# Time-Varying Comovement of Foreign Exchange Markets: A GLS-Based Time-Varying Model Approach

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## Abstract

**:**

## 1. Introduction

## 2. Model

#### 2.1. Exchange Rate Dynamics and Cointegration

#### 2.2. The Vector Error-Correction Model

#### 2.3. The Time-Varying VEC Model

#### 2.4. The Degree of Market Comovement

#### 2.5. Confirming Our Assumption of Constant Cointegrating Vectors

## 3. Data

## 4. Empirical Results

#### 4.1. Preliminaries

#### 4.2. The Time-Varying Model

## 5. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ADF | Augmented Dickey-Fuller |

GLS | Generalized least squares |

MAIC | modified Akaike information criterion |

MBIC | Modified Bayesian information criterion |

MCMC | Markov chain Monte Carlo |

OLS | Ordinary least squares |

STAR | Smooth transition autoregressive |

TV-VEC | time-varying vector error correction |

VEC | Vector error correction |

## Appendix A. Vector Error-Correction Model

**Case**

**1:**

**Case**

**2:**

**Case**

**3:**

## Appendix B. Parameter Constancy Test

## Appendix C. Time-Varying VEC Model

## Appendix D. Degree of Market Comovement

## References

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Mean | SD | Min | Max | ADF-GLS | Lags | $\widehat{\mathit{\varphi}}$ | $\mathcal{N}$ | ||
---|---|---|---|---|---|---|---|---|---|

Level | |||||||||

${S}_{CA}$ | $0.2175$ | $0.1452$ | $-0.0454$ | $0.4697$ | $-1.2364$ | 1 | $0.9948$ | 308 | |

${F}_{CA}$ | $0.2180$ | $0.1449$ | $-0.0447$ | $0.4699$ | $-1.2470$ | 1 | $0.9947$ | ||

${S}_{JP}$ | $4.6918$ | $0.1486$ | $4.3396$ | $5.0352$ | $-1.6915$ | 1 | $0.9902$ | ||

${F}_{JP}$ | $4.6899$ | $0.1483$ | $4.3393$ | $5.0344$ | $-1.6844$ | 1 | $0.9903$ | ||

${S}_{UK}$ | $-0.4965$ | $0.0905$ | $-0.7279$ | $-0.3387$ | $-2.9910$ ** | 1 | $0.9700$ | ||

${F}_{UK}$ | $-0.4953$ | $0.0900$ | $-0.7269$ | $-0.3373$ | $-2.9969$ ** | 1 | $0.9698$ | ||

First Difference | |||||||||

$\Delta {S}_{CA}$ | $0.0005$ | $0.0167$ | $-0.0626$ | $0.1083$ | $-11.8715$ *** | 0 | $0.3648$ | 307 | |

$\Delta {F}_{CA}$ | $0.0005$ | $0.0167$ | $-0.0626$ | $0.1077$ | $-11.8337$ *** | 0 | $0.3676$ | ||

$\Delta {S}_{JP}$ | $-0.0007$ | $0.0261$ | $-0.1095$ | $0.0823$ | $-11.5044$ *** | 0 | $0.3948$ | ||

$\Delta {F}_{JP}$ | $-0.0007$ | $0.0261$ | $-0.1094$ | $0.0831$ | $-11.5039$ *** | 0 | $0.3948$ | ||

$\Delta {S}_{UK}$ | $0.0004$ | $0.0228$ | $-0.0583$ | $0.1066$ | $-12.2763$ *** | 0 | $0.3385$ | ||

$\Delta {F}_{UK}$ | $0.0004$ | $0.0227$ | $-0.0585$ | $0.1080$ | $-12.2221$ *** | 0 | $0.3424$ |

Eigenvalues | Max Eigen | Trace | |
---|---|---|---|

None | 0.1352 | 44.45 ** | 100.19 * |

At most 1 | 0.0826 | 26.39 | 55.74 |

${\mathit{S}}_{\mathit{C}\mathit{A}}$ | ${\mathit{F}}_{\mathit{C}\mathit{A}}$ | ${\mathit{S}}_{\mathit{J}\mathit{P}}$ | ${\mathit{F}}_{\mathit{J}\mathit{P}}$ | ${\mathit{S}}_{\mathit{U}\mathit{K}}$ | ${\mathit{F}}_{\mathit{U}\mathit{K}}$ | ||
---|---|---|---|---|---|---|---|

Difference | |||||||

${S}_{CA}$ | $-0.4209$ | $-0.2370$ | $6.0713$ | $6.1933$ | $2.0254$ | $2.1560$ | |

$\left[1.8453\right]$ | $\left[1.8414\right]$ | $\left[4.5597\right]$ | $\left[4.5475\right]$ | $\left[4.4885\right]$ | $\left[4.4183\right]$ | ||

${F}_{CA}$ | $0.7703$ | $0.5885$ | $-6.0537$ | $-6.1739$ | $-1.6558$ | $-1.7854$ | |

$\left[1.8608\right]$ | $\left[1.8560\right]$ | $\left[4.5810\right]$ | $\left[4.5684\right]$ | $\left[4.4883\right]$ | $\left[4.4181\right]$ | ||

${S}_{JP}$ | $1.6242$ | $1.6664$ | $-0.7974$ | $-0.5514$ | $1.0198$ | $0.9973$ | |

$\left[1.8997\right]$ | $\left[1.8902\right]$ | $\left[3.3395\right]$ | $\left[3.3475\right]$ | $\left[3.0337\right]$ | $\left[3.0269\right]$ | ||

${F}_{JP}$ | $-1.6031$ | $-1.6460$ | $1.1054$ | $0.8580$ | $-0.9913$ | $-0.9692$ | |

$\left[1.9188\right]$ | $\left[1.9093\right]$ | $\left[3.3492\right]$ | $\left[3.3573\right]$ | $\left[3.0485\right]$ | $\left[3.0412\right]$ | ||

${S}_{UK}$ | $0.1174$ | $0.0667$ | $-3.9585$ | $-4.1291$ | $-6.0241$ | $-5.8113$ | |

$\left[2.1271\right]$ | $\left[2.1253\right]$ | $\left[3.1513\right]$ | $\left[3.1505\right]$ | $\left[3.5704\right]$ | $\left[3.5750\right]$ | ||

${F}_{UK}$ | $-0.1774$ | $-0.1263$ | $3.9517$ | $4.1228$ | $6.2729$ | $6.0556$ | |

$\left[2.1210\right]$ | $\left[2.1200\right]$ | $\left[3.1599\right]$ | $\left[3.1593\right]$ | $\left[3.6079\right]$ | $\left[3.6115\right]$ | ||

Level | |||||||

Constant | $-0.0334$ | $-0.0334$ | $0.2006$ | $0.1995$ | $0.1467$ | $0.1452$ | |

$\left[0.0470\right]$ | $\left[0.0468\right]$ | $\left[0.0680\right]$ | $\left[0.0680\right]$ | $\left[0.0470\right]$ | $\left[0.0467\right]$ | ||

${S}_{CA}$ | $0.8047$ | $0.9465$ | $3.3757$ | $3.3233$ | $2.2904$ | $2.2845$ | |

$\left[1.1232\right]$ | $\left[1.1317\right]$ | $\left[2.6344\right]$ | $\left[2.6460\right]$ | $\left[2.2176\right]$ | $\left[2.2168\right]$ | ||

${F}_{CA}$ | $-0.8390$ | $-0.9808$ | $-3.3467$ | $-3.2952$ | $-2.2552$ | $-2.2494$ | |

$\left[1.1254\right]$ | $\left[1.1337\right]$ | $\left[2.6427\right]$ | $\left[2.6543\right]$ | $\left[2.2191\right]$ | $\left[2.2181\right]$ | ||

${S}_{JP}$ | $1.3304$ | $1.2953$ | $-1.0766$ | $-0.9938$ | $-1.5576$ | $-1.5626$ | |

$\left[1.0113\right]$ | $\left[1.0109\right]$ | $\left[1.3641\right]$ | $\left[1.3615\right]$ | $\left[1.1244\right]$ | $\left[1.1277\right]$ | ||

${F}_{JP}$ | $-1.3189$ | $-1.2837$ | $1.0302$ | $0.9477$ | $1.5173$ | $1.5226$ | |

$\left[1.0135\right]$ | $\left[1.0131\right]$ | $\left[1.3695\right]$ | $\left[1.3668\right]$ | $\left[1.1194\right]$ | $\left[1.1226\right]$ | ||

${S}_{UK}$ | $-2.0125$ | $-2.0557$ | $-1.4713$ | $-1.4990$ | $-1.8191$ | $-1.7604$ | |

$\left[1.5610\right]$ | $\left[1.5579\right]$ | $\left[2.2110\right]$ | $\left[2.2115\right]$ | $\left[2.3796\right]$ | $\left[2.3631\right]$ | ||

${F}_{UK}$ | $2.0478$ | $2.0910$ | $1.4478$ | $1.4764$ | $1.7452$ | $1.6862$ | |

$\left[1.5691\right]$ | $\left[1.5658\right]$ | $\left[2.2266\right]$ | $\left[2.2271\right]$ | $\left[2.3855\right]$ | $\left[2.3688\right]$ | ||

${\overline{R}}^{2}$ | $0.0984$ | $0.1005$ | $0.1032$ | $0.1026$ | $0.1918$ | $0.1900$ | |

${L}_{C}$ | $65.5233$ *** |

^{2}, and “L

_{C}” denotes the Hansen [10,11] joint L

_{C}statistic with variance. (2) Newey andWest [19] robust standard errors are in brackets. (3) “***” indicates statistically significant at 1% level. (4) R version 4.0.5 was used to compute the estimates and the statistics.

${\mathit{S}\mathit{u}\mathit{p}\mathit{Q}}^{1}$ | ${\mathit{S}\mathit{u}\mathit{p}\mathit{Q}}^{2}$ | WQ | SQ | |
---|---|---|---|---|

Test Stats | 8.55 *** | 10.81 *** | 8.55 *** | 15.94 *** |

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**MDPI and ACS Style**

Ito, M.; Noda, A.; Wada, T. Time-Varying Comovement of Foreign Exchange Markets: A GLS-Based Time-Varying Model Approach. *Mathematics* **2021**, *9*, 849.
https://doi.org/10.3390/math9080849

**AMA Style**

Ito M, Noda A, Wada T. Time-Varying Comovement of Foreign Exchange Markets: A GLS-Based Time-Varying Model Approach. *Mathematics*. 2021; 9(8):849.
https://doi.org/10.3390/math9080849

**Chicago/Turabian Style**

Ito, Mikio, Akihiko Noda, and Tatsuma Wada. 2021. "Time-Varying Comovement of Foreign Exchange Markets: A GLS-Based Time-Varying Model Approach" *Mathematics* 9, no. 8: 849.
https://doi.org/10.3390/math9080849