# Sovereign Risk Indices and Bayesian Theory Averaging

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Bayesian Theory Averaging

#### 2.1. BTA and Linear Gaussian Regression

#### 2.2. Multivariate BTA and Generalized Regression Models

## 3. Using BTA to Construct Sovereign Risk Indices

#### 3.1. Data Outline

#### 3.2. Results

#### 3.3. Investigating the Multiple Response Framework

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Full Algorithm Details

- ${M}_{1},\cdots ,{M}_{T}$ the models associated with theories 1 through T
- ${\mathit{\beta}}_{1},\cdots ,{\mathit{\beta}}_{T}$, the coefficient vectors associated with each theory. Note that by construction ${\beta}_{jt}=0$ when $j\notin {M}_{t}$
- ${\gamma}_{1},\cdots ,{\gamma}_{R}$ the theory-scaling vectors for each outcome equation r. A ${\gamma}_{tr}$ can be set to zero, indicating that theory-t is not currently relevant for outcome equation r. For purposes of identification if multiple ${\gamma}_{tr}$ are non-zero for a given t, we set ${\gamma}_{tr}=1$ for whichever r is smallest.
- ${\mathit{I}}_{1},\cdots ,{\mathit{I}}_{T}$ the latent theory index vectors (each of length n) where ${I}_{it}$ is the current state of the theory t index for observation i. By convention if ${\gamma}_{tr}=0$ for all r then ${I}_{it}=0$ for all i.
- ${\nu}_{1},\cdots ,{\nu}_{T}$, the random effect precision terms
- Global parameters ${\mathit{\alpha}}_{r}$ in the R outcome equations

- Gibbs sampling, relevant for updating ${\mathit{\beta}}_{t}$ and ${\nu}_{t}$
- Conditional Bayes Factors, which are used to update the theory-level models ${M}_{t}$
- Metropolis-Hastings via Laplacian calculations of the log posterior density which are used, in turn, to update theory indices ${\mathit{I}}_{t}$, global parameters ${\mathit{\alpha}}_{r}$ and those theory-scaling parameters ${\gamma}_{tr}$ which are neither constrained to zero or one.
- Reversible Jump Methods for alternating ${\gamma}_{tr}$ between being 0 or in $\mathbb{R}$. Note that the moves here become especially detailed–though primarily in the sense of bookkeeping–when ${\gamma}_{tr}$ is currently set to 1, or if ${\gamma}_{tr}$ is currently zero and r is smaller than all other non-zero ${\gamma}_{t{r}^{\prime}}$. Finally, this becomes a joint reversible jump move when the model move will either turn-on or shut-off the theory entirely, as both ${\gamma}_{tr}$ and ${\mathit{I}}_{r}$ will be affected.

#### Appendix A.1. Gibbs Sampling Updates

#### Appendix A.2. Conditional Bayes Factors to Update M_{t}

#### Appendix A.3. Metropolis-Hastings Updates via Laplacian Expansions

#### Appendix A.3.1. Logistic Regression

#### Appendix A.3.2. Bayesian Quantile Regression

#### Appendix A.3.3. GEV Regression

#### Appendix A.3.4. Updating Theory Indices

#### Appendix A.4. Updating Theory Inclusion Parameters via Reversible Jump

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Theory | Short Variable Name | Description |
---|---|---|

Insolvency | MAC | Market access to capital markets, dummy |

Insolvency | IMF | IMF lending dummy |

Insolvency | CAY | Current account balance, in % of GDP |

Insolvency | ResG | Reserves growth/change in % |

Insolvency | XG | Export growth in % |

Insolvency | WX | Export in USD billions |

Insolvency | TEDX | Total external debt to exports, in % |

Insolvency | MG | Import growth, in % |

Insolvency | FDIY | Foreign direct investment to GDP, in % |

Insolvency | FDIG | Change in % of foreign direct investment inflows |

Insolvency | TEDY | Total external debt to GDP, in % |

Insolvency | SEDY | Short term external debt to GDP, in % |

Insolvency | PEDY | Public external debt to GDP, in % |

Insolvency | OPEN | Exports and imports over GDP, in % |

Illiquidity | STDR | Short term debt to reserves |

Illiquidity | M2R | M2 to reserves |

Illiquidity | DSER | Debt service on long term debt to reserves |

Macroeconomic | DOil | Oil producing dummy |

Macroeconomic | RGRWT | Real (inflation adjusted) GDP change in % |

Macroeconomic | OVER | Exchange rate residual over liner trend |

Macroeconomic | UST | US treasury bill |

Political | PR | Index of political rights, 1 (most free) to 7 (least free) |

Political | History | Number of past defaults |

Systemic | Cont_tot | Number of defaults in the world |

Systemic | Cont_area | Number of defaults in the region the country is part of |

Default | Inflation | Unemployment | Devaluation | |
---|---|---|---|---|

Insolvency | 1 | 0.643 | 0.006 | 0.036 |

Illiquidity | 1 | 0.367 | 1 | 0.061 |

Macroeconomic | 1 | 1 | 1 | 0.114 |

Political | 1 | 1 | 1 | 0.052 |

Systemic | 1 | 0.996 | 1 | 0.036 |

Default | Inflation | Unemployment | Devaluation | |
---|---|---|---|---|

Insolvency | 1 | 0.451 | 0 | 0 |

Illiquidity | 1 | 0.124 | −0.598 | −0.002 |

Macroeconomic | 1 | 2.24 | 1.775 | 0.007 |

Political | 1 | 2.499 | −1.644 | −0.002 |

Systemic | 1 | 1.422 | 0.946 | 0 |

Probability | Conditional Mean | |
---|---|---|

IMF | 0.037 | −0.024 |

CAY | 0.04 | 0.025 |

ResG | 0.991 | 0.082 |

XG | 0.985 | 0.099 |

WX | 0.023 | 0.003 |

TEDX | 0.084 | −0.043 |

MG | 1 | −0.397 |

FDIY | 0.994 | −0.103 |

FDIG | 1 | −0.249 |

TEDY | 0.087 | −0.052 |

SEDY | 0.1 | 0.06 |

PEDY | 0.084 | −0.039 |

OPEN | 0.268 | −0.054 |

Probability | Conditional Mean | |
---|---|---|

STDR | 0.045 | −0.026 |

M2R | 0.971 | 0.082 |

DSER | 0.054 | 0.04 |

Probability | Conditional Mean | |
---|---|---|

DOil | 0.05 | 0.028 |

RGRWT | 1 | −0.466 |

OVERN | 0.022 | 0.011 |

UST | 1 | 0.175 |

Probability | Conditional Mean | |
---|---|---|

PR | 1 | 0.095 |

History | 1 | 0.213 |

Probability | Conditional Mean | |
---|---|---|

Cont_tot | 1 | 0.529 |

Cont_area | 1 | 0.215 |

Year | Insolvency | Default | Inflation | Unemployment | Devaluation | |
---|---|---|---|---|---|---|

Gabon | 1995 | −13.683 | 0 | 36.116 | NA | NA |

Moldova | 1994 | −6.704 | 0 | 35.749 | NA | NA |

Korea, Rep. | 2009 | −6.157 | 0 | 4.704 | 3.2 | 0.071 |

Portugal | 1995 | −5.664 | 0 | 5.214 | 6.713 | 0.018 |

Gabon | 1981 | −5.543 | 0 | 12.34 | NA | NA |

Trinidad and Tobago | 1987 | 4.407 | 0 | 7.694 | 9.37 | NA |

Tunisia | 1988 | 4.437 | 1 | 8.226 | NA | NA |

Sri Lanka | 2009 | 4.448 | 0 | 22.564 | 5.22 | 0.012 |

Niger | 1983 | 4.703 | 1 | 11.642 | NA | NA |

Haiti | 1979 | 4.864 | 0 | −2.674 | NA | NA |

Year | Illiquidity | Default | Inflation | Unemployment | Devaluation | |
---|---|---|---|---|---|---|

Burundi | 1991 | −5.391 | 0 | 7.002 | 0.48 | NA |

Pakistan | 1976 | −5.223 | 0 | 20.905 | 1.7 | NA |

Bangladesh | 1997 | −5.154 | 0 | 2.377 | 2.51 | NA |

Malaysia | 2007 | −5.079 | 0 | 3.609 | 3.32 | 0.012 |

Indonesia | 1977 | −5.07 | 0 | 19.859 | 1.92 | NA |

Jamaica | 1986 | 7.266 | 0 | 25.673 | 33.39 | NA |

Lesotho | 2009 | 7.311 | 0 | 10.721 | 35.46 | NA |

Lesotho | 1998 | 7.329 | 0 | −100 | 37.94 | NA |

Gabon | 2002 | 8.162 | 1 | 2.138 | NA | NA |

Jamaica | 1981 | 9.208 | 1 | 27.308 | 35.51 | NA |

**Table 11.**Posterior correlation matrix of theory indices. This table mainly shows that the indices have the desirable property of low dependence between one another.

Insolvency | Illiquidity | Macroeconomic | Political | Systemic | |
---|---|---|---|---|---|

Insolvency | 1 | −0.015 | 0.024 | −0.012 | 0.002 |

Illiquidity | −0.015 | 1 | −0.061 | 0.066 | −0.021 |

Macroeconomic | 0.024 | −0.061 | 1 | −0.161 | 0.081 |

Political | −0.012 | 0.066 | −0.161 | 1 | −0.095 |

Systemic | 0.002 | −0.021 | 0.081 | −0.095 | 1 |

**Table 12.**Comparison of out of sample scores for each dependent variable. This table shows that, with the exception the default variable, there is significant reduction in predictive loss when using a joint modeling framework. Asterisks indicate greater than 99% significance of differences based on a permutation test.

Variable | Joint Model | Single Model | Ratio Joint to Single |
---|---|---|---|

Default | 0.046 | 0.046 | 1.003 |

Unemployment | 19.992 | 22.765 | 0.878 * |

Inflation | 2.362 | 2.658 | 0.889 * |

Devaluation | 3.528 | 5.464 | 0.646 * |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Lenkoski, A.; Aanes, F.L.
Sovereign Risk Indices and Bayesian Theory Averaging. *Econometrics* **2020**, *8*, 22.
https://doi.org/10.3390/econometrics8020022

**AMA Style**

Lenkoski A, Aanes FL.
Sovereign Risk Indices and Bayesian Theory Averaging. *Econometrics*. 2020; 8(2):22.
https://doi.org/10.3390/econometrics8020022

**Chicago/Turabian Style**

Lenkoski, Alex, and Fredrik L. Aanes.
2020. "Sovereign Risk Indices and Bayesian Theory Averaging" *Econometrics* 8, no. 2: 22.
https://doi.org/10.3390/econometrics8020022