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Nonlinear Time Series and Neural-Network Models of Exchange Rates between the US Dollar and Major Currencies^{ †}

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## Abstract

**:**

## 1. Introduction

## 2. Research Methods

#### 2.1. Data Set and Econometric Models

#### Data Sets

#### 2.2. Data Characteristics

#### 2.3. Econometric Methods

## 3. Empirical Results

#### 3.1. Nonlinear Time Series Analysis

#### 3.2. Further Analysis Using Neural Nets

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Time series plots of the base series and their logarithmic differences. (

**a**) US -EURO; (

**b**) US-CHINA; (

**c**) US-JAPAN; (

**d**) US-UK.

**Figure 3.**SETAR analysis of US$ - Euro returns; (

**a**) Euro SETAR Residuals and ACF; (

**b**) MI Euro and lag −1, 0; (

**c**) lag 1 −1 SETAR Euro and Regime Switching.

**Figure 5.**Residual plots Neural Network Regression Analysis; (

**a**) Euro; (

**b**) China; (

**c**) Japan; (

**d**) UK.

KPSS Test | Probability | Fractional Integration (Whittle Estimator) | Z Statistic | Probability | |
---|---|---|---|---|---|

EURO - US Dollar exchange rate | 4.1066 | 0.01 * | 1.01789 | 21.156 | 0.0000 |

CHINESE Yuan - US Dollar exchange rate | 25.1896 | 0.01 * | 1.101 | 22.865 | 0.0000 |

JAPANESE YEN - US Dollar exchange rate | 8.4585 | 0.01 * | 1.0163 | 13.985 | 0.0000 |

UK Pound - US Dollar exchange rate | 13.8446 | 0.01 * | 1.032 | 21.463 | 0.0000 |

Country | Symbol | Abbreviations |
---|---|---|

EURO | EURET | EURO exchange rate return |

CHINA | CHRET | CHINESE exchange rate return |

JAPAN | JPRET | JAPANESE exchange rate return |

UK | UKRET | UK exchange rate return |

Statistics | EURET | CHRET | JPRET | UKRET |
---|---|---|---|---|

Mean | −0.0037 | −0.0094 | 0.0037 | −0.0063 |

Median | 0.000 | −0.0024 | 0.0084 | 0.0065 |

Maximum | 4.621 | 1.816 | 3.342 | 4.4348 |

Minimum | −3.003 | −0.998 | −5.216 | −4.9662 |

Skewness | 0.188 | 1.637 | −0.326 | −0.3404 |

Excess Kurtosis | 3.028 | 33.897 | 5.159 | 6.57086 |

Standard Deviation | 0.638 | 0.119 | 0.664 | 0.6194 |

Coefficient of Variation | 173.97 | 12.615 | 178.89 | 97.827 |

Euro | Intercept | F Smooth Terms V1 | F Smooth Terms V10 | AIC | MAPE | R-sq.(adj) |
---|---|---|---|---|---|---|

AAR | −0.00394 | 1.7249 | 2.4575* | −2244 | 104.5% | 0.00629 |

SETAR model ( 2 regimes) | Constant L | phiL.1 | phiL.2 | |||

Low regime | −0.00972614 | 0.02389115 | ||||

Constant H | phiH.1 | phiH.2 | ||||

High regime | 0.2307366 ** | −0.0153295 | −0.2220478** | |||

Threshold | Value | Propn. in high | Propn. in low | |||

Z(t) = + (0) X(t)+ (1)X(t-1) | 0.5448 | 15.6% | 84.4% | −2258 | 106.1% | |

NNET time series model | 2-3-1 network with 13 weights | −2317 | 102.9% | |||

LSTAR model | Constant L | phiL.1 | phiL.2 | |||

Low regime | −2.39025525 | −0.02832705 | −0.87307328 | |||

Constant H | phiH.1 | phiH.2 | ||||

High regime | 4.08892959 | 0.07937599 | 0.23125994 | |||

smoothing parameter | gamma = 0.8042 | |||||

Threshold | Value | |||||

Z(t) = + (0) X(t) + (1) X(t-1) | -0.4226 | −2259 | 106% | |||

Random Walk(1) | Constant | slope coefficient | ||||

—0.00387743 | 0.0176781 | 118% | −0.000087 | |||

Random Walk (20) lags | 108% |

China | Intercept | F Smooth Terms V1.0 | F Smooth Terms V1.1 | AIC | MAPE | R-sq.(adj) |
---|---|---|---|---|---|---|

AAR | −0.0094677 | 33.4181 *** | 3.4645 *** | −10854 | 122.6% | 0.078 |

SETAR model ( 2 regimes) | Constant L | phiL.1 | phiL.2 | |||

Low regime | −0.00972614 | 0.02389115 | ||||

Constant H | phiH.1 | phiH.2 | ||||

High regime | 0.2307366 ** | −0.0153295 | −0.2220478 ** | |||

Threshold | Value | Propn. in high | Propn. in low | |||

Z(t) = + (0) X(t)+ (1)X(t-1) | −0.04467 | 73.3% | 26.7% | −10695.75 | 116.9% | |

NNET time series model | 2-3-1 network with 13 weights | −10870.93 | 121.8% | |||

LSTAR model | Constant L | phiL.1 | phiL.2 | |||

Low regime | −0.1336682 | −0.2158839 | −0.4292838 | |||

Constant H | phiH.1 | phiH.2 | ||||

High regime | 0.1294817 | 0.1725717 | 0.3706611 | |||

smoothing parameter | gamma = 23.85 | |||||

Threshold | Value | |||||

Z(t) = + (0) X(t) + (1) X(t-1) | −0.4226 | −10691.80 | 117.8% | |||

Random walk (1)) | Constant | Slope coefficient | ||||

—0.0100210 *** | —0.0591522 *** | 100.2% | 0.003095 | |||

Random walk (20) | 121.38% |

Japan | Intercept | F Smooth Terms V1.0 | F Smooth Terms V1.1 | AIC | MAPE | R-sq.(adj) |
---|---|---|---|---|---|---|

AAR | 0.0038522 | 5.4933 ** | 4.0491 | −2048 | 104.5% | 0.00859 |

SETAR model ( 2 regimes) | Constant L | phiL.1 | phiL.2 | |||

Low regime | −0.00972614 | 0.02389115 | ||||

Constant H | phiH.1 | phiH.2 | ||||

High regime | 0.2307366 ** | −0.0153295 | −0.2220478 ** | |||

Threshold | Value | Propn. in high | Propn. in low | |||

Z(t) = + (0) X(t)+ (1)X(t-1) | −0.04467 | 73.3% | 26.7% | −10695.75 | 116.9% | |

NNET time series model | 2-3-1 network with 13 weights | −2081.341 | 101.9 % | |||

LSTAR model | Constant L | phiL.1 | phiL.2 | |||

Low regime | −0.1092538 | −0.1124170 | −0.0582778 | |||

Constant H | phiH.1 | phiH.2 | ||||

High regime | 0.12426792 | 0.11504397 | 0.01657785 | |||

smoothing parameter | gamma = 100 | |||||

Threshold | Value | |||||

Z(t) = + (0) X(t) + (1) X(t-1) | −0.7085 | −2048 | 104.9% | |||

Random walk(1) | Constant | Slope coefficient | ||||

0.00366027 | —0.0170771 | 88.92% | −0.000108 | |||

Random walk (20) | 104.44% |

UK | Intercept | F Smooth Terms V1.0 | F Smooth Terms V1.1 | AIC | MAPE | R-sq.(adj) |
---|---|---|---|---|---|---|

AAR | −0.0069404 | 3.6884 ** | 1.0387 | −2382 | 103.6% | 0.00687 |

SETAR model ( 2 regimes) | Constant L | phiL.1 | phiL.2 | |||

Low regime | 0.14175779 * | −0.04871131 | 0.14579643 * | |||

Constant H | phiH.1 | phiH.2 | ||||

High regime | −0.02411116 ** | 0.02448354 | 0.04441084 | |||

Threshold | Value | Propn. in high | Propn. in low | |||

Z(t) = + (0) X(t)+ (1)X(t-1) | −0.3935 | 78.09% | 21.91% | −2406.445 | 109.06 % | |

NNET time series model | 2-3-1 network with 13 weights | −2415.012 | 106.2 % | |||

LSTAR model | Constant L | phiL.1 | phiL.2 | |||

Low regime | 0.135255* | −0.048852 | 0.141031 * | |||

Constant H | phiH.1 | phiH.2 | ||||

High regime | −0.157804 ** | 0.073844 | −0.099545 | |||

smoothing parameter | gamma = 100 | |||||

Threshold | Value | |||||

Z(t) = + (0) X(t) + (1) X(t-1) | −0.397965 | −2403.007 | 106.8% | |||

Random walk(1) | Constant | Slope coefficient | ||||

—0.00630317 | 0.00808887 | 89.29% | −0.000334 | |||

Random walk (20) | 110.28% |

Euro (model 1) |

Y1[t] = 0.0848635 + EURET[t-2]×EURET[t-2], cubert×(-0.0136538) + EURET[t-2]×EURET[t-3]×(−0.0357626) + EURET[t-3]×EURET[t-6], cubert×0.0464996 + time×EURET[t-8], cubert×2.72731e-05 + EURET[t-4]×EURET[t-11]×(−0.0678379) + cycle×0.0028167 + EURET[t-8], cubert×EURET[t-10], cubert×0.0583482 |

Euro (model 2) |

Y1 = 0.000426977 – LUKRET×N9×0.694354 + N9×1.15686 |

N9 = 0.0127939 – LEURET×LCHRET×0.237853 – LEURET^2×0.0444379 |

China (model 1) |

China Model 1 Y1 = 0.00936077 + N76×1.02217 + N118×1.0439N118 = −0.00973411 + LJPRET×0.0164911 – LJPRET×LEURET, cubert×0.00848911 - LEURET, cubert×0.0206867 |

N118 = −0.00973411 + LJPRET*0.0164911 – LJPRET×LEURET, cubert×0.00848911 - LEURET, cubert×0.0206867 |

N76 = 0.00455715 - LCHRET×0.202939 + LCHRET^2×0.446615 + LCHRET, cubert×0.0261334 – LCHRET, cubert^2×0.142591 |

Euro | China | Japan | UK | |
---|---|---|---|---|

Model fit | 2006 observations | 2006 observations | 2006 observations | 2006 observations |

Mean absolute error | 0.4578 | 0.0676 | 0.4666 | 0.4465 |

Root mean square error | 0.6319 | 0.1167 | 0.6597 | 0.6245 |

Coefficient of Determination ( ${R}^{2}$) | 0.0068 | 0.1000 | 0.0045 | 0.0011 |

Predictions | 501 observations | 501 observations | 501 observations | 501 observations |

Mean absolute error | 0.4818 | 0.0712 | 0.5036 | 0.4311 |

Root mean square error | 0.6649 | 0.1167 | 0.6751 | 0.5945 |

Coefficient of Determination ( ${R}^{2}$) | −0.0383 | 0.1125 | 0.0003 | 0.0039 |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons by Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Allen, D.E.; McAleer, M.; Peiris, S.; Singh, A.K. Nonlinear Time Series and Neural-Network Models of Exchange Rates between the US Dollar and Major Currencies. *Risks* **2016**, *4*, 7.
https://doi.org/10.3390/risks4010007

**AMA Style**

Allen DE, McAleer M, Peiris S, Singh AK. Nonlinear Time Series and Neural-Network Models of Exchange Rates between the US Dollar and Major Currencies. *Risks*. 2016; 4(1):7.
https://doi.org/10.3390/risks4010007

**Chicago/Turabian Style**

Allen, David E., Michael McAleer, Shelton Peiris, and Abhay K. Singh. 2016. "Nonlinear Time Series and Neural-Network Models of Exchange Rates between the US Dollar and Major Currencies" *Risks* 4, no. 1: 7.
https://doi.org/10.3390/risks4010007