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C H A P T E R 4 Basic Estimation Techniques 151

The statistical analyses (or, if you wish, econometrics) that we are going to use in this text are really rather simple. Our two major objectives are also simple: 1. We want you to be able to set up a regression equation that could subsequently be estimated by using one of the readily available regression packages. 2. We want you to be able touse the output of a regression to examine those economic issues that are of interest to the manager of an enterprise. Hence, in terms of the field of study known as econometrics, we will concentrate our attention on helping you avoid what are called specification errors. In simple terms, this means that we will show you how to set up an estimation equation that is appropriate for the use to which itis to be put. Specification errors—such as excluding important explanatory variables or using an inappropriate form for the equation—are serious; they can result in the estimates being biased. In addition to specification errors, there are other problems that can be encountered in regression analysis. These problems, which are more difficult than the material we want to cover in this text, arereviewed briefly in the appendix to this chapter. 4.8 SUMMARY
This chapter set forth the basic principles of regression analysis: estimation and testing for statistical significance. The emphasis of the chapter was on explaining how to interpret the results of regression analysis, rather than on the mathematics of regression analysis. A mathematical derivation of the statistical techniques ispresented in the appendix at the end of the chapter. The simple linear regression model relates a dependent variable to a single explanatory variable in a linear fashion: Y a bX. The parameter a is the Y-intercept: the value of Y when X is 0. The parameter b is the slope of the regression line; it measures the rate of change in Y as X changes ( Y/ X). Because the variation in Y is affected not only byvariation in X but also by various random effects, we cannot predict exactly the actual value of Y. Thus you should interpret the regression equation as giving the average or expected value of Y for any particular value of X. Parameter estimates are obtained by choosing values of a and b that minimize the sum of the squared residuals. The residual is the difference between the actual value of ˆ Yand the fitted value of Y (Yi Yi ). This method of estimating a and b is called the method of least-squares. The ˆ ˆ estimated regression line, Y a bX, is called the sample regression line. The sample regression line is an estimate of the true regression line. ˆ ˆ The estimates a and b do not, in general, equal the true ˆ ˆ values of a and b. Because a and b are computed from the data in therandom sample, the estimates themselves are random variables. Statisticians have shown that the distribution of values that the estimates might take is centered around the true value of the parameter. An estimator is unbiased if the mean value of the estimator is equal to the true value of the parameter. The method of least-squares produces unbiased estimates of a and b. It is the randomness of theparameter estimates that necessitates testing for statistical significance. Just because ˆ the estimate b is not equal to 0 does not mean the true value of b is not actually equal to 0. Even when b does equal 0, it is still possible that the sample of Y and X valˆ ues will produce a least-squares estimate b that is different from 0. It is necessary to determine statistically if there is sufficientevidence in the sample to indicate that Y is truly related to X (i.e., b 0). This is called testing for statistical significance. A t-test can be used to test for statistical significance of parameter estimates. To test for statistical significance of an individual parameter estimate, the researcher must first determine the level of significance for the test. The significance level of a test is...