Linear regression — the quant workhorse

Linear regression fits the best straight line through a cloud of points, expressing one variable as a weighted sum of others plus an error term:

y = b0 + b1*x1 + b2*x2 + ... + error

It is the most-used model in all of quant finance, and ForecastingStocks itself uses linear regression for its short-horizon forecasts alongside its deep-learning model.

What the pieces mean

• b0 — the intercept, the value of y when all inputs are zero.
• b1, b2, ... — the coefficients, how much y changes per unit of each input.
• error — everything the line cannot explain.

The fit is found by minimising the sum of squared errors — ordinary least squares (OLS).

Reading a regression honestly

• R-squared tells you the fraction of variance the model explains, from 0 to 1. In finance, a daily-return R-squared of even 0.05 can be tradeable; do not expect the 0.9 values seen in physics.
• Statistical significance (the p-value of a coefficient) tells you whether a relationship is likely real or just noise in this sample.

The famous example: beta

Regressing a stock's returns on the market's returns gives a slope called beta — how much the stock moves for a 1 percent market move. Beta above 1 means amplified market moves; below 1, dampened. This single regression underlies the Capital Asset Pricing Model and most factor models that follow.

The danger

A regression with enough inputs will fit any in-sample data beautifully and predict the future terribly. Each extra variable buys in-sample fit at the cost of out-of-sample reliability. That tension — fit versus generalisation — is the theme of this entire track.