What does "$R^2$ estimates the combined dispersion against the single dispersion of the observed and predicted series" mean? In a paper I came across the description of $R^2$ as "it estimates the combined dispersion against the single dispersion of the observed and predicted series". I am not able to understand this statement. I understand that $R^2$ is the square of the correlation coefficient of predicted and observed values.
 A: Dispersion broadly refers to how broad or narrow a distribution (either theoretical or observed) stretches. Variance is a common measure of dispersion.
$R^2$ has another interpretation as the percentage of variance explained:

In this form $R^2$ is expressed as the ratio of the explained variance (variance of the model's predictions, which is $SS_{reg} / n$) to the total variance (sample variance of the dependent variable, which is $SS_{tot} / n$).

It seems (without further context) that the authors of your paper are reporting the $R^2$ as a measure of model fit:

$R^2$ is a statistic that will give some information about the goodness of fit of a model. In regression, the $R^2$ coefficient of determination is a statistical measure of how well the regression line approximates the real data points. An $R^2$ of 1 indicates that the regression line perfectly fits the data.

If one takes "combined dispersion" in the quotation to refer to the dispersion of observed values, "combined" referring to both modeled values and residual error not accounted for in the model, and "single dispersion" to refer to the dispersion of the modeled series, then "observed and predicted series" correspond to $SS_{tot}/n$ and $SS_{tot}/n$ above.
