The data used in this study (Table 1) were obtained from quinquennial WDI statistics for the 1977-2017 interval from the five aforementioned countries in West Africa (Figure 1). When data were missing, the entire rows were eliminated, and for this reason there is a total observation number of 32 (Table 2).

A partial least square regression (PLSR) method is recommended when the number of observations is limited compared to the dependent \ independent variables. This method is used widely in spectroscopy to fit quantitative dimensions 9, discriminate qualitative classes 10, or to solve economic problems ^{11, 12}. A different approach, with latent variables, which is usually applied in the socio-economic field, is the PLS path method 13. In the present paper, a multilinear regression (MLR), a PLSR and a principal component analysis (PCA) have been used, as taken from the XLSTAT 2019.4.1 (Addinsoft SARL USA, New York, NY, USA) package. The PLSR models were fitted to the GD ratio for the five countries, using all of the four main independent variables (P, D, A, M) as well as their relative ratios to P (PG, PD, PA). A single MLR model was fitted to the overall countries, with GD as a dependent variable and PG, PD, PA, and M as independent ones. The PCA was applied to the whole dataset to find clusters among the countries.

The GD, G to D ratio (Figure 3). On average, a rise of 0.33±0.09 Y^-1(R² 0.28; P 0.00083) is observed over the years, but a decline can be observed for the 1977-1997 interval and a negative shift for GMB, which makes its performance worse from 2007, similarly to SEN, whose trend, however, becomes regularized in 2017. The greatest improvement is observed for GNB. On average, a yearly linear regression of +0.33 Y^-1 in the GD ratio is reported, with a mean value of 14.94, corresponding to a growth of +2.2% Y^-1 in the GD ratio, but not in G. However, it should be pointed out that the trend only becomes favorable in the new millennium.

The PD, P to D ratio (Figure 4). The value of the ratio started to increase from 2000, as it is driven by the performances of SEN, GMB and GNB. The yearly linear regression is 0.052±0.009 Y^-1(R²0.55; P 0.0001), which results in a mean value of 0.778, corresponding to a growth of +6.6% Y^-1. However, it should be pointed out that the trend only becomes favorable in the new millennium.

The PG, P to G ratio (Figure 5). The ratio of P to G generally increases, and this increase begins to accelerate from 2002, albeit more regularly in GMB and SEN than in GNB and MLI.

The PA, P to A ratio (Figure 6). On average, the P-to-A ratio rises almost regularly by 0.13 per year from 1977, except for MRT, albeit with great differences in the starting levels among the countries, where SEN and MLI surpass the others.

The M, Net Migration flux (Figure 7). Dramatic MLI events occurred in 1987 and 2012. A large increase in migration can be observed from SEN from 1992 to 2012, which eventually reduces. The GNB trend is oriented toward reduced fluxes. The GMB, from the previous host country, is therefore marked by a stable share of estimated migrants.

A, Agriculture employment (Figure 8). Somewhat different conditions characterize the GNB and MLI, which show values of over 65% for 2017, for MRT and around 30% for SE and GMB. The A decay is minimum for GNB (-0.167 % Y^-1), intermedium for GMB (-0.387 % Y^-1), for MLI ( -0.419 % Y^-1) and for MRT (-0.467 % Y^-1), while it is maximum for SEN (-0.724 % Y^-1).

P, Personal remittances (Figure 9, Figure 10). Starting from 2000, theSenegalese people increase their personal remittances and reach a nearly three-fold value forMLI in 2017 and a nearly fifteen-fold value for the other three (Figure 6). As far as the GMB of the minor countries is concerned, a steady increase vs. GNB and MRT can be observed from 2007 (Figure 7).

D- Net bilateral aid flows (Figure 11). GMB and GNB are mostly entropic, and MRT shows a slow flexus: these three countries are clustered in the D-to-G ratio (see below). SEN and MLI show similar trends with the relative maxima in 1992 and 2012 and with the minima in 2000, but show divergent final trends because SEN D is lowered while D is growing in MLI.

The PCA shown in Figure 12 highlights that the active observations are clustered into three aggregates, namely, GMB, GNB and MRT (as Cluster I), MLI (as Cluster II) and SEN as (Cluster III). Among the variables in the first quadrant, GD neighbors PG and PD, as opposed to A and DG displayed in the third quadrant, while P, PA, G and D are positioned in the second quadrant. Migration (negative sign) is the only active variable in the fourth quadrant and distinguishes the members of Cluster I because of their reduced fluxes.

1Values are considered significant at P 0.05 > |0.372| r MD.G = -0.359

The solutions of the complete models are reported in Table 4, Table 5 for the three envisaged clusters.

The coefficient of the three clusters are reported in Table 6 as Polydromic coefficients for the G-to-D ratio.

MLI shows a great benefit from the M and A factors in cluster II (active Country).

Country SEN excels for the maximum reaction to D decrease and shows high scores for the PG ratio in cluster III (reactive Country).

Countries GMB, GNB and MRT show a favorable reaction to a D decrease and negative influences from a higher A in cluster I (entropic Country).

In short, as far as Migration is concerned, an increase shows a positive response to MLI (Coeff. -0.29: active Country) and a slightly negative one to the others (0.20). Agriculture employment is apparently more favorable to the G-to-D ratio in MLI-active (0.16), but unfavorable to the others.

Importantly, the consequences of a decrease in D appears more favorable to the G-to-D ratio for SEN (-0.45: reactive Country), slightly favorable for GMB, GNB, MRT (-0.28) and not favorable at all for MLI (-0.03).

The G-to-D ratio is responsible for increasing Personal remises for MLI (0.31) and SEN (0.27).

The effects of the migration (as net fluxes) returns on P are delayed over the years, thus the favorable raw Pearson correlations of M with P (-0.309 not significant) cannot account for time-lag effects. According to Rozelle et al 17, remittances are a positive function of migration.

As far as D and M are concerned, the correlation table reflects a general greater effort (D) to deter future migration (r D,M -0.489), where almost all the negative M net fluxes are ligated with a higher income from D. It should be noted that the DM correlation decreases slightly to a value that was recalculated at a parity of G (r D,M.G -0.359). Clemens and Postel 18 (2017) reached a similar conclusion from a wider panel (2678 observations in the1995-2013 interval) and observed that the relationship between D (X axis) and the Log of Emigration Rate (Y) was negative, even for different levels of Income. The same Authors pointed out that the capacity of aid to deter migration is limited at best. Aid can only encourage economic growth, employment and security to a limited extent. Beyond this, successful development in almost all formerly-poor countries has produced an increase in emigration. Moreover, Lanati and Thiele 19 wrote about the impact of foreign aid on revisited migration and found a negative correlation between the total aid a country obtains and its emigration rate. This evidence was also confirmed in most of the important empirical studies, similar to the results of Berthélemy et al 20, who studied the connection between the total aid received in a country and migration for a large cross section of developing countries.

In our analyses, we confirm that, in a long-term framework, migration acts like a magnet for donor aid. However, because of the weak and non-linear influence of migration on the GD ratio, our MLR model puts the partial coefficient of migration in the last position, with a nearly zero coefficient, and accounting for only 0.5% of the variance, thus supporting the independence of the average of G to D from migration.

It should be recalled that the average trend for the GD ratio, in current values, shows increasing values (+0.33 Y^-1) with a parabolic acceleration over the years, and the level is somewhat higher for SEN (21.42; +68% vs. the others). The aid for development naturally appears in the national balances, since the variations in G in fact depend on D for more than 76%, while an even greater influence (over 88%) can be observed for the P source (Table 5). In this work, the importance of the P-to-D ratio for the GD donor efficiency has emerged (nearly 70%). The PD ratio is below 1 for many years in almost all the considered countries. However, after the first returns from migrants, the critical threshold is exceeded to a great extent, which is a sign that a real economic takeoff has begun. In fact, after maximizing overall, it appears that the variation in D raw efficiency, in the framework of the endogenous variates censed every five years, mainly depends on the PD ratio for about 70%, that is, a greater value than the P with D itself (R²52%, Table 5). Thus, the main key for assessing economic efficiency appears to be the balance of the personal remittances with donor aid: a shift up of the numerator (P) as well as a shift down of the denominator (D) is ceteris paribus an advantage in the global raw efficiency chain. The yearly linear regression of +6.6% Y^-1,and even more for the P-to-D ratio in the new millennium, may be a quite optimistic result, but this sample allows us to explain a different response from the three Clusters as a result of the variations in the D values. In fact, our aim was to discern countries according to their aptitude against variations in D. From this perspective, the polydromic GD coefficients for D distinguished the three clusters, enhancing a greater resilience for SEN (-0.45) than for GMB, GNM and MRT (-0.28) and for MLI (-0.03), which is a still active country as regard higher employment in agriculture.

On the other hand, Alemu and Lee 21 revealed that middle-income African countries tend to experience a greater impact on their economic growth from direct foreign investment and natural resource revenues, and oil exports in particular. In fact, the growth experienced in the five countries since the millennium seems to be due not only to exogenous forces, such as Donor aid and Personal remittances, but also to endogenous economic factors. Our study is in agreement with that of Minasyan et al 22,23, whose empirical results supported the hypothesis that higher remittances paid by donor countries strengthened the growth effects of foreign aid, especially where a more people-dedicated dual channel can be developed and the remises were dedicated to such solid familiar investments as land, building and culture. A concept that Clement and Postel (2017) also outlined is that donors could achieve a greater impact by leveraging on foreign aid in order not to deter migration, but to shape it for mutual benefits.