Regression Analysis for Traders: Beta, Factors and Pitfalls

Regression Analysis

Regression models how one variable relates to others. Learn linear regression, beta, R squared, multiple regression for factors, hedge ratios and common pitfalls.

Intermediate3 min readUpdated 3 Oct 2026

Lesson 19 of 46

Regression analysis estimates how one variable, such as a stock's return, relates to one or more other variables, such as the market's return or a set of factors. It is one of the most widely used tools in finance: it produces betas, hedge ratios, factor exposures and alpha estimates, and it underpins many forecasting models. Understanding how regression works, and how it can mislead, is essential for quantitative trading and risk management.

Simple linear regression#

y = α + β × x + ε

Term	Meaning
y	Dependent variable (for example, a stock's return)
x	Independent variable (for example, the market's return)
α (intercept)	Value of y when x is zero; in finance, often called alpha
β (slope)	Change in y for a one unit change in x
ε (error)	The part of y not explained by x

Ordinary least squares (OLS) chooses α and β to minimise the sum of squared errors.

β = Cov(x, y) / Var(x)

Worked example: market beta#

Multiple regression#

With several explanatory variables:

y = α + β1 x1 + β2 x2 + ... + βk xk + ε

Factor models use multiple regression to measure exposure to market, size, value, momentum and other factors. A fund's "alpha" is often defined as the intercept after controlling for these factors. See Factor Models.

Uses in trading#

Use	Example	Lesson
Beta and hedging	Hedge a stock with index futures	Alpha and Beta
Hedge ratios for pairs	Regress one stock's price on another's	Pairs Trading
Factor exposures	Measure a portfolio's tilt to value or momentum	Factor Models
Performance attribution	Separate skill from factor returns	P&L and Performance Attribution
Forecasting	Predict returns from signals	Signal Discovery
Cost models	Estimate market impact from order size	Market Impact

Assumptions and checks#

Assumption	Problem if violated	Remedy
Linear relationship	Biased estimates	Transform variables, add terms
Independent errors	Understated standard errors	Newey West standard errors. See Autocorrelation and Partial Autocorrelation
Constant error variance	Unreliable inference	Robust (heteroskedasticity consistent) standard errors
No strong multicollinearity	Unstable coefficients	Remove or combine correlated variables
Stationary data	Spurious results	Use returns, not prices; test for cointegration. See Stationarity, Differencing and Unit Roots
No extreme outliers	Distorted fit	Robust regression. See Outliers and Robust Statistics

Spurious regression#

Regressing one trending price series on another can produce high R² and "significant" coefficients even when the series are unrelated. Clive Granger and Paul Newbold showed this in 1974. Use returns or test for cointegration when working with prices. See Cointegration.

Overfitting#

Adding more variables always increases in sample R², even if they are noise. Use adjusted R², information criteria, regularisation (such as ridge and lasso regression) and out of sample tests to avoid overfitting. See Overfitting and Curve Fitting and Model Evaluation and Cross-Validation.

Frequently asked questions#

What is regression analysis in trading?#

A statistical method that estimates how one variable, such as a stock's return, relates to others, such as market or factor returns.

What does beta mean in a regression?#

The slope coefficient: how much the dependent variable changes, on average, for a one unit change in the independent variable.

What is a spurious regression?#

A regression that shows a strong relationship between unrelated variables, often caused by regressing trending, non stationary series on each other.

Next, learn a general method for fitting models in Maximum Likelihood.

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Next lessonMaximum LikelihoodMaximum likelihood estimation finds the model parameters that make observed data most probable. Learn the idea, simple examples, its use in GARCH and its limits.