## 3 Materials and Methods
In the first case: \(11:f\Big{(}x,\ (W^{l}\mathbb{e}\mathbb{R}^{Ml\otimes Ml-1})+u^{l}\in\mathbb{R}^{Ml }=\sigma^{n}=\delta\big{(}W^{n}\sigma^{1}\big{)}=\delta\big{(}W^{n}\delta \big{(}W^{l}x\big{)}\big{)}\)[11] If \(\delta\) is a linear function \(\to\ \delta\Big{(}u^{I}\Big{)}=k_{l}u^{I}\) where \(f(x,\gamma)=k_{n}(W^{n}\mathbb{r}x+\hat{\sigma}_{1})+\hat{\sigma}_{n}=\ \hat{W}x+\hat{n}\). If we choose, in an arbitrary way, to use a non-linear activation function, we have: \(12:\delta(x)=\frac{1}{1+\exp(-x)}\) \(13:\delta^{\prime}=\delta(x)(1-u(x))\) \(14:\tanh(x)=\frac{e^{-x}-e^{-x}}{e^{-x}+e^{-x}}\) \(15:tanh^{\prime}(x)=1-\tanh^{2}(x)\)  Thus, with rectified linear unit, we have: \(16:Relu(x)=\left\{ \begin{array}{l}x\ for\ x>0\\ 0\ for\ x<=0\end{array} \right.\) In our NN, MSE will be: \(17:MSE(\tau)=\frac{1}{2N}\sum_{n=1}^{N}\|f(x_{n},\ \tau)-t_{n}\|_{2}^{2}\) In [17]: \(18:\|f(x_{n},\ \tau)-t_{n}\|_{2}^{2}=\left(l_{2,n}\right)^{2}\to\|V\|_{2}^{2}= \sum_{k=1\ or\ n}^{n}V_{k}^{2}\) Now, the Log-Likelihood (LL) will be: \(19:LL(\lfloor x_{n}\rfloor=\ \frac{1}{N}\sum\limits_{n=1}^{N}\,\underset{or\ n=n}{ \sum}\,logP\ (x_{n})\)
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**Algorithm 1** Algorithms terms to describe neural networks process We expanded the observations through the quadratic transformations (_LRCAS_, _LAGTPS_, and _LGAOS_) and first differences of each variable (_LRCADF_, _LAGTPDF_, and _LRCADF_). In this way, our NN has worked on over 21,000 data and has guaranteed us better ML results. After, we test, as robustness checks, the PVAR results obtained with panel data methodologies through the ANNs analyses, using the Oryx 2.0.8 software. We choose to use this ANNs experiment for the following reasons. NNs work in parallel and are, therefore, able to process a lot of data simultaneously and autonomously. In contrast, in standard or econometric statistical processes, each data is treated individually and/or in time series. Even though each neuron is relatively slow, the parallelism partly explains the faster speed of the brain in performing tasks that require the simultaneous processing of a large number of data. In essence, it is a sophisticated statistical system with excellent noise immunity; if some units of the system were to malfunction, the network as a whole would have reduced performance but would hardly encounter a system crash. The characteristics of a NN model that we use are the following:  
1. The development of the \"neuron system\" is distributed over many elements. In other words, many neurons do the same thing;
2. An address identifies each data of the algorithm used (a number), which is used to retrieve the knowledge necessary to perform a certain task;
3. ANNs, unlike standard econometric models and their software, do not have to be programmed to perform a task. ANNs learn independently based on experience or with the help of an external instructor. Therefore, we used the same dataset as the PVAR econometric modeling; however, we expanded the observations through the quadratic transforms (_LRCAS_, _LAGTPS_, and _LGAOS_) and firsts differences of each variable (_LRCADF_, _LAGTPD_,\(F\) and _LRCADF_). In this way, our neural network has worked on over 21,000 data and has guaranteed us a better ML results. Assuming the variable redundancy process's validity in a neural network, we obtain seven inputs (_LRCA_, _LRCAS_, _LAGTPS_, _LGA_, _LRCADF_, _LAGTPDF_, and _LRCADF_) on 10 nodes with two targets (_LAGTP_ and _LGAO_).