Dynamic panel data models are an important part of medical, social, and economic studies.
A distinctive feature of these models is the presence of a lagged dependent variable as a descriptive variable. The estimation problem in these models arises from the correlation between the lagged dependent variable and the current error component. Recently, penalized quantile regression has been considered for the analysis of dynamic panel data. In this paper, first, a quantile regression model with a lasso-like penalty on random effects is presented for dynamic panel data assuming the dependence of random effects and initial observations. This model will also be examined assuming the independence between random effects and initial observations. Both models are analyzed from the perspective of Bayesian statistics. Since in these two methods, the posterior distribution of parameters cannot be obtained in a closed form, the full conditional posterior distributions of the parameters are calculated and the Gibbs sampling algorithm is used for inference. To compare the efficiency of the presented Bayesian methods with conventional methods, a simulation study was conducted, and finally, the method of using the models will be explained in the form of a practical example.