# Money Neutrality, Monetary Aggregates and Machine Learning
## 3 Empirical Results  
We proceeded in three steps. The first step was to use the SVM to produce the best autoregressive model AR(q) in forecasting the Eurocoin index. The AR(q) model is the simplest model structure where lagged values of the dependent variable are used to forecast the current value and is defined as:  
\[M_{t}=\sum_{j=1}^{n}x_{jt}, \tag{14}\]  
where \(\mathbf{X}\) is the Eurocoin index, q the maximum number of lags and \(\varphi_{i}\) the parameter vector of the lags to be estimated. We used q = 1-15 and tested the corresponding directional accuracy with respect to the Eurocoin index. The best AR model was the one that included six lags for both, linear and RBF kernel; the AR model is not overfitted, since the in-sample and out-of-sample accuracies are high and close (Figures 4 and 5). Overall, the best AR model is the one equipped with the RBF kernel with in-sample accuracy of 84.62% and out-of-sample accuracy 79.49%. In the second step, we built structural directional forecasting models by augmenting the best AR(6) with the monetary aggregates and their lags, and we evaluated their ability to forecast the Eurocoin index. Figure 4: AR models for directional forecasting of Eurocoin index, linear kernel, in-sample and out-of-sample forecasting accuracy. Finally, we attempt to provide evidence on the issue of money neutrality. We create and test the forecasting accuracy of six structural models by augmenting the best AR models with the inclusion one-by-one of all simple sum and Divisia monetary aggregates. If the augmented model outperforms, in terms of directional forecasting accuracy, the corresponding AR one, then this is interpreted as evidence against money neutrality. Money matters as it is mentioned in the relevant literature. In the opposing case, where the augmented models are unable to outperform the AR ones, we have evidence of money neutrality. The Eurocoin is an index that measures the real GDP growth rate in the euro area. Thus, by providing evidence that it can be forecasted more accurately with the inclusion of the monetary aggregates is interpreted as direct evidence that money affects not only the nominal, but the real economic activity as well. In this context, we can also compare the forecasting ability of the simple sum and the Divisia monetary aggregates. If money is not neutral, then we expect that the Divisia will be superior to the simple sum ones in terms of forecasting ability. The Divisia are constructed in a theoretically \"correct\" procedure following aggregation theory. The simple sum ones are mere sums of the included assets. These results are presented in Table 1 for the linear kernel and in Table 2 for the RBF one. We can observe from Table 1 that the inclusion of the monetary aggregates in the structural models improved the in-sample forecasting accuracy of the AR(6) one for all six aggregates. The out-of-sample forecasting accuracy was improved for the simple sum M1, Divisia M1, Divisia M2, and Divisia M3, while it remained the same as the AR model for the simple sum M2 and the simple sum M3. Thus, the augmented models improve the directional forecasting of the Eurocoin index -as they are compared to the AR ones- providing evidence against money neutrality. \begin{table}
\begin{tabular}{l c c c c c c c} \hline  & **SS M1** & **D M1** & **SS M2** & **D M2 Lag** & **SS M3** & **D M3** & **AR Lag 6** \\  & **Lag 4** & **Lag 10** & **Lag 5** & **12** & **Lag 3** & **Lag 9** & **AR Lag 6** \\ \hline 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\% \\ \hline \end{tabular}
\end{table} 
Table 1: Out-of-sample and in-sample directional forecasting accuracy of the simple sum and Divisia monetary aggregates: Linear kernel. Figure 5: AR models for directional forecasting of Eurocoin index, RBF kernel, in-sample and out-of-sample forecasting accuracy. In Table 2, we present the results for the RBF kernel.