Measuring Yield Curve Forecast Error with Integrated Squared Error

Summary

The document considers how to evaluate forecasts of an entire yield curve when the usual pointwise R-squared measure does not directly fit the task. The answer proposes extending mean squared error to curves by integrating the squared difference between the predicted and observed yields over the tenor range, then dividing by the maximum tenor. This is the mean squared L2 distance between the two functions, treating each point along the curve as part of an overall forecast error.

The response gives a mathematical definition but does not explain how this error relates to explained variance or how to construct an R-squared-like statistic for curves. It also leaves implementation choices open, including tenor weighting, discretization, units, and whether different maturities should have equal importance. The suggested loss offers a direct curve-wide error measure, but the excerpt provides no empirical comparison or guidance on choosing an evaluation metric for a particular forecasting objective.

Key ideas

• A forecast of a yield curve can be compared with the observed curve using integrated squared error.
• The proposed metric averages squared forecast deviations over the tenor interval.
• The measure is a curve-wide analogue of mean squared error, not an R-squared calculation.
• Tenor weighting and discretization choices are not specified.

Tags

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


# $R^{2}$ 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).

## Answer by Brian B (score 3, accepted)

https://quant.stackexchange.com/a/36892

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,

$$ MSE = \frac{1}{T}\int_0^T \left|p(t) - y(t)\right|^2 dt $$

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.