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What is Regression?

Regression models how a response variable moves with one or more predictors. In trading it is the workhorse behind hedge ratios, factor models and risk decomposition.

The simple linear model
Y=β0+β1X+εY = \beta_0 + \beta_1 X + \varepsilon

A line plus a noise term. Almost every question in regression is about what that error is allowed to do.

β1\beta_1 is the slope: the expected change in YY per unit change in XX. β0\beta_0 is the intercept. ε\varepsilon is everything the model does not explain.

The trading interpretation

Regress a stock's returns on the market's and the slope is its beta, which is simultaneously three things:

  • A measure of systematic risk.
  • The hedge ratio: short β\beta units of index per unit of stock to neutralise market exposure.
  • The decomposition of returns into market-driven and idiosyncratic components.

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