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2024 | OriginalPaper | Chapter

3. The Multiple Regression Model

Author : Valérie Mignon

Published in: Principles of Econometrics

Publisher: Springer Nature Switzerland

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Abstract

Regression analysis consists in studying the dependence of a variable (the explained variable) on one or more other variables (the explanatory variables). This chapter presents the multiple regression model, which is a linear model comprising a single equation linking an explained variable to several explanatory variables. Since the parameters of this model are unknown, they must be estimated to quantify the relationship between the dependent and the explanatory variables. The chapter presents the most frequently used estimation method, i.e., the ordinary least squares (OLS) method. It also establishes the properties of the OLS estimators, describes the various tests on the regression coefficients, and presents key indicators such as the (adjusted) coefficient of determination. All the concepts are illustrated thanks to several empirical applications.

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previous chapter The Simple Regression Model
next chapter Heteroskedasticity and Autocorrelation of Errors
Appendix
Available only for authorised users
Footnotes
1
This chapter calls upon various notions of matrix algebra. In Appendix 3.1, readers will find the elements of matrix algebra necessary to understand the various developments here.
 
2
Matrices and vectors are written in bold characters. This notation convention will be used throughout the book.
 
3
In the sense that the matrix of explanatory variables is assumed to be unchanged whatever the sample of observations.
 
4
We will see later that such an assumption implies that there is no collinearity between the explanatory variables.
 
5
We consider the \(\left ( i+1\right )\)th element and not the ith since the first element of the matrix \(\left ( \boldsymbol {X}^{\prime }\boldsymbol {X}\right )^{-1}\) relates to the constant term. Obviously, if the variables are centered, it is appropriate to choose the ith element.
 
6
Recall that the first element of the matrix corresponds to the constant term, which explains why we consider the \(\left ( i+1\right )\)th element and not the ith.
 
7
The first element of the matrix corresponding to the constant \(\alpha .\)
 
8
Strictly speaking, this is known as Kullback-Leibler information (see Kullback and Leibler, 1951).
 
9
See also Akaike (1969, 1974).
 
10
In this case, the OLS and maximum likelihood estimators are equivalent.
 
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Metadata
Title
The Multiple Regression Model
Author
Valérie Mignon
Copyright Year
2024
Book
Principles of Econometrics

Print ISBN: 978-3-031-52534-6

Electronic ISBN: 978-3-031-52535-3

Copyright Year: 2024

DOI
https://doi.org/10.1007/978-3-031-52535-3_3

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