F P Mechel

Helmholtz Resonators with Added Porous Absorbers

Keywords:

Helmholtz resonators often are combined with porous ma terials in order to increase the bandwidth of their reso

nances. A previous article [START_REF] Mechel | About Perforated Metal Sheets on Porous Absorbers[END_REF] has treated perforated met al sheet covers on porous absorber layers. The results from this can be applied for nonnal sound incidence directly to Helmholtz resonators in which absorber materials fill up the resonator volume totally, and in the case where there is an air gap between the resonator plate and the absorb er material in the resonator chamber. Thus, with the por ous material behind the resonator plate, it only remains to discuss the situation when a thin absorber layer is in immediate contact with the resonator plate, an arrange ment which is often applied. The performance of the com bination of Helmholtz resonators with absorber layers be comes significantly different if the absorber layer is in front of the resonator plate, in particular the high-frequen cy characteristics are changed. Exact field solutions are derived for resonator arrangements with slit-shaped ori fices (the extension to circular orifices can be made by methods indicated). The influence of the absorber ar rangement on the wall admittance of a grid of resonators is discussed. and numerical examples are shown. It is demonstrated how the maximum sound attenuation in ducts lined with the described absorber arrangements can be achieved.

Helmholtz-Resonatoren mit porosen Absorbern

Zusammenfassung Helmholtz Resonatoren werden oft mit porosen Materialien kombiniert, um die Bandbreite ihrer Resonanzen zu ver groBern. In einem friiheren Artikel [START_REF] Mechel | About Perforated Metal Sheets on Porous Absorbers[END_REF] wurde die Kombina tion von Lochblech-Abdeckungen mit Absorberschichten be handelt. Die Ergebnisse von dort konnen fiir senkrechten Schalleinfall direkt auf Helmholtz Resonatoren tibertragen werden, bei welchen das Absorbermaterial entweder das Re sonatorvol umen vollstandig ausfiillt oder ein Luftspalt zwi schen der Resonatorplatte und der Absorberschicht in der Re sonatorkammer angeordnet ist. So bleibt bei einer Absorber-

Introduction

Helmholtz resonators are often combined in technical applications with porous absorber materials in order to Prof. F. P. Mechel, Landhausstr. 12, D-71120 Grafenau, Germany.

schicht hinter der Resonatorplatte nur noch der Fall zu be handeln, daB eine dtinne Absorberschicht unmittelbaren Kon takt mit der Resonatorplatte hat, eine vielfach verwendete Anordnung. Das Verhalten einer Kombination aus Absor berschicht und Helmholtz Resonator wird vollig anders, wenn sich die Absorberschicht auf der Frontseite der Reso natorplatte befindet, insbesondere werden die hochfrequen ten Eigenschaften verandert. Exakte FeldlOsungen werden hergeleitet fiir das Beispiel von Helmholtz Resonatoren mit schlitzformigen Offnungen (die Obertragung auf Resonato ren mit runden Offnungen wird skizziert). Der EinfluB der Absorberanordnung auf die Wandimpedanz eines Resona torgitters wird diskutiert und numerische Beispiele werden vorgefiihrt. Es wird gezeigt, wie man in Kanalen, welche mit den beschriebenen Absorberanordnungen ausgekleidet sind, die maximale Schalldampfung erreichen kann.

Resonateurs de Helmholtz a absorbants poreux ajoutes

Sommaire

On ajoute souvent des materiaux poreux aux resonateurs de Helmholtz, dans le but d'accroitre la bande passante autour de leurs resonances. Un precedent article [START_REF] Mechel | About Perforated Metal Sheets on Porous Absorbers[END_REF] a traite le cas de plaques de metal perforees recouvrant des couches d' absor bants poreux. Les resultats de cette etude s' appliquent direc tement, dans l'hypothese de !'incidence normale, aux reso nateurs de Helmholtz completement emplis de materiaux ab sorbants, et dans le cas ou ii y a une couche d'air entre la plaque et I' absorbant contenu dans le resonateur. II ne reste done plus qu'a examiner la situation ou, en plus du materiau poreux place derriere la plaque du resonateur, on a dispose une fine couche absorbante au contact immectiat de la plaque, ce qui est un arrangement d'usage frequent. Les performances d'un resonateur ainsi equipe sont nettement differentes, sur tout en haute frequence. On etablit des solutions exactes du champ pour des resonateurs a orifices en forme de fente, et I' on indique Jes methodes qui permettent I' extension au cas des orifices circulaires. On discute de !'influence de ce type d'absorbeurs sur !'admittance de paroi d'un reseau de reso nateurs, et I' on fournit des exemples numeriques. On en de duit la fa9on d 'obtenir un maximum d' attenuation du son dans des tuyaux tapisses de tels absorbeurs.

increase the bandwidth of the resonator and to improve the stability of the resonator against flow-induced whis tling if the resonators are applied in flow ducts. Helm holtz resonators are locally reactive because of the rigid side walls of their resonator chambers. So they can quite easily be introduced into design computations for attenu ation in silencers or for absorption and scattering from absorbing walls if their input admittance is known ana lytically. It is the main purpose of this contribution to give such analytical descriptions for the combination of Helmholtz resonators with porous layers.

In a previous paper, [START_REF] Mechel | About Perforated Metal Sheets on Porous Absorbers[END_REF], the analysis of perforated cover sheets on porous absorber layers was described. As a result of the assumption of normal sound incidence ap plied there, those methods and results can immediately be applied to Helmholtz resonators with absorber material in the resonator volume. Two situations were discussed previously: first, the absorber material totally fills up the resonator volume; second, there is an air gap between the back side of the resonator plate and the absorber layer which itself extends to the rigid termination of the res onator chamber. Although the ranges of variables and parameters in the numerical examples described there mainly were selected for perforated covers, they also ap ply to Helmholtz resonators. The principal results of that paper were, that the inner end correction of the resonator neck becomes complex and describes a flow resistance if the absorber material is in contact with the resonator plate, and that small air gaps between the resonator plate and the absorber material modify the end correction to wards the usual inner end correction of empty resonator chambers. The situations treated in the former paper are depicted in Fig. 1, graphs a) and b).

An interesting and important question arises from the described results. It is common practice to increase the losses of Helmholtz resonators by applying thin porous layers, such as wire mesh or gauze, to one side of the resonator plate, and it is common in the literature to take these layers into account in the computations just by adding their flow resistance to the impedance of the neck, without any modification of the end correction taken from the empty resonator. This seems to be in contradic tion with the results mentioned above, therefore the ar rangement of Fig. 1,c) will be studied here.

Helmholtz resonators with rigid resonator plates show useful values of their wall admittance only in the range of their resonances. Generally they are useful as absorbers for low frequencies only. In fact, at high frequencies the mass reactance in the necks becomes so large that the total arrangement tends towards the behaviour of a rigid wall. If one is interested in a combination of low-frequen cy resonant absorption with a high-frequency broad band absorption, then an arrangement such as that shown in Fig. 1,d) may be a solution.

In this paper we shall apply the conventions for sym bols and methods used in the former article. Thus we can refer to that paper instead of repeating common equa tions. As previously a time fa ctor e jwr is assumed which gives a positive sign to the reactive component of the admittance if it is resilient, and a negative sign if it is mass-like.

Absorber layer behind the necks

The system considered first is that of Fig. 1,c). The parameters and the coordinates are shown in Fig. 2. A plane sound wave is incident on an arrangement of Helmholtz resonators with slit-shaped orifices of width a, with a separation distance L, in a resonator plate of thickness d in front of a resonator volume of depth t. A porous absorber layer of thickness s is arranged behind the resonator plate. The material of that layer is charac terized by its porosity (J a ' characteristic propagation con stant I:i and wave impedance Z3, which in normalized forms are represented by I:i n = I:i/ k0, Z an = Z3/ Z0, where k0 and Z0 are the free-field wave number and wave impedance of air respectively. The material characteris tics can be computed either from regressions through measured data or, preferably, from model theories which avoid physically unreal results for extreme values of the absorber variable E = Qo f/ 5, (here Qo is the density of air, f the frequency, and 5 the (measured) flow resistivity). In many applications, the thickness s is small not only against the wavelength ).0, but also compared to all other dimensions of the resonator when the porous layer is a wire mesh, a thin felt or glass fibre cloth, and then it is better to characterize it by its flow resistance R (normal ized to Z0). We shall treat this case by the transitions--+ 0 in such a way that the value of R = 5 s/Z0 is kept con stant.

In Fig. 2 we have subdivided the sound field into zones I to IV. Assuming an amplitude A e of the incident plane wave, we constitute the sound pressure in the zones as a superposition of higher modes from which the axial par ticle velocity vx is derived.

Zone I:

P1 (x, y)

= A e e -j kox + L bn A n e1"xCOS ( 1JnY ), n=O Z ( ) _ Aj ko x Zone III:

• " S: Y n A YnX ( ) (1
Pm (x,y) = L bn(Dne-E "x + E n e+E"x)cos (17 n y), n=O Z o Vm x (x,y)

Zone IV:

(3)

P1v (x, y) = L bJn cosh ( Y n (x -d -t)) •cos ('1 n y), (4

) n=O

The factor J n = 1 for n = 0, and J n = 2 for n > 0. The summations run from m, n = 0 to m, n = oo. The wave numbers and propagation constants are:

(5)

The limit mg = INT (2aj).0) (with l-0 the free-field wave length) separates cut-on modes in the slit (m �mg) from cut-off modes (m >mg). These (exact) formulations satisfy the boundary condition of zero normal particle velocity at the rigid surfaces of the resonator and in the planes of symmetry.

The sets of unknown amplitudes A n; Bm, C m; Dn, E n; Fn are determined by the boundary conditions for the sound pressures p and for the axial particle velocities vx at the interfaces between the zones. In principle it is an extrac tion of Fourier components from a spatial Fourier series by which the above field representations can be inter preted. In the course of this procedure integrals will ap pear of the type:

1 +a/2 sm,n := -s cos (2m1t y/a)cos (11 ny)dy a -a/ 2 _ _ � [ sin (m 7t -n 7t a/L) + sin (m 7t + n 7t a/L) J . 2 m1t -n1ta/L m1t + nrra/L ' m, n>O -( -1 r na/L . , 2 sin (nrra/L); m, n > O, m =F na/L 7t m--(n a/L) sin (n 7t a/L) m = 0, n � 0 n rr a/L = O ; m > 0, n == 0 =1; m=n=O = .!.• 2 ' m = na/L =FO , (6)
for which we introduce the abbreviation s m , n with the special values: S0.0 = 1; Sm.o = 0 and So .n =(sin (n 1t a/L))/ (n rr a/L).

Matching the particle velocities at the interfaces be tween the zones will give the equations:

k z c � _!!_ ( D e -E:"( d + s J _ E e + t:n( d + s > ) J;.Z ak o n n = -j l!: Fnsinh (y n (t -s)) . k o (7) (9)
Matching the sound pressure delivers: 1 with the coefficients:

-(Bm+C m )=b o ,m A e+ L bnSm,n A n ,
(18) a Y. k b 1 + e-2 jxm d a _ • _ _!!_ ""6....!!. s s + __.!!!_:!!. m,n -J L k _ L... i m,i n,i s: - 1 -----2 -j x_ m _ d ' 0 1 = 0 }\ u m e ( 19
)
After the solution of this system for x n -' Y,, _' the quan tities B m , C m are computed from eq. ( 17), and then the other amplitudes from eq. ( 7), ( 8) and (9). Thus the sound fields in the zones are known.

The volume of computation is vastly reduced by the approximate assumption that only plane waves exist in the resonator necks. Then the system described in eq. ( 18

The final quantity sought is the input impedance Z or, preferably, the input admittance G = 1/Z of the absorber arrangement. These are p/v-impedances or v/p-admit tances averaged over the necks and rigid sections of the resonator plate. We name them planar impedance or ad mittance in distinction to the slit impedance Zs or admit-tance Gs = 1/Zs. If the resonator repeat length Li s small compared to the wave length A.0, then the simple relation G = a/L • Gs holds.

We start with the slit impedance Zsh at the exit plane of the neck. Then the impedance Zsv in the entrance plane of the neck is obtained by a simple impedance transfor mation using the neck as a transformer line. The rear orifice impedance z s h is defined by, and can be computed from:

Zsh <Pu (d, y))a Yo + Z0 Z0 <vnx(d,y))a Y0_ e -j k o d x 1 + e-2j k o d = 2 _Q_=._ ----- 1 _ e-2j k o d y0_ 1 _ e-2 jk0 d '
with the result:

(23)

The first term in the brackets (together with Z an in front of the brackets) is just the planar input impedance of the absorber layer, reduced to the slit impedance by the fac tor a/ L and transformed into the exit plane of the neck. Thus the second term in the brackets (with the factor in front of the brackets) is the impedance of the oscillating mass in the rear orifice plane:

The first term in the brackets of (24) can be written:

1 +r0 e -2 r a s 1 +jZ an tanh (I'ans)tan (k o (t -s))

1-r0 e-2 r a s tanh (I'ans) + jZ an tan (k0(t-s)) (26) Now the transition s -+ 0 mentioned above can be per formed in two ways. In the first way we let Gan' Zan be constant. The fractions in (24) then become: . " (si n (i rr a/L)) 2 k0 h ( ) ]

1 + r 0 e -2 r a s I j ---_ ---= 2 -= r - s = --cot (k0t),
+ 2J £..... -cot Yi t . i>O i rr a/L Yi
This is just the rear orifice impedance for an empty resonator chamber. Thus the absorber layer "vanishes" by this kind of transition: s -+ 0.

In the second way of making the transition we let R = S s/Z0 be constant. Then, remembering that the in put impedance of an absorber layer, with a pressure re lease termination for low frequencies and/or thin layers, is just the flow resistance of the layer, Z a • tanh (I'a s) � S s, we may treat R �Zan Fas as being constant, and allow Fas -+ 0 in the arguments of the exponential and of the trigonometric functions. This gives: and:

1 + r 0 e -2 r a s I 1 -r 0 e -2 r a s s -0 1 + j R tan (k0 t) j Zan tan (k0 t) (29) 1 -2 e• s 1 1 I k ( 30
)
+r i e

•.
+ri . 0ssi

h( )

= -- =J--COt y.t. 1 -r i e -2e , s s -o 1 -r i s -o R }' i i
The result of an insertion into (24) is for a thin absorber layer behind the neck plate:

z sh a [ . k -= -R -J cot ( 0 t) Z0 L (31) (si n (i rr a/ L)) 2 k0 J + 2j L: . -coth (y i t) . i>O zrr a/L Yi
The interpretation of this is very simple. First: the flow resistance R is just added to the planar input impedance of the resonator volume (first two terms in the brackets); second: the reactance of the oscillating mass, i.e. the inner end correction, remains that of the empty chamber. A numerical study of the transition s -+ 0 shows that the relevant condition is s �a. For s smaller than about a/2 one can apply eq. ( 31) as an approximation with the inner end correction for empty resonator volumes. For s >a one should apply the more complicated eq. ( 25) with the inner end correction modified by the absorber layer. The resonance remains unchanged if R is purely resistive, but thin layers may also be excited into oscillation by the frictional forces. In most cases we can take this into con sideration by applying an effective resistance R eff which is composed of the pure resistance R = S s/Z0, and the surface mass density m = Q m s of the layer according to:

1 jw m•Ss R eff = - z . � 0Jcvm+ .::::,s j k0 s • Qm/Q0 • R jk0s • Qm/Q0 + R ( 32
)
For k0 s • Qm/Q o � R (the layer becomes effectively rigid by its mass rectance) we obtain R alone; for k0 s • Qm/(!0 � R we obtain an impervious thin foil with only a mass rectance. This, for example, can be applied to down-tune the resonator.

It is evident that the principal results derived here for a slit resonator also apply to resonators with circular necks.

Absorber layer in front of the necks

The discussion of the arrangement of Fig. 1,d) would be an avoidable repetition of the above results if the ab sorber layer is very thin. However, there are some differ ent points of interest in this arrangement. Near the reso nances, the absorber layer has a low-impedance termi nation. Its sound absorption then can be much higher than with a rigid termination. This fact can be applied to achieve an increase in the sound absorption of the layer at low frequencies. At high frequencies, when the mass reactance of the resonator necks become large and the resonator plate effectively rigid, there still remains the absorption of the absorber layer. The arrangement of the layer and the numbering of the zones (which is chosen as shown, with the intention of retaining as much as possible unchanged from the previous section) are shown in Fig. 3.

The fields established in the zones are:

Zone I: = L bn Fn cosh ( Yn (x -d -t)) cos (11 n y), (36) n=O

p i (x, y) = Ace -j ko x + L bn A n eYn x cos (17 n y), n=O Z ( ) _ A -j kox • " i;: Yn A YnX ( ) (33
The wave numbers and propagation constants are the same as in eq. ( 5).

The boundary conditions in the planes x = -s, x = 0, x = d for the particle velocity v x give (in that order) with the coefficients sm.n from eq. ( 6):

Yn a x m • d + • d -F sinh (y t) = J • -" -S (B e -Jxm -C e JXm ). k n n L � k m, n m m O m -0 O (39)
The boundary conditions for the sound pressure yield:

(Bo+ Co)= L <\So,n (D n +E n ), (39a)
(B0 e -jko d + C0 e +jko d ) = L bnSo .nFn cosh (y n t). (42a) n=O The aim of the next step is again the derivation of a system of linear equations for the unknown ampli tudes. One equation can be easily derived from eq. (39a) and eq. (42a) for the quantities X o ± = B0 ± C0, Yo±= Bo e-jko d ± Co e -jko d : This gives, fr om eq. ( 23), the impedance Zsh of the rear orifice which, as expected, is the impedance for the empty resonator chamber. The derivation of a second equation is somewhat more complicated. We define U n ±, v:i± in analogy with X 0 ±, Y0 ±:

for which the following hold: ei;., s 1 + e -2 i:., s

Y.i + = -2 U n - -z + Y.i - -zi: s
1 -e i:., s

1 -e "

Then we find from eq. (38 a):

and with eq. (41 a) this gives:

X0+ = [ -jcot ( k0d)] • X0_ + [j/sin ( k0 d)] • Y0_ = L bnS O .n • U n + = L bnS O ,n n =O n=O b s + n O,n , V sinh ( c; n s) n -' (47)
where the first line stems fr om eq. ( 16), the second line from eq. (41 a) and ( 45), and eq. ( 46) is introduced into the third line. We still have to reduce V n _ to X o-and Y0 _.

This derivation starts with eq. ( 37) from which we obtain: This, inserted into eq. ( 47), gives the second required equation:

V -b Z A +jko s • --r: Z Y n A -y.
sinh ( (i s) + Zan cosh ((is)

The eqs. ( 43) and (50) can be explicitly solved for X 0 _ and Y0_, from which all amplitudes can be computed. ---+---+---f---l---1----+--+------- resonance is tuned to still lower frequencies, as in Fig. 6 a and 6 b, the sound absorption in the first maximum will usually be decreased. The diameter of the additional loop can be influenced by the ratio a/L. The diameter generally becomes smaller, the closer the original curve is to the matching point at the onset of the additional loop. This is quite plausible, because the absorber layer must be rather thick and/or must have a high flow resistivity to approach the match ing point. Under such conditions, the absorber layer -in an acoustical sense -becomes infinitely thick, and then it t t 0.8 --rU--rV--rW--rX--rY--rZ--r[--r\--r]--r^ .,,.----0.2 r-----r---r.--r/--r0--r1--r2--r3--r4--r5--r6 -0 . 4 t-----+---+-----+--+---+----+--+------ becomes less important whether the termination of the layer has a high or a low impedance.

Up to now we have considered optimizations with respect to sound absorption. The goal of optimization here is the point of impedance matching Z0 G = 1 (for normal sound incidence and sufficiently large absorber areas). The aim of optimization becomes quite different from that for sound absorbers in silencers. This shall be explained shortly. These explanations hold for locally reacting duct linings. A Helmholtz resonator with a front side absorber layer is not precisely locally reacting, but it can be approximately treated as locally reacting if the flow resistance of the layer is high enough ( 5 s/Z0 > 2) and/or the layer is thin (s < L). Since Cremer's famous paper [START_REF] Cremer | Theorie der Luftschall-Dampfung im Rechteck kanal mit schluckender Wand und das sich dabei ergebende hochste DampfungsmaB[END_REF] the design computa tions for lined silencers assume one mode of the sound wave propagating in the duct, namely the least attenu ated mode, and the absorbent lining is designed such that the attenuation of this mode becomes as high as possible. We consider a two-dimensional rectangular duct along the x-axis having a width 2 h along the y-axis and being coated by locally reacting absorbers with a wall admit tance G at y = ±h. A sound wave symmetrical with re spect to y, which also describes the least attenuated mode, can be written as:

p (x,y) = A • e -r x • cos (c:y), j o p . c: -r x ( ) v (x,y) = ---= -J A• --e • cos c:y . Y k o Z oDY k o Z o (53)
The propagation constants rand c: are connected to each other by the wave equation:

(54)

The boundary condition of agreement between the field admittance and the wall admittance at y = ± h leads to the characteristic equation for c: h:

c: h • tan ( c: h) = j k0 h • Z0 G = j U, (55)
where we have introduced the quantity U = k0 h • Z0 G, which we name ab sorber function, because it is the quan tity which contains the information about the absorber lining. There exists an infinite manifold of solutions c:m h, m = 0, 1, 2, ... as a consequence of the periodicity of tan (c: h); they correspond to the possible modes (56)

A usual measure for the sound attenuation is the sound level decrease D h in dB along a distance L\x = h. This is given by D h = 8.68 • Re {I' m h}. The decrease per meter is then L\L = Dh/h dB, with h in meters.

The modal solutions c:mh cover the U-plane as an infin ity of leaves of a Riemann surface. The boundaries be tween the solutions are the branch cuts between adjacent leaves. Succeeding modal solutions have the branch point, which is the end of the branch cut between them, as a common solution.

Cremer has shown that the least attenuated mode be longs to the first two orders, m = 0, 1, and that it attains its maximum attenuation in the branch point u = ub between these two modes. The branch points Ub and (c: h)b for the first 20 modes were computed in [START_REF] Mechel | Modal solutions in rectangular ducts lined with locally reacting absorbers[END_REF] Thus, an optimization goal for silencer absorbers is to design the wall admittance of the absorber such that the absorber function U = k0 h • Z0 G attains the value of Ub and keeps it over as wide a frequency band as possible. There exist, however, some principal dif ficulties. The first problem lies in the fact that the ab sorber must be rather reactive. The tan <5 of G must be tan b =Im {G}/Re {G} =Im {Ub}/Re {Ub} = 0.801 which is a value characteristic for reactive absorbers. Reactive absorbers generally have a pronounced frequency depen-dence. One type of frequency dependence would be fa vourable; in order that U can be constant in a frequency band, the admittance Gi n that band should be inversely proportional to the frequency. This is the frequency de pendence of a mass-type reactance. However, the positive sign of Im { Ub} indicates a spring-type reactance. It be comes plausible from these facts, that a passive absorber can reach Ub only by some kind of resonance.

The next problem is, how precisely the branch point must be reached. To find an answer to that question we show a 3-D-plot of the attenuation Dh around Ub taking a fixed value for the frequency. We choose k0 h = 0.597 which belongs to a frequency about two octaves below the ray formation limit (that is the frequency above which the sound begins to propagate in the duct as a ray which is barely influenced by the duct walls, and above which, therefore, the sound attenuation generally decreases; see also Fig. 8b. The exact value of k0 h is not important for the present question). Fig. 7 shows the distribution of Dh around Ub. The curves for the least attenuated mode are thick; the curves for the next higher mode are thin. One sees that the attenuation Dh steeply decreases if U devi ates from U b towards the real axis of U, and that there is a saddle point on the other side of Ub on which the attenuation becomes nearly as high as in the branch point Ub itself. An important design rule can be derived from these facts: if it is not possible that the curve of U of a given absorber in the U-plane can exactly pass through Ub, then it is better that the curve passes behind Ub rather than in front of it (as seen from the side of the real axis of U).

A final point is a suitable representation of intermedi ate results of design computations. It would be a too great an amount of computational work to solve eq. ( 55) for each design guess in order to see whether the maxi mum attenuation is reached. On the other hand, in 55); for methods of solution see [START_REF] Mechel | Modal solutions in rectangular ducts lined with locally reacting absorbers[END_REF]) is plotted in Fig. 8 b (as a full line) against the product f • h [Hz• m]. The dashed curve is the maxi mum attenuation of the least attenuated duct mode if the value of the absorber function U would be Ub for all frequencies. One sees that the maximum attenuation is approached indeed at two frequencies, but one also realizes that the design is not good, because the max ima are reached in a frequency range beyond the limit of ray formation, in which the attenuation principally de creases. As all parameters are non-dimensional (and the abscissa f • h comes from k0 h for the sound velocity c0 = 343.3 [m/s] of air at room temperature), we can tune the maxima to some wanted frequencies.

The loop in Fig. 8 a, which crosses the branch point, comes from the Helmholtz resonance. It is possible to combine the Helmholtz resonance with an absorber layer resonance in order to achieve a triple crossing of the branch point. Fig. 9 a shows a plot of the UL curve for an absorber arrangement with a front-side absorber layer and slit resonators behind it which realizes this idea. The L/ h ratio is L/ h = 0.52. With (s + d + t)/ L = 2.125, the lining-to-duct width ratio is (s + d + t)/ h = 1.105, which is a quite normal ratio for absorber linings. Fig. 9 b shows the corresponding attenuation curve. The first maximum is tuned down by the increase of the thickness ratio. The two other maxima do not completely reach the theoreti cal maximum because their U curve crosses the saddle in Fig. 7 behind the branch point. It would be possible to bring them closer to the branch point by a fine-tuning of the parameters, but the intention was here to demon strate the effect of a crossing over the Dh-U-saddle.

Concluding remarks

It was shown above (and in [START_REF] Mechel | About Perforated Metal Sheets on Porous Absorbers[END_REF]) how the near fields of the resonator orifices, which are characterized by the end corrections of the orifices, are influenced by porous ab sorber layers behind or in front of the resonator plate. This influence was demonstrated for the special cases of normal sound incidence and slit resonators. The restriction to normal sound incidence can be re laxed without any modification (as an approximation) in the case of absorber layers behind the resonator plate, because the absorber arrangement is locally reacting. The modification of the external end correction by an oblique sound incidence generally can be neglected as long as the period length Lis small compared to the wave length ).0. If L approaches ).0/2 or exceeds that value, then the absorber arrangement must be treated as a periodic structure as was shown in [START_REF] Mechel | AuBere Schallfelder[END_REF]. For the arrangement with the absorber layer in front of the resonators, an oblique sound incidence can be taken into account by a replace ment of the first term in the third line of ( 51 ), which is the (planar) input impedance of an absorber layer with a rigid termination for normal sound incidence, by the corre sponding impedance for oblique sound incidence with a polar angle e: Z an Ze/ Z 0 = --coth (�s •cos e1), cos e1

(57) wherein e 1 is the (complex) angle of the refracted wave in the absorber layer given by: (58)

Another geometry of interest, other than that of slit resonators, is that of Helmholtz resonators with circular necks arranged in a grid with square or hexagonal res onator volumes. Such arrangements are treated analyti cally in [START_REF] Mechel | Innere Schallfelder, Strukturen[END_REF]. There the absorber surface is subdivided into cylindrical cells with a cell radius R and one cell concen tric to each orifice with radius a. The cell radius R is selected such that the cell cross-sections just cover the absorber area, i.e. the porosity (a/R) 2 is that of the real absorber. The (approximate) replacement of the square or hexagonal resonator volumes by cylindrical resonator volumes of the same cross-sectional area is based on the experience that the exact shape of the polygon in a polyg onal grid becomes unimportant if R/ }.0 � 1. Then the sound fields in the zones of a cell are synthesized by radial mode sums, and the mode amplitudes are determined as above. The main modifications as compared to the results for slit resonators are the replacement of the porosity a/L of the slit resonators by (a/ R) 2 for cylindrical resonators and further of cos (c; y) by J0 (c; r) and of sin (c; y) by J1 (c; r), with J0,J1 being Bessel functions.

Fig. 1 .

 1 Fig. 1. Treated combinations of porous absorber layers with Helmholtz resonators; a), b): in the fo rmer article; c), d): in this article.

Fig. 2 .

 2 Fig. 2. Helmholtz resonators in a linear grid with additional losses from a porous layer in zone III behind the resonator plate.

  e -JX m d + Cme +Jxm d ) (11) = L bnSm, n(Dn e -t: n d + Ene + t: n d), n=O with the Kronecker symbol <> m.n = 0 or 1 depending on m =t= norm= n . Elimination of F n from eq. (9) and eq. (12) gives: with the modal reflection factors: i(X m + + Xm_) = i(Ym + + Ym_)e + jxmd C m = i(X m+ -Xm_) = i( Ym + -Ym_)ejxmd. (17) For these auxiliary quantities one obtains two coupled inhomogeneous linear systems of equations with m = 0, 1, 2, ... from the above boundary equations: oc; L: a m.nxn-+cm Ym-= bm , n=O L: dm. n Y,, _ + cmxm-= o , n=O

  ) reduces to the two equations a o .o Xo -+Co Yo -= 2 Co Xo -+ do .o Yo -= 0, o = J-LL i-: S o.i-JCO t(k0 ) ,

  . e -2 e; s l j s- 1 -2 e ; s = --_:coth (y J) . 1 -r i e s-0 Fan z a n y i This gives the rear orifice impedance: z sh a [ . k -= --J cot ( 0

Fig. 3 .

 3 Fig. 3. Helmholtz resonators in a linear grid with additional losses from a porous layer in zone III in fr ont of the resonator plate.

  e -J xrnd+C e+Jxmd) = " b S F cosh (y t) (42) the assumption of only plane waves in the res onator necks, which we mostly apply in what follows for reasons of simplicity, i.e. with B m =C m= 0 for m > 0, these matching conditions simplify to: (01 t) = j -S (B e -J 0 -C e 1 0 )

1 +

 1 Y0_ [cot ( k0 d) + � L: bnS cf.n coth (y n t)] L n=O Y n/ko X0_ = 0.

[ a k0 1 ++) 2

 12 , s n--O .n an c e + J an an k n e 0 _ s: A + jk 0 s ( z • r z Y n ) • r z Y n . V u o, n c e JI'.in Z ank --L I'.in Z an-So.n . h( coth (e. s) • V0_ J (48) in which the second line comes from eq. (40a) and the third line from eq. (38 a). The solution for V n-is: V.-= [ 2 <> o .• z,. A, e + ;,,, • h( ) X oc;n SID c;n S /[ 1 -j r;. z,. �coth(e.s)J. (49)

  The planar input impedance (normalized to Z0) of the arrangement is then:(Pm ( -s , y ) ) L = z Vo+ Z o ( Vm x ( -s, Y )) Lan Vo -2ej kos cosh (�s)-a/L• X0_ = Z an 2 e j kos sinh ( I's )+ a/L Z • in the last line is just the (normalized) input impedance of a rigidly backed absorber layer of thickness s. The second term thus is a correction term as a conse quence of the backing by the Helmholtz resonator.

4 .

 4 Fig.4a shows an example of the curve of the (planar) normalized input admittance Z0 G of an absorber layer backed by Helmholtz resonators in the complex admit tance plane, with a running frequency variable Lj).0. The dashed curve is for the absorber layer with a rigid back ing; the continuous curve is for the absorber layer with Helmholtz resonators. The depth of the resonators is relatively small with s/L = 1; t/L = 0.075. (The other shape parameters are indicated in the graph.) The sum mation index of the sums runs up to n0b = 10. The opti mum for the sound absorption under normal sound inci dence (which was assumed here for simplicity) lies in the matching point Z0 G = 1. The absorber layer alone (dashed curve) would, for higher frequencies, trace other circles which would contract towards the matching point. The Helmholtz resonators add a loop to the original curve which is directed towards the inner side of the original loop and which approaches the optimum condi tions for sound absorption already at lower frequencies. This can be seen immediately from Fig.4 b, where the sound absorption coefficient r:J. is plotted against L/ ).0 for the absorber layer with rigid backing (dashed curve) and with the Helmholtz resonators from Fig.4a(full line). The absorption coefficient is computed as:4Z0G' r:J. = ---------(1 + Z0 G') 2 + (Z0 G") 2 (52)from the components of Z0 G = Z0 G' + j • Z0 G". The starting point for the additional loop can be con trolled by those parameters which influence the reso nance frequency of the Helmholtz resonators. In Fig.5a and 5 b the resonance frequency was lowered mainly by increasing t/L. The maxima of the sound absorption have about the same heights with and without the resonators, respectively; the first maximum with the resonators is shifted downwards by about one octave. If the Helmholtz

Fig. 4a .

 4a Fig. 4a. Curve for varying L/1.0 of the input admittance Z0 G of an arrangement as in Fig. 3. Full line: Absorber layer backed by resonators; dashed: absorber layer with a rigid backing.

Fig. 4 b

 4 Fig. 4 b. Frequency curves of the absorption coefficients corre sponding to the plots of Fig. 4 a.

Fig. 5 a

 5 Fig. 5 a. As in Fig. 4 a, but with the resonators tuned to a lower fr equency.Fig. 5 b. Absorption curves for Fig. 5 a.

Fig

  Fig. 6 a. As in Fig. 4 a and 5 a, but with a still lower tuning of the resonators. Fig. 6 b. Absorption curves for Fig. 6 a.

  . The values needed here are Ub = 2.05998 + j • 1.65061, and the solu tion ( c: h)b = 2. 1062 + j • 1.12536 of eq. (55) belonging to this value of the absorber function Ub. If in eq. (54) ( k0h) 2 �Ic:h1 2 , i.e. for low frequencies, the attenuation Dh in the branch point attains the extremely high value Dh � 18.3 dB (which corresponds in a duct with h = 0.1 [m] to L\L � 185 [dB]!).

Fig. 7

 7 Fig. 7. 3 D-plot of the attenuation Dh in dB over the complex plane of the absorber function U of a lined duct with that ab sorber function around the branch point Ub between the first two duct modes. Thick lines: least attenuated duct mode; thin lines: next higher mode.

U

  = k0 h • Z0 G the length h of the (half) duct width is not a dimension of the absorber. In order to design an absorber without this length h in the first design steps, one takes any characteristic length of absorber, let us say L, and plots the curve of UL = k0 L • Z0 G in the plane of UL (see Fig. 8 a). Since UL= ( L/ h). U, the branch point ub is shifted by the factor ( L/ h) into the branch point ubL = ( L/ h). Ub. This means that the branch points ub L for ratios L/ h < 1 lie on a straight line between the origin of the plane UL and Ub, and branch points for ratios L/ h > 1 lie on a continuation of the straight line beyond Ub.

Fig. 8 a

 8 Fig. 8 a shows a plot of UL = k0 L • Z 0 G in the complex plane of UL for an absorber arrangement consisting of an absorber layer with slit resonators as a termination. The parameters are indicated in the graph. The value of Ub is indicated in the graph as a dot near the upper border, and the straight line (dash-dotted) connecting it with the orig in is the locus for branch points for ratios L/h < 1. With a value L/ h = 0.282 the branch point is passed by the curve at two frequencies. The thickness ratio of the total arrangement is (s + d + t)/ L = 1.75, and the ratio (s + d + t)/ h = 0.4935 of the absorber thickness to the duct width is rather low. The attenuation Dh of the least attenuated duct mode with this absorber arrangement (which follows from a solution of eq. (55); for methods of solution see [5]) is plotted in Fig. 8 b (as a full line) against

Fig

  Fig. 8 a. Curve for varying L/ A.0 of the absorber fu nction k0 L • Z0 G for an absorber layer backed by Helmholtz res onators. Point at the upper border: branch point Uh for L/h = 1; dash-dotted line: locus of the branch points for other values of the ratio L/h.

Fig. 8 b

 8 Fig. 8 b. Curve of the attenuation Dh for the arrangement of Fig. 8 a (full line). Dashed line: Theoretical maximum attenua tion if the absorber function would keep the value Uh of the branch point for all frequencies. The upper decline of this curve comes from the ray formation in the duct.

R=0. 43 Fig. 9

 439 Fig.9a. As in Fig.8a, but now the Helmholtz resonance com bined with a layer resonance in order to achieve a triple crossing of the branch point.

Fig. 9 b

 9 Fig. 9 b. Attenuation curve belonging to Fig. 9 a.


Here you have three posters for three military recruitment campaigns: Navy, Air force and Army. But it is only for the context because my main concern, now, will be retention on one side and pay policy on the other one.

My point of departure was first a report of the Court of the Accounts, dated 2019, about the execution of the 2018 defence budget. It showed a surplus of the military pay budget of 148 million Euros, due to the retention and recruitment simultaneous problems of the ministry of defence.

At the same moment, the parliament discussed the project of military programming law for the years 2019 to 2025, with this mention "the new military pay policy will simplify the allowance scheme, so contributing to the appeal of the military career by clarifying the pay structure, especially its compensatory part".

The first question set by these quotations is to know if the retention problems denounced by the Court of Accounts are only those of the 2018 budget, of if they have existed for a longer period.

Symmetrically, it is interesting to know if an "ancient" military pay policy has existed during this "longer" period, especially to increase the retention.

If the answer is yes, it would be correct to say that this first pay policy was not effective and justified a "new" policy.

For these 2 parts, I shall use a (modest) quantitative approach, with official data of the ministry of defence, given in the "Defence social assessments" published from 1990 to 2020, that is during thirty years.

In the third part, I shall try to discuss the ability of the measures included in the NMPP to retain the military. In this case there is no quantitative approach possible, of course, and I will use the presentation of NMPP by the ministry themselves.

It seems useful to begin with a global view of the ministry of defence workforce during the period under examination. Of course, for the following part, the green and blue zones will be out of the scope.

As you can see, the dominant impression is a dramatic reduction of the global number, with two interesting events: first, the end of the draft between 1997 and 2001; second, the transfer of the Gendarmerie Nationale to the ministry of internal affairs in 2009.

To have an image of the military workforce management, I have calculated a Recruitment to Departure ratio. It shows the main periods of the recent history of the French military policy.

First an increase of the recruitment number because of the Kuwait war; then a new increase to compensate the end of the draft. After 1999 we find a global problem of budget deficit and the need to "control the wage bill". Finally, after 2014, there is the acknowledgement that France is at war with terrorism; and needs men and women to wage this war. This first index is certainly interesting, but it cannot render the idea of professional fidelity.

Of course, the variable « departure » seems to be the good one to study the question of retention or, presented otherwise, the one of fidelity. But all the departures are not retention failures.

On this table the column "Departure without a right to perceive a pension", immediately or after a delay, seems the good one to render this idea of failure, because "without pension" means very early, and perhaps too early to amortize the training cost.

In 2007, the rules changed. Now all the military people leaving the service have a right to perceive a pension, of course proportional to the service length. It is the reason why the table changed also. The question is to find what length is equivalent to the preceding "without pension".

Il appears that under 15 years of service, the pension is delayed. Of course, the sacrifice is less heavy than before, but it is a sacrifice, nevertheless. And it appears too, that the number of people leaving under 15 years of service in 2007, is almost equal to the number of people leaving without pension in 2006.

Consequently, it seems possible to build a homogenous series with the two characters "without pension" and "after 15 years of service or less".

This diagram shows that the period when the index is growing is the same as the one when the ratio recruitments-departures is above 1.

So, the explanation of this growth is certainly in terms of decreasing marginal quality of the enlistments. Nothing surprising in that because it is proving the legitimacy of a selection process.

As everybody knows, there are two parts in the pay: a rank-related part, and a compensatory one. Consequently, a pay policy can be applied alternatively or simultaneously to the corresponding parts of the total pay bill of the ministry, the rank-related one and the compensatory one.

The problem is that in the thirty Defence social assessments there are data only for the total pay bill. This variable is treated "per capita" here.

Happily, there are also data for the compensatory and rank-related bills in the three last assessments.

Then, for both composing series, the only solution is to retropolate them from the known data, by applying respective evolution laws.

The simulated rank-related bill per capita is determined by two explaining variables: first the pay rate and then the workforce structure per rank.

For the whole French "fonction publique", the wage rate is the index point value. Since 2017, this value is equal to 56.2323.

Then, every civil or military servant is identified by a certain number of index points. For instance, a colonel with less than one year of seniority "weighs" eight hundred and ten index points. Consequently, he earns 45 548 euros per year with his rank-related pay.

At this point on can remark that the uniqueness of the wage rate for military and civil servants make difficult to use the rank-related pay for a military pay policy, because of the inevitability of a diffusion effect. So, a military pay policy, or a civil one of course, must be a compensatory pay policy.

Here we have the pay rate effect.

To take into account the structure effect on the rank-related pay bill, one has to calculate a strength index, weighted by the corresponding pays. The Defence social assessment gives average pays only for the most recent part of the period. The solution was to calculate an "inverted" Laspeyre index, that is weighted with the respective average pays of the terminal year, rather than the initial one. This is the effect of the structure per rank.

And now we have a simulated rank-related pay bill So, we can calculate the compensatory pay bill values by difference And finally, we have the proportion of the compensatory pay bill in the total pay bill.

If we superpose the compensatory pay proportion curve on the rate of infidelity index curve, we have something resembling a relation. On fourteen data we find a correlation coefficient at 0.96. So, we can suspect that the compensatory pay policy launched after two thousand and six was a reaction, somewhat delayed, to the growth of military infidelity. But this policy doesn't seem to have been effective because the infidelity rate remains at a high level and, perhaps still growing during the four last years. This statement would be a good reason to launch a new military pay policy.

In fact, if an infidelity exists, it is the infidelity of the non-commissioned members as it appears on this diagram representing the turn-over of each category. So, if the new policy is to be effective, it must convince this specific category. Number five doesn't concern NCM, by definition Numbers 6, 7 and 8, perhaps, because NCM also have access to training and career improvement. In that case, the longer the career, the higher the skill marketable in the civil world.

But globally, one can remain perplexed!

  Previously, there were one hundred and seventy-four allowances and bonuses. Next year it would be only eight, certainly a simplification. But are they the good ones? Number one existed already under another name Number two is interesting essentially for large families renting premisses in the free-price sector, not generally the case of NCM Number three, yes. The problem of absence concerns any military person's partner. Number four doesn't concern most of NCM, because they generally remain in the same garrison.


Jean-Pierre Allegret
email: allegret@gate.cnrs.fr

Alain Sand-Zantman

Does a Monetary Union protect against foreign shocks? An assessment of Latin American integration using a Bayesian VAR

Keywords: Business Cycles, OCA, Bayesian VAR, Latin American Countries JEL Classification: C32, E32, F42

This paper analyses the monetary consequences of the Latin-American trade integration process. We consider a sample of five countries -Argentina, Brazil, Chile, Mexico and Uruguay-spanning the period 1991-2007. The main question raised pertains to the feasibility of a monetary union between L.A. economies. To this end, we study whether this set of countries is characterized by business cycle synchronization with the occurrence of common shocks, a strong similarity in the adjustment process and the convergence of policy responses. We focus especially our attention on two points. First, we try to determine to what extent international disturbances influence the domestic business cycles through trade and/or financial channels. Second, we analyze the impact of the adoption of different exchange rate regimes on the countries' responses to shocks. All these features are the main issues in the literature relative to regional integration and OCA process.

Introduction

The 1990s were characterized by an intensification of Regional Trade Agreements in the Americas. The main agreements are the Southern Common Market (MERCOSUR) -signed in 1991 between Argentina, Brazil, Paraguay, Uruguay (and more recently Venezuela), with Bolivia, Chili, Peru, Colombia, and Ecuador as associates-and the North American Free Trade Area (NAFTA) -signed in 1994 between Canada, Mexico and the United States with more and more agreements with other Latin American countries (LACs) (Chili, Peru, Ecuador…) 1 . Since 1994, a Free Trade Area Agreement for the Americas has been discussed, as an extension of the NAFTA. In the spirit of [START_REF] Eichengreen | The Monetary Consequences of a Free Trade Area of the Americas[END_REF], this paper analyses the monetary consequences of this trade integration process. We consider a sample of five countries -Argentina, Brazil, Chile, Mexico and Uruguay-that account for some 70 per cent of the region's GDP spanning the period 1991-Q1 -2007-Q1. The paper assembles a quarterly dataset (see Appendix 1 for data sources) including both main international macroeconomic variables -such as the GDP of the Group of seven countries and the world commodities prices index-and main domestic variables -such as GDP and real exchange rates. In addition, we consider variables based on the literature dedicated to the sudden stop problem [START_REF] Calvo | On the Empirics of Sudden Stops: The Relevance of Balance-Sheet Effects[END_REF]: the Emerging Economy spread index and the foreign exchange reserves.

The main question raised in this paper refers to the feasibility of a monetary union between these countries. To this end, we study whether this set of countries is characterized by business cycle synchronization with the occurrence of common shocks, a strong similarity in the adjustment process and the convergence of policy responses. We focus especially our attention on two points. First, we try to determine to what extent international disturbances influence the domestic business cycles through trade and/or financial channels. Second, we analyze the impact of the adoption of different exchange rate regimes on the countries' responses to shocks. As shown in appendix 2, studied countries adopted very different exchange rate regimes over the 1991-2007 period. While at the beginning, the set of countries ranged from hard peg (Argentine currency board) to intermediate regimes, at the end, it exhibited a clear switch toward floating regimes.

The present paper is linked to two separate strands of literature. The first one, dedicated to the debate of monetary union versus dollarization, includes numerous papers analyzing the situation of Central and Latin American countries relative to the United States 2 . Empirical studies suggest that dollarization is not an obvious solution, even for Mexico. For instance, Karas (2003) finds that Mexican output fluctuations have been negatively correlated with the American fluctuations. According to [START_REF] Hallwood | An Assessment of the Case for Monetary Union or Official Dollarization in Five Latin American Countries[END_REF], Brazilian, Chilean and Uruguayan permanent shocks are correlated with Argentina suggesting that a monetary union could be a better solution than dollarization. The second strand of literature analyzes the sources of business cycles fluctuations in emerging countries. Two lessons from this literature are especially interesting for our purpose. On the one hand, a large body of studies suggests that the main source of fluctuations originated from external factors. [START_REF] Aiolfi | Common Factors in Latin America's Business Cycles[END_REF] considering a sample of four LACs 3 -identify the presence of a common regional factor.

Taking into account the weak intra-regional trade integration, this result suggests that the regional business cycle (major turning points are common to the four countries) is driven by external variables and common external shocks. On the other hand, the financial channelbased on international interest rates for instance-seems more significant than the trade channel in understanding the influence of external shocks on domestic business cycle fluctuations in LACs 4 .

The remainder of this paper is organized as follows. Section 2 explains the methodology adopted in this paper and founded on Bayesian structural VAR models. Section 3 presents the macroeconomic variables included in the VAR, the results and the policy implications. Section 4 concludes.

Methodology of the Study

This paper rests on two important methodological points: on the one hand, we propose a way to take into account the structural breaks affecting LACs over the period and, on the other hand, we use a Bayesian structural VAR.

Non-Stationarity and Structural Breaks: the Special Case of Emerging Economies

The emerging economies case -and more especially LACs-is not the simplest for time series methodology. Indeed, this group of countries exhibits numerous structural breaks -e.g. the end of hyperinflation and the dramatic increase in commodities prices after 2000-and/or changes in policy regimes such as the exchange rate regimes collapses that result in new 2 . See for instance [START_REF] Salvatore | Which Countries in the Americas should Dollarize?[END_REF], [START_REF] Corbo | Is It Time for a Common Currency for the Americas?[END_REF], [START_REF] Alesina | Optimal Currency Areas[END_REF], Karas (2003), [START_REF] Larrain | Regional Currencies Versus Dollarization: Options for Asia and the Americas[END_REF], [START_REF] Hallwood | An Assessment of the Case for Monetary Union or Official Dollarization in Five Latin American Countries[END_REF][START_REF] Allegret | Disentangling Business Cycles and Macroeconomic Policy in Mercosur: a VAR and Unobserved Components Model Approaches[END_REF][START_REF] Allegret | Modeling the Impact of Real and Financial Shocks on Mercosur: the Role of the Exchange Rate Regime[END_REF]. 3 . Argentina, Brazil, Chile, and Mexico over the period . See also [START_REF] Canova | The Transmission of US Shocks to Latin America[END_REF] and [START_REF] Maćkowiak | External Shocks, U.S. Monetary Policy and Macroeconomic Fluctuations in Emerging Markets[END_REF]. 4 . See [START_REF] Ahmed | Sources of Economic Fluctuations in Latin America and Implications for Choice of Exchange Rate Regimes[END_REF], [START_REF] Canova | The Transmission of US Shocks to Latin America[END_REF], and [START_REF] İsterholm | The Effect of External Conditions on Growth in Latin America[END_REF]. monetary policy frameworks. Since the seminal works of [START_REF] Nelson | Trends and Random Walks in Macroeconomic Time Series: Some Evidence and Implications[END_REF], most macroeconomic time series in level are considered unit root process. On the same sample, Nelson and Plosser, Perron (1989) challenged this interpretation, indicating that most macroeconomic variables are trend stationary, coupled with structural breaks. Looking at the Latin-American macroeconomic time series, we assert the same hypothesis: the econometricians had to take into account structural breaks due to non-random external as well as internal shocks and policy regime changes. The right way to deal with this question consists (in the Perron procedure) of testing for unit roots in the presence of structural change at a known date. If the date of the break is uncertain, other tests are available [START_REF] Vogelsang | Additional Tests for a Unit Root Allowing for a Break in the Trend Function at an Unknown Time[END_REF]Perron, 1998, or Zivot and[START_REF] Zivot | Further Evidence on the Great Crash, the Oil Price Shock and the Unit Root Hypothesis[END_REF] on common softwares. However, as shown in Le [START_REF] Bihan | Tests de rupture : une application au PIB tendanciel français[END_REF] all these procedures are powerless when the number and the date of the break are unknown. Overall, the combination of a short sample and multiple breaks weaken the break diagnosis compared to the following unit root test.

We choose a rougher but probably more secure method. First we identify the noticeable breaks of the figures5 as being the well-known historical ones (due for instance to balance of payments crisis or policy regime switches): the results are displayed in Appendix 3. As particular (and generally deterministic) events, these breaks can hardly be considered as the N.I.D. stochastic innovations of a random walk. Second, in order to stationarize the macroeconomic series, we clean them from the various deterministic trends and intercept leaps, using simple time trends and dummy variables. We finish with a common A.D.F. test, finding all series stationary. Thus, we can exclude any cointegration relationship but a VAR in level is an available alternative to the VECM one. We choose a recursive semi-structural approach for a VAR in level of the detrended series.

A Bayesian Structural VAR

Undeniably, the sample is short and the number of variables fairly high. In this case, [START_REF] Litterman | Techniques of Forecasting Using Vector Autoregressions[END_REF][START_REF] Litterman | Techniques of Forecasting Using Vector Autoregressions[END_REF] suggests specifying blurred restrictions on the mean and variance of the coefficients instead of brutal "ad hoc" exclusions. As [START_REF] Doan | User's GuideRats version 7, Estima[END_REF] concludes, "in a vector autoregression, we must concern ourselves not only with the lags of the dependent variables, but also with the lags of the other endogenous variables. Because of the stability conditions, we have some pretty good information about the size of lag coefficients in a simple autoregression. However, it's not so clear what the sizes of the coefficients on the other variables should be, and these depend, in part, on the relative scales of the variables involved". As indicated by [START_REF] Canova | Methods for Applied Macroeconomic Research[END_REF], priors on the mean and variance of the variable allow one to deal with over parametrization.

The choice of priors is the simplest one: overweighting the first lags of endogenous variables of each equation. Although a fine tuning prior is unrealistic, a deeper investigation is necessary to provide a better assessment of the consequences of innovations but it could be time-wasting.

In the same way, this version uses a semi structural BVAR. Using a Bayesian justifies once more avoiding structural orthogonalization: [START_REF] Canova | Methods for Applied Macroeconomic Research[END_REF] shows that the combination of Bayesian methods and structural hypothesis is not the simplest one, particularly for economies characterised by a succession of policy regimes.

The Model and the Results

Variables selection

Our variables are based on the traditional ones for VARs analyzing external shocks and macroeconomic packages in open economies and on the literature dedicated to the sudden stop problem.

Each domestic VAR includes three external variables. As real external shocks, we consider (i) the Gross Domestic Product for the G7 (noted LGDPG7) and (ii) the world commodities prices excluding oil (noted WCPNO). Our choice to exclude oil from our commodities prices index is due to the fact that some LACs (for instance Brazil and Mexico) are both producers and consumers of oil.

The Emerging Economy spread index of J.P. Morgan (EMBI) 6 accounts for the international financial shock. Many studies choose US interest rates or international interest rates -such as LIBOR-to estimate the impact of external financial shocks on emerging markets. We prefer to use the EMBI in order to disentangle monetary policy shocks and financial shocks. Further, over our sample period, the EMBI does not seem significantly influenced by LIBOR, confirming the González-Rozada and Levy-Yeyati (2005) results which show that spreads are 6 We merged two time series: the EMBI for the period 1991Q1-1997Q4 and the EMBI+ from 1998Q1. As indicated in [START_REF] Cunningham | Emerging Economy Spread Indices and Financial Stability[END_REF], the main differences between these indices are (i) the number of financial instruments embodied (the EMBI tracks returns and spreads on Brady Bonds and some other restructured sovereign debts, the EMBI+ tracks returns on a wider range of instruments), (ii) the number of countries (11 for the EMBI, 16 for the EMBI+). However, in both indices the weight of the LAC is very important (respectively 83.8% and 70.2%). Amongst the LAC, both Argentina and Brazil account for 47.6% of the EMBI+. In 1999, J.P.Morgan released a new index, the EMBIG (for "global") embodying more countries (27) and more titles. In this last index, LAC decreased to 61.5%.

determined by global factors 7 . [START_REF] Uribe | Country Spreads and Emerging Countries: Who Drives Whom?[END_REF] analyze the respective influence of US interest rates and EMBI shocks on the macroeconomic fluctuations in a sample of seven emerging countries covering the period 1994-2001. An important finding is that EMBI shocks exacerbate the US interest rate shocks, implying a strong macroeconomic volatility in the studied emerging countries.

For the domestic variables, we took the foreign reserves (FOREX) as proxy for the balance of payments, the gross domestic product (GDP), the consumption prices index (CPI), the nominal money market interest rate (R) and the real effective exchange rate 8 (ER). [START_REF] Calvo | On the Empirics of Sudden Stops: The Relevance of Balance-Sheet Effects[END_REF] stress that sudden stop episodes are characterized by both international reserves losses and sharp current account reversals. The former increases the country's vulnerability to shocks while the latter leads to output and employment contractions.

Quarterly balance of payments data are not reliable and subject to sizable revisions. As a result, our VARs do not include current account data. As a proxy for sudden stop problems, we choose to include central bank's foreign exchange reserves. In order to test the robustness of the results, we substitute the deseasonalized exports-imports ratio to FOREX. This ratio represents a proxy for the intertemporal constraint of the current account: a decrease in capital inflows imposes the reduction of absorption in order to increase exports and decrease imports.

Interestingly, the results do not change significantly. As a result, we prefer to consider only the FOREX variables in order to avoid some interpretation difficulties owing to the fact that the ratio obeys in part competitive factors, and not exclusively financial factors.

The Model

The model is tested separately for each LAC. The number of lags -two in each model-has been selected using the common set of criteria and tests. As the inverse roots of the AR polynomial lie in the unit circle, VARs satisfy the stability condition.

The following order of Choleski factorization is deduced from our theoretical interpretation of the contemporary correlation matrix of the reduced form residuals of each country model and from block exogeneity Wald tests. 7 . We perform different experiments in our VARs: first, we include both LIBOR and EMBI; second, we include only LIBOR. Results do not significantly change. Granger causality tests do not exhibit relations between EMBI and LIBOR.

8 . An increase (decrease) in the real exchange rate means real depreciation (appreciation).

1(1,1) 1(1,8) 1( , ) 1(8,1) 1(8,8) _ 7( ) _ 7( _ ( ) _ ( ) _ _ ( ) _ _ ( ) _ _ ( ) _ _ ( ) _ _ ( ) i j C C CYC LGDPG t CYC GDPG t CYC LWCPNO t CYC EMBI t C CYC Li GDP t CYC Li CPI t CYC Li FOREX t CYC i R t C C CYC Li ER t -                         +                             2(1,1) 2(1,8) 2( , ) 2(8,1) 2(8,8) 1) _ ( 1) _ ( 1) _ _ ( 1) _ _ ( 1) _ _ ( 1) _ _ ( 1) _ _ ( 1) i j CYC LWCPNO t CYC EMBI t CYC Li GDP t CYC Li CPI t CYC Li FOREX t CYC i R t CYC Li ER t C C C C C     -     -   -   +   -   -     -     -                             _ 7 _ _ _ _ _ _ _ _ 7( 2) _ ( 2) _ ( 2) _ _ ( 2) _ _ ( 2) _ _ ( 2) _ _ ( 2) _ _ ( 2)
-     -     -   -   +   -   -     -     -   _ _ _ _ _ Li FOREX CYC i R CYC Li ER e e                            
with = A for Argentina, B for Brazil, C for Chile, M for Mexico, and U for Uruguay.

External variables are considered the most exogenous. We assume that real external variables are predetermined relative to external financial ones. In addition, we consider that the GDP of the G7 countries exerts an influence on commodities prices through a demand effect. For domestic variables, different plausible orders have been experimented. They do not significantly change the results.

Econometric findings

Using this framework, we combine the impulse response functions (tracing out the time paths of the effects of pure shocks on the set of variables, see Appendix 4), and the forecast error variance decomposition (indicating the proportion of the movements in a sequence due to its own shocks versus to the other variables, see Appendix 5). This allows us to assess the degree of similarity in the reactions of macroeconomic variables to shocks amongst the set of countries. At the same time, we get a first outline of the specific -versus common-economic consequences of shocks in terms of spontaneous adjustments, as well as in terms of policy responses 9 .

Responses of domestic variables to external shocks: is transmission real or financial?

In all studied countries, GDP fluctuations are significantly influenced by foreign variables.

Our results show that in Argentina, Brazil, and Chile foreign variables explain at least 29% of the GDP variance decompositions after 16 quarters. In Mexico and Uruguay, the shares are 16.1% and 20% respectively. Above all, no domestic variables -except GDP itself-exert a greater influence than foreign innovations in all countries.

As expected, GDP increases after a shock on GDPG7. The positive influence of GDPG7 means that improvement (vs degradation) of the business cycle in G7 countries can result in 9 . All results mentioned in the text but not displayed are available upon request to the authors.

an increase (vs slowdown) of growth in LACs. The weak influence of GDPG7 in Brazil rests on the fact that this economy is a more closed economy relative to other studied countries.

In all countries except Uruguay, GDP increases after a shock on commodities prices (WCPNO) confirming the importance of commodities in LACs' business cycles. Not only are contemporaneous responses significant and positive but they continue to have significant persistent effects.

A large body of empirical literature dedicated to the business cycle in LACs stresses that growth in these economies follows international capital flows. More precisely, these studies suggest that the behavior of capital inflows is pro-cyclical: they tend to increase when growth in LACs improves. As a result, we can expect a significant influence of EMBI shocks on GDP during the period on our sample of countries. We find that GDP decreases after a shock on EMBI. The magnitude of the GDP response is important in Argentina and Uruguay, and to a lesser extent in Mexico. The Chilean case is particularly interesting. While Chilean spread stayed substantially below EMBI+ or Latin American spread over the period, its GDP responds negatively to EMBI shock. Even if the response is weakly significant from a statistical point of view, this result suggests that this type of shock is global, i.e. affects all countries, even economies benefiting from low idiosyncratic risk premiums.

Our findings confirm [START_REF] Allegret | Modeling the Impact of Real and Financial Shocks on Mercosur: the Role of the Exchange Rate Regime[END_REF] with regard to the specific sensitivity of Argentina to EMBI shock. During the first half of the 90s, Argentina was one of the main borrowers in international capital markets enjoying very favorable financing conditions, while in the second half of the decade the economy, it suffered from a sudden-stop of capital inflows. In addition, the monetary policy constraints due to the currency board limited the ability of authorities to react in the face of EMBI shocks, inducing strong and ample macroeconomic variability.

Overall, LACs differ according to the respective influence of trade and financial channels.

Two groups of countries can thus be distinguished: in the first one, including Brazil, Chile and Mexico, trade channels predominate and in the second group, composed of Argentina and Uruguay, financial channels exert the main influence on GDP variances. In addition, as suggested by the Chilean experience, international financial shocks have a global nature.

The relevance of the sudden stop

Two main points characterize the sudden stop literature. First, external factors exert a decisive influence on capital inflows into emerging markets. Second, depreciation results in contractionary output in emerging markets whereas it produces traditional expansionist effects in industrialized countries [START_REF] Calvo | Fixing for Your Life[END_REF]. Indeed, exchange rate crises in emerging markets are followed by a sudden stop to capital inflows. These countries suffer from reserve losses and severe reversal in the current account deficit. Such reversal is based on a major decline in aggregates.

In order to assess the relevance of the sudden stop literature, we determine what variablesforeign or domestic, real or financial-exert the main influence on FOREX included in our VARs as a proxy of international capital flows. The theoretical prediction is that international financial shocks, here the EMBI shock-are the main variables influencing FOREX in our five countries. In addition, we analyze the influence of FOREX on other domestic variables.

According to the sudden stop literature, a negative shock on FOREX must lead to a contraction in GDP.

Interestingly, from the sudden stop literature standpoint, FOREX is influenced by international variables, and more specifically by financial variables. Thus, the international financial shock produces the expected effects when significant. An increase in the spreadmeaning degradation in the financial conditions for emerging countries-leads to a decrease in FOREX in Argentina and Brazil, and to a lesser extent in Chile after 3 quarters. Variance decompositions support the significant influence of EMBI on the behavior of FOREX. In Argentina, Brazil and Chile, EMBI innovations explain 15.4%; 8.2% and 7.8% of the FOREX variance. In each of the three countries, EMBI is its main explanatory variable. In addition, we find that FOREX does not respond to interest shocks. In other words, increasing domestic interest rates is insufficient to induce the accumulation of international reserves. Such a result is in accordance with the sudden stop literature that suggests that FOREX responds more to global shocks than domestic ones.

A FOREX shock generates few domestic fluctuations. This deceptive result does not necessarily contradict the sudden stop literature. Indeed, as stressed by Izquierdo et al. (2007), episodes of financial volatility tend to produce effects on real variables only in short-run. A plausible explanation is that the more significant effects of sudden stop on domestic variables are absorbed extremely rapidly, within one or two quarters. VAR models in levels are not well-equipped to detect these types of changes. Indeed, such models analyze the responses of macroeconomic variables to shocks of standard magnitude (usually one standard deviation),

and not to unusual disturbances proper to crisis episodes. In addition, the main purpose of VAR models is not to identify crisis events. Crisis episodes are relevant only if they lead to structural breaks in the studied macroeconomic series.

Finally, we see that GDP decreases in the aftermath of the real depreciation. Three competing explanations are advanced in this paper. First, real depreciations may be indicative of economic activity slowdown in the medium-term. This interpretation is not confirmed by the behavior of the FOREX variable. Indeed, while a negative response of FOREX to a real depreciation is expected -due to capital outflows -we observe in fact insignificant reactions of this variable. Real exchange rates innovations do not explain a significant share of FOREX variance in the five countries. Second, an alternative of the negative relationship between GDP and real depreciations rests on the presence of negative balance sheet effects.

The inability to borrow abroad in local currency -the so-called original sin-and the dollarization of the domestic economy can induce currency mismatch in the balance sheets of public and private agents. Using the degree of dollarization estimated by [START_REF] Reinhart | Addicted to Dollars[END_REF], such explanation seems relevant only in Argentina and Uruguay 10 . Finally, a last explanation, most likely relevant in countries with low degrees of dollarization (Brazil, Chile and Mexico) suggests that the negative correlation between GDP and real exchange rate depreciations stems from shocks that induce both a real depreciation and a decline in GDP.

Negative terms of trade shocks result in such negative correlation.

Credibility matters

Given the importance of the inflationary history of numerous LACs, the responses of interest rates to innovations in consumption prices are particularly significant. Responses of interest rates are especially important to consider because they allow us to discriminate between credible and less credible countries. In Argentina and Chile, interest rates decrease or do not react after a CPI shock. In these two countries, inflation expectations are well anchored by the monetary regime in place in each country. Recall that from 1991 to 2001, Argentina had experienced a currency board arrangement while Chile had adopted an inflation targeting framework since 1991. In countries with soft pegs and a monetary policy not based on an inflation targeting framework -as Brazil, Mexico and Uruguay-inflation expectations are imperfectly anchored. So, in such countries, shocks on prices induce higher fluctuations.

Variance decompositions exhibit a clear picture. In Argentina and Chile, CPI innovations explain a mild or marginal share of the interest rates variances, respectively 8.7 and 0.2 contemporaneously; and 12.8 and 1.8 after 16 quarters, while in Brazil the respective shares 10 . According to the authors, Argentina and Brazil belong to Type I dollarization in which domestic and external liability dollarization co-exist; Uruguay is a dollarized economy of Type II where dollarization is predominantly of a domestic nature; and Chile and Mexico are Type III dollarization: the main part of debt in foreign currencies is external. Degrees of dollarization differ among our countries: high in Argentina (index 20 on a scale that goes from 0 to 30) and Uruguay ( 21), but weak in Brazil (7), Chile (7) and Mexico (5).

are 70.9% and 68.2%. In this latter country interest rate innovations explain only a weak share of the CPI variance (0% contemporaneously and 10.2% after 16 quarters). As expected, CPI shocks lead to real exchange rate appreciations. In comparison with countries enjoying imperfect monetary credibility, the responses of the real exchange rates are short-lived in Argentina and Chile.

Policy recommendations

Recently, a set of publications displayed a critical balance of stabilization and structural reforms in emerging economies, and particularly in the LA ones. In line with a large stream of critics on the "Washington-Consensus-style" reforms, the contribution of Hausmann, Rodrik, and Velasco ( 2005) rejects the idea of a unified paradigm altogether, advocating idiosyncratic policies adjusted to specific features and constraints. But these conclusions are not shared by all observers: for Zettelmeyer ( 2006) the unusually high macroeconomic volatility of the last decades is a common feature of the main part of LACs, in spite of recent improvements in macroeconomic management and structural reforms. [START_REF] Singh | Latin America: Sustaining Reforms and Growth" Remarks by Anoop Director of the Western Hemisphere Department International Monetary Fund Delivered at investors' meetings in Lima[END_REF] explains this vulnerability by a combination of domestic and foreign factors. From the domestic side, he listed mainly fiscal rigidities, weak institutions, low productivity, strong inequalities and unemployment or insufficient market competition. The consequences are excess demand, inflation, financial instability, external imbalance and unofficial dollarization.

These characteristics limit the scope for macroeconomic management faced with external shocks (unstable terms of trade or sudden stops of capital flows). Further, from an optimal currency area perspective such volatility could hinder regional integration, increasing the costs of policy coordination.

Our outcomes are in line with this evidence, entailing two main policy implications. First, foreign variables bring on a near-common business cycle in the region: LACs tend to react similarly to the same foreign shocks whatever the exchange rate regime. Above all, we do not detect significant adverse asymmetric external shocks among our studied countries even if different degrees and types of dollarization imply that LACs are more or less sensitive to international financial shocks. In other words, most external shocks are common to LACs and characterize global shocks. Contrary to several studies, we also find that real channels seem as important as financial ones in explaining the influence of foreign variables on domestic ones in our sample. Faced with such a common fate, an important question is to determine to what extent a monetary union may insulate against such shocks. Using probit panel regressions to investigate whether countries forming a monetary union have a lower occurrence of sudden stop episodes and of current account reversal episodes, and whether they are better able to absorb external shocks, [START_REF] Edwards | Monetary Unions, External Shocks and Economic Performance: a Latin American Perspective[END_REF] finds that belonging to a currency union has not lowered the probability of a sudden stop or a current account reversal, and external shocks have been amplified in currency union countries.

Another way to face adverse foreign shocks is to favour more exchange rate flexibility in order to use them as bumper. Besides, since 2002 LACs exhibit a clear shift to more flexible exchange rates. Obviously, it leads to higher bilateral exchange rate volatility and as such, it may be unfavourable to economic and financial integration. But this stage could be a transitory process, necessary to put the house in order if during this adjustment, LACs strengthen policy coordination. Institutional frameworks for this purpose already exist: for instance, in the Mercosur case, the Protocol of Ushuaia (1997) established a permanent structure dedicated to coordination. Targets and procedures intended to allow the convergence of public deficits and the debt ratios were defined. A high-level macroeconomic Group of surveillance equivalent to the Ecofin council in the European Union was created. In 2002 the project of the Monetary Institute for Mercosur was launched. Clearly, policy coordination improves the ability of LACs to respond to external shocks, but coordination does not suffice:

LACs may create a regional monetary fund in order to use it in case of adverse external shocks. The boom in commodity prices -that began in 2002-is an exceptional opportunity to favor the accumulation of foreign exchange reserves dedicated to this fund. The main point is that such a regional fund, already envisaged in East and South-East Asia, may function as a mutual insurance mechanism11 . But is such a mechanism optimal in case of symmetric shocks? It is dubious. First because international financial markets could better allocate risks between countries in which the business cycles are asymmetric than in a set of countries sharing the same business cycle (except perhaps in case of high transactions costs differentials) (see [START_REF] Obstfeld | International Capital Mobility in the 1990s[END_REF] 12 . Second because the strong correlation between country risk ratings and risk of contagion effects in regional zones as the Mercosur can be explain by the imperfect information prevailing in international financial market: each time one of the partners -and particularly one of the largest-was overtaken by a crisis, the most prevalent reaction amongst the other members was a fear of spillovers. As noted by [START_REF] Machinea | Macroeconomic Coordination in Latin America: Does It Have a Future?[END_REF], the usual response of policy makers in other countries has been to send out signal that differentiates them from their distressed neighbor in an effort to influence the mainstream perception on financial markets 13 . Obviously, the contagion issue is linked with the credibility one.

The question of domestic credibility is the second policy issue of our paper, in particular monetary policy credibility. Indeed, our estimates do not allow us to distinguish countries according to their exchange rate regimes. A better distinction to analyze responses to similar shocks is based on the different degrees of credibility. As is well-known, LACs have a long history of inflation resulting in fragile inflation expectations anchors. Since the beginning of 1990s, credibility gains have been impressive in LACs. The main point is that, as suggested by the endogenous OCA approach, differences in authorities' credibility may represent an impediment to the convergence targets. To favor economic policy convergence, it seems important that LACs adopt a similar anchor. Shock asymmetry between the LACs and the United States implies that the US dollar is not a good candidate. As a result, LACs need an alternative anchor to dollarization to avoid "the perennial misuse of monetary policy" by their central banks (Corbo, 2001: 246). Trade diversity from a geographic point of view may favor a peg to a common currency basket. But such a peg lacks transparency. An alternative solution, suggested by Eichengreen and Taylor (2004 and more recently by [START_REF] Rose | A Stable International Monetary System Emerges: Bretton Woods Revisited[END_REF], would be for LACs to adopt a similar inflation target in the conduct of their monetary policy.

Brazil, Chile and Mexico have already adopted such a monetary framework. Inflation targeting provides two main advantages that are especially important for LACs. First, central banks with inflation targeting tend to strengthen their credibility. Second, not only is such monetary regime more viable than fixed or intermediate exchange rate arrangements, but it also favors exchange rate stability. In other words, it seems possible to decrease bilateral exchange rate volatility without an explicit coordination mechanism that is very difficult to implement in the area. A similar monetary framework -less ambitious relative to policy coordination-may lead to less volatility. Recent evidence exhibits mixed results. Brazil, Chile and Mexico have experienced better inflation performance than Argentina and Uruguay, two countries that have adopted a monetary aggregate target in the aftermath of their exchange rate regime collapse. From this standpoint, inflation targeting has improved economic policy credibility in significant LACS. But at the same time, Argentine and Uruguayan experiences suggest that the authorities of these two countries follow a de facto managed floating regime 13 And the authors explain: "Their reason for doing so is that the adverse effects to be expected in the short run have been of such magnitude, and the decision-making horizon for governmental and private-sector actors has shortened to such a point that the ratio between the perceived costs and benefits of integration has grown drastically worse".

in which their exchange rates are confined within a very narrow range. As a consequence, these multiples objectives damage the credibility of their central banks leading to a slowdown of their inflation convergence towards the Brazilian and the Mexican indexes (the best performers). Such a policy does not favor economic policy convergence between LACS.

Credibility gains may also favor the financial structure convergence among LACs. As our results above suggest, the lack of financial structure convergence -mainly due to different degrees and types of dollarization-is an additional factor explaining the slow economic policy convergence. Indeed, dollarized countries such as Argentina and Uruguay are especially sensitive to specific shocks (external financial shocks inducing real exchange rates depreciations) limiting their ability to converge with other LACs from an economic policy point of view. More credible monetary policies may lead to a decrease in the degree of dollarization, reducing the currency mismatch and the external financial vulnerabilities of LACs. Indeed, as stressed by [START_REF] Levy-Yeyati | Financial Dollarization: Evaluating the Consequences[END_REF], dollarization is a rational response of economic agents to expectations of high and volatile inflation. Recent trends are promising.

Except Uruguay, and to a lesser extent Argentina, all studied countries have significantly reduced their currency mismatch since 2001 (IMF, 2007: 35). [START_REF] Cowan | Sovereign Debt in the Americas: New Data and Stylized Facts[END_REF] show that not only domestic public debt follows a significant dedollarization trend, but also that some

LACs are now able to borrow in domestic currency in international capital markets decreasing the magnitude of the original sin14 .

Concluding remarks

Our results converge to indicate that LACs are influenced by foreign variables, either real ones for Brazil, Chile, and Mexico, or financial ones for Argentina and Uruguay. Our attempt to test the relevance of the sudden stop literature leads us to mixed conclusions. If our proxy of international capital inflows -the FOREX variables-is significantly explained by foreign financial variables, the analysis of domestic variable responses to FOREX shocks does not follow the predictions of common knowledge.

We need to be cautious in the interpretation of our results. On the one hand, our analysis does not take into account the real convergence process. [START_REF] Camarero | Monetary Union and Productivity Differences in Mercosur Countries[END_REF] study such a process by considering productivity differences among Mercosur countries. Over the period 1960-1999, the authors find the presence of a "Mercosur club" meaning a real convergence process. Regional integration has played a significant role in this process. As suggested by [START_REF] Camarero | Monetary Union and Productivity Differences in Mercosur Countries[END_REF], real convergence raises the question of the level of the exchange rate chosen by each country at the entering date in the monetary union. Indeed, as most LACs know, during a catching-up process, their equilibrium exchange rates may change. On the other hand, our VAR models ignore regional interdependencies despite the fact that spillovers within the region exert a significant influence on the LACs' business cycles. For instance, Uruguayan economic activity depends mainly on Argentine and Brazilian business cycles. In addition, some empirical studies suggest that financial market interdependencies may explain exchange rates movements within the region 15 .

Figures display for each The results display the fraction of variance in a given variable (in %) after 16 quarters due to foreign and domestic shocks indicated in columns.

  Figures display for each country the responses of GDP to GDPG7, WPNCO and EMBI shocks respectively. Argentina: shocks GDPG7 WPNCO EMBI Brazil: shocks GDPG7 WPNCO EMBI

. To this end, we use Chow tests.

A detailed analysis of insurance mechanisms for LACs in beyond the scope of this paper. For an overview, see World Bank (2007).

As noted by the World Bank, the problem overcomes the LAC area as for any mutual found or market insurance "the diversification margin is relatively limited in light of the high correlation of credit risk within the emerging market class, limiting the potential size of credible coverage and increasing its costs",(World Bank, 2007: 22-23).

According to the authors, the size of the economy and the degree of development of domestic financial markets exert an influence on this trend. This latter factor is in part linked with macroeconomic policy credibility.

Chili:

The Chilean economy is particularly sensible to international financial mayhem: so, the main break is due to the Asian Crisis, in 1997, hitting all the variables except the GDP. But the uncertainty following the Argentinean crisis is perceptible as much on the Exchange Rate as on the Interest Rate.

Mexico:

Obviously, the Currency Crash of 1994-95 hit all the real and nominal variables, beginning in the last quarter of 1994 with the Foreign Reserves, the Interest Rate, and then hurting the Exchange Rate, the CPI, and the GDP in 1995.

Uruguay:

The introduction of structural breaks in the case of Uruguay could be discussed. Although some shocks are obviously non-random ones, the high frequency of macro-fluctuations in the Uruguayan case makes break detection difficult. However, two shocks are clearly perceptible, with a break on the GDP (due to the Brazilian Currency Crash at the end of 1998) and a break on all the macroeconomic variables (except the CPI) after the Argentinean Crisis of 2002.

Appendix 4 Forecast Error Impulse Responses of One Standard Deviation (Innovations ±2 SE) 1-Are International Transmission Real or Financial?


Dennis Van Der Meer

Simon Camal

George Kariniotakis

Automatic Feature Selection and Forecast Combination to Enhance and Generalize Renewable Energy Forecasting

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Objectives

Methods

The study presents a forecast model chain that automatically selects and weighs features from diverse input sources and where the probabilistic forecasts are optimally combined.

The proposed approach outperforms the same forecast model without feature selection and is substantially faster. Although nonlinear forecast combination outperforms linear forecast combination, naïve forecast combination is recommended to every forecaster.

Conclusions

Results

Replace with QR code Replace with QR code

• Feature selection results in positive skill when compared to AnEn forecast model without feature selection.

• Generating 6 forecasts with feature selection is approximately 60% faster than generating 1 forecast without feature selection.

• Nonlinear forecast combination (SLP and BLP) performs slightly better than linear forecast combination (OLP and TLP).

• Nevertheless, naïve forecast combination (OLP) is competitive and straightforward to implement.

• QRF and combination methods perform similarly for the first two hours, then QRF performs better.

A forecast model chain is proposed that comprises automatic feature selection and feature weighting by means of filters based on mutual information in tandem with linear and nonlinear forecast combination techniques.

Automatically select features to relieve the curse of dimensionality and avoid overfitting.

Combine probabilistic forecasts to mitigate uncertainty induced by feature selection

Improve the accuracy and reliability of forecasts to increase their value in decision-making.

Acknowledgments

The present research was carried as part of the Smart4RES Project (European Union's Horizon 2020, No. 864337 https://www.smart4res.eu