# Data Cloning Estimation and Identification of a Medium-Scale DSGE Model

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## Abstract

**:**

## 1. Introduction

## 2. Estimation and Identification of DSGE Models

#### 2.1. Estimation

#### 2.2. Identification

## 3. The Data Cloning Method

## 4. Model and Estimation Details

## 5. Results

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Standardized maximum eigenvalue of the posterior covariance matrix. Note: figure plots the standardized maximum eigenvalue of data cloning posterior the covariance matrix estimates against the $1/K$ expected value for a well-identified model.

Parameter | Definition | Prior | Optimization | Single-Sample (MCMC) | 5 Clones | 10 Clones | 25 Clones | Benchmark | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Density | Mean | Std | Mode | Mean | Std | Mean | Std | Mean | Std | Mean | Std | Mean | ||

$\psi $ | Investment adj. cost. | N | 4.00 | 1.50 | 5.49 | 5.588 | 2.241 | 6.851 | 1.091 | 6.004 | 0.694 | 5.974 | 1.044 | 5.74 |

${\sigma}_{c}$ | Inv. elats. intert. subst. | N | 1.50 | 0.38 | 1.47 | 1.435 | 0.328 | 1.401 | 0.327 | 1.406 | 0.243 | 1.468 | 0.061 | 1.38 |

h | Consump. habit | B | 0.70 | 0.10 | 0.70 | 0.705 | 0.102 | 0.794 | 0.071 | 0.771 | 0.032 | 0.759 | 0.054 | 0.71 |

${\xi}_{w}$ | Calvo wage | B | 0.50 | 0.10 | 0.73 | 0.692 | 0.163 | 0.896 | 0.024 | 0.855 | 0.067 | 0.838 | 0.121 | 0.71 |

${\sigma}_{l}$ | Elast. labour supply | N | 2.00 | 0.75 | 1.67 | 1.655 | 1.259 | 3.791 | 0.433 | 3.267 | 0.731 | 3.398 | 1.852 | 1.83 |

${\xi}_{p}$ | Calvo price | B | 0.50 | 0.10 | 0.68 | 0.676 | 0.125 | 0.784 | 0.048 | 0.737 | 0.039 | 0.695 | 0.076 | 0.66 |

${\iota}_{w}$ | Index. of wages | B | 0.50 | 0.15 | 0.56 | 0.543 | 0.277 | 0.476 | 0.145 | 0.495 | 0.141 | 0.582 | 0.037 | 0.58 |

${\iota}_{p}$ | Index. of prices | B | 0.50 | 0.15 | 0.24 | 0.26 | 0.202 | 0.273 | 0.092 | 0.276 | 0.073 | 0.265 | 0.039 | 0.24 |

$\Psi $ | Capital utilization | B | 0.50 | 0.15 | 0.40 | 0.411 | 0.211 | 0.171 | 0.122 | 0.199 | 0.102 | 0.226 | 0.388 | 0.54 |

$\Phi $ | Fixed cost | N | 1.25 | 0.12 | 1.65 | 1.643 | 0.171 | 1.581 | 0.095 | 1.591 | 0.065 | 1.575 | 0.078 | 1.6 |

${r}_{\pi}$ | Response to inflation | N | 1.50 | 0.25 | 1.98 | 2.015 | 0.375 | 2.131 | 0.316 | 1.936 | 0.072 | 1.949 | 0.193 | 2.04 |

$\rho $ | Interest rate smooth. | N | 0.75 | 0.10 | 0.82 | 0.82 | 0.052 | 0.872 | 0.016 | 0.854 | 0.018 | 0.851 | 0.021 | 0.81 |

${r}_{y}$ | Response to output | N | 0.13 | 0.05 | 0.09 | 0.095 | 0.052 | 0.138 | 0.069 | 0.123 | 0.029 | 0.118 | 0.007 | 0.08 |

${r}_{\Delta y}$ | Response to output gap | N | 0.13 | 0.05 | 0.22 | 0.222 | 0.064 | 0.191 | 0.013 | 0.185 | 0.027 | 0.192 | 0.047 | 0.22 |

$\overline{\pi}$ | Steady state inflation | G | 0.63 | 0.10 | 0.67 | 0.679 | 0.157 | 0.534 | 0.121 | 0.604 | 0.107 | 0.604 | 0.065 | 0.78 |

100 $({\beta}^{-1}-1)$ | Discount factor | G | 0.25 | 0.10 | 0.21 | 0.241 | 0.203 | 0.232 | 0.167 | 0.186 | 0.041 | 0.195 | 0.038 | 0.16 |

$\overline{l}$ | Steady state hours worked | N | 0.00 | 2.00 | 0.40 | 0.296 | 2.249 | 2.351 | 1.921 | 0.876 | 1.135 | 1.038 | 1.222 | 0.53 |

100 $(\gamma -1)$ | Trend growth | N | 0.40 | 0.10 | 0.44 | 0.435 | 0.035 | 0.451 | 0.011 | 0.456 | 0.025 | 0.455 | 0.011 | 0.43 |

$\alpha $ | Share of capital | N | 0.30 | 0.05 | 0.32 | 0.314 | 0.092 | 0.354 | 0.106 | 0.371 | 0.031 | 0.356 | 0.067 | 0.19 |

$\delta $ | Depreciation rate | n.a. | 0.025 | n.a. | n.a. | n.a. | n.a. | n.a. | n.a. | n.a. | n.a. | n.a. | n.a. | n.a. |

${g}_{y}$ | Government/Output ratio | n.a. | 0.18 | n.a. | n.a. | n.a. | n.a. | n.a. | n.a. | n.a. | n.a. | n.a. | n.a. | n.a. |

${\varphi}_{w}$ | Wage mark-up | n.a. | 1.5 | n.a. | n.a. | n.a. | n.a. | n.a. | n.a. | n.a. | n.a. | n.a. | n.a. | n.a. |

${\u03f5}_{w}$ | Kimball (wage) | n.a. | 10 | n.a. | n.a. | n.a. | n.a. | n.a. | n.a. | n.a. | n.a. | n.a. | n.a. | n.a. |

${\u03f5}_{p}$ | Kimball (price) | n.a. | 10 | n.a. | n.a. | n.a. | n.a. | n.a. | n.a. | n.a. | n.a. | n.a. | n.a. | n.a. |

${\rho}_{ga}$ | Tech. shock to gov. spending | N | 0.50 | 0.25 | 0.60 | 0.587 | 0.204 | 0.553 | 0.097 | 0.644 | 0.116 | 0.625 | 0.036 | 0.52 |

${\rho}_{a}$ | AR nonspecific technology | B | 0.50 | 0.20 | 0.95 | 0.948 | 0.0354 | 0.974 | 0.027 | 0.968 | 0.014 | 0.967 | 0.005 | 0.95 |

${\rho}_{b}$ | AR risk premium | B | 0.50 | 0.20 | 0.17 | 0.212 | 0.1957 | 0.219 | 0.183 | 0.217 | 0.034 | 0.209 | 0.014 | 0.22 |

${\rho}_{g}$ | AR government spending | B | 0.50 | 0.20 | 0.97 | 0.973 | 0.0196 | 0.997 | 0.001 | 0.996 | 0.016 | 0.995 | 0.019 | 0.97 |

${\rho}_{i}$ | AR investment | B | 0.50 | 0.20 | 0.74 | 0.747 | 0.134 | 0.647 | 0.083 | 0.671 | 0.101 | 0.681 | 0.045 | 0.71 |

${\rho}_{m}$ | AR monetary policy | B | 0.50 | 0.20 | 0.12 | 0.142 | 0.1294 | 0.042 | 0.049 | 0.121 | 0.095 | 0.121 | 0.031 | 0.15 |

${\rho}_{p}$ | AR price mark-up | B | 0.50 | 0.20 | 0.90 | 0.879 | 0.1326 | 0.999 | 0.001 | 0.999 | 0.075 | 0.999 | 0.016 | 0.89 |

${\rho}_{w}$ | AR wage mark-up | B | 0.50 | 0.20 | 0.97 | 0.959 | 0.0366 | 0.972 | 0.029 | 0.969 | 0.004 | 0.967 | 0.011 | 0.96 |

${\mu}_{p}$ | MA price mark-up | B | 0.50 | 0.20 | 0.77 | 0.723 | 0.2217 | 0.966 | 0.013 | 0.952 | 0.115 | 0.927 | 0.081 | 0.69 |

${\mu}_{w}$ | MA wage mark-up | B | 0.50 | 0.20 | 0.87 | 0.809 | 0.1694 | 0.945 | 0.048 | 0.914 | 0.026 | 0.904 | 0.029 | 0.84 |

${\sigma}_{a}$ | Std. technology shock | IG | 0.10 | 2.00 | 0.43 | 0.434 | 0.0506 | 0.472 | 0.029 | 0.445 | 0.011 | 0.446 | 0.026 | 0.45 |

${\sigma}_{b}$ | Std. risk premium shock | IG | 0.10 | 2.00 | 0.24 | 0.237 | 0.0606 | 0.243 | 0.052 | 0.241 | 0.038 | 0.243 | 0.003 | 0.23 |

${\sigma}_{g}$ | Std. government spending | IG | 0.10 | 2.00 | 0.51 | 0.516 | 0.0519 | 0.52 | 0.016 | 0.504 | 0.061 | 0.512 | 0.011 | 0.53 |

${\sigma}_{i}$ | Std. investment shock | IG | 0.10 | 2.00 | 0.43 | 0.435 | 0.0688 | 0.469 | 0.079 | 0.458 | 0.085 | 0.461 | 0.011 | 0.56 |

${\sigma}_{m}$ | Std. monetary policy | IG | 0.10 | 2.00 | 0.24 | 0.244 | 0.1014 | 0.231 | 0.011 | 0.233 | 0.006 | 0.228 | 0.009 | 0.24 |

${\sigma}_{\pi}$ | Std. price mark-up shock | IG | 0.10 | 2.00 | 0.14 | 0.141 | 0.0328 | 0.145 | 0.016 | 0.139 | 0.014 | 0.133 | 0.004 | 0.14 |

${\sigma}_{w}$ | Std. wage mark-up shock | IG | 0.10 | 2.00 | 0.24 | 0.235 | 0.0359 | 0.231 | 0.024 | 0.217 | 0.008 | 0.224 | 0.028 | 0.24 |

Parameters | ${\mathit{s}}_{1}^{*}$ | ${\mathit{s}}_{2}^{*}$ | ${\mathit{s}}_{3}^{*}$ | ${\mathit{s}}_{5}^{*}$ | ${\mathit{s}}_{10}^{*}$ | ${\mathit{s}}_{25}^{*}$ |
---|---|---|---|---|---|---|

$\Psi $ | 1.000 | 0.389 | 0.373 | 0.582 | 0.445 | 1.841 |

${\sigma}_{l}$ | 1.000 | 0.441 | 0.348 | 0.344 | 0.33 | 1.471 |

${\rho}_{g}$ | 1.000 | 0.178 | 0.142 | 0.081 | 0.132 | 1.004 |

${\sigma}_{w}$ | 1.000 | 0.406 | 0.434 | 0.685 | 0.328 | 0.799 |

${\xi}_{w}$ | 1.000 | 0.176 | 0.174 | 0.151 | 0.318 | 0.738 |

$\alpha $ | 1.000 | 0.428 | 0.545 | 1.149 | 0.495 | 0.733 |

${r}_{\Delta y}$ | 1.000 | 0.338 | 0.361 | 0.207 | 0.515 | 0.733 |

${\xi}_{p}$ | 1.000 | 0.339 | 0.343 | 0.382 | 0.409 | 0.61 |

$\overline{l}$ | 1.000 | 0.621 | 0.575 | 0.853 | 1.046 | 0.543 |

h | 1.000 | 0.383 | 0.294 | 0.7 | 0.127 | 0.534 |

${\sigma}_{a}$ | 1.000 | 0.284 | 0.533 | 0.588 | 0.792 | 0.519 |

${r}_{\pi}$ | 1.000 | 0.352 | 0.571 | 0.843 | 0.225 | 0.516 |

$\psi $ | 1.000 | 0.429 | 0.423 | 0.486 | 0.206 | 0.466 |

$\Phi $ | 1.000 | 0.488 | 0.611 | 0.556 | 0.441 | 0.456 |

$\overline{\pi}$ | 1.000 | 0.529 | 0.375 | 0.763 | 1.059 | 0.413 |

$\rho $ | 1.000 | 0.299 | 0.329 | 0.316 | 0.193 | 0.397 |

${\mu}_{p}$ | 1.000 | 0.097 | 0.086 | 0.059 | 0.087 | 0.362 |

${\rho}_{i}$ | 1.000 | 0.426 | 0.452 | 0.621 | 0.555 | 0.341 |

100 $(\gamma -1)$ | 1.000 | 0.314 | 0.257 | 0.311 | 0.226 | 0.311 |

${\rho}_{w}$ | 1.000 | 0.265 | 0.295 | 0.803 | 0.532 | 0.306 |

${\rho}_{m}$ | 1.000 | 0.326 | 0.333 | 0.383 | 0.912 | 0.238 |

${\sigma}_{g}$ | 1.000 | 0.425 | 0.483 | 0.312 | 0.676 | 0.204 |

${\iota}_{p}$ | 1.000 | 0.485 | 0.288 | 0.455 | 0.433 | 0.194 |

${\sigma}_{c}$ | 1.000 | 0.425 | 0.465 | 0.999 | 0.536 | 0.187 |

100 $({\beta}^{-1}-1)$ | 1.000 | 0.705 | 0.525 | 0.825 | 0.226 | 0.186 |

${\rho}_{ga}$ | 1.000 | 0.357 | 0.497 | 0.475 | 0.328 | 0.178 |

${\mu}_{w}$ | 1.000 | 0.108 | 0.109 | 0.285 | 0.324 | 0.171 |

${\sigma}_{i}$ | 1.000 | 0.302 | 0.773 | 1.156 | 1.315 | 0.175 |

${\rho}_{a}$ | 1.000 | 0.209 | 0.333 | 0.779 | 0.581 | 0.163 |

${r}_{y}$ | 1.000 | 0.316 | 0.728 | 1.324 | 0.277 | 0.148 |

${\iota}_{w}$ | 1.000 | 0.503 | 0.384 | 0.523 | 0.628 | 0.135 |

${\sigma}_{p}$ | 1.000 | 0.307 | 0.371 | 0.496 | 0.368 | 0.128 |

${\rho}_{p}$ | 1.000 | 0.027 | 0.017 | 0.007 | 0.002 | 0.122 |

${\sigma}_{m}$ | 1.000 | 0.359 | 0.116 | 0.116 | 0.146 | 0.088 |

${\rho}_{b}$ | 1.000 | 0.392 | 0.386 | 0.937 | 0.325 | 0.072 |

${\sigma}_{b}$ | 1.000 | 0.402 | 0.419 | 0.871 | 0.712 | 0.061 |

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**MDPI and ACS Style**

Chaim, P.; Laurini, M.P.
Data Cloning Estimation and Identification of a Medium-Scale DSGE Model. *Stats* **2023**, *6*, 17-29.
https://doi.org/10.3390/stats6010002

**AMA Style**

Chaim P, Laurini MP.
Data Cloning Estimation and Identification of a Medium-Scale DSGE Model. *Stats*. 2023; 6(1):17-29.
https://doi.org/10.3390/stats6010002

**Chicago/Turabian Style**

Chaim, Pedro, and Márcio Poletti Laurini.
2023. "Data Cloning Estimation and Identification of a Medium-Scale DSGE Model" *Stats* 6, no. 1: 17-29.
https://doi.org/10.3390/stats6010002