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Introduction to Regression Analysis
COURSE SYLLABUS
Course Description
Course Objectives
Course Outline
Book Guide
References
I INTRODUCTION
1
Preliminaries
1.1
Review of Matrix Theory
Basic Concepts
Matrix Operations
Special Matrices
Invertibility and Singularity
Eigenvalues and Eigenvectors
Linear Dependence and Ranks
Matrix Calculus
R outputs
1.2
Review of Statistical Inference
Basic Definitions
Variance-Covariance Matrix and Correlation Matrix
The Normal Distribution
Approaches to Inference
1.3
The Model Building Process
Types of Models
General Classification of Stochastic Models
Purpose of Modelling
Types of Variables in a Regression Problem
Types of Data
Steps in the Model-Building Process
1.4
Measures of Correlation
Pearson’s
r
Spearman’s
ρ
Kendall’s
τ
ϕ
Coefficient
Contingency Coefficient
Other measures of association if at least one is categorical
1.5
Overview of Regression
2
Introduction to Regression and Linear Models
2.1
Historical Origin of the term “Regression”
2.2
Uses of Regression Analysis
2.3
Classification of Regression Models
In terms of distributional assumptions
In terms of types of dependent and independent variables
2.4
The Linear Model
Justification of the Error Term
Linear Model in Matrix Form
Illustrations for the Case of 2 Independent Variables
2.5
Assumptions
Types of Error Assumptions
Linear Regression Variable Assumptions
Important Features of the Model
II MODEL BUILDING
3
Estimation Procedures
3.1
Method of Least Squares
The Normal Equations
The OLS Estimator
Exercises
3.2
Other Estimation Methods
3.3
Interpretation of the Coefficients
4
Least Squares Theory and Analysis of Variance
4.1
Fitted Values and Residuals
4.2
Analysis of Variance
Sum of Squares
Mean Squares
Sampling Distribution of the Sum of Squares while Assuming Same Means of
Y
i
|
x
ANOVA Table and F Test for Regression Relation
4.3
Estimation of the Error Variance
4.4
Coefficient of Multiple Determination
4.5
Properties of the Estimators of the Regression Parameters
Best Linear Unbiased Estimator of
\boldsymbol{\beta}
Maximum Likelihood Estimator of
\boldsymbol{\beta}
and
\sigma^2
UMVUE of
\boldsymbol{\beta}
and
\sigma^2
4.6
Confidence Interval Estimation for Slope Parameters
4.7
Hypothesis Testing for Individual Slope Parameters
5
Model Evaluation in Multiple Linear Regression
5.1
Preliminaries
Full and Reduced Models
Extra Sum of Squares
General Linear Test
5.2
Hypothesis Tests on Model Parameters
Testing all slope parameters equal to 0
Testing one slope parameter equal to 0
Testing several slope parameters equal to 0
Testing a slope parameter equal to 0 sequentially
5.3
Summary of the General Linear Tests
5.4
Coefficient of Partial Determination
6
Variable Selection Procedures and Standardized Coefficients
6.1
All-Possible-Regression Approach (APR)
6.2
Prediction Error Sum of Squares Procedure
6.3
Automatic Search Procedures (ASP)
Forward Selection Procedure
Backward Elimination Procedure
Stepwise Selection Procedure
6.4
Standardized Coefficients
7
Inference on Mean Response and Prediction of New Observations
7.1
Inference about the mean response
Point Estimation
Confidence Interval Estimation
Hypothesis Testing
7.2
Prediction of a new observation
7.3
The Inverse Prediction Problem
7.4
Assessment of Predictive Ability of Model
Cross Validation using Train and Test Sets
Predicted
R^2
8
Dummy Variables
8.1
Dichotomous Independent Variables
Interaction Effect
Testing for Parallelism
8.2
Polytomous Independent Variables
8.3
Regime-Switching Models
III DIAGNOSTIC CHECKING
9
Assessing General Diagnostic Plots and Testing for Linearity
9.1
Residual Plots
Partial Regression Plots
Partial Residual Plots
9.2
Effects and Usual Causes of Nonlinearity
9.3
Testing for Nonlinearity
9.4
Remedial Measures
10
Nonnormality and Heteroskedasticity
10.1
Effects and Usual Causes of Non-Normality and Heteroskedasticity
10.2
Testing for Nonnormality
10.3
Testing for Heteroskedasticity
10.4
Remedial Measures
10.5
Exercise
11
Autocorrelation
11.1
Effects and Usual Causes of Autocorrelation
11.2
Testing for Autocorrelation
11.3
Remedial Measures
11.4
Exercise
12
Multicollinearity
12.1
Effects and Usual Causes of Multicollinearity
12.2
Testing for Multicollinearity
12.3
Remedial Measures
12.4
Exercise
13
On Outliers and Influential Observations
13.1
Effects and Usual Causes of Outliers and Influential Observations
13.2
Testing for Outliers and Influential Observations
13.3
Remedial Measures
13.4
Exercise
UP School of Statistics
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STAT 136: Introduction to Regression Analysis
CHAPTER 13
On Outliers and Influential Observations
13.1
Effects and Usual Causes of Outliers and Influential Observations
13.2
Testing for Outliers and Influential Observations
13.3
Remedial Measures
13.4
Exercise