On Bootstrap Inference for Quantile Regression Panel Data: A Monte Carlo Study
Abstract
:1. Introduction
2. The Model and Estimator
2.1. The Model
2.2. Fixed-Effects Quantile Regression Estimator
3. Inference
3.1. The Bootstrap
3.2. Practical Implementation
- (i)
- Obtain the resampled data .
- (ii)
- Estimate .
4. Monte Carlo
4.1. Design
- Random-effects model: , where both and are standard Gaussian in the location shift model and in the location-scale shift model3. This model has a covariate that is not correlated with the unobserved FE, and thus, the random-effects model is correct.
- Fixed-effects model: , where is standard Gaussian in the location shift model and in the location-scale shift model. In this model, x is correlated with the unobserved FE.
4.2. Results
Cross-Sectional | Temporal | Cross-Sectional and Temporal | ||||||||
---|---|---|---|---|---|---|---|---|---|---|
N | T | |||||||||
, | ||||||||||
10 | 5 | 0.154 | 0.156 | 0.168 | 0.074 | 0.062 | 0.064 | 0.020 | 0.016 | 0.030 |
10 | 10 | 0.160 | 0.192 | 0.172 | 0.058 | 0.046 | 0.056 | 0.012 | 0.016 | 0.030 |
10 | 50 | 0.160 | 0.172 | 0.158 | 0.074 | 0.056 | 0.076 | 0.008 | 0.010 | 0.014 |
25 | 5 | 0.144 | 0.136 | 0.128 | 0.078 | 0.062 | 0.072 | 0.020 | 0.026 | 0.034 |
25 | 10 | 0.152 | 0.128 | 0.130 | 0.070 | 0.040 | 0.056 | 0.024 | 0.020 | 0.018 |
25 | 50 | 0.126 | 0.136 | 0.116 | 0.040 | 0.054 | 0.052 | 0.012 | 0.020 | 0.022 |
50 | 5 | 0.122 | 0.108 | 0.112 | 0.072 | 0.064 | 0.076 | 0.002 | 0.006 | 0.010 |
50 | 10 | 0.088 | 0.098 | 0.112 | 0.074 | 0.056 | 0.056 | 0.010 | 0.012 | 0.010 |
50 | 50 | 0.108 | 0.100 | 0.112 | 0.054 | 0.052 | 0.060 | 0.008 | 0.008 | 0.008 |
, | ||||||||||
10 | 5 | 0.162 | 0.168 | 0.166 | 0.070 | 0.062 | 0.090 | 0.026 | 0.030 | 0.022 |
10 | 10 | 0.146 | 0.152 | 0.166 | 0.072 | 0.064 | 0.076 | 0.032 | 0.016 | 0.014 |
10 | 50 | 0.170 | 0.162 | 0.160 | 0.068 | 0.054 | 0.060 | 0.018 | 0.016 | 0.026 |
25 | 5 | 0.110 | 0.126 | 0.142 | 0.066 | 0.078 | 0.076 | 0.024 | 0.022 | 0.024 |
25 | 10 | 0.130 | 0.128 | 0.128 | 0.050 | 0.038 | 0.070 | 0.014 | 0.004 | 0.012 |
25 | 50 | 0.132 | 0.134 | 0.142 | 0.066 | 0.080 | 0.094 | 0.026 | 0.018 | 0.008 |
50 | 5 | 0.130 | 0.120 | 0.102 | 0.094 | 0.070 | 0.068 | 0.012 | 0.008 | 0.012 |
50 | 10 | 0.130 | 0.124 | 0.094 | 0.052 | 0.046 | 0.078 | 0.014 | 0.014 | 0.012 |
50 | 50 | 0.108 | 0.082 | 0.090 | 0.068 | 0.066 | 0.072 | 0.006 | 0.018 | 0.010 |
, | ||||||||||
10 | 5 | 0.182 | 0.154 | 0.148 | 0.070 | 0.060 | 0.066 | 0.022 | 0.022 | 0.022 |
10 | 10 | 0.154 | 0.162 | 0.150 | 0.050 | 0.052 | 0.056 | 0.014 | 0.010 | 0.018 |
10 | 50 | 0.174 | 0.156 | 0.190 | 0.076 | 0.062 | 0.058 | 0.014 | 0.032 | 0.028 |
25 | 5 | 0.126 | 0.124 | 0.122 | 0.072 | 0.060 | 0.068 | 0.012 | 0.008 | 0.006 |
25 | 10 | 0.148 | 0.122 | 0.100 | 0.038 | 0.024 | 0.044 | 0.012 | 0.010 | 0.010 |
25 | 50 | 0.136 | 0.128 | 0.116 | 0.058 | 0.074 | 0.072 | 0.020 | 0.016 | 0.020 |
50 | 5 | 0.124 | 0.102 | 0.124 | 0.074 | 0.060 | 0.078 | 0.026 | 0.024 | 0.044 |
50 | 10 | 0.118 | 0.136 | 0.132 | 0.060 | 0.044 | 0.068 | 0.024 | 0.016 | 0.018 |
50 | 50 | 0.114 | 0.110 | 0.102 | 0.074 | 0.086 | 0.078 | 0.030 | 0.016 | 0.024 |
Cross-Sectional | Temporal | Cross-Sectional and Temporal | ||||||||
---|---|---|---|---|---|---|---|---|---|---|
N | T | |||||||||
, | ||||||||||
10 | 5 | 0.190 | 0.178 | 0.180 | 0.130 | 0.088 | 0.076 | 0.016 | 0.016 | 0.032 |
10 | 10 | 0.158 | 0.174 | 0.158 | 0.090 | 0.074 | 0.088 | 0.020 | 0.030 | 0.042 |
10 | 50 | 0.150 | 0.136 | 0.154 | 0.096 | 0.074 | 0.072 | 0.030 | 0.018 | 0.018 |
25 | 5 | 0.148 | 0.132 | 0.140 | 0.122 | 0.080 | 0.130 | 0.054 | 0.028 | 0.042 |
25 | 10 | 0.122 | 0.116 | 0.134 | 0.094 | 0.058 | 0.088 | 0.030 | 0.012 | 0.022 |
25 | 50 | 0.118 | 0.114 | 0.122 | 0.096 | 0.082 | 0.100 | 0.016 | 0.028 | 0.022 |
50 | 5 | 0.138 | 0.150 | 0.152 | 0.136 | 0.064 | 0.154 | 0.048 | 0.020 | 0.066 |
50 | 10 | 0.120 | 0.128 | 0.092 | 0.098 | 0.052 | 0.076 | 0.024 | 0.016 | 0.034 |
50 | 50 | 0.102 | 0.128 | 0.122 | 0.082 | 0.076 | 0.090 | 0.026 | 0.016 | 0.034 |
, | ||||||||||
10 | 5 | 0.178 | 0.190 | 0.186 | 0.108 | 0.088 | 0.124 | 0.040 | 0.028 | 0.046 |
10 | 10 | 0.136 | 0.152 | 0.152 | 0.060 | 0.054 | 0.098 | 0.014 | 0.012 | 0.018 |
10 | 50 | 0.196 | 0.154 | 0.160 | 0.076 | 0.060 | 0.082 | 0.036 | 0.044 | 0.034 |
25 | 5 | 0.146 | 0.138 | 0.148 | 0.078 | 0.090 | 0.148 | 0.022 | 0.018 | 0.068 |
25 | 10 | 0.118 | 0.136 | 0.128 | 0.080 | 0.062 | 0.090 | 0.020 | 0.016 | 0.038 |
25 | 50 | 0.138 | 0.130 | 0.140 | 0.094 | 0.064 | 0.086 | 0.024 | 0.018 | 0.024 |
50 | 5 | 0.136 | 0.122 | 0.152 | 0.096 | 0.100 | 0.240 | 0.034 | 0.026 | 0.100 |
50 | 10 | 0.106 | 0.110 | 0.126 | 0.094 | 0.080 | 0.100 | 0.018 | 0.012 | 0.050 |
50 | 50 | 0.128 | 0.086 | 0.120 | 0.082 | 0.064 | 0.080 | 0.020 | 0.016 | 0.016 |
, | ||||||||||
10 | 5 | 0.210 | 0.178 | 0.196 | 0.126 | 0.064 | 0.084 | 0.044 | 0.022 | 0.040 |
10 | 10 | 0.210 | 0.166 | 0.174 | 0.092 | 0.072 | 0.112 | 0.044 | 0.020 | 0.046 |
10 | 50 | 0.172 | 0.160 | 0.166 | 0.074 | 0.076 | 0.082 | 0.020 | 0.020 | 0.018 |
25 | 5 | 0.158 | 0.118 | 0.144 | 0.100 | 0.064 | 0.132 | 0.044 | 0.020 | 0.056 |
25 | 10 | 0.128 | 0.124 | 0.150 | 0.088 | 0.046 | 0.096 | 0.050 | 0.026 | 0.044 |
25 | 50 | 0.126 | 0.130 | 0.148 | 0.072 | 0.076 | 0.068 | 0.016 | 0.020 | 0.030 |
50 | 5 | 0.172 | 0.146 | 0.128 | 0.148 | 0.040 | 0.136 | 0.048 | 0.010 | 0.056 |
50 | 10 | 0.116 | 0.112 | 0.130 | 0.142 | 0.060 | 0.114 | 0.020 | 0.002 | 0.022 |
50 | 50 | 0.154 | 0.102 | 0.148 | 0.086 | 0.084 | 0.102 | 0.016 | 0.018 | 0.024 |
Cross-Sectional | Temporal | Cross-Sectional and Temporal | ||||||||
---|---|---|---|---|---|---|---|---|---|---|
N | T | |||||||||
, | ||||||||||
10 | 5 | 0.174 | 0.156 | 0.148 | 0.082 | 0.060 | 0.068 | 0.034 | 0.026 | 0.030 |
10 | 10 | 0.152 | 0.172 | 0.152 | 0.058 | 0.032 | 0.038 | 0.022 | 0.012 | 0.022 |
10 | 50 | 0.140 | 0.148 | 0.188 | 0.076 | 0.072 | 0.082 | 0.024 | 0.018 | 0.012 |
25 | 5 | 0.110 | 0.108 | 0.122 | 0.080 | 0.066 | 0.082 | 0.016 | 0.010 | 0.016 |
25 | 10 | 0.136 | 0.136 | 0.138 | 0.060 | 0.048 | 0.076 | 0.014 | 0.014 | 0.024 |
25 | 50 | 0.116 | 0.126 | 0.112 | 0.060 | 0.070 | 0.072 | 0.016 | 0.010 | 0.006 |
50 | 5 | 0.126 | 0.112 | 0.108 | 0.094 | 0.052 | 0.072 | 0.026 | 0.022 | 0.020 |
50 | 10 | 0.106 | 0.114 | 0.112 | 0.056 | 0.038 | 0.040 | 0.010 | 0.020 | 0.020 |
50 | 50 | 0.136 | 0.132 | 0.102 | 0.090 | 0.084 | 0.072 | 0.020 | 0.020 | 0.010 |
, | ||||||||||
10 | 5 | 0.188 | 0.172 | 0.158 | 0.088 | 0.072 | 0.070 | 0.008 | 0.010 | 0.012 |
10 | 10 | 0.182 | 0.160 | 0.160 | 0.064 | 0.068 | 0.076 | 0.016 | 0.022 | 0.026 |
10 | 50 | 0.156 | 0.160 | 0.128 | 0.074 | 0.072 | 0.078 | 0.014 | 0.016 | 0.016 |
25 | 5 | 0.124 | 0.122 | 0.112 | 0.098 | 0.066 | 0.062 | 0.032 | 0.032 | 0.030 |
25 | 10 | 0.156 | 0.136 | 0.126 | 0.056 | 0.044 | 0.066 | 0.022 | 0.008 | 0.012 |
25 | 50 | 0.140 | 0.140 | 0.124 | 0.058 | 0.072 | 0.062 | 0.016 | 0.012 | 0.016 |
50 | 5 | 0.100 | 0.108 | 0.146 | 0.078 | 0.058 | 0.072 | 0.016 | 0.014 | 0.016 |
50 | 10 | 0.096 | 0.110 | 0.110 | 0.044 | 0.026 | 0.050 | 0.012 | 0.012 | 0.012 |
50 | 50 | 0.122 | 0.110 | 0.088 | 0.064 | 0.064 | 0.084 | 0.018 | 0.020 | 0.016 |
, | ||||||||||
10 | 5 | 0.176 | 0.166 | 0.154 | 0.090 | 0.078 | 0.096 | 0.028 | 0.018 | 0.016 |
10 | 10 | 0.130 | 0.144 | 0.176 | 0.056 | 0.052 | 0.056 | 0.014 | 0.006 | 0.014 |
10 | 50 | 0.190 | 0.166 | 0.146 | 0.072 | 0.060 | 0.062 | 0.028 | 0.014 | 0.018 |
25 | 5 | 0.128 | 0.090 | 0.110 | 0.072 | 0.042 | 0.070 | 0.016 | 0.024 | 0.030 |
25 | 10 | 0.120 | 0.118 | 0.108 | 0.052 | 0.062 | 0.062 | 0.016 | 0.012 | 0.010 |
25 | 50 | 0.136 | 0.146 | 0.130 | 0.080 | 0.084 | 0.064 | 0.020 | 0.020 | 0.018 |
50 | 5 | 0.130 | 0.134 | 0.110 | 0.072 | 0.058 | 0.068 | 0.024 | 0.022 | 0.028 |
50 | 10 | 0.132 | 0.120 | 0.104 | 0.052 | 0.044 | 0.056 | 0.020 | 0.012 | 0.010 |
50 | 50 | 0.120 | 0.118 | 0.122 | 0.060 | 0.052 | 0.070 | 0.012 | 0.014 | 0.020 |
Cross-Sectional | Temporal | Cross-Sectional and Temporal | ||||||||
---|---|---|---|---|---|---|---|---|---|---|
N | T | |||||||||
, | ||||||||||
10 | 5 | 0.166 | 0.150 | 0.184 | 0.102 | 0.074 | 0.122 | 0.044 | 0.036 | 0.032 |
10 | 10 | 0.206 | 0.170 | 0.164 | 0.100 | 0.066 | 0.096 | 0.020 | 0.012 | 0.038 |
10 | 50 | 0.172 | 0.150 | 0.188 | 0.080 | 0.080 | 0.104 | 0.030 | 0.018 | 0.016 |
25 | 5 | 0.130 | 0.116 | 0.144 | 0.142 | 0.074 | 0.128 | 0.030 | 0.012 | 0.038 |
25 | 10 | 0.126 | 0.118 | 0.126 | 0.094 | 0.052 | 0.086 | 0.038 | 0.020 | 0.044 |
25 | 50 | 0.152 | 0.118 | 0.134 | 0.078 | 0.070 | 0.068 | 0.010 | 0.010 | 0.010 |
50 | 5 | 0.130 | 0.150 | 0.132 | 0.162 | 0.076 | 0.146 | 0.046 | 0.022 | 0.048 |
50 | 10 | 0.132 | 0.112 | 0.134 | 0.106 | 0.076 | 0.140 | 0.030 | 0.010 | 0.018 |
50 | 50 | 0.112 | 0.110 | 0.122 | 0.078 | 0.062 | 0.076 | 0.020 | 0.014 | 0.026 |
, | ||||||||||
10 | 5 | 0.184 | 0.142 | 0.172 | 0.084 | 0.084 | 0.130 | 0.028 | 0.040 | 0.072 |
10 | 10 | 0.158 | 0.162 | 0.186 | 0.072 | 0.054 | 0.098 | 0.014 | 0.020 | 0.050 |
10 | 50 | 0.188 | 0.156 | 0.180 | 0.086 | 0.086 | 0.096 | 0.026 | 0.032 | 0.032 |
25 | 5 | 0.158 | 0.138 | 0.146 | 0.112 | 0.078 | 0.190 | 0.024 | 0.032 | 0.072 |
25 | 10 | 0.130 | 0.150 | 0.112 | 0.086 | 0.052 | 0.100 | 0.020 | 0.014 | 0.040 |
25 | 50 | 0.110 | 0.104 | 0.106 | 0.070 | 0.076 | 0.072 | 0.034 | 0.016 | 0.012 |
50 | 5 | 0.136 | 0.130 | 0.166 | 0.102 | 0.084 | 0.254 | 0.032 | 0.028 | 0.086 |
50 | 10 | 0.120 | 0.132 | 0.136 | 0.078 | 0.072 | 0.150 | 0.016 | 0.008 | 0.036 |
50 | 50 | 0.122 | 0.122 | 0.116 | 0.066 | 0.086 | 0.094 | 0.020 | 0.014 | 0.014 |
, | ||||||||||
10 | 5 | 0.200 | 0.164 | 0.170 | 0.108 | 0.048 | 0.130 | 0.034 | 0.014 | 0.024 |
10 | 10 | 0.150 | 0.138 | 0.184 | 0.098 | 0.058 | 0.106 | 0.034 | 0.018 | 0.038 |
10 | 50 | 0.158 | 0.148 | 0.138 | 0.104 | 0.086 | 0.106 | 0.026 | 0.022 | 0.024 |
25 | 5 | 0.140 | 0.140 | 0.146 | 0.146 | 0.082 | 0.136 | 0.050 | 0.020 | 0.050 |
25 | 10 | 0.132 | 0.136 | 0.142 | 0.110 | 0.038 | 0.102 | 0.036 | 0.016 | 0.026 |
25 | 50 | 0.100 | 0.106 | 0.142 | 0.104 | 0.066 | 0.092 | 0.034 | 0.008 | 0.036 |
50 | 5 | 0.156 | 0.104 | 0.146 | 0.192 | 0.068 | 0.178 | 0.072 | 0.014 | 0.064 |
50 | 10 | 0.118 | 0.116 | 0.120 | 0.138 | 0.052 | 0.124 | 0.044 | 0.012 | 0.036 |
50 | 50 | 0.122 | 0.148 | 0.140 | 0.098 | 0.076 | 0.108 | 0.036 | 0.020 | 0.008 |
5. Conclusions
Acknowledgments
Author Contributions
Conflicts of Interest
References
- R. Koenker. “Quantile Regression for Longitudinal Data.” J. Multivar. Anal. 91 (2004): 74–89. [Google Scholar] [CrossRef]
- K. Kato, A.F. Galvao, and G. Montes-Rojas. “Asymptotics for Panel Quantile Regression Models with Individual Effects.” J. Econom. 170 (2012): 76–91. [Google Scholar] [CrossRef]
- A.F. Galvao. “Quantile regression for dynamic panel data with fixed effects.” J. Econom. 164 (2011): 142–157. [Google Scholar] [CrossRef]
- A.F. Galvao, C. Lamarche, and L. Lima. “Estimation of Censored Quantile Regression for Panel Data with Fixed Effects.” J. Am. Stat. Assoc. 108 (2013): 1075–1089. [Google Scholar] [CrossRef]
- A.F. Galvao, and L. Wang. “Efficient Minimum Distance Estimator for Quantile Regression Fixed Effects Panel Data.” J. Multivar. Anal. 133 (2015): 1–26. [Google Scholar] [CrossRef]
- M. Buchinsky. “Estimating the Asymptotic Covariance Matrix for Quantile Regression Models: A Monte Carlo Study.” J. Econom. 68 (1995): 303–338. [Google Scholar] [CrossRef]
- J. Hahn. “Bootstrapping Quantile Regression Estimators.” Econom. Theory 11 (1995): 105–121. [Google Scholar] [CrossRef]
- J.L. Horowitz. “Bootstrap Methods for Median Regression Models.” Econometrica 66 (1998): 1327–1351. [Google Scholar] [CrossRef]
- X. Feng, X. He, and J. Hu. “Wild bootstrap for quantile regression.” Biometrika 98 (2011): 995–999. [Google Scholar] [CrossRef] [PubMed]
- H. Wang, and X. He. “Detecting Differential Expressions in GeneChip Microarray Studies.” J. Am. Stat. Assoc. 102 (2007): 104–112. [Google Scholar] [CrossRef]
- J. Abrevaya, and C.M. Dahl. “The Effects of Birth Inputs on Birthweight: Evidence From Quantile Estimation on Panel Data.” J. Bus. Econ. Stat. 26 (2008): 379–397. [Google Scholar] [CrossRef]
- C. Lamarche. “Penalized Quantile Regression Estimation for a Model with Endogenous Individual Effects.” J. Econom. 157 (2010): 396–408. [Google Scholar] [CrossRef]
- G. Kapetanios. “A Bootstrap Procedure for Panel Datasets with Many Cross-Sectional Units.” Econom. J. 11 (2008): 377–395. [Google Scholar] [CrossRef]
- A.C. Cameron, and P.K. Trivedi. Microeconometrics: Methods and Applications. Cambridge, MA, USA: Cambridge University Press, 2005. [Google Scholar]
- R. Koenker, and K. Hallock. “Quantile Regression: An Introduction.” University of Illinois at Urbana-Champaign, Champaign-Urbana, IL, USA, Unpublished work. 2000. [Google Scholar]
- R. Koenker, and G.W. Bassett. “Regression Quantiles.” Econometrica 46 (1978): 33–49. [Google Scholar] [CrossRef]
- D.A. Freedman. “Bootstrapping Regression Models.” Ann. Stat. 9 (1981): 1218–1228. [Google Scholar] [CrossRef]
- 1An alternative model was developed by [1], where the individual effects are the same across quantiles. In most applications, the time series dimension T is relatively small compared to the number of individuals N. Therefore, it might be difficult to estimate a τ-dependent distributional individual effect. The restriction of the individual effects, η, to be independent of the specific quantile, τ, is implemented by estimating the model for several quantiles simultaneously.
- 2The work in [1] introduced a general approach to estimate quantile regression models for panel data with FE that may be subject to shrinkage by regularization methods. It is well know that the optimal estimator for the random effects Gaussian model involves shrinking the individual effects toward a common value. When there is an intercept in the model, this common value can be taken to be the conditional central tendency of the response at a point determined by the centering of the other covariates. In the quantile regression model, this would be some corresponding conditional quantile of the response. Particularly, when N is large relative to T, shrinkage may be advantageous in controlling the variability introduced by the large number of estimated individual-specific parameters.
- 3This is to avoid non-linearities in the linear quantile functions, which arise if x can take both positive and negative values.
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Galvao, A.F.; Montes-Rojas, G. On Bootstrap Inference for Quantile Regression Panel Data: A Monte Carlo Study. Econometrics 2015, 3, 654-666. https://doi.org/10.3390/econometrics3030654
Galvao AF, Montes-Rojas G. On Bootstrap Inference for Quantile Regression Panel Data: A Monte Carlo Study. Econometrics. 2015; 3(3):654-666. https://doi.org/10.3390/econometrics3030654
Chicago/Turabian StyleGalvao, Antonio F., and Gabriel Montes-Rojas. 2015. "On Bootstrap Inference for Quantile Regression Panel Data: A Monte Carlo Study" Econometrics 3, no. 3: 654-666. https://doi.org/10.3390/econometrics3030654