data usually does, and the plain z-score only when it does not
- For a pair or spread trade, test for cointegration (Engle-Granger or Johansen) before trading the spread's z-score, and read the half-life to see whether a reversion is fast enough to trade
- For hedging, use beta for the size of the hedge and r-squared for how much of the position it actually covers
- For whether a series trends or reverts, read the Hurst exponent and the autocorrelation together; a Hurst value above 0.5 with positive autocorrelation says trend, below 0.5 with negative autocorrelation says reversion
- Prefer a rolling window long enough for the statistic to be stable and short enough to track a changing market; there is no window that does both, so state the choice

## Statistical & Quantitative indicators at a glance

| Indicator | What it measures | How it is read |
| --- | --- | --- |
| [Rolling Mean](https://www.getswoopr.com/indicators/statistical-and-quantitative/rolling-mean/) | Rolling mean is the arithmetic average of the latest N observations, recomputed each bar as the window advances. | It is the baseline level of the window. |
| [Rolling Median](https://www.getswoopr.com/indicators/statistical-and-quantitative/rolling-median/) | Rolling median is the middle value of the latest N observations when sorted. | Compared with a rolling mean, a rolling median that differs materially from the mean indicates a skewed window or outliers. |
| [Rolling Standard Deviation](https://www.getswoopr.com/indicators/statistical-and-quantitative/rolling-standard-deviation/) | Rolling standard deviation is the square root of the sample variance of the latest N observations. | Higher readings mean wider recent dispersion, which for returns is higher volatility. |
| [Rolling Variance](https://www.getswoopr.com/indicators/statistical-and-quantitative/rolling-variance/) | Rolling variance is the sample variance of the latest N observations, recomputed each bar. | Larger values mean wider dispersion. |
| [Covariance](https://www.getswoopr.com/indicators/statistical-and-quantitative/covariance/) | Covariance measures how two series move together. | The sign shows the direction of co-movement and the size depends on the units of both series. |
| [Correlation Coefficient](https://www.getswoopr.com/indicators/statistical-and-quantitative/correlation-coefficient/) | The Pearson correlation coefficient measures the strength and direction of the linear relationship between two variables, scaled to lie between -1 and +1. | +1 is a perfect increasing linear relation, -1 a perfect decreasing one and 0 no linear relation. |
| [Log Correlation](https://www.getswoopr.com/indicators/statistical-and-quantitative/log-correlation/) | Log correlation is the correlation between two series measured on logarithmic returns, ln(P_t / P_(t-1)), rather than on price levels or simple returns. | It is read like any correlation, from -1 to +1. |
| [Rolling Correlation](https://www.getswoopr.com/indicators/statistical-and-quantitative/rolling-correlation/) | Rolling correlation is the Pearson correlation coefficient between two series, recomputed over a moving window of the latest N observations. | Values range from -1 to +1. |
| [Beta](https://www.getswoopr.com/indicators/statistical-and-quantitative/beta/) | Beta measures how sensitively an asset's returns move with a benchmark's returns. | A beta of 1 means the asset has moved one-for-one with the benchmark in the window, above 1 means larger swings and below 1 smaller ones. |
| [Alpha](https://www.getswoopr.com/indicators/statistical-and-quantitative/alpha/) | Jensen's alpha is the average return of an asset or portfolio above what the capital asset pricing model predicts given its beta. | Positive alpha means the return exceeded what its market exposure implied, and negative alpha means it fell short. |
| [R-Squared](https://www.getswoopr.com/indicators/statistical-and-quantitative/r-squared/) | R-squared, the coefficient of determination, is the share of the variation in a series that a regression explains. | Values run from 0 to 1 for a regression with an intercept. |
| [Linear Regression](https://www.getswoopr.com/indicators/statistical-and-quantitative/linear-regression/) | The linear regression value is the end point of a least-squares straight line fitted to the latest N prices. | Price above the line means price is above its fitted trend and below means under it. |
| [Linear Regression Slope](https://www.getswoopr.com/indicators/statistical-and-quantitative/linear-regression-slope/) | The linear regression slope is the per-bar change of the least-squares line fitted to the latest N values. | Positive values indicate a rising fitted trend and negative values a falling one, with larger magnitude meaning a steeper path. |
| [Linear Regression Intercept](https://www.getswoopr.com/indicators/statistical-and-quantitative/linear-regression-intercept/) | The linear regression intercept is the value at which the least-squares line crosses the vertical axis at the start of the fitting window. | Its value depends on where x is counted from, so it is meaningful only together with the slope and the x convention. |
| [Linear Regression Angle](https://www.getswoopr.com/indicators/statistical-and-quantitative/linear-regression-angle/) | The linear regression angle is the angle, in degrees, of the fitted regression line, computed as the arctangent of its slope. | Positive angles are rising trends and negative angles falling ones, and an angle near 0 is a flat trend. |
| [Regression Residual](https://www.getswoopr.com/indicators/statistical-and-quantitative/regression-residual/) | A regression residual is the difference between an observed value and the value fitted by a regression line. | A positive residual means the actual value is above the fitted line and a negative one below. |
| [Time Series Forecast](https://www.getswoopr.com/indicators/statistical-and-quantitative/time-series-forecast/) | Time series forecast, or TSF, fits a least-squares straight line to the latest N values and extends that line one bar ahead. | TSF above the current price means the fitted trend points above where price sits, and below means the opposite. |
| [Z-Score](https://www.getswoopr.com/indicators/statistical-and-quantitative/z-score/) | The z-score states how many standard deviations a value lies from the mean of a reference sample, usually a rolling window of recent observations. | A z-score of 0 means the value equals the mean, and +2 or -2 means it is two standard deviations away. |
| [Robust Z-Score](https://www.getswoopr.com/indicators/statistical-and-quantitative/robust-z-score/) | The robust z-score, also called the modified z-score, replaces the mean and standard deviation with the median and the median absolute deviation. | The 0.6745 factor makes the score comparable with a standard z-score for normal data. |
| [Percentile](https://www.getswoopr.com/indicators/statistical-and-quantitative/percentile/) | A percentile is the value below which a given percentage of observations fall. | Rolling high percentiles of a measure such as volatility or drawdown set an empirical band that does not assume a normal distribution. |
| [Percent Rank](https://www.getswoopr.com/indicators/statistical-and-quantitative/percent-rank/) | Percent rank states where the latest value stands within the distribution of its lookback window, as the share of window values that are lower. | Values near 100 percent mean the current reading is at the top of its recent range, and values near 0 percent mean it is at the bottom. |
| [Skewness](https://www.getswoopr.com/indicators/statistical-and-quantitative/skewness/) | Skewness measures the asymmetry of a distribution around its mean. | Zero means a symmetric distribution, negative values signal a heavier left tail, and positive values a heavier right tail. |
| [Kurtosis](https://www.getswoopr.com/indicators/statistical-and-quantitative/kurtosis/) | Kurtosis measures how heavy the tails of a distribution are compared with a normal distribution. | Positive excess kurtosis means fatter tails than a normal curve, so large moves occur more often than a normal model suggests. |
| [Autocorrelation](https://www.getswoopr.com/indicators/statistical-and-quantitative/autocorrelation/) | Autocorrelation is the correlation of a series with a lagged copy of itself. | Positive values mean moves tend to persist, negative values mean they tend to reverse, and zero means no linear dependence at that lag. |
| [Partial Autocorrelation](https://www.getswoopr.com/indicators/statistical-and-quantitative/partial-autocorrelation/) | Partial autocorrelation at lag k measures the correlation between a series and its value k periods earlier after removing the linear effect of the intermediate lags. | Values outside roughly plus or minus 1.96 divided by the square root of the sample size are conventionally treated as statistically different from zero. |
| [Hurst Exponent](https://www.getswoopr.com/indicators/statistical-and-quantitative/hurst-exponent/) | The Hurst exponent H summarizes long-range dependence in a series. | H near 0.5 describes a random-walk-like series, H above 0.5 persistent behavior where moves tend to continue, and H below 0.5 anti-persistent behavior where moves tend to reverse. |
| [Shannon Entropy](https://www.getswoopr.com/indicators/statistical-and-quantitative/shannon-entropy/) | Shannon entropy measures the uncertainty in a distribution of outcomes. | Zero means all observations fall in one bin. |
| [Approximate Entropy](https://www.getswoopr.com/indicators/statistical-and-quantitative/approximate-entropy/) | Approximate entropy (ApEn), introduced by Steve Pincus, quantifies the regularity of a time series. | Values near zero indicate a highly regular series, and larger values indicate more irregularity. |
| [Sample Entropy](https://www.getswoopr.com/indicators/statistical-and-quantitative/sample-entropy/) | Sample entropy, introduced by Richman and Moorman, estimates the irregularity of a time series. | Low values mean the series is regular and self-similar, so sequences tend to repeat. |
| [Half-Life of Mean Reversion](https://www.getswoopr.com/indicators/statistical-and-quantitative/half-life-of-mean-reversion/) | The half-life of mean reversion is the expected time for a deviation from the long-run mean to shrink by half. | Short half-lives mean deviations close quickly, long half-lives mean slow reversion. |
| [Augmented Dickey-Fuller Statistic](https://www.getswoopr.com/indicators/statistical-and-quantitative/augmented-dickey-fuller-statistic/) | The augmented Dickey-Fuller (ADF) statistic tests the null hypothesis that a series has a unit root, meaning it is non-stationary. | More negative values are stronger evidence against a unit root. |
| [KPSS Statistic](https://www.getswoopr.com/indicators/statistical-and-quantitative/kpss-statistic/) | The KPSS statistic, from Kwiatkowski, Phillips, Schmidt and Shin, tests the null hypothesis that a series is stationary, in contrast with unit-root tests whose null is non-stationarity. | Larger values are evidence against stationarity. |
| [Cointegration Statistic](https://www.getswoopr.com/indicators/statistical-and-quantitative/cointegration-statistic/) | The Engle-Granger cointegration statistic tests whether two non-stationary series share a stable long-run relationship. | More negative statistics are stronger evidence that the residual is stationary. |
| [Johansen Cointegration Rank](https://www.getswoopr.com/indicators/statistical-and-quantitative/johansen-cointegration-rank/) | The Johansen procedure tests how many independent long-run equilibrium relationships (the cointegration rank) exist among several non-stationary series. | A rank of 0 means no cointegration, a rank between 1 and n-1 means that many stable linear combinations exist, and each is a candidate spread. |
| [Ornstein-Uhlenbeck Mean Reversion Speed](https://www.getswoopr.com/indicators/statistical-and-quantitative/ornstein-uhlenbeck-mean-reversion-speed/) | In an Ornstein-Uhlenbeck process a series is pulled back toward a long-run mean at a rate theta. | A positive theta means the series is pulled back toward its mean, and a larger value means faster reversion. |
| [Kalman Filter Estimate](https://www.getswoopr.com/indicators/statistical-and-quantitative/kalman-filter-estimate/) | The Kalman filter estimate is the filter's current best estimate of a hidden state, such as the underlying level or hedge ratio, combining a model prediction with each new noisy observation according to their relative uncertainties. | The estimate follows the data but smoothed, with the weight on new observations (the Kalman gain) set by the ratio of uncertainty. |
| [Kalman Filter Residual](https://www.getswoopr.com/indicators/statistical-and-quantitative/kalman-filter-residual/) | The Kalman filter residual, also called the innovation, is the gap between the latest observation and the filter's prediction of it made before seeing that observation. | If the model is right, standardized innovations are centered on zero with a standard deviation near 1 and no autocorrelation. |
| [Mahalanobis Distance](https://www.getswoopr.com/indicators/statistical-and-quantitative/mahalanobis-distance/) | Mahalanobis distance measures how far a point is from the center of a multivariate distribution, taking into account the variances and correlations of the variables. | Larger distance means the point is more unusual relative to the usual pattern of the variables. |
| [Principal Component Score](https://www.getswoopr.com/indicators/statistical-and-quantitative/principal-component-score/) | A principal component score is the coordinate of an observation along a principal component axis. | A large positive or negative score means the observation sits far out along that component's direction. |
| [Rolling PCA Factor](https://www.getswoopr.com/indicators/statistical-and-quantitative/rolling-pca-factor/) | A rolling PCA factor is the first principal component of a group of related series, such as asset returns, estimated on a moving window. | The factor series often resembles a common market or sector driver, and its explained-variance share shows how tightly the group moves together. |
| [Signal-to-Noise Ratio](https://www.getswoopr.com/indicators/statistical-and-quantitative/signal-to-noise-ratio/) | Signal-to-noise ratio compares the size of a signal with the size of the random variation around it. | Higher absolute values mean the average move is large relative to the scatter, so direction is more consistent. |
| [Coefficient of Variation](https://www.getswoopr.com/indicators/statistical-and-quantitative/coefficient-of-variation/) | The coefficient of variation is the standard deviation divided by the mean. | A higher value means more relative variability. |
| [Median Absolute Deviation](https://www.getswoopr.com/indicators/statistical-and-quantitative/median-absolute-deviation/) | Median absolute deviation, or MAD, measures dispersion as the median of the absolute distances from the sample median. | A larger MAD means a wider typical spread around the median. |
| [Interquartile Range](https://www.getswoopr.com/indicators/statistical-and-quantitative/interquartile-range/) | The interquartile range, or IQR, is the distance between the third and first quartiles, so it covers the middle half of the observations in the window. | A larger value means the central half of the data is more spread out. |
| [Winsorized Z-Score](https://www.getswoopr.com/indicators/statistical-and-quantitative/winsorized-z-score/) | A winsorized z-score is a z-score computed after extreme values in the sample have been clipped to chosen percentile limits, for example the 5th and 95th. | It reads like a standard z-score, but extreme readings are capped at the clipping limits, so the largest possible value is bounded. |
| [Exponential Z-Score](https://www.getswoopr.com/indicators/statistical-and-quantitative/exponential-z-score/) | An exponential z-score standardizes a value using an exponentially weighted mean and exponentially weighted variance, so recent observations count more than older ones. | It reads like a regular z-score, in standard deviations from the weighted mean. |
| [Cross-Sectional Z-Score](https://www.getswoopr.com/indicators/statistical-and-quantitative/cross-sectional-z-score/) | A cross-sectional z-score standardizes one asset's value against all other assets on the same date, rather than against its own history. | Positive values rank the asset above the typical member of the universe on the metric and negative values below. |
| [Cross-Sectional Percentile Rank](https://www.getswoopr.com/indicators/statistical-and-quantitative/cross-sectional-percentile-rank/) | Cross-sectional percentile rank places an asset within the ordered list of all assets in the universe on a given date, expressed as a percentage. | High percentiles mean the asset scores near the top of the universe on the metric, low percentiles near the bottom. |
| [Residual Z-Score](https://www.getswoopr.com/indicators/statistical-and-quantitative/residual-z-score/) | A residual z-score standardizes the error left over after a regression fit. | A large positive value means the series is above what the relationship implies, and a large negative one means below. |
| [Pair Spread Z-Score](https://www.getswoopr.com/indicators/statistical-and-quantitative/pair-spread-z-score/) | The pair spread z-score standardizes the spread between two related assets by its own mean and standard deviation. | A large positive score means the first leg is rich relative to the second compared with its recent norm, and a large negative score means the opposite. |
| [Distance Correlation](https://www.getswoopr.com/indicators/statistical-and-quantitative/distance-correlation/) | Distance correlation, introduced by Szekely, Rizzo and Bakirov, measures dependence between two variables of any shape and is zero only if they are independent. | Values lie between 0 and 1, with 0 meaning independence and 1 a strong dependence. |
| [Mutual Information](https://www.getswoopr.com/indicators/statistical-and-quantitative/mutual-information/) | Mutual information measures how much knowing one variable reduces uncertainty about another. | Zero indicates independence, and larger values indicate stronger dependence of any shape. |
| [Fractal Dimension](https://www.getswoopr.com/indicators/statistical-and-quantitative/fractal-dimension/) | The fractal dimension of a price path measures how jagged it is. | Lower values indicate a more directional, trending path and higher values a choppier one. |

Each row is the opening sentence of that indicator's own definition and reading, so the table says exactly what the profile says.






## Guides in This Category



- [Rolling Mean](https://www.getswoopr.com/indicators/statistical-and-quantitative/rolling-mean/)
- [Rolling Median](https://www.getswoopr.com/indicators/statistical-and-quantitative/rolling-median/)
- [Rolling Standard Deviation](https://www.getswoopr.com/indicators/statistical-and-quantitative/rolling-standard-deviation/)
- [Rolling Variance](https://www.getswoopr.com/indicators/statistical-and-quantitative/rolling-variance/)
- [Covariance](https://www.getswoopr.com/indicators/statistical-and-quantitative/covariance/)
- [Correlation Coefficient](https://www.getswoopr.com/indicators/statistical-and-quantitative/correlation-coefficient/)
- [Log Correlation](https://www.getswoopr.com/indicators/statistical-and-quantitative/log-correlation/)
- [Rolling Correlation](https://www.getswoopr.com/indicators/statistical-and-quantitative/rolling-correlation/)
- [Beta](https://www.getswoopr.com/indicators/statistical-and-quantitative/beta/)
- [Alpha](https://www.getswoopr.com/indicators/statistical-and-quantitative/alpha/)
- [R-Squared](https://www.getswoopr.com/indicators/statistical-and-quantitative/r-squared/)
- [Linear Regression](https://www.getswoopr.com/indicators/statistical-and-quantitative/linear-regression/)
- [Linear Regression Slope](https://www.getswoopr.com/indicators/statistical-and-quantitative/linear-regression-slope/)
- [Linear Regression Intercept](https://www.getswoopr.com/indicators/statistical-and-quantitative/linear-regression-intercept/)
- [Linear Regression Angle](https://www.getswoopr.com/indicators/statistical-and-quantitative/linear-regression-angle/)
- [Regression Residual](https://www.getswoopr.com/indicators/statistical-and-quantitative/regression-residual/)
- [Time Series Forecast](https://www.getswoopr.com/indicators/statistical-and-quantitative/time-series-forecast/)
- [Z-Score](https://www.getswoopr.com/indicators/statistical-and-quantitative/z-score/)
- [Robust Z-Score](https://www.getswoopr.com/indicators/statistical-and-quantitative/robust-z-score/)
- [Percentile](https://www.getswoopr.com/indicators/statistical-and-quantitative/percentile/)
- [Percent Rank](https://www.getswoopr.com/indicators/statistical-and-quantitative/percent-rank/)
- [Skewness](https://www.getswoopr.com/indicators/statistical-and-quantitative/skewness/)
- [Kurtosis](https://www.getswoopr.com/indicators/statistical-and-quantitative/kurtosis/)
- [Autocorrelation](https://www.getswoopr.com/indicators/statistical-and-quantitative/autocorrelation/)
- [Partial Autocorrelation](https://www.getswoopr.com/indicators/statistical-and-quantitative/partial-autocorrelation/)
- [Hurst Exponent](https://www.getswoopr.com/indicators/statistical-and-quantitative/hurst-exponent/)
- [Shannon Entropy](https://www.getswoopr.com/indicators/statistical-and-quantitative/shannon-entropy/)
- [Approximate Entropy](https://www.getswoopr.com/indicators/statistical-and-quantitative/approximate-entropy/)
- [Sample Entropy](https://www.getswoopr.com/indicators/statistical-and-quantitative/sample-entropy/)
- [Half-Life of Mean Reversion](https://www.getswoopr.com/indicators/statistical-and-quantitative/half-life-of-mean-reversion/)
- [Augmented Dickey-Fuller Statistic](https://www.getswoopr.com/indicators/statistical-and-quantitative/augmented-dickey-fuller-statistic/)
- [KPSS Statistic](https://www.getswoopr.com/indicators/statistical-and-quantitative/kpss-statistic/)
- [Cointegration Statistic](https://www.getswoopr.com/indicators/statistical-and-quantitative/cointegration-statistic/)
- [Johansen Cointegration Rank](https://www.getswoopr.com/indicators/statistical-and-quantitative/johansen-cointegration-rank/)
- [Ornstein-Uhlenbeck Mean Reversion Speed](https://www.getswoopr.com/indicators/statistical-and-quantitative/ornstein-uhlenbeck-mean-reversion-speed/)
- [Kalman Filter Estimate](https://www.getswoopr.com/indicators/statistical-and-quantitative/kalman-filter-estimate/)
- [Kalman Filter Residual](https://www.getswoopr.com/indicators/statistical-and-quantitative/kalman-filter-residual/)
- [Mahalanobis Distance](https://www.getswoopr.com/indicators/statistical-and-quantitative/mahalanobis-distance/)
- [Principal Component Score](https://www.getswoopr.com/indicators/statistical-and-quantitative/principal-component-score/)
- [Rolling PCA Factor](https://www.getswoopr.com/indicators/statistical-and-quantitative/rolling-pca-factor/)
- [Signal-to-Noise Ratio](https://www.getswoopr.com/indicators/statistical-and-quantitative/signal-to-noise-ratio/)
- [Coefficient of Variation](https://www.getswoopr.com/indicators/statistical-and-quantitative/coefficient-of-variation/)
- [Median Absolute Deviation](https://www.getswoopr.com/indicators/statistical-and-quantitative/median-absolute-deviation/)
- [Interquartile Range](https://www.getswoopr.com/indicators/statistical-and-quantitative/interquartile-range/)
- [Winsorized Z-Score](https://www.getswoopr.com/indicators/statistical-and-quantitative/winsorized-z-score/)
- [Exponential Z-Score](https://www.getswoopr.com/indicators/statistical-and-quantitative/exponential-z-score/)
- [Cross-Sectional Z-Score](https://www.getswoopr.com/indicators/statistical-and-quantitative/cross-sectional-z-score/)
- [Cross-Sectional Percentile Rank](https://www.getswoopr.com/indicators/statistical-and-quantitative/cross-sectional-percentile-rank/)
- [Residual Z-Score](https://www.getswoopr.com/indicators/statistical-and-quantitative/residual-z-score/)
- [Pair Spread Z-Score](https://www.getswoopr.com/indicators/statistical-and-quantitative/pair-spread-z-score/)
- [Distance Correlation](https://www.getswoopr.com/indicators/statistical-and-quantitative/distance-correlation/)
- [Mutual Information](https://www.getswoopr.com/indicators/statistical-and-quantitative/mutual-information/)
- [Fractal Dimension](https://www.getswoopr.com/indicators/statistical-and-quantitative/fractal-dimension/)





## What limits do Statistical & Quantitative indicators share?

Every statistic here assumes the window it was computed on describes the next observation, and financial series change regime without notice, so correlations measured in calm markets rise in crashes and a cointegration that held for years can break. A rolling window makes each statistic a function of an arbitrary length, stationarity tests have a known false-positive rate and low power in short samples, the Hurst exponent is notoriously noisy in windows of a few hundred observations, and a z-score on returns treats a fat-tailed distribution as if it were normal, understating how often large readings occur.

## Frequently Asked Questions

What is a z-score in trading?

A z-score is the number of standard deviations a value sits from the mean of a window, computed as the value minus the rolling mean divided by the rolling standard deviation. A reading of plus two means the value is two standard deviations above its recent average, which under a normal distribution would occur about 2.3 percent of the time; financial returns have fatter tails, so such readings occur more often than that.

What does cointegration mean?

Two price series are cointegrated when each wanders without a fixed level but a fixed combination of them, such as one minus a multiple of the other, stays around a stable mean. Engle and Granger defined the test in 1987 and Johansen extended it to several series. A cointegrated pair is the basis of a pairs trade, since the spread between them can be expected to return to its mean.

What is the Hurst exponent?

The Hurst exponent measures whether a series tends to continue in its direction, revert, or move randomly, on a scale where 0.5 is a random walk, above 0.5 is trending and below 0.5 is mean-reverting. Harold Hurst derived it in 1951 from Nile flood records. It needs long windows to estimate reliably and is sensitive to the method used.

What is the half-life of mean reversion?

The half-life is the time a mean-reverting series is expected to take to close half the gap between its current value and its long-run mean, computed from the speed parameter of an Ornstein-Uhlenbeck fit. A spread with a half-life of ten days can be traded on a horizon of weeks; one with a half-life of a year is a position, not a trade.





[Swoopr Editorial Team](https://www.getswoopr.com/authors/swoopr-editorial-team/) produces independent investment education and research content.



Corrections: [corrections policy](https://www.getswoopr.com/legal/corrections-policy/). Standards: [editorial standards](https://www.getswoopr.com/legal/editorial-policy/).