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Quantile Regression ISQS 5349 – Regression Analysis Spring 2014

Quantile Regression ISQS 5349 – Regression Analysis Spring 2014. Laurie Corradino Daniela Sanchez March 13, 2014. What is Quantile Regression?.

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Quantile Regression ISQS 5349 – Regression Analysis Spring 2014

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  1. Quantile RegressionISQS 5349 – Regression Analysis Spring 2014 Laurie Corradino Daniela Sanchez March 13, 2014

  2. What is Quantile Regression? • A form of regression analysis designed to estimate models for the conditional median or other conditional quantile functions of the predictor variable (Y) against the covariates (X’s). • Different slopes/rates of change (β’s) for different quantilesof the response variable (Y) distribution.

  3. Background • Boscovich proposed median regression in the 18th century. • Laplace and Edgeworth further investigated that idea. • Mosteller and Tukey (1977) first stated that functions could be fitted to describe parts of the response variable (y) distribution aside from simply the mean of the distribution. • Quantile regression (other than median) is the work of Roger Koenker and Gilbert Bassett (1978) – University of Illinois.

  4. What is a Quantile?

  5. OLS vs. Quantile Regression (Hao and Naiman, 2007; Koenker, 2000)

  6. OLS vs. Quantile Regression (Cade and Noon, 2003; Haoand Naiman, 2007)

  7. Quantile Regression Graph adapted from Fitzenberger (2012)

  8. Quantile Regression

  9. Quantile Regression – March Madness Example

  10. March Madness Example Continued • Why Quantile Regression? • Teams’ consistencies (different variances). • Teams’ performance non-symmetric (non-normal distributions). • Very high and low scoring games occur (outliers). • Predictions for certain gambling opportunities may necessitate predictions for parts of the score distribution aside from the mean. • Caveats later controlled for: • Positive/negative momentum (correlated/dependent errors). • Single game scores for both teams usually similar (dependent errors).

  11. March Madness Example Implementation • Data on 2,940 games for 232 Division I NCAA teams • 199 quantiles calculated for each team • Using past data, score predictions made for each pair of teams in the tournament at each of the 199 quantiles Note: this model assumes independence of errors which is unlikely in reality. More in-depth analysis using more advanced statistical and quantile regression techniques and survival analysis are used in the paper to deal with such issues.

  12. R-Code (He and Wei, 2005); Quantile Regression - R

  13. Comparison of More Common Algorithm Methods (He and Wei, 2005); Quantile Regression – R; Susmel

  14. Methods of Calculating Standard Errors Summary.rq(object, se=“ ”…) or Summary(object,se=“ ”…) For a discussion of the methods and their relative advantages/disadvantages see http://www.econ.uiuc.edu/~roger/research/rqci/rqci.pdf (He and Wei, 2005); Quantile Regression – R; Susmel

  15. Other Quantile Regression Applications • Applications • Engineering: Building energy consumption vs. temperature/weather and varying levels of end uses (NREL) - Henzeet al. (2014) • Upper and lower control limits desired • Marketing: Tourist spending patterns vs. various spending stimuli (e.g. length of stay, job type, gender, age, etc.) - Lew and Ng (2012) • Market segmentation desired • Accounting/Finance: - Earnings vs. firm size, financial leverage, and R&D expenditures - Li and Wang (2011) • Prior research inconclusive regarding effect of factors on earnings

  16. On a Practical Note • Is CEO total compensation associated with firm size? • I examine CEO Total Compensation as a function of Total Assets. • Y = CEO Total Compensation S&P1500 firms • X = Total Assets (size proxy) • Merged 2012 data downloaded from COMPUSTAT and EXECUCOMP. • Total Compensation data is in thousands • Total Assets data is in millions

  17. Quantile Regression (Koenker and Hallock, 2001)

  18. Quantile Regression: tau = .50 Intercept tau = .50 Centercept tau = .50 • The intercept is a centercept and estimates the quantile function of Total CEO Compensation conditional on mean Total Assets at each particular quantile.

  19. Interpreting Coefficients? • The same way as ordinary regression coefficients. • The total asset quantile coefficients are positively associated with total compensation.

  20. Conclusions

  21. References • Cade, B. S., & Noon, B. R. (2003). A gentle introduction to quantile regression for ecologists. Frontiers in Ecology and the Environment, 1(8), 412-420. http://www.fort.usgs.gov/products/publications/21137/21137.pdf • Fitzenberger, Bernd (2012). Quantile Regression. Universität Linz. http://www.econ.jku.at/members%5CDerntl%5Cfiles%5CPHD%5CFitzenberger_QuantileRegression.pdf • Hao, L., & Naiman, D. Q. (2007). Quantile regression (No. 149). Sage. http://www.sagepub.com/upm-data/14855_Chapter3.pdf • He, X., & Wei, W. (2005). Tutorial on Quantile Regression. Cached page: http://webcache.googleusercontent.com/search?q=cache:-IugoWaFOXoJ:epi.univ-paris1.fr/servlet/com.univ.collaboratif.utils.LectureFichiergw%3FID_FICHE%3D27872%26OBJET%3D0008%26ID_FICHIER%3D83379+&cd=1&hl=en&ct=clnk&gl=us • Koenker, R., & Bassett Jr, G. (1978). Regression quantiles. Econometrica: Journal of the Econometric Society, 33-50. • Koenker, R. W. (2000). Quantile Regression, article prepared for the statistics section of the International Encyclopedia of the Social Sciences. University of Illinois: Urbana-Champaign, IL. http://www.econ.uiuc.edu/~roger/research/rq/rq.pdf • Koenker, R., & Hallock, K. (2001). Quantile regression. Journal of Economic Perspectives, 15(4), 143-156. http://www.econ.uiuc.edu/~roger/research/rq/QRJEP.pdf • Koenker, R., & Bassett Jr, G. W. (2010). March Madness, Quantile Regression Bracketology, and the Hayek Hypothesis. Journal of Business & Economic Statistics, 28(1). http://www.econ.uiuc.edu/~roger/research/bracketology/MM.pdf • Koenker, R. (2011). “Quantile Regression: A Gentle Introduction.” University of Illinois Urbana- Champaign. http://www.econ.uiuc.edu/~roger/courses/RMetrics/L1.pdf • Quantile Regression – R Documentation for Package ‘quantreg’ version 4.30. http://svitsrv25.epfl.ch/R-doc/library/quantreg/html/rq.html • Susmel, Rauli. “Lecture 10 Robust and Quantile Regression.” Bauer College of Business University of Houston. http://www.bauer.uh.edu/rsusmel/phd/ec1-25.pdf

  22. References for Noted Discipline-Specific Applications • Henze, G. P., Pless, S., Petersen, A., Long, N., & Scambos, A. T. (2014). Control Limits for Building Energy End Use Based on Engineering Judgment, Frequency Analysis, and Quantile Regression. http://www.nrel.gov/docs/fy14osti/60020.pdf • Lew, A. A., & Ng, P. T. (2012). Using quantile regression to understand visitor spending. Journal of Travel Research, 51(3), 278-288. http://jtr.sagepub.com.lib-e2.lib.ttu.edu/content/51/3/278.full.pdf+html • Li, M., & Hwang, N. (2011). Effects of Firm Size, Financial Leverage and R&D Expenditures on Firm Earnings: An Analysis Using Quantile Regression Approach. Abacus, 47(2), 182-204. doi:10.1111/j.1467-6281.2011.00338.x http://eds.a.ebscohost.com.lib-e2.lib.ttu.edu/ehost/pdfviewer/pdfviewer?sid=91bf3ebd-6f4d-42dd-bb3b-e4818335144b%40sessionmgr4005&vid=2&hid=4110

  23. Questions? Thank You!

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