# Quantile Regression

## Applications on Experimental and Cross Section Data Using EViews

I Gusti Ngurah Agung

The Ary Suta Center

Jakarta, Indonesia

Dedicated to my wife Anak Agung Alit Mas, our children Martiningsih, Ratnaningsing, and Darma Putra, as well as all our generation

# Preface

This book presents various Quantile Regressions (QR), based on the cross‐section and experimental data. It has been found that the equation specification or the estimation equation of the LS‐Regression or Mean‐Regression (MR) can be applied directly for the Quantile‐Regression. Hence, this book can be considered as an extension or a modification of all Mean‐Regressions presented in all books, and papers, such as Agung (2008, 2009b, 2011a,b), Gujarati (2003), Wooldridge (2002), Huitema (1980), Kementa (1980), and Neter and Wasserman (1974).

In addition, compare to the Mean‐Regression (MR), the Quantile‐Regression is a robust regression having critical advantages over the MR, mainly for the robustness to outliers, no normal distribution assumption, and it can present more complete distribution of the objective, criterion or dependent random variable, using the linear programing estimation method (Davino et al. 2014; Koenker 2005; Koenker and Hallock 2001), As a more detail comparison between the Conditional QR and the Condional MR, see Appendix A.

The models presented in this book in fact are the extension or modification of all mean‐regression presented in my second book: “Cross Section and Experimental Data Analysis Using EViews” (Agung 2011a). For this reason, it is recommended the readers to use also the models in the book to conduct the quantile‐regression analysis, using their own data sets.

This book contains ten chapters.

Chapter 1 presents the applications of the test for medians of any response or criterion variable Yi, by series/group of categorical variables, numerical categorical or the ranks of a numerical variable.

Chapter 2 presents the applications of One‐Way and Two‐Way ANOVA Quantile‐Regressions. In addition, the application of the object Quantile Process having three alternative options are introduced. As the modification of the parametric DID (Difference‐In‐Differences) of the means of any variable Yi by two factors or categorical variables, this chapter presents the DID of the Quantile(τ) of any variable Yi, by two categorical variables, starting with two dichotomous variables or 2 × 2 factorial QR. It has been well known that the value of a DID is representing the two‐way interaction effect of the corresponding two factors, indicating the effect of a factor on the criterion variable Yi depends on the other factor. Then a special 2 × 3 factorial QR to demonstrate how to compute its DID.

Chapter 3 presents N‐Way ANOVA Quantile‐Regressions. Specific for N = 3, alternative equation specifications are presented, starting with a 2 × 2 × 2 factorial ANOVA QR without an intercept, to show how compute conditional two‐way interactions factors, and a three‐way interaction factor, which should be tested using the Wald Test. As an extension the 2 × 2 × 2 factorial ANOVA QR without an intercept, a 2 × 2 × 3 factorial ANOVA QR without an intercepts to show how to compute the 3‐way interaction factor. Then selected I × J × K factorial ANOVA QRs with an intercepts are presented using alternative equation specifications, to show the advantages in using each equation specification.

Chapter 4 presents quantile regressions based on bivariate numerical variable (Xi,Yi), starting with the simplest linear quantile‐regression. Then it is extended quadratic QR and alternative polynomial QRs, and alternative logarithmic QRs with lower and upper bounds.

Chapter 5 presents various QRs based on triple variables (X1t,X2t,Y1t) as the extension or modification of each model presented in chapter 4. Three alternative path diagrams based on the triple variables are presented, as the guide to defined equation specifications of alternative QRs. An additive QR is presented as the simplest QR, which is extended to semi‐logarithmic and translog QRs, with the examples presented based on an experimental data in Data_Faad.wf1. Then they are extended to interaction QRs. As additional illustrative examples, the statistical results of alternative QRs are presented based on MLogit.wf1, which I consider as a special data in EViews work file. In addition special Quantile Slope Equality Test also is presented as an illustration.

Chapter 6 presents various QRs with multi numerical predictors. Four alternative path diagrams based on the variables (X1,X2,X3,Y1) are presented as a guide to define specific equations of alternative QRs., such as additive, two‐way interaction and three‐way interaction QRs. The Data_Faad.wf1 having four numerical variables, namely X1, X2, Y1, and Y2, can be used to replace the causal or up‐and‐down relationships presented in Figure 6.1. Hence, this chapter presents statistical results of additive, two‐way interaction and three‐way interaction QRs of Y2 on Y1, X1 and X2 with its possible reduced QRs, which are develop using the manual multistage selection method (MMSM) and the trial‐and error method, since the STEPWISE method is not applicable for the QR. As additional empirical statistical results, selected sets of four variables in Mlogit.wf1 are used.

Furthermore, more advanced statistical analyses, such as the application of Quantile Process, Residual Analysis, Stability Test, and the “Static Forecast Method”, are presented, with special notes and comments.

Chapter 7 presents various QRs with te ranks of numerical variables as the predictors or independent variables, which is considered as nonparametric‐quantile‐ regressions (NPQR). They are the modification of the QRs having numerical independent variables presented in previous chapters, which are defined as the semiparametric‐quantile‐regressions. Statistical results of alternative QRs are presented based on the MCycle.wf1, Data_Faad.wf1, and Mlogit.wf1.

Chapter 8 presents illustrative examples on Heterogeneous Quantile Regressions (HQRs) based on experimental data. In this chapter it is introduced a symbol CF = Cell‐Factor to represent one or more categorical variable, which also can be generated using one or more numerical variables Then various HQRs based on (X1,Y1) are presented, starting with the simplest HQR of Y1 on a numerical variable X1 by a dichotomous factor or a dummy variable using three alternative equation specification. As the extension of HLQRs, Heterogeneous Quadratic QR and Heterogeneous Polynomial QRs are presented. Then they are extended to Heterogeneous Linear QR (HLQR) and Heterogeneous Polynomial QR (HPQR) using one or more factors. Furthermore, they are extended to Additive and Two‐Way‐Interaction HQRs based on numerical variables (X1, X2, Y1) and (X1, X2, X3, Y1). Whenever the CF is generated, based on the independent variable X1, then we have a Peace‐Wise‐QR of Y1 on X1 or X2 by CF.

Chapter 9 presents various QRs, as the applications of selected QRs presented in previous chapters, based on selected sets of variables in CPS88.wf1, which is a special work‐file in EViews 10.

Chapter 10 presents various QRs based on a health data set, specifically QRs of a baby latent variable (BLV) on selected sets of eight selected set of mother indicators in BBW.wf1. The BLV is generated using three baby indicators, namely BBW = baby birth weight, FUNDUS, and MUAC = mid upper arm circumference. Two special One‐Way ANOVA‐QRs of BLV on @Expand(@Round(MW) or @Expand(AGE), and selected types of QRs, which have been presented in previous chapters. As additional QRs, a mother latent variable (MLV) is generated based on four mother indicators, namely based on the two ordinal variables ED (Education level) and SE (Social Economic levels), and two dummy variables SMOKING, which is defined SMOKING = 1 for the smoking mothers, and NO_ANC, which is defined as NO_ANC = 1 for the mothers don't have antenatal care. Then outputs of selected QRs of BLV on MLV and other mother indicators are presented as illustrative empirical the full latent variable QRs.

I wish to express my gratitude to the graduate School of Management, Faculty of Economics and Business, University of Indonesia for providing rich intellectual environment while I was at the University of Indonesia from 1987 to 2018, that were indispensable for supporting in writing this text. I also would like to thank The Ary Suta Center, Jakarta for providing intellectual environment and facilities, while I am an advisor from 2008 till now.

Finally, I would like to thank the reviewers, editors, and all staffs at John Wiley for their hard works in getting this book to a publication.