A Kolmogorov-Smirnov Based Test for Comparing the Predictive Accuracy of Two Sets of Forecasts
Abstract
:1. Introduction
2. Theoretical Foundation
2.1. The Kolmogorov-Smirnov (KS) Test
2.2. Testing for Statistically Significant Differences between the Distribution of Two Sets of Forecast Errors
2.3. Testing for the Lower Stochastic Error
3. Simulation Results
3.1. Size of the Test
h | Error Distribution | Test | n = 8 | n = 16 | n = 32 | n = 64 | n = 128 | n = 256 | n = 512 |
---|---|---|---|---|---|---|---|---|---|
1 | Gaussian | DM | 8.4 | 9.6 | 9.7 | 10.1 | 9.9 | 10.4 | 10.6 |
Gaussian | KSPA | 8.6 | 9.4 | 8.9 | 9.6 | 8.4 | 9.4 | 8.6 | |
Uniform | KSPA | 9.1 | 8.9 | 8.6 | 9.4 | 8.9 | 8.9 | 8.5 | |
Cauchy | KSPA | 9.0 | 9.1 | 8.4 | 9.2 | 8.5 | 8.9 | 8.6 | |
Student’s t | KSPA | 8.5 | 9.4 | 9.3 | 9.5 | 9.0 | 8.7 | 8.6 | |
2 | Gaussian | DM | 16.4 | 14.2 | 12.2 | 11.2 | 10.8 | 10.5 | 10.3 |
Gaussian | KSPA | 9.0 | 9.5 | 8.5 | 9.2 | 8.6 | 9.1 | 8.4 | |
Uniform | KSPA | 9.1 | 9.4 | 8.9 | 9.8 | 8.8 | 9.2 | 8.8 | |
Cauchy | KSPA | 9.3 | 9.5 | 9.0 | 9.3 | 8.8 | 9.4 | 9.0 | |
Student’s t | KSPA | 8.7 | 9.3 | 9.1 | 9.1 | 8.4 | 9.7 | 8.9 | |
3 | Gaussian | DM | 18.1 | 18.5 | 14.3 | 12.2 | 10.7 | 10.8 | 10.9 |
Gaussian | KSPA | 8.6 | 9.6 | 8.7 | 9.2 | 8.7 | 9.1 | 9.1 | |
Uniform | KSPA | 8.7 | 9.8 | 9.0 | 9.2 | 8.6 | 9.4 | 8.7 | |
Cauchy | KSPA | 8.4 | 9.4 | 9.3 | 9.7 | 8.7 | 9.5 | 8.7 | |
Student’s t | KSPA | 8.2 | 9.7 | 8.8 | 9.5 | 8.9 | 9.1 | 8.6 | |
4 | Gaussian | DM | 16.3 | 19.8 | 16.1 | 13.4 | 11.5 | 10.9 | 11.0 |
Gaussian | KSPA | 8.5 | 9.4 | 8.3 | 8.9 | 8.6 | 9.2 | 9.0 | |
Uniform | KSPA | 8.7 | 9.6 | 8.6 | 9.2 | 9.4 | 9.6 | 9.1 | |
Cauchy | KSPA | 8.4 | 9.4 | 9.0 | 9.4 | 9.6 | 9.7 | 8.7 | |
Student’s t | KSPA | 8.7 | 9.1 | 8.8 | 9.9 | 8.7 | 9.7 | 8.8 | |
5 | Gaussian | DM | 12.9 | 19.9 | 17.8 | 14.9 | 12.2 | 11.1 | 11.0 |
Gaussian | KSPA | 8.4 | 9.4 | 8.9 | 9.4 | 8.3 | 9.7 | 8.3 | |
Uniform | KSPA | 8.2 | 9.2 | 8.7 | 9.1 | 8.4 | 9.3 | 8.9 | |
Cauchy | KSPA | 8.8 | 9.6 | 8.5 | 9.5 | 9.0 | 8.8 | 8.9 | |
Student’s t | KSPA | 8.4 | 9.3 | 9.1 | 9.9 | 9.1 | 9.6 | 8.6 | |
6 | Gaussian | DM | 10.6 | 19.8 | 18.8 | 16.0 | 12.9 | 11.4 | 11.2 |
Gaussian | KSPA | 8.6 | 9.5 | 8.9 | 9.5 | 8.6 | 9.1 | 9.0 | |
Uniform | KSPA | 8.7 | 9.4 | 8.8 | 9.1 | 8.4 | 9.2 | 8.3 | |
Cauchy | KSPA | 8.9 | 9.8 | 9.1 | 9.9 | 8.5 | 9.2 | 8.6 | |
Student’s t | KSPA | 8.7 | 9.3 | 8.8 | 9.4 | 9.0 | 9.8 | 9.1 | |
7 | Gaussian | DM | 9.9 | 18.2 | 19.5 | 16.8 | 13.6 | 11.6 | 11.4 |
Gaussian | KSPA | 8.6 | 9.5 | 9.3 | 8.9 | 8.8 | 9.3 | 9.0 | |
Uniform | KSPA | 8.4 | 9.0 | 8.7 | 9.9 | 9.0 | 9.1 | 8.7 | |
Cauchy | KSPA | 8.5 | 9.2 | 8.7 | 9.1 | 9.0 | 9.4 | 8.9 | |
Student’s t | KSPA | 8.8 | 9.1 | 9.0 | 9.0 | 8.6 | 8.8 | 9.2 | |
8 | Gaussian | DM | - | 17.4 | 20.2 | 18.0 | 13.8 | 11.9 | 11.4 |
Gaussian | KSPA | - | 9.3 | 8.6 | 9.1 | 8.5 | 9.5 | 8.7 | |
Uniform | KSPA | - | 9.5 | 8.7 | 9.8 | 9.0 | 9.7 | 8.7 | |
Cauchy | KSPA | - | 9.5 | 8.3 | 9.2 | 8.8 | 8.9 | 8.9 | |
Student’s t | KSPA | - | 9.7 | 8.3 | 9.6 | 8.6 | 9.1 | 9.1 | |
9 | Gaussian | DM | - | 15.1 | 20.2 | 19.0 | 14.7 | 12.4 | 11.6 |
Gaussian | KSPA | - | 9.5 | 8.6 | 9.2 | 8.5 | 9.4 | 8.8 | |
Uniform | KSPA | - | 9.4 | 9.0 | 9.7 | 8.0 | 9.5 | 8.9 | |
Cauchy | KSPA | - | 9.8 | 8.6 | 8.9 | 8.6 | 9.4 | 8.8 | |
Student’s t | KSPA | - | 9.1 | 8.6 | 9.2 | 8.9 | 9.6 | 9.0 | |
10 | Gaussian | DM | - | 14.0 | 20.2 | 19.1 | 15.1 | 12.6 | 11.8 |
Gaussian | KSPA | - | 9.2 | 8.9 | 9.3 | 8.7 | 9.7 | 9.0 | |
Uniform | KSPA | - | 9.2 | 8.7 | 9.8 | 8.7 | 9.1 | 9.4 | |
Cauchy | KSPA | - | 9.2 | 8.8 | 9.7 | 9.1 | 9.5 | 9.3 | |
Student’s t | KSPA | - | 9.3 | 8.8 | 9.0 | 8.7 | 9.1 | 8.6 |
3.2. Power of the Test
Combinations | Test | n = 8 | n = 16 | n = 32 | n = 64 | n = 128 | n = 256 | n = 512 |
---|---|---|---|---|---|---|---|---|
Case 1 | DM | 7.3 | 17.5 | 31.9 | 37.3 | 39.3 | 40.3 | 40.9 |
KSPA | 19.6 | 35.8 | 61.0 | 91.7 | 99.9 | 100.0 | 100.0 | |
Case 2 | DM | 5.2 | 13.4 | 26.5 | 35.4 | 39.5 | 41.0 | 40.8 |
KSPA | 15.9 | 25.8 | 42.0 | 75.3 | 97.6 | 100.0 | 100.0 | |
Case 3 | DM | 59.3 | 96.0 | 99.7 | 100.0 | 100.0 | 100.0 | 100.0 |
KSPA | 65.1 | 92.0 | 100.0 | 100.0 | 100.0 | 100.0 | 100.0 | |
Case 4 | DM | 91.6 | 99.7 | 100.0 | 100.0 | 100.0 | 100.0 | 100.0 |
KSPA | 97.3 | 100.0 | 100.0 | 100.0 | 100.0 | 100.0 | 100.0 |
4. Empirical Evidence
4.1. Scenario 1: Tourism Series
Test | Two-Sided (p-Value) | One-Sided (p-Value) |
---|---|---|
Modified DM | <0.01 * | N/A |
KSPA | <0.01 * | <0.01 * |
4.2. Scenario 2: Accidental Deaths Series
Test | Two-Sided (p-Value) | One-Sided (p-Value) |
---|---|---|
DM | 0.04 * | N/A |
Modified DM | N/A | N/A |
KSPA | 0.03 * | 0.02 * |
4.3. Scenario 3: Trade Series
Test | Two-Sided (p-Value) | One-Sided (p-Value) |
---|---|---|
Modified DM | 0.30 | N/A |
KSPA | 0.17 | 0.08 * |
5. Conclusions
Supplementary Materials
Supplementary File 1Acknowledgments
Author Contributions
Conflicts of Interest
Appendix: R Code for the KSPA Test
# Install and load the "stats" package in R. install.packages("stats") library(stats) # Input the forecast errors from two models. Let Error1 show errors from the model with the lower error based on some loss function. Error1<-scan() Error2<-scan() # Convert the raw forecast errors into absolute values or squared values depending on the loss function. abs1<-abs(Error1) abs2<-abs(Error2) sqe1<-(Error1)^2 sqe2<-(Error2)^2 # Perform the KSPA tests for distinguishing between the predictive accuracy of forecasts from the two models*. # Two-sided KSPA test: ks.test(abs1,abs2) # One-sided KSPA test: ks.test(abs1,abs2, alternative = c("greater")) OPTIONAL GRAPHS FOR MORE INFORMATION # Draw histograms for the forecast errors from each model. par(mfrow=c(1,2)) hist(abs1, xlab="Model 1 Absolute Errors", main="") hist(abs2, xlab="Model 2 Absolute Errors",main="") # Plot the cdf of forecast errors from each model*. plot(ecdf(abs1),do.points=T,col="red",xlim=range(abs1,abs2),main="") plot(ecdf(abs2),do.points=T,col="blue",add=TRUE, main="") legend("bottomright",legend=c("Model 1 Absolute Errors","Model 2 Absolute Errors"), lty=1, col=c("red","blue")) #NOTE: *Replace abs1 and abs2 with sqe1 and sqe2 as appropriate.
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Hassani, H.; Silva, E.S. A Kolmogorov-Smirnov Based Test for Comparing the Predictive Accuracy of Two Sets of Forecasts. Econometrics 2015, 3, 590-609. https://0-doi-org.brum.beds.ac.uk/10.3390/econometrics3030590
Hassani H, Silva ES. A Kolmogorov-Smirnov Based Test for Comparing the Predictive Accuracy of Two Sets of Forecasts. Econometrics. 2015; 3(3):590-609. https://0-doi-org.brum.beds.ac.uk/10.3390/econometrics3030590
Chicago/Turabian StyleHassani, Hossein, and Emmanuel Sirimal Silva. 2015. "A Kolmogorov-Smirnov Based Test for Comparing the Predictive Accuracy of Two Sets of Forecasts" Econometrics 3, no. 3: 590-609. https://0-doi-org.brum.beds.ac.uk/10.3390/econometrics3030590