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Chapter 13: Regression Analysis

Test Statistics

Although the least squares estimators of the true parameters of economic relationships are frequently "best" estimates they are not necessarily good estimates. Very often economic data is so heavily affected by errors and the modeling process so uncertain that the regression results cannot be relied upon. Thus it is imperative to have some means of appraising the accuracy and reliability of the least squares estimators. This function is performed by the regression test statistics. The following is a listing with brief descriptions of some of the more important of the test statistics.

1) The coefficient of determination (R²) measures "goodness of fit", the proportion of variance in Y that is explained by the regression equation.

R² is a measure of the overall explanatory power of the regression equation. It ranges in value from 0 to 1. Low values suggest the omission of important causal variables raising the possibility that the coefficients of the included variables may be biased. M

2) The Standard Error of Estimate,

SEE is the standard deviation of the errors, corrected for degrees of freedom, and is used to determine the probability of errors of any given size. Note that degrees of freedom = n - # of variables(v) - 1.

3) The standard errors of the regression coefficients â and are written as and respectively, where

The t statistic is used to determine the probability of obtaining the value of observed purely by chance when the true value of b is smaller than or equal to any given value b* (which may be set equal to zero) for degrees of freedom less than 30. Computer software for running regressions and computing the test statistics described above is widely available. The spreadsheet program Lotus 123 has this capability (select the command sequence Data, Regression) as do statistical packages such as Minitab, SSPS, TSP, and SAS. The last two have the extended capabilities discussed below in the section on Econometric Modeling.

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