# Refined Measures of Dynamic Connectedness based on Time-Varying Parameter Vector Autoregressions

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### TVP-VAR

## 3. Monte Carlo Simulation

## 4. Data and Summary Statistics

## 5. Empirical Illustration

#### 5.1. Dynamic Total Connectedness

#### 5.2. Net Total and Net Pairwise Directional Connectedness

#### 5.3. Sensitivity Analysis

#### 5.3.1. Prior Sensitivity Analysis

#### 5.3.2. Forgetting Factor Sensitivity Analysis

#### 5.4. Forecast Performance

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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1. | Although there is in fact a wealth of literature regarding TVP-VAR models (see, inter alia, Primiceri 2005; Cogley and Sargent 2005; Koop and Korobilis 2013, 2014; Del Negro and Primiceri 2015; Petrova 2019) we do not focus on the TVP-VAR framework specifically, but we are rather concerned with utilising the TVP-VAR framework in order to improve the accuracy of the dynamic connectedness measures. |

2. | Both the code for Monte Carlo simulation and the results of different rolling-windows are available upon request. |

**Figure 5.**Dynamic total connectedness. the darker the series, the smaller the window size, and vice versa (25, 26, …, 274, 275).

**Figure 10.**Prior sensitivity analysis: Net total and net pairwise directional connectedness measures.

**Figure 14.**Confidence intervals: Net total and net pairwise directional connectedness measures of uncertainty.

${\mathit{DV}}_{11}$ | ${\mathit{DV}}_{12}$ | ${\mathit{DV}}_{21}$ | ${\mathit{DV}}_{22}$ | |
---|---|---|---|---|

(1) | (2) | (3) | (4) | |

Outlier | 0.025 *** | 0.016 *** | 0.012 *** | 0.027 *** |

(0.0003) | (0.0002) | (0.0002) | (0.0003) | |

Structural Break | 0.033 *** | 0.040 *** | 0.032 *** | 0.028 *** |

in Parameters | (0.0005) | (0.001) | (0.0005) | (0.0004) |

EUR | GBP | JPY | CHF | |
---|---|---|---|---|

Mean | 0.056 | 0.14 | −0.156 | −0.141 |

Variance | 5.996 | 5.988 | 7.121 | 7.785 |

Skewness | 0.124 | 0.411 *** | −0.320 *** | 0.057 |

(0.244) | (0.000) | (0.003) | (0.591) | |

$Excess$ | 0.270 | 1.932 *** | 0.794 *** | 0.913 *** |

$Kurtosis$ | (0.196) | (0.000) | (0.003) | (0.001) |

$JB$ | 2.913 | 95.694 *** | 22.557 *** | 18.375 *** |

(0.233) | (0.000) | (0.000) | (0.000) | |

$ERS$ | −6.171 *** | −6.605 *** | −5.372 *** | −6.446 *** |

(0.000) | (0.000) | (0.000) | (0.000) | |

$Q\left(20\right)$ | 57.561 *** | 64.935 *** | 79.655 *** | 42.297 *** |

(0.000) | (0.000) | (0.000) | (0.000) | |

${Q}^{2}\left(20\right)$ | 18.874 ** | 59.908 *** | 30.270 *** | 12.111 |

(0.028) | (0.000) | (0.000) | (0.300) | |

Unconditional Correlation | ||||

EUR | $1.000$ | $0.701$ | $0.466$ | $0.868$ |

GBP | $0.701$ | $1.000$ | $0.328$ | $0.634$ |

JPY | $0.466$ | $0.328$ | $1.000$ | $0.544$ |

CHF | $0.868$ | $0.634$ | $0.544$ | $1.000$ |

TVP-VAR | |||||
---|---|---|---|---|---|

TO (i) | EUR | GBP | JPY | CHF | FROM (i) |

EUR | 40.1 | 18.2 | 10.9 | 30.8 | 59.9 |

GBP | 23.8 | 48.3 | 7.4 | 20.5 | 51.7 |

JPY | 15.1 | 9.0 | 57.5 | 18.3 | 42.5 |

CHF | 30.7 | 16.3 | 12.5 | 40.5 | 59.5 |

Contribution TO others | 69.5 | 43.6 | 30.9 | 69.6 | 213.6 |

NET directional connectedness | 9.6 | -8.1 | −11.6 | 10.1 | TCI |

NPSO transmitter | 2 | 1 | 0 | 3 | 53.4 |

50-Month Rolling-Window VAR | |||||

EUR | 40.0 | 19.8 | 9.9 | 30.2 | 60.0 |

GBP | 24.6 | 47.4 | 7.6 | 20.5 | 52.6 |

JPY | 13.9 | 8.9 | 60.0 | 17.3 | 40.0 |

CHF | 30.3 | 17.5 | 11.8 | 40.4 | 59.6 |

Contribution TO others | 68.7 | 46.2 | 29.3 | 68.0 | TCI |

NET directional connectedness | 8.8 | -6.4 | −10.7 | 8.4 | 53.0 |

NPDC transmitter | 3 | 1 | 0 | 2 | |

100-month Rolling-Window VAR | |||||

EUR | 39.8 | 20.0 | 9.1 | 31.2 | 60.2 |

GBP | 25.7 | 46.4 | 6.5 | 21.4 | 53.6 |

JPY | 13.3 | 8.2 | 60.6 | 17.8 | 39.4 |

CHF | 31.2 | 17.8 | 11.3 | 39.7 | 60.3 |

Contribution TO others | 70.2 | 46.0 | 26.9 | 70.4 | TCI |

NET directional connectedness | 10.0 | -7.6 | −12.5 | 10.1 | 53.4 |

NPDC transmitter | 3 | 1 | 0 | 2 | |

200-Month Rolling-Window VAR | |||||

EUR | 39.3 | 20.3 | 8.6 | 31.8 | 60.7 |

GBP | 26.2 | 46.1 | 5.7 | 21.9 | 53.9 |

JPY | 13.4 | 7.7 | 60.9 | 18.1 | 39.1 |

CHF | 31.9 | 18.1 | 10.9 | 39.1 | 60.9 |

Contribution TO others | 71.5 | 46.1 | 25.3 | 71.8 | TCI |

NET directional connectedness | 10.8 | -7.8 | −13.9 | 10.9 | 53.7 |

NPDC transmitter | 3 | 1 | 0 | 2 |

${\mathit{\kappa}}_{1},{\mathit{\kappa}}_{2}$ | EUR | GBP | JPY | CHF | $\overline{\mathit{MAPE}}$ | EUR | GBP | JPY | CHF | $\overline{\mathbf{MAPE}}$ |
---|---|---|---|---|---|---|---|---|---|---|

1-Step Ahead Forecast | 2-Step Ahead Forecast | |||||||||

0.99,0.99 | $0.740$ | $0.688$ | $0.751$ | $0.713$ | $0.723$ | $1.001$ | $0.883$ | $\mathbf{0.951}$ | $0.935$ | $0.943$ |

0.99,0.98 | $0.739$ | $0.686$ | $0.751$ | $0.711$ | $0.722$ | $1.000$ | $0.881$ | $\mathbf{0.951}$ | $0.934$ | $0.941$ |

0.99,0.97 | $0.738$ | $0.684$ | $0.751$ | $0.710$ | $0.721$ | $0.999$ | $0.878$ | $\mathbf{0.951}$ | $0.933$ | $0.940$ |

0.99,0.96 | $\mathbf{0.737}$ | $\mathbf{0.682}$ | $\mathbf{0.750}$ | $\mathbf{0.709}$ | $\mathbf{0.720}$ | $\mathbf{0.998}$ | $\mathbf{0.876}$ | $\mathbf{0.951}$ | $\mathbf{0.932}$ | $\mathbf{0.939}$ |

0.98,0.99 | $0.746$ | $0.688$ | $0.757$ | $0.721$ | $0.728$ | $1.010$ | $0.888$ | $0.956$ | $0.940$ | $0.949$ |

0.98,0.98 | $0.745$ | $0.687$ | $0.757$ | $0.720$ | $0.727$ | $1.008$ | $0.885$ | $0.956$ | $0.939$ | $0.947$ |

0.98,0.97 | $0.743$ | $0.686$ | $0.757$ | $0.719$ | $0.726$ | $1.006$ | $0.882$ | $0.955$ | $0.938$ | $0.945$ |

0.98,0.96 | $0.742$ | $0.684$ | $0.756$ | $0.719$ | $0.725$ | $1.004$ | $0.881$ | $0.954$ | $0.938$ | $0.944$ |

0.97,0.99 | $0.756$ | $0.687$ | $0.763$ | $0.727$ | $0.733$ | $1.025$ | $0.893$ | $0.962$ | $0.946$ | $0.956$ |

0.97,0.98 | $0.754$ | $0.686$ | $0.763$ | $0.726$ | $0.732$ | $1.021$ | $0.890$ | $0.962$ | $0.944$ | $0.954$ |

0.97,0.97 | $0.752$ | $0.686$ | $0.762$ | $0.725$ | $0.731$ | $1.018$ | $0.889$ | $0.961$ | $0.943$ | $0.952$ |

0.97,0.96 | $0.751$ | $0.685$ | $0.760$ | $0.725$ | $0.730$ | $1.015$ | $0.887$ | $0.960$ | $0.942$ | $\mathbf{0.951}$ |

0.96,0.99 | $0.767$ | $0.692$ | $0.770$ | $0.734$ | $0.741$ | $1.039$ | $0.904$ | $0.969$ | $0.953$ | $0.966$ |

0.96,0.98 | $0.765$ | $0.690$ | $0.770$ | $0.732$ | $0.739$ | $1.035$ | $0.900$ | $0.969$ | $0.950$ | $0.963$ |

0.96,0.97 | $0.762$ | $0.687$ | $0.769$ | $0.731$ | $0.737$ | $1.031$ | $0.896$ | $0.968$ | $0.948$ | $0.961$ |

0.96,0.96 | $0.760$ | $0.685$ | $0.767$ | $0.730$ | $0.735$ | $1.027$ | $0.894$ | $0.967$ | $0.947$ | $0.959$ |

RW 50 | $0.779$ | $0.713$ | $0.782$ | $0.752$ | $0.757$ | $1.057$ | $0.932$ | $0.994$ | $0.974$ | $0.989$ |

RW 100 | $0.740$ | $0.704$ | $0.759$ | $0.724$ | $0.732$ | $1.005$ | $0.902$ | $0.963$ | $0.936$ | $\mathbf{0.951}$ |

RW 200 | $0.741$ | $0.687$ | $0.754$ | $0.719$ | $0.725$ | $1.004$ | $0.884$ | $0.957$ | $0.947$ | $0.948$ |

3-Step Ahead Forecast | 6-Step Ahead Forecast | |||||||||

0.99,0.99 | $1.208$ | $1.036$ | $\mathbf{1.143}$ | $1.106$ | $1.123$ | $\mathbf{1.673}$ | $1.424$ | $1.639$ | $\mathbf{1.453}$ | $\mathbf{1.547}$ |

0.99,0.98 | $1.208$ | $1.034$ | $1.144$ | $1.105$ | $1.122$ | $\mathbf{1.673}$ | $1.423$ | $1.639$ | $\mathbf{1.453}$ | $\mathbf{1.547}$ |

0.99,0.97 | $\mathbf{1.207}$ | $1.031$ | $\mathbf{1.143}$ | $\mathbf{1.104}$ | $1.122$ | $\mathbf{1.673}$ | $1.422$ | $\mathbf{1.638}$ | $\mathbf{1.453}$ | $\mathbf{1.547}$ |

0.99,0.96 | $\mathbf{1.207}$ | $\mathbf{1.029}$ | $\mathbf{1.143}$ | $\mathbf{1.104}$ | $\mathbf{1.121}$ | $\mathbf{1.673}$ | $\mathbf{1.421}$ | $\mathbf{1.638}$ | $1.454$ | $\mathbf{1.547}$ |

0.98,0.99 | $1.220$ | $1.041$ | $1.151$ | $1.111$ | $1.131$ | $1.686$ | $1.429$ | $1.650$ | $1.458$ | $1.556$ |

0.98,0.98 | $1.218$ | $1.039$ | $1.150$ | $1.110$ | $1.129$ | $1.684$ | $1.428$ | $1.647$ | $1.458$ | $1.554$ |

0.98,0.97 | $1.216$ | $1.037$ | $1.149$ | $1.109$ | $1.128$ | $1.683$ | $1.427$ | $1.645$ | $1.458$ | $1.553$ |

0.98,0.96 | $1.215$ | $1.035$ | $1.148$ | $1.109$ | $1.127$ | $1.682$ | $1.426$ | $1.644$ | $1.459$ | $1.553$ |

${\mathit{\kappa}}_{1},{\mathit{\kappa}}_{2}$ | EUR | GBP | JPY | CHF | $\overline{\mathbf{MAPE}}$ | EUR | GBP | JPY | CHF | $\overline{\mathbf{MAPE}}$ |
---|---|---|---|---|---|---|---|---|---|---|

9-Step Ahead Forecast | 12-Step Ahead Forecast | |||||||||

0.99,0.99 | $\mathbf{2.032}$ | $1.746$ | $1.989$ | $\mathbf{1.703}$ | $\mathbf{1.867}$ | $2.358$ | $2.011$ | $\mathbf{2.301}$ | $\mathbf{1.917}$ | $2.147$ |

0.99,0.98 | $\mathbf{2.032}$ | $1.744$ | $1.988$ | $1.704$ | $\mathbf{1.867}$ | $\mathbf{2.357}$ | $2.009$ | $\mathbf{2.301}$ | $1.918$ | $\mathbf{2.146}$ |

0.99,0.97 | $\mathbf{2.032}$ | $1.743$ | $\mathbf{1.987}$ | $1.705$ | $\mathbf{1.867}$ | $\mathbf{2.357}$ | $2.007$ | $\mathbf{2.301}$ | $1.919$ | $\mathbf{2.146}$ |

0.99,0.96 | $\mathbf{2.032}$ | $\mathbf{1.742}$ | $1.988$ | $1.706$ | $\mathbf{1.867}$ | $\mathbf{2.357}$ | $\mathbf{2.005}$ | $\mathbf{2.301}$ | $1.920$ | $\mathbf{2.146}$ |

0.98,0.99 | $2.047$ | $1.755$ | $1.999$ | $1.708$ | $1.877$ | $2.374$ | $2.026$ | $2.309$ | $1.925$ | $2.158$ |

0.98,0.98 | $2.045$ | $1.753$ | $1.996$ | $1.709$ | $1.876$ | $2.372$ | $2.022$ | $2.307$ | $1.925$ | $2.157$ |

0.98,0.97 | $2.044$ | $1.751$ | $1.995$ | $1.710$ | $1.875$ | $2.371$ | $2.018$ | $2.307$ | $1.926$ | $2.156$ |

0.98,0.96 | $2.043$ | $1.749$ | $1.994$ | $1.711$ | $1.874$ | $2.370$ | $2.015$ | $2.307$ | $1.927$ | $2.155$ |

0.97,0.99 | $2.062$ | $1.770$ | $2.011$ | $1.712$ | $1.889$ | $2.392$ | $2.045$ | $2.320$ | $1.930$ | $2.172$ |

0.97,0.98 | $2.059$ | $1.765$ | $2.007$ | $1.711$ | $1.885$ | $2.389$ | $2.038$ | $2.317$ | $1.930$ | $2.168$ |

0.97,0.97 | $2.056$ | $1.762$ | $2.004$ | $1.711$ | $1.883$ | $2.386$ | $2.033$ | $2.315$ | $1.930$ | $2.166$ |

0.97,0.96 | $2.054$ | $1.759$ | $2.003$ | $1.712$ | $1.882$ | $2.384$ | $2.030$ | $2.315$ | $1.931$ | $2.165$ |

0.96,0.99 | $2.080$ | $1.795$ | $2.027$ | $1.722$ | $1.906$ | $2.413$ | $2.075$ | $2.337$ | $1.943$ | $2.192$ |

0.96,0.98 | $2.073$ | $1.785$ | $2.020$ | $1.718$ | $1.899$ | $2.406$ | $2.063$ | $2.329$ | $1.939$ | $2.184$ |

0.96,0.97 | $2.069$ | $1.779$ | $2.016$ | $1.716$ | $1.895$ | $2.402$ | $2.055$ | $2.326$ | $1.937$ | $2.180$ |

0.96,0.96 | $2.066$ | $1.775$ | $2.015$ | $1.715$ | $1.893$ | $2.399$ | $2.049$ | $2.326$ | $1.936$ | $2.177$ |

RW 50 | $2.170$ | $1.940$ | $2.137$ | $1.815$ | $2.016$ | $2.546$ | $2.284$ | $2.518$ | $2.078$ | $2.357$ |

RW 100 | $2.111$ | $1.775$ | $2.032$ | $1.754$ | $1.918$ | $2.487$ | $2.050$ | $2.353$ | $1.996$ | $2.222$ |

RW 200 | $2.045$ | $1.749$ | $2.005$ | $1.732$ | $1.883$ | $2.375$ | $2.018$ | $2.316$ | $1.948$ | $2.164$ |

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## Share and Cite

**MDPI and ACS Style**

Antonakakis, N.; Chatziantoniou, I.; Gabauer, D.
Refined Measures of Dynamic Connectedness based on Time-Varying Parameter Vector Autoregressions. *J. Risk Financial Manag.* **2020**, *13*, 84.
https://doi.org/10.3390/jrfm13040084

**AMA Style**

Antonakakis N, Chatziantoniou I, Gabauer D.
Refined Measures of Dynamic Connectedness based on Time-Varying Parameter Vector Autoregressions. *Journal of Risk and Financial Management*. 2020; 13(4):84.
https://doi.org/10.3390/jrfm13040084

**Chicago/Turabian Style**

Antonakakis, Nikolaos, Ioannis Chatziantoniou, and David Gabauer.
2020. "Refined Measures of Dynamic Connectedness based on Time-Varying Parameter Vector Autoregressions" *Journal of Risk and Financial Management* 13, no. 4: 84.
https://doi.org/10.3390/jrfm13040084