Triple the Gamma—A Unifying Shrinkage Prior for Variance and Variable Selection in Sparse State Space and TVP Models
Abstract
:1. Introduction
2. The Triple Gamma as a Prior for Variance Parameters
2.1. Motivation and Definition
2.2. Properties of the Triple Gamma Prior
 (a)
 It has following representation as a localglobal shrinkage prior:$$\begin{array}{cc}\hfill & {\sqrt{\theta}}_{j}{\psi}_{j}^{2},{\kappa}_{B}^{2}\sim \mathcal{N}\left(0,\frac{2}{{\kappa}_{B}^{2}}{\psi}_{j}^{2}\right),\phantom{\rule{1.em}{0ex}}{\psi}_{j}^{2}{a}^{\xi},{c}^{\xi}\sim \mathrm{F}\left(2{a}^{\xi},2{c}^{\xi}\right).\hfill \end{array}$$
 (b)
 The marginal prior $p({\sqrt{\theta}}_{j}{\varphi}^{\xi},{a}^{\xi},{c}^{\xi})$ takes the following form with ${\varphi}^{\xi}=\frac{2{c}^{\xi}}{{\kappa}_{B}^{2}{a}^{\xi}}$,$$\begin{array}{c}\hfill p({\sqrt{\theta}}_{j}{\varphi}^{\xi},{a}^{\xi},{c}^{\xi})=\frac{\Gamma ({c}^{\xi}+\frac{1}{2})}{\sqrt{2\pi {\varphi}^{\xi}}B({a}^{\xi},{c}^{\xi})}U\left({c}^{\xi}+\frac{1}{2},\frac{3}{2}{a}^{\xi},\frac{{\theta}_{j}}{2{\varphi}^{\xi}}\right),\end{array}$$$$\begin{array}{c}\hfill U\left(a,b,z\right)=\frac{1}{\Gamma \left(a\right)}{\int}_{0}^{\infty}{e}^{zt}{t}^{a1}{(1+t)}^{ba1}dt.\end{array}$$
 (a)
 For $0<{a}^{\xi}<0.5$ and small values of ${\sqrt{\theta}}_{j}$,$$\begin{array}{c}\hfill p({\sqrt{\theta}}_{j}{\varphi}^{\xi},{a}^{\xi},{c}^{\xi})=\frac{\Gamma (\frac{1}{2}{a}^{\xi})}{\sqrt{\pi}{\left(2{\varphi}^{\xi}\right)}^{{a}^{\xi}}B({a}^{\xi},{c}^{\xi})}{\left(\frac{1}{{\sqrt{\theta}}_{j}}\right)}^{12{a}^{\xi}}+O\left(1\right).\end{array}$$
 (b)
 For ${a}^{\xi}=0.5$ and small values of ${\sqrt{\theta}}_{j}$,$$\begin{array}{c}\hfill p({\sqrt{\theta}}_{j}{\varphi}^{\xi},{a}^{\xi},{c}^{\xi})=\frac{1}{\sqrt{2\pi {\varphi}^{\xi}}B(0.5,{c}^{\xi})}\left(log{\theta}_{j}+log\left(2{\varphi}^{\xi}\right)\psi ({c}^{\xi}+0.5)\right)+O({\theta}_{j}log{\theta}_{j}),\end{array}$$where $\psi (\xb7)$ is the digamma function.
 (c)
 For ${a}^{\xi}>0.5$,$$\begin{array}{c}\hfill {\displaystyle \underset{{\sqrt{\theta}}_{j}\to 0}{lim}}p({\sqrt{\theta}}_{j}{\varphi}^{\xi},{a}^{\xi},{c}^{\xi})=\frac{\Gamma ({c}^{\xi}+\frac{1}{2})\Gamma ({a}^{\xi}\frac{1}{2})}{\sqrt{2\pi {\varphi}^{\xi}}\Gamma \left({c}^{\xi}\right)\Gamma \left({a}^{\xi}\right)}.\end{array}$$
 (d)
 As ${\sqrt{\theta}}_{j}\to \infty $,$$\begin{array}{c}\hfill p({\sqrt{\theta}}_{j}{\varphi}^{\xi},{a}^{\xi},{c}^{\xi})=\frac{\Gamma ({c}^{\xi}+\frac{1}{2}){\left(2{\varphi}^{\xi}\right)}^{{c}^{\xi}}}{\sqrt{\pi}B({a}^{\xi},{c}^{\xi})}{\left(\frac{1}{{\sqrt{\theta}}_{j}}\right)}^{2{c}^{\xi}+1}\left[1+O\left(\frac{1}{{\theta}_{j}}\right)\right].\end{array}$$
2.3. Relation of the Triple Gamma to Other Shrinkage Priors
2.4. Using the Triple Gamma for Variance Selection in TVP Models
3. Shrinkage Profiles and BMALike Behavior
3.1. Shrinkage Profiles
3.2. BMAType Behaviour
4. MCMC Algorithm
Algorithm 1. MCMC inference for TVP models under the triple gamma prior. 
Choose starting values for all global shrinkage parameters $({a}^{\tau},{c}^{\tau},{\lambda}_{B}^{2},{a}^{\xi},{c}^{\xi},{\kappa}_{B}^{2})$ and local shrinkage parameters ${\{{\stackrel{\u02c7}{\tau}}_{j}^{2},{\stackrel{\u02c7}{\lambda}}_{j}^{2},{\stackrel{\u02c7}{\xi}}_{j}^{2},{\stackrel{\u02c7}{\kappa}}_{j}^{2}\}}_{j=1}^{d}$, and repeat the following steps:

5. Applications to TVPVARSV Models
5.1. Model
5.2. A Brief Sketch of the TVPVARSV MCMC Algorithm
Algorithm 2. MCMC inference for TVPVARSV models under the triple gamma prior. 
Choose starting values for all global and local shrinkage parameters in prior (31) for each equation and repeat the following steps: 
For $i=1,\dots ,m$, update all the unknowns in the ith equation:

5.3. Illustrative Example with Simulated Data
5.4. Modeling Area Macroeconomic and Financial Variables in the Euro Area
6. Conclusions
Author Contributions
Conflicts of Interest
Appendix A. Proofs
Appendix B. Details on the MCMC Scheme
Appendix C. Posterior Paths for the Simulated Data
Appendix D. Application
Appendix D.1. Data Overview
Variable  Abbreviation  Description  Tcode 

Real output  YER  Gross domestic product (GDP) at market prices in millions of Euros, chain linked volume, calendar and seasonally adjusted data, reference year 1995.  1 
Prices  YED  GDP deflator, index base year 1995. Defined as the ratio of nominal and real GDP.  1 
Shortterm interest rate  STN  Nominal shortterm interest rate, Euribor 3month, percent per annum  2 
Investment  ITR  Gross fixed capital formation in millions of Euros, chain linked volume, calendar and seasonally adjusted data, reference year 1995.  1 
Consumption  PCR  Individual consumption expenditure in millions of Euros, chain linked volume, calendar and seasonally adjusted data, reference year 1995.  1 
Exchange rate  EEN  Nominal effective exchange rate, Euro area19 countries visàvis the NEER38 group of main trading partners, index base Q1 1999.  1 
Unemployment  URX  Unemployment rate, percentage of civilian work force, total across age and sex, seasonally adjusted, but not working day adjusted.  2 
Appendix D.2. Posterior Paths
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1  Let ${f}_{{\sqrt{\theta}}_{j}}\left(x\right)$ and ${F}_{{\sqrt{\theta}}_{j}}\left(x\right)$ be, respectively, the pdf and cdf of the random variable ${\sqrt{\theta}}_{j}$. The cdf ${F}_{{\theta}_{j}}\left(x\right)$ of the random variable ${\theta}_{j}$ is given by
$$\begin{array}{c}\hfill {F}_{{\theta}_{j}}\left(x\right)=\mathrm{Pr}({\theta}_{j}\le x)=\mathrm{Pr}(\sqrt{x}\le {\sqrt{\theta}}_{j}\le \sqrt{x})={F}_{{\sqrt{\theta}}_{j}}\left(\sqrt{x}\right){F}_{{\sqrt{\theta}}_{j}}(\sqrt{x})=2{F}_{{\sqrt{\theta}}_{j}}\left(\sqrt{x}\right),\end{array}$$
$$\begin{array}{c}\hfill {f}_{{\theta}_{j}}\left(x\right)=\frac{\mathrm{d}\phantom{\rule{0.166667em}{0ex}}{F}_{{\theta}_{j}}\left(x\right)}{\mathrm{d}\phantom{\rule{0.166667em}{0ex}}x}={f}_{{\sqrt{\theta}}_{j}}\left(\sqrt{x}\right)/\sqrt{x}.\end{array}$$

2  Note that the $X\sim \mathcal{BP}\left(a,b\right)$distribution has pdf
$$f\left(x\right)=\frac{1}{B(a,b)}\frac{{x}^{a1}}{{(1+x)}^{a+b}}.$$
Furthermore, $Y=X/(1+X)$ follows the $\mathcal{B}\left(a,b\right)$distribution. 
3  The pdf of a $\mathrm{SBeta}2\left(a,c,\varphi \right)$distribution reads:
$$\begin{array}{c}\hfill f\left(x\right)=\frac{1}{{\varphi}^{a}B(a,c)}\phantom{\rule{0.166667em}{0ex}}{x}^{a1}\phantom{\rule{0.166667em}{0ex}}{(1+x/\varphi )}^{(a+c)},\end{array}$$

4  Using (3), we obtain the following prior for ${\rho}_{j}=1/(1+{\psi}_{j}^{2})$ by the law of transformation of densities:
$$\begin{array}{c}\hfill p\left({\rho}_{j}\right)={\displaystyle \frac{1}{\Gamma \left({a}^{\xi}\right)}}{\left(\frac{{a}^{\xi}{\kappa}_{B}^{2}}{2}\right)}^{{a}^{\xi}}{(1{\rho}_{j})}^{{a}^{\xi}1}{{\rho}_{j}}^{({a}^{\xi}+1)}exp\left(\left(\frac{1{\rho}_{j}}{{\rho}_{j}}\right)\frac{{a}^{\xi}{\kappa}_{B}^{2}}{2}\right).\end{array}$$

5  The pdf of the $\mathcal{GIG}\left(p,a,b\right)$distribution is given by
$$\begin{array}{c}\hfill {\displaystyle f\left(x\right)=\frac{{(a/b)}^{p/2}}{2{K}_{p}\left(\sqrt{ab}\right)}{x}^{p1}{e}^{\frac{1}{2}(ax+b/x)},}\end{array}$$

Prior for ${\sqrt{\mathit{\theta}}}_{\mathit{j}}$  ${\mathit{a}}^{\mathit{\xi}}$  ${\mathit{c}}^{\mathit{\xi}}$  ${\mathit{\kappa}}_{\mathit{B}}^{2}$  ${\mathit{\varphi}}^{\mathit{\xi}}$  

$\mathcal{N}\left(0,{\psi}_{j}^{2}\right),{\psi}_{j}^{2}\sim GG\left({a}^{\xi},{c}^{\xi},{\varphi}^{\xi}\right)$  normalgammagamma  ${a}^{\xi}$  ${c}^{\xi}$  $\frac{2{c}^{\xi}}{{\varphi}^{\xi}{a}^{\xi}}$  ${\varphi}^{\xi}$ 
$\mathcal{N}\left(0,\frac{1}{{\kappa}_{j}}1\right),{\kappa}_{j}\sim \mathcal{TPB}\left({a}^{\xi},{c}^{\xi},{\varphi}^{\xi}\right)$  generalized beta mixture  ${a}^{\xi}$  ${c}^{\xi}$  $\frac{2{c}^{\xi}}{{\varphi}^{\xi}{a}^{\xi}}$  ${\varphi}^{\xi}$ 
$\mathcal{N}\left(0,{\psi}_{j}^{2}\right),{\psi}_{j}^{2}\sim \mathrm{SBeta}2\left({a}^{\xi},{c}^{\xi},{\varphi}^{\xi}\right)$  hierarchical scaled beta2  ${a}^{\xi}$  ${c}^{\xi}$  $\frac{2{c}^{\xi}}{{\varphi}^{\xi}{a}^{\xi}}$  ${\varphi}^{\xi}$ 
$\mathcal{DE}\left(0,\sqrt{2}\phantom{\rule{0.166667em}{0ex}}{\psi}_{j}\right),{\psi}_{j}^{2}\sim \mathcal{G}\left({c}^{\xi},\frac{1}{{\lambda}^{2}}\right)$  normalexponentialgamma  1  ${c}^{\xi}$  $2{\lambda}^{2}{c}^{\xi}$  $\frac{1}{{\lambda}^{2}}$ 
$\mathcal{N}\left(0,{\tau}^{2}{\psi}_{j}^{2}\right),{\psi}_{j}\sim {t}_{1}$  Horseshoe  $\frac{1}{2}$  $\frac{1}{2}$  $\frac{2}{{\tau}^{2}}$  ${\tau}^{2}$ 
$\mathcal{N}\left(0,\frac{1}{{\kappa}_{j}}1\right),{\kappa}_{j}\sim \mathcal{B}\left(1/2,1\right)$  StrawdermanBerger  $\frac{1}{2}$  1  4  1 
$\mathcal{N}\left(0,{\tau}^{2}{\tilde{\xi}}_{j}\right),{\tilde{\xi}}_{j}\sim \mathcal{G}\left({a}^{\xi},{a}^{\xi}\right)$  double gamma  ${a}^{\xi}$  ∞  $\frac{2}{{\tau}^{2}}$   
$\mathcal{N}\left(0,{\tau}^{2}{\tilde{\xi}}_{j}\right),{\tilde{\xi}}_{j}\sim \mathcal{E}\left(1\right)$  Lasso  1  ∞  $\frac{2}{{\tau}^{2}}$   
${t}_{\nu}\left(0,{\tau}^{2}\right)$  halft  ∞  $\frac{\nu}{2}$  $\frac{2}{{\tau}^{2}}$   
${t}_{1}\left(0,{\tau}^{2}\right)$  halfCauchy  ∞  $\frac{1}{2}$  $\frac{2}{{\tau}^{2}}$   
$\mathcal{N}\left(0,{B}_{0}\right)$  normal  ∞  ∞  $\frac{2}{{B}_{0}}$   
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Cadonna, A.; FrühwirthSchnatter, S.; Knaus, P. Triple the Gamma—A Unifying Shrinkage Prior for Variance and Variable Selection in Sparse State Space and TVP Models. Econometrics 2020, 8, 20. https://doi.org/10.3390/econometrics8020020
Cadonna A, FrühwirthSchnatter S, Knaus P. Triple the Gamma—A Unifying Shrinkage Prior for Variance and Variable Selection in Sparse State Space and TVP Models. Econometrics. 2020; 8(2):20. https://doi.org/10.3390/econometrics8020020
Chicago/Turabian StyleCadonna, Annalisa, Sylvia FrühwirthSchnatter, and Peter Knaus. 2020. "Triple the Gamma—A Unifying Shrinkage Prior for Variance and Variable Selection in Sparse State Space and TVP Models" Econometrics 8, no. 2: 20. https://doi.org/10.3390/econometrics8020020