R Squared Statistics

RATS reports several variations of the \(R^2\) statistic. The centered \(R^2\) (labeled "Centered R^2" in the output) is calculated as:

\(1 - \frac{{{\bf{e'e}}}}{{{\bf{\tilde y'\tilde y}}}}\)

where e is the vector of the residuals across the sample and \({{\bf{\tilde y}}}\) is the vector of deviations from the mean of the dependent variable. It is always less than or equal to 1. However, it may be less than 0 in the following situations:

1.Model estimated does not include an intercept.

2.Model is estimated by instrumental variables.

3.Model is estimated by seemingly unrelated regressions.

The 2nd and 3rd cases can produce negative values even with an intercept in the regression because the estimation method does not attempt to minimize the sum of squared residuals. Note that RATS only computes and displays \(R^2\) when the objective function is least squares. Although people sometimes report \(R^2\) when estimating a model by (for instance) instrumental variables, this is really not good practice.

The adjusted \(R^2\) (labeled "R-Bar^2" in the output) is \(R^2\) adjusted for degrees of freedom. This is computed as

\({\bar R^2} = 1 - \left( {\frac{{{\bf{e'e}}/\left( {T - K} \right)}}{{{\bf{\tilde y'\tilde y}}/\left( {T - 1} \right)}}} \right)\)

where \(T\) is the number of observations and \(K\) the number of regressors (including the CONSTANT if present). If the regression isn't full rank (due to restrictions, or because you included perfectly collinear variables), the \(K\) is adjusted to match the rank of the regressors. The adjusted \(R^2\) can be negative in any type of model.