# Money Neutrality, Monetary Aggregates and Machine Learning

^{*}

## Abstract

**:**

## 1. Introduction

_{jt}is one of the n components of the monetary aggregate M

_{t}

_{,}implying that all financial assets contribute equally to the money total and all assets are perfect substitutes.

_{jt}+ w

_{j,t−1}) for j = 1, …, n, where

_{jt}is the user cost of asset j, derived in [7],

_{jt}is the market yield on the jth asset and R

_{t}is the yield available on a ‘benchmark’ asset that is held only to carry wealth between multiperiods [8] (For more details regarding the Divisia approach to monetary aggregation see [9].).

## 2. Methodology and Data

#### 2.1. Support Vector Machines

**x**

_{i}∈ℝ

^{m}(i = 1, 2…, n) belonging to two classes y

_{i}∈ {−1, +1}.

**w**is the parameter vector, and b is the bias (Figure 1). So ${y}_{i}f\left({\mathit{x}}_{i}\right)>0,\forall i$.

_{i}≥ 0, ∀i, and a parameter C, describing the desired tolerance to classification errors. The solution to the problem of identifying the optimal separator can be dealt with through the Lagrange relaxation procedure of the following equation:

_{1}, α

_{2}, …, α

_{n}are the non-negative Lagrange multipliers.

- The linear:$${K}_{1}\left({\mathbf{x}}_{1},{\mathbf{x}}_{2}\right)={\mathit{x}}_{1}^{T}{\mathit{x}}_{2},$$
- The RBF:$${K}_{2}\left({\mathit{x}}_{1},{\mathit{x}}_{2}\right)={e}^{-\gamma |\left|{\mathit{x}}_{1}-{\mathit{x}}_{2}\right|{|}^{2}},$$

#### 2.2. The Data

## 3. Empirical Results

**X**is the Eurocoin index, q the maximum number of lags and φ

_{i}the parameter vector of the lags to be estimated.

## 4. Conclusion

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Hyperplane selection and support vectors. The SVs (represented with the pronounced black contour) define the margins that are represented with the dashed lines and outline the separating hyperplane represented by the continuous line.

**Figure 2.**The non-separable two-class scenario in the data space (

**left**) and the separable case in the feature space after the projection (

**right**).

**Figure 3.**Overview of a three-fold cross validation for a given set of values of the model’s parameters. Each fold is used as a testing sample, while the rest are used for training the model. The model is evaluated by the average forecasting accuracy for each set of parameters over the k-folds.

**Figure 4.**AR models for directional forecasting of Eurocoin index, linear kernel, in-sample and out-of-sample forecasting accuracy.

**Figure 5.**AR models for directional forecasting of Eurocoin index, RBF kernel, in-sample and out-of-sample forecasting accuracy.

**Table 1.**Out-of-sample and in-sample directional forecasting accuracy of the simple sum and Divisia monetary aggregates: Linear kernel.

SS M1 Lag 4 | D M1 Lag 10 | SS M2 Lag 5 | D M2 Lag 12 | SS M3 Lag 3 | D M3 Lag 9 | AR Lag 6 | |
---|---|---|---|---|---|---|---|

out-of-sample accuracy | 82.05% | 82.05% | 76.92% | 82.05% | 76.92% | 79.49% | 76.92% |

in-sample accuracy | 85.26% | 85.90% | 85.26% | 87.18% | 83.33% | 83.97% | 82.69% |

**Table 2.**Out-of-sample and in-sample directional forecasting accuracy of the simple sum and Divisia monetary aggregates: Radial basis function (RBF) kernel.

SS M1 Lag 2 | D M1 Lag 2 | SS M2 Lag 2 | D M2 Lag 2 | SS M3 Lag 7 | D M3 Lag 7 | AR Lag 6 | |
---|---|---|---|---|---|---|---|

out-of-sample accuracy | 58.97% | 58.97% | 61.54% | 61.54% | 74.36% | 64.10% | 79.49% |

in-sample accuracy | 75.00% | 75.00% | 77.56% | 77.56% | 74.36% | 76.92% | 84.62% |

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**MDPI and ACS Style**

Gogas, P.; Papadimitriou, T.; Sofianos, E.
Money Neutrality, Monetary Aggregates and Machine Learning. *Algorithms* **2019**, *12*, 137.
https://doi.org/10.3390/a12070137

**AMA Style**

Gogas P, Papadimitriou T, Sofianos E.
Money Neutrality, Monetary Aggregates and Machine Learning. *Algorithms*. 2019; 12(7):137.
https://doi.org/10.3390/a12070137

**Chicago/Turabian Style**

Gogas, Periklis, Theophilos Papadimitriou, and Emmanouil Sofianos.
2019. "Money Neutrality, Monetary Aggregates and Machine Learning" *Algorithms* 12, no. 7: 137.
https://doi.org/10.3390/a12070137