# Structural Panel Bayesian VAR with Multivariate Time-Varying Volatility to Jointly Deal with Structural Changes, Policy Regime Shifts, and Endogeneity Issues

## Abstract

**:**

## 1. Introduction

## 2. Econometric Model

`i`) ${Y}_{im,t}$ is an $NM\xb71$ vector of outcomes to be predicted for each i for a given m. (

`ii`) ${A}_{im,j\varrho}$ is an $NM\xb7NM$ matrix of coefficients for each pair of countries $(i,j)$ for a given m, with $m=1,\dots ,M$, and ${Y}_{im,t-\lambda}$ is an $NM\xb71$ vector of observed lagged variables for each i for a given m to address economic–financial issues. More precisely, stacking for m, I decompose it in ${Y}_{i,t-\lambda}={\left[{Y}_{i,t-\lambda}^{{\underset{\_}{o}}^{\prime}},{Y}_{i,t-\lambda}^{{\underset{\_}{c}}^{\prime}}\right]}^{\prime}$, with ${Y}_{i,t-\lambda}^{{\underset{\_}{o}}^{\prime}}$ denoting lagged outcomes to capture the persistence and ${Y}_{i,t-\lambda}^{{\underset{\_}{c}}^{\prime}}$ including lagged control variables such as general economic–financial conditions. (

`iii`) ${B}_{iq,j\varpi}$ is an $NQ\xb7NQ$ matrix of coefficients for each pair of countries $(i,j)$ for a given q, with $q=1,\dots ,Q$, and ${W}_{iq,t-\lambda}$ is an $NQ\xb71$ vector including a set of (directly) observed lagged variables for each i for a given q to evaluate economic–institutional interdependencies (such as additional transmission channels). (

`iv`) ${\ddot{B}}_{i\tilde{q},j\ddot{\varpi}}$ is an $N\tilde{Q}\xb7N\tilde{Q}$ matrix of coefficients for each pair of countries $(i,j)$ for a given $\tilde{q}$, with $\tilde{q}=1,\dots ,\tilde{Q}$, and ${\ddot{W}}_{i\tilde{q},t-\lambda}$ is an $N\tilde{Q}\xb71$ vector including a set of additional (directly) observed lagged variables for each i for a given $\tilde{q}$ to evaluate policy implications and interactions (such as policy tools). (

`v`) ${C}_{i\xi ,j\ddot{\phi}}$ is an $N\Xi \xb7N\Xi $ matrix of coefficients for each pair of countries $(i,j)$ for a given $\xi $, with $\xi =1,\dots ,\Xi $, and ${Z}_{i\xi ,t-\lambda}$ is an $N\Xi \xb71$ vector including a set of observed lagged proxy7 variables for each i for a given $\xi $ to address further economic–financial linkages (e.g., territorial competitiveness and infrastructural system) and economic–institutional implications (e.g., competitiveness developments and macroeconomic imbalances). Here, all variables in the system are endogenous and time-varying.

`variable-specific`effects), specific to a country given a set of observable and directly measured variables (

`country-specific`effects), common among countries and sectors (

`common`effects), and specific to additional time-varying effects among countries and sectors directly affecting the outcomes in ${Y}_{t}$ (

`misspecified`effects due to endogeneity issues). Such a reparametrization has three appealing features. $\left(i\right)$ First, it reduces the problem of estimating high dimensional (potential) combinations of time-varying coefficients into the problem of estimating a smaller number of loadings on some linear combinations of the right-hand variables of (1). $\left(ii\right)$ Second, since the loadings of the SNLR model are observable and time-varying linear combinations of the right-hand variables of (1), an estimable hierarchical structure is feasible and suitable for policy purposes and strategies. $\left(iii\right)$ Third, representing the main thrust of this study, the parsimonious SNRL is able to identify additional common or heterogeneous effects between different countries and sectors that vary over time and directly affect the variables of interest in ${Y}_{t}$.

- Model I (${M}_{I}$): A benchmark model with no change-points, denoting the ‘
`General Case`’.Here, ${\eta}_{t}\sim N(0,\dot{\Sigma})$ and depends on the only disturbances contained in ${u}_{t}$, with $\dot{\Sigma}=diag({\Sigma}_{e}^{{}^{\prime}},{\Sigma}_{e}^{{}^{\prime}},\dots ,{\Sigma}_{e}^{{}^{\prime}})$. The ${M}_{I}$ would corresponds to the standard SPBVAR, with ${h}_{0}=0$, ${h}_{t}={h}_{t-1}$, and ${\Sigma}_{t}=\Sigma $. - Model II (${M}_{II}$): A benchmark model with change-points in the only log-volatilities, denoting the ’
`Special Case`’.Here, ${\eta}_{t}\sim N(0,\tilde{\Sigma})$ and depends on the only disturbances contained in ${E}_{t}$, with $\tilde{\Sigma}={\sigma}_{t}$. The ${M}_{II}$ refers to the case of structural breaks because of (potential) unmodeled dynamics12 in ${\gamma}_{t}$, with ${h}_{0}\ne 0$, ${h}_{t}$ evolving over time, ${\Sigma}_{t}\ne \Sigma $, and uncorrelatedness between the innovations ${\eta}_{t}$ and ${\tilde{v}}_{t}$. - Model III (${M}_{III}$): A benchmark model with change-points in either time-varying parameters or log-volatilities, denoting the ’
`Full Case`’.Here, ${\eta}_{t}\sim N(0,\ddot{\Sigma})$ and depends on the disturbances contained in ${v}_{t}$ and ${u}_{t}$, with $\ddot{\Sigma}={\sigma}_{t}\xb7diag({\Sigma}_{e}^{{}^{\prime}},{\Sigma}_{e}^{{}^{\prime}},\dots ,{\Sigma}_{e}^{{}^{\prime}})$. The ${M}_{III}$ refers to the case of structural breaks because of both unmodeled dynamics and policy regime shifts, with ${h}_{0}\ne 0$, ${\Sigma}_{t}\ne \Sigma $, and ${h}_{t}$ evolving over time.

`General Case`’ (${M}_{I}$) with no change-points and ${M}_{{k}^{*}}$ refers to all possible model solutions according to the ’

`Special Case`’ (${M}_{II}$) or the ’

`Full Case`’ (${M}_{III}$). The higher lBF denotes the final solution having higher Posterior Model Probabilities (PMPs)15 according to a generalized version of the Kass and Raftery (1995)’s scale of evidence:

#### 2.1. Model Features

## 3. Dynamic Analysis

#### 3.1. Hierarchical Prior Setups and Assumptions

#### 3.2. Posterior Distributions and MCMC Implementations

#### 3.2.1. Conditional Likelihood and Kalman Filter Technique for Time-Varying Parameters

#### 3.2.2. Metropolis–Hastings Algorithm for ${h}_{it}$

#### 3.3. Analytical Integration for Integrating out the Time-Varying Volatilities

#### 3.3.1. Expectation Step (E-Step)

#### 3.3.2. Maximization Step (M-step)

`E-step`: Compute ${\Phi}_{\beta}$, $\tilde{\beta}$, and $\overline{\theta}$ given the current value ${h}_{it}^{\nu -1}$, with $\nu $ denoting the $\nu -th$ iteration.`M-step`: Maximise $\Psi \left(h\right|{h}^{\nu -1})$ with respect to h by the N-R method. That is,$${h}^{\nu}=\underset{h}{argmax}\phantom{\rule{1.em}{0ex}}\Psi \left(h\right|{h}^{\nu -1}).$$- Compute ${g}_{\Psi}$ and ${\mathcal{H}}_{\Psi}$ from ${\Psi}_{\beta}$, $\tilde{\beta}$, and $\overline{\theta}$ obtained in $\left(A\right)$, and set $h={h}^{(\overline{m}-1,\nu -1)}$.
- Update ${h}^{(\overline{m},\nu -1)}={h}^{(\overline{m}-1,\nu -1)}-\left({\mathcal{H}}_{\Psi}^{-1}\xb7{g}_{\Psi}\right)$.
- Repeat steps $\left(A\right)$–$\left(D\right)$ until some convergence criterion is met at the OARs in (9). Thus, terminate the iteration and set ${h}^{\nu}={h}^{(\overline{m},\nu -1)}$, denoting that a certain change-point among time-varying coefficient vectors and log-volatilities has been assessed correctly.

## 4. Data Description and Empirical Model

`Full Case`’ (${M}_{III}$), where structural changes and policy regime shifts hold in either time-varying parameters or log-volatilities (see Table 2). It accounts for: two of the country-specific factors (${\chi}_{7t}{\widehat{\beta}}_{7t}$ and ${\chi}_{8t}{\widehat{\beta}}_{8t}$); the cross-country variable-specific factor (${\chi}_{9t}{\widehat{\beta}}_{9t}$) belonging to the variable groups ${M}_{v3}$ and ${M}_{v4}$; and the common factor (${\chi}_{10t}{\widehat{\beta}}_{10t}$) belonging to the common group ${M}_{c2}$. All remaining empirical results embrace the ’

`Special Case`’ (${M}_{II}$), except for two factors concerning the ’

`General Case`’ (${M}_{I}$). They correspond to the first two country-specific indicators (${\chi}_{1t}{\widehat{\beta}}_{1t}$ and ${\chi}_{2t}{\widehat{\beta}}_{2t}$). These findings highlight the performance and then the potential of the SPBVAR-MTV model pointing out that: $\left(i\right)$ change-points and policy regime shifts need to be taken into account when dealing with macroeconomic–financial linkages in multicountry dynamic panel setups; $\left(ii\right)$ multiple structural changes in time-varying log-volatilities occur when evaluating international transmission channels and policy implications among countries and sectors in both the real and the financial dimensions; and $\left(iii\right)$ change-points and policy regime shifts in either time-varying coefficients or log-volatilities occur when accounting for economic–institutional implications to investigate unobserved heterogeneity and misspecified dynamics among country- and variable-specific factors and common features.

## 5. Macroeconomic-Financial Linkages with Structural Changes and Policy Regime Shifts: A Counterfactual Assessment

#### 5.1. International Spillovers and Policy Issues among CEWE Economies

#### 5.2. Unobserved Heterogeneity and Misspecified Dynamics Accounting for Additional Time-Variant Factors

`pseudo-shock`’ in the short term to catch up with the economic growth of the other euro partecipants31 (inward spillovers). Second, structural–institutional implications along with policy reforms affect the intensity (or volatility) of spillover effects in CEE and even more in BLS countries—because of larger current account deficits and lower real economic convergence—via international transmission channels, that allow in turn financial shocks to spill over. Third, persistent cross-country heterogeneity during monetary policy regimes emphasizes that the fairly well synchronized business cycles among emerging and advanced economies might be unlikely, mainly on account of triggering events in the long run. Thus, the increasing need of consistent reforms of the international financial system to accelerate well-suited financial integration in developing countries. These findings are against existing studies that support similarity across business cycles in CEWE economies because of dealing with too short periods. For instance, they consider up to seven years or less, implying that only a single business cycle would be covered by the available data.

#### 5.3. Policy Interactions, Common Features, and Contagion Measures among Countries and Sectors

`pseudo-shock`’ among CEE and BLS economies because of larger fiscal adjustments—mainly in the last two decade—to catch up with the economic growth of the other advanced EA economies (from net receivers to net senders).

`credit`) and countercyclical fiscal policies are not common in countries with higher credit risk (inward spillovers in

`cpi`). Compared to Romer and Romer (2018), countries with lower debt-to-GDP (larger debit sustainability) tend to use fiscal policy more aggressively during crisis (inward spillovers in

`mkt`per WE). $\left(iii\right)$ Third, jointly dealing with endogeneity and policy issues, the results contrast the previous findings (Figure 7c). During dramatic structural breaks, either advanced or emerging economies would address unconventional monetary policy measures increasing monetary policy space to help central banks meet their output and inflation goals (inward spillovers in

`cpi`), mitigating limitations to monetary transmission that may hamper the provision of credit where it is most needed (outward spillovers in

`credit`), and supporting liquidity in financial markets or expanding fiscal space (inward spillovers in

`mkt`). Even if low-income countries with more developed capital markets and effective transmission via interest rates should be more likely to benefit more from unconventional monetary policy, the results show smaller benefits than in advanced economies (lower spillover effects). Thus, in emerging economies, countercyclical fiscal policy has been conducted but with delay. The limit to resort to fiscal policy during recession in low-income countries is due to their limited ability in using traditional monetary tools. These findings reflect the recent reports on the conduct of monetary policy during the coronavirus pandemic crisis (e.g., European Central Bank, 19 October 2020 and International Monetary Fund, 23 September 2020). In the next section, based on the full estimation sample, I emphasize the results conducting a further analysis on the economic outlook amid the COVID-19 pandemic shock.

#### 5.4. Lessons and Matters for Future Policy Efforts

`quasi-flexible`’ policies should be conducted in order to ensure in a not-too-distant future: $\left(i\right)$ the restoration of the confidence in financial systems, still recovering from the recent financial crisis; $\left(ii\right)$ higher homogeneity across countries’ responses in real economy given an unexpected financial shock so as to safeguard the inter-country real convergence; and $\left(iii\right)$ stronger cross-correlations among CEWE economies when facing international shocks transmission.

`quasi-flexible`’ policies stand for coordinated structural policy actions among foreign and domestic sectors along with more pointed fiscal adjustments according to country-specific requirements. Furthermore, the analysis highlights that, in case of a noteworthy unexpected shock in real economy, outward government benefits would be really beneficial for supporting the European integration and boosting the output to potential growth. Thus, the need of examining international spillovers accounting for both model misspecification problems and implied volatility changes.

## 6. Estimating Procedure through Monte Carlo Simulations

`General Case`’), with no structural breaks and volatility changes (hereafter ’unobserved effects’); and ${M}_{III}$ (’

`Full Case`’), with unobserved effects in either time-varying parameters or log-volatilities. The idea is to highlight the thrust of the estimation method of performing better conditional forecasts when studying macroeconomic–financial linkages among countries and sectors with a high dimensional estimation sample covering—for example—triggering events (unobserved effects). Then, I compare the estimation method to the following related works: $\left(i\right)$ multicountry Bayesian VAR (BVAR) as in Canova andCiccarelli (2009); $\left(ii\right)$ multicountry Panel Bayesian VAR (PBVAR) as in Ciccarelliet al. (2018); $\left(iii\right)$ Structural Panel Bayesian VAR (SPBVAR) as in Pacifico (2019b); and $\left(iv\right)$ Large Bayesian VAR with Stochastic Volatility (LBVAR-SV) as in Carrieroet al. (2019)34.

`(i)`${\beta}_{1t}^{\underset{\_}{S}}={({\beta}_{1t,1},{\beta}_{1t,2},{\beta}_{1t,3},{\beta}_{1t,4},{\beta}_{1t,5},{\beta}_{1t,6})}^{\underset{\_}{S}}$ denoting variable-specific effects for ${m}^{\underset{\_}{S}}=1$ and ${m}^{\underset{\_}{S}}=2$ stacked in six (theoretical) variable groups: (${M}_{v1}^{\underset{\_}{S}}$,${M}_{v2}^{\underset{\_}{S}}$), investigating (theoretical) macroeconomic–financial linkages given ${M}^{\underset{\_}{S}}$; (${M}_{v3}^{\underset{\_}{S}}$,${M}_{v4}^{\underset{\_}{S}}$), evaluating (theoretical) economic–financial issues with policy shifts given ${M}^{\underset{\_}{S}}$ and ${\tilde{Q}}^{\underset{\_}{S}}$; and (${M}_{v5}^{\underset{\_}{S}}$,${M}_{v6}^{\underset{\_}{S}}$), jointly dealing with (theoretical) not directly observed and measured factors (hereafter, ’additional factors’) and policy changes given ${M}^{\underset{\_}{S}}$, ${Q}^{\underset{\_}{S}}$, ${\tilde{Q}}^{\underset{\_}{S}}$, and ${\Xi}^{\underset{\_}{S}}$.

`(ii)`${\beta}_{2t}^{\underset{\_}{S}}={({\beta}_{2t,1},{\beta}_{2t,2},{\beta}_{2t,3})}^{\underset{\_}{S}}$ denoting common effects stacked in three (theoretical) common groups: ${M}_{c1}^{\underset{\_}{S}}$, containing the only supposed variables ${M}^{\underset{\_}{S}}$ given N to address (theoretical) macroeconomic–financial linkages; ${M}_{c2}^{\underset{\_}{S}}$, containing the supposed variables ${M}^{\underset{\_}{S}}$ and ${\tilde{Q}}^{\underset{\_}{S}}$ given N to investigate (theoretical) economic–financial issues with policy shifts; and ${M}_{c3}^{\underset{\_}{S}}$, containing all supposed variables ${M}^{\underset{\_}{S}}$, ${Q}^{\underset{\_}{S}}$, ${\tilde{Q}}^{\underset{\_}{S}}$, and ${\Xi}^{\underset{\_}{S}}$ given N to jointly deal with (theoretical) additional factors and policy changes. Thus, stacking for indices and variables, the supposed SPBVAR-MTV in (1) and SNLR in (6) assume the form:

## 7. Concluding Remarks

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Data Collection

Variable | Data Description |
---|---|

General Government Spending | Financial accounts for general government spending. |

Gross Fixed Capital Formation | Investments of fixed assets at current prices. |

GDP Growth Rate | It is calculated as: $Log\left(\frac{GD{P}_{it,j}}{GD{P}_{it-1,j}}\right)$. |

Inflation | It is calculated from the Consumer Price Index. |

Bank Leverage | It is calculated as Loan (L) to Deposit (D) ratio. |

Credit Growth | It is calculated as: $100\xb7\left(\frac{({L}_{t}/{P}_{t})-({L}_{t-4}/{P}_{t-4})}{{L}_{t-4}/{P}_{t-4}}\right)$. |

Bilateral Flows of Trade | Exports and imports in goods and services. |

Financial Transactions | Financial Assets on the total economy. |

Interest Rate | International interest policy rate. |

Public Debt | Non-financial accounts for general government debt. |

Current Account Balance | Non-financial accounts for general government net. |

Financial Consumption Expenditure | Total general government expenditure at current prices. |

Private Sector Consumption | Private consumption expenditure. |

Change of Unemployment Rate | Growth rate of the unemployment rate as percentage. |

Nominal Labour Cost | It is defined as the ratio of labour costs to labour productivity. |

Household price index | Real household per capita index. |

Productivity | It corresponds to logarithm of the real GDP per capita. |

Here, general government spending, gross fixed capital formation, bilateral flows of trade, financial transactions, public debt, current account balance, financial consumption, private consumption, and unemployment rate are weighted for the GDP. |

## Notes

1. | |

2. | |

3. | See, for instance, Gelfand and Dey (1994). |

4. | It would correspond to the conditional density of the data given the log-volatilities, but marginal of the time-varying parameters. |

5. | Overfitting and thus overestimation of effect sizes refers to a common problem in Bayesian Model Averaging since more complex models will always provide a somewhat better fit to the data than simpler models, where the ’complexity’ stands (for example) for the number of unknown parameters. See, for instance, Pacifico (2020b). |

6. | |

7. | A proxy variable is an easily measurable variable used in place of a variable that cannot be directly measured. |

8. | |

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

10. | Its form would be similar to the parsimonious Seemingly Unrelated Regression (SUR) model (homoskedastic VAR) developed in the literature (see, for instance, Canova and Ciccarelli (2009) and Ciccarelli et al. (2018)), but adapted to a hierarchical time-varying multicountry structural setup. |

11. | 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. |

12. | They denote both unobserved heterogeneity and misspecification problems. |

13. | See Section 3.2 for further detail. |

14. | |

15. | |

16. | |

17. | See, for instance, Levine and Casella (2014). |

18. | Analytical integration for integrating out the time-varying log-volatilities are explained in depth in Section 3.2. |

19. | See, for instance, Carter and Kohn (1994). |

20. | |

21. | See Section 3.2.2. |

22. | See Section 3.3. |

23. | See, for instance, Jacquier et al. (2002). |

24. | See, for instance, Roberts and Rosenthal (2001). |

25. | |

26. | See, for instance, Kroese et al. (2011). |

27. | Czech Republic (CZ), Hungary (HU), Estonia (EE), Latvia (LV), Lithuania (LT), Poland (PO), Slovak Republic (SK), and Slovenia (SL). |

28. | Austria (AT), Belgium (BE), France (FR), Germany (DE), Ireland (IE), Italy (IT), Portugal (PT), and Spain (ES). |

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

30. | It stands for the Baltic States (EE, LV, and LT). |

31. | They refer to the advanced and then WE countries. |

32. | World Health Organization, 11 March 2020. |

33. | See, e.g., Dell’Ariccia et al. (2018); Guerrieri et al. (2020); Romer and Romer (2018), and Bernanke (2020). |

34. | See, e.g., Banbura et al. (2010); Carriero et al. (2015); Giannone et al. (2015), and Koop (2013) for recent studies on LBVAR models and time variation in their volatilities; and Clark (2011); Clark and Ravazzolo (2015a), and Primiceri (2005) concerning studies on the importance of time variation in the volatilities modeled as stochastic volatility. |

35. | Monte Carlo simulations would work well up to 35,000 data. Compared to higher sample (≥35,000), the associated computational costs seem to become quite expensive. In that case, whether one would be interested to improve the empirical analysis accounting for larger country indices and sets of variables, but ensuring consistent posterior estimates, a simple solution would be fixing the time (T) as necessary (e.g., according to the economic–financial and policy issues to be addressed). |

36. | The idea is to highlight the performance of the estimation method by conducting a simulated experiment preserving the empirical findings (such as presence of cross-unit interdependencies, commonality, and dynamic feedback and interactions related to (theoretical) unobserved effects). |

37. | The convergence has been found by amounting to about 1.5 draws per (theoretical) regression parameter. |

38. | Generally, the conditional projection in density forecasts is the one that the model would have obtained over the same period conditionally on the actual path of unexpected dynamics for that period. |

39. | Generally, the unconditional projection in density forecasts 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. |

## References

- Banbura, Marta, Domenico Giannone, and Lucrezia Reichlin. 2010. Large bayesian vector autoregressions. Journal of Applied Econometrics 25: 71–92. [Google Scholar] [CrossRef]
- Bernanke, Ben S. 2020. The new tools of monetary policy. American Economic Review 110: 943–83. [Google Scholar] [CrossRef][Green Version]
- Canova, Fabio, and Matteo Ciccarelli. 2009. Estimating multicountry var models. International Economic Review 50: 929–59. [Google Scholar] [CrossRef][Green Version]
- Canova, Fabio, Matteo Ciccarelli, and Eva Ortega. 2007. Similarities and convergence in g7 cycles. Journal of Monetary Economics 54: 850–78. [Google Scholar] [CrossRef][Green Version]
- 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][Green Version]
- Canova, Fabio, and Luca Gambetti. 2009. Structural changes in the us economy: Is there a role for monetary policy? Journal of Economic Dynamics and Control 33: 477–90. [Google Scholar] [CrossRef][Green Version]
- Carriero, Andrea, Todd E. Clark, and Massimiliano Marcellino. 2015. Bayesian vars: specification choices and forecast accuracy. Journal of Applied Econometrics 30: 46–73. [Google Scholar] [CrossRef]
- Carriero, Andrea, Todd E. Clark, and Massimiliano Marcellino. 2019. Large bayesian vector autoregressions with stochastic volatility and non-conjugate priors. Journal of Econometrics 212: 137–54. [Google Scholar] [CrossRef]
- Carter, Chris K., and Robert Kohn. 1994. On gibbs sampling for state space models. Biometrika 81: 541–53. [Google Scholar] [CrossRef]
- Cette, Gilbert, John Fernald, and Benoît Mojon. 2016. The pre-great recession slowdown in productivity. European Economic Review 88: 3–20. [Google Scholar] [CrossRef][Green Version]
- Chan, Joshua C. C., and Angelia L. Grant. 2015. Pitfalls of estimating the marginal likelihood using the modified harmonic mean. Economics Letters 131: 29–33. [Google Scholar] [CrossRef]
- Chan, Joshua C. C., and Ivan Jeliazkov. 2009. Efficient simulation and integrated likelihood estimation in state space models. International Journal of Mathematical Modelling and Numerical Optimisation 1: 101–20. [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]
- Clark, Todd E. 2009. Is the great moderation over? an empirical analysis. Economic Review 94: 5–42. [Google Scholar]
- Clark, Todd E. 2011. Real-time density forecasts from bayesian vector autoregressions with stochastic volatility. Journal of Business and Economic Statistics 29: 327–41. [Google Scholar] [CrossRef]
- Clark, Todd E., and Francesco Ravazzolo. 2015. Macroeconomic forecasting performance under alternative specifications of time-varying volatility. Journal of Applied Econometrics 30: 551–75. [Google Scholar] [CrossRef]
- Cogley, Timothy, Giorgio E. Primiceri, and Thomas J. Sargent. 2010. Inflation-gap persistence in the us. American Economic Journal: Macroeconomic 2: 43–69. [Google Scholar] [CrossRef][Green Version]
- 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][Green Version]
- Coibion, Olivier, and Yuriy Gorodnichenko. 2012. Why are target interest rate changes so persistent? American Economic Journal: Macroeconomics 4: 126–62. [Google Scholar] [CrossRef][Green Version]
- Curcio, Domenico, Rosa Cocozza, and Antonio Pacifico. 2020. Do global markets imply common fear? Rivista Bancaria-Minerva Bancaria 2020: 1–24. [Google Scholar]
- D’Agostino, Antonello, Luca Gambetti, and Domenico Giannone. 2013. Macroeconomic forecasting and structural change. Journal of Applied Econometrics 28: 82–101. [Google Scholar] [CrossRef][Green Version]
- Daly, Mary C., John G. Fernald, Óscar Jordá, and Fernanda Nechio. 2016. Shocks and Adjustments. Working Paper 2013-32. San Francisco: Federal Reserve Bank of San Francisco. [Google Scholar]
- Dell’Ariccia, Giovanni, Pau Rabanal, and Damiano Sandri. 2018. Unconventional monetary policies in the euro area, japan, and the united kingdom. Journal of Economic Perspectives 32: 147–72. [Google Scholar] [CrossRef][Green Version]
- Foerster, Andrew, and Christian Matthes. 2020. Learning about Regime Change. Working Paper 2020-15. San Francisco: Federal Reserve Bank of San Francisco. [Google Scholar]
- Frühwirth-Schnatter, Sylvia, and Helga Wagner. 2008. Marginal likelihoods for non-gaussian models using auxiliary mixture sampling. Computational Statistics and Data Analysis 52: 4608–24. [Google Scholar] [CrossRef]
- Gelfand, Alan E., and Dipak K. Dey. 1994. Bayesian model choice: Asymptotics and exact calculations. Journal of the Royal Statistical Society: Series B 56: 501–14. [Google Scholar]
- Giannone, Domenico, Michele Lenza, and Giorgio E. Primiceri. 2015. Prior selection for vector autoregressions. The Review of Economics and Statistics 97: 436–51. [Google Scholar] [CrossRef][Green Version]
- Guerrieri, Veronica, Guido Lorenzoni, Ludwig Straub, and Iván Werning. 2020. Macroeconomic Implications of covid-19: Can Negative Supply Shocks Cause Demand Shortages? NBER Working Paper 26918. Cambridge: National Bureau of Economic Research. [Google Scholar]
- Jacquier, Eric, Nicholas G. Polson, and Peter E. Rossi. 2002. Bayesian analysis of stochastic volatility. Journal of Business and Economic Statistics 20: 69–87. [Google Scholar] [CrossRef]
- Kadiyala, Rao K., and Sune Karlsson. 1997. Numerical methods for estimation and inference in bayesian var models. Journal of Applied Econometrics 12: 99–132. [Google Scholar] [CrossRef]
- Kallianiotis, Ioannis N. 2019. Monetary policy, real cost of capital, financial markets and the real economic growth. Journal of Applied Finance & Banking 9: 75–118. [Google Scholar]
- Kass, Robert E., and Adrian E. Raftery. 1995. Bayes factors. Journal of American Statistical Association 90: 773–95. [Google Scholar] [CrossRef]
- Koop, Gary. 1996. Parameter uncertainty and impulse response analysis. Journal of Econometrics 72: 135–49. [Google Scholar] [CrossRef]
- Koop, Gary. 2013. Forecasting with medium and large bayesian vars. Journal of Applied Econometrics 28: 177–203. [Google Scholar] [CrossRef][Green Version]
- Koop, Gary, and Dimitris Korobilis. 2016. Model uncertainty in panel vector autoregressive models. European Economic Review 81: 115–31. [Google Scholar] [CrossRef][Green Version]
- Koop, Gary, Roberto Leon-Gonzalez, and Rodney W. Strachan. 2009. On the evolution of the monetary policy transmission mechanism. Journal of Economic Dynamics and Control 33: 997–1017. [Google Scholar] [CrossRef]
- Korobilis, Dimitris. 2016. Prior selection for panel vector autoregressions. Computational Statistics and Data Analysis 101: 110–20. [Google Scholar] [CrossRef][Green Version]
- Kroese, Dirk P., and Joshua C. C. Chan. 2014. Statistical Modeling and Computation. New York: Springer. [Google Scholar]
- Kroese, Dirk P., Thomas Taimre, and Zdravko I. Botev. 2011. Handbook of Monte Carlo Methods. New York: John Wiley and Sons. [Google Scholar]
- Krolzig, Hans-Martin. 1997. Markov Switching Vector Autoregressions: Modelling, Statistical Inference and Application to Business Cycle Analysis. Berlin: Springer. [Google Scholar]
- Krolzig, Hans-Martin. 2000. Predicting Markov-Switching Vector Autoregressive Processes. Nuffield College Economics Working Papers, 2000-WP31. Oxford: Nuffield College. [Google Scholar]
- Levine, Richard A., and George Casella. 2014. Implementations of the monte carlo em algorithm. Journal of Computational and Graphical Statistics 10: 422–39. [Google Scholar] [CrossRef][Green Version]
- Liu, Yuelin, and James Morley. 2014. Structural evolution of the u.s. economy. Journal of Economic Dynamics and Control 42: 50–68. [Google Scholar] [CrossRef][Green Version]
- McLachlan, Geoffrey J., Thriyambakam Krishnan, and Shu K. Ng. 2012. The em Algorithm. Handbook of Computational Statistics. Berlin and Heidelberg: Springer. [Google Scholar]
- Pacifico, Antonio. 2019a. International co-movements and business cycles synchronization across advanced economies: A spbvar evidence. International Journal of Statistics and Probability 8: 68–85. [Google Scholar] [CrossRef]
- Pacifico, Antonio. 2019b. Structural panel bayesian var model to deal with model misspecification and unobserved heterogeneity problems. Econometrics 7: 1–24. [Google Scholar] [CrossRef][Green Version]
- Pacifico, Antonio. 2020a. Fiscal implications, misspecified dynamics, and international spillover effects across europe: A time-varying multicountry analysis. International Journal of Statistics and Economics 21: 18–40. [Google Scholar]
- Pacifico, Antonio. 2020b. Robust open bayesian analysis: Overfitting, model uncertainty, and endogeneity issues in multiple regression models. Econometric Reviews 40: 148–76. [Google Scholar] [CrossRef]
- Primiceri, Giorgio E. 2005. Time varying structural vector autoregressions and monetary policy. Review of Economic Studies 72: 821–52. [Google Scholar] [CrossRef]
- Roberts, Gareth O., and Jeffrey S. Rosenthal. 2001. Optimal scaling for various metropolis-hastings algorithms. Statistical Science 16: 351–67. [Google Scholar] [CrossRef]
- Romer, Christina D., and David H. Romer. 2018. Phillips lecture-why some times are different: macroeconomic policy and the aftermath of financial crises. Economica 85: 1–40. [Google Scholar] [CrossRef]
- Sims, Christopher A., and Tao Zha. 2006. Were there regime switches in u.s. monetary policy? The American Economic Review 96: 54–81. [Google Scholar] [CrossRef][Green Version]
- Steele, Brian M. 1996. A modified em algorithm for estimation in generalized mixed models. Biometrics 52: 1295–310. [Google Scholar] [CrossRef] [PubMed]
- Stock, James H., and Mark W. Watson. 2012. Disentangling the Channels of the 2007–2009 Recession. NBER Working Paper Series. Cambridge: National Bureau of Economic Research, vol. 18094. [Google Scholar]
- Tjalling, Ypma J. 1995. Historical development of the newton-raphson method. SIAM Review 37: 531–51. [Google Scholar]

**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. 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. They account for ${\chi}_{3t}{\widehat{\beta}}_{3t}$ (plot

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

**b**) cross-country indicators, where ${\widehat{\beta}}_{3t}$ and ${\widehat{\beta}}_{4t}$ 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. They account for the variable-specific indicators ${\chi}_{9t,1}{\widehat{\beta}}_{9t,1}$ and ${\chi}_{9t,2}{\widehat{\beta}}_{9t,2}$ during ISE (plot

**a**) and ZIRE (plot

**b**) regimes, where ${\widehat{\beta}}_{9t,{M}_{\dot{v}}}$’s are posterior means with $\dot{v}=v1,v2$.

**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. They account for the cross-country indicators ${\chi}_{5t}{\widehat{\beta}}_{5t}$ (plot

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

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

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

**d**), where ${\widehat{\beta}}_{5}$, ${\widehat{\beta}}_{6t}$, ${\widehat{\beta}}_{7t}$, and ${\widehat{\beta}}_{8t}$ 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. They account for the variable-specific indicators ${\chi}_{9t,3}{\widehat{\beta}}_{9t,3}$ and ${\chi}_{9t,4}{\widehat{\beta}}_{9t,4}$ during ISE (plot

**a**) and ZIRE (plot

**b**) regimes, where ${\widehat{\beta}}_{9t,{M}_{\ddot{v}}}$’s are posterior means with $\ddot{v}=v3,v4$.

**Figure 6.**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. They account for the variable-specific indicators ${\chi}_{9t,3}{\widehat{\beta}}_{9t,3}$ and ${\chi}_{9t,4}{\widehat{\beta}}_{9t,4}$ during crisis (plot

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

**b**) periods, where ${\widehat{\beta}}_{9t,{M}_{\ddot{v}}}$’s are posterior means with $\ddot{v}=v3,v4$.

**Figure 7.**Systemic contributions of the productivity given a $1\%$ shock to financial markets are drawn as standard deviations of the variables in the system. They account for the cross-country indicators ${\chi}_{2t}{\widehat{\beta}}_{2t}$ (plot

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

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

**c**), where ${\widehat{\beta}}_{2t}$, ${\widehat{\beta}}_{6t}$, and ${\widehat{\beta}}_{8t}$ are posterior means.

**Figure 8.**Generalized Entropy index according to the productivity growth from $1994q4$ to $2021q1$ is drawn. It corresponds to the Theil’s Entropy and is computed by weighing the GDP with the population in terms of proportions with respect to the total. The conditional projections of each variable drawn in the SPBVAR-MTV in (1) have been used to perform forecasting from $2019q1$ to $2021q1$.

**Figure 9.**The plot draws (theoretical) density forecast combinations for outcomes ${Y}_{t}^{\underset{\_}{S}}$ given N for the indicators that account for additional factors and policy shifts with unobserved effects. They are: ${\chi}_{1t,5}^{\underset{\_}{S}}{\widehat{\beta}}_{1t,5}^{\underset{\_}{S}}$ (’theoretical’ real economy); ${\chi}_{1t,6}^{\underset{\_}{S}}{\widehat{\beta}}_{1t,6}^{\underset{\_}{S}}$ (’theoretical’ financial markets); and ${\chi}_{2t,3}^{\underset{\_}{S}}{\widehat{\beta}}_{2t,3}^{\underset{\_}{S}}$ (’theoretical’ total economy). They correspond to (theoretical) conditional (blue line) and unconditional (purple line) projections of each supposed variable drawn in (82).

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

$LG{B}_{\pi}$ | 16573 | 1649 | 0.00 |

${P}_{\pi}$ | 837.3 | 1261 | 0.30 |

$ML{E}_{f}$ | 67.44 | 10 | 0.00 |

_{π}stands for a Multivariate Ljung-Box Test of the series, with lags π = 30; P

_{π}refers to the Portmanteau (Asymptotic) Test on the residuals, with lags π = 30; MLE

_{f}is the Marginal (Conditional) Likelihood Estimation Test obtained through the Schwartz approximation, with f = 10.

Time-Varying Factors | ‘General Case’ (${\mathit{M}}_{\mathit{I}}$) | ‘Special Case’ (${\mathit{M}}_{\mathit{I}\mathit{I}}$) | ‘Full Case’ (${\mathit{M}}_{\mathit{I}\mathit{I}\mathit{I}}$) |
---|---|---|---|

${\chi}_{1t}{\widehat{\beta}}_{1t}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}{\chi}_{2t}{\widehat{\beta}}_{2t}$ | lBF >10 | $2\le lBF\le 6$ | $0\le lBF\le 2$ |

${\chi}_{3t}{\widehat{\beta}}_{3t}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}{\chi}_{4t}{\widehat{\beta}}_{4t}$ | $6\le lBF\le 10$ | lBF >10 | $2\le lBF\le 6$ |

${\chi}_{5t}{\widehat{\beta}}_{5t}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}{\chi}_{6t}{\widehat{\beta}}_{6t}$ | $0\le lBF\le 2$ | lBF >10 | $6\le lBF\le 10$ |

${\chi}_{7t}{\widehat{\beta}}_{7t}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}{\chi}_{8t}{\widehat{\beta}}_{8t}$ | $0\le lBF\le 2$ | $6\le lBF\le 10$ | lBF >10 |

${\chi}_{9,1t}{\widehat{\beta}}_{9,1t}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}{\chi}_{9,2t}{\widehat{\beta}}_{9,2t}$ | $6\le lBF\le 10$ | lBF >10 | $2\le lBF\le 6$ |

${\chi}_{9,3t}{\widehat{\beta}}_{9,3t}\phantom{\rule{3.33333pt}{0ex}},\phantom{\rule{3.33333pt}{0ex}}{\chi}_{9,4t}{\widehat{\beta}}_{9,4t}$ | $0\le lBF\le 2$ | $6\le lBF\le 10$ | lBF >10 |

${\chi}_{10,1t}{\widehat{\beta}}_{10,1t}$ | $6\le lBF\le 10$ | lBF >10 | $2\le lBF\le 6$ |

${\chi}_{10,2t}{\widehat{\beta}}_{10,2t}$ | $0\le lBF\le 2$ | $6\le lBF\le 10$ | lBF >10 |

Theoretical Factors | General Case (${\mathit{M}}_{\mathit{I}}$) | Full Case (${\mathit{M}}_{\mathit{III}}$) | |||
---|---|---|---|---|---|

BVAR | PBVAR | SPBVAR | LBVAR-SV | SPBVAR-MTV | |

${\widehat{\beta}}_{1t,1}^{\underset{\_}{S}}$ | 0.062 * | 0.058 * | 0.023 ** | 0.026 ** | 0.022 ** |

${\widehat{\beta}}_{1t,2}^{\underset{\_}{S}}$ | 0.052 * | 0.047 ** | 0.022 ** | 0.021 ** | 0.018 ** |

${\widehat{\beta}}_{1t,3}^{\underset{\_}{S}}$ | 0.214 | 0.145 | 0.057 * | 0.015 ** | 0.009 *** |

${\widehat{\beta}}_{1t,4}^{\underset{\_}{S}}$ | 0.112 | 0.131 | 0.044 ** | 0.028 ** | 0.017 ** |

${\widehat{\beta}}_{1t,5}^{\underset{\_}{S}}$ | 0.301 | 0.225 | 0.051 * | 0.015 ** | 0.007 *** |

${\widehat{\beta}}_{1t,6}^{\underset{\_}{S}}$ | 0.201 | 0.173 | 0.057 * | 0.031 ** | 0.012 ** |

${\widehat{\beta}}_{2t,1}^{\underset{\_}{S}}$ | 0.034 ** | 0.031 ** | 0.010 ** | 0.006 *** | 0.003 *** |

${\widehat{\beta}}_{2t,2}^{\underset{\_}{S}}$ | 0.093 * | 0.076 * | 0.042 ** | 0.012 ** | 0.008 *** |

${\widehat{\beta}}_{2t,3}^{\underset{\_}{S}}$ | 0.307 | 0.214 | 0.107 | 0.092 * | 0.021** |

_{I}and M

_{III}. The best model solution accounts for the only (theoretical) factors highlighted in bold and corresponds to the highest log Bayes Factor (lBF > 10) according to the generalized version of the Kass and Raftery (1995)’s scale of evidence in (53). The significance codes stand for: (*) significance at 10%; (**) significance at 5%; and (***) significance at 1%.

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## Share and Cite

**MDPI and ACS Style**

Pacifico, A.
Structural Panel Bayesian VAR with Multivariate Time-Varying Volatility to Jointly Deal with Structural Changes, Policy Regime Shifts, and Endogeneity Issues. *Econometrics* **2021**, *9*, 20.
https://doi.org/10.3390/econometrics9020020

**AMA Style**

Pacifico A.
Structural Panel Bayesian VAR with Multivariate Time-Varying Volatility to Jointly Deal with Structural Changes, Policy Regime Shifts, and Endogeneity Issues. *Econometrics*. 2021; 9(2):20.
https://doi.org/10.3390/econometrics9020020

**Chicago/Turabian Style**

Pacifico, Antonio.
2021. "Structural Panel Bayesian VAR with Multivariate Time-Varying Volatility to Jointly Deal with Structural Changes, Policy Regime Shifts, and Endogeneity Issues" *Econometrics* 9, no. 2: 20.
https://doi.org/10.3390/econometrics9020020