# $R^{2}$ Measure for Functions (Yield Curves)

• I am used to applying $R^{2}$ (relative explained variance) as a measure for point estimates.
• I am now confronted with forecasting the whole of the yield curve and would like to see what fraction of variance in the actual market yield curves is explained by the forecast model. I am slightly unsure how to proceed as I am no longer dealing with a single point, but rather the whole curve (or, if discretised, a collection of points).

I'm not sure you are truly asking the right questions about this project, but the concept of mean squared error is easily extended to continuous curves. Let's say our maximum yield curve tenor is $T$, that we have a prediction $p(t)$ and a true value $y(t)$. Then we define our mean squared error as the mean $L^2$ distance,
$$MSE = \frac{1}{T}\int_0^T \left|p(t) - y(t)\right|^2 dt$$
• Would it not be more correct to say "we define our root mean squared error as the $L^{2}$ distance?" – A.L. Verminburger Nov 15 '17 at 9:29