Introduction to Econometrics
Chapter 6 : The linear regression model with several explanatory variable
By Lund University
In this chapter, we allow for several explanatory variables. We begin by setting up the linear regression with several explanatory variables including the assumptions that we need to make. As in the simpler model with one explanatory variable, the main focus is on estimating the beta-parameters. However, we will no longer be able to present general formulas, such as the OLS formula for our beta-estimates. To do this, we need matrix algebra which is outside the scope of this course. Instead, we rely on the fact that they have been correctly programmed into software such as Excel, EVies, Stata and more. Once we have fully understood the general linera regression model, we move on to inference.
The linear regression model and OLS
We begin this chapter by extending the linear regression model allowing for several explanatory variables. For example, if we have three explanatory variables then, including the intercept, we will have four unknown beta parameters. We will use the symbol k to denote the number of unknown beta parameters. The OLS principle for estimating the beta parameters will still work but the mathematics will become more complicated and is best done using matrices. However, we can always feed data into software and get the OLS estimates from the software. The fundamental assumption introduced in chapter 3, exogeneity, will be discussed and we will conclude that the OLS estimator will be unbiased and consistent under this assumption. Further, the OLS estimator will be best if the error terms are homoscedastic.
Linear regression with several explanatory variables
OLS
The properties of the OLS estimator
Inference in the linear regression model with several explanatory variables
We begin by looking at the t-test which we use to test a single restriction. In a linear regression model with several explanatory variables it is common to consider hypotheses involving several restrictions. Such hypotheses can be tested using an F test. Finally we look at confidence intervals when we have many explanatory variables.
Hypothesis testing, one restriction – the t-test
Hypothesis testing, several restrictions – the F-test
Confidence intervals in the LRM
Multicollinearity and forecasting
this section contains two unrelated topics. We begin by looking at multicollinearity, a problem where the explanatory variables are highly correlated. Presence of multicollinearity makes it difficult to estimate individual beta parameters. Forecasting will allow us to predict the value of the dependent variable for given values of the explanatory variables even when the observation is not part of our sample.