# Conditional Granger Causality and Genetic Algorithms in VAR Model Selection

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. The Vector Autoregression Model and Its Limitations

_{0}is the vector of intercepts, ${A}_{i}$ is the k × k matrix of linear coefficients for the i’th lagged observations of y and ${\epsilon}_{t}$ represents a vector of independent error terms.

^{2}. This issue has been dubbed “the curse of dimensionality” or the tendency of a model to produce biased coefficients when the number of its variables increases [5].

#### 1.2. Coefficient Shrinkage Methods

^{2}parameters to a few hyperparameters. The prior mean of the coefficients determines to what level the shrinkage should be done, whereas the prior variance decides by how much the shrinkage will be done. Other hyperparameters decide the importance ranking between own lags, other variable lags and exogenous factors. In the original Minnesota prior methodology, the tightness of the prior was set by maximizing a pre-sample of the data and for that set, minimizing the out-of-sample forecast error.

#### 1.3. Model Selection Methods

## 2. Methodology

#### 2.1. Competing Approaches to VAR Model Selection

#### 2.1.1. Unrestricted VAR and the Identification Problem

_{t}is a (k × 1) vector of independent and standard normally distributed errors, ε~N(0,1). In the structural form of the VAR, each term of the equation has a specific form with a corresponding functional meaning; thus, B

_{0}is a full rank (k × k) matrix with ones on the main diagonal and possible none zero off-diagonal elements. From a non-statically viewpoint, B

_{0}describes the contemporaneous causality between variables in the system. c

_{0}is a (k × 1) vector of constants (bias), or mean values the variables, ε

_{t}is the vector of structural shock or innovations. One important feature of these error terms under the structural form of the VAR is that their covariance matrix Σ = E(ε

_{t}ε

_{t}’) is diagonal.

_{0}, the contemporaneous causality effect matrix that cannot be uniquely identified, requiring k(k − 1)/2 zero restrictions in order to render the OLS estimation of the system possible after variable reordering. Moreover, in the structural VAR, the error terms are correlated with the regressors. To solve this issue, both sides of the equation are multiplied by the inverse of B

_{0}, leading to the reduced form of the VAR:

_{0}

^{−}

^{1}c

_{0}with c, B

_{0}

^{−}

^{1}B

_{p}with A

_{p}and B

_{0}

^{−1}ε

_{t}with e, the reduced form of (3) can be reinterpreted as:

_{t}ε

_{t}’) is no longer diagonal. If B

_{0}was known, then, going backwards, the system could be rewritten in its structural form and estimated. However, there are many possible values for B

_{0}so that its inverse, multiplied with e, yields an identity matrix. If we instead aim at measuring the covariance between the error terms of Ω = E(ε

_{t}ε

_{t}’), the lower triangular part of the matrix has (k

^{2}/2) + k positions to estimate with k

^{2}unknowns in the B

_{0}matrix, so the system is undetermined. Faced with these issues, economists resort to short-run restrictions, sign restrictions, long-run restrictions, Bayesian restrictions or variable ordering depending on the degree of exogeneity.

#### 2.1.2. Conditional Granger Causality VAR

_{t}(A|B) be the least-squares prediction of a stationary stochastic process A at time t, given all available information in set B, where B can incorporate past measurements of one or multiple stochastic variables. Given this definition, the least-squares autoregressive prediction for A can be:

- ${P}_{t}(A|\overline{A})$ if only the past of A is considered;
- ${P}_{t}(A|{U}_{t})$ if all available information in the measurable past universe (${U}_{t}$) is incorporated;
- ${P}_{t}(A|{U}_{t}-B)$ if the perspective of causality from B to A is to be tested.

_{2i}and β

_{2i}are asymptotically zero. If instead the variables are not independent Equation (6) can be rewritten as

_{2i}should be uniformly zero and, as a consequence, the variance of the terms ε

_{1t}and ε

_{2t}is identical and therefore ${F}_{A\to C|B}$ is equal to 1. The above reasoning can be extended to systems of more than three variables where the causality is conditioned relative to combinations of multiple time series.

- (1)
- Identify for each variable of the dataset which are the individual lags and variables that Granger cause it. These are called “ancestors”.
- (2)
- After compiling the first list of “ancestors”, each one is tested for significance in a multivariate VAR by incorporating all possible combinations of two, then three, four, etc. “ancestors”. If during this testing process the coefficient significance becomes null, the candidate “ancestor” is dropped.
- (3)
- The previous procedure iterates until all possible testing is completed, leaving only the most resilient “ancestors”.

#### 2.1.3. Lasso VAR

_{j}are said to have unbiased estimators $\widehat{\beta}$ if $E\left[\widehat{\beta}\right]=\beta $, where $\widehat{\beta}={c}_{1}{y}_{1}+{c}_{2}{y}_{2}+\dots +{c}_{n}{y}_{n}$, a linear combination of only observed variables. In practice, bias is always present in the estimation results due to unaccounted factors that lead to heteroskedacity of errors and correlations between error terms.

#### 2.1.4. Genetic VAR

^{2}) + n genes, representing the lagged coefficients and the constant terms, where n is the number of variables in the system. If one coefficient is insignificant, then its corresponding gene value should be zero, meaning that the gene is inactive; otherwise, the gene expression would equal the true value of the coefficient. The purpose of the genetic algorithm is to select those individuals who have a corresponding genotype ensuring the best fitness. Just like in classical VAR estimation, the fitness/objective function should be the maximum likelihood of the model. Other fitness functions like Akaike or Bayesian criteria could be employed, but in order to compare the GA methodology with other well-known procedures, consistency must be maintained with respect to the maximum likelihood objective.

#### 2.2. Performance Assessment Criteria

#### 2.2.1. Performance Assessment of Competing Algorithms through Simulation

^{2}) for l lags and n variables, the simulations were capped at a maximum thirty variables and eight lags. Usually, VAR modeling is limited size if short or long run restrictions are not imposed on the coefficient matrix and on the covariance matrix. Moving beyond a large number of variables generates bias risk in OLS or maximum likelihood estimations.

#### 2.2.2. Performance Assessment of Competing Algorithms on Empirical Data Sets

#### 2.3. Utility of Model Shrinkage and Selection in Explaining Networks

## 3. Results and Discussion

#### 3.1. Simulation Performance Results

#### 3.1.1. Number of Estimated Model Parameters

#### 3.1.2. VAR Log Likelihood

#### 3.1.3. Forecast Error

#### 3.1.4. Model Information Criteria

#### 3.2. Empirical Performance Results

#### 3.2.1. US Economy Data Set

#### 3.2.2. Euro Area Dataset

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 10.**Results of genetic optimization. (

**a**) Scatter plot of solution population at each generation (

**b**) Individual score vs. population score.

Number of lags | ||||||
---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | ||

Number of model variables | 1 | 1 | 3 | 7 | 15 | 31 |

2 | 3 | 15 | 63 | 255 | 1023 | |

3 | 7 | 63 | 511 | 4095 | 32,767 | |

4 | 15 | 255 | 4095 | 65,535 | 1,048,575 | |

5 | 31 | 1023 | 32,767 | 1,048,575 | 33,554,431 | |

6 | 63 | 4095 | 262,143 | 16,777,215 | 107 × 10^{7} | |

7 | 127 | 16,383 | 2,097,151 | 268 × 10^{6} | 344 × 10^{8} | |

8 | 255 | 65,535 | 16,777,215 | 429 × 10^{7} | 110 × 10^{10} | |

9 | 511 | 262,143 | 134 × 10^{6} | 687 × 10^{8} | 352 × 10^{11} | |

10 | 1023 | 1,048,575 | 107 × 10^{7} | 110 × 10^{10} | 113 × 10^{13} |

Criterion | Source | Decision |
---|---|---|

Likelihood of the model parameters given the data | [52] | Higher is better |

Number of estimated parameters | Lower is better | |

Mean of squared errors for forecasting 5% of the dataset | Lower is better | |

Akaike information criterion | [41] | Lower is better |

Likelihood of the model parameters given the data | [42] | Lower is better |

FRED Series | Description |
---|---|

GDP | Gross Domestic Product (USD billions, Quarterly) |

GDPDEF | Gross Domestic Product Implicit Price Deflator |

COE | Paid Compensation of Employees (USD billions, Quarterly) |

HOANBS | Nonfarm Business Sector Hours of All Persons |

FEDFUNDS | Effective Federal Funds Rate (Annualized, Percent, Monthly) |

PCEC | Personal Consumption Expenditures (USD billions, Quarterly) |

GPDI | Gross Private Domestic Investment (USD billions, Quarterly) |

Indicator | Measure |
---|---|

Money market interest rates | Rates on money markets, 3-month rates |

Euro/USD exchange rates | |

Gross domestic product at market prices | Millions of euros |

Real labor productivity per person | GDP/ Total employment, all industries, in persons |

Nominal unit labor cost based on persons | Ratio of labor costs to labor productivity |

Employment rate | Number of persons aged 20 to 64 in employment by the total population of the same age group |

Government consolidated gross debt | Total gross debt at nominal value outstanding at the end of the year (percentage of GDP) |

Global oil price | Crude Oil Prices: West Texas Intermediate (WTI) |

Number of variables | Number of Lags | |||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Unrestricted | Conditional Granger | Evolutionary | ||||||||||||||||

1 | 2 | 3 | 4 | 5 | 6 | 1 | 2 | 3 | 4 | 5 | 6 | 1 | 2 | 3 | 4 | 5 | 6 | |

4 | 16 | 32 | 48 | 64 | 80 | 96 | 3 | 2 | 4 | 2 | 4 | 1 | 4 | 5 | 10 | 10 | 13 | 16 |

6 | 36 | 72 | 108 | 144 | 180 | 216 | 2 | 4 | 7 | 5 | 3 | 3 | 10 | 11 | 17 | 25 | 37 | 56 |

8 | 64 | 128 | 192 | 256 | 320 | 384 | 4 | 8 | 2 | 5 | 2 | 6 | 10 | 27 | 50 | 64 | 82 | 122 |

10 | 100 | 200 | 300 | 400 | 500 | 10 | 8 | 5 | 8 | 7 | 10 | 20 | 53 | 85 | 128 | 202 | 232 | |

12 | 144 | 288 | 432 | 576 | 6 | 9 | 18 | 9 | 6 | 17 | 31 | 83 | 142 | 239 | 377 | 226 | ||

14 | 196 | 392 | 588 | 784 | 8 | 20 | 15 | 10 | 13 | 15 | 43 | 143 | 210 | 742 | 272 | 212 |

Number of Variables | Number of LAGS | |||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Unrestricted | Conditional Granger | Evolutionary | ||||||||||||||||

1 | 2 | 3 | 4 | 5 | 6 | 1 | 2 | 3 | 4 | 5 | 6 | 1 | 2 | 3 | 4 | 5 | 6 | |

4 | −0.98 | −0.96 | −0.96 | −0.93 | −0.93 | −0.89 | −0.98 | −1.01 | −1.01 | −1.04 | −1.04 | −1.1 | −1.04 | −1.03 | −1.03 | −1.02 | −1.04 | −1.01 |

6 | −0.96 | −0.95 | −0.92 | −0.89 | −0.87 | −0.81 | −1.01 | −1 | −1.03 | −1.06 | −1.08 | −1.14 | −1.03 | −1.04 | −1.05 | −1.05 | −1.06 | −1.04 |

8 | −0.95 | −0.93 | −0.88 | −0.86 | −0.79 | −0.66 | −1 | −1.04 | −1.1 | −1.1 | −1.15 | −1.25 | −1.04 | −1.03 | −1.02 | −1.04 | −1.06 | −1.09 |

10 | −0.95 | −0.91 | −0.87 | −0.79 | −0.62 | −1 | −1.06 | −1.09 | −1.15 | −1.32 | −1.15 | −1.05 | −1.03 | −1.05 | −1.06 | −1.06 | −0.85 | |

12 | −0.94 | −0.88 | −0.83 | −0.68 | −1.02 | −1.09 | −1.11 | −1.27 | −1.27 | −1.85 | −1.04 | −1.03 | −1.06 | −1.05 | −0.73 | −0.15 | ||

14 | −0.93 | −0.87 | −0.74 | −0.19 | −1.02 | −1.09 | −1.2 | −2.02 | −1.67 | −1.05 | −1.04 | −1.04 | −1.07 | −0.79 | −0.33 | −0.95 |

Number of variables | Number of Lags | |||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Unrestricted | Conditional Granger | Evolutionary | ||||||||||||||||

1 | 2 | 3 | 4 | 5 | 6 | 1 | 2 | 3 | 4 | 5 | 6 | 1 | 2 | 3 | 4 | 5 | 6 | |

4 | 1 | 1 | 0.8 | 1.1 | 1.3 | 1.4 | 1 | 1 | 1.2 | 1 | 0.9 | 1 | 1 | 1 | 1 | 0.9 | 0.8 | 0.6 |

6 | 1 | 1 | 1.1 | 1.2 | 1.4 | 1.3 | 1 | 1.1 | 1 | 1 | 0.8 | 0.9 | 1 | 0.9 | 0.9 | 0.9 | 0.8 | 0.8 |

8 | 1 | 1.2 | 0.9 | 1.6 | 1 | 1.7 | 1 | 1 | 1.1 | 0.7 | 1.1 | 0.7 | 1 | 0.9 | 1 | 0.7 | 0.9 | 0.5 |

10 | 1 | 1.3 | 1.3 | 1.6 | 1.9 | 1 | 0.8 | 0.9 | 0.9 | 0.6 | 1.2 | 1 | 0.8 | 0.8 | 0.5 | 0.5 | 0.8 | |

12 | 1 | 1 | 1.1 | 1.7 | 1 | 1 | 1 | 0.6 | 0.8 | 0.8 | 1 | 1 | 0.9 | 0.6 | 1.2 | 1.2 | ||

14 | 1.1 | 0.8 | 1.2 | 2.5 | 1 | 1 | 1 | 0.4 | 0.8 | 1.1 | 0.9 | 1.2 | 0.8 | 0.2 | 1.2 | 0.9 |

Number of Variables | Number of Lags | |||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Unrestricted | Conditional Granger | Evolutionary | ||||||||||||||||

1 | 2 | 3 | 4 | 5 | 6 | 1 | 2 | 3 | 4 | 5 | 6 | 1 | 2 | 3 | 4 | 5 | 6 | |

4 | 1 | 1 | 1.1 | 1.1 | 1.2 | 1.2 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | 1.1 | 1 | 1 | 0.9 | 0.9 | 0.9 |

6 | 1 | 1.1 | 1.1 | 1.2 | 1.2 | 1.2 | 1 | 0.9 | 0.9 | 0.9 | 0.8 | 0.9 | 1 | 1 | 1 | 0.9 | 0.9 | 0.9 |

8 | 1 | 1.1 | 1.2 | 1.2 | 1.3 | 1.2 | 0.9 | 0.9 | 0.9 | 0.8 | 0.8 | 0.8 | 1 | 1 | 0.9 | 0.9 | 0.9 | 1 |

10 | 1.1 | 1.1 | 1.2 | 1.3 | 1.2 | 0.9 | 0.9 | 0.8 | 0.8 | 0.8 | 0.9 | 1 | 1 | 1 | 1 | 1 | 1.1 | |

12 | 1.1 | 1.2 | 1.2 | 1.2 | 0.9 | 0.9 | 0.8 | 0.8 | 1 | 1.8 | 1 | 1 | 1 | 1 | 1 | 0.2 | ||

14 | 1.1 | 1.2 | 1.2 | 0.7 | 0.9 | 0.8 | 0.8 | 1 | 1.6 | 0.9 | 1 | 1 | 1 | 1.3 | 0.4 | 1.1 |

Indicator | Unrestricted VAR | Simplified VAR | Conditional Granger Search | Lasso Regression | GA Variable Selection |
---|---|---|---|---|---|

Log-Likelihood | 7489 | 7294 | 7286 | 7420 | 7860 |

Model number of parameters | 196 | 32 | 16 | 71 | 121 |

Mean squared error | 1.02% | 0.88% | 0.94% | 0.92% | 0.88% |

Akaike criterion | −14,571 | −14,510 | −14,527 | −14,684 | −15,465 |

Bayesian criterion | −13,879 | −14,377 | −14,448 | −14,418 | −15,022 |

Indicator | Unrestricted VAR | Simplified VAR | Conditional Granger Search | Lasso Regression | GA Variable Selection |
---|---|---|---|---|---|

Log-Likelihood | 143 | −228 | −223 | −80 | 12 |

Model number of parameters | 324 | 70 | 17 | 90 | 149 |

Mean squared error | 0.24 | 0.45 | 0.21 | 0.20 | 0.22 |

Akaike criterion | 380 | 615 | 497 | 358 | 293 |

Bayesian criterion | 1114 | 789 | 555 | 576 | 650 |

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**MDPI and ACS Style**

Marica, V.G.; Horobet, A.
Conditional Granger Causality and Genetic Algorithms in VAR Model Selection. *Symmetry* **2019**, *11*, 1004.
https://doi.org/10.3390/sym11081004

**AMA Style**

Marica VG, Horobet A.
Conditional Granger Causality and Genetic Algorithms in VAR Model Selection. *Symmetry*. 2019; 11(8):1004.
https://doi.org/10.3390/sym11081004

**Chicago/Turabian Style**

Marica, Vasile George, and Alexandra Horobet.
2019. "Conditional Granger Causality and Genetic Algorithms in VAR Model Selection" *Symmetry* 11, no. 8: 1004.
https://doi.org/10.3390/sym11081004