# Granger Causality on forward and Reversed Time Series

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. The Standard Granger Causality Test (GC)

#### 2.2. Predictive Error Test for Granger Causality (PEGC)

#### 2.3. Modification of Time-reversed Granger Causality Test (mTRGC)

## 3. Data and Experimental Setup

**Causal independence ($X\perp Y$)**$$\begin{array}{cc}\hfill x\left(t\right)=& 0.5x(t-1)+{\u03f5}_{x}\left(t\right)\hfill \\ \hfill y\left(t\right)=& ay(t-1)+{\u03f5}_{y}\left(t\right),\hfill \end{array}$$**Unidirectional causal connection ($X\to Y$)**$$\begin{array}{cc}\hfill x\left(t\right)=& 0.5x(t-1)+{\u03f5}_{x}\left(t\right)\hfill \\ \hfill y\left(t\right)=& 0.5y(t-1)+{c}_{1}x(t-1)+{\u03f5}_{y}\left(t\right),\hfill \end{array}$$**Bidirectional causal connection ($X\leftrightarrow Y$)**$$\begin{array}{cc}\hfill x\left(t\right)=& 0.5x(t-1)+0.5y(t-1)+{\u03f5}_{x}\left(t\right)\hfill \\ \hfill y\left(t\right)=& 0.5y(t-1)+{c}_{2}x(t-1)+{\u03f5}_{y}\left(t\right),\hfill \end{array}$$

- Condition A (normal distribution): The predictive errors ${\u03f5}_{x}$, ${\u03f5}_{y}$ were independent normally distributed random variables with zero mean and with the variance ${\sigma}_{x}^{2}=0.5$ and ${\sigma}_{y}^{2}={\sigma}_{x}^{2}\ast \{0.25,0.5,0.75,1,1.25,1.5,1.75\}$ (i.e., ${\sigma}_{y}^{2}$ was a multiple of ${\sigma}_{x}^{2}$), respectively.
- Condition B (uniform distribution): The predictive errors ${\u03f5}_{x}$, ${\u03f5}_{y}$ were independent uniformly distributed random variables in intervals $[{a}_{x},{b}_{x}]$, $[{a}_{y},{b}_{y}]$, respectively. The distribution parameters for ${\u03f5}_{x}$ were: ${a}_{x}=-\sqrt{3}/2$, and ${b}_{x}=-{a}_{x}$. The distribution parameters for ${\u03f5}_{y}$ were: ${a}_{y}={a}_{x}\ast \{0.25,0.5,0.75,1,1.25,1.5,1.75\}$, and ${b}_{y}=-{a}_{y}$.
- Condition C (triangular distribution): The predictive errors ${\u03f5}_{x}$, ${\u03f5}_{y}$ were independent triangular-distributed random variables. The triangular distribution parameters for ${\u03f5}_{x}$ were: lower limit ${a}_{x}=-2$, upper limit ${b}_{x}=-{a}_{x}/2$ and mode ${c}_{x}={b}_{x}$. The triangular distribution parameters for ${\u03f5}_{y}$ were: lower limit ${a}_{y}={a}_{x}\ast \{0.25,0.5,0.75,1,1.25,1.5,1.75\}$, upper limit ${b}_{y}=-{a}_{y}/2$ and mode ${c}_{y}={b}_{y}$.
- Condition D (a mixture of normal distributions): Both predictive errors ${\u03f5}_{x}$, ${\u03f5}_{y}$ were generated from a mixture of two normal distributions. The error term ${\u03f5}_{x}$ was generated from a distribution where the probability of drawing from the normal distribution $N(1,5({\sigma}_{x}^{2}-1/4)/9)$ was 1/5 and from the normal distribution $N(-1/4,10({\sigma}_{x}^{2}-1/4)/9)$ was 4/5, where ${\sigma}_{x}^{2}=0.5$. The error term ${\u03f5}_{y}$ was generated from a distribution where the probability of drawing from the normal distribution $N(1,5({\sigma}_{y}^{2}-1/4)/9)$ was 1/5 and from the normal distribution $N(-1/4,10({\sigma}_{y}^{2}-1/4)/9)$ was 4/5, where ${\sigma}_{y}^{2}={\sigma}_{x}^{2}\ast \{0.75,1,1.25,1.5,1.75\}$.
- Condition E (moving average): The predictive errors ${\u03f5}_{x}$, ${\u03f5}_{y}$ were defined as ${\u03f5}_{x}\left(t\right)=0.5{\xi}_{x}(t-1)+{\xi}_{x}\left(t\right)$, ${\u03f5}_{y}\left(t\right)=0.5{\xi}_{y}(t-1)+{\xi}_{y}\left(t\right)$, respectively. The variables ${\xi}_{x}$, ${\xi}_{y}$ were independent normally distributed with zero mean and with the variance ${\sigma}_{x}^{2}=0.4$ and ${\sigma}_{y}^{2}={\sigma}_{x}^{2}\ast \{0.25,0.5,0.75,1,1.25,1.5,1.75\}$, respectively.
- Condition F (quadratic moving average): The predictive errors ${\u03f5}_{x}$, ${\u03f5}_{y}$ were defined as ${\u03f5}_{x}\left(t\right)=0.5{\xi}_{x}^{2}(t-1)-0.5{\xi}_{x}^{2}\left(t\right)$, ${\u03f5}_{y}\left(t\right)=0.5{\xi}_{y}^{2}(t-1)-0.5{\xi}_{y}^{2}\left(t\right)$, respectively. The variables ${\xi}_{x}$, ${\xi}_{y}$ were independent normally distributed with zero mean and with the variance ${\sigma}_{x}^{2}=\sqrt{0.5}$ and ${\sigma}_{y}^{2}={\sigma}_{x}^{2}\ast \{0.25,0.5,0.75,1,1.25,1.5,1.75\}$, respectively.
- Condition G (correlation): The predictive errors ${\u03f5}_{x}$, ${\u03f5}_{y}$ were correlated, with $cov({\u03f5}_{x},{\u03f5}_{y})$ = 0.1. Like in the condition A, the error terms were normally distributed variables with zero mean and with the variance ${\sigma}_{x}^{2}=0.5$ and ${\sigma}_{y}^{2}={\sigma}_{x}^{2}\ast \{0.25,0.5,0.75,1,1.25,1.5,1.75\}$, respectively.

## 4. Results

#### 4.1. GC Results

#### 4.2. PEGC Results

#### 4.3. mTRGC Results

#### 4.4. mTRGC* Results

#### 4.5. GC+mTRGC*

## 5. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AR | Autoregressive model |

VAR | Vector autoregressive model |

GC | Standard Granger causality test |

PEGC | Predictive error test of Granger causality |

TRGC | Time-reversed Granger causality test |

mTRGC | Modification of time-reversed Granger causality test |

mTRGC* | Modification of time-reversed Granger causality (non-statistical) test |

FPR | False-positive rate |

FNR | False-negative rate |

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**Figure 1.**Rates of false detections obtained by GC on time series of length $T=300$ generated with normally distributed errors (condition A) for unidirectionally causally connected variables $(X\to Y)$: (

**a**) false positive rates observed on original time series, (

**b**) false negative rates observed on original time series, (

**c**) false positive rates observed on time-reversed series, (

**d**) false negative rates observed on time-reversed series.

**Figure 2.**Correlation for time series of length $T=300$ generated with independent normally distributed errors (condition A) for unidirectionally connected variables $(X\to Y)$: (

**a**) correlation of variables, (

**b**) correlation of predictive error elements fitted by VAR on original time series, (

**c**) correlation (multiplied by −1) of predictive error elements fitted by VAR on time-reversed series.

**Table 1.**False positive rates (in %) for causally independent $\left(X\perp Y\right)$ variables. The results for eight discussed testing procedures (

**inv**—results in the time-reversed series) are presented, with the worst (more than $3\%$) FPR highlighted in bold.

Condition for ${\mathit{\u03f5}}_{\mathit{x}}$, ${\mathit{\u03f5}}_{\mathit{y}}$ | Sample Size | GC | inv GC | PEGC | inv PEGC | mTRGC | inv mTRGC | mTRGC* | GC +mTRGC* | |
---|---|---|---|---|---|---|---|---|---|---|

A | 300 | FPR | 2.6 | 2.6 | 0.6 | 0.6 | 0 | 0 | 49.9 | 2.5 |

3000 | FPR | 2.5 | 2.5 | 0.6 | 0.6 | 0 | 0 | 49.8 | 2.5 | |

B | 300 | FPR | 2.7 | 2.6 | 1.1 | 0.7 | 0 | 0 | 50.1 | 2.6 |

3000 | FPR | 2.5 | 2.5 | 1.1 | 0.6 | 0 | 0 | 50 | 2.5 | |

C | 300 | FPR | 2.6 | 2.6 | 0.9 | 0.6 | 0 | 0 | 50 | 2.5 |

3000 | FPR | 2.5 | 2.5 | 0.9 | 0.6 | 0 | 0 | 49.9 | 2.4 | |

D | 300 | FPR | 2.7 | 2.6 | 0.6 | 0.7 | 0 | 0 | 50.1 | 2.6 |

3000 | FPR | 2.5 | 2.5 | 0.6 | 0.6 | 0 | 0 | 49.9 | 2.4 | |

E | 300 | FPR | 2.7 | 2.7 | 2.4 | 2.3 | 0 | 0 | 50.1 | 2.6 |

3000 | FPR | 2.5 | 2.5 | 3.9 | 3.8 | 0 | 0 | 50 | 2.5 | |

F | 300 | FPR | 3.4 | 3.5 | 1.7 | 1 | 0 | 0 | 50 | 3.4 |

3000 | FPR | 3.2 | 3.2 | 5.9 | 1.8 | 0 | 0 | 50 | 3.1 | |

G | 300 | FPR | 2.6 | 17.2 | 0.7 | 3.2 | 0 | 2.6 | 34.7 | 2.2 |

3000 | FPR | 2.6 | 62.3 | 0.6 | 27.6 | 0 | 43.2 | 13.2 | 1.1 |

**Table 2.**False positive rates and false negatives rates (in %) for unidirectionally causally connected $(X\to Y)$ variables. The results for eight discussed testing procedures (

**inv**—results in the time-reversed series) are presented, with the worst (more than $3\%$) FPR highlighted in bold.

Condition for ${\mathit{\u03f5}}_{\mathit{x}}$, ${\mathit{\u03f5}}_{\mathit{y}}$ | Sample Size | GC | inv GC | PEGC | inv PEGC | mTRGC | inv mTRGC | mTRGC* | GC +mTRGC* | |
---|---|---|---|---|---|---|---|---|---|---|

A | 300 | FPR | 2.2 | 11.7 | 0.7 | 1.8 | 0 | 0 | 3.4 | 0.3 |

FNR | 10.3 | 10.3 | 22.2 | 22.3 | 14 | 14.2 | 3.5 | 10.3 | ||

3000 | FPR | 2.3 | 48.1 | 0.9 | 16 | 0 | 0 | 0.7 | 0.1 | |

FNR | 2.5 | 2.5 | 6.4 | 6.5 | 4 | 4 | 0.7 | 2.5 | ||

B | 300 | FPR | 2.2 | 12.3 | 1.3 | 2.2 | 0 | 0 | 3.6 | 0.3 |

FNR | 10.7 | 10.7 | 31.6 | 22 | 14.8 | 15.4 | 3.6 | 10.7 | ||

3000 | FPR | 2.3 | 47 | 1.6 | 17.8 | 0 | 0 | 0.8 | 0.1 | |

FNR | 2.7 | 2.7 | 9.7 | 6.3 | 4 | 4 | 0.8 | 2.7 | ||

C | 300 | FPR | 2.3 | 12.2 | 1.1 | 2 | 0 | 0 | 3.7 | 0.3 |

FNR | 10.7 | 10.7 | 28.2 | 22.5 | 15.4 | 15.4 | 3.7 | 10.8 | ||

3000 | FPR | 2.4 | 46.9 | 1.3 | 17.4 | 0 | 0 | 0.8 | 0.1 | |

FNR | 2.6 | 2.6 | 8.7 | 6.5 | 4 | 4 | 0.8 | 2.7 | ||

D | 300 | FPR | 2.3 | 8.3 | 0.6 | 1.4 | 0 | 0 | 4.1 | 0.4 |

FNR | 12 | 12 | 26.5 | 25.4 | 18 | 17.2 | 4.1 | 12 | ||

3000 | FPR | 2.4 | 40.4 | 0.7 | 11 | 0 | 0 | 0.9 | 0.1 | |

FNR | 3 | 3 | 7.9 | 7.4 | 4.8 | 4.8 | 0.9 | 3 | ||

E | 300 | FPR | 2.4 | 6.7 | 2.3 | 3.8 | 0 | 0 | 4.2 | 0.4 |

FNR | 12.5 | 12.4 | 21 | 21.1 | 17.4 | 17.4 | 4.3 | 12.5 | ||

3000 | FPR | 2.6 | 36.3 | 4 | 19.7 | 0 | 0 | 1 | 0.1 | |

FNR | 3.4 | 3.4 | 5.9 | 5.8 | 5.2 | 5.2 | 1 | 3.4 | ||

F | 300 | FPR | 16.1 | 9.3 | 1.9 | 2 | 0 | 0 | 5.9 | 0.7 |

FNR | 15.9 | 15.9 | 17.7 | 39.9 | 24.2 | 24.4 | 5.9 | 16 | ||

3000 | FPR | 51.7 | 52.5 | 22.6 | 19.1 | 0 | 0 | 1.5 | 0.2 | |

FNR | 4.4 | 4.4 | 5.3 | 12.5 | 6.2 | 6.2 | 1.5 | 4.5 | ||

G | 300 | FPR | 2.4 | 34.1 | 0.8 | 7.6 | 0 | 0 | 3.4 | 0.3 |

FNR | 10.7 | 10 | 23.6 | 21.7 | 15.4 | 14.2 | 3.7 | 10.8 | ||

3000 | FPR | 2.4 | 73.5 | 1 | 51.2 | 0 | 0 | 0.7 | 0.1 | |

FNR | 2.6 | 2.4 | 6.7 | 6.3 | 4.4 | 3.8 | 0.8 | 2.6 |

**Table 3.**False negatives rates (in %) for bidirectionally causally connected $(X\leftrightarrow Y)$ variables. The results for eight discussed testing procedures (

**inv**—results in the time-reversed series).

Condition for ${\mathit{\u03f5}}_{\mathit{x}}$, ${\mathit{\u03f5}}_{\mathit{y}}$ | Sample Size | GC | inv GC | PEGC | inv PEGC | mTRGC | inv mTRGC | mTRGC* | GC +mTRGC* | |
---|---|---|---|---|---|---|---|---|---|---|

A | 300 | FNR | 9.7 | 13.7 | 24.7 | 28.6 | 62.4 | 62 | 50 | 50 |

3000 | FNR | 1.8 | 5.1 | 5.8 | 9.1 | 53 | 53 | 50 | 50 | |

B | 300 | FNR | 12.7 | 21.5 | 42.4 | 34.5 | 57.9 | 57.9 | 50 | 50.1 |

3000 | FNR | 2 | 8.2 | 11 | 14 | 52.3 | 52.6 | 50 | 50 | |

C | 300 | FNR | 12.8 | 21.6 | 38.3 | 35.5 | 57.9 | 57.5 | 50 | 50.1 |

3000 | FNR | 2 | 8.3 | 9.3 | 14.2 | 51.9 | 51.9 | 50 | 50 | |

D | 300 | FNR | 11.2 | 15.9 | 25.5 | 27.9 | 58.4 | 58.4 | 50 | 50 |

3000 | FNR | 2.3 | 6.6 | 7.2 | 10.9 | 52.1 | 52.1 | 50 | 50 | |

E | 300 | FNR | 13 | 16.4 | 25.2 | 28.7 | 62 | 62.8 | 50 | 50 |

3000 | FNR | 3.3 | 6.2 | 5.8 | 8.7 | 53 | 53 | 50 | 50 | |

F | 300 | FNR | 33.7 | 34.4 | 34.7 | 61.7 | 66.2 | 65.8 | 50 | 50.6 |

3000 | FNR | 2.1 | 9 | 5.6 | 21.9 | 54.5 | 54.9 | 50 | 50 | |

G | 300 | FNR | 10.3 | 19.9 | 27.6 | 34.1 | 61.3 | 60.9 | 50 | 50 |

3000 | FNR | 2 | 7.3 | 6 | 13.8 | 53 | 53 | 50 | 50 |

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**MDPI and ACS Style**

Chvosteková, M.; Jakubík, J.; Krakovská, A.
Granger Causality on forward and Reversed Time Series. *Entropy* **2021**, *23*, 409.
https://doi.org/10.3390/e23040409

**AMA Style**

Chvosteková M, Jakubík J, Krakovská A.
Granger Causality on forward and Reversed Time Series. *Entropy*. 2021; 23(4):409.
https://doi.org/10.3390/e23040409

**Chicago/Turabian Style**

Chvosteková, Martina, Jozef Jakubík, and Anna Krakovská.
2021. "Granger Causality on forward and Reversed Time Series" *Entropy* 23, no. 4: 409.
https://doi.org/10.3390/e23040409