Section 6.1: Multiple Regression Models

1. State the assumptions of the multiple linear regression (MLR) model as a system of equations when the distribution of the noise term is unspecified beyond its mean and variance-covariance.
2. State the assumptions of the multiple linear regression model as matrix-vector equation when the distribution of the noise term is unspecified beyond its mean and variance-covariance.
3. Sketch a drawing of the main parts of the multiple linear regression model with two predictors, including:
   • The population regression plane.
   • The expected value of the response.
   • The observed value of the response.
   • The error term.
4. State the assumptions of the multiple linear regression with Gaussian noise (MLRGN) model.

Section 6.8: Diagnostics and Remedial Measures

1. Explain the rationale for using the statistical properties of the residuals from a fitted multiple linear regression model as a proxy for the population errors.
2. State the relevant residual diagnostic plots to generate after fitting a multiple linear regression model.
3. Relate each residual diagnostic plot from the previous learning objective to the associated assumption of the multiple linear regression with Gaussian noise (MLRGN) model.
4. Given a residual diagnostic plot, determine whether the plot indicates a departure from the assumptions of MLRGN.

Properties of the Ordinary Least Squares Estimators Under MLR and MLRGN (Lecture Notes for Lecture 14)

1. State the point estimator we will use for the population coefficients \(\boldsymbol{\beta}\).
2. State the point estimator we will use for the population noise variance \(\sigma_{\epsilon}^{2}\).
3. State the mean and variance-covariance matrix of the ordinary least squares estimator \(\mathbf{b}\) under the MLR model.
4. State the variance of the coefficient estimator \(b_{j}\) under the MLR model.
5. Describe how the variance of the coefficient estimator for the coefficient parameter for the j-th predictor under the MLR model varies with:
   • The sample size.
   • The population noise variance.
   • The standard deviation of \(j\)-th predictor.
   • The correlation between the \(j\)-th predictor and the other predictors used in the model.
6. Define the multiple \(R^2\) between a predictor variable and the other predictor variables in the model.
7. Reason about what makes the multiple \(R^2\) between a predictor variable and the other predictor variables in the model large (\(\approx 1\)) or small (\(\approx 0\)).
8. State the sampling distribution of the coefficient estimators under the multiple linear regression with Gaussian noise (MLRGN) model.
9. State the sampling distribution of the studentized coefficient estimators under the multiple linear regression with Gaussian noise (MLRGN) model.

R (Lecture Notes for Lecture 14)

1. Create residual diagnostic plots “by-hand” using functions from ggformula.
2. Create residual diagnostic plots using gf_residuals_versus_predictors from the MUsaic package.