Comparing Return Estimates with Realized Returns Using the KS Test

Summary

The document asks how to compare a year of asset returns with CAPM return estimates, after using a chi-squared measure. It suggests the two-sample Kolmogorov–Smirnov test, which compares empirical cumulative distribution functions and can detect differences in their location and shape.

The answer provides no worked example, test statistic, or calculation of a p-value; it points to a separate explanation and notes that an R implementation exists. The suggestion concerns whether the two samples have different distributions, rather than directly measuring average forecast error or testing whether each estimate matches its corresponding daily return. The document does not discuss assumptions or adjustments for dependence in time-series data, so those would need separate consideration before applying the test to daily financial observations.

Key ideas

• The two-sample Kolmogorov–Smirnov test compares empirical distributions of two samples.
• It is sensitive to differences in both distribution location and shape.
• A distributional comparison does not directly measure paired daily forecast errors.
• The document gives no implementation details or treatment of time-series dependence.

Tags

# Measure difference between estimations and historic returns


# Measure difference between estimations and historic returns












For every day in a year, I have the return on an asset and the CAPM estimation for the return.

I want to measure the average difference between the set of returns and set of estimations.

So far, I have applied Chi Squared but I want to use other measurements as well.

Is there anything equivalent to Chi_Squared which is normally used for measuring the difference between returns and estimations of a financial model?

## Answer by Richi Wa (score 1)

https://quant.stackexchange.com/a/18225

You can apply the Kolmogorov-Smirnov test. I simply quote from the entry:

"The two-sample K–S test is one of the most useful and general nonparametric methods for comparing two samples, as it is sensitive to differences in both location and shape of the empirical cumulative distribution functions of the two samples."

There is an R-implementation too.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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