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Probability, Statistics and Financial Data / Statistical inference

Linear Regression

A regression of stock excess returns on market excess returns estimates the stock's factor exposure. The slope describes a conditional relationship, not a trading guarantee. This lesson derives ordinary least squares, explains its projection geometry, and calculates uncertainty in a fitted beta. Residual checks show when the usual model assumptions need revision.
Learning objectives. By the end of this note you should be able to:
  • state the simple linear model and derive β^0,β^1\hat\beta_0,\hat\beta_1 by minimising RSS;
  • interpret OLS as projection onto span{1,x}\mathrm{span}\{\mathbf{1},\mathbf{x}\} in one paragraph;
  • form a tt-interval / test for β1\beta_1 under normal errors (or large nn);
  • compute R2R^2, read residual plots for curvature / heteroskedasticity / outliers;
  • fit and interpret a stock-on-market regression by hand on a small sample.

In this lesson

  1. The model
  2. Ordinary least squares
  3. Geometry in one paragraph
  4. Inference
  5. R2R^2 and residual checks
  6. Connection to conditional expectation and MLE
  7. Worked factor example

Before you start

The full lesson, with worked examples and exercise solutions, is for members.

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