# Return and Risk of Pairs Trading Using a Simulation-Based Bayesian Procedure for Predicting Stable Ratios of Stock Prices

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## Abstract

**:**

## 1. Introduction

## 2. Pairs Trading: Implicit and Conditional Statistical Arbitrage

## 3. Bayesian Analysis of the Cointegration Model Under Linear and Orthogonal Normalization

#### 3.1. The Encompassing and Jeffreys’ Framework for Prior Specification and Posterior Simulation

#### 3.2. Bayes Factors

## 4. Empirical Application

## 5. Conclusions and Topics for Further Research

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A: Posterior Distribution and Sampling Algorithm Under Encompassing Prior

#### Sampling Algorithm Based on Diffuse Prior on Π and Nesting

- Draw ${\Sigma}^{i+1}$ from ${p}_{LEC}(\Sigma |Y)$.
- Draw ${\Pi}^{i+1}$ from ${p}_{LEC}(\Pi |\Sigma ,Y)$.
- Compute ${\alpha}^{i+1},{\beta}^{i+1},{\lambda}^{i+1}$ from ${\Pi}^{i+1}$ using the singular value decomposition.
- Accept ${\Sigma}^{i+1}$, ${\alpha}^{i+1}$ and ${\beta}^{i+1}$ with probability $min\left(\frac{w({\alpha}^{i+1},{\beta}^{i+1},{\lambda}^{i+1},{\Sigma}^{i+1})}{w({\alpha}^{i},{\beta}^{i},{\lambda}^{i},{\Sigma}^{i})},1\right)$.

## Appendix B: Jacobian of the Transformation from Π to $(\alpha ,\beta ,\Sigma )$ under Orthogonal Normalization

## Appendix C: Tables for Ten Pairs of Stocks

**Table 4.**Performance evaluation measure $Profitability$ in (22)–(23), which is the ratio of cumulative income to the average daily capital absorption (in %), under a normal distribution for the innovations. The meaning of the ten pairs of stocks is indicated in Table 1. Outperforming normalization in boldface.

$\mathit{k}\mathbf{=}\mathbf{0}$ (Directional Accuracy) | $\mathit{k}\mathbf{=}\mathbf{1}$ (Accuracy) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

CSA | ISA | CSA | |||||||||||||||||

0% | 20% | 30% | 40% | 50% | 60% | 0% | 20% | 30% | 40% | 50% | 60% | ||||||||

linear normalization of β | |||||||||||||||||||

AA-OSG | 19.5 | 25.2 | 15.5 | 23.2 | 25.5 | 40.1 | 22.5 | 4.2 | 2.1 | 3.5 | 2.1 | 7 | 5.2 | ||||||

DUK-IBM | 7.3 | 17 | 16.2 | 22.9 | 16.7 | 74.3 | 6.5 | 0.9 | 1.8 | 1.5 | 2.5 | 2.3 | 7.5 | ||||||

DUK-OSG | −3 | 10.5 | 15.9 | 47.7 | 46.2 | 82.7 | 24.9 | 2.9 | 4.3 | 4.9 | 9.2 | 11.4 | 16.1 | ||||||

NI-NSC | 24.9 | 35.9 | 40.4 | 33.5 | 33.9 | 38.8 | 7.1 | 13.2 | 19.1 | 25 | 25.8 | 33.8 | 37.4 | ||||||

NI-OSG | 11.2 | 19.9 | 22.3 | 34.1 | 47.9 | 70.6 | 33.2 | 3.4 | 4.8 | 5.7 | 7.7 | 10 | 14.7 | ||||||

CNP-OSG | 14.8 | 24.8 | 32 | 29.7 | 23 | 37.3 | 10.3 | 1.6 | 2.1 | 2.7 | 2.8 | 3.5 | 8.8 | ||||||

MO-UPS | −5.8 | −5.9 | 10.1 | 23.3 | 30.5 | 16.6 | 19.8 | 1.4 | 1.5 | 3 | 4.3 | 5.2 | 4.6 | ||||||

NI-R | 39.2 | 32.9 | 22.9 | 14.4 | 20 | 13.4 | 24.1 | 17.8 | 15.2 | 8.3 | 4.7 | 3.8 | 3.3 | ||||||

NI-UNP | 11.2 | 9.1 | 8.3 | 18.6 | 31.6 | 44 | 6.4 | 0.9 | 1.3 | 1.6 | 2.8 | 4.7 | 6.9 | ||||||

NI-UTX | 12.9 | 9.5 | 9.4 | 18.3 | 32.1 | 48.6 | 6.5 | 1.1 | 1.5 | 1.9 | 3 | 4.7 | 7.7 | ||||||

orthogonal normalization of β | |||||||||||||||||||

AA-OSG | 108.7 | 138.6 | 145.2 | 155.2 | 167.4 | 181.2 | 133.3 | 229 | 293.3 | 324.6 | 358.6 | 397.1 | 428.2 | ||||||

DUK-IBM | 23.6 | 36.3 | 32.6 | 60.7 | 82.3 | 364 | 14.9 | 19.7 | 34.7 | 36.4 | 65.9 | 112.9 | 626.1 | ||||||

DUK-OSG | 80.9 | 85.7 | 85.3 | 124.3 | 136.5 | 59.7 | 62.2 | 77.5 | 96.5 | 100.7 | 130.7 | 174.4 | 12.1 | ||||||

NI-NSC | 46.4 | 54.8 | 59.8 | 81.2 | 81.5 | 80.9 | 17.5 | 32.3 | 44.4 | 56.4 | 82.1 | 95.1 | 132.8 | ||||||

NI-OSG | 67.1 | 100.3 | 94.2 | 113.3 | 160.4 | 150.2 | 65.5 | 74 | 96.6 | 108.9 | 130.2 | 164.4 | 103.5 | ||||||

CNP-OSG | 25.5 | 21.4 | 16.9 | 17.4 | 6.4 | 9.2 | 8.8 | 3.4 | 2.7 | 1.8 | 2.8 | 1.1 | 2.1 | ||||||

MO-UPS | 0.2 | 4.4 | 2.8 | 15.6 | 32 | 66.7 | 17.9 | 1 | 1.1 | 1.3 | 2.8 | 4.6 | 7.5 | ||||||

NI-R | 23.7 | 12.9 | 2.5 | −10.2 | −9.4 | 12.4 | 17.2 | 9.7 | 7.7 | 5.5 | 1 | 1.5 | 2.6 | ||||||

NI-UNP | 14.2 | 7.5 | 1.5 | 16.8 | 30.7 | 21.6 | 5.8 | 2 | 1 | 1.1 | 2.7 | 4.7 | 3.8 | ||||||

NI-UTX | 180.7 | 186.1 | 195.3 | 191.3 | 147.1 | 176.1 | 118 | 229.2 | 252.3 | 265.7 | 269.4 | 225.1 | 288.1 |

$\mathit{k}\mathbf{=}\mathbf{0}$ (Directional Accuracy) | $\mathit{k}\mathbf{=}\mathbf{1}$ (Accuracy) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

CSA | ISA | CSA | |||||||||||||||||

0% | 20% | 30% | 40% | 50% | 60% | 0% | 20% | 30% | 40% | 50% | 60% | ||||||||

linear normalization of β | |||||||||||||||||||

AA-OSG | 24.5 | 31.1 | 18.8 | 24.7 | 28.8 | 41.6 | 24.6 | 2.5 | 3.2 | 3 | 3.5 | 4 | 6.4 | ||||||

DUK-IBM | 10.5 | 16.5 | 15.4 | 25.3 | 17.8 | 95.4 | 7 | 1.3 | 2.4 | 2.1 | 3.2 | 3.3 | 14.9 | ||||||

DUK-OSG | 13.8 | 27 | 35.8 | 45.2 | 47.7 | 82.6 | 24.7 | 2.8 | 4.5 | 5.1 | 7.1 | 8 | 11.7 | ||||||

NI-NSC | 41.5 | 53.1 | 73.2 | 71.3 | 67.6 | 79.8 | 17.8 | 34.8 | 48.2 | 70.4 | 87.7 | 93.8 | 106.3 | ||||||

NI-OSG | 29 | 35.7 | 42.8 | 51 | 77 | 86.8 | 34.1 | 4.8 | 6.9 | 8 | 9.9 | 13.7 | 18 | ||||||

CNP-OSG | 17.1 | 24.3 | 34.2 | 17.7 | 26.4 | 61.2 | 10.3 | 1.5 | 2.2 | 3 | 1.7 | 3.1 | 11.6 | ||||||

MO-UPS | 9.9 | 7.9 | 20.8 | 39.8 | 44 | 48.7 | 20.5 | 2.3 | 2.2 | 3.7 | 5.6 | 6.1 | 5.6 | ||||||

NI-R | 32.4 | 28.1 | 24.6 | 4.2 | 11.5 | 14.8 | 23.9 | 17.4 | 15.9 | 16.2 | 1.5 | 2.6 | 3.3 | ||||||

NI-UNP | 4.9 | 8.3 | 7.4 | 18.6 | 31.5 | 50.2 | 6.5 | 0.9 | 1.7 | 1.7 | 2.4 | 4.3 | 8.2 | ||||||

NI-UTX | 19.7 | 17.2 | 16.8 | 23 | 25.2 | 29.8 | 6.6 | 4.5 | 3.9 | 2.9 | 2.7 | 3 | 3.5 | ||||||

orthogonal normalization of β | |||||||||||||||||||

AA-OSG | 118.3 | 150.9 | 170.1 | 182.5 | 192.7 | 213.2 | 141.1 | 245.1 | 314 | 351.8 | 391.9 | 427.4 | 466.3 | ||||||

DUK-IBM | 25.9 | 29.7 | 26.6 | 63.1 | 98.4 | 350.3 | 14.9 | 19.8 | 30.4 | 35 | 65.3 | 97.7 | 608.5 | ||||||

DUK-OSG | 78.4 | 84.2 | 96.7 | 115.3 | 148.5 | 101.9 | 60.5 | 73.7 | 90.5 | 102.5 | 121.3 | 133.5 | 16.4 | ||||||

NI-NSC | 43.5 | 54.6 | 58.2 | 69.9 | 53 | 16.1 | 12.8 | 23.4 | 41 | 41.5 | 54.8 | 25.2 | 7.8 | ||||||

NI-OSG | 55.5 | 77 | 74.3 | 101.5 | 152.1 | 119.6 | 54.8 | 53.6 | 69.9 | 80.8 | 114 | 145.1 | 100.2 | ||||||

CNP-OSG | 29.7 | 26.9 | 18.9 | 20.8 | 15.4 | 6.6 | 9.7 | 5.8 | 7.3 | 2.5 | 3.3 | 2.9 | 2.9 | ||||||

MO-UPS | 11.1 | 4.9 | 2.3 | 15.4 | 30.7 | 68.9 | 18.5 | 1.7 | 1.5 | 1.5 | 2.9 | 4.6 | 7.7 | ||||||

NI-R | 33.4 | 22.1 | 13.8 | 2.6 | -12.4 | 1.7 | 16.9 | 9.3 | 6.7 | 5 | 0.7 | 1.3 | 1.8 | ||||||

NI-UNP | 16 | 15 | 9.2 | 35.4 | 40.4 | 48.8 | 6.5 | 2.9 | 3.4 | 3.4 | 6.9 | 7.8 | 11.3 | ||||||

NI-UTX | 190.9 | 198.8 | 206.6 | 211.3 | 196.5 | 213.4 | 127.5 | 248.5 | 272.8 | 289.7 | 299.3 | 293.8 | 333.8 |

$\mathit{k}\mathbf{=}\mathbf{0}$ (Directional Accuracy) | $\mathit{k}\mathbf{=}\mathbf{1}$ (Accuracy) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

CSA | ISA | CSA | |||||||||||||||||

0% | 20% | 30% | 40% | 50% | 60% | 0% | 20% | 30% | 40% | 50% | 60% | ||||||||

linear normalization of β | |||||||||||||||||||

AA-OSG | 0.42 | 0.43 | 0.42 | 0.46 | 0.41 | 0.39 | 0.48 | 0.49 | 0.48 | 0.43 | 0.46 | 0.42 | 0.39 | ||||||

DUK-IBM | 0.46 | 0.44 | 0.46 | 0.43 | 0.36 | 0 | 0.49 | 0.55 | 0.55 | 0.52 | 0.5 | 0.5 | 0 | ||||||

DUK-OSG | 0.55 | 0.48 | 0.49 | 0.41 | 0.44 | 0.36 | 0.5 | 0.58 | 0.5 | 0.52 | 0.43 | 0.44 | 0.36 | ||||||

NI-NSC | 0.4 | 0.38 | 0.37 | 0.43 | 0.53 | 0.54 | 0.48 | 0.5 | 0.42 | 0.4 | 0.48 | 0.53 | 0.54 | ||||||

NI-OSG | 0.46 | 0.43 | 0.43 | 0.4 | 0.4 | 0.36 | 0.42 | 0.48 | 0.44 | 0.44 | 0.41 | 0.4 | 0.36 | ||||||

CNP-OSG | 0.42 | 0.34 | 0.29 | 0.3 | 0.29 | 0.33 | 0.44 | 0.48 | 0.39 | 0.34 | 0.33 | 0.29 | 0.33 | ||||||

MO-UPS | 0.52 | 0.52 | 0.47 | 0.42 | 0.39 | 0.42 | 0.5 | 0.58 | 0.58 | 0.53 | 0.46 | 0.43 | 0.46 | ||||||

NI-R | 0.38 | 0.39 | 0.42 | 0.45 | 0.45 | 0.45 | 0.49 | 0.42 | 0.44 | 0.47 | 0.49 | 0.47 | 0.47 | ||||||

NI-UNP | 0.41 | 0.45 | 0.43 | 0.45 | 0.42 | 0.33 | 0.44 | 0.46 | 0.49 | 0.43 | 0.45 | 0.42 | 0.33 | ||||||

NI-UTX | 0.42 | 0.47 | 0.44 | 0.48 | 0.46 | 0.33 | 0.45 | 0.5 | 0.49 | 0.44 | 0.48 | 0.46 | 0.33 | ||||||

orthogonal distribution of β | |||||||||||||||||||

AA-OSG | 0.45 | 0.42 | 0.42 | 0.4 | 0.39 | 0.37 | 0.4 | 0.51 | 0.48 | 0.49 | 0.45 | 0.45 | 0.4 | ||||||

DUK-IBM | 0.46 | 0.42 | 0.46 | 0.4 | 0.36 | 0 | 0.49 | 0.54 | 0.48 | 0.52 | 0.43 | 0.36 | 0 | ||||||

DUK-OSG | 0.4 | 0.44 | 0.44 | 0.38 | 0.42 | 0.36 | 0.4 | 0.49 | 0.52 | 0.51 | 0.45 | 0.48 | 0.36 | ||||||

NI-NSC | 0.36 | 0.36 | 0.35 | 0.35 | 0.39 | 0.46 | 0.44 | 0.44 | 0.42 | 0.39 | 0.39 | 0.43 | 0.54 | ||||||

NI-OSG | 0.39 | 0.32 | 0.36 | 0.36 | 0.31 | 0.27 | 0.37 | 0.44 | 0.39 | 0.41 | 0.39 | 0.34 | 0.23 | ||||||

CNP-OSG | 0.4 | 0.39 | 0.4 | 0.42 | 0.42 | 0.4 | 0.45 | 0.44 | 0.41 | 0.42 | 0.45 | 0.42 | 0.4 | ||||||

MO-UPS | 0.52 | 0.52 | 0.54 | 0.49 | 0.44 | 0.31 | 0.53 | 0.58 | 0.58 | 0.6 | 0.56 | 0.5 | 0.34 | ||||||

NI-R | 0.45 | 0.46 | 0.5 | 0.54 | 0.57 | 0.51 | 0.5 | 0.49 | 0.52 | 0.56 | 0.61 | 0.63 | 0.56 | ||||||

NI-UNP | 0.44 | 0.46 | 0.54 | 0.5 | 0.46 | 0.47 | 0.46 | 0.52 | 0.52 | 0.59 | 0.57 | 0.54 | 0.53 | ||||||

NI-UTX | 0.22 | 0.25 | 0.22 | 0.23 | 0.26 | 0.29 | 0.41 | 0.35 | 0.34 | 0.32 | 0.32 | 0.35 | 0.38 |

$\mathit{k}\mathbf{=}\mathbf{0}$ (Directional Accuracy) | $\mathit{k}\mathbf{=}\mathbf{1}$ (Accuracy) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

CSA | ISA | CSA | |||||||||||||||||

0% | 20% | 30% | 40% | 50% | 60% | 0% | 20% | 30% | 40% | 50% | 60% | ||||||||

linear normalization of β | |||||||||||||||||||

AA-OSG | 0.45 | 0.46 | 0.49 | 0.46 | 0.42 | 0.37 | 0.48 | 0.48 | 0.48 | 0.49 | 0.46 | 0.42 | 0.37 | ||||||

DUK-IBM | 0.45 | 0.48 | 0.49 | 0.46 | 0.43 | 0 | 0.5 | 0.54 | 0.55 | 0.53 | 0.5 | 0.43 | 0 | ||||||

DUK-OSG | 0.51 | 0.46 | 0.46 | 0.38 | 0.44 | 0.36 | 0.49 | 0.54 | 0.47 | 0.48 | 0.41 | 0.44 | 0.36 | ||||||

NI-NSC | 0.37 | 0.32 | 0.27 | 0.39 | 0.41 | 0.4 | 0.48 | 0.45 | 0.38 | 0.36 | 0.43 | 0.47 | 0.47 | ||||||

NI-OSG | 0.41 | 0.39 | 0.39 | 0.38 | 0.32 | 0.33 | 0.44 | 0.45 | 0.4 | 0.4 | 0.39 | 0.34 | 0.36 | ||||||

CNP-OSG | 0.45 | 0.38 | 0.33 | 0.34 | 0.29 | 0.25 | 0.46 | 0.5 | 0.42 | 0.39 | 0.42 | 0.33 | 0.25 | ||||||

MO-UPS | 0.49 | 0.49 | 0.47 | 0.41 | 0.42 | 0.42 | 0.49 | 0.53 | 0.55 | 0.53 | 0.48 | 0.51 | 0.46 | ||||||

NI-R | 0.42 | 0.41 | 0.44 | 0.49 | 0.5 | 0.47 | 0.48 | 0.45 | 0.44 | 0.48 | 0.53 | 0.52 | 0.49 | ||||||

NI-UNP | 0.45 | 0.45 | 0.4 | 0.4 | 0.38 | 0.25 | 0.45 | 0.54 | 0.51 | 0.46 | 0.47 | 0.46 | 0.25 | ||||||

NI-UTX | 0.43 | 0.44 | 0.45 | 0.43 | 0.41 | 0.4 | 0.5 | 0.49 | 0.48 | 0.48 | 0.47 | 0.44 | 0.44 | ||||||

orthogonal normalization of β | |||||||||||||||||||

AA-OSG | 0.46 | 0.41 | 0.39 | 0.37 | 0.35 | 0.34 | 0.42 | 0.49 | 0.43 | 0.42 | 0.38 | 0.37 | 0.36 | ||||||

DUK-IBM | 0.42 | 0.43 | 0.44 | 0.34 | 0.23 | 0 | 0.45 | 0.51 | 0.52 | 0.52 | 0.38 | 0.23 | 0 | ||||||

DUK-OSG | 0.39 | 0.41 | 0.4 | 0.37 | 0.27 | 0.18 | 0.4 | 0.51 | 0.52 | 0.51 | 0.46 | 0.37 | 0.27 | ||||||

NI-NSC | 0.29 | 0.32 | 0.32 | 0.35 | 0.4 | 0.5 | 0.43 | 0.39 | 0.37 | 0.32 | 0.31 | 0.35 | 0.42 | ||||||

NI-OSG | 0.44 | 0.4 | 0.38 | 0.39 | 0.32 | 0.38 | 0.42 | 0.5 | 0.45 | 0.44 | 0.43 | 0.35 | 0.43 | ||||||

CNP-OSG | 0.37 | 0.38 | 0.35 | 0.38 | 0.36 | 0.4 | 0.44 | 0.43 | 0.42 | 0.38 | 0.38 | 0.36 | 0.4 | ||||||

MO-UPS | 0.52 | 0.53 | 0.54 | 0.49 | 0.43 | 0.3 | 0.5 | 0.55 | 0.55 | 0.56 | 0.51 | 0.43 | 0.3 | ||||||

NI-R | 0.42 | 0.45 | 0.49 | 0.53 | 0.57 | 0.55 | 0.5 | 0.5 | 0.5 | 0.51 | 0.56 | 0.59 | 0.55 | ||||||

NI-UNP | 0.43 | 0.47 | 0.48 | 0.4 | 0.44 | 0.4 | 0.47 | 0.46 | 0.47 | 0.48 | 0.4 | 0.44 | 0.4 | ||||||

NI-UTX | 0.24 | 0.25 | 0.24 | 0.25 | 0.25 | 0.24 | 0.49 | 0.38 | 0.4 | 0.37 | 0.37 | 0.38 | 0.36 |

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**Table 1.**Ten pairs of stocks with highest Bayes factor of model with Π having rank 1 (cointegration) versus model with Π having rank 0 (two random walks) under both the linear normalization and the orthogonal normalization (among stocks in the Dow Jones Composite Average index, using daily closing prices recorded over the period of 1 January 2009 until 30 June 2009).

AA | - | OSG | : | ALCOA Inc. | - | Overseas Shipholding Group, Inc. |

CNP | - | OSG | : | CenterPoint Energy, Inc. | - | Overseas Shipholding Group, Inc. |

DUK | - | IBM | : | Duke Energy Corp. | - | International Business Machines Corp. |

DUK | - | OSG | : | Duke Energy Corp. | - | Overseas Shipholding Group, Inc. |

MO | - | UPS | : | Altria Group, Inc. | - | United Parcel Service, Inc. |

NI | - | NSC | : | NiSource, Inc. | - | Norfolk Southern Corp. |

NI | - | OSG | : | NiSource, Inc. | - | Overseas Shipholding Group, Inc. |

NI | - | R | : | NiSource, Inc. | - | Ryder System, Inc. |

NI | - | UNP | : | NiSource, Inc. | - | Union Pacific Corp. |

NI | - | UTX | : | NiSource, Inc. | - | United Technologies Corp. |

**Table 2.**Performance evaluation measure $Profitability$ in (22)–(23) (in %). Average over 10 pairs of assets.

CSA ($\mathit{k}\mathbf{=}\mathbf{0}$, Directional Accuracy) | ||||||||||

$\mathbf{\xi}$ | ||||||||||

0% | 20% | 30% | 40% | 50% | 60% | |||||

linear normalization | ||||||||||

normal | 12.52 | 17.08 | 19.72 | 26.94 | 31.32 | 47.37 | ||||

Student’s t | 19.87 | 24.23 | 30.11 | 32.9 | 38.74 | 61.03 | ||||

orthogonal normalization | ||||||||||

normal | 57.10 | 64.8 | 63.61 | 76.56 | 83.49 | 112.2 | ||||

Student’s t | 60.27 | 66.41 | 67.67 | 81.78 | 91.53 | 114.05 | ||||

ISA | CSA ($k=1$, accuracy) | |||||||||

ξ | ||||||||||

0% | 20% | 30% | 40% | 50% | 60% | |||||

linear normalization | ||||||||||

normal | 15.42 | 4.80 | 5.73 | 6.07 | 6.98 | 8.82 | 11.89 | |||

Student’s t | 16.82 | 7.81 | 9.77 | 12.57 | 13.53 | 15.32 | 20.34 | |||

orthogonal normalization | ||||||||||

normal | 46.11 | 67.78 | 83.03 | 90.24 | 104.62 | 118.09 | 160.68 | |||

Student’s t | 46.32 | 68.38 | 83.75 | 91.37 | 106.04 | 113.93 | 155.67 |

CSA ($\mathit{k}\mathbf{=}\mathbf{0}$, Directional Accuracy) | ||||||||||

$\mathbf{\xi}$ | ||||||||||

0% | 20% | 30% | 40% | 50% | 60% | |||||

linear normalization | ||||||||||

normal | 0.45 | 0.43 | 0.42 | 0.42 | 0.42 | 0.35 | ||||

Student’s t | 0.44 | 0.42 | 0.41 | 0.41 | 0.40 | 0.32 | ||||

orthogonal normalization | ||||||||||

normal | 0.40 | 0.40 | 0.42 | 0.41 | 0.40 | 0.34 | ||||

Student’s t | 0.39 | 0.40 | 0.40 | 0.39 | 0.36 | 0.33 | ||||

ISA | CSA ($k=1$, accuracy) | |||||||||

ξ | ||||||||||

0% | 20% | 30% | 40% | 50% | 60% | |||||

linear normalization | ||||||||||

normal | 0.47 | 0.51 | 0.48 | 0.45 | 0.45 | 0.44 | 0.35 | |||

Student’s t | 0.48 | 0.50 | 0.47 | 0.46 | 0.46 | 0.44 | 0.34 | |||

orthogonal normalization | ||||||||||

normal | 0.45 | 0.48 | 0.46 | 0.48 | 0.46 | 0.45 | 0.37 | |||

Student’s t | 0.46 | 0.47 | 0.47 | 0.45 | 0.42 | 0.39 | 0.35 |

^{1.}We also considered models with a constant term inside the cointegration relationship and/or drift terms in the model equation. The inclusion of such terms did not change the conclusions of our paper.^{2.}In general we can add restrictions to the normalization restriction ${\beta}^{\prime}\beta ={I}_{r}$ in order to uniquely identify the $r\times r$ matrix ${\beta}_{1}$ given the $(n-r)\times r$ matrix ${\beta}_{2}$, where $\beta ={\left({\beta}_{1}^{\prime}\phantom{\rule{0.277778em}{0ex}}{\beta}_{2}^{\prime}\right)}^{\prime}$. For example, if we assume that $n=2$, $r=1$, then we can add the restriction ${\beta}_{1}\ge 0$, so that only ${\beta}_{1}=\sqrt{1-{\beta}_{2}^{2}}$ satisfies ${\beta}_{1}^{2}+{\beta}_{2}^{2}=1$. Note that without that restriction ${\beta}_{1}\ge 0$ we could have ${\beta}_{1}=\sqrt{1-{\beta}_{2}^{2}}$ or ${\beta}_{1}=-\sqrt{1-{\beta}_{2}^{2}}$, so that in that case ${\beta}_{1}$ would not be uniquely identified by ${\beta}_{2}$.

© 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**

Ardia, D.; Gatarek, L.T.; Hoogerheide, L.; Van Dijk, H.K. Return and Risk of Pairs Trading Using a Simulation-Based Bayesian Procedure for Predicting Stable Ratios of Stock Prices. *Econometrics* **2016**, *4*, 14.
https://doi.org/10.3390/econometrics4010014

**AMA Style**

Ardia D, Gatarek LT, Hoogerheide L, Van Dijk HK. Return and Risk of Pairs Trading Using a Simulation-Based Bayesian Procedure for Predicting Stable Ratios of Stock Prices. *Econometrics*. 2016; 4(1):14.
https://doi.org/10.3390/econometrics4010014

**Chicago/Turabian Style**

Ardia, David, Lukasz T. Gatarek, Lennart Hoogerheide, and Herman K. Van Dijk. 2016. "Return and Risk of Pairs Trading Using a Simulation-Based Bayesian Procedure for Predicting Stable Ratios of Stock Prices" *Econometrics* 4, no. 1: 14.
https://doi.org/10.3390/econometrics4010014