Description Usage Arguments Author(s) References Examples

This function implements the idea of Bayesian Lasso quantile regression using a likelihood function that is based on the asymmetric Laplace distribution (Rahim, 2016). The asymmetric Laplace error distribution is written as a scale mixture of normal distributions as in Reed and Yu (2009). This function implements the Bayesian lasso for linear quantile regression models by assigning scale mixture of normal (SMN) priors on the parameters and independent exponential priors on their variances. We propose an alternative Bayesian analysis of the Bayesian lasso problem reported in Li, et al. (2010). A Gibbs sampling algorithm for the Bayesian Lasso quantile regression is constructed by sampling the parameters from their full conditional distributions.

1 | ```
BLqr(x,y, tau = 0.5, runs = 11000, burn = 1000, thin=1)
``` |

`x` |
Matrix of predictors. |

`y` |
Vector of dependent variable. |

`tau` |
The quantile of interest. Must be between 0 and 1. |

`runs` |
Length of desired Gibbs sampler output. |

`burn` |
Number of Gibbs sampler iterations before output is saved. |

`thin` |
thinning parameter of MCMC draws. |

Rahim Alhamzawi

[1] Alhamzawi, R. (2016). Bayesian variable selection in quantile regression using asymmetric Laplace distribution. Working paper.

[2] Reed, C. and Yu, K. (2009). A partially collapsed Gibbs sampler for Bayesian quantile regression. *Technical Report*. Department of Mathematical Sciences, Brunel
University. URL: http://bura.brunel.ac.uk/bitstream/2438/3593/1/fulltext.pdf.

[3] Li, Q., Xi, R. and Lin, N. (2010). Bayesian regularized quantile regression. Bayesian Analysis, 5(3): 533-56.

1 2 3 4 5 6 7 8 9 10 11 12 |

```
Call:
Brq.formula(formula = y ~ 0 + x, tau = 0.5, method = "BLqr",
runs = 5000, burn = 1000)
tau:[1] 0.5
Estimate L.CredIntv U.CredIntv
x1 4.864536182 4.7261717 5.01827356
x2 -0.053635411 -0.1892678 0.08707872
x3 0.035960866 -0.1006203 0.16142960
x4 0.033672880 -0.1100871 0.18965683
x5 0.010339228 -0.1347425 0.16298730
x6 0.017849028 -0.1467269 0.19443873
x7 0.006955432 -0.1436218 0.13857486
x8 -0.008872732 -0.1385060 0.12853875
===== Model selection based on credible intervals ======
# #
# Author: Rahim Alhamzawi #
# Contact: rahim.alhamzawi@qu.edu.iq #
# July, 2018 #
# #
=========================================================
Estimate
x1 4.864536
x2 0.000000
x3 0.000000
x4 0.000000
x5 0.000000
x6 0.000000
x7 0.000000
x8 0.000000
```

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