# Binary Quantile Regression

### Description

This function is used to fit a quantile regression model when the response is binary.

### Usage

1 2 3 |

### Arguments

`formula` |
an object of class |

`x` |
the design matrix. |

`y` |
the response variable. |

`tau` |
quantile to be estimated. |

`data` |
an optional data frame, list or environment (or object coercible by as.data.frame to a data frame) containing the variables in the model. If not found in data, the variables are taken from environment(formula), typically the environment from which lqm is called. |

`weights` |
an optional vector of weights to be used in the fitting process. Should be NULL or a numeric vector - not yet implemented. |

`contrasts` |
an optional list. See the |

`normalize` |
character specifying the type of normalization of the coefficients: if "last" (default), the last coefficient is set equal to 1; if "all", the vector of all coefficients except the intercept has norm equal to 1. |

`control` |
list of control parameters of the fitting process. See |

`fit` |
logical flag. If |

### Details

A binary quantile regression model is fitted as linear specification of the quantile function of a latent response variable (Manski 1975, 1985). The function `rqbin.fit`

calls the Fortran routine `simann.f`

implementing the simulated annealing algorithm of Goffe et al (1994) â€“ original code by William Goffe, modified by Gregory Kordas. Normalization is necessary for estimation to be possible. The normalization proposed by Horowitz (1992) assumes that there is a continuous regressor independent of the (latent) error and the corresponding regression coefficient is constrained to be equal to 1. Therefore, the user must ensure that the last term in `formula`

or the last column in the matrix `x`

corresponds to such regressor. If the argument `normalize = "all"`

, then the normalization proposed by Manski (1975) is applied so that the norm of the vector with all the 'slopes' (i.e., excluding the intercept), is equal to 1.

### Value

a list of class `rq.bin`

containing the following components

`coefficients` |
a vector of coefficients. |

`logLik` |
the logâ€“likelihood. |

`opt` |
details on optimization. |

`call` |
the matched call. |

`term.labels` |
names for theta. |

`terms` |
the terms object used. |

`nobs` |
the number of observations. |

`edf` |
the numer of parameters (minus 1 if normalize is |

`rdf` |
the number of residual degrees of freedom. |

`tau` |
the estimated quantile(s). |

`x` |
the model matrix. |

`y` |
the model response. |

`weights` |
the weights used in the fitting process (a vector of 1's if |

`levels` |
factors levels. |

`control` |
list of control parameters used for optimization (see |

`normalize` |
type of normalization. |

### Author(s)

Marco Geraci

### References

Goffe WL, Ferrier GD, Rogers J. Global optimization of statistical functions with simulated annealing. Journal of Econometrics 1994;60(1):65-99. Code retrieved from http://econpapers.repec.org/software/wpawuwppr/9406001.htm.

Kordas G. Smoothed binary regression quantiles. Journal of Applied Econometrics 2006;21(3):387-407. Code retrieved from http://qed.econ.queensu.ca/jae/2006-v21.3/kordas/.

Horowitz JL. A Smoothed Maximum Score Estimator for the Binary Response Model. Econometrica 1992;60(3):505-531.

Manski CF. Maximum score estimation of the stochastic utility model of choice. Journal of Econometrics 1975;3(3):205-228.

Manski, CF. Semiparametric analysis of discrete response: Asymptotic properties of the maximum score estimator. Journal of Econometrics 1985;27(3):313-333.