Introduction to Econometrics

Chapter 4 : 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 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

Symbols in the LRM

LRM, averages

Unconditional expectation of the error term

Understanding the LRM

LRM: Interpretation of the OLS estimator

Fitted values and residuals in the LRM

LRM, simulation

The difference between the error term and the residual

The statistical formula for b2

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

Unbiasedness and consistency in the LRM

Proving that the slope OLS estimator is unbiased

Proving that the intercept OLS estimator is unbiased

Heteroscedasticity