DEAUtils

Utilities for Data Envelopment Analysis (DEA)

This repository will host various utilities that one may need when doing a DEA analysis.

If you use any of these functions in your own work, then please acknowledge my work whenever possible.

Nonparametric kernel density test for testing equality of densities

Currently, it contains a MATLAB implementation of the nonparametric kernel density test of Li et al. (2009) in the function "litest2009". This function can be used as follows:

[h,p,Tn,exitflag,bw] = litest2009(X,Y,alpha,nboot,bwmethod)

where X and Y are i.i.d samples of some dimension p, but possibly with different sample sizes. alpha is the desired significance level (default = 0.05), nboot the number of bootstrap samples (default = 2000) and bwmethod the bandwidth selection method (default = 'lscv'). The null hypothesis states that both densities of samples X and Y are equal. Bandwidth tuning is performed using either least-squares ('lscv') or maximum likelihood ('mlcv') cross-validation. The function returns 5 arguments:

• h the result of the hypothesis test: h = 0 means the null hypothesis cannot be rejected at the given significance level alpha. h = 1 means the null hypothesis is rejected at the given significance level.
• p the p-value obtained by a bootstrapping procedure using nboot bootstrap iterations. See Li et al.(2009) for details.
• Tn the computed test statistic.
• exitflag the exitflag from the optimization of the bandwidth bw.
• bw the bandwidth used in the kernel estimation.

The function might print this warning message during computation while still finishing successfully:

Exiting: Maximum number of function evaluations has been exceeded - increase MaxFunEvals option. Current function value: XXXXX

The warning message originates from optimizing the bandwidth smoothing parameter bw over an unbounded interval. The warning message indicates that this search does not converge fast enough, so the optimization is restarted from the rule-of-thumb bandwidth but now limiting the search interval. The result should still be reliable though. The warning message is printed to give full transparency to the user.

Simar & Zelenyuk (2006) propose a modification of this test for use with DEA efficiency scores. The function "smootheffscorebeforelitest" implements their "Algorithm II". See Simar & Zelenyuk (2006) for details. The function needs to be called before using the above mentioned litest2009 as follows:

TES = smootheffscorebeforelitest(TE,ninputs,noutputs,alpha,effval)

where the arguments are:

• TE: (k x 1) vector of DEA efficiency scores;
• ninputs: number of inputs;
• noutputs: number of outputs;
• alpha: significance level of the Li test (optional, default = 5%);
• effval: value that indicates an efficient DMU (optional, default = 1.0);

The function returns the "smoothed" efficiency scores in the vector TES. The inefficient scores remain untouched and only the efficient scores are "smoothed". These "smoothed" efficiency scores can then be used as arguments X and Y in litest2009.

Note that productivity scores need not be smoothed, because these are not bounded!

Limitations

The litest2009 implementation can only handle continuous data at the moment. Furthermore, the test is for unconditional densities only. If there is any interest then I might implement support for categorical data and conditional densities in the future.

Acknowledgements

I gratefully acknowledge Qianying Jin for reporting strange results in some cases when using the least-squares cross-validation method. This led me to add the maximum likelihood cross-validation as a bandwidth selection method.

References

• Qi Li, Esfandiar Maasoumi & Jeffrey S. Racine (2009) "A nonparametric test for equality of distributions with mixed categorical and continuous data", Journal of Econometrics, 148, 186-200, DOI: http://dx.doi.org/10.1016/j.jeconom.2008.10.007
• Léopold Simar & Valentin Zelenyuk (2006) "On Testing Equality of Distributions of Technical Efficiency Scores", Econometric Reviews, 25:4, 497-522, DOI:10.1080/07474930600972582