doi:10.1007/s11749-023-00895-6engCastillo-Mateo, JorgeGelfand, Alan E.Asín, JesúsCebrián, Ana C.Abaurrea, JesúsBayesian joint quantile autoregressionART-2024-135633Quantile regression continues to increase in usage, providing a useful alternative to customary mean regression. Primary implementation takes the form of so-called multiple quantile regression, creating a separate regression for each quantile of interest. However, recently, advances have been made in joint quantile regression, supplying a quantile function which avoids crossing of the regression across quantiles. Here, we turn to quantile autoregression (QAR), offering a fully Bayesian version. We extend the initial quantile regression work of Koenker and Xiao (J Am Stat Assoc 101(475):980–990, 2006. https://doi.org/10.1198/016214506000000672) in the spirit of Tokdar and Kadane (Bayesian Anal 7(1):51–72, 2012. https://doi.org/10.1214/12-BA702). We offer a directly interpretable parametric model specification for QAR. Further, we offer a pth-order QAR(p) version, a multivariate QAR(1) version, and a spatial QAR(1) version. We illustrate with simulation as well as a temperature dataset collected in Aragón, Spain.2024http://zaguan.unizar.es/record/12942310.1007/s11749-023-00895-6http://zaguan.unizar.es/record/129423oai:zaguan.unizar.es:129423info:eu-repo/grantAgreement/ES/DGA/E46-20Rinfo:eu-repo/grantAgreement/ES/MICINN/PID2020-116873GB-I00info:eu-repo/grantAgreement/EUR/MICINN/TED2021-130702B-I00Test 33 (2024), 335–357byhttps://creativecommons.org/licenses/by/4.0/deed.esinfo:eu-repo/semantics/openAccess