80 The Basics of financial economeTrics
the quality of the model with respect to the given data (diagnosis of the
model).■ (^) There are criteria for diagnosing the quality of a model. The tests used
involve statistical tools from inferential statistics. The estimated regres-
sion errors play an important role in these tests and the tests accord-
ingly are based on the three assumptions about the error terms.
■ (^) The first test is for the statistical significance of the multiple coefficient
of determination, which is the ratio of the sum of squares explained by
the regression and the total sum of squares.
■ (^) If the standard deviation of the regression errors from a proposed model
is found to be too large, the fit could be improved by an alternative
specification. Some of the variance of the errors may be attributable to
the variation in some independent variable not considered in the model.
■ (^) An analysis of variance test is used to test for the statistical significance
of the entire model.
■ (^) Because one can artificially increase the original R^2 by including addi-
tional independent variables into the regression, one will not know
the true quality of the model by evaluating the model using the same
data. To deal with this problem, the adjusted goodness-of-fit measure
or adjusted R^2 is used. This measure takes into account the number of
observations as well as the number of independent variables.
■ (^) To test for the statistical significance of individual independent vari-
ables, a t-test is used.
■ (^) To test for the statistical significance of a set or group of independent
variables, an F-test is used.