A De Cecco

K Le Calvez

B Sacépé

C B Winkelmann

H Courtois

Interplay Between Electron Over-Heating and ac Josephson Effect

We study the response of high-critical current proximity Josephson junctions to a microwave excitation. Electron over-heating in such devices is known to create hysteretic dc voltage-current characteristics. Here we demonstrate that it also strongly influences the ac response. The interplay of electron over-heating and ac Josephson dynamics is revealed by the evolution of the Shapiro steps with the microwave drive amplitude. Extending the Resistively Shunted Josephson junction model by including a thermal balance for the electronic bath coupled to phonons, a strong electron over-heating is obtained.

A normal metal (N) coupled to two superconducting electrodes (S) constitutes a Josephson junction, that is, a device capable of sustaining a dissipationless supercurrent [START_REF] Likharev | Dynamics of Josephson Junctions and Circuits[END_REF][START_REF] Courtois | [END_REF][3]. The small normal state resistance and leadto-lead capacitance of SNS junctions make these strongly overdamped in the RCSJ model [START_REF] Likharev | Dynamics of Josephson Junctions and Circuits[END_REF], meaning that the quantum phase dynamics is intrinsically non-hysteretic. Still, hysteresis is observed in the voltage-current (V -I) characteristics of high-critical current SNS junctions due to electronic over-heating associated to the sudden onset of dissipation when the bias exceeds the critical current [4]. Besides, under a microwave excitation at frequency ν, the junction characteristics display voltage plateaus at V n = nhν/2e with n an integer [5]. These so-called Shapiro steps are due to the phase-locking of the supercurrent oscillations at the Josephson frequency 2eV /h to the microwave. Shapiro steps have been frequently used for studying the phase dynamics of a variety of Josephson junctions [6][7][8][9][10]. This includes in particular junctions based on novel materials [11][12][13][14][15][16][17][18][19][20][21] where the absence of odd steps can be the signature of topological transport. Still, the interplay between the electron over-heating and the ac Josephson dynamics has been so far overlooked.

In this paper, we discuss the response of high-critical current proximity Josephson junctions to a microwave excitation. We demonstrate the prominent role of electronic over-heating in the Shapiro steps map. Low-index steps can be masked by the switching to the resistive state. A simple model explains this behavior as well as the observation of a sharp discontinuity in the measured critical current when the ac current is increased.

We have fabricated Nb-Au-Nb junctions using a lift-off lithography technique based on an Al-Mo metallic bilayer as a shadow mask [22], which avoids the deterioration of Nb superconducting properties by organic contamination. Conventional e-beam lithography, followed by a dry etch of the Mo top layer and a wet etch of the Al bottom layer, produces a locally suspended Mo mask. A shadow evaporation of Au (N island) and Nb (S leads) is performed through this mask at opposite angles. The edge roughness of the structures (see Fig. 1c inset) arises from the granularity of the Mo mask. The junctions were all about 210 nm wide, the separation between the Nb electrodes ranging from 180 to 500 nm, while the Au part was about 200 nm longer in order to ensure a good overlap with each electrode. The normal-state resistance of the junctions can be related to a diffusion constant D in Au of about 100 cm 2 /s. The critical temperature of the Nb electrodes is 8.5 K. Table 1 lists the main device parameters for the different junctions investigated.

Transport measurements were performed at temperatures down to 100 mK. Fig. 1a-b show V-I characteristics of a typical sample. At low temperature, a strong superconducting branch is observed (Fig. 1a), with a large critical current I c exceeding 100 µA. This is achieved owing to the short length of the Au bridge and the high interface transparencies. Above I c , the V-I characteristic switches to the ohmic branch, characterized by the normal-state resistance R N . When the current is lowered again, the V -I characteristic remains on the ohmic branch down to the so-called retrapping current I r < I c . This pronounced hysteretic behaviour is of thermal origin and is typical of SNS junctions with a large critical current density [4]. The retrapping current can be roughly thought of as the value of the critical current at a biasdependent, higher electronic temperature T * , determined by the thermal balance between the dissipated Joule heat and the coupling to the phonon bath. Conversely, T * is also the temperature scale above which I r and I c merge (see Fig. 1c) and the behavior of the junction becomes non-hysteretic (Fig. 1b).

The temperature dependence of the critical current is displayed in Fig. 1c. The properties of long SNS junctions depend only little on the superconducting electrodes' energy gap ∆, but are mainly governed by the normal island's Thouless energy [START_REF] Courtois | [END_REF]8] E Th = hD/L 2 ≪ ∆, where D is the diffusion constant and L the length of N. Within the Usadel equations framework and assuming k B T > E Th , the critical current I c follows [23]:

eR N I c = η E Th 32 3 + 2 √ 2 2πk B T E Th 3/2 exp - 2πk B T E Th .
(1) Here the phenomenological parameter η < 1 describes a reduction in the critical current related, for instance, to non-ideal interfaces. A very good fit to the data is obtained in all devices, with η always exceeding 0.6, see Table 1. The effective length L eff = hD/E Th associated to the fit Thouless energy E fit Th exceeds the mere separation L between the electrodes. It should indeed include about twice the superconducting coherence length since Andreev reflections take place in S over such a length [23]. The dependence of the critical current on a perpendicular magnetic field (Fig. 1d) shows both a quasi-gaussian monotonic decay associated to dephasing and a oscillatory part arising from interference effects [24,25].

N o L RN E fit Th L eff η I 0 c I 0 r T * (nm) (Ω) (µeV) (nm) (µA) (µA) (K) J1 225
We now turn to the effect of a microwave excitation on the V-I characteristics. Microwave signals in the frequency range 1 to 26 GHz and with a power P µ were applied to the shielded cavity containing the sample. As the impedance of our samples is small compared to the estimated line impedance at high frequency, the samples are still current-biased in the microwave regime. The color map of Fig. 2a shows the differential resistance dV /dI (obtained by numerical derivation) as a function of the dc bias current I and the microwave current I µ ∝ P 1/2 µ at a frequency ν = 6 GHz. The supercurrent branch and the Shapiro steps (up to an index exceeding 10) appear as dark regions, with zero differential resistance. At large excitation amplitudes, Fig. 2a map is symmetric in I and the Shapiro steps' widths oscillate with the microwave excitation amplitude, producing a well-known pattern [START_REF] Likharev | Dynamics of Josephson Junctions and Circuits[END_REF]. At small microwave current I µ , the hysteresis appears through the asymmetry of the map with respect to I.

Strikingly, several low-index steps do not appear in the (positive) current branch corresponding to a current increasing from zero to above the critical current I c . Individual V-I characteristics clearly demonstrate (Fig. 2bd) that the absence of these steps stems from the voltage directly jumping from zero up to about R N I c , which corresponds to the ohmic branch. Steps corresponding to voltages below R N I c thus cannot be detected. This behavior here is clearly distinct from the discussion of recent experiments on Josephson junctions based on topological insulators, in which odd-index Shapiro steps are predicted to be absent.

Let us now consider the behavior of the critical current as a function of the microwave current amplitude I µ . In a current bias picture, the microwave irradiation adds adiabatically an oscillatory excursion I µ to the bias current I, so that the current oscillates between I -I µ and I + I µ [9]. In a quasi-static regime, switching to the resistive state then occurs at a lower critical current I c -I µ . Once the junction has switched, it remains in the resistive state due to hysteresis. One expects therefore a linear suppression of I c with increasing I µ , as seen in Fig. 2a. We attribute the slight downward deviation from linear behavior to a small increase in the electronic temperature at high microwave power. On the retrapping branch, a similar decay of the retrapping current I r with the microwave current I µ is observed.

In order to provide a quantitative description, we need to consider the energy relaxation from the normal metal electronic population. In the present temperature range, electron-phonon scattering is the dominant mechanism. The related coupling power between electrons at a temperature T e and phonons at a temperature T ph is P e-ph = ΣU (T 5 e -T 5 ph ), where Σ is the material-dependent cou- pling constant and U is the metal volume. Considering a retrapping temperature T * of 1 to 3 K, the related rate τ -1 e-ph ≈ 0.16×T 3 GHz in Au [26] is in the low GHz range (≈ 0.4 GHz in J2 at T * = 1.4 K for instance). In most of the frequency range investigated here, the thermal relaxation is thus slow compared to the microwave (τ -1 e-ph < ν) so that the electronic temperature can be considered as almost constant with time at a given (I,I µ ) bias point.

We consider a Resistively Shunted Junction (RSJ) model [START_REF] Likharev | Dynamics of Josephson Junctions and Circuits[END_REF] with a current bias. The time-dependent current i(t) = I + I µ sin(2πνt) through the junction is considered as the sum of the ohmic current v/R and the Josephson current I c sin ϕ, where ϕ is the phase difference across the junction:

i(t) = I + I µ sin(2πνt) = I c sin ϕ + v/R. (2)
The time-dependent voltage v relates to the timederivative of the phase as v(t) = h φ/2e from the sec-ond Josephson relation. From Eq. ( 2), the phase dynamics can be modeled as that of a massless particle of position ϕ in a tilted washboard potential U (ϕ) = -h[I c cos ϕ+i(t)]/2e. The potential slope is proportional to the current bias i(t): its average is thus determined by I and it oscillates with an amplitude given by I µ . For large enough I or I µ , the particle rolls down the slope.

The Shapiro steps at voltage values V n = n hν/2e correspond to the particle hopping down by n minima during one microwave period. We can write the instantaneous Joule power:

p(t) = i(t) • v(t) = I c h 2e φ sin ϕ + 1 R h 2e φ 2 . (3)
The first term relates to the change in the Josephson energy. It is zero in average and does not contribute to the average power P dissipated over one cycle. Only the second term, which is the Joule power across the junction resistance, dissipates. It can be non-zero in average even though the average voltage V is zero. At this point, we now introduce a heat balance where the dissipated power P is balanced by the electronphonon coupling power P e-ph . The related temperature elevation acts on the phase dynamics through the temperature dependence of the critical current following Eq. ( 1). Using Eqs. ( 2) and ( 3) and taking the volume U as a free parameter, we have numerically calculated the timedependence of the phase, the dissipated power and the ensuing electronic temperature T e for every (I,I µ ) bias point, which gives access to the related dc voltage drop V . Fig. 2e shows the calculated differential resistance for device J2's parameters. For the best fit, the volume U was chosen as 10 times the physical volume. This can be explained by both the inverse proximity effect in the leads in the vicinity of the N-S interface and the thermal conductance of the leads between the N island and the N metal regions of the leads. A semi-quantitative agreement for the differential resistance between Fig. 2a and 2e is readily seen.

The associated temperature map of Fig. 3a highlights the importance of dissipation in the ac phase dynamics in SNS junctions. Strikingly, the electronic temperature varies significantly as a function of the microwave current bias: the temperature increases from the bath temperature of 0.1 K up to above 2 K. On the first Shapiro step, the temperature is already of about 1 K. Even for zero dc current I and hence zero average voltage V , electrons in are significantly overheated at large microwave drives, see Fig. 3a inset. The Shapiro steps structure appears also on the temperature map, as can be seen in Fig. 3a and more clearly in Fig. 3b as wiggles in every curve, especially the curve [START_REF] Courtois | [END_REF].

Both the data (Fig. 2a) and the calculations (Fig. 2e) exhibit a sudden drop of the critical current I c as the retrapping current I r approaches zero at a microwave current I µ = I * (of about 0.04 in Fig. 2a). As shown in Fig. 3a inset, the microwave amplitude I µ at this point is approximately equal to the critical current I c (T e ) at the electronic temperature at zero bias, so that no supercurrent can be established due to the electronic overheating at this point of the V-I characteristics. The numerical solution of the phase dynamics illustrates this precisely: while at an ac drive below I * the particle oscillates in a single washboard valley, above I * it hops forth between two valleys [27]. Let us note that this feature would not appear if the microwave amplitude would be swept at a fixed (and low) dc bias current.

We have obtained a good agreement between experiment and calculation at every investigated microwave frequency up to 26 GHz. At higher frequency, as dissipation increases with excitation frequency ν as ν 2 , the electron over-heating is much increased in the superconducting state (V = 0). The two currents I c (I µ ) and I r (I µ ) are thus seen to merge before I r approaches zero [27] and no discontinuity is observed.

The interplay of electron over-heating and ac Josephson dynamics discussed here is prominent in junctions that are hysteretic in dc. Indeed, when driving our same devices to non-hysteretic conditions, either by applying a magnetic field (Fig. 4a) or increasing the temperature (Fig. 4b), the usual Shapiro pattern is recovered [START_REF] Likharev | Dynamics of Josephson Junctions and Circuits[END_REF]. This is evidenced by the good agreement of the width of the n = 0 step with the standard Bessel function expression (white dotted lines in Fig. 4a andb). The crossover between the overheating regime and the isothermal be- haviors depends not only on the junction critical current amplitude but also on the coupling to the thermal bath.

In summary, we have evidenced that electron overheating is of paramount influence in the microwave response of Josephson junctions. Exploring the microwave response of Josephson junctions involves variable electronic temperatures, which is of uttermost importance for the complete analysis of devices based on new materials like topological conductors.

Supplemental Materials: Interplay Between Electron Over-Heating and ac Josephson Effect

In this supplemental part, we provide some more details about the numerical calculation of the differential resistance map under microwave irradiation, as well as the calculation results supporting the explanation for the discontinuity observed in the differential resistance map. We also discuss additional experimental data obtained at different values of the microwave frequency.

NUMERICAL CALCULATIONS

Our model is an extension of the RSJ model with the temperature dependence of the critical current (Eq. ( 1) of the main text), the electronic temperature T e in N being governed by the thermal balance between the Joule power and the coupling to phonons. We here neglect other heat transport channels out of N, which will eventually lead to a somewhat lower temperature.

A sharp discontinuity in the differential conductance map at 6 GHz is described in the main text, see Fig. 2 of the main paper. Fig. S1 shows both a zoom of the map and the time-dependence of the phase at two points at the same dc current bias, but at an ac current signal amplitude just below and above the threshold. While the mean voltage is zero in every case, the phase's excursion is larger in amplitude at larger ac signal. It exceeds 2π in the latter case, meaning that the effective particle travels over two neighboring minima of the potential landscape. The related dissipation makes the electronic temperature rise and the effective critical current drop, resulting in a sharp change in the differential conductance map.

The agreement with experimental data at the same frequency is remarkable at every microwave frequency investigated. Fig. 2 in the main paper displays data and calculation at 6 GHz, while Fig. S2 and Fig. S3 in this supplementary materials display similar information at 8.8 GHz and 24.2 GHz.

Here we also provide the map of the temperature derivative with respect to the current bias, see Fig. S4. This plot highlights the appearance in the temperature map of the Shapiro steps structure.

BEHAVIOR AT HIGHER FREQUENCY

In the higher frequency regime, we observe the appearance of fractional Shapiro steps at voltages V n,m = (n/m) hν/2e, where n and m are integers. This appears as thin zero-differential resistance regions at intermediate positions compared to the integer steps in Fig. S3a, and short steps in Fig. S3b-d Voltage is normalized to hν/2e. (e) Calculated differential resistance map, including thermal effects, see text for details. The related calculated temperature is displayed in (f), its derivative with respect to bias current is in (g).

At higher frequency, the dissipation at a given amplitude I µ is larger. The dominating effect of the microwave irradiation on the electronic population appears in Fig. S3 through the merging of the critical current and the retrapping current above an intermediate level of ac current bias I µ > 0.02. As a consequence, no discontinuity is observed in the differential resistance map. Still, the n = 1 step is missing at low microwave excitation.

I 0 c and I 0 r

 0 the values of the critical/retrapping currents respectively at T → 0 and in the absence of magnetic field and microwave excitations. T * is defined by Ic(T * ) = I 0 r .

FIG. 1 .

 1 FIG. 1. V-I characteristics of device J3 measured at temperatures of (a) 100 mK and (b) 4.2 K. The arrows indicate the sweeping direction of the current bias. (c) Temperature dependence of the critical current Ic (black dots) and the retrapping current Ir (red dots) for sample J3. The green dotted line represents a fit to Eq. (1). Inset: AFM image of a typical Nb-Au-Nb junction. (d) Magnetic-field dependence of the critical current Ic (black dots) and the retrapping current Ir (red dots) (T = 280 mK, device J1). Steps in current values appearing at large magnetic field are artifacts.

FIG. 2 .

 2 FIG. 2. (a) Differential resistance map as a function of the dc current bias I and the microwave excitation amplitude Iµ for a frequency ν = 6 GHz (device J2, T bath = 100 mK). The white arrow indicates the sweeping direction of the dc current bias. The top black arrows point to the individual V-I curves taken at microwave drives of (b) 0.020 (c) 0.035 (d) 0.056 (a.u.). Voltage is normalized to hν/2e. (e) Calculated differential resistance map, including thermal effects.

FIG. 3 .

 3 FIG. 3. (a) Calculated electronic temperature map corresponding to the situation of Fig. 2, i.e. for device J2 at T bath = 100 mK. Inset: Line cut of this map showing the electronic temperature at zero dc bias current as a function of the microwave current and calculated value of the critical current at this electronic temperature. The dashed line indicates the Ic(Te) = Iµ correspondence. (b) Line cuts of the map (a) showing the dependence of the electronic temperature as a function of the dc bias current I at different values of the microwave current Iµ indicated by arrows at the top of the map.

FIG. 4 .

 4 FIG. 4. Differential resistance maps as a function of I and Iµ for ν = 6 GHz in non-hysteretic conditions, (a) at base temperature and under a magnetic field of 100 mT (device J2), and (b) at a bath temperature of about 4.2 K (device J3, no magnetic field). The white lines represent the usual Bessel-function-like amplitude of the step n = 0.

  . Fractional Shapiro steps were already observed in similar devices [S1-S3]. Multiple Andreev Reflections as well as non-thermal out-of-equilibrium energy distribution function can contribute to this phenomena. As they are not included in our model, fractional steps are absent from the calculation results.

FIG. S1 .

 S1 FIG. S1. (a) Zoom of the differential resistance map as a function of I and Iµ at ν = 6 GHz (device J2, T bath = 100 mK) shown in Fig. 2e in the main paper. (b) Schematics showing the extent of the phase excursion at the two bias points. (c) Calculated time dependence of the phase at the bias points indicated by the crosses in (a). The dc current is zero in both cases.

2 FIG.

 2 FIG. S2. (a) Differential resistance map as function of the dc current bias I and the microwave excitation amplitude Iµ for a frequency ν = 8.8 GHz (device J2, T bath = 100 mK). The white arrow indicates the sweeping direction of the dc current bias. The top black arrows point to the individual V-I curves taken at microwave drives of (b) 0.010 (c) 0.043 (d) 0.072 (a.u.).Voltage is normalized to hν/2e. (e) Calculated differential resistance map, including thermal effects, see text for details. The related calculated temperature is displayed in (f), its derivative with respect to bias current is in (g).

FIG. S3 .

 S3 FIG. S3. (a) Differential resistance map as a function of I and Iµ ν = 24.2 GHz (device J2, T = 100 mK). The white arrow indicates the sweeping direction of the dc current bias. The top black arrows point to the individual V-I curves taken at microwave drives of (b) 0.010 (c) 0.028 (d) 0.049 (a.u.). Voltage is normalized to hν/2e. (e) Calculated differential resistance map, including thermal effects, see text for details. The related calculated temperature is displayed in (f), its derivative with respect to bias current is in (g).

  FIG. S4. Calculated map of the temperature derivative with respect to the current bias, including thermal effects, see text for details, corresponding to the case of Fig. 2 of the main paper, i.e. for sample J2 at a bath temperature of 100 mK.

We acknowledge financial support from the ANR contract "Nanoquartets" and the LANEF project "UHV-NEQ". Samples were fabricated at the Nanofab platform at CNRS, Grenoble. We thank A. Nabet and D. van Zanten for help in the experiments, S. Samaddar and J. P. Pekola for discussions.


Hermann Djédjé

Michaël Goujon
email: michael.goujon@udamail.fr

Bertrand Laporte
email: bertrand.laporte@udamail.fr

Samuel Guerineau
email: samuel.guerineau@udamail.fr

Catherine Araujo Bonjean

Mario Mansour

Dora Benedek

Ernesto Crivelli

Sanjeev Gupta

Patrick Guillaumont

Véronique Thelen

Simone Bertoli

Diallo Ibrahima Thierry Yogo

Rasmané Ouedraogo

Keywords: Foreign aid, Taxation, Revenues, Tax data, Nonlinearities, Africa. JEL codes F35, H20, C23, O55

We explore the heterogeneous effects together with the transmission channels of aid on tax revenues in 47 African countries over 1990-2011 using a panel smooth threshold regression model and two alternative tax datasets from IMF and ICTD. We find that aid enhances tax revenues with decreasing returns for a threshold of 6.3% and 23% of GNI for total taxes and non-resource taxes respectively. Aid effect varies across countries and over time, but, on average, is positive. Moreover, we evidence that aid conditions the impact of the level of development, trade, institutions and resource wealth on tax.

Introduction

Financing development remains a great challenge as witnessed at the UN Third International Conference held in Addis Ababa on 13-16 July 2015 where were advocated the use of all sources of finance -public, private, domestic and international -together with the implementation of effective policies. The mobilization of these different sources of finance raises the question of whether they are independent, or complements or substitutes. Recently, the relation between aid and tax in developing countries has particularly received a great attention. The question is of great importance as these countries are highly reliant on foreign aid and short of domestic resources to support the funding of pro poor spending. Sub-Sahara African countries are a very illustrative case, being the most reliant on foreign aid (aid to gdp of 10.8% on average) and, with a tax to GDP ratio of 16%, far to reach the 20% target as suggested by the United Nations to achieve the Millennium Development Goals (UN, 2011). The issue is hence whether aid discourages domestic resources mobilization in the recipient countries. It is obviously a big challenge to give a clear-cut answer. Indeed, aid is directed to governments which use it to finance expenditure which in turn can promote tax compliance and growth. A big concern is how the recipient governments behave in presence of heavy reliance on aid. If all the state policy outcomes depend on foreign assistance, the possibility to be more responsive to donors than to the citizens is likely acute. This may weaken a proper development of internal democratic institutions and leads to a low level of tax compliance and revenues.

No consensus emerges from the existing empirical literature. [START_REF] Benedek | Foreign Aid and Revenue: Still a Crowding Out Effect?[END_REF] and [START_REF] Gupta | Foreign Aid and Revenue Response Does the Composition of Aid Matter?[END_REF] show that aid harms tax effort. [START_REF] Ouattara | Aid, debt and fiscal policies in Senegal[END_REF], [START_REF] Teera | Tax performance: a comparative study[END_REF] find that aid has no effect on tax revenues while [START_REF] Clist | Aid and tax revenue: signs of a positive effect since the 1980s[END_REF], [START_REF] Clist | Foreign Aid and Domestic Taxation: Multiple Sources, One Conclusion[END_REF] and [START_REF] Carter | Does Foreign Aid Displace Domestic Taxation?[END_REF] find some positive effects. Besides the debate on data quality, three main arguments can explain this ambiguity. First, aid composition matters. Aid loans may have positive effects as they are to be repaid unlike grants for which negative effects are expected. But empirical results on aid composition are also mixed [START_REF] Gupta | Foreign Aid and Revenue Response Does the Composition of Aid Matter?[END_REF][START_REF] Benedek | Foreign Aid and Revenue: Still a Crowding Out Effect?[END_REF][START_REF] Clist | Aid and tax revenue: signs of a positive effect since the 1980s[END_REF][START_REF] Clist | Foreign Aid and Domestic Taxation: Multiple Sources, One Conclusion[END_REF]. Second, heterogeneities among countries play a critical role. In particular, countries with sound policies and institutions tend to exhibit a positive link between aid and tax revenues (Brun et al., 2011b,a). However, [START_REF] Yohou | Heterogeneous Aid Effects on Tax Revenues: Accounting for Government Stability in WAEMU Countries[END_REF] show that it would be misleading to not strongly account for the heterogeneities among countries even if they share identical macroeconomic policies.

Third, some authors debate the econometric specification and emphasize that results are sensitive to samples [START_REF] Moss | An Aid-Institutions Paradox? A Review Essay on Aid Dependency and State Building in Sub-Saharan Africa[END_REF][START_REF] Moore | How does taxation affect the quality of governance?[END_REF][START_REF] Carter | Does Foreign Aid Displace Domestic Taxation?[END_REF][START_REF] Clist | Foreign Aid and Domestic Taxation: Multiple Sources, One Conclusion[END_REF][START_REF] Alonso | Does aid hinder tax efforts? More evidence[END_REF]. In fact, most of studies suggest that the aid-tax relationship is nonlinear by introducing a square term of aid. In particular, the echoed works of [START_REF] Benedek | Foreign Aid and Revenue: Still a Crowding Out Effect?[END_REF] and [START_REF] Gupta | Foreign Aid and Revenue Response Does the Composition of Aid Matter?[END_REF] as well as [START_REF] Clist | Aid and tax revenue: signs of a positive effect since the 1980s[END_REF] show a positive sign for the square terms of overall ODA and grants against a negative sign for loans. In other words, overall aid and grants present increasing returns meaning that their negative effects may be offset if their levels are sufficient. By contrast, the positive effect of loans will be offset if they are too important. In more recent papers, [START_REF] Morrissey | Aid and Taxation: Exploring the relationship using new data[END_REF] and [START_REF] Clist | Foreign Aid and Domestic Taxation: Multiple Sources, One Conclusion[END_REF] find exceptional large levels of aid respectively at 69% and 65% of GDP beyond which the effect of aid becomes negative. This paper offers a unique but simple empirical threshold framework that addresses thoroughly the issues of heterogeneities and the form of aid-tax relationship. It brings three major innovations focusing on 47 African countries over the period 1990-2011. First, drawing from the previous studies, we argue that the impact of aid on tax revenues is not homogenous, but depends on the degree of aid reliance.

However, the assessment of the heterogeneous effects here differs significantly from that of existing studies. As noted above existing studies model nonlinearity by a square term or sub samples. By doing so, they explicitly assume that the nonlinear impact of aid is constant through the entire time span and homogenous between countries. This is not helpful to catch for example some effects of a learning-by-doing process in aid schemes management and the efficiency of some particular policies implemented at a certain point of time. We do believe that the impact of aid for the same country can gradually improve, as the aid allocation and management improve. One important example in recent years is the framework created by the mechanism of the HIPC initiative which has enhanced to a certain extent the conditionality related to good governance and has improved the targeting of areas to be primarily funded. Therefore, we reasonably assume that aid effects are not constant over time and across countries. We then depart from the existing literature by using a Panel Smooth Threshold Regression (PSTR) approach to account for such time and individual heterogeneous aid effects.

Second the PSTR is also helpful to address the issue of the usually-evoked but less explored mediating effect of aid through the other tax potential factors. As suggested by [START_REF] Carter | Does Foreign Aid Displace Domestic Taxation?[END_REF] and [START_REF] Morrissey | Aid and Taxation: Exploring the relationship using new data[END_REF], aid has some potential indirect effects through some variables, generally included in aid-tax equation, that are related to tax bases and collection efficiency (income per capita, institutions, imports etc.). This idea is also supported by the aid effectiveness literature. This suggests that the single estimate of aid may hide some part of the real effect of aid on tax revenues.

Third, existing aid-tax studies test the relationship between aid and total tax revenue not exclusive of resource revenues. This is an imperfect approximation since the main question of aid-tax relationship is how aid modulates state-society relationships represented by the state reliance on non-resource tax revenue. We then take advantage of the two advanced and competing tax datasets from International Monetary Fund (IMF) and International Center for Taxation and Development (ICTD). Instead of focus on the dispute about their superiority over each other, we rather hang on their original definitions to highlight the theoretical relationship of interest. While IMF dataset documents the total tax revenue, the dataset from ICTD conversely provides data on total tax excluding resource revenue.

Our main finding is that aid has a country specific and time-varying effect depending on the deviation from a certain threshold level of aid estimated at 6.3% and 23% of GNI with IMF total tax revenue and ICTD non-resource revenues, respectively. Beyond these thresholds, aid has a positive effect with diminishing returns for larger aid. For the entire sample, it results in a positive effect. Considering ICTD non-resource taxes, our results support the evidence that larger aid enhances the positive effects of trade openness and of the level of development but hampers that of institutions while worsening the resource curse. Conversely with IMF total tax revenues, larger aid lowers the positive effect of the level of development while increases those of trade openness and of the manufacture sector.

The remainder of the paper is organized as follow. Section 2 reviews the aid-tax literature. Section 3 describes data and methodology. Section 4 presents the results. Section 5 concludes.

Related literature

The recent studies on aid-tax specifically use a single relationship in which aid is one of the tax determinants beside the degree of openness, the level of development, the quality of institutions, the level of inflation etc. The results are very controversial. [START_REF] Gupta | Foreign Aid and Revenue Response Does the Composition of Aid Matter?[END_REF] and [START_REF] Benedek | Foreign Aid and Revenue: Still a Crowding Out Effect?[END_REF] find that aid lowers tax revenue on the one hand. On the other hand, grants are pervasive whereas loans are efficient because they must be repaid. These findings are strongly challenged by [START_REF] Clist | Foreign Aid and Domestic Taxation: Multiple Sources, One Conclusion[END_REF]Morrissey (2011), Clist (2014) and [START_REF] Morrissey | Aid and Taxation: Exploring the relationship using new data[END_REF] who provide no evidence of a negative impact of neither aid nor grants. They question also the quality of tax data used by the two former works and find that their conclusion are driven by a break due to data compilation from multiple sources. By using different datasets including those of [START_REF] Gupta | Foreign Aid and Revenue Response Does the Composition of Aid Matter?[END_REF] and [START_REF] Benedek | Foreign Aid and Revenue: Still a Crowding Out Effect?[END_REF], they conclude that results of [START_REF] Gupta | Foreign Aid and Revenue Response Does the Composition of Aid Matter?[END_REF] and [START_REF] Benedek | Foreign Aid and Revenue: Still a Crowding Out Effect?[END_REF] are not robust. Using dataset of [START_REF] Mansour | Revenue mobilization in sub-saharan Africa: challenges from Globalization[END_REF], [START_REF] Bhushan | Aid and Taxation: Is Sub-Saharan Africa Different?[END_REF] find that aid does not significantly impact tax in general and in sub-Saharan Africa over the period 1972-2008. One prior concern about the ability of foreign aid to promote tax performances tends to emphasize on its potential harmful effects on the recipient institutions. Taxation is the core of state building since it demands more accountability from state towards its citizen. Foreign assistance can then reduce the dependence of recipient governments on their own citizen [START_REF] Knack | Sovereign rents and quality of tax policy and administration[END_REF]. It results in an increase in the accountability towards the funders rather than towards the citizen and a more probable social contract deterioration. To the extent that institutions matter for tax collection, foreign aid then hinders tax performances. [START_REF] Bräutigam | Foreign Aid, Institutions, and Governance in Sub-Saharan Africa[END_REF] give econometric support that aid damages the quality of governance and the tax collection while [START_REF] Knack | Sovereign rents and quality of tax policy and administration[END_REF] shows that aid reduces governments' incentives to develop efficient tax system. [START_REF] Azam | Aid dependence reconsidered[END_REF] provide theoretical evidence that aid discourages governments to adopt good policies and to develop an efficient system of public revenue and expenditure in the long run.

In spite of these detrimental effects, aid may also contribute to institutional building and tax performances. The aid-disbursement conditionalities on good governance and economic reforms and associated technical assistance in tax administration may be successful. [START_REF] Tavares | Does foreign aid corrupt?[END_REF] finds that aid decreases corruption. [START_REF] Brun | IMF programs and tax effort What role for institutions in Africa?[END_REF] find that IMF programs have a positive impact on the public revenue mobilization through improvements of the technical administrations in charge of the definition and implementation of the tax reforms.

However, it tends to be broadly established that these positive effects emerge when the recipient governments adopt good practices and have clear development agendas [START_REF] Bräutigam | Foreign Aid, Institutions, and Governance in Sub-Saharan Africa[END_REF]. In contrast of the previous studies, the ad hoc assumption is implicitly made that the quality of institutions and policies of the recipient countries are the main driver of aid effectiveness, the reverse causality being controlled by an instrumental variable approach. Hence, [START_REF] Brun | IMF programs and tax effort What role for institutions in Africa?[END_REF] find that IMF programs work better if the quality of domestic administration is good. Especially, they argue that the programs are less effective in sub-Saharan Africa countries because of the poor quality of their institutions. [START_REF] Gupta | Foreign Aid and Revenue Response Does the Composition of Aid Matter?[END_REF] exhibit that decline in revenues completely offsets the increase in grants in countries experiencing high levels of corruption. [START_REF] Benedek | Foreign Aid and Revenue: Still a Crowding Out Effect?[END_REF] come to similar results and show that poor institutional quality worsens the negative effect of aid on tax revenues. Focusing on WAEMU countries through a PSTR approach, [START_REF] Yohou | Heterogeneous Aid Effects on Tax Revenues: Accounting for Government Stability in WAEMU Countries[END_REF] show that aid efficiency requires a minimum level of government stability.

The empirical literature is aware that the inconclusiveness on aid-tax relationship is due to shortcomings in data quality on tax revenues and econometric specification. Even if IMF, OECD and think thanks like ICTD regularly update and correct existing data, primary data quality remains a challenge given low domestic institutional development.

Various improvements of the econometric specification have been suggested. [START_REF] Carter | Does Foreign Aid Displace Domestic Taxation?[END_REF] points that empirical studies should markedly control for the endogeneity of aid and heterogeneities both in aid effects and in countries' tax structures. Some previous studies deal with some of these concerns. [START_REF] Benedek | Foreign Aid and Revenue: Still a Crowding Out Effect?[END_REF] decompose total tax revenues in various components: value added tax (VAT), excise and income tax and trade tax. This does not change their main conclusion that aid has harmful effects on taxation except the positive impact on trade taxes. The results of [START_REF] Morrissey | Aid and Taxation: Exploring the relationship using new data[END_REF] are based on the non resource tax revenues while [START_REF] Bhushan | Aid and Taxation: Is Sub-Saharan Africa Different?[END_REF] also use the tax to GDP ratio excluding both trade and resource taxes. Control for endogeneity has been done in many ways with contrasting results. Brun et al. (2011a) and [START_REF] Brun | IMF programs and tax effort What role for institutions in Africa?[END_REF] use an instrument based on [START_REF] Tavares | Does foreign aid corrupt?[END_REF], and find a positive effect, while [START_REF] Benedek | Foreign Aid and Revenue: Still a Crowding Out Effect?[END_REF] concludes to a negative effect out of a GMM estimation. Using the initial income and population and lagged values of aid as instruments in 2SLS estimations also produces a negative effect in [START_REF] Gupta | Foreign Aid and Revenue Response Does the Composition of Aid Matter?[END_REF] and [START_REF] Bräutigam | Foreign Aid, Institutions, and Governance in Sub-Saharan Africa[END_REF] while [START_REF] Bhushan | Aid and Taxation: Is Sub-Saharan Africa Different?[END_REF] do not find any statistically effect. The issue of endogeneity remains quite unsolved as this is the case in the broader aid effectiveness literature, as detailed in [START_REF] Guillaumont | Aid Effectiveness for Poverty Reduction: Lessons from Cross-country Analyses, with a Special Focus on Vulnerable Countries[END_REF].

Regarding heterogeneities, countries present diverse experiences in terms of supporting tax policy and tax administration reforms which can affect aid-tax relationship [START_REF] Benedek | Foreign Aid and Revenue: Still a Crowding Out Effect?[END_REF]Prichard et al., 2013a). Three strategies are actually employed to control heterogeneity in aid-tax relationship. The first one is single country study using time series. This strategy yields also mixed results. [START_REF] Franco-Rodriguez | Aid and the public sector in Pakistan: evidence with endogenous aid[END_REF] find a negative relationship between aid and tax effort for Pakistan. [START_REF] Franco-Rodriguez | Recent developments in fiscal response with an application to Costa Rica[END_REF] and [START_REF] Osei | The fiscal effects of aid in Ghana[END_REF] shows a positive effect of aid on tax in Costa Rica and Ghana respectively. This approach is limited by the low number of observations and the difficulty to make a comparison among countries which is essential in aid effectiveness literature. The second and most used strategy is panel data to account for both the individual and temporal dimensions. The third is PSTR. We proceed with a formalization of the ways heterogeneity is taken into account in the literature, to show the advantage of the PSTR approach following [START_REF] Destais | Economic development and energy intensity: a panel data analysis[END_REF].

Let us consider a simple cross-section model of the relationship between aid and taxation like that used by [START_REF] Bräutigam | Foreign Aid, Institutions, and Governance in Sub-Saharan Africa[END_REF] and [START_REF] Knack | Sovereign rents and quality of tax policy and administration[END_REF]:

tax i = α + βaid i + X i + ϵ i (1) i = 1, ...N .
where tax i is the taxation index for the i th country of the sample of N countries for the average period studied and aid the aid measure. X i are control variables, the other determinants of tax. α and β are constants for the period and ϵ i is i.i.d.(0, σ 2 ϵ ). This simple model exhibits that the (long term) impact of aid is the same for all the countries and given by β. This strong assumption can be partially circumvented by introducing dummy variables and pooling countries by classes. Another drawback of cross-section model is that the number of observations is limited. Panel data analysis, by contrast, makes use of larger samples and reduces the problem of multicollinearity among the explanatory variables and the effects of missing or unobserved variables [START_REF] Hsiao | Analysis of panel data[END_REF].

A simple form of panel specification is:

tax it = α + βaid it + X it + ϵ it (2)
where ϵ it is i.i.d.(0, σ 2 ϵ ); t denotes time period with t = 1, ...T . However, this model does not still allow for heterogeneity. Heterogeneity of the constant α can be introduced through country fixed effect.

tax it = α i + βaid it + X it + ϵ it (3)
The individual effects α i capture the influence of country-specific time-invariant variables like tax system administration capacity or structural characteristics. However, this allows heterogeneity of the average level of tax only.

These panel specifications can then be improved by including dummy variables for subgroups of countries sharing some specific economic characteristics or institutional arrangements. [START_REF] Gupta | Foreign Aid and Revenue Response Does the Composition of Aid Matter?[END_REF] add a dummy for oil exporters. [START_REF] Benedek | Foreign Aid and Revenue: Still a Crowding Out Effect?[END_REF] and [START_REF] Bhushan | Aid and Taxation: Is Sub-Saharan Africa Different?[END_REF] include a full set of time dummies. [START_REF] Brun | IMF programs and tax effort What role for institutions in Africa?[END_REF] interact their variable of interest (IMF program) with a dummy for sub-Saharan African countries.

A more general accounting for heterogeneity is to split the sample into subgroups. As robustness tests, regressions are then run on these sub-samples to catch whether the effects differ from those of the whole of sample and across the sub-samples. The guiding criteria are either the membership to a specific region, or the countries are situated above or under a certain cutoff of a specific economic or institutional variable or time period. [START_REF] Gupta | Foreign Aid and Revenue Response Does the Composition of Aid Matter?[END_REF] split their sample by quartiles according to ICRG corruption index. Their results suggest that grants have a stronger negative effect on tax effort in the most corrupt countries. Adopting the same approach, [START_REF] Benedek | Foreign Aid and Revenue: Still a Crowding Out Effect?[END_REF] distinguish the type of aid, and country groups by income and regions. For instance, the resulting estimations show a negative relationship for total aid and ODA grants in Africa and Asia and Pacific. Loans have significant negative effects only in Africa.

Restricting their sample over the period 1985-2005[START_REF] Clist | Aid and tax revenue: signs of a positive effect since the 1980s[END_REF] take into account a structural break in the aid-tax relationship, giving grants a statistically significant and positive effect. [START_REF] Morrissey | Aid and Taxation: Exploring the relationship using new data[END_REF] find evidence that grants have a positive impact in the cases of Sub-Saharan Africa and Latin America and Caribbean. Second, splitting the sample into two sub-samples, 1980-1990 and 1990-2010 they come to the same conclusion of [START_REF] Clist | Aid and tax revenue: signs of a positive effect since the 1980s[END_REF].

The sub-sample approach with panel data provides more insight on the heterogeneity in aid effects on tax revenues. Nevertheless, the previous critics against the use of dummy variables are also valid for the sub-sample technique. It estimates an average effect of aid for a particular sub-group. Moreover,the choice of the cutoff is often discretionary and arbitrary. It does not allow a country to move from a group to another. It implies that the distribution of each group is the same relative to the chosen threshold variable. This strong assumption may result in a meaningless estimate of aid impact if, for instance, the degree of political commitment in reforms differs across countries of the same group and over time. In addition, [START_REF] Hansen | Threshold effects in non-dynamic panels: Estimation, testing, and inference[END_REF] shows that the ad hoc sample splitting may pose serious inference problems.

Another approach to account for the heterogeneous effect of aid is to allow for nonlinearity in the form of increasing or decreasing returns. Determining potential aid thresholds from which aid effect shifts has important policy implications. Indeed, if aid has a positive effect on taxation but with decreasing returns above a certain cutoff of aid, then the absorptive capacity of aid is limited. The resulting straight policy implication is to not exceed this cutoff. Adversely, a negative effect with increasing returns means that countries need a big push in aid to reverse the effect. The opposition between absorption capacity and big push is at the core of the debate on aid-development relationship. It is usual to check such nonlinearities by including aid and its square term 10 in the regressions as follows:

tax it = α + βaid it + ρaid 2 it + X it + ϵ it (4)
For instance, in [START_REF] Gupta | Foreign Aid and Revenue Response Does the Composition of Aid Matter?[END_REF], and in [START_REF] Benedek | Foreign Aid and Revenue: Still a Crowding Out Effect?[END_REF], the square term coefficient is positive for total aid and grants, but rarely significant for loans. By contrast, Clist (2014) find a positive effect with decreasing returns for all types of aid (total, grants and loans). [START_REF] Morrissey | Aid and Taxation: Exploring the relationship using new data[END_REF] find decreasing returns for grants and total aid against positive ones for loans. However, both studies find that the scope of decreasing returns is modest since the net effect of aid becomes negative only at exceptional turning points of 69% of GDP [START_REF] Morrissey | Aid and Taxation: Exploring the relationship using new data[END_REF] and 65% of GDP [START_REF] Clist | Foreign Aid and Domestic Taxation: Multiple Sources, One Conclusion[END_REF].

Moreover, the use of the quadratic specification has a number of caveats, since the nonlinearity shape is arbitrary and do not allow modeling multiple turning points.

Lastly, the aid-tax literature is inconclusive also because aid could affect both directly and indirectly tax revenues via several potential factors. As we state above policies associated with aid may potentially affect tax rates, bases and tax collection. According to [START_REF] Morrissey | Aid and Taxation: Exploring the relationship using new data[END_REF], right-hand side variables in tax equation that aim at proxying tax base like agriculture and industry shares, GDP, imports and exports cannot adequately account for these policy effects. [START_REF] Morrissey | Aid and Taxation: Exploring the relationship using new data[END_REF] and [START_REF] Carter | Does Foreign Aid Displace Domestic Taxation?[END_REF] show how it is hard to make distinction between behavioral, conditionality/policy and technical assistance effects. For instance, the liberalization reforms, supported by aid conditionality, have induced tax reductions but also a growth revival which may potentially generate tax revenue increase.

The empirical results on these potential channels are mixed. A typical illustration is that of the level of development, proxied by GDP or GDP per capita. Economic growth, human capital development and infrastructures are engines of development and they are likely supported by aid through budget support. However, impacts of aid on these factors are likely to be nonlinear and heterogeneous across countries [START_REF] Elbadawi | External aid: help or hindrance to export orientation in Africa?[END_REF][START_REF] Fielding | Aid and Dutch Disease in Sub-Saharan Africa[END_REF][START_REF] Dalgaard | On aid, growth and good policies[END_REF][START_REF] Hadjimichael | Sub-saharan Africa Growth, savings, and investment[END_REF][START_REF] Kourtellos | Is the relationship between aid and economic growth nonlinear?[END_REF][START_REF] Kalyvitis | Are Aid Flows Excessive or Insufficient? Estimating the Growth Impact of Aid in Threshold Regressions[END_REF][START_REF] Guillaumont | Accounting for vulnerability of African countries in performance based aid allocation[END_REF]. This obviously cannot be captured by the estimate of a single average value of the aid coefficient [START_REF] Gomanee | Aid and growth in Sub-Saharan Africa: accounting for transmission mechanisms[END_REF].

In short, as stated by [START_REF] Carter | Does Foreign Aid Displace Domestic Taxation?[END_REF] simple panel specifications cannot capture nonlinearities and heterogeneities in direct and indirect impacts of aid on tax. Thus, while Carter (2013) uses a Pooled Mean Group (PMG)estimators that allows for hetero-geneities in the short run responses and intercepts but imposes a common long run relationship, we here adopt a more general approach with the PSTR estimator on a reduced-form model.

Data and specification

Model specification

We use a Panel Smooth Transition Regression (PSTR) specification that was recently developed by [START_REF] González | Panel smooth transition regression models[END_REF] and [START_REF] Fok | A multi-level panel STAR model for US manufacturing sectors[END_REF]. It is a generalization of the Panel Threshold Regression (PTR hereafter) approach of [START_REF] Hansen | Threshold effects in non-dynamic panels: Estimation, testing, and inference[END_REF] in which the coefficients of a reduced-form model are function of the value of a transition variable. A number of recent papers have used this approach to examine how the individual coefficients of an interest variable vary smoothly over time according to a threshold variable. On aid-tax issue, only [START_REF] Yohou | Heterogeneous Aid Effects on Tax Revenues: Accounting for Government Stability in WAEMU Countries[END_REF] have used PSTR. Considering a basic case of two regimes and a single transition function, the PSTR model takes the following form:

tax it = µ i + β 0 x it + β 1 x it g(A it ; γ; A) + u it (5)
The vector x it includes the aid variable A and the other traditional factors. β 0 and β 1 are two vectors of parameters to be estimated. They capture the direct impact of x on tax and the indirect, conditional on aid, non-linear, impact of x on tax, respectively. µ i denotes the fixed individual effects while u it the idiosyncratic error and g is the transition function.

The transition function is a continuous function of the threshold variable given by the following logistic function [START_REF] Granger | Modelling Non-Linear Economic Relationships[END_REF][START_REF] González | Panel smooth transition regression models[END_REF]:

g(A it ; γ; A) =   1 + exp(-γ m ∏ ȷ=1 (A it -A ȷ))   -1 (6)
where A (A 1 , . . . , A m ) ′ is an m-dimensional vector of location (threshold) parameters and the estimated term γ measures the slope of the transition function or the sharpness of the regime shift. The effect of x on tax revenue for country i at time t results from its direct and aid-conditional effects. It is given by:

∂tax it ∂x it = β 0 + β 1 g(A it ; γ; A); g ∈ [0, 1] (7)
However, the expression of aid effect is slightly different from that of the equation 7 since the threshold variable q it is a function of aid. It equals to:

∂tax it ∂A it = β 0 + β 1 g(A it ; γ; A) + β 1 ∂g(A it ; γ; A) ∂A it A it (8)
As equations 7 and 8 show, the size of coefficients β 0 and β 1 cannot be directly interpreted at the first stage of the estimation only their respective signs make sense because there are as many values of each estimates between β 0 and β 0 + β 1 as the number of country-year observations. However, when the value of the slope γ equals zero, the transition function reduces to a constant. The model becomes a standard linear model with individual effects, i.e. constant and homogeneous coefficients. If γ tends towards infinity, the transition function becomes an indicator function and the PSTR model in (1) reduces to the two-regime PTR model of [START_REF] Hansen | Threshold effects in non-dynamic panels: Estimation, testing, and inference[END_REF] in the case where m = 1, for instance. When m ≻ 1 and γ tends to infinity, the number of regimes remains two but the function switches between zero and one [START_REF] Colletaz | Threshold effects of the public capital productivity: an international panel smooth transition approach[END_REF].

Estimation and tests of specification

The estimation procedure consists in eliminating the individual effects µ i by removing country-specific means and applying nonlinear least squares to the transformed model. [START_REF] González | Panel smooth transition regression models[END_REF] propose the following procedure: i) testing the linearity against the PSTR model (or testing homogeneity against the PSTR alternative), ii) determining the number, r, of transition functions, that means the number of extreme regimes which is equal to r + 1. The test of homogeneity in PSTR model can be done by testing: H 0 : γ = 0 or H 0 : β 1 = 0. However under the null hypothesis, the tests are non-standard since the PSTR model contains unidentified nuisance parameters. This identification problem is circumvented by replacing g(aid it ; γ; aid) by its firstorder Taylor expansion around γ = 0 and to test an equivalent hypothesis based on the auxiliary regression:

tax it = µ i + β * 0 x it + β * 1 x it A it + ... + β * m x it A m it + u m it (9)
The linearity test against PSTR is then equivalent to testing

H * o : β * 1 = ... = β * m = 0.
The corresponding F-statistic is then defined by:

LM F = (SSR 0 -SSR 1 )/mk SSR 0 /(T N -N -mk) ∼ F (mk, T N -N -m(k + 1)) (10)
where SSR 0 is the panel sum of squared residuals under H 0 , and SSR 1 , the panel sum of squared residuals with regimes. k is the number of explanatory variables, T is the number of years and N the number of countries. The test of homogeneity is also used to determine sequentially the number of transitions. The null hypothesis that the model is linear at a predetermined significance level α is tested. If it is rejected, a two-regime PSTR model is estimated. If the two-regime at turn is rejected a threeregime is estimated. The testing procedure continues until the first acceptance of the null hypothesis of no remaining heterogeneity. At each step of the sequential procedure, the significance level must be reduced by a constant factor 0 ≺ τ ≺ 1 in order to avoid excessively large models.

There remains a crucial question about the issue of endogeneity caused by a simultaneity or double-causality since low tax level countries may then benefit from higher aid. The use of instrumental variables in aid-tax studies provides contrasting results according to the instruments used as argued by [START_REF] Clist | Aid and tax revenue: signs of a positive effect since the 1980s[END_REF]. We avoid such sensitivity to instruments by not using any instrumentation for a couple of reasons. First, [START_REF] Fouquau | The Feldstein-Horioka puzzle: a panel smooth transition regression approach[END_REF] and [START_REF] Béreau | Currency misalignments and growth: a new look using nonlinear panel data methods[END_REF] argue that a PSTR limits the potential endogeneity issue since there is a specific value of the parameter for each level of threshold variable. For comparison purpose, they use the generalized method moments (GMM) estimation with interaction terms and a PSTR corrected for endogeneity and a "standard" PSTR. They find that the results derived from the corrected PSTR are close to those of non corrected PSTR and both are superior to the former. Second and most importantly, [START_REF] Yu | Inconsistency of 2SLS estimators in threshold regression with endogeneity[END_REF] and [START_REF] Yu | Threshold Regression with Endogeneity[END_REF] demonstrate that in threshold regressions, both the threshold point and the threshold effect do not need any instrumentation in order to be identified. The need for extra randomness in the regressors is averted by the discontinuity structure of the cutoff itself which supplies identifying information for the parametric coefficients. In particular, the former author shows that 2SLS estimators in threshold models with endogeneity are inconsistent. Obviously, it would be imprudent to generalize this conclusion to PSTR model even if the latter and the instrumental variable threshold models are close to the pioneer model of [START_REF] Hansen | Threshold effects in non-dynamic panels: Estimation, testing, and inference[END_REF]. Consequently, we lag aid (as suggest [START_REF] Clist | Aid and tax revenue: signs of a positive effect since the 1980s[END_REF]) and all the variables. The reference paper of [START_REF] González | Panel smooth transition regression models[END_REF] adopts a similar approach.

Data and the model variables

Most existing studies use either total government revenues or total tax revenues as the left-hand side variable. However, the real concern of the political economy literature is how aid does affect the state-citizen-relationship. Then, reliance on non-resource tax revenues is rather the most appropriate measure [START_REF] Prichard | Taxation, Non-Tax Revenue and Democracy: New Evidence Using New Cross-Country Data[END_REF]. In addition, there is no theoretical argument that aid leads a government to cut natural resource revenues.

We use two tax effort measures from the two advanced competing datasets from the fiscal affairs department of International Monetary Fund kindly provided by [START_REF] Benedek | Foreign Aid and Revenue: Still a Crowding Out Effect?[END_REF] (IMF henceforth) and from the newly government revenue dataset developed by [START_REF] Prichard | The ICTD Government Revenue Dataset[END_REF] through the International Center for Taxation and Development (ICTD henceforth). ICTD dataset builders claim that they have greatly improved data in terms of coverage and accuracy in comparison with the dataset used by [START_REF] Benedek | Foreign Aid and Revenue: Still a Crowding Out Effect?[END_REF] especially. However, the updated IMF tax dataset has also benefited from critical improvements taking advantage of the updated work of [START_REF] Mansour | A Tax Revenue Dataset for Sub-Saharan Africa: 1980-2010[END_REF]. Moreover, tax data definitions differ between the two datasets. IMF tax variable is the ratio of total tax revenues excluding social contributions and grants as percentage of GDP while ICTD tax variable additionally excludes non tax revenues (mainly composed of resource revenues). To our knowledge, only [START_REF] Morrissey | Aid and Taxation: Exploring the relationship using new data[END_REF] make use of non-resource government revenues data.

Regarding the right hand side variables, we follow the standard empirical literature on aid-tax issue [START_REF] Gupta | Determinants of tax revenue efforts in developing countries[END_REF][START_REF] Gupta | Foreign Aid and Revenue Response Does the Composition of Aid Matter?[END_REF][START_REF] Teera | Tax performance: a comparative study[END_REF][START_REF] Morrissey | Aid and Taxation: Exploring the relationship using new data[END_REF][START_REF] Yohou | Heterogeneous Aid Effects on Tax Revenues: Accounting for Government Stability in WAEMU Countries[END_REF], and we select the most prominent mediating factors. Hence the set of variables includes the quality of institutions, the level of development, the sectoral composition of the economy, the trade openness and the resource rent wealth.

Strong institutions are beneficial to tax collection and implementation of tax reforms in a sustained effort. Good institutions can create lower and bearable tax rates while expanding the bases through growth notably, improving the collection efficiency. On the contrary, in a weak institutions country, where the level of corruption is high for instance, the trust in government may collapse leading to lower incentives to pay taxes. The noxious effects of rent-seeking on tax performances are well established. Aid conditionality including some obligations to run specific tax reforms or to improve governance may then potentially strengthen the positive effects of the level of institutional quality on tax capacity. The institutional quality variable is taken from [START_REF] Kunčič | Institutional quality dataset[END_REF] who has summarized more than 30 institutional quality indexes around three major components including legal, political and economic institutions. We use the average of the three sub-indices. We then avoid an arbitrary choice of a specific institutional quality indicator as [START_REF] Gupta | Foreign Aid and Revenue Response Does the Composition of Aid Matter?[END_REF], [START_REF] Benedek | Foreign Aid and Revenue: Still a Crowding Out Effect?[END_REF] and [START_REF] Brun | IMF programs and tax effort What role for institutions in Africa?[END_REF] have done.

GNI per capita is a proxy of the level of development of a country. Higher income countries has a more monetized economy and efficient tax administration [START_REF] Agbeyegbe | Trade liberalization, exchange rate changes, and tax revenue in Sub-Saharan Africa[END_REF]. A higher level of development is also positively associated with a higher relative demand for income elastic public goods and services [START_REF] Bird | Tax effort in developing countries and high income countries: The impact of corruption, voice and accountability[END_REF] and so with a higher capacity and compliance to pay taxes. Even if this is always debated, we assume that aid can promote growth and, given that it supports many administration tax reforms and assistance, that aid can affect the impact of the national income on tax.

The sectoral composition of the economy is usually captured by two variables: the share of value added of industry and/or value added of agriculture in GDP. Given that the statistical distribution of the industry ratio is mostly influenced by the distribution of mining and oil in the African sample, and the high correlation of this ratio with other control variables, we prefer to use the share of manufacturing excluding mining and oil in the total value added and the share of agricultural value added in the total value added. Data are from The United Nations Statistics Division (UNSD) of the Department of Economic and Social Affairs (DESA) (UNDATA henceforth). It is usually assumed that the agriculture or rural sector is more difficult to tax than the industrial (or urban) sector. A higher agriculture (industrial) share is then negatively (positively) correlated with tax. Besides the controversies on absorptive capacity and the effects on private investment, aid may sustain the domestic effort in expanding those sectors without creating additional government revenues. Simply, government may substitute aid for taxes raised from these sectors in order to encourage their development. This is not bad per se since it may satisfy the final objective of aid. However it is of concern when it is a systematic response to lobbies claiming tax exemptions notably when the government expects a higher support from donors. For these reasons, we assume that larger aid may hinder the contribution of sectoral composition to tax effort.

Trade openness, measured by the ratio of the sum of imports and exports to GDP, is expected to be positively related to tax, as trade taxes would be easier to collect than other taxes.This enhancing effect might somewhat crumble with the liberalization and fiscal transition reforms accompanying aid. [START_REF] Agbeyegbe | Trade liberalization, exchange rate changes, and tax revenue in Sub-Saharan Africa[END_REF] have provided an interesting explanation on this issue. Trade liberalization has involved the replacement of quantitative barriers with import duties which would have generated higher trade tax revenues depending on the level of duties and impact on imports value. Subsequent fiscal transition reforms have reduced import duties and trade tax revenues in favor of VAT and other domestic taxes. The net effect of aid on tax through liberalization and reforms, trade openness, tax and customs administration, is then uncertain.

Natural resource wealth is alternatively captured by the ratio of non-tax revenue to GDP and the sum of oil and gas rents ratios to GDP. A strand of the resource curse literature gives a similar role to aid and natural resources. The resource curse literature suggests that resource windfalls impede democracy and thus deteriorate the social contract leading to tax collection, lower the incentives to build state capacity for collecting taxes and providing services [START_REF] Knack | Sovereign rents and quality of tax policy and administration[END_REF]. One can expect that aid and technical assistance, and associated tax reforms for instance, may influence the effect of natural resources on tax, for a given level of natural resources.

Regarding the aid variable, we select the usual ratio of aid-to-gross national income. It refers to the total net official disbursement flows (net of repayment of principal) according to the Development Assistance Committee (DAC) definition of official development assistance. Data on aid, GNI per capita, oil-gas rents and trade variables are from the World Development Indicators (WDI) of the World Bank. Constrained by data availability, our study covers an unbalanced panel of 47 out of the 54 African countries over the period 1990-2011. 1

Descriptive statistics

Table 6 in appendix reports the main descriptive statistics. The average tax level in Africa is relatively low in comparison with the other developing countries and OECD countries. It stands at about 18 % and 13% of GDP for the total tax revenues (Tax-IMF) and the non-resource tax revenues (Tax-ICTD) respectively. High values of standard deviations indicate a huge variability in our sample. Meanwhile, aid reliance is high standing at more than 10% of GNI. As for tax revenues, the standard deviations of aid variables also indicate important disparities across countries. Aid-tax relationship also differs greatly at a first glance. Some countries show low aid reliance and poor nonresource tax revenues (Algeria, Angola) but high total tax performances. Benin and Central African Republic have almost the same aid reliance (about 11% of GNI) but exhibit contrasting total tax performances (about 14% and 8% of GDP respectively).

Figure 1 plots the simple relationship between aid and tax revenue or change in tax revenue according to the degree of aid reliance for both tax effort measures. Roughly imposed here, high (low) aid reliance countries are those where aid-to-GNI ratio is higher (lower) than the sample average. The figure shows that high reliance is negatively associated with tax-to-GDP level, but positively with tax-to-GDP growth, signaling 1 Algeria, Angola, Benin, Botswana, Burkina Faso, Burundi, Cabo Verde, Cameroon, Central African Republic, Chad, Comoros, Congo Democratic Republic, Congo Republic, Cote d'Ivoire, Egypt Arab Republic, Equatorial Guinea, Ethiopia, Gabon, Gambia, The Ghana, Guinea, Guinea-Bissau, Kenya, Lesotho, Madagascar, Malawi, Mali, Mauritania, Mauritius, Morocco, Mozambique, Namibia, Niger, Nigeria, Rwanda, Senegal, Seychelles, Sierra Leone, South Africa, Sudan, Swaziland, Tanzania, Togo, Tunisia, Uganda, Zambia, Zimbabwe. Excluding countries are mostly conflict countries: Djibouti, Eritrea, Liberia, Libya, Sao Tome and Principe, Somalia and South Sudan. the complexity of the relationship. This may suggest that initially aid-reliant countries are not necessarily the least performing in improving tax collection through time.

Consequently, accounting for time-varying relationship between aid and tax may be a requisite.

Pairwise correlations from table 7 show that total tax revenue (IMF) is positively correlated with its usual determinants, except with aid and agriculture. Yet, the nonresource tax variable (ICTD) is negatively associated with natural resource wealth. Aid tends to be associated with the other tax determinants, except agriculture import. The high correlation between some tax determinants signals a potential multicollinearity bias.

Tax maximizing overall aid rate

We start with simple PSTR regressions before testing for aid-conditional channels of the impact of aid on tax. Details of these exploratory results are not reported to save space and are available on request. First, a simple PSTR regression of tax on aid only evidences a non-linearity but estimated coefficients are insignificant throughout the regimes, suggesting the need to include additional control variables. Second, we add control variables (openness, GNI per capita, manufacture, agriculture, institutions, resources) in a simple way, with no interaction with aid. Results still suggest a very marginally significant non-linearity. Third, we gradually enter aid variable and interaction between aid and controls. In some cases, we do not find any non-linearity.

Table 1 then reports the conclusive results of the fourth step, where we add interaction between aid and each control, alternatively (maintaining the others as simple controls), for both ICTD and IMF measures of tax. We here exclude single aid variable to explore only indirect conditional to aid controlling for the effects of the other tax factors. Roughly, we find that increasing aid improves the effect of trade openness and of GNI per capita on tax, but worsens the effect of resource wealth. Generally, the other controls have the expected coefficient signs. Openness, GNI per capita and institutions affect positively and significantly tax effort while agriculture and manufacture effects are statistically insignificant. As expected, resource wealth increases IMF-total taxes and lowers ICTD-non-resource taxes.

We now use the general PSTR specification presented above where all right-hand side variable impacts are conditional on aid thresholds. We used various specifications, log-log, semi-log and so on, to check the sensitivity of our results to specification and/or outliers. The results are similar to those reported in Tables 2 to 4.

For both tax variables, 5 to 6 specifications are proposed to check the robustness of the results. The first specification is the most standard specification without resource wealth as control. Following [START_REF] Treviño | Resource Dependence and Fiscal Effort in Sub-Saharan Africa[END_REF], the second specification includes resource wealth. The third and fourth specifications address [START_REF] Clist | Aid and tax revenue: signs of a positive effect since the 1980s[END_REF] suggestion to disentangling and replacing trade by import and export openness, respectively (alternatively to avoid multicollinearity). We do not report estimates for IMF-tax revenue with exports that have no economic sense. The following specification excludes the "single" aid variable from regressions to keep sight on potential shifts on the other determinants effects if the direct aid effect is not accounted for. The last one deals with the issue of potential collinearity posed by the agriculture variable. .849 (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) .589 (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) Table 2 reports the results for linearity and no remaining nonlinearity tests with one location parameter (m = 1) because the tests of the existence of two thresholds are rejected. 2 Whatever the tax effort measure and the specification considered, the three tests strongly reject the null hypothesis of linearity at the level of 1% between tax revenues and aid. It then questions the accuracy of most of the recent aid-tax empirical findings that do not show such a relationship, especially for Africa, like those of [START_REF] Bhushan | Aid and Taxation: Is Sub-Saharan Africa Different?[END_REF], [START_REF] Benedek | Foreign Aid and Revenue: Still a Crowding Out Effect?[END_REF], [START_REF] Clist | Foreign Aid and Domestic Taxation: Multiple Sources, One Conclusion[END_REF] and [START_REF] Clist | Aid and tax revenue: signs of a positive effect since the 1980s[END_REF]. Moreover, tests do not reject the null hypothesis of one transition function against two for all specifications and tax effort variables. They indicate that the nonlinearity or cross-country heterogeneity and time variability of aid-tax revenues relationship are properly featured by a small number of extreme regimes.

H 1 : r = 1 versus H 1 : r = 2 1.
H 1 : r = 1 versus H 1 : r = 2
Figure 2 displays the graphs of the transition functions for both tax revenues from the equation 5. Except for the IMF-model 2, the relative low values of the slope parameters show that the transition function is suitably featured by a PSTR structure. However, results from ICTD-models record the lowest values of the slope parameters suggesting that aid effects are more gradual for non-resource tax revenues compared to total tax revenues. From IMF-models, the slope parameter of the transition function ranks between 1.5 and 2.6 except in the second specification which exhibits the exceptional value of 1192.1. From the ICTD models, the lowest and the highest slope values are 0.20 and 1.49 respectively. Tables 3 and4 report the estimates of the final PSTR models for IMF and ICTD tax variables (IMF-models and ICTD-models), respectively. As noted above, at this step only the signs of the estimated parameters are interpretable, not their values. For each specification, β 0 measures the direct effect on tax of the corresponding variable whereas the aid-conditional effect is measured by β 1 . A positive (negative) value of β 1 indicates an increase (a decrease) in the aid or the channel impact with the value of the threshold variable.

Our results point three major lessons. First, the nonlinearity between aid and tax collection cannot be reduced to a limited number of regimes with different estimates. Second, they confirm that the approaches consisting in grouping countries, using subsamples or modeling the nonlinearity by a quadratic term of aid are unsatisfactory. They heavily underestimate the pattern of the heterogeneities across countries and over time. Third, the nonlinearity between aid and tax must account for the transmission channels through the effect of aid on the relationship between tax and its other determinants.

The estimated threshold stands at about 6.3% of GNI for the total tax revenue against 23% for non-resource tax revenue. For both tax effort measures, the aid coefficient β 0 is significantly positive while β 1 is significantly negative excepted for ICTDmodels 2, 3 and 6. This implies that the positive effect of aid decreases when the flows of aid increase with respect to the estimated threshold. Interestingly, the turning point in the aid-non-resource tax relationship is relatively high compared to that of the aidtotal tax relationship (as table 6 shows), suggesting that only a substantial high level of aid higher than the sample average of the aid-to-GNI ratio can dampen the social bargaining between state and citizen.

However, irrespective of the tax effort measure, our results do not support the evidence that the impact of aid on tax shifts at exceptionally high and implausible levels of aid as found by [START_REF] Clist | Foreign Aid and Domestic Taxation: Multiple Sources, One Conclusion[END_REF] and [START_REF] Morrissey | Aid and Taxation: Exploring the relationship using new data[END_REF].

Regarding the indirect effects, they vary depending on the channel considered and to some extent on the tax measure. Meanwhile the signs associated with β 0 for each factor are in line with the literature and our preliminary results. Trade openness has a statistically positive effect on both tax measures at 1 or 5% level. Its aid-conditional effect exhibits a positive and significant sign across the specifications. Similar results emerge from specifications with import openness but not with export openness that does not show a robust effect. Our results confirm those of [START_REF] Morrissey | Aid and Taxation: Exploring the relationship using new data[END_REF] and support the assumption of "aid absorption" mechanism as argued by [START_REF] Aiyar | Where Did All the Aid Go? An Empirical Analysis of Absorption and Spending[END_REF] and [START_REF] Carter | Does Foreign Aid Displace Domestic Taxation?[END_REF]. Aid causes import directly when donors and recipients pay for imports and indirectly when the dollars are sold by the recipient central bank to domestic importers.

Results on agriculture are less robust. While not significant in the IMF-models, a significant positive effect is found in only two ICTD-models. By contrast, manufacture shows robust effect with IMF-models. Its effect is negative but higher aid induces a positive effect. This little evidence of the sectoral effect may result from a combination of domestic policies that grants a lot of tax exemptions and aid support toward the private sector. If the former policies tend to inhibit the contribution of the private sector to tax, the second by encouraging its development allows the private sector to be compliant with taxes.

We find that higher GNI per capita enhances significantly tax performances irrespective of the tax measure. However, the aid-conditional effect of GNI per capita is positive on non-resource tax but negative on total tax. Institutional quality also affects positively tax revenues. However, there is no evidence of any effect of the institutional quality in upper aid regimes when considering total tax revenues. By contrast, increase in aid tends to lower institution effect on non-resource tax. A higher resource wealth is associated with higher total tax revenues and this effect is not affected with larger aid flows. Conversely, higher resource wealth harms non-resource tax significantly and this negative effect is also significantly worsened when aid increases in respect with the estimated threshold. Hence our results indicate that the arguments of [START_REF] Bräutigam | Foreign Aid, Institutions, and Governance in Sub-Saharan Africa[END_REF], [START_REF] Knack | Sovereign rents and quality of tax policy and administration[END_REF] and [START_REF] Azam | Aid dependence reconsidered[END_REF] that larger aid and/or resource rents are noxious to institutions in promoting taxation hold only when one considers the non-resource tax revenues. Our results are also in line with the findings of [START_REF] Treviño | Resource Dependence and Fiscal Effort in Sub-Saharan Africa[END_REF] that exhibit a negative effect of resource wealth on taxation for Sub-Saharan Africa.

So far the PSTR helps to highlight the heterogeneous aid effects across countries and over time according to aid deviation from its estimated threshold. We now turn to an exploration of country-specific results where the actual impact of aid is determined by the actual location of the country, given by the actual aid receipts, which can change over time, relatively to aid threshold.

Country-specific elasticities

Figure 3 displays GNI per capita, openness, institutions and resources elasticities of tax that are functions of the level of aid-to-GNI from IMF-model 5 and ICTD-model 3 using equation 7. Table 5 reports for each country the average over the period of these estimated elasticities. 3For the entire group and the entire period, the aid-elasticity of tax, given the observed levels of aid, is positive, equaling 0.33 for IMF-total tax and 0.03 for ICTDnon-resource tax. At least 74% of the countries of our sample has registered a positive effect of aid on domestic revenue mobilization. This result is in line with some existing findings especially [START_REF] Gupta | Determinants of tax revenue efforts in developing countries[END_REF] and [START_REF] Clist | Foreign Aid and Domestic Taxation: Multiple Sources, One Conclusion[END_REF] that evidence a small positive effect. However, 12 and 7 out of the 47 countries exhibit a negative elasticity with IMF-model and ICTD model respectively. Source: Authors' estimations However these averaged elasticities hide changes over time as Figure 4 illustrates for 6 countries. The negative aid-elasticities in Tanzania, Rwanda, Guinea-Bissau are mainly due to the past inefficiencies that recent progress can not curb. The aid-elasticity for Morocco tends to be constant, that for Cabo Verde is positive since 1999 while that for Sierra Leone is very erratic. Similar analysis with the other factor-elasticities of tax are derived from equation 7. Except institutions and resource wealth in IMF-model which exhibit homogenous elasticity across countries when β 1 statistically equals 0, Table 5 highlights heterogeneous effects. On average, an increase of 1% of GNI per capita is associated with a 0.13% or a 0.03% increase in tax in IMF and ICTD-model respectively.

A 1-point higher quality of institutions is associated with a 0.23 point and 0.07 point increase in tax in IMF-total tax and ICTD-non-resource tax respectively. While resource wealth-elasticity of IMF-total tax is homogenous and positive across countries, the resource-wealth-elasticity of ICTD-non-resource tax is heterogeneous, depends on aid levels compared to thresholds, but is negative for every countries, with a sample average of -0.15 (a 1% increase in resource-wealth lowers non-resource tax by 0.15%). In sum, in African countries, the levels of aid are such that they do not hinder direct effects of aid and transmission channels effects to tax, except in the case of resource wealth to non-resource tax. We do not report here our preliminary results based on the grants-loans decomposition of aid as this is not the primary focus of this paper, but they do not alter significantly our main conclusions and are available on request.

Conclusion

We examine the effects of foreign aid on tax performances for 47 African countries over 1990-2011 using two tax revenue measures: total tax revenue from IMF and nonresource tax revenues from ICTD. As originally discussed, the ICTD non-resource tax measure would better portray the State-citizens contract and then the State bargaining trade-off between tax and aid.

We assume that the effects of aid on tax vary across countries and over time and that they depend on aid levels. We also explore the impact of aid on the effects of other usual determinants on tax.

Using Panel Smooth Threshold Regressions, our results show a huge heterogeneity in aid effects across African countries and over time. Aid has an enhancing effect but with diminishing returns when its level becomes increasingly high relative to the estimated threshold of aid. The estimated turning points are 6.3% and 23% for total taxes and non-resource taxes respectively. Computing the country-specific aid-elasticities of tax, we find that, on average and in the majority of African countries, aid has a positive effect on domestic tax mobilization. We however find more reasonable results in line with the theoretical expectations with ICTD-non-resource tax variable, suggesting that the choice of the tax measure is quite important. For instance, larger aid beyond specific thresholds strengthens the positive effect of the level of development and trade openness (including import) and induces a positive effect of agriculture although marginally. Meanwhile, larger aid is reducing the institutions and worsening the resource curse effects on tax. However, the estimated thresholds are high compared to the actual flows so that aid has not substantially shifted the first impact (in the first regime) of each factor.

In short, our results indicate that the negative effect found in the literature is overestimated and does not match with the case of African countries when one strongly accounts for country and time-varying heterogeneities as well as transmission channel. Providing also the best interpretation of the relationship of interest is also helpful to suggest appropriate policy recommendations.

Figure 1 :

 1 Figure 1: Aid reliance and tax performances over 1990-2011

Figure2

  Figure 2: Transition functions

Figure 3 :

 3 Figure 3: Factor elasticities of tax

Figure 4 :

 4 Figure 4: Time-varying country-specific aid-elasticities of ICTD-non-resource tax, selected countries

Table 1 :

 1 Aid, tax factors and Tax effort: Individual variable conditional effect

	Channel	Openness	GNI per capita	Resource revenue
	Model	Tax-ICTD	Tax-IMF	Tax-ICTD	Tax-IMF	Tax-ICTD	Tax-IMF
	T hreshold (ODA/GN I)	0.7303***	17.2082***	0.8393***	13.8132***	6.9195***	-
	Slope parameter (γ)	1605.52*** 95.5630***	0.4996***	47.0784*** 3128.3407***	-
	β 0 : Channel	-0.0438***	0.0049	0.7636	4.6099***	-0.1636***	-
		(0.0154)	(0.0084)	(0.6564)	(0.8043)	(0.0306)
	β 1 : Channel*g(A it ; γ; A)	0.0537***	0.0111**	1.5695***	0.4780***	-0.0398*	-
		(0.0149)	(0.0056)	(0.3616)	(0.1797)	(0.02133)	-
	Control variables:
	Openness			0.0077*	0.0156**	0.0089**	-
				(0.0041)	(0.00627)	(0.0041)	-
	GNI per capita (log)	2.3130***	5.0332***			2.5847***	-
		(0.5422)	(0.7886)			(0.5554)	-
	Manufacture	-0.0752	-0.0469	-0.0662	-0.0378	-0.0860*	-
		(0.0475)	(0.0679)	(0.0475)	(0.0680)	(0.0477)	-
	Agriculture	0.0354	0.0205	0.0324	0.0226	0.0361	-
		(0.0297)	(0.0447)	(0.0296)	(0.04466)	(0.0299)	-
	Institutions (log)	2.4268***	2.9304***	2.3932***	2.8588***	2.4865***	-
		(0.4471)	(0.6551)	(0.4459)	(0.6550)	(0.4490)	-
	Resource taxes	-0.1976***		-0.2212***			-
		(0.0250)		(0.0255)
	Oil and Gas		0.1624***		0.1582***		-
			(0.0251)		(0.0250)
	AIC criterion	1.8469	2.5484	1.8407	2.5449	1.8572	-
	Schwartz criterion	1.8938	2.5962	1.8876	2.5927	1.9041	-
	Number of observations	926	905	926	905	926	-

Notes: Independent variables are lagged. * significant at 10%, **significant at 5%, *** significant at 1%. Standard deviation in parentheses.

Table 2 :

 2 Tests for linearity and remaining nonlinearity

	Variable			Tax-IMF					Tax-ICTD
	Model	1	2	3	4	5	1	2	3	4	5	6
	Number of location parameters	1	1	1	1	1	1	1	1	1	1	1
	Wald test
	H 0 : r = 0 versus H 1 : r = 1	31.861 36.175 35.288 35.317 35.965	84.360 58.538	55.048	62.129	68.008	58.722
		(0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000) (0.000)	(0.000) (0.000) (0.000)
	H 1 : r = 1 versus H 1 : r = 2	9.978	11.569 11.615	7.510	9.030	5.128	3.000	3.301	3.744	6.579	2.100
		(0.126) (0.116) (0.114) (0.185) (0.172) (0.527) (0.885) (0.856)	(0.879) (0.474) ( 0.910)
	Fischer test
	H 0 : r = 0 versus H 1 : r = 1	5.191	5.068	4.939	6.936	5.884	14.561	8.405	7.873	7.828	9.867	9

Table 3 :

 3 PSTR aid-IMF total taxes (% of GDP) in African countries All independent variables are lagged. *, **, *** significant at 10%, 5% and 1% respectively. Mauritania is excluded because of a critical level of missing data. The dependent variable is the log of the ratio of total tax revenue to GDP. Standard deviation in parentheses.

		IMF1	IMF2	IMF3	IMF4	IMF5
	Threshold (ODA/GNI)	7.1857	6.3433	6.317	6.5244	6.3467
	Slope	1.7665	1192.1	2.4144	1.5119	2.6448
		β0	β1	β0	β1	β0	β1	β0	β1	β0	β1
	Aid	0.0096**	-0.0090*	0.0146***	-0.0140***	0.0112**	-0.0106**			0.0122**	-0.0116**
		(0.0047)	(0.0046)	(0.0049)	(0.0049)	( 0.0051)	(0.0051)			(0.0048	(0.0047)
	Openness	0.0006***	0.0004	0.0007***	0.0006**			0.0008***	0.0006*	0.0008***	0.0006*
		(0.0002)	(0.0003)	(0.0003)	(0.0003)			( 0.0003)	(0.0003)	(0.0003	(0.0003)
	Import					0.0009**	0.0002
						(0.0004)	(0.0004)
	GNI per capita (log)	0.2064***	-0.0870***	0.1774***	-0.0742**	0.1614***	-0.0504	0.1492***	-0.0509	0.1682***	-0.0623*
			( 0.0357)	(0.0311)	(0.0351)	(0.0316)	(0.0360)	(0.0295)	( 0.0363)	(0.0307	(0.0360)
	Manufacture	-0.0191***	0.0164***	-0.0185***	0.0159***	-0.0195*** 0.0170*** -0.0207*** 0.0181*** -0.0195 ***	0.0169***
		(0.0037)	(0.0036)	(0.0037)	(0.0035)	( 0.0039)	(0.0038)	( 0.0039)	( 0.0039)	( 0.0039	(0.0038)
	Agriculture	-0.0013	-0.0007	0.0009	0.0001	0.0004	-0.0001
		(0.0016)	(0.0012)	(0.0755)	( 0.0666)	(0.0017)	(0.0012)
	(log)	0.2121***	0.0681	0.2171***	0.0355	0.2530***	0.0009	0.2674***	-0.0469	0.2337***	0.0047
		(0.0768)	(0.0701)	(0.0017)	(0.0011)	(0.0764)	(0.0683)	(0.0743)	( 0.0627)	(0.0758	( 0.0669)
	Oil and Gas			0.0035***	0.0004	0.0036***	0.0012	0.0030***	0.0010	0.0032***	0.0008
				(0.0008)	(0.0008)	(0.0008)	(0.0009)	(0.0008)	(0.0008)	(0.0008)	(0.0008)
	AIC	-4.687	-4.780	-4.756	-4.776	-4.779
	BIC	-4.616	-4.700	-4.675	-4.716	-4.708
	Observations	905	905	905		905		905
	Note:

Table 4 :

 4 PSTR aid-ICTD non-resource taxes (% of GDP) in African countries

		ICTD1	ICTD2	ICTD3	ICTD4	ICTD5	ICTD6
	Threshold (ODA/GNI)	24.6485	23.6777	23.0085	23.4754	26.2719	23.782
	Slope	0.2895	0.8452	1.4941	0.9626	0.2066	0.7767
		β0	β1	β0	β1	β0	β1	β0	β1	β0	β1	β0	β1
	Aid	0.0549*	-0.0466	0.0491**	-0.0405	0.0494**	-0.0534**	0.0509**	-0.0465*			0.0484**	-0.0343
		( 0.0294 )	(0.0284)	( 0.0235)	(0.0259)	(0.0236)	(0.0266)	( 0.0236)	( 0.0260)			(0.0235)	( 0.0253)
	Openness	0.0094**	0.0812***	0.0114***	0.0657***							0.0118***	0.0622***
		(0.0038)	(0.0097)	( 0.0035)	(0.0075)							(0.0035)	(0.0074)
	Import					0.0130***	0.0759***	0.0096**	0.0723***	0.0055	0.0655***
						(0.0032)	(0.0086)	(0.0041)	(0.0097)	(0.0044)	(0.0112)
	Export							0.0174	0.0416	0.0174	0.0815*
								(0.0166)	( 0.0273)	(0.0174)	(0.0441)
	per capita (log)	1.8806***	2.2019	2.8707***	3.7619***	2.8653***	3.4410***	2.8672***	3.5630***	2.3731***	6.5873***	2.9659***	4.4805***
		( 0.4630)	(1.5872)	( 0.5163)	(1.0818)	(0.5019)	(0.9912)	(0.5025)	(1.0628)	(0.5004)	(1.6654)	(0.5231)	(0.9792)
	Manufacture	-0.0972	-0.0026	-0.0935	-0.0653	-0.0888	-0.0206	-0.0951	-0.0510	-0.1046*	-0.0630	-0.0901	-0.0787
			(0.0822)	( 0.0627)	(0.0604)	(0.0627)	(0.0549)	(0.0602)	(0.0603)	(0.0603)	(0.0785)	(0.0632)	(0.0576)
	Agriculture	-0.0104	0.1253***	-0.0156	0.0402	-0.0213	0.0410*	-0.0131	0.0402	-0.0196	0.0060
			(0.0432)	(0.0209)	(0.0262)	(0.0210)	(0.0247)	(0.0220)	(0.0257)	(0.0228)	(0.0428)
	Institutions (log)	7.3385*** -9.7585***	7.1767***	-8.5804***	7.2763***	-7.6487***	7.0225***	-8.1543***	7.5883***	-12.3667***	7.1376***	-8.6366***
			(1.9303)	(1.5781)	(1.4928)	(1.5917)	(1.3531)	(1.5706)	(1.4932)	(1.6452)	(2.1368)	(1.5716)	(1.5082)
	Resource revenues			-0.1454*** -0.2575*** -0.1392***	-0.1063*	-0.1510***	-0.2043**	-0.1475***	-0.3712***	-0.1449*** -0.3161***
				(0.0247)	(0.0649)	(0.0254)	(0.0551)	(0.0315)	(0.0817)	(0.0325)	(0.1098)	(0.0247)	(0.0591)
	AIC	1.815	1.776	1.781	1.781	1.779	1.771
	BIC	1.885	1.855	1.86		1.87	1.858	1.841
	Observations		926		926	926		926		926		926
	Note: All independent variables are lagged. *, **, *** significant at 10%, 5% and 1% respectively. Standard deviation in parentheses.

Table 5 :

 5 Country-specific average factor-elasticities of tax over 1990-2011 The elasticities are calculated according to the formula of the equation 7, except for aid (equation 8) and results from IMF-model 5 and ICTD-model 3 regressions. Elasticities of institutions and resource wealth with IMF are invariant since the associated β 1 are insignificant.

		Aid	Openness	GNI pc	Institutions	Resource wealth
	Country	IMF	ICTD	IMF	ICTD	IMF	ICTD	IMF	ICTD	IMF	ICTD
	Algeria	1.2200	0.0494	0.0800 0.0130 0.1682 0.0287 0.2337 0.0728	0.3200	-0.1392
	Angola	0.3069	0.0273	0.1000 0.0147 0.1474 0.0294 0.2337 0.0710	0.3200	-0.1416
	Benin	0.0100	0.0494	0.1399 0.0130 0.1060 0.0287 0.2337 0.0728	0.3200	-0.1392
	Botswana	1.0234	0.0494	0.0822 0.0130 0.1659 0.0287 0.2337 0.0728	0.3200	-0.1392
	Burkina Faso	-0.0775	0.0236	0.1395 0.0149 0.1064 0.0295 0.2337 0.0709	0.3200	-0.1418
	Burundi	-0.0640 -0.0474 0.1378 0.0596 0.1082 0.0498 0.2337 0.0258	0.3200	-0.2044
	Cabo verde	0.0600	0.0193	0.1400 0.0304 0.1059 0.0365 0.2337 0.0552	0.3200	-0.1636
	Cameroon	0.1856	0.0494	0.0937 0.0130 0.1540 0.0287 0.2337 0.0728	0.3200	-0.1392
	Central African Republic	-0.2556	0.0483	0.1346 0.0131 0.1115 0.0287 0.2337 0.0727	0.3200	-0.1393
	Chad	-0.5582	0.0492	0.1305 0.0130 0.1158 0.0287 0.2337 0.0728	0.3200	-0.1392
	Comoros	-0.1868	0.0425	0.1375 0.0165 0.1085 0.0302 0.2337 0.0692	0.3200	-0.1441
	Congo, Dem. Rep.	0.5121	0.0276	0.1105 0.0176 0.1366 0.0307 0.2337 0.0681	0.3200	-0.1457
	Congo, Rep.	0.4674	0.0282	0.1055 0.0226 0.1417 0.0330 0.2337 0.0631	0.3200	-0.1527
	Cote d'Ivoire	0.2122	0.0451	0.1076 0.0132 0.1395 0.0287 0.2337 0.0726	0.3200	-0.1395
	Egypt, Arab Rep.	0.9684	0.0494	0.0885 0.0130 0.1594 0.0287 0.2337 0.0728	0.3200	-0.1392
	Equatorial Guinea	0.6587	0.0255	0.0998 0.0307 0.1476 0.0367 0.2337 0.0549	0.3200	-0.1640
	Ethiopia	-0.1606	0.0490	0.1390 0.0130 0.1070 0.0287 0.2337 0.0727	0.3200	-0.1392
	Gabon	1.2044	0.0494	0.0801 0.0130 0.1681 0.0287 0.2337 0.0728	0.3200	-0.1392
	Gambia, The	-0.0738	0.0470	0.1262 0.0165 0.1202 0.0302 0.2337 0.0693	0.3200	-0.1440
	Ghana	-0.2058	0.0494	0.1256 0.0130 0.1208 0.0287 0.2337 0.0728	0.3200	-0.1392
	Guinea	-0.3444	0.0494	0.1263 0.0130 0.1201 0.0287 0.2337 0.0728	0.3200	-0.1392
	Guinea-Bissau	0.0600	-0.0019 0.1400 0.0521 0.1059 0.0464 0.2337 0.0334	0.3200	-0.1939
	Kenya	0.5758	0.0494	0.0997 0.0130 0.1477 0.0287 0.2337 0.0728	0.3200	-0.1392
	Lesotho	-0.2010	0.0494	0.1151 0.0130 0.1318 0.0287 0.2337 0.0728	0.3200	-0.1392
	Madagascar	0.0954	0.0330	0.1322 0.0194 0.1140 0.0315 0.2337 0.0664	0.3200	-0.1481
	Malawi	0.0600	-0.0557 0.1400 0.0505 0.1059 0.0457 0.2337 0.0349	0.3200	-0.1918
	Mali	0.0600	0.0268	0.1400 0.0173 0.1059 0.0306 0.2337 0.0684	0.3200	-0.1452
	Mauritania		0.0092		0.0241		0.0337		0.0616		-0.1548
	Mauritius	1.2198	0.0494	0.0800 0.0130 0.1682 0.0287 0.2337 0.0728	0.3200	-0.1392
	Morocco	1.0802	0.0494	0.0805 0.0130 0.1677 0.0287 0.2337 0.0728	0.3200	-0.1392
	Mozambique	0.0600	-0.0865 0.1400 0.0627 0.1059 0.0512 0.2337 0.0227	0.3200	-0.2088
	Namibia	0.8069	0.0494	0.0832 0.0130 0.1649 0.0287 0.2337 0.0728	0.3200	-0.1392
	Niger	0.0575	0.0317	0.1400 0.0163 0.1059 0.0302 0.2337 0.0694	0.3200	-0.1439
	Nigeria	0.8926	0.0494	0.0845 0.0130 0.1635 0.0287 0.2337 0.0728	0.3200	-0.1392
	Rwanda	0.0600	-0.0105 0.1400 0.0280 0.1059 0.0355 0.2337 0.0576	0.3200	-0.1603
	Senegal	-0.2932	0.0494	0.1388 0.0130 0.1072 0.0287 0.2337 0.0728	0.3200	-0.1392
	Seychelles	0.6605	0.0494	0.0876 0.0130 0.1603 0.0287 0.2337 0.0728	0.3200	-0.1392
	Sierra Leone	0.0600	0.0010	0.1400 0.0449 0.1059 0.0431 0.2337 0.0407	0.3200	-0.1838
	South Africa	1.2200	0.0494	0.0800 0.0130 0.1682 0.0287 0.2337 0.0728	0.3200	-0.1392
	Sudan	0.5363	0.0494	0.0921 0.0130 0.1556 0.0287 0.2337 0.0728	0.3200	-0.1392
	Swaziland	1.2063	0.0494	0.0801 0.0130 0.1681 0.0287 0.2337 0.0728	0.3200	-0.1392
	Tanzania	0.0599	-0.0097 0.1400 0.0235 0.1059 0.0334 0.2337 0.0622	0.3200	-0.1539
	Togo	0.4104	0.0494	0.1157 0.0130 0.1311 0.0287 0.2337 0.0728	0.3200	-0.1392
	Tunisia	1.2198	0.0494	0.0800 0.0130 0.1682 0.0287 0.2337 0.0728	0.3200	-0.1392
	Uganda	0.0598	0.0443	0.1400 0.0165 0.1059 0.0302 0.2337 0.0692	0.3200	-0.1441
	Zambia	-0.1818 -0.0295 0.1360 0.0345 0.1100 0.0384 0.2337 0.0510	0.3200	-0.1694
	Zimbabwe	0.3491	0.0494	0.1066 0.0130 0.1406 0.0287 0.2337 0.0728	0.3200	-0.1392
	Mean	0.3269	0.0303	0.1147 0.0202 0.1322 0.0319 0.2337 0.0655	0.3200	-0.1494
	Notes:

Études et Documents n° 8, CERDI, 2016

As we use log of tax revenue for IMF-models, the coefficients of the independent variables have been multiplied by 100, except GNI per capita and institution which are already in log to ease comparison between the size of the coefficients. For ICTD-models, the coefficients GNI per capita and institution have been divided by 100.

Hermann Djédjé Yohou

This work was supported by the LABEX IDGM+ (ANR-10-LABX-14-01) within the program "Investissements d'Avenir" operated by the French National Research Agency (ANR). http://www.cerdi.org/

A Appendix