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Bayesian Quantile Regression Analysis for Bivariate Vector Autoregressive Models with an Application to Financial Time Series

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Abstract

To capture the conditional correlations between bivariate financial responses at different quantile levels, this paper considers the Bayesian quantile regression for bivariate vector autoregressive models. With the well known location-scale mixture representation for the asymmetric Laplace distribution, a working likelihood is obtained. By introducing the latent variables, a new Gibbs sampling algorithm is developed for drawing the posterior samples for the parameters and latent variables. The numerical simulation implies that the Gibbs sampling algorithm converges fast and the Bayesian quantile estimators perform well. Finally, a real example is given to discuss the relationship between the Canadian dollar to U.S. dollar exchange rate and long term annual interest rate of Canada.

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Acknowledgements

This work is supported by National Natural Science Foundation of China (No. 11901053), Natural Science Foundation of Jilin Province (No. 20220101038JC, 20210101149JC), Postdoctoral Foundation of Jilin Province (No. 2023337), Scientific Research Project of Jilin Provincial Department of Education (No. JJKH20220671KJ, JJKH20230665KJ), Humanities and Social Science Research Project of Jilin Provincial Department of Education (No. JJKH20220649SK).

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  1. School of Mathematics and Statistics, Changchun University of Technology, Changchun, 130012, China

    Kai Yang, Luan Zhao, Qian Hu & Wenshan Wang

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  1. Kai Yang
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  2. Luan Zhao
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  3. Qian Hu
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  4. Wenshan Wang
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Correspondence to Kai Yang.

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Appendix: The full conditional distributions

Appendix: The full conditional distributions

The detailed calculations of (3.11) to (3.15) are as follows:

(1) The full conditional density of \( z_{jt} |\varvec{y},\varvec{\beta } \sim \mathcal {G I G}\left( \frac{1}{2},\delta _{jt}, \gamma _{j}\right) \), since

$$\begin{aligned}&\pi (z_{jt} |\varvec{y},\varvec{\beta }) \\ \propto&z_{jt}^{-\frac{1}{2}} \exp \left\{ -\frac{(y_{jt}-a_{j0}-\sum _{i=1}^p\sum _{k=1}^2 a_{jk}^{(i)}y_{k,t-i}-\xi _{j}z_{jt})^{2}}{2 \sigma _{j}^{2} z_{jt}}-z_{jt}\right\} \\ \propto&z_{jt}^{-\frac{1}{2}} \exp \left\{ -\frac{1}{2}\left( \frac{(y_{jt}-a_{j0}-\sum _{i=1}^p\sum _{k=1}^2 a_{jk}^{(i)}y_{k,t-i})^{2}}{\sigma _{j}^{2}} z_{jt}^{-1}+\frac{\xi _{j}^{2}}{\sigma _{j}^{2}} z_{jt}\right) -z_{jt}\right\} \\ \propto&z_{jt}^{-\frac{1}{2}} \exp \left\{ -\frac{1}{2}\left( \frac{(y_{jt}-a_{j0}-\sum _{i=1}^p\sum _{k=1}^2 a_{jk}^{(i)}y_{k,t-i})^{2}}{\sigma _{j}^{2}} z_{jt}^{-1}+ \left( \frac{\xi _{j}^{2}}{\sigma _{j}^{2}}+2\right) z_{jt}\right) \right\} \\ \propto&z_{jt}^{-\frac{1}{2}} \exp \left\{ -\frac{1}{2}\left( \delta _{jt}^{2} z_{jt}^{-1}+ \gamma _{j}^{2} z_{jt}\right) \right\} , \end{aligned}$$
(A.1)

where \(\delta _{jt}^{2} =(y_{jt}-a_{j0}-\sum _{i=1}^p\sum _{k=1}^2 a_{jk}^{(i)}y_{k,t-i})^{2}/\sigma _{j}^{2}\), and \(\gamma _{j}^{2} =2+{\xi _{j}^{2}}/{\sigma _{j}^{2}}\).

(2) The full conditional density of \( \varvec{\beta }_{0} |\varvec{y}, \varvec{Z}_{1},\varvec{Z}_{2}, \varvec{\beta }_{11},\varvec{\beta }_{12},\cdots ,\varvec{\beta }_{p1},\varvec{\beta }_{p2} \sim \mathcal {MVN} (\hat{\varvec{\beta }}_{0}, \hat{\varvec{A}}_{00})\), since

$$\begin{aligned}&\pi \left( \varvec{\beta }_{0} |\varvec{y},\varvec{Z}_{1},\varvec{Z}_{2}, \varvec{\beta }_{11},\varvec{\beta }_{12},\cdots ,\varvec{\beta }_{p1},\varvec{\beta }_{p2} \right) \\ \propto&\exp \left\{ -\frac{1}{2}\left( \sum _{t=1}^{n} \left( \left( \tilde{\varvec{y}}_{t0}-\varvec{\beta }_{0}\right) ^{\textsf {T}} \varvec{\Sigma }_{t}^{-1} \left( \tilde{\varvec{y}}_{t0}-\varvec{\beta }_{0}\right) \right) +\left( \varvec{\beta }_{0}-\varvec{C}_{0 0}\right) ^{\textsf {T}} \varvec{\Sigma }_{00}^{-1}\left( \varvec{\beta }_{0}-\varvec{C}_{00}\right) \right) \right\} \\ \propto&\exp \left\{ -\frac{1}{2}\left( -2\left( \sum _{t=1}^{n}\tilde{\varvec{y}}_{t0}^{\textsf {T}} \varvec{\Sigma }_{t}^{-1}+\varvec{C}_{00}^{\textsf {T}} \varvec{\Sigma }_{00}^{-1}\right) \varvec{\beta }_{0} +\varvec{\beta }_{0}^{\textsf {T}}\left( \sum _{t=1}^{n}\varvec{\Sigma }_{t}^{-1}+\varvec{\Sigma }_{00}^{-1}\right) \varvec{\beta }_{0}\right) \right\} \\ \propto&\exp \left\{ -\frac{1}{2}\left( \varvec{\beta }_{0}-\hat{\varvec{\beta }}_{0} \right) ^{\textsf {T}} \varvec{A}_{00}^{-1}\left( \varvec{\beta }_{0}-\hat{\varvec{\beta }}_{0}\right) \right\} , \end{aligned}$$
(A.2)

where \( \hat{\varvec{A}}_{00}^{-1} = \sum _{t=1}^{n}\varvec{\Sigma }_{t}^{-1}+\varvec{\Sigma }_{00}^{-1}\), \( \hat{\varvec{\beta }}_{0} =\left( \left( \sum _{t=1}^{n}\tilde{\varvec{y}}_{t0}^{\top } \varvec{\Sigma }_{t}^{-1}+\varvec{C}_{00}^{\top } \varvec{\Sigma }_{00}^{-1}\right) \hat{\varvec{A}}_{00}\right) ^{\top } \), and \( \tilde{\varvec{y}}_{t0}=\varvec{y}_{t}-\sum _{i=1}^p \sum _{j=1}^2\) \({\varvec{X}_{j,t-i}\varvec{\beta }_{ij}}-\varvec{\xi }\varvec{z}_{t}. \)

(3) The full conditional density of \(\varvec{\beta }_{ij} |\varvec{Z}_{1},\varvec{Z}_{2}, \varvec{y}, \varvec{\beta }_{-ij}\sim \mathcal {MVN} (\hat{\varvec{\beta }}_{ij}, \hat{\varvec{A}}_{ij})\), since

$$\begin{aligned}&\pi \left( \varvec{\beta }_{ij} |\varvec{Z}_{1},\varvec{Z}_{2}, \varvec{y},\varvec{\beta }_{-ij} \right) \\ \propto&\exp \left\{ -\frac{1}{2}\left( \sum _{t=1}^{n} \left( \left( \tilde{\varvec{y}}_{tij}-{X_{j,t-i}\varvec{\beta }_{ij}}\right) ^{\textsf {T}} \varvec{\Sigma }_{t}^{-1} \left( \tilde{\varvec{y}}_{tij}-{X_{j,t-i}\varvec{\beta }_{ij}}\right) \right) +\left( \varvec{\beta }_{ij}-\varvec{C}_{i1}\right) ^{\textsf {T}} \varvec{\Sigma }_{ij}^{-1}\left( \varvec{\beta }_{ij}-\varvec{C}_{ij}\right) \right) \right\} \\ \propto&\exp \left\{ -\frac{1}{2}\left( -2\left( \sum _{t=1}^{n}\tilde{\varvec{y}}_{tij}^{\textsf {T}} \varvec{\Sigma }_{t}^{-1}X_{j,t-i}+\varvec{C}_{ij}^{\textsf {T}} \varvec{\Sigma }_{ij}^{-1}\right) \varvec{\beta }_{ij} +\varvec{\beta }_{ij}^{\textsf {T}}\left( \sum _{t=1}^{n}X_{j,t-i}^{\textsf {T}}\varvec{\Sigma }_{t}^{-1}X_{j,t-i}+\varvec{\Sigma }_{ij}^{-1}\right) \varvec{\beta }_{ij}\right) \right\} \\ \propto&\exp \left\{ -\frac{1}{2}\left( \varvec{\beta }_{ij}-\hat{\varvec{\beta }}_{ij} \right) ^{\textsf {T}} \varvec{A}_{ij}^{-1}\left( \varvec{\beta }_{ij}-\hat{\varvec{\beta }}_{ij}\right) \right\} , \end{aligned}$$
(A.3)

where

\(\hat{\varvec{A}}_{ij}^{-1} = \sum _{t=1}^{n}X_{j,t-i}^{\top }\varvec{\Sigma }_{t}^{-1}X_{j,t-i}+\varvec{\Sigma }_{ij}^{-1}\), \( \hat{\varvec{\beta }}_{ij} =\left( \left( \sum _{t=1}^{n}\tilde{\varvec{y}}_{tij}^{\top } \varvec{\Sigma }_{t}^{-1}X_{j,t-i}+\varvec{C}_{ij}^{\top } \varvec{\Sigma }_{ij}^{-1}\right) \hat{\varvec{A}}_{ij}\right) ^{\top }, \) and \( \tilde{\varvec{y}}_{tij}=\varvec{y}_{t}-\varvec{\beta }_{0}- \sum _{k=1}^p\sum _{s=1}^2{X_{s,t-k}\varvec{\beta }_{ks}+X_{j,t-i}\varvec{\beta }_{ij}}-\varvec{\xi } \varvec{z}_{t}. \)

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Yang, K., Zhao, L., Hu, Q. et al. Bayesian Quantile Regression Analysis for Bivariate Vector Autoregressive Models with an Application to Financial Time Series. Comput Econ (2023). https://doi.org/10.1007/s10614-023-10498-w

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