Introduction to Econometrics

Chapter 3 : The linear regression model with one explanatory variable

By Lund University

This chapter formalizes the most important model in econometrics, the linear regression model. The entire chapter is restricted to a special case, nameley when you have only one explanatory variable. The key assumtion of the linear regression model, exogeneity is introduced. Then, the OLS formula from chapter 1 is reinterpreted as an estimator of unknown parameters in the linear regression model. This chapter also introduces the variance of the OLS estimator under an important set of assumptions, the Gauss-Markov assumptions. The chapter concludes with inference in the linear regression model, specifically discussing hypothesis testing and confidence intervals.

The linear regression model

The first section of this chapter is devoted to describing the linear regression model. In the linear regression model a dependent variable is explained partly as a linear function of an explanatory variable and partly by an error term. The parameters of the linear function, the beta parameters, are viewed as unknown. If we estimate these parameters using the OLS formula then we have what is called the OLS estimator.

The linear regression model (LRM)

LRM with an exogenous explanatory variable

The OLS estimator

The properties of the OLS estimator

in order to evaluate the usefulness of an estimator, we introduced to properties held by good estimators, unbiasedness and consistency. It turns out that the OLS estimator has favorable properties as long as the explanatory variable is exogenous. However, it is possible to find many different estimators with these properties. Therefore, we need some method of distinguishing between them. To do that, we begin by assuming that the error terms are homoscedastic, that is, they all have the same variance. With this assumption, we can find the variance of the OLS estimators and show that the OLS estimator has the lowest variance among all linear unbiased estimators. This result is called the Gauss Markov theorem.

When are the OLS estimators unbiased and consistent?

Homoscedasticity, heteroscedasticity and the Gauss-Markov assumptions

The variance of the OLS estimators

Estimating σ2

Estimating the variance of the OLS estimators

The Gauss-Markov theorem

Some distributions

In order to prepare for the final section of this chapter, the section dealing with inference in the linear regression model, we need to discuss a couple of families of random variables. Specifically, we will look at the chi-square distribution, the t-distribution, and the F distribution and we will also look at the concept critical value.

The chi-square distribution

The t-distribution

The F-distribution

Critical values

Inference in the linear regression model

this section is an introduction to inference in the linear regression model. We will begin by looking at hypothesis testing as a general idea in statistics followed by hypothesis testing in the linear regression model. In this section we will only look at the t-test where we test if one of the unknown parameters is equal to some given value. Hypothesis testing is closely related to confidence intervals and we will look at confidence intervals for the beta parameters of the linear regression model.

Introduction to hypothesis testing

Hypothesis testing in the LRM: The t-test

Confidence intervals in the LRM