This chapter set forth the basic principles of regression analysis: estimation and assessment of statistical significance. We emphasized how to interpret the results of regression analysis, rather than focusing on the mathematics of regression analysis.

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 (ΔY /ΔX ). Parameter estimates can be obtained by using the method of least-squares, leaving the actual computation for computer software packages.

The estimated parameter values _{<a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=jpg::::/sites/dl/free/0073402818/459491/eq2.jpg','popWin', 'width=66,height=96,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif"> (1.0K)</a>} and _{<a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=jpg::::/sites/dl/free/0073402818/459491/eq1.jpg','popWin', 'width=66,height=96,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif"> (1.0K)</a>} do not, in general, equal the true values of a and b. Because _{<a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=jpg::::/sites/dl/free/0073402818/459491/eq2.jpg','popWin', 'width=66,height=96,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif"> (1.0K)</a>} and _{<a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=jpg::::/sites/dl/free/0073402818/459491/eq1.jpg','popWin', 'width=66,height=96,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif"> (1.0K)</a>} are computed from the data in the random sample, the estimates themselves are random variables. The randomness of the parameter estimates necessitates assessing the statistical significance of estimated parameters. Just because the estimate is not equal to 0 does not mean the true value of b is not actually equal to 0. It is necessary to determine statistically whether there is sufficient evidence in the sample to indicate that Y is truly related to X (i.e., b ≠ 0). This is called testing for statistical significance. Two ways to assess the statistical significance of the estimated parameters are by using t-tests and p-values. Most regression software now calculates a p-value for each parameter estimate. The p-value gives the exact (or minimum) level of significance for a parameter estimate.

The R² and F-statistics, both of which are computed by the regression software, can be used to measure how well the sample regression line fits the data. The R² statistic, which measures the fraction of the total variation in Y that is explained by the variation in X, ranges in value from 0 (the regression equation explains none of the variation in Y) to 1 (the regression equation explains all the variation in Y).

Many economic relations of interest to managers are nonlinear in nature. Two types of nonlinear models presented in this chapter are (1) quadratic regression models and (2) log-linear regression models. The quadratic regression model is appropriate when the curve fitting the scatter diagram is either <a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=jpg::::/sites/dl/free/0073402818/459485/union.jpg','popWin', 'width=59,height=79,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif"> (0.0K)</a>-shaped or inverted-<a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=jpg::::/sites/dl/free/0073402818/459485/union.jpg','popWin', 'width=59,height=79,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif"> (0.0K)</a>-shaped. The log-linear model for two explanatory variables takes the form Y = aX^bZ^c. A particularly useful feature of this specification is that the parameters b and c are elasticities.

All the statistics needed for regression analysis are automatically computed when using a computerized regression routine. The purpose of this chapter was to show you how to interpret and use the regression statistics produced by the computer.