The term R-squared refers to the fraction of variance explained by a model, but--what
is the relevant variance that demands explanation? We have seen by now that there
are many transformations that may be applied to a variable before it is used as a
dependent variable in a regression model: deflation, logging, seasonal adjustment,
differencing. All of these transformations will change the variance and may also
change the units in which variance is measured. Deflation and logging may
dramatically change the units of measurement, while seasonal adjustment and
differencing generally reduce the variance significantly when properly applied.
Therefore, if the dependent variable in the regression model has already been
transformed in some way, it is possible that much of the variance has already been
"explained" merely by the choice of an appropriate transformation. Seasonal
adjustment obviously tries to explain the seasonal component of the original variance,
while differencing tries to explain changes in the local mean of the series over time.
With respect to which variance should R-squared be measured--that of the original
series, the deflated series, the seasonally adjusted series, and/or the differenced series?
This question does not always have a clear-cut answer, and as we will see below,
there are usually several reference points that may be of interest in any particular case.