# Disentangling Civilian and Military Spending Shocks: A Bayesian DSGE Approach for the US Economy

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## Abstract

**:**

## 1. Introduction

## 2. The Model

#### 2.1. Households

#### 2.1.1. Asset Holders

#### 2.1.2. Non-Asset Holders

#### 2.2. Firms

#### 2.3. Fiscal Policy

#### 2.3.1. Total Government Spending

#### 2.3.2. Non-Military and Military Expenditures

#### 2.3.3. Financing Mechanism of Public Expenditure

#### 2.4. Monetary Policy

#### 2.5. General Equilibrium and Aggregation

## 3. Estimating the Model

#### 3.1. Data Description

#### 3.2. Prior Distributions of the Parameters

#### 3.3. Posterior Estimates of the Parameters

## 4. Analysing the Effects of Different Public Spending Shocks on the Economy

#### 4.1. Model with Aggregate Government Spending

#### 4.2. Model with Non-Military and Military Expenditures

## 5. Robustness Analysis: Different Assumptions about the Taylor Rule

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Maximization Problems of the Model

## Appendix B. Steady States

## Appendix C. The Log-Linearized Model

## Appendix D. Diagnostic Tests

#### Appendix D.1. Prior and Posterior Distributions

**Figure A1.**Total government spending model. Notes: In the above graphs, the grey lines represent the prior distributions, whereas the black lines correspond to the posterior distributions.

**Figure A2.**Non-military and military spending model. Notes: In the above graphs, the grey lines represent the prior distributions, whereas the black lines correspond to the posterior distributions.

#### Appendix D.2. Monte Carlo Markov Chain Univariate Diagnostics

**Figure A3.**Total government spending model: S1 (1954:Q3–1979:Q2). Notes: In the above graphs, the blue lines represent the 80% interval range based on the pooled draws from all sequences, whereas the red lines indicate the mean interval based on the draws of the individual sequences. The first column shows the convergence diagnostics for the 80% interval. The second and the third column with labels denote an estimate of the same statistics for the second and third central moments.

**Figure A4.**Total government spending model, S2 (1983:Q1–2008:Q2). Notes: In the above graphs, the blue lines represent the 80% interval range based on the pooled draws from all sequences, whereas the red lines indicate the mean interval based on the draws of the individual sequences. The first column shows the convergence diagnostics for the 80% interval. The second and the third column with labels denote an estimate of the same statistics for the second and third central moments.

**Figure A5.**Non-military and military spending model: S1 (1954:Q3–1979:Q2). Notes: In the above graphs, the blue lines represent the 80% interval range based on the pooled draws from all sequences, whereas the red lines indicate the mean interval based on the draws of the individual sequences. The first column shows the convergence diagnostics for the 80% interval. The second and the third column with labels denote an estimate of the same statistics for the second and third central moments.

**Figure A6.**Non-military and military spending model: S2 (1983:Q1–2008:Q2). Notes: In the above graphs, the blue lines represent the 80% interval range based on the pooled draws from all sequences, whereas the red lines indicate the mean interval based on the draws of the individual sequences. The first column shows the convergence diagnostics for the 80% interval. The second and the third column with labels denote an estimate of the same statistics for the second and third central moments.

#### Appendix D.3. Multivariate Convergence Diagnostics

**Figure A7.**Total government spending model. Notes: In the above graphs, the diagnostics is based on the range of the posterior likelihood function.

**Figure A8.**Non-military and military spending model. Notes: In the above graphs, the diagnostics is based on the range of the posterior likelihood function.

#### Appendix D.4. Smoothed Shocks

**Figure A9.**Total government spending model. Notes: In the above graphs, the black line represents the estimate of the smoothed structural shocks.

**Figure A10.**Non-military and military spending model. Notes: In the above graphs, the black line represents the estimate of the smoothed structural shocks.

#### Appendix D.5. Historical and Smoothed Variables

**Figure A11.**Total government spending model. Notes: In the above graphs, the dotted black lines indicate the observed data. The red lines indicate the estimates of the smoothed variables.

**Figure A12.**Non-military and military spending model. Notes: In the above graphs, the dotted black lines indicate the observed data. The red lines indicate the estimates of the smoothed variables.

#### Appendix D.6. Parameters’ Identification

**Figure A13.**Total government spending model. Notes: In the above graphs, blue bars indicate the identification strength of the parameters based on their prior means, whereas orange bars denote the identification strength of the parameters based on their standard deviations.

**Figure A14.**Non-military and military spending model. Notes: In the above graphs, blue bars indicate the identification strength of the parameters based on their prior means, whereas orange bars denote the identification strength of the parameters based on their standard deviations.

## Appendix E. Estimated Impulse Response Functions

**Figure A15.**Total government spending shock. Notes: The above graphs show the responses of the key variables together with their 95% confidence intervals.

**Figure A16.**Non-military spending shock. Notes: The above graphs show the responses of the key variables together with their 95% confidence intervals.

**Figure A17.**Military spending shock. Notes: The above graphs show the responses of the key variables together with their 95% confidence intervals.

## Appendix F. Benchmark Model vs. DSGE-VARs

**Table A1.**Comparison between the benchmark model and DSGE-VARs: model with total government spending.

Sub-Sample 1 | ||
---|---|---|

Marginal Log Density | Bayes Factor vs. Benchmark Model | |

DSGE-VAR (1) | −189.714 | exp[27.449] |

DSGE-BVAR(2) | −175.595 | exp[13.331] |

DSGE-BVAR (3) | −170.572 | exp[8.308] |

DSGE-BVAR (4) | −171.844 | exp[9.579] |

Benchmark Model | −162.264 | exp[0.000] |

Sub-Sample 2 | ||

Marginal Log Density | Bayes Factor vs. Benchmark Model | |

DSGE-VAR (1) | −156.199 | exp[−10.742] |

DSGE-BVAR (2) | −154.106 | exp[−12.835] |

DSGE-BVAR (3) | −156.258 | exp[−10.684] |

DSGE-BVAR (4) | −171.844 | exp[4.902] |

Benchmark Model | −166.941 | exp[0.000] |

**Table A2.**Comparison between the benchmark model and DSGE-VARs: model with non-military and military expenditures.

Sub-Sample 1 | ||
---|---|---|

Marginal Log Density | Bayes Factor vs. Benchmark Model | |

DSGE-VAR (1) | −372.712 | exp[46.058] |

DSGE-BVAR (2) | −368.905 | exp[42.252] |

DSGE-BVAR (3) | −368.800 | exp[42.146] |

DSGE-BVAR (4) | −361.313 | exp[34.659] |

Benchmark Model | −326.653 | exp[0.000] |

Sub-Sample 2 | ||

Marginal Log Density | Bayes Factor vs. Benchmark Model | |

DSGE-VAR (1) | −347.561 | exp[36.952] |

DSGE-BVAR (2) | −329.899 | exp[19.290] |

DSGE-BVAR (3) | −336.235 | exp[25.626] |

DSGE-BVAR (4) | −326.873 | exp[16.264] |

Benchmark Model | −310.609 | exp[0.000] |

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1 | |

2 | |

3 | All the estimations were done with Dynare (http://www.dynare.org/). |

4 | All the relative figures are reported in Appendix D together with prior and posterior distributions of the parameters estimated with Bayesian methods. |

5 | In Appendix F, Table A1 and Table A2 compare the different DSGE-VAR models against the benchmark models, reporting their marginal log densities and Bayes factors. |

6 | In Appendix E, we report the estimated IRFs and their relative error bands for all three public spending shocks in both sub-samples. |

7 | In order to further assess the different contribution of fiscal spending shocks on aggregate output, we also performed the forecast error variance decomposition for 1, 4, 10, and 30 quarters ahead (Albonico et al. 2019). Our results indicated that fiscal spending shocks had larger contributions on GDP during the post-financial liberalisation period. Moreover, we found that non-military spending shocks contributed to output changes more than military spending shocks. |

**Figure 1.**Total government spending shock. Notes: Simulated 1% increase in total government spending. Parameters are set according to their estimated values. The blue lines indicate the responses of the estimated model for S1, whereas the red lines denote the responses of the estimated model in S2.

**Figure 2.**Non-military spending shock. Notes: Simulated 1% increase in non-military spending. Parameters are set according to their estimated values. The blue lines indicate the responses of the estimated model for S1, whereas the red lines denote the responses of the estimated model in S2.

**Figure 3.**Military spending shock. Notes: Simulated 1% increase in military spending. Parameters are set according to their estimated values. The blue lines indicate the responses of the estimated model for S1, whereas the red lines denote the responses of the estimated model in S2.

**Figure 4.**Alternative assumptions on the Taylor rule. Notes: In the above graphs, the black lines denote the IRFsin the presence of the actual U.S. monetary policy, whereas the green lines indicate the IRFs associated with the counterfactual monetary policy.

Parameter | S1 (1954:Q3–1979:Q2) | S2 (1983:Q1–2008:Q2) |
---|---|---|

(a) Model with Total Government Spending | ||

$\beta $ | $0.99$ | $0.99$ |

${G}_{Y}$ | $0.28$ | $0.18$ |

$\tau $ | $0.30$ | $0.30$ |

${\varphi}_{g}$ | $0.17$ | $0.64$ |

$\eta $ | $0.51$ | $0.71$ |

$\alpha $ | $0.75$ | $0.75$ |

$\sigma $ | $2.00$ | $2.00$ |

N | $0.25$ | $0.25$ |

(b) Model with Non-Military and Military Expenditures | ||

$\beta $ | $0.99$ | $0.99$ |

${G}_{Y}$ | $0.28$ | $0.18$ |

${M}_{Y}$ | $0.10$ | $0.06$ |

$N{M}_{Y}$ | $0.18$ | $0.12$ |

${\varphi}_{g}$ | $0.17$ | $0.64$ |

$\eta $ | $0.51$ | $0.71$ |

$\tau $ | $0.30$ | $0.30$ |

$\alpha $ | $0.75$ | $0.75$ |

$\sigma $ | $2.00$ | $2.00$ |

N | $0.25$ | $0.25$ |

Parameter | Prior Distribution | Prior Mean | Prior St. Dev. | ||
---|---|---|---|---|---|

(a) Model with Total Government Spending | |||||

S1 | S2 | S1 | S2 | ||

${\rho}^{R}$ | Beta | 0.65 | 0.65 | 0.10 | 0.10 |

${r}_{\pi}$ | Gamma | 1.50 | 1.50 | 0.10 | 0.10 |

${r}_{y}$ | Gamma | 0.10 | 0.10 | 0.05 | 0.05 |

$\lambda $ | Gamma | 0.50 | 0.30 | 0.01 | 0.01 |

(b) Model with Non-Military and Military Expenditure | |||||

S1 | S2 | S1 | S2 | ||

${\rho}^{R}$ | Beta | 0.65 | 0.65 | 0.10 | 0.10 |

${r}_{\pi}$ | Gamma | 1.50 | 1.50 | 0.10 | 0.10 |

${r}_{y}$ | Gamma | 0.10 | 0.10 | 0.05 | 0.05 |

$\lambda $ | Gamma | 0.50 | 0.30 | 0.01 | 0.01 |

Parameter | Prior Distribution | Prior Mean | Prior St. Dev. | ||
---|---|---|---|---|---|

(a) Model with Total Government Spending | |||||

S1 | S2 | S1 | S2 | ||

${\rho}^{G}$ | Beta | 0.70 | 0.70 | 0.20 | 0.20 |

${\rho}^{\pi}$ | Beta | 0.70 | 0.70 | 0.20 | 0.20 |

${\sigma}_{G}$ | Inverse-Gamma | 0.01 | 0.01 | $Inf.$ | $Inf.$ |

${\sigma}_{\pi}$ | Inverse-Gamma | 0.01 | 0.01 | $Inf.$ | $Inf.$ |

${\sigma}_{R}$ | Inverse-Gamma | 0.01 | 0.01 | $Inf.$ | $Inf.$ |

(b) Model with Non-Military and Military Expenditure | |||||

S1 | S2 | S1 | S2 | ||

${\rho}^{NM}$ | Beta | 0.70 | 0.70 | 0.20 | 0.20 |

${\rho}^{M}$ | Beta | 0.70 | 0.70 | 0.20 | 0.20 |

${\rho}^{\pi}$ | Beta | 0.70 | 0.70 | 0.20 | 0.20 |

${\sigma}_{NM}$ | Inverse-Gamma | 0.01 | 0.01 | $Inf.$ | $Inf.$ |

${\sigma}_{M}$ | Inverse-Gamma | 0.01 | 0.01 | $Inf.$ | $Inf.$ |

${\sigma}_{\pi}$ | Inverse-Gamma | 0.01 | 0.01 | $Inf.$ | $Inf.$ |

${\sigma}_{R}$ | Inverse-Gamma | 0.01 | 0.01 | $Inf.$ | $Inf.$ |

Parameter | Posterior Mean | Confidence Interval | Posterior Mean | Confidence Interval | ||
---|---|---|---|---|---|---|

S1 (1954:Q3–1979:Q2) | S2 (1983:Q1–2008:Q2) | |||||

(a) Model with Total Government Spending | ||||||

${\rho}^{R}$ | $0.3240$ | $0.2576$ | $0.3894$ | $0.3961$ | $0.3362$ | $0.4557$ |

${r}_{\pi}$ | $1.5330$ | $1.3677$ | $1.6937$ | $1.4920$ | $1.3314$ | $1.6513$ |

${r}_{y}$ | $0.1396$ | $0.0363$ | $0.2355$ | $0.1286$ | $0.0340$ | $0.2237$ |

$\lambda $ | $0.4484$ | $0.4390$ | $0.4559$ | $0.2898$ | $0.2743$ | $0.3051$ |

(b) Model with Non-Military and Military Expenditures | ||||||

${\rho}^{R}$ | $0.2419$ | $0.1647$ | $0.3208$ | $0.3664$ | $0.2969$ | $0.4362$ |

${r}_{\pi}$ | $1.5194$ | $1.3634$ | $1.6816$ | $1.4835$ | $1.3201$ | $1.6416$ |

${r}_{y}$ | $0.1183$ | $0.0290$ | $0.2009$ | $0.1252$ | $0.0300$ | $0.2142$ |

$\lambda $ | $0.4488$ | $0.2998$ | $0.5384$ | $0.2901$ | $0.2745$ | $0.3053$ |

Parameter | Posterior Mean | Confidence Interval | Posterior Mean | Confidence Interval | ||
---|---|---|---|---|---|---|

S1 (1954:Q3–1979:Q2) | S2 (1983:Q1–2008:Q2) | |||||

(a) Model with Total Government Spending | ||||||

${\rho}^{G}$ | $0.8231$ | $0.7340$ | $0.9116$ | $0.7628$ | $0.6653$ | $0.8607$ |

${\rho}^{\pi}$ | $0.9629$ | $0.9305$ | $0.9980$ | $0.9580$ | $0.9223$ | $0.9966$ |

${\sigma}_{G}$ | $0.4954$ | $0.4390$ | $0.5520$ | $0.3155$ | $0.2819$ | $0.3492$ |

${\sigma}_{\pi}$ | $0.2270$ | $0.1624$ | $0.2927$ | $0.2921$ | $0.1976$ | $0.3900$ |

${\sigma}_{R}$ | $1.3924$ | $1.1659$ | $1.6141$ | $1.2582$ | $1.0998$ | $1.4162$ |

(b) Model with Non-Military and Military Expenditures | ||||||

${\rho}^{NM}$ | $0.6152$ | $0.4861$ | $0.7413$ | $0.8049$ | $0.7192$ | $0.8934$ |

${\rho}^{M}$ | $0.9291$ | $0.8830$ | $0.9785$ | $0.8394$ | $0.7552$ | $0.9236$ |

${\rho}^{\pi}$ | $0.9601$ | $0.9264$ | $0.9975$ | $0.9564$ | $0.9196$ | $0.9962$ |

${\sigma}_{NM}$ | $0.5104$ | $0.4501$ | $0.5688$ | $0.3273$ | $0.2919$ | $0.3624$ |

${\sigma}_{M}$ | $0.9652$ | $0.8502$ | $1.0741$ | $0.8604$ | $0.7671$ | $0.9534$ |

${\sigma}_{\pi}$ | $0.2101$ | $0.1482$ | $0.2689$ | $0.2894$ | $0.1936$ | $0.3869$ |

${\sigma}_{R}$ | $1.4603$ | $1.2293$ | $1.6907$ | $1.2815$ | $1.1146$ | $1.4454$ |

© 2019 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/).

## Share and Cite

**MDPI and ACS Style**

Lorusso, M.; Pieroni, L.
Disentangling Civilian and Military Spending Shocks: A Bayesian DSGE Approach for the US Economy. *J. Risk Financial Manag.* **2019**, *12*, 141.
https://doi.org/10.3390/jrfm12030141

**AMA Style**

Lorusso M, Pieroni L.
Disentangling Civilian and Military Spending Shocks: A Bayesian DSGE Approach for the US Economy. *Journal of Risk and Financial Management*. 2019; 12(3):141.
https://doi.org/10.3390/jrfm12030141

**Chicago/Turabian Style**

Lorusso, Marco, and Luca Pieroni.
2019. "Disentangling Civilian and Military Spending Shocks: A Bayesian DSGE Approach for the US Economy" *Journal of Risk and Financial Management* 12, no. 3: 141.
https://doi.org/10.3390/jrfm12030141