## 1. Introduction

## 2. Econometric Model and Specifications

#### 2.1. Model Estimation

#### 2.2. Model Features

## 3. Dynamic Analysis

#### 3.1. Hierarchical Structure for Time-Varying Coefficient Vectors

#### 3.2. Prior Assumptions

#### 3.3. Posterior Distributions and MCMC Implementation

## 4. Empirical Application

#### 4.1. The Data and the Empirical Model

#### 4.2. Structural Spillovers and Shock Transmission

#### 4.3. Heterogeneity and Interactions during the Crisis Period and Post-Crisis Consolidation

## 5. Concluding Remarks

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Agudze, Komla Mawulom, Monica Billio, Roberto Casarin, and Francesco Ravazzolo. 2018. Markov Switching Panel with Network Interaction Effects. CAMP Working Paper Series No 1/2018. Oslo, Norway: Norwegian Business School. [Google Scholar]
- Albert, James H., and Siddhartha Chib. 1993. Bayes inference via gibbs sampling of autoregressive time series subject to markov mean and variance shifts. Journal of Business and Economic Statistics 11: 1–15. [Google Scholar]
- Beetsma, Roel, and Massimo Giuliodori. 2011. The effects of government purchase shocks: Review and estimates for the eu. Economic Journal 121: F4–F32. [Google Scholar] [CrossRef]
- Billio, Monica, Roberto Casarin, Francesco Ravazzolo, and Herman K. Van Dijk. 2016. Interconnections between eurozone and us booms and busts: A bayesian panel markov-switching var model. Journal of Applied Econometrics 31: 1352–70. [Google Scholar] [CrossRef]
- Buti, Marco, Daniele Franco, and Hedwig Ongena. 1998. Fiscal discipline and flexibility in emu: The implementation of the stability and growth pact. Oxford Review of Economic Policy 14: 81–97. [Google Scholar] [CrossRef]
- Canova, Fabio, and Matteo Ciccarelli. 2004. Forecasting and turning point predictions in a bayesian panel var model. Journal of Econometrics 120: 327–59. [Google Scholar] [CrossRef]
- Canova, Fabio, and Matteo Ciccarelli. 2009. Estimating multicountry var models. International Economic Review 50: 929–59. [Google Scholar] [CrossRef]
- Canova, Fabio, Matteo Ciccarelli, and Eva Ortega. 2007. Similarities and convergence in g7 cycles. Journal of Monetary Economics 54: 850–78. [Google Scholar] [CrossRef]
- Canova, Fabio, Matteo Ciccarelli, and Eva Ortega. 2012. Do institutional changes affect business cycles? Journal of Economic Dynamics and Control 36: 1520–33. [Google Scholar] [CrossRef]
- Canova, Fabio, and Jane Marrinan. 1997. Sources and propagation of international output cycles: Common shocks or transmission? Journal of International Economics 46: 133–66. [Google Scholar] [CrossRef]
- Carter, Chris K., and Robert Kohn. 1994. On gibbs sampling for state space models. Biometrika 81: 541–53. [Google Scholar] [CrossRef]
- Chib, Siddhartha. 1995. Marginal likelihood from the gibbs output. Journal of the American Statistical Association 90: 1313–21. [Google Scholar] [CrossRef]
- Chib, Siddhartha. 1996. Calculating posterior distributions and model estimates in markov mixture models. Journal of Econometrics 75: 79–97. [Google Scholar] [CrossRef]
- Chib, Siddhartha, and Edward Greenberg. 1995a. Hierarchical analysis of sur models with extensions to correlated serial errors and time-varying parameter models. Journal of Econometrics 68: 409–31. [Google Scholar] [CrossRef]
- Chib, Siddhartha, and Edward Greenberg. 1995b. Understanding the metropolis-hastings algorithm. The American Statistician 49: 327–35. [Google Scholar]
- Chib, Siddhartha, and Ivan Jeliazkov. 2001. Marginal likelihood from the metropolis–hastings output. Journal of the American Statistical Association 96: 270–81. [Google Scholar] [CrossRef]
- Ciccarelli, Matteo, Eva Ortega, and Maria T. Valderrama. 2018. Commonalities and cross-country spillovers in macroeconomic-financial linkages. Journal of Macroeconomics 16: 231–75. [Google Scholar] [CrossRef]
- Ciccarelli, Matteo, and Alessandro Rebucci. 2007. Measuring contagion using a bayesian time-varying coefficient model. Journal of Financial Econometrics 5: 285–320. [Google Scholar] [CrossRef]
- Cogley, Timothy, and Thomas J. Sargent. 2005. Drifts and volatilities: Monetary policy and outcomes in the post wwii u.s. Review of Economic Dynamics 8: 262–302. [Google Scholar] [CrossRef]
- Crespo-Cuaresma, Jesus, and Octavio Fernandez-Amadorb. 2013. Business cycle convergence in emu: A first look at the second moment. Journal of Macroeconomics 37: 265–84. [Google Scholar] [CrossRef]
- De Mol, Christine, Domenico Giannone, and Lucrezia Reichlin. 2008. Forecasting using a large number of predictors: Is bayesian regression a valid alternative to principal components? Journal of Econometrics 146: 318–28. [Google Scholar] [CrossRef]
- Dees, Stephane, Filippo di Mauro, M. Hashem Pesaran, and L. Vanessa Smith. 2007. Exploring the international linkages of the euro area: A global var analysis. Journal of Applied Econometrics 22: 1–38. [Google Scholar] [CrossRef]
- Degiannakis, Stavros, David Duffy, George Filis, and Alexandra Livada. 2016. Business cycle synchronisation in emu: Can fiscal policy bring member-countries closer? Economic Modelling 52: 551–63. [Google Scholar] [CrossRef]
- Doan, Thomas, Robert Litterman, and Christopher Sims. 1984. Forecasting and conditional projection using realistic prior distributions. Econometric Review 3: 1–100. [Google Scholar] [CrossRef]
- Eichengreen, Barry, and Charles Wyplosz. 1998. The stability pact: More than a minor nuisance? Economic Policy 13: 65–113. [Google Scholar] [CrossRef]
- Facchini, François, Mickael Melki, and Andrew Pickering. 2017. Labour costs and the size of government. Oxford Bulletin of Economics and Statistics 79: 251–75. [Google Scholar] [CrossRef]
- Forni, Mario, Marc Hallin, Marco Lippi, and Lucrezia Reichlin. 2000. The generalized dynami factor model: Identification and estimation. The Review of Economics and Statistics 82: 540–54. [Google Scholar] [CrossRef]
- Giannone, Domenico, Marta Banbura, and Lucrezia Reichlin. 2009. Large bayesian vector auto regressions. Journal of Applied Econometrics 25: 71–92. [Google Scholar]
- Gordon, Roger H., and A. Bovenberg. 1996. Why is capital so immobile internationally? possible explanations and implications for capital income taxation. The American Economic Review 86: 1057–75. [Google Scholar]
- Kadiyala, K. Rao, and Sune Karlsson. 1997. Numerical methods for estimation and inference in bayesian var models. Journal of Applied Econometrics 12: 99–132. [Google Scholar] [CrossRef]
- Kaufmann, Sylvia. 2010. Dating and forecasting turning points by bayesian clustering with dynamic structure: A suggestion with an application to austrian data. Journal of Applied Econometrics 25: 309–44. [Google Scholar] [CrossRef]
- Kaufmann, Sylvia. 2015. K-state switching models with time-varying transition distributions. does loan growth signal stronger effects of variables on inflation? Journal of Econometrics 187: 82–94. [Google Scholar] [CrossRef]
- Koop, Gary. 1996. Parameter uncertainty and impulse response analysis. Journal of Econometrics 72: 135–49. [Google Scholar] [CrossRef]
- Krolzig, Hans-Martin. 1997. Markov Switching Vector Autoregressions: Modelling, Statistical Inference and Application to Business Cycle Analysis. Berlin, Germany: Springer. [Google Scholar]
- Krolzig, Hans-Martin. 2000. Predicting Markov-Switching Vector Autoregressive Processes. Nuffield College Economics Working Papers 2000-WP31. Oxford, UK: Nuffield College Oxford University. [Google Scholar]
- Lane, Philip R., and Gian Maria Milesi-Ferretti. 2007. The external wealth of nations mark ii: Revised and extended estimates of foreign assets and liabilities, 1970–2004. Journal of International Economics 73: 223–50. [Google Scholar] [CrossRef]
- Litterman, Robert B. 1985. Ljung, l. and soderstrom, t. Automatica 21: 25–38. [Google Scholar]
- Litterman, Robert B. 1986. Forecasting with bayesian vector autoregressions five years of experience. Journal of Business and Economic Statistics 4: 25–38. [Google Scholar]
- Mastrogiacomo, Mauro, Nicole M. Bosch, Miriam D. Gielen, and Egbert L. Jongen. 2017. Heterogeneity in labour supply responses: Evidence from a major tax reform. Oxford Bulletin of Economics and Statistics 79: 769–96. [Google Scholar] [CrossRef]
- Pesaran, Hashem M., and Yongcheol Shinb. 1998. Generalized impulse response analysis in linear multivariate models. Economics Letters 58: 17–29. [Google Scholar] [CrossRef][Green Version]
- Pesaran, M. Hashem, Til Schuermann, and Scott M. Weiner. 2004. Modelling regional interdependencies using a global error correcting macroeconometric model. Journal of Business and Economic Statistics 22: 129–62. [Google Scholar] [CrossRef]
- Reinhart, Carmen M., and Kenneth S. Rogoff. 2009. The aftermath of financial crises. American Economic Review 99: 466–72. [Google Scholar] [CrossRef]
- Sims, Christopher A., and Tao Zha. 1998. Bayesian methods for dynamic multivariate models. International Economic Review 39: 949–68. [Google Scholar] [CrossRef]
- Sims, Christopher A., and Tao Zha. 2006. Were there regime switches in U.S. monetary policy? American Economic Review 96: 54–81. [Google Scholar] [CrossRef]
- Sorensen, Bent E., and Oved Yosha. 1998. International risk sharing and european monetary unification. Journal of International Economics 45: 211–38. [Google Scholar] [CrossRef]

1 | Hidden or Latent factors are variables that are not directly observed but are rather inferred from other variables that are observed and, hence, directly measured. |

2 | To be more precise, if the elements of ${A}_{t}\left(L\right)$, ${B}_{t}\left(L\right)$, and ${C}_{t}\left(L\right)$ are stacked over i, it is possible to obtain matrices that are not block-diagonal for at least some l. |

3 | The vec operator transforms a matrix into a vector by stacking the columns of the matrix, one underneath the other. |

4 | A proxy variable is an easily measurable variable that is used in place of a variable that cannot be (directly) measured or is difficult to measure. |

5 | The Wishart distribution is a multivariate extension of ${\chi}^{2}$ distribution and, in Bayesian statistics, corresponds to the conjugate prior of the inverse-covariance matrix of a multivariate normal random vector. |

6 | The Gamma Distribution is a two-parameter family of continuous probability distributions that provides the probabilities of occurrence of different possible outcomes in an experiment. |

7 | For instance, the Minnesota priors are based on an approximation that involves replacing ${\Sigma}_{e}$ with an estimate, ${\widehat{\Sigma}}_{e}$. See, e.g., Doan et al. (1984) and Litterman (1986). |

8 | These implementations do not allow for the use of the Minnesota prior since its covariance matrix is written in terms of blocks that vary across equations. |

9 | See, e.g., Chib and Greenberg (1995b). |

10 | The $weight{s}_{it,j}$ component corresponds to the sum of $rweight{s}_{it,j}$ and $fweight{s}_{it,j}$. |

11 | Own computations. |

12 | The conditional projection for output growth is the one that the model would have obtained over the same period conditionally on the actual path of unexpected shock for that period. |

13 | The unconditional projection is the one that the model would obtain for output growth for that period only on the basis of historical information, and it is consistent with a model-based forecast path for the other variables. |

**Figure 1.**Systemic Contributions of the $productivity$ given a $1\%$ shock to real and financial dimensions are drawn as standard deviations of the variables in the system and in year-on-year growth rates. They account for ${\chi}_{1t}{\widehat{\beta}}_{1t}$ (plot

**a**) and ${\chi}_{2t}{\widehat{\beta}}_{2t}$ (plot

**b**) cross-country indicators, where ${\widehat{\beta}}_{1t}$ and ${\widehat{\beta}}_{2t}$ are posterior means.

**Figure 2.**Systemic Contributions of the $productivity$ given a $1\%$ shock to real and financial dimensions are drawn as standard deviations of the variables in the system and in year-on-year growth rates. They account for ${\chi}_{3t}{\widehat{\beta}}_{3t}$ (plot

**a**), ${\chi}_{4t}{\widehat{\beta}}_{4t}$ (plot

**b**), ${\chi}_{5t}{\widehat{\beta}}_{5t}$ (plot

**c**), and ${\chi}_{6t}{\widehat{\beta}}_{6t}$ (plot

**d**) cross-country indicators, where ${\widehat{\beta}}_{3t}$, ${\widehat{\beta}}_{4t}$, ${\widehat{\beta}}_{5t}$, and ${\widehat{\beta}}_{6t}$ are posterior means.

**Figure 3.**Systemic Contributions of the $productivity$ given a $1\%$ shock to real and financial dimensions are drawn as standard deviations of the variables in the system and in year-on-year growth rates. They account for ${\chi}_{7t,1}{\widehat{\beta}}_{7t,1}$ (plot

**a**), ${\chi}_{7t,2}{\widehat{\beta}}_{7t,2}$ (plot

**a**), ${\chi}_{7t,3}{\widehat{\beta}}_{7t,3}$ (plot

**b**), and ${\chi}_{7t,4}{\widehat{\beta}}_{7t,4}$ (plot

**b**) variable-specific indicators, where ${\widehat{\beta}}_{7t,{M}_{v}}$’s are posterior means.

**Figure 4.**Systemic Contributions of the $productivity$ given a $1\%$ shock to real and financial dimensions are drawn as standard deviations of the variables in the system and in year-on-year growth rates. They account for ${\chi}_{8t,1}{\widehat{\beta}}_{8t,1}$ (plot

**b**), ${\chi}_{8t,2}{\widehat{\beta}}_{8t,2}$ (plot

**a**), ${\chi}_{8t,3}{\widehat{\beta}}_{8t,3}$ (plot

**a**), and ${\chi}_{8t,4}{\widehat{\beta}}_{8t,4}$ (plot

**b**) common indicators, where ${\widehat{\beta}}_{8t,{M}_{c}}$’s are posterior means.

**Figure 5.**Systemic Contributions of the $productivity$ given a $1\%$ shock to real and financial dimensions are drawn as standard deviations of the variables in the system and in year-on-year growth rates, focusing on the recent financial crisis (plots

**a**,

**c**) and post-crisis consolidation (plots

**b**,

**d**) periods. They account for ${\chi}_{5t}{\widehat{\beta}}_{5t}$ and ${\chi}_{6t}{\widehat{\beta}}_{6t}$ cross-country indicators, where ${\widehat{\beta}}_{5t}$ and ${\widehat{\beta}}_{6t}$ are posterior means.

**Figure 6.**The figure draws CF of the productivity, general government debt, real GDP growth rate, current account balance, general government spending, and the generalized entropy index from $1999q1$ to $2020q2$. The latter corresponds to Theil’s Entropy and is computed by weighing the GDP with the population in terms of proportions with respect to the total. It can be viewed as a measure of divergence and economic inequality. Here, forecasts from $2016q1$ to $2020q2$ correspond to conditional projections of each variable drawn in the $SPBVAR\left(1\right)$.

**Figure 7.**Systemic Contributions of the $productivity$ given a $1\%$ shock to real and financial dimensions are drawn as standard deviations of the variables in the system and in year-on-year growth rates, focusing on the recent financial crisis (plot

**a**) and post-crisis consolidation period (plot

**b**). They account for ${\chi}_{7t,3}{\widehat{\beta}}_{7t,3}$ and ${\chi}_{7t,4}{\widehat{\beta}}_{7t,4}$ variable-specific indicators, where ${\widehat{\beta}}_{7t,3}$ and ${\widehat{\beta}}_{7t,4}$ are posterior means.

**Figure 8.**Systemic Contributions of the productivity given a $1\%$ shock to real and financial dimensions are drawn as standard deviations of the variables in the system and in year-on-year growth rates, focusing on the recent financial crisis (plots

**a**,

**c**) and post-crisis consolidation (plots

**b**,

**d**) periods. They account for ${\chi}_{8t,1}{\widehat{\beta}}_{8t,1}$, ${\chi}_{8t,2}{\widehat{\beta}}_{8t,2}$, ${\chi}_{8t,3}{\widehat{\beta}}_{8t,3}$, and ${\chi}_{8t,4}{\widehat{\beta}}_{8t,4}$ common indicators, where ${\widehat{\beta}}_{8t,{M}_{c}}$’s are posterior means.

Test | Test Statistics | Degrees of Freedom | p-Value |
---|---|---|---|

$LG{B}_{\pi}$ | 5665 | 2430 | 0.00 |

${P}_{\pi}$ | 526.51 | 540 | 0.6531 |

$ML{E}_{f}$ | 29.41 | 12 | 0.00342 |

Shock/Response | ${\mathit{y}}_{1}$ | ${\mathit{y}}_{2}$ | … | ${\mathit{y}}_{\mathit{n}}$ | To Others |
---|---|---|---|---|---|

${y}_{1}$ | $I{R}_{{y}_{1}\to {y}_{1}}$ | $I{R}_{{y}_{1}\to {y}_{2}}$ | … | $I{R}_{{y}_{1}\to {y}_{n}}$ | ${\sum}_{j=1}^{N}I{R}_{{y}_{1}\to {y}_{j}}$$j\ne 1$ |

${y}_{2}$ | $I{R}_{{y}_{2}\to {y}_{1}}$ | $I{R}_{{y}_{2}\to {y}_{2}}$ | … | $I{R}_{{y}_{2}\to {y}_{n}}$ | ${\sum}_{j=1}^{N}I{R}_{{y}_{2}\to {y}_{j}}$$j\ne 2$ |

⋮ | ⋮ | ⋮ | ⋱ | ⋮ | ⋮ |

${y}_{n}$ | $I{R}_{{y}_{n}\to {y}_{1}}$ | $I{R}_{{y}_{n}\to {y}_{2}}$ | … | $I{R}_{{y}_{n}\to {y}_{n}}$ | ${\sum}_{j=1}^{N}I{R}_{{y}_{n}\to {y}_{j}}$$j\ne n$ |

From Others | ${\sum}_{j=1}^{N}I{R}_{{y}_{j}\to {y}_{1}}$ | ${\sum}_{j=1}^{N}I{R}_{{y}_{j}\to {y}_{2}}$ | … | ${\sum}_{j=1}^{N}I{R}_{{y}_{j}\to {y}_{n}}$ | ${\sum}_{j=1}^{N}(I{R}_{{y}_{i}\to {y}_{j}}-I{R}_{{y}_{j}\to {y}_{i}})$ |

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