# Detecting Common Bubbles in Multivariate Mixed Causal–Noncausal Models

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## Abstract

**:**

## 1. Introduction

## 2. Multivariate Mixed Causal–Noncausal Models

#### 2.1. Common Bubbles in VMAR(r,s)

**Definition**

**1.**

**Proposition**

**1.**

#### 2.2. Testing for Common Bubbles

## 3. Monte Carlo Analysis

## 4. Common Bubbles in Commodity Indices?

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Notes

1 | This is the restricted linear form that is used in the ML estimation. Gourieroux and Jasiak (2017) have proposed an alternative approach based on roots inside and outside the unit circle of an autoregressive polynomial. |

2 | The “axLik” package in R offers a routine for maximizing a given likelihood function with various optimization algorithms. We used the Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm. |

3 | Optimization algorithms to maximize the Student’s t multivariate likelihood function are known to be sensitive to starting values and might easily reach local maxima. Since our focus is not on accurate estimation of the models but instead on the detection of commonalities, in order to speed up convergence, we follow previous contributions by employing either the true coefficient matrices when the estimated model correctly imposes k CBs; otherwise, we use an approximation of them with a rank different from $(n-k)$. |

4 | Results for other tests, such as 1 vs. 2 when the true rank is 2 for instance, are available upon request. |

5 | Recall from footnote 3 that we employ as starting values an approximation of the true coefficient matrices when the estimated model has a wrong number of CBs. This entails that when the true rank is 3, estimating the restricted models with rank 1 or 2 might encounter convergence issues. This could imply an overestimation of the frequencies displayed in the 2 vs. 3 and 1 vs. 3 when the true rank is 3. |

6 | Note that Hannan-Quin information criterion $HQC=2Kln\left(ln\left(T\right)\right)-2\phantom{\rule{0.166667em}{0ex}}\mathrm{ln}\left(\widehat{L}\right)$ performs exactly in between BIC and AIC both under the null and under the alternative. We thus omitted this to save space, but results are available upon request. |

7 | Including agricultural raw materials, such as includes timber, cotton, wool, rubber, and leather. |

8 | Includes crude oil, natural gas, coal and propane. |

9 | Data are retrieved from the IMF database. They are price indices with base year 2016. |

10 | We used various starting values to account for the bimodality of the coefficients (see Bec et al. 2020, for more details). |

11 | We fixed the starting values for the correlation matrix $\Sigma $ and the degrees of freedom $\lambda $ and performed 100 MLEs based on random lead and lag coefficient matrices fulfilling stationary conditions. |

12 | We also used 100 combinations of starting values to make sure we obtained the best-fitting models. |

## References

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$\mathsf{\Phi}=\left[\begin{array}{cc}0.5& 0.1\\ 0.2& 0.3\end{array}\right]$ | $\mathsf{\Sigma}=\left[\begin{array}{cc}4& 0.5\\ 0.5& 1\end{array}\right]$ |

$T=\left\{500,\phantom{\rule{0.277778em}{0ex}}1\phantom{\rule{0.166667em}{0ex}}000\right\}$ | |

$\lambda =\left\{1.5,\phantom{\rule{0.277778em}{0ex}}3\right\}$ | |

$\mathsf{\Psi}=\left\{\begin{array}{cc}\left[\begin{array}{cc}0.3& 0.25\\ 0.6& 0.5\end{array}\right]=\left[\begin{array}{c}1\\ 2\end{array}\right]\left[\begin{array}{cc}0.3& 0.25\end{array}\right]\hfill & ({H}_{0}:\phantom{\rule{0.166667em}{0ex}}\mathrm{CB})\hfill \\ \\ \left[\begin{array}{cc}0.1& 0.4\\ 0.6& 0.5\end{array}\right]\hfill & ({H}_{1}:\phantom{\rule{0.166667em}{0ex}}\mathrm{no}\phantom{\rule{4.pt}{0ex}}\mathrm{CB})\hfill \end{array}\right.$ |

$\lambda =3$ | |||||||
---|---|---|---|---|---|---|---|

T = 500 | T = 1000 | ||||||

DGP | LR test | BIC | AIC | LR test | BIC | AIC | |

With CB (rank 1) | 0.946 | 0.989 | 0.838 | 0.944 | 0.993 | 0.834 | |

Without CB (rank 2) | 0.999 | 0.994 | 1.000 | 1.000 | 1.000 | 1.000 | |

$\lambda =1.5$ | |||||||

T = 500 | T = 1000 | ||||||

DGP | LR test | BIC | AIC | LR test | BIC | AIC | |

With CB (rank 1) | 0.913 | 0.968 | 0.779 | 0.914 | 0.977 | 0.783 | |

Without CB (rank 2) | 0.999 | 0.999 | 0.999 | 1.000 | 1.000 | 1.000 |

$\mathsf{\Phi}=\left[\begin{array}{ccc}0.5& 0.1& 0.2\\ 0.2& 0.3& 0.1\\ 0.1& 0.4& 0.6\end{array}\right]$ | $\Sigma =\left[\begin{array}{ccc}2& 0.5& 0.5\\ 0.5& 1& 0.5\\ 0.5& 0.5& 4\end{array}\right]$ |

$T=\left\{500,\phantom{\rule{0.277778em}{0ex}}1\phantom{\rule{0.166667em}{0ex}}000\right\}$ | |

$\lambda =\left\{1.5,\phantom{\rule{0.277778em}{0ex}}3\right\}$ | |

$\mathsf{\Psi}=\left\{\begin{array}{cc}\left[\begin{array}{ccc}0.3& 0.1& 0.1\\ 0.2& 0.3& 0.4\\ 0.7& 0.35& 0.4\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 0& 1\\ 2& 0.5\end{array}\right]\left[\begin{array}{ccc}0.3& 0.1& 0.1\\ 0.2& 0.3& 0.4\end{array}\right]\hfill & ({H}_{0}:\phantom{\rule{0.166667em}{0ex}}1\phantom{\rule{4.pt}{0ex}}\mathrm{CB}\phantom{\rule{4.pt}{0ex}}\mathrm{feature})\hfill \\ \\ \left[\begin{array}{ccc}0.15& 0.25& 0.4\\ 0.3& 0.5& 0.8\\ 0.075& 0.125& 0.2\end{array}\right]=\left[\begin{array}{c}1\\ 2\\ 0.5\end{array}\right]\left[\begin{array}{ccc}0.15& 0.25& 0.4\end{array}\right]\hfill & ({H}_{0}:\phantom{\rule{0.166667em}{0ex}}2\phantom{\rule{4.pt}{0ex}}\mathrm{CB}\phantom{\rule{4.pt}{0ex}}\mathrm{features})\hfill \\ \\ \left[\begin{array}{ccc}0.3& 0.2& 0.1\\ 0.2& 0.5& 0.4\\ 0.7& 0.125& 0.2\end{array}\right]\hfill & ({H}_{1}:\phantom{\rule{0.166667em}{0ex}}\mathrm{no}\phantom{\rule{4.pt}{0ex}}\mathrm{CB}\phantom{\rule{4.pt}{0ex}}\mathrm{feature})\hfill \end{array}\right.$ |

$\lambda =3$ | |||||||||
---|---|---|---|---|---|---|---|---|---|

T = 500 | T = 1000 | ||||||||

$rank\left(\mathsf{\Psi}\right)$ | Rank test | LR | BIC | AIC | LR | BIC | AIC | ||

2 | 2 vs 3 | 0.944 | 0.984 | 0.817 | 0.951 | 0.992 | 0.843 | ||

1 | 1 vs 3 | 0.919 | 1.000 | 0.871 | 0.933 | 1.000 | 0.883 | ||

3 | 2 vs 3 | 0.695 | 0.481 | 0.855 | 0.932 | 0.802 | 0.970 | ||

1 vs 3 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | |||

$\lambda =1.5$ | |||||||||

T = 500 | T = 1000 | ||||||||

$rank\left(\mathsf{\Psi}\right)$ | Rank test | LR | BIC | AIC | LR | BIC | AIC | ||

2 | 2 vs. 3 | 0.915 | 0.972 | 0.775 | 0.907 | 0.978 | 0.776 | ||

1 | 1 vs 3 | 0.857 | 0.998 | 0.774 | 0.860 | 0.999 | 0.783 | ||

3 | 2 vs. 3 | 0.997 | 0.994 | 0.999 | 1.000 | 1.000 | 1.000 | ||

1 vs. 3 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

Variable | Estimated Coefficients | ||||
---|---|---|---|---|---|

Multiplicative | Linear | ||||

$\mathit{\varphi}$ | $\mathit{\psi}$ | $\mathit{\lambda}$ | ${\mathit{b}}_{1}$ | ${\mathit{b}}_{2}$ | |

Food & Beverage | 0.38 | 0.85 | 3.70 | 0.29 | 0.64 |

log(Food & Beverage) | 0.34 | 0.86 | 5.47 | 0.26 | 0.67 |

Industrial inputs | 0.43 | 0.87 | 1.66 | 0.31 | 0.63 |

log(Industrial inputs) | 0.42 | 0.89 | 4.62 | 0.31 | 0.65 |

Fuel (energy) | 0.87 | 0.44 | 2.20 | 0.63 | 0.32 |

log(Fuel) | 0.83 | 0.48 | 4.95 | 0.59 | 0.34 |

${B}_{1}$ | ${B}_{-1}$ | $\Omega $ | $\lambda $ |
---|---|---|---|

Food and Indus | |||

$\left[\begin{array}{cc}0.28& 0.01\\ 0.26& 0.27\end{array}\right]$ | $\left[\begin{array}{cc}0.65& -0.02\\ -0.11& 0.65\end{array}\right]$ | $\left[\begin{array}{cc}1.32& 0.16\\ 0.16& 3.35\end{array}\right]$ | 2.49 |

Food and Fuel | |||

$\left[\begin{array}{cc}0.35& 0.05\\ 0.47& 0.52\end{array}\right]$ | $\left[\begin{array}{cc}0.55& -0.04\\ -0.40& 0.40\end{array}\right]$ | $\left[\begin{array}{cc}1.42& 0.87\\ 0.87& 12.90\end{array}\right]$ | 3.01 |

Indus and Fuel | |||

$\left[\begin{array}{cc}0.29& 0.01\\ -0.11& 0.47\end{array}\right]$ | $\left[\begin{array}{cc}0.63& 0.03\\ 0.09& 0.48\end{array}\right]$ | $\left[\begin{array}{cc}2.22& 1.50\\ 1.50& 7.17\end{array}\right]$ | 1.67 |

Food, Indus, and Fuel | |||

$\left[\begin{array}{ccc}0.27& 0.01& 0.03\\ 0.27& 0.25& 0.02\\ 0.30& -0.10& 0.56\end{array}\right]$ | $\left[\begin{array}{ccc}0.64& -0.02& -0.02\\ -0.16& 0.66& -0.01\\ -0.27& 0.12& 0.37\end{array}\right]$ | $\left[\begin{array}{ccc}1.34& 0.12& 0.55\\ 0.12& 3.29& 2.20\\ 0.55& 2.20& 10.02\end{array}\right]$ | 2.28 |

${B}_{1}$ | ${B}_{-1}$ | ${10}^{3}\Omega $ | $\lambda $ |

Food and Indus | |||

$\left[\begin{array}{cc}0.25& 0.01\\ 0.22& 0.24\end{array}\right]$ | $\left[\begin{array}{cc}0.69& -0.03\\ -0.12& 0.70\end{array}\right]$ | $\left[\begin{array}{cc}0.27& 0.03\\ 0.03& 0.46\end{array}\right]$ | 6.30 |

Food and Fuel | |||

$\left[\begin{array}{cc}0.25& 0.02\\ 0.16& 0.38\end{array}\right]$ | $\left[\begin{array}{cc}0.67& -0.02\\ -0.14& 0.55\end{array}\right]$ | $\left[\begin{array}{cc}0.25& 0.06\\ 0.06& 1.21\end{array}\right]$ | 5.23 |

Indus and Fuel | |||

$\left[\begin{array}{cc}0.26& 0.04\\ -0.09& 0.56\end{array}\right]$ | $\left[\begin{array}{cc}0.67& -0.01\\ 0.09& 0.37\end{array}\right]$ | $\left[\begin{array}{cc}0.42& 0.24\\ 0.24& 1.19\end{array}\right]$ | 4.77 |

Food, Indus, and Fuel | |||

$\left[\begin{array}{ccc}0.88& -0.17& -0.02\\ -0.04& 0.27& 0.07\\ -0.04& 0.06& 0.58\end{array}\right]$ | $\left[\begin{array}{ccc}0.21& 0.15& 0.00\\ 0.13& 0.76& -0.08\\ 0.02& 0.05& 0.33\end{array}\right]$ | $\left[\begin{array}{ccc}0.32& 0.02& 0.07\\ 0.02& 0.51& 0.26\\ 0.07& 0.26& 1.35\end{array}\right]$ | 6.15 |

Levels | ||||||

Food | Indus | Fuel | Rank test | LRT | BIC | AIC |

∎ | ∎ | 1 vs. 2 | 25.93 | 20.04 | 23.93 | |

∎ | ∎ | 1 vs. 2 | 59.96 | 54.07 | 57.96 | |

∎ | ∎ | 1 vs. 2 | 70.49 | 64.59 | 68.49 | |

∎ | ∎ | ∎ | 2 vs. 3 | 16.26 | 10.37 | 14.26 |

1 vs. 3 | 88.12 | 64.55 | 80.12 | |||

Logs | ||||||

Food | Indus | Fuel | Rank test | LRT | BIC | AIC |

∎ | ∎ | 1 vs. 2 | 16.04 | 10.15 | 14.04 | |

∎ | ∎ | 1 vs. 2 | 34.36 | 28.47 | 32.36 | |

∎ | ∎ | 1 vs. 2 | 46.05 | 40.16 | 44.05 | |

∎ | ∎ | ∎ | 2 vs. 3 | 15.81 | 9.92 | 13.81 |

1 vs. 3 | 75.01 | 51.44 | 67.01 |

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## Share and Cite

**MDPI and ACS Style**

Cubadda, G.; Hecq, A.; Voisin, E.
Detecting Common Bubbles in Multivariate Mixed Causal–Noncausal Models. *Econometrics* **2023**, *11*, 9.
https://doi.org/10.3390/econometrics11010009

**AMA Style**

Cubadda G, Hecq A, Voisin E.
Detecting Common Bubbles in Multivariate Mixed Causal–Noncausal Models. *Econometrics*. 2023; 11(1):9.
https://doi.org/10.3390/econometrics11010009

**Chicago/Turabian Style**

Cubadda, Gianluca, Alain Hecq, and Elisa Voisin.
2023. "Detecting Common Bubbles in Multivariate Mixed Causal–Noncausal Models" *Econometrics* 11, no. 1: 9.
https://doi.org/10.3390/econometrics11010009