# On the Validity of Tests for Asymmetry in Residual-Based Threshold Cointegration Models

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Models and Tests

- (1)
- Estimate the long-run equilibrium equation to obtain ${\widehat{\beta}}_{0}$, ${\widehat{\beta}}_{1}$ and the cointegration residuals ${\widehat{z}}_{t}$. Conduct the F-test for asymmetry based on the MTAR/SETAR model and save ${F}_{apt}$.
- (2)
- Estimate the symmetric model$$\Delta {z}_{t}=\varrho \phantom{\rule{0.166667em}{0ex}}{z}_{t-1}+\sum _{j=1}^{k}{\gamma}_{j}\phantom{\rule{0.166667em}{0ex}}\Delta {z}_{t-j}{\epsilon}_{t}$$
- (3)
- Draw randomly from the residuals ${\widehat{\epsilon}}_{t}$ to obtain a bootstrap sample ${\epsilon}_{t}^{b}$.
- (4)
- Generate the bootstrap cointegration residuals series as $\Delta {z}_{t}^{b}=\widehat{\varrho}{z}_{t-1}^{b}+{\sum}_{j=1}^{k}{\widehat{\gamma}}_{j}\phantom{\rule{0.166667em}{0ex}}\Delta {z}_{t-j}^{b}+{\epsilon}_{t}^{b}$ and use $({z}_{1}^{b},\dots ,{z}_{k}^{b})=({\widehat{z}}_{1},\dots ,{\widehat{z}}_{k})$ as initial observations.
- (5)
- Generate the bootstrap variable ${y}_{t}^{b}={\widehat{\beta}}_{0}+{\widehat{\beta}}_{1}{x}_{t}+{z}_{t}^{b}$.
- (6)
- Estimate the long-run equilibrium equation for ${y}_{t}^{b}$ and ${x}_{t}$ and re-estimate the MTAR/SETAR model to compute the bootstrap F-statistic, ${F}_{apt}^{b}$.
- (7)
- Repeat (2) to (6) sufficiently often to obtain the empirical distribution of ${F}_{apt}^{b}$. Compute the p-value for ${F}_{apt}$ based on the bootstrap distribution.

## 3. Simulations

## 4. Application: (A)symmetric Fuel Price Transmissions

## 5. Summary and Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Caner, Mehmet, and Bruce E. Hansen. 2001. Threshold Autoregression with a Unit Root. Econometrica 69: 1555–96. [Google Scholar] [CrossRef]
- Chan, Kung-Sik. 1993. Consistency and Limiting Distribution of the Least Squares Estimator of a Threshold Autoregressive Model. The Annals of Statistics 21: 520–33. [Google Scholar] [CrossRef]
- Cook, Steven, Sean Holly, and Paul Turner. 1999. The power of tests for non-linearity: The case of Granger-Lee asymmetry. Economics Letters 62: 155–59. [Google Scholar] [CrossRef]
- Enders, Walter, and Clive W. J. Granger. 1998. Unit-Root Tests and Asymmetric Adjustment with an Example using the Term Structure of Interest Rates. Journal of Business & Economic Statistics 16: 304–11. [Google Scholar] [CrossRef]
- Enders, Walter, and Pierre L. Siklos. 2001. Cointegration and Threshold Adjustment. Journal of Business & Economic Statistics 19: 166–76. [Google Scholar] [CrossRef]
- Engle, Robert F., and Clive W. J. Granger. 1987. Co-Integration and Error Correction: Representation, Estimation and Testing. Econometrica 55: 251–76. [Google Scholar] [CrossRef]
- Frey, Giliola, and Matteo Manera. 2007. Econometric Models of Asymmetric Price Transmission. Journal of Economic Surveys 21: 349–415. [Google Scholar] [CrossRef]
- Galeotti, Marzio, Alessandro Lanza, and Matteo Manera. 2003. Rockets and feathers revisited: An international comparison on European gasoline markets. Energy Economics 25: 175–90. [Google Scholar] [CrossRef]
- Giacomini, Raffaella, Dimitris N. Politis, and Halbert White. 2013. A Warp-Speed Method for Conducting Monte Carlo Experiments Involving Bootstrap Estimators. Econometric Theory 29: 567–89. [Google Scholar] [CrossRef]
- Godby, Rob, Anastasia M. Lintner, Thanasis Stengos, and Bo Wandschneider. 2000. Testing for asymmetric pricing in the Canadian retail gasoline market. Energy Economics 22: 349–68. [Google Scholar] [CrossRef]
- Grasso, Margherita, and Matteo Manera. 2007. Asymmetric error correction models for the oil-gasoline price relationship. Energy Policy 35: 156–77. [Google Scholar] [CrossRef]
- Hansen, Bruce E. 1996. Inference when a Nuisance Parameter is not identified under the Null Hypothesis. Econometrica 64: 413–30. [Google Scholar] [CrossRef]
- Hansen, Bruce E. 1999. Testing for linearity. Journal of Economic Surveys 13: 551–76. [Google Scholar] [CrossRef]
- Honarvar, Afshin. 2010. Modeling of Asymmetry between Gasoline and Crude Oil Prices: A Monte Carlo Comparison. Computational Economics 36: 237–62. [Google Scholar] [CrossRef]
- Karoglou, Michail, and Bruce Morley. 2012. Purchasing power parity and structural instability in the US/UK exchange rate. Journal of International Financial Markets, Institutions and Money 22: 958–72. [Google Scholar] [CrossRef]
- Lee, Oesook, and Dong Wan Shin. 2000. On geometric ergodicity of the MTAR process. Statistics & Probability Letters 48: 229–37. [Google Scholar] [CrossRef]
- Li, Guodong, Bo Guan, Wai Keung Li, and Philip L. H. Yu. 2015. Hysteretic autoregressive time series models. Biometrika 102: 717–23. [Google Scholar] [CrossRef]
- Mohammadi, Hassan. 2011. Market integration and price transmission in the U.S. natural gas market: From the wellhead to end use markets. Energy Economics 33: 227–35. [Google Scholar] [CrossRef]
- Norman, Stephen. 2008. Systematic small sample bias in two regime SETAR model estimation. Economics Letters 99: 134–38. [Google Scholar] [CrossRef]
- Payne, James E., and George A. Waters. 2008. Interest rate pass through and asymmetric adjustment: Evidence from the federal funds rate operating target period. Applied Economics 40: 1355–62. [Google Scholar] [CrossRef]
- Perdiguero-Garcia, Jordi. 2013. Symmetric or asymmetric oil prices? A meta-analysis approach. Energy Policy 57: 389–97. [Google Scholar] [CrossRef]
- Petruccelli, Joseph D., and Samuel W. Woolford. 1984. A Threshold AR(1) Model. Journal of Applied Probability 21: 270–86. [Google Scholar] [CrossRef]
- Simioni, Michel, Frédéric Gonzales, Patrice Guillotreau, and Laurent Le Grel. 2013. Detecting Asymmetric Price Transmission with Consistent Threshold along the Fish Supply Chain. Canadian Journal of Agricultural Economics 61: 37–60. [Google Scholar] [CrossRef]
- Teräsvirta, Timo. 1994. Specification, Estimation, and Evaluation of Smooth Transition Autoregressive Models. Journal of the American Statistical Association 89: 208–18. [Google Scholar] [CrossRef]
- Thompson, Mark A. 2006. Asymmetric adjustment in the prime lending-deposit rate spread. Review of Financial Economics 15: 323–29. [Google Scholar] [CrossRef]
- Tong, Howell. 1978. On a Threshold Model, Pattern Recognition, and Signal Processing. Amsterdam: Sijthoff and Noordhoff. [Google Scholar]
- Van Dijk, Dick, Timo Teräsvirta, and Philip Hans Franses. 2002. Smooth Transition Autoregressive Models—A Survey of Recent Developments. Econometric Reviews 21: 1–47. [Google Scholar] [CrossRef]
- Von Cramon-Taubadel, Stephan, and Jochen Meyer. 2000. Asymmetric Price Transmission: Fact of Artefact? Working Paper. Göttingen, Germany: University of Göttingen, pp. 1–22. [Google Scholar]
- Zhu, Ke, Wai Keung Li, and Philip L. H. Yu. 2017. Buffered Autoregressive Models with Conditional Heteroscedasticity: An Application to Exchange Rates. Journal of Business and Economic Statistics 35: 528–42. [Google Scholar] [CrossRef]

1 | |

2 | While Honarvar (2010) allows for more types of asymmetric cointegration than the restrictive model used in this paper and suggests a different method for estimation and testing to account for this, it appears that the suggested method retains the implicit conditioning on evidence for cointegration. Thus, while the author aims at improving the ability to detect asymmetry, the problem of excessive rejections of the null of symmetry, invalidating the test, might persist in this approach. |

3 | See Caner and Hansen (2001) for a more detailed discussion in the context of MTAR processes with a unit root. |

4 | Our results also hold for univariate MTAR/SETAR models where the test for asymmetry depends on a primary unit root test. Simulation results can be obtained from the author upon request. |

**Figure 2.**

**Left panel**: $\varrho =-0.5$,

**right panel**: $\varrho =-0.1$. Horizontal dotted line: Nominal level of significance (5% level). Vertical dotted line: Critical value of the cointegration test (5% level). Solid line: Empirical size of the test for asymmetry.

**Figure 3.**

**Left panel**: $\varrho =-0.5$,

**right panel**: $\varrho =-0.1$. Horizontal dotted line: Nominal level of significance (5% level). Vertical dotted line: Critical value of the cointegration test (5% level). Solid line: Empirical size of the test for asymmetry.

MTAR | T | SETAR | T | ||||
---|---|---|---|---|---|---|---|

$\mathit{q}$ | 100 | 200 | 400 | $\mathit{q}$ | 100 | 200 | 400 |

0.25 | 0.162 | 0.185 | 0.200 | 0.25 | 0.022 | 0.022 | 0.027 |

0.20 | 0.192 | 0.218 | 0.20 | 0.042 | 0.044 | 0.052 | |

0.15 | 0.227 | 0.257 | 0.277 | 0.15 | 0.073 | 0.081 | 0.090 |

0.10 | 0.260 | 0.303 | 0.327 | 0.10 | 0.121 | 0.135 | 0.148 |

0.05 | 0.312 | 0.361 | 0.401 | 0.05 | 0.196 | 0.223 | 0.240 |

$\mathit{\varrho}=-0.5$ | MTAR | SETAR | ||||
---|---|---|---|---|---|---|

T | 100 | 200 | 400 | 100 | 200 | 400 |

$\alpha $ = 10% | 0.103 | 0.101 | 0.102 | 0.109 | 0.105 | 0.100 |

5% | 0.053 | 0.053 | 0.052 | 0.058 | 0.052 | 0.053 |

1% | 0.010 | 0.011 | 0.010 | 0.012 | 0.009 | 0.012 |

${\mathit{\varrho}}^{+}=-\mathbf{0.75},{\mathit{\varrho}}^{-}=-\mathbf{0.5},\mathit{\tau}=\mathbf{0}$ | ||||||

T | 100 | 200 | 400 | 100 | 200 | 400 |

$\alpha $ = 10% | 0.254 | 0.403 | 0.638 | 0.226 | 0.370 | 0.619 |

5% | 0.162 | 0.288 | 0.513 | 0.143 | 0.254 | 0.502 |

1% | 0.059 | 0.129 | 0.284 | 0.045 | 0.095 | 0.259 |

${\mathit{\varrho}}^{+}=-\mathbf{0.75},{\mathit{\varrho}}^{-}=-\mathbf{0.25},\mathit{\tau}=\mathbf{0}$ | ||||||

T | 100 | 200 | 400 | 100 | 200 | 400 |

$\alpha $ = 10% | 0.553 | 0.821 | 0.977 | 0.456 | 0.752 | 0.962 |

5% | 0.424 | 0.725 | 0.954 | 0.339 | 0.640 | 0.929 |

1% | 0.215 | 0.503 | 0.860 | 0.151 | 0.394 | 0.810 |

$\mathit{\varrho}=-0.5$ | MTAR | SETAR | ||||
---|---|---|---|---|---|---|

T | 100 | 200 | 400 | 100 | 200 | 400 |

$\alpha $ = 10% | 0.082 | 0.091 | 0.090 | 0.109 | 0.101 | 0.101 |

5% | 0.038 | 0.047 | 0.042 | 0.055 | 0.051 | 0.052 |

1% | 0.007 | 0.010 | 0.008 | 0.010 | 0.010 | 0.011 |

${\mathit{\varrho}}^{+}=-\mathbf{0.75},{\mathit{\varrho}}^{-}=-\mathbf{0.5},\mathit{\tau}=\mathbf{0}$ | ||||||

T | 100 | 200 | 400 | 100 | 200 | 400 |

$\alpha $ = 10% | 0.232 | 0.446 | 0.748 | 0.198 | 0.300 | 0.500 |

5% | 0.144 | 0.327 | 0.640 | 0.119 | 0.197 | 0.375 |

1% | 0.042 | 0.145 | 0.410 | 0.034 | 0.067 | 0.183 |

${\mathit{\varrho}}^{+}=-\mathbf{0.75},{\mathit{\varrho}}^{-}=-\mathbf{0.25},\mathit{\tau}=\mathbf{0}$ | ||||||

T | 100 | 200 | 400 | 100 | 200 | 400 |

$\alpha $ = 10% | 0.595 | 0.896 | 0.996 | 0.440 | 0.730 | 0.954 |

5% | 0.466 | 0.833 | 0.992 | 0.322 | 0.607 | 0.914 |

1% | 0.241 | 0.643 | 0.961 | 0.133 | 0.361 | 0.780 |

Panel (a): MTAR ($\mathit{\tau}=0$) | |||||

${\mathit{\varrho}}^{+}$ | ${\mathit{\varrho}}^{-}$ | ${\mathit{F}}_{\mathit{CI}}$ | ${\mathit{F}}_{\mathit{apt}}$ | $\mathit{p}$-Value | |

US | $-0.069$ | $-0.075$ | $8.450$ *** | 0.022 | 0.882 (0.428) |

CAN | $-0.052$ | $-0.105$ | $8.453$ *** | 1.822 | 0.178 (0.162) |

FRA | $-0.191$ | $-0.205$ | $23.880$ *** | 0.059 | 0.808 (0.793) |

GBR | $-0.197$ | $-0.112$ | $18.360$ *** | 2.599 | 0.108 (0.092) |

GER | $-0.120$ | $-0.165$ | $13.730$ *** | 0.691 | 0.406 (0.363) |

ITA | $-0.234$ | $-0.167$ | $23.300$ *** | 1.277 | 0.259 (0.242) |

Panel (b): SETAR ($\mathit{\tau}=\mathbf{0}$) | |||||

${\mathit{\varrho}}^{+}$ | ${\mathit{\varrho}}^{-}$ | ${\mathit{F}}_{\mathit{CI}}$ | ${\mathit{F}}_{\mathit{apt}}$ | $\mathit{p}$-Value | |

US | $-0.081$ | $-0.058$ | $8.949$ *** | 0.455 | 0.500 (0.252) |

CAN | $-0.090$ | $-0.062$ | $7.794$ ** | 0.519 | 0.472 (0.220) |

FRA | $-0.190$ | $-0.208$ | $23.910$ *** | 0.097 | 0.756 (0.612) |

GBR | $-0.170$ | $-0.133$ | $17.220$ *** | 0.486 | 0.486 (0.278) |

GER | $-0.125$ | $-0.159$ | $13.580$ *** | 0.385 | 0.535 (0.285) |

ITA | $-0.195$ | $-0.211$ | $22.650$ *** | 0.070 | 0.792 (0.663) |

Panel (c): MTAR (τ *) | |||||

${\mathit{\varrho}}^{+}$ | ${\mathit{\varrho}}^{-}$ | ${\mathit{F}}_{\mathit{CI}}$ | ${\mathit{F}}_{\mathit{apt}}$ | $\mathit{p}$-Value | |

US | $-0.113$ | $-0.057$ | $9.587$ *** | 1.847 | 0.175 (0.735) |

CAN | $-0.047$ | $-0.144$ | $10.160$ *** | 5.333 | 0.021 (0.172) |

FRA | $-0.318$ | $-0.171$ | $25.370$ *** | 4.361 | 0.037 (0.195) |

GBR | $-0.326$ | $-0.111$ | $22.840$ *** | 11.750 | 0.001 (0.013) |

GER | $-0.106$ | $-0.191$ | $14.180$ *** | 2.460 | 0.118 (0.532) |

ITA | $-0.338$ | $-0.163$ | $25.220$ *** | 6.405 | 0.012 (0.097) |

Panel (d): SETAR (τ *) | |||||

${\mathit{\varrho}}^{+}$ | ${\mathit{\varrho}}^{-}$ | ${\mathit{F}}_{\mathit{CI}}$ | ${\mathit{F}}_{\mathit{apt}}$ | $\mathit{p}$-Value | |

US | $-0.106$ | $-0.037$ | $10.750$ *** | 4.074 | 0.044 (0.092) |

CAN | $-0.111$ | $-0.052$ | $8.596$ ** | 2.311 | 0.129 (0.247) |

FRA | $-0.176$ | $-0.240$ | $23.610$ *** | 1.209 | 0.272 (0.540) |

GBR | $-0.178$ | $-0.123$ | $17.060$ *** | 1.070 | 0.302 (0.588) |

GER | $-0.109$ | $-0.194$ | $14.120$ *** | 2.355 | 0.126 (0.230) |

ITA | $-0.187$ | $-0.228$ | $21.930$ *** | 0.462 | 0.497 (0.862) |

© 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**

Schild, K.-H.; Schweikert, K.
On the Validity of Tests for Asymmetry in Residual-Based Threshold Cointegration Models. *Econometrics* **2019**, *7*, 12.
https://doi.org/10.3390/econometrics7010012

**AMA Style**

Schild K-H, Schweikert K.
On the Validity of Tests for Asymmetry in Residual-Based Threshold Cointegration Models. *Econometrics*. 2019; 7(1):12.
https://doi.org/10.3390/econometrics7010012

**Chicago/Turabian Style**

Schild, Karl-Heinz, and Karsten Schweikert.
2019. "On the Validity of Tests for Asymmetry in Residual-Based Threshold Cointegration Models" *Econometrics* 7, no. 1: 12.
https://doi.org/10.3390/econometrics7010012