Unit 7 Ordinary Least Squares

The previous unit concluded without providing a complete answer. It identified the target, the population regression function, introduced a model for individual deviations, and established a criterion for selecting among candidate lines: minimising the sum of squared residuals.

This unit implements these steps by deriving the two estimators, clarifying their guarantees and limitations, evaluating their precision, and testing related claims.

The unit answers six questions.

1. Which line minimises the sum of squared residuals?
2. What does that line guarantee, and what does it merely appear to guarantee?
3. How much of the variation in \(y\) has it accounted for?
4. Is \(\hat{\beta}_1\) centred on \(\beta_1\), and how much does it vary?
5. What happens when the errors are not equally spread?
6. Is the slope we found distinguishable from zero?

And then the question the mechanics cannot answer: what does that slope actually mean?