What is Linear Regression?

Linear regression is one of the most widely used techniques in statistics and finance: a method for modelling the relationship between variables and using it to make predictions. From estimating how marketing spend drives sales to measuring a stock's sensitivity to the market, linear regression turns scattered data into a usable relationship. This guide explains what linear regression is, how it works, the assumptions behind it, and how finance uses it — in plain language. It's a core quantitative-methods topic, featured in qualifications like the FRM.

What is linear regression?

Linear regression is a method for estimating the relationship between a dependent variable (the outcome you want to explain or predict) and one or more independent variables (the factors you think drive it). It does this by fitting a straight line through the data that best captures the relationship.

When there's a single driver, it's called simple linear regression, and the line takes the familiar form y = a + bx: a is the intercept (the value of y when x is zero) and b is the slope (how much y changes for a one-unit change in x). When several drivers are involved, it's multiple linear regression, with a separate coefficient for each. The slope coefficients are the heart of the model — they quantify the effect of each driver.

A simple example

Suppose a company wants to understand how advertising affects sales. It plots monthly advertising spend (the independent variable) against sales (the dependent variable) and fits a regression line. If the slope comes out at 4, it means that, on average, every extra £1 of advertising is associated with £4 of additional sales. The intercept estimates baseline sales with no advertising at all. The company can then use the fitted line to predict the sales it might expect from a planned advertising budget — the essence of how regression turns past data into a forward-looking estimate.

How does it work?

Regression finds the "best-fitting" line using a principle called ordinary least squares (OLS). For any candidate line, each data point sits some distance above or below it; these gaps are the residuals. OLS chooses the line that makes the sum of the squared residuals as small as possible — squaring ensures positive and negative gaps don't cancel out and penalises larger errors more heavily. The result is the line that, overall, sits closest to the data.

A key output is R-squared, which measures how much of the variation in the dependent variable the model explains, on a scale from 0 to 1. A higher R-squared means the model accounts for more of what's going on — though, importantly, a good fit doesn't prove that one variable causes the other.

The assumptions behind it

Linear regression is reliable only when its underlying assumptions broadly hold. The main ones are that the relationship is genuinely linear, that the residuals are independent of one another (no autocorrelation), that they have constant variance, and that they are roughly normally distributed. When these break down — common with financial time-series data — the model's estimates and especially its measures of statistical significance can become misleading, so checking the assumptions is part of doing regression properly.

How finance uses linear regression

• Estimating beta. A stock's beta — its sensitivity to market movements — is the slope from regressing the stock's returns on the market's returns.
• Forecasting. Modelling how revenue responds to drivers such as price, advertising or economic indicators, then projecting forward.
• Risk modelling. Identifying and quantifying the factors that drive returns or losses in a portfolio.
• Valuation and analysis. Testing relationships between financial variables to inform decisions.

Why it matters for finance professionals

Linear regression is the workhorse of quantitative analysis — simple enough to interpret, yet powerful enough to underpin serious financial modelling. Understanding both how it works and where its assumptions can fail is essential for anyone who works with data: it's the difference between drawing a sound conclusion and being misled by a relationship that isn't really there. It's a foundational skill across finance, risk and analytics, and a regularly examined topic in professional qualifications.

Frequently asked questions

What is linear regression used for?

Modelling the relationship between a dependent variable and one or more independent variables, in order to explain or predict outcomes — from forecasting sales to estimating a stock's beta.

What is the difference between simple and multiple regression?

Simple linear regression uses one independent variable; multiple linear regression uses several, each with its own coefficient measuring its effect on the outcome.

What does R-squared tell you?

How much of the variation in the dependent variable the model explains, on a scale from 0 to 1. A higher value means a better fit — but it doesn't prove that one variable causes the other.

Does linear regression prove causation?

No. It measures association and predictive relationships. A strong fit can reflect coincidence or a hidden common cause, so causation requires additional reasoning and evidence.

Build your quant skills with Learnsignal

Linear regression is the gateway to data analysis and financial modelling. Learnsignal's tutor-led courses, including the FRM, develop the quantitative-methods understanding that topics like this build on — with clear teaching that makes the statistics genuinely click.