Bayesian quantile regression

Yu, Keming; Moyeed, Rana

doi:10.1016/s0167-7152(01)00124-9

Cited by 794 publications

(679 citation statements)

References 9 publications

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“…This result allows (maximum) likelihood estimation, and has motivated Bayesian solutions for this problem, as proposed in Yu and Moyeed (2001), Tsionas (2003), Yu and Zhang (2005) and Geraci and Bottai (2007), and subsequently extended in Chen et al (2011). These designs all involve MCMC computational methods due to the non-standard form of the posterior resulting from the skewed-Laplace likelihood.…”

Section: Estimation and Forecast Evaluationmentioning

confidence: 99%

“…This during the 2008-09 GFC; evaluate how the crisis affected risk management practices, forecasts of VaR and daily capital charges; and discuss diagnostic checking of VaR methods. Further, we adapt the Bayesian estimation methods in Yu and Moyeed (2001), exploiting the link between quantile estimation and the Skewed-Laplace distribution, first discussed by Koenker and Machado (1999), to the range of models in the CAViaR family in a systematic way, conducting a comparison with the frequentist estimation of Engle and Manganelli (2004), in regards to the forecasts produced from these models.…”

Section: Introductionmentioning

confidence: 99%

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Forecasting Value-at-Risk using nonlinear regression quantiles and the intra-day range

Chen

Gerlach

Hwang

et al. 2012

International Journal of Forecasting

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Value-at-Risk (VaR) is commonly used for financial risk measurement. It has recently become even more important, especially during the 2008-09 global financial crisis. We propose some novel nonlinear threshold conditional autoregressive VaR (CAViaR) models that incorporate intra-day price ranges. Model estimation and inference are performed using the Bayesian approach via the link with the Skewed-Laplace distribution. We examine how a range of risk models perform during the 2008-09 financial crisis, and evaluate how the crisis affects the performance of risk models via forecasting VaR. Empirical analysis is conducted on five Asia-Pacific Economic Cooperation stock market indices as well as two exchange rate series. We examine violation rates, back-testing criteria, market risk charges and quantile loss function values to measure and assess the forecasting performance of a variety of risk models. The proposed threshold CAViaR model, incorporating range information, is shown to forecast VaR more efficiently than other models, across the series considered, which should be useful for financial practitioners.

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Section: Estimation and Forecast Evaluationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Forecasting Value-at-Risk using nonlinear regression quantiles and the intra-day range

Chen

Gerlach

Hwang

et al. 2012

International Journal of Forecasting

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“…Recently, an asymmetric Laplace distribution (ALD), which is characterized by a peak at the mode and thick tails, has been used in Bayesian quantile regression for error distribution (Yu and Zhang, 2005;Yu and Moyeed, 2001). The connection between maximizing a likelihood function composed of independently distributed ALD and minimisation of the objective function in quantile regression was shown by Yu and Moyeed (2001) and used to develop the likelihood ratio test for quantile regression (Koenker and Machado, 1999), and to apply quantile regression to longitudinal data (Geraci and Bottai, 2007). In the mean regression, a similar connection exists between MLE based on normal distribution and least-squares estimation.…”

Section: Introductionmentioning

confidence: 99%

Mixed-Effects Models for Conditional Quantiles with Longitudinal Data

Liu¹,

Bottai²

2009

The International Journal of Biostatistics

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We propose a regression method for the estimation of conditional quantiles of a continuous response variable given a set of covariates when the data are dependent. Along with fixed regression coefficients, we introduce random coefficients which we assume to follow a form of multivariate Laplace distribution. In a simulation study, the proposed quantile mixed-effects regression is shown to model the dependence among longitudinal data correctly and estimate the fixed effects efficiently. It performs similarly to the linear mixed model at the central location when the regression errors are symmetrically distributed, but provides more efficient estimates when the errors are over-dispersed. At the same time, it allows the estimation at different locations of conditional distribution, which conveys a comprehensive understanding of data. We illustrate an application to clinical data where the outcome variable of interest is bounded within a closed interval.

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“…and Q τ (ε τ t |F t−1 ) = 0. According to Koenker and Machado (1999) and Yu and Moyeed (2001), the quantile regression estimatorβ β β(τ ) can be interpreted as a quasi-MLE assuming that ε τ t follows an asymmetric Laplace distribution having the density function:…”

Section: Encompassing Caviar Modelsmentioning

confidence: 99%

Forecasting value-at-risk by encompassing CAViaR models via information criteria

Lee¹,

Noh²

2013

Journal of the Korean Data and Information Science Society

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This paper proposes a new method of VaR forecasting using the conditional autoregressive VaR (CAViaR) models and information criteria. Instead of using a single CAViaR model, we propose to utilize several candidate CAViaR models during a forecasting period. By adopting the Akaike and Bayesian information criteria for quantile regression, we can update not only parameter estimates but also the CAViaR specifications. We also propose extended CAViaR models with a constant location parameter. An empirical study is provided to examine the performance of the proposed method. The results suggest that our method shows more stable performance than those using a single specification.

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Bayesian quantile regression

Cited by 794 publications

References 9 publications

Forecasting Value-at-Risk using nonlinear regression quantiles and the intra-day range

Forecasting Value-at-Risk using nonlinear regression quantiles and the intra-day range

Mixed-Effects Models for Conditional Quantiles with Longitudinal Data

Forecasting value-at-risk by encompassing CAViaR models via information criteria

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