Linear models: Inference

We now understand the least squares estimator \(\boldsymbol{\widehat{\beta}}\) from geometric and algebraic points of view. In Unit 2, we switch to a probabilistic perspective to derive inferential statements for linear models, in the form of hypothesis tests and confidence intervals. In order to facilitate this, we will assume that the error terms are normally distributed:

\[ \boldsymbol{y} = \boldsymbol{X} \boldsymbol{\beta} + \boldsymbol{\epsilon}, \quad \text{where} \ \boldsymbol{\epsilon} \sim N(\boldsymbol{0}, \sigma^2 \boldsymbol{I}_n). \] We first establish some building blocks necessary for linear models inference, primarily related to manipulating the normal distribution (Chapter 7). Then, we discuss univariate and multivariate hypothesis testing in linear models (Chapter 8), as well as the power of these hypothesis tests (Chapter 9). We then move on to the construction of confidence intervals and confidence regions (Chapter 10). We conclude with a discussion of practical considerations (Chapter 11) and an R demo (Chapter 12).