24 The Basics of financial economeTrics
R^2 takes on values in the interval [0,1]. The meaning of R^2 = 0 is that
there is no discernable linear relationship between x and y. No variation
in y is explained by the variation in x. Thus, the linear regression makes
little sense. If R^2 = 1, the fit of the line is perfect. All of the variation in y
is explained by the variation in x. In this case, the line can have either a
positive or negative slope and, in either instance, expresses the linear rela-
tionship between x and y equally well.^6 Then, all points (xi,yi) are located
exactly on the line.^7
As an example, we use the monthly return data from the previous exam-
ple. Employing the parameters b = 1.0575 and a = 0.0093 for the regression
yˆt estimates, we obtain SST = 0.5259, SSR = 0.2670, and SSE = 0.2590. The
R^2 = 0.5076 (0.2670/0.5259). For the weekly fit, we obtain, SST = 0.7620,
SSR = 0.4420, and SSE = 0.3200 while got daily fit we have SST= 0.8305,
SSR = 0.4873, and SSE = 0.3432. The coefficient of determination is R^2 =
0.5800 for weekly and R^2 = 0.5867 for daily.
relationship between Coefficient of Determination
and Correlation Coefficient
Further analysis of the R^2 reveals that the coefficient of determination is just
the squared correlation coefficient, rx,y, of x and y. The consequence of this
equality is that the correlation between x and y is reflected by the goodness-
of-fit of the linear regression. Since any positive real number has a positive
and a negative root with the same absolute value, so does R^2. Hence, the
extreme case of R^2 = 1 is the result of either rx,y = –1 or rx,y = 1. This is
repeating the fact mentioned earlier that the linear model can be increasing
or decreasing in x. The extent of the dependence of y on x is not influenced
by the sign. As stated earlier, the examination of the absolute value of rx,y is
important to assess the usefulness of a linear model.
With our previous example, we would have a perfect linear relationship
between the monthly S&P 500 (i.e., x) and the monthly GE stock returns
(i.e., y), if say, the GE returns were y = 0.0085 + 1.1567x. Then R^2 = 1 since
all residuals would be zero and, hence, the variation in them (i.e., SSE would
be zero, as well).
(^6) The slope has to be different from zero, however, since in that case, there would be
no variation in the y-values. As a consequence, any change in value in x would have
no implication on y.
(^7) In the next chapter we introduce another measure of goodness-of-fit called the
adjusted R2. This measure takes into account not only the number of observations
used to estimate the regression but also the number of independent variables.