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7 Appendix: Test Regressors in Arti cial Regressions

# 7 Appendix: Test Regressors in Arti cial Regressions

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Chapter 1

Regression Models

1.1 Introduction

Regression models form the core of the discipline of econometrics. Although

econometricians routinely estimate a wide variety of statistical models, using

many different types of data, the vast majority of these are either regression

models or close relatives of them. In this chapter, we introduce the concept of

a regression model, discuss several varieties of them, and introduce the estimation method that is most commonly used with regression models, namely, least

squares. This estimation method is derived by using the method of moments,

which is a very general principle of estimation that has many applications in

econometrics.

The most elementary type of regression model is the simple linear regression

model, which can be expressed by the following equation:

yt = β1 + β2 Xt + ut .

(1.01)

The subscript t is used to index the observations of a sample. The total number of observations, also called the sample size, will be denoted by n. Thus,

for a sample of size n, the subscript t runs from 1 to n. Each observation

comprises an observation on a dependent variable, written as yt for observation t, and an observation on a single explanatory variable, or independent

variable, written as Xt .

The relation (1.01) links the observations on the dependent and the explanatory variables for each observation in terms of two unknown parameters, β1

and β2 , and an unobserved error term, ut . Thus, of the five quantities that

appear in (1.01), two, yt and Xt , are observed, and three, β1 , β2 , and ut , are

not. Three of them, yt , Xt , and ut , are specific to observation t, while the

other two, the parameters, are common to all n observations.

Here is a simple example of how a regression model like (1.01) could arise in

economics. Suppose that the index t is a time index, as the notation suggests.

Each value of t could represent a year, for instance. Then yt could be household consumption as measured in year t, and Xt could be measured disposable

income of households in the same year. In that case, (1.01) would represent

what in elementary macroeconomics is called a consumption function.

Copyright c 1999, Russell Davidson and James G. MacKinnon

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7 Appendix: Test Regressors in Arti cial Regressions

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