2) Basic Ideas of Linear Regression: The Two-Variable Model
In this chapter we introduced some fundamental ideas of regression analysis. Starting with the key concept of the population regression function (PRF), we developed the concept of linear PRF. This book is primarily concerned with linear PRFs, that is, regressions that are linear in the parameters regardless of whether or not they are linear in the variables. We then introduced the idea of the stochastic PRF and discussed in detail the nature and role of the stochastic error term u. PRF is, of course, a theoretical or idealized construct because, in practice, all we have is a sample(s) from some population. This necessitated the discussion of the sample regression function (SRF). We then considered the question of how we actually go about obtaining the SRF. Here we discussed the popular method of ordinary least squares (OLS) and presented the appropriate formulas to estimate the parameters of the PRF. We illustrated the OLS method with a fully worked-out numerical example as well as with several practical examples. Our next task is to find out how good the SRF obtained by OLS is as an estimator of the true PRF. We undertake this important task in Chapter 3.
3) The Two-Variable Model: Hypothesis Testing
In Chapter 2 we showed how to estimate the parameters of the two-variable linear regression model. In this chapter we showed how the estimated model can be used for the purpose of drawing inferences about the true population regression model. Although the two-variable model is the simplest possible linear regression model, the ideas introduced in these two chapters are the foundation of the more involved multiple regression models that we will discuss in ensuing chapters. As we will see, in many ways the multiple regression model is a straightforward extension of the two-variable model.
4) Multiple Regression: Estimation and Hypothesis Testing
In this chapter we considered the simplest of the multiple regression models, namely, the three-variable linear regression model—one dependent variable and two explanatory variables. Although in many ways a straightforward extension of the two-variable linear regression model, the three-variable model introduced several new concepts, such as partial regression coefficients, adjusted and unadjusted multiple coefficient of determination, and multicollinearity. Insofar as estimation of the parameters of the multiple regression coefficients is concerned, we still worked within the framework of the classical linear regression model and used the method of ordinary least squares (OLS). The OLS estimators of multiple regression, like the two-variable model, possess several desirable statistical properties summed up in the Gauss-Markov property of best linear unbiased estimators (BLUE). With the assumption that the disturbance term follows the normal distribution with zero mean and constant variance σ2, we saw that, as in the two-variable case, each estimated coefficient in the multiple regression follows the normal distribution with a mean equal to the true population value and the variances given by the formulas developed in the text. Unfortunately, in practice, σ2 is not known and has to be estimated. The OLS estimator of this unknown variance is . But if we replace σ2 by , then, as in the two-variable case, each estimated coefficient of the multiple regression follows the t distribution, not the normal distribution. The knowledge that each multiple regression coefficient follows the t distribution with d.f. equal to (n – k), where k is the number of parameters estimated (including the intercept), means we can use the t distribution to test statistical hypotheses about each multiple regression coefficient individually. This can be done on the basis of either the t test of significance or the confidence interval based on the t distribution. In this respect, the multiple regression model does not differ much from the...
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