T Auger

L L Wang

Duane D Johnson

X Gong

First-principles study of grain-boundary wetting in Fe-Σ5(013)[100] tilt boundary

Keywords:

We report an ab-initio study of grain-boundary wetting (GBW) in three systems (Fe/Zn, Fe/Pb & Fe/Bi) on a Σ5 (013)[100] tilt grain boundary. An energy-based approach is pursued and adapted to the study of liquid-metal/ solid-metal interfaces. We provide a sound calculation of the wettability parameter S (i.e., defined as the minimum energy difference between the unbroken initial state and final state with liquid-metal/solid-metal interfaces). Solid/liquid interfaces are modelled by ab-initio molecular dynamics, also allowing a study of structural effects in the liquid phase near the solid interface. We show how surface energies for solid-metal/ liquid-metal interfaces need a special treatment for the liquid-metal chemical potential due to the change in crystallinity near the interface. The computation of the wettability criterion for the three systems correctly predicts the GBW trend known experimentally for these systems. The computations are compared with smallscale TEM experiments on GBW of two systems (Fe/Pb and Fe/Bi) and it is shown to provide reasonable results. This approach offers new perspectives in predicting GBW trends across various systems.

Introduction

Grain boundaries (GB) of metallic polycrystalline materials can suffer from grain-boundary wetting (GBW) when in contact with low melting point liquid metals [START_REF] Straumal | Grain boundary phase transitions and their influence on properties of polycrystals[END_REF]. This is observed in several binary systems where the polycrystalline solid phase evolves spontaneously towards the replacement of grain boundaries with an intergranular film of liquid metal of nanometer-sized thickness. The nanoscale intergranular layer spontaneously forms as a first wetting step before eventually becoming much larger by dissolution diffusion phenomena. Another process where one observes GBW is by alloying several elements (often immiscible) followed by high-temperature annealing above the disordering or phase decomposition temperature [START_REF] Bian | Kinetic and structural insights into the grain boundary phase transitions in Ni-Bi alloys[END_REF]. After thermal annealing, the thermodynamic equilibrium state is reached with GB segregation of impurities or as a complexion of the GB [START_REF] Luo | The role of a bilayer interfacial phase on liquid metal embrittlement[END_REF]. Generally, when GBW occurs, in some cases above a transition temperature, it leads to detrimental mechanical properties with brittle fracture at the intergranular film. The known couples are Al/Sn, Cu/In, Al/Pb-Sn, Al/Ga, Al/Sn-Ga, Cu/Bi, Fe-Si/Zn, Mo/Ni, W/Ni, Zn/Sn and Ni/Bi [START_REF] Straumal | Grain boundary phase transitions and their influence on properties of polycrystals[END_REF][START_REF] Bian | Kinetic and structural insights into the grain boundary phase transitions in Ni-Bi alloys[END_REF]. The kinetics of such spontaneous processes can be truly astounding with GB penetration speed reaching hundreds of micrometers per second in the Al/Ga case or leading to disastrous brittle fracture or complete ruin of an otherwise ductile material like Cu/Bi [START_REF] Hugo | The kinetics of gallium penetration into aluminum grain boundaries-in situ TEM observations and atomistic models[END_REF][START_REF] Wolski | Grain boundary diffusion and wetting in the analysis of intergranular penetration[END_REF]. With ferromagnetic bcc alloys, a well-known case of GBW is Fe-5 %Si/Zn above 650 • C [START_REF] Rabkin | Penetration of tin and zinc along tilt grain boundaries 43 • [100]in Fe-5 at.% Si alloy: premelting phase transition?[END_REF]. At the same time, GBW does not spontaneously occur in other systems such as Fe/Pb [START_REF] Monchoux | Microstucture evolution and interfacial properties in the Fe-Pb system[END_REF]. No experimental data has been reported up to now on the Fe/Bi couple. Instead these former systems when tested against liquid-metal embrittlement (LME) require additional mechanical strain in order to fracture in a brittle manner at grain boundaries; it is then not proven that GBW systematically plays a role in LME in particular for that class. The conventional analysis behind this major difference in GB wettability behavior is supposed to be captured by the energetics of the interface being driven towards GBW if energetically favorable, while the classical LME behavior only occurs as a result of the reduction of cohesion by adsorption of liquid metal at the crack tip [START_REF] Kolman | A review of recent advances in the understanding of liquid metal embrittlement[END_REF]. In the GBW case, transport of liquid species precedes as a requirement or as an aggravating factor for LME.

Wettability of interfaces is classically described energetically using the energy difference between the final (broken) and initial (unbroken) states [START_REF] De Gennes | Wetting: statics and dynamics[END_REF]. For partial wetting, the deviation from equilibrium is a relevant measure of the driving force of the liquid spreading evolution [START_REF] Benhassine | Nonreactive spreading at hightemperature revisited for metal systems via molecular dynamics[END_REF]. For the GBW process, the quantity of interest is the wettability parameter, which is denoted as S and defined as the energy difference between two times the solid-liquid surface energy (eventually depending upon the liquid metal chemical potential) and the GB energy, i.e.:

S(μ LM ) 2γ Fe LM SL (μ LM ) γ GB (1)
In case of a negative energy difference (S<0), the system should evolve towards the formation of an intergranular film characterized by two solid-liquid interfaces, i.e., a GBW transition. For a positive energy difference (S>0), the GB is deemed more favorable energetically. In such case, the solid-liquid system should not evolve from an initial state characterized by a triple line and the GB is to remain bonded unless one provides sufficient mechanical strain energy to allow for a brittle fracture dissipation induced by adsorption (i.e. standard LME process).

The major difficulty in studying this issue is to correctly assess solidliquid interfaces at the atomic level (on the structural as well as energetical point of view). Experimentally it is not feasible to assess the solidliquid interfacial energetics for all the systems that one would like to be able to investigate, with a few exceptions like aluminum-based systems [START_REF] Roth | The temperature dependence of the mechanical properties of aluminum alloys containing low-melting-point inclusions[END_REF]. Indeed, one could turn towards wetting studies on a surface by the sessile drop technique to measure solid-liquid energetics, but this does not allow the investigation of systems showing total wetting. In that case, only an upper solid-liquid interfacial energy limit can be inferred, due to the absence of a triple junction with a non-null Young's angle. Therefore, only partial wetting cases can in principle be interpreted.

Modelling a solid-liquid interface by ab-initio simulations is therefore a reasonable alternative. The systems of interest here are chosen to give unequivocal response. In this respect, describing GBW or its nonoccurrence is very desirable as it would validate the ability of ab-initio methods to predict these behaviors (at least when extrapolated at 0 K). This requires one to model the solid/liquid interfaces by ab-initio molecular dynamics (AIMD) in a first step, then also allowing a study of structuring effects in the liquid phase in the vicinity of a solid surface.

Studying the liquid structure by AIMD (static or dynamic properties) is a well-established technique where Pb and Bi for example have been modelled since the nineties [START_REF] Hafner | Structural and electronic properties of the liquid polyvalent elements. IV. The pentavalent semimetals and trends across the periodic table[END_REF][START_REF] Kresse | Ab initio molecular dynamics: recent progresses and limitations[END_REF]. AIMD has been shown to be able to reproduce several important experimental features of the liquid dynamics (structural and electronic levels) [START_REF] Souto | Ab initio molecular dynamics study of the static, dynamic, and electronic properties of liquid Bi near melting using real-space pseudopotentials[END_REF]. Pb and Zn are considered as simple liquids with a conventional molecular behavior. On the other hand, Bi has a complex molecular structure in the liquid state with either a double structure [START_REF] Davidović | Disordered structure of molten monatomic metals, semimetals and semiconductor[END_REF] or a remnant from a Peierls' instability even in the liquid state [START_REF] Mayo | Short range order in liquid pnictides[END_REF].

For such purpose, the selected systems are the Fe/Zn, Fe/Pb and Fe/ Bi binaries. The Fe/Zn system has a clear GBW transition above 650 • C, which is of concern in industrial context for welding of galvanized steel sheets [START_REF] Rabkin | Penetration of tin and zinc along tilt grain boundaries 43 • [100]in Fe-5 at.% Si alloy: premelting phase transition?[END_REF]. On the other hand, experiments indicate that the Fe/Pb binary (and supposedly also Fe/Bi) does not display such spontaneous transition [START_REF] Monchoux | Microstucture evolution and interfacial properties in the Fe-Pb system[END_REF], while Fe can be strongly embrittled by liquid Pb under tensile loading [START_REF] Dmukhovskaya | Effect of lead on the fracture of Armco iron[END_REF]. In the context of the established LME concern for Fe/Pb-Bi systems, the question of their GBW trend is of prime importance.

An ab-Initio study is here carried out of liquid-metal (LM)/solidmetal (SM) interfaces (Fe(013)/LM) in the three systems (Fe/Zn, Fe/Pb & Fe/Bi). An energetic approach of GBW is then pursued and adapted for a Σ5(013)[100] tilt GB in iron taken an example of a general GB. The calculation of the wettability parameters S for each system is pursued and a subsequent discussion of GBW is carried out.

Computational details

The first-principles scheme utilized to model solid-liquid interfaces uses AIMD and total-energy calculations within the density functional theory (DFT) framework, as implemented in the Vienna Atomic Simulation Package (VASP, version 5.4.4) [START_REF] Blochl | Projector augmented-wave method[END_REF][START_REF] Kresse | Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set[END_REF][START_REF] Kresse | Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set[END_REF] using plane-wave basis set and projector-augmented wave (PAW) pseudo-potentials, within a Generalized-Gradient Approximation (GGA) via the PBE exchange-correlation functional [START_REF] Perdew | Generalized gradient approximation made simple[END_REF]. The Monkhorst-Pack scheme was systematically used for k-point sampling of the Brillouin zone [START_REF] Monkhorst | Special points for Brillouin-zone integrations[END_REF]. The cutoff energy for plane-wave basis was set to 520 eV for all calculations. The electronic minimizations were carried out using a first-order Methfessel-Paxton smearing of 0.05 eV [START_REF] Methfessel | High-precision sampling for Brillouin-zone integration in metals[END_REF]. Extra valence electron d-states were included for Pb (5d 10 6s 2 6p 2 ) and Bi (5d 10 6s 2 6p 3 ), while the standard electronic configuration was used for Zn (3d 10 4s 2 ) and Fe (3d 7 4s 1 ). Spin-polarized calculations were used for all calculations. For each supercell, the correct alignment of Fe spins (ferromagnetic state) was checked at every MD step (relaxation and final convergence).

The ground states for the elements considered in this work (Fe, Pb, Bi and Zn) as modelled by DFT-GGA are reported in the table below. Our results are consistent with previous studies for these elements [START_REF] Bauer | A first principles investigation of zinc induced embrittlement at grain boundaries in bcc iron[END_REF][START_REF] Gonze | First-principles study of As, Sb, and Bi electronic properties[END_REF][START_REF] Lejaeghere | Error estimates for solid-state density-functional theory predictions: an overview by means of the ground-state elemental crystals[END_REF]. For Bi, not including the spin-orbit (SO) coupling is known not to affect charge density computations and therefore it was not included in computations with heavy elements [START_REF] Gonze | First-principles study of As, Sb, and Bi electronic properties[END_REF]. Including SO effects for Pb largely reduces the difference between the experimental measurement and the first-principles value of cohesive energy [START_REF] Lejaeghere | Error estimates for solid-state density-functional theory predictions: an overview by means of the ground-state elemental crystals[END_REF]. It does not affect the surface energies computations.

To model a solid-liquid interface one runs an AIMD simulation of the liquid in a slab geometry in contact with an iron solid surface at high temperature. Several liquid-like configurations can then be generated to be used as a starting point for further structural relaxation. The GB chosen here is a Σ5[100](013) Coincident Site Lattice (CSL) type GB (symmetric tilt of 36.8 • around <001>). Therefore, an iron {013} surface was chosen to face the liquid phase. It is taken to be representative of a general GB from the energy point of view. The configuration allowing to model a solid-liquid interface is a liquid layer sandwiched between an iron slab with a normal along a <013> direction. 20 layers of six iron atoms (so a total M of 120 iron atoms) are used in an orthorhombic supercell of 0.849 nm x 0.895 nm x 3.579 nm size (axis oriented along , [00-1] and [310] directions) with periodic boundary conditions. The number of liquid atoms (hereafter N atoms) is adjusted for each liquid metal so as to reach a density close to the equilibrium one as well as ensuring that the external pressure after relaxation remains below 5 kBar. With this constraint, the Fe/Pb supercell, in addition to Fe atoms, contains 40 Pb atoms while the Fe/Bi one contains 36 Bi atoms and the Fe/Zn one contains 86 Zn atoms. The supercell transverse size allows for a liquid channel nearly 1.8 nm thick (a "nano-channel").

The calculation sequence is the following: First, up to 60 ps of total MD time with a 5 fs time step were simulated for each system using a Nosé thermostat to regulate the temperature at 1000 K. All the atoms of Fe are held fixed during the MD run. Only the gamma point is used to sample the Brillouin zone at this stage. The average force felt by each atom, the energy, pressure and the temperature are recorded at every step to be able to select a configuration labelled as a "liquid" configuration. So, at a late time of the MD simulation, where the effect of the initial configuration is deemed erased, a supercell configuration is selected based upon a force (local minimum of the average force felt by all atoms) or an energy criterion (local minimum of the cell energy) with simultaneously a low pressure within the supercell.

Second, relaxation of the supercell towards fully relaxed forces (average force < 0.02 eV/Å -1 ) is carried out while checking that the external pressure remains lower than 5 kBar (This is the criterion that was used to select the number of liquid-metal atoms in the supercell). The 12 inner layers of (013) of Fe are held fixed at the equilibrium Fe unit cell while 4 outer layers of Fe are allowed to relax at each interface (periodic boundary conditions ensure that there are two solid-liquid interfaces in the cell). The k-points mesh used to relax is 4 × 4 × 1. Some initial configurations led to a full recrystallization of the liquid during relaxation, so they were further disregarded and the process repeated with another starting configuration until a satisfying structural configuration exempt from a full crystallization was obtained. Lastly, upon structural relaxation convergence, a final calculation is carried out using a finer k-points mesh (8 × 8 × 2) to ensure that the energy calculation (E 0 ) is well k-point converged.

The solid-liquid interface energy (in J/m 2 ) is then obtained as a function of the chemical potentials (M Fe atoms and N liquid-metal atoms) in the following way:

γ Fe ML SL (μ LM ) 1 2A (E 0 Mμ Fe Nμ LM ) (2)
with A 0.759855 nm 2 (0.849 nm x 0.895 nm), M 120. E 0 is the converged cell energy, the iron chemical potential is taken from the bulk state. The chemical potential to be considered for the liquid metal is estimated by comparing the crystalline state versus a "liquid" or amorphous-like state. Here the strategy is to calculate the range of chemical potential spanned by a transition from the crystalline state to an amorphous one. This is justified as the liquid state evolves from a layered structure close to the iron (310) surface compared to a more amorphous structural organization a few layers away from it. This leads to a dependence of surface energies to the chemical potential between these two extreme cases. The discussion on grain-boundary wetting will be led using the same approach.

Experimental grain-boundary wetting analysis for Fe-Bi cast alloy

Estimates of the solid-liquid interfacial energy are available for Fe/ Zn and Fe/Pb. For Fe/Zn, based on fit of the hydrostatic pressure dependence of GBW and estimates by Miedema et al., it has been considered that γ SL ≈ 0.3 ± 0.1 J m 2 [START_REF] Straumal | Pressure influence on the grain boundary wetting phase transition in Fe-Si alloys[END_REF]. For Fe/Pb, using annealed Fe/Pb (873 K) sintered powder samples data observed by TEM [START_REF] Monchoux | Microstucture evolution and interfacial properties in the Fe-Pb system[END_REF], it has been evaluated using the dihedral angle technique at γ SL ≈ 1.2 J m 2 [START_REF] Martin | Liquid-metal-induced fracture mode of martensitic T91 steels[END_REF]. To provide for an experimental estimate of the solid-liquid interface energy for the remaining system of interest here (i.e., Fe/Bi), the methodology of studying the equilibrium dihedral angle at a junction between a grain boundary and a liquid inclusion was pursued similarly to previous studies [START_REF] Monchoux | Microstucture evolution and interfacial properties in the Fe-Pb system[END_REF][START_REF] Martin | Liquid-metal-induced fracture mode of martensitic T91 steels[END_REF]. An Fe-10wt% Bi alloy was prepared by vacuum-induction melting. The residual oxygen in the ingot was measured to be 0.001248wt%. A small sample was cut from the ingot and then sealed in a quartz tube under high vacuum, followed by annealing in a furnace at 350 • C for 2 h (above the melting temperature of bismuth). Afterwards, the tube was taken from the furnace and immediately threw into ice water to quench the sample. The triple junction at a random GB in contact with an inclusion of bismuth was characterized by FIB machining of TEM lamella (see supplementary). The dihedral wetting angle at a random GB was consistently measured at 102 • for three triple lines.

Results

Liquid structure

The purpose of investigating the liquid structure is twofold. One would like first to check that the liquid phase can be simulated properly. One criterion is that the pair distribution function should be reasonably reproduced giving access to realistic configurations in the liquid phase by the modelling scheme. Second, one needs to reasonably estimate the range of chemical potential to consider an estimate of solid-liquid interface energy. For such purpose, realistic configurations need to be modelled and used to define this energy. The liquid structure was investigated for Bi and Pb using a cubic supercell (1.666 × 1.666 × 1.666 nm 3 for Bi and 1.6 × 1.6 × 1.6 nm 3 for Pb) filled with 125 atoms (Bi or Pb, respectively) in periodic boundary conditions. A cubic supercell (1.165 × 1.165 × 1.165 nm 3 ) with 100 atoms was used for Zn.

The corresponding density is 9.384 g⋅cm -3 for liquid Bi as compared with an experimental density of 9.87 g⋅cm -3 at 700 K, while the corresponding density is 10.503 g⋅cm -3 for liquid Pb as compared with an experimental density of 10.5 g⋅cm -3 at 700 K. For liquid Zn the corresponding density is 6.69 g⋅cm -3 (compared with a 6.87 g⋅cm -3 experimental density at 723 K). The AIMD was carried out in the canonical ensemble (NVT) with a Nosé-Hoover thermostat. The static properties are investigated via the pair distribution function that is computed over the supercell for N atoms using the following expression:

ρ 0 g(r) 1 N 〈 ∑ i ∑ j δ { r ( r i r j )} 〉 δ(r) (3)
where ρ 0 is the average number density. The coordination number for number of first neighbors is then computed classically using:

n 1 ∫ rmin 0 4πρ 0 g(r)dr (4)
The AIMD reproduces fairly well the pair distribution functions as measured by x-ray diffraction (Fig. 1a for liquid Bi, Fig. 1b for liquid Pb and Fig. 1c for liquid Zn) [START_REF] Waseda | The Structure of Non-Crystalline Materials: Liquids and Amorphous Solids[END_REF]. In the AIMD, the first-neighbor peak distance is found at 0.334 nm for Pb, 0.319 nm for Bi and 0.253 nm for Zn. Previous work by AIMD found similar first-neighbor peak distance [START_REF] Kresse | Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set[END_REF]. Experimentally, the first-neighbor peak is measured at 0.33 nm for Pb [START_REF] Waseda | The Structure of Non-Crystalline Materials: Liquids and Amorphous Solids[END_REF], at 0.325 nm for liquid Bi [START_REF] Dahlborg | On the structure of liquid bismuth[END_REF] and 0.26 nm for liquid Zn [START_REF] Waseda | The Structure of Non-Crystalline Materials: Liquids and Amorphous Solids[END_REF].

For Pb, the agreement is quite good with a first-neighbor coordination number approaching 11, characteristic of a simple liquid. A similar conclusion is reached for liquid Zn, where the AIMD coordination number of 11.5 also signals a simple liquid. For Bi, a shoulder is observed on the right side from the first peak as reported by many previous experiments [START_REF] Souto | Ab initio molecular dynamics study of the static, dynamic, and electronic properties of liquid Bi near melting using real-space pseudopotentials[END_REF]. The first-neighbor coordination number found for Bi is between 7 and 7.5 (r max 0.405 nm). Our modelling of liquid Bi compares well with previous AIMD modelling [START_REF] Souto | Ab initio molecular dynamics study of the static, dynamic, and electronic properties of liquid Bi near melting using real-space pseudopotentials[END_REF]. The shoulder in the pair distribution function of Bi is indicative of the fact that Bi is not a simple liquid but belongs to the class of complex liquids with structural order. The liquid description is therefore sufficient to allow a calculation of the chemical potential of "liquid" Bi, Pb or Zn species.

A liquid configuration with low average force as well as low internal cell pressure was selected and further relaxed using a conjugate gradient algorithm. The Brillouin zone was sampled using a 3 × 3 × 3 k-points grid. The convergence criterion was all force amplitudes lying below 0.02 eV⋅Å -1 threshold. Some trials resulted in recrystallization. When such an evolution was detected, the configuration was no longer considered and disregarded. A final higher density Brillouin zone sampling (7 × 7 × 7 k-points) was carried out to ensure that convergence is reached within 1 meV.

These results can be used to qualitatively estimate the heat of fusion of each of these three elements in a rudimentary approach. Table 2 summarizes the calculations: one can see a relatively good agreement for the enthalpy of fusion for Pb and Bi. The agreement is less good for Zn but remains acceptable.

Solid-liquid interfaces

Fig. 2a-2c show the projected atomic-density profile for Fe/Pb, Fe/Bi and Fe/Zn interfaces respectively as given by the following:

ρ(z) 〈N z 〉 L x L y Δz (5)
where <N z > is the time average of the number of particles between z and z+Δz. L x and L y are lateral sizes of the simulation box. The narrow peaks on the right-hand side come from the rigid position of Fe atoms. On the left-hand side of the plot, one observes the density fluctuations of the liquid (symmetric due to the two Fe(013) interfaces). Near the interface, a layering in the liquid phase can be seen for the three systems extending at most up to three or four layers with an exponential decay moving away from the interface. The correlation length for the decay is larger for Fe/Pb than for Fe/Bi or Fe/Zn, indicating a deeper influence for Pb at the Fe(013) interface. This layering is typical of the one found in atomistic studies of solid-liquid interfaces [START_REF] Geysermans | Molecular dynamics study of the solid-liquid interface[END_REF]. The liquid structure transition from a statistically average ordered state near the Fe surface to an amorphous structure further away. This implies that the chemical potential of the LM varies within a band fixed by the liquid state difference compared with the solid state and its exact value remains unknown. 2). Note that the interval for the chemical potential is restricted to these physical states of the LM species instead of being taken arbitrarily large like in some prior studies [START_REF] Bauer | A first principles investigation of zinc induced embrittlement at grain boundaries in bcc iron[END_REF].

Solid-liquid surface energies and wettability

The projected density of state is plotted in Fig. 4a-c for three interfaces (respectively, Fe/Pb, Fe/Bi and Fe/Zn) for bulk and interfacial atoms. The density of state is broadened for Fe at the interface in the three couples (plotted in red), while there is a noticeable hybridization of the liquid metals d bands with Fe (plotted in blue). Fe majority spin DOS at Fe/Zn interface are shifted to higher energy than those at Fe/Bi and Fe/Pb interfaces (Fig. 4-d).

The charge density difference is defined by the following:

ρ diff (x, y, z) ρ Fe(013)/LM (x, y, z) ρ LM (x, y, z) ρ Fe(013) (x, y, z) (

The charge-density difference plots for the three interfaces are shown in Fig. 5. There is a charge accumulation (in yellow) on the liquid side of the first layer of adsorbed Pb and Bi. This indicates that there is some degree of charge transfer across the interface in these two cases. For the Fe/Zn interface, there is more charge depletion (in cyan) from Fe than in the Fe/Pb or Fe/Bi cases, agreeing with Fe PDOS. No charge accumulation (yellow) is observed on the other side of Zn, unlike Pb and Bi. This is largely responsible for the larger drop in surface energy in contact with Zn than with Pb or Bi. The charge transfer remains contained in the interface plane for that Fe surface orientation.

Discussion

In immiscible systems, such as Fe/Pb or Fe/Bi with partial wetting behavior, liquid inclusions will be expelled from solid solution after elaboration as droplets at GB after processing or heat treatment. There will be a triple junction characterized by a dihedral angle, so that at equilibrium one can infer the ratio of grain boundary interfacial energy relative to the solid-liquid surface energy by:

γ GB 2γ SL cos ( θ 2 ) (7)
Provided that one can exclude liquid species diffusion at the GB and Fig. 1. AIMD-derived pair distribution function g(r) for the liquid state: (a) Bi at 1000 K compared with x-ray data from Dahlborg (at 566 K) [START_REF] Dahlborg | On the structure of liquid bismuth[END_REF] and Waseda (at 577 K) [START_REF] Waseda | The Structure of Non-Crystalline Materials: Liquids and Amorphous Solids[END_REF]; (b) Pb at 1000 K compared with x-ray data from Waseda (at 823 K) [START_REF] Waseda | The Structure of Non-Crystalline Materials: Liquids and Amorphous Solids[END_REF]; (c) Zn at 800 K compared with x-ray data from Waseda (at 833 K) [START_REF] Waseda | The Structure of Non-Crystalline Materials: Liquids and Amorphous Solids[END_REF]. A sample configuration is shown in insert. (color online).

mitigate these issues [START_REF] Ahmadian | Interstitial Segregation has the Potential to Mitigate Liquid Metal Embrittlement in Iron[END_REF][START_REF] Scheiber | The influence of alloying on Zn liquid metal embrittlement in steels[END_REF].

Fig. 3

 3 Fig.3presents the calculation of solid-liquid surface energies as a function of the LM chemical potential variation (Δμ LM μ Bulk μ LM ) for the Fe(013)/Pb, Fe(013)/Bi and Fe(013)/Zn interfaces. The chemical potential scale is limited to the relevant scale between the LM chemical potential defined by the liquid state and the chemical potential defined by the solid ground state (see Table2). Note that the interval for the chemical potential is restricted to these physical states of the LM species instead of being taken arbitrarily large like in some prior studies[START_REF] Bauer | A first principles investigation of zinc induced embrittlement at grain boundaries in bcc iron[END_REF].The projected density of state is plotted in Fig.4a-cfor three interfaces (respectively, Fe/Pb, Fe/Bi and Fe/Zn) for bulk and interfacial atoms. The density of state is broadened for Fe at the interface in the three couples (plotted in red), while there is a noticeable hybridization of the liquid metals d bands with Fe (plotted in blue). Fe majority spin

Acknowledgments

This work was granted access to the HPC resources of the IDRIS computing facility under the allocation 2020-[A0090912063] (GENCI access). TA acknowledges the use of the Cassiopée Arts et Métiers Institute of Technology HPC Center made available for conducting part of the research reported in this paper. DDJ and LLW acknowledge support from Ames National Laboratory and Iowa State University by the U. S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Science and Engineering Division. Ames Laboratory is operated for the U.S. DOE by Iowa State University under contract DE-AC02-07CH11358. XG thanks the support from the National Natural Science Foundation of China, China (Grant No. 52271063 and U21B2066).

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Supplementary materials

Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.actamat.2023.119635.


A Dermoune
email: azzouz.dermoune@univ-lille1.fr

B Djehiche

N Rahmania
email: nadji.rahmania@univ-lille1.fr

ESTIMATION OF THE SMOOTHING PARAMETERS IN THE HPMV FILTER BY

Keywords: Mathematics Subject Classification 2000: 62M10, 62G15 adaptive estimation, Gaussian process, Hodrick-Prescott filter, noiseto-signal ratio, orthogonal parametrization, output gap, potential output, detrending, unobserved components models

We suggest an optimality criterion, for choosing the best smoothing parameters for an extension of the so-called Hodrick-Prescott Multivariate (HPMV) filter. We show that this criterion admits a whole set of optimal smoothing parameters, to which belong the widely used noise-to-signal ratios. We also propose explicit consistent estimators of these noise-to-signal ratios, which in turn yield a new performant method to estimate the output gap.

Introduction

To estimate unobserved economic variables such as the potential output, the non-accelerating inflation rate of unemployment (NAIRU), and the real interest rate, three approaches are usually advocated. These include structural methods which rely on economic modeling, filtering methods which apply statistical devices, and methods which combine both and to which belongs the so-called Hodrick-Prescott Multivariate (HPMV) filter, suggested in Laxton and Tetlow [START_REF] Laxton | A simple multivariate filter for the measurement of potentiel output[END_REF]. The HPMV filter is the standard univariate Hodrick-Prescott (HP) filter (Leser [START_REF] Leser | A simple method of trend construction[END_REF]) augmented with relevant economic information, such that the Phillips curve, Okun's law, production function (see for example Proietti et al. [START_REF] Proietti | Estimating potential output and the output gap for the Euro area: a model-based production function approach[END_REF]) and other indicators such as the output capacity utilization (see Bell [START_REF] Bell | Alternative method to estimate unobserved variables[END_REF], Laxton and Tetlow [START_REF] Laxton | A simple multivariate filter for the measurement of potentiel output[END_REF], Richardson et al. [START_REF] Richardson | The concept, policy use of structural unemployment: estimating a time varying Nairu across 21 OECD countries[END_REF], Cotis et al. [START_REF] Cotis | Estimates of potential output: Benefits and Pitfalls from a policy perspective[END_REF], Boone [START_REF] Boone | Signal extraction for nonstationary time series[END_REF], Chagny and Lemoine [START_REF] Chagny | Ecart de production dans la zone Euro: Une estimation par le filtre de Hodrick-Prescott multivarie[END_REF], Benes and N'Diaye [START_REF] Benes | A multivariate filter for measuring potentiel Output and the NAIRU[END_REF], Trimbur [START_REF] Trimbur | Detrending economic time series: a Bayesian generalization of the Hodrick-Prescott filter[END_REF], and the references therein).

The univariate HP filter captures the output processes, without taking into account any structure or dynamics of other important macroeconomic variables such as inflation rate. Moreover, it suffers from a number of deficiencies such as end-points uncertainty in estimating the output gap. Indeed, since trends are estimated as two-sided moving averages of the data, with future outcomes used to condition the current value, at the end of the sample, where these future values are not available, the filter does not have the benefit of hindsight to infer the current trend value. This means that the precision of the trend estimates deteriorates markedely right when those estimates are needed most to prepare a forecast or make judgments as to appropriate settings of the policy instrument (see [START_REF] Boone | Signal extraction for nonstationary time series[END_REF] and [START_REF] Proietti | Estimating potential output and the output gap for the Euro area: a model-based production function approach[END_REF] for more details). The main idea behind the HPMV filter is thus to be able to correct this deficiency by including more information on the majore macroeconomic variables than just the available data of the variable of interest.

More specifically, the HPMV filter seeks to estimate the unobserved variable (y t , t = 1, . . . , T ) which minimizes

(1.1) T ∑ t=1 (x t -y t ) 2 + α 1 T -2 ∑ t=1 (y t+2 -2y t+1 + y t ) 2 + α 2 ξ 2 t
for appropriately chosen positive parameters α 1 and α 2 . This is a basic HP filter augmented with the residuals ξ from the following estimated economic relationship (the Phillips Curve): x * t = βy t + dX t + ξ t , where, x * t is another explanatory variable that can be explained by the unobserved variable y, and X is an exogenous variable affected by the parameter d.

There are two different approaches to the question of choosing the smoothing parameters α 1 and α 2 . The first approach is to consider the free choice of the smoothing parameters as an advantage. The features of the data can be explored by varying them (see Richardson et al. [START_REF] Richardson | The concept, policy use of structural unemployment: estimating a time varying Nairu across 21 OECD countries[END_REF] and Trimbur [START_REF] Trimbur | Detrending economic time series: a Bayesian generalization of the Hodrick-Prescott filter[END_REF] for instance). The other view often used by macreconomists suggests that the choice of α 1 and α 2 should be on the basis of priors about the relative variance of each of the terms in (1.1); the higher the variance of the term the lower weight it receives in the minimization process (see Laxton-Tetlow [START_REF] Laxton | A simple multivariate filter for the measurement of potentiel output[END_REF] and Boone [START_REF] Boone | Signal extraction for nonstationary time series[END_REF]).

The main result of this paper is an optimality criterion for selecting the best smoothing parameters and construct consistent estimators of them.

In turn, this will be the basis for improving estimation methods of the potential output gap which is a powerful concept in the models of many inflation targeting central banks, which rely on this "unobserved measure" in its policy making.

The optimal smoothing parameters for the HPMV filter are pairs of positive parameters (α 1 , α 2 ) which minimize the difference (using the Euclidean norm) between ŷ((α 1 , α 2 ), (x, z)), that solves (1.1), and

E[ y| x], α * = arg min (α 1 ,α 2 ) ∥E[ y| (x, z)] -ŷ((α 1 , α 2 ), (x, z))∥ 2 ,
where, z = x * -dX.

The organisation of this paper is as follows. Section 2 describes the HPMV filter, and suggests an optimality criterion for choosing the best smoothing parameters, where we show that there is a whole set of such optimal solutions. We also show that the noise-to-signal ratios used in the macroeconomic literature are optimal as well. In Section 3, we apply the recent results obtained in Dermoune et al. [START_REF] Dermoune | A consistent estimator of the smoothing parameter in the Hodrick-Prescott filter[END_REF] and [START_REF] Dermoune | Multivariate extension of the Hodrick-Prescott filter. Optimality and characterization[END_REF] to find consistent estimators of these noise-to-signal ratios. Finally, in Section 4 we present calibration and simulation results that highlight the performance of these estimators. We apply our estimators to analyse the performance of the potential output of the US GDP using US inflation as explanatory variable. The data sets correspond to the period 2002-2009. Compared with the values of the smoothing parameters suggested in the classical HP filter and the HPMV of Laxton-Tetlow [START_REF] Laxton | A simple multivariate filter for the measurement of potentiel output[END_REF], our results show higher performance in estimating the potential output gap, notably at the end-points.

The Hodrick-Prescott Multivariate filter

As a general principle, a macroeconomic time series can be decomposed into its seasonal variation, a 'business-cycle' component , irregular shortterm movements, and its long-term trend (or potential output) component. It is standard practice for economic series to be seasonally adjusted. The univariate HP filter is then applied to the seasonally adjusted series to obtaine cyclical residual, also called the potential output gap, which is an estimate of the combined cyclical and irregular component of the series.

Let x = (x 1 , . . . , x T ) ∈ R T be a time series of observables. The HP filter decomposes x into a nonstationary potential output y ∈ R T and a cyclical component, the potential output gap (noise term) u ∈ R T :

(2.1)

x = y + u.
Given a smoothing parameter α > 0, this decomposition of x is obtained by minimizing the weighted sum of squares

(2.2) ∥x -y∥ 2 + α∥D 2 y∥ 2
with respect to y, where, for a ∈ R T , ∥a∥ 2 = ∑ T i=1 a 2 i . Here, D 2 y is the potential output disturbance obtained by acting the second order forward shift operator D 2 on y = (y 1 , y 2 , . . . , y T ):

D 2 y t := (y t+2 -y t+1 ) -(y t+1 - y t ), t = 1, 2, . . . , T -2, or, equivalently, D 2 y t := 2( y t+2 +yt 2 -y t+1 ), t = 1, 2, . . . , T -2,
measuring the deviation between the value of the output at t + 1, y t+1 and the linear interpolation between y t and y t+2 .

In vector form,

(2.3) P y(t) = D 2 y t , t = 1, . . . , T -2,
where, the shift operator P is the following (T -2) × T -matrix (2.4)

P :=     1 -2 1 . . . . . . 0 0 1 -2 1 . . . 0 . . . . . . . . . . . . 0 . . . . . . 1 -2 1     .
The first term in (2.2) measures a goodness-of-fit by minimizing the deviation between the potential output y t and the observation x t and the second term is a measure of the degree-of-smoothness which penalizes decelerations in growth rate of the trend component, by minimizing the deviation between the value y t+1 and the linear interpolation between y t and y t+2 .

The Hodrick-Prescott Multivariate filter (HPMV in short) seeks to estimate the unobserved variable y as a solution to the following minimization problem (2.5) arg min

y (||x -y|| 2 + α 1 ||P y|| 2 + α 2 ||ξ|| 2 ),
given the following dynamics:

(2.6)

     x = y + u, z := x * -dX = βy + ξ, P y = v,
where, x * is another explanantory variable and X is an exogenous variable affected by the parameter d. Moreover, the noise term u and the signal terms (2.7)

  u ξ v   ∼ N (0, Σ), with covariance matrix Σ :=   σ 2 u I T 0 0 0 σ 2 ξ I T 0 0 0 σ 2 v I T -2   .
Since P is of rank T -2, the signal v = P y does not determine a unique y but rather yields an orthogonal parameterization given by the set of solutions y := {P ′ (P P ′ ) -1 v + Zγ; γ ∈ R 2 }, where, the T × 2-matrix Z satisfies (2.8)

P Z = 0, Z ′ Z = I 2 .
In view of (2.6), the time series (x, z) can be represented in terms of (u, v, ξ) as (2.9)

{ x = u + P ′ (P P ′ ) -1 v + Zγ, z = ξ + βP ′ (P P ′ ) -1 v + βZγ, for some γ ∈ R 2 .
Hence,

(2.10)

  x z y   ∼ N     Z βZ Z   γ, Σ xzy   ,
with covariance matrix

Σ xzy =   σ 2 u I T + σ 2 v Q βσ 2 v Q σ 2 v Q βσ 2 v Q σ 2 ξ I T + β 2 σ 2 v Q βσ 2 v Q σ 2 v Q βσ 2 v Q σ 2 v Q   ,
where, Q := P ′ (P P ′ ) -1 (P P ′ ) -1 P is a symmetric matrix that together with P and Z satisfies the following properties.

(2.11)

P ′ (P P ′ ) -1 P + ZZ ′ = I T , QZ = 0, Z ′ Z = I T ,
and, for any λ > 0,

(2.12)

Q[λI T + Q] -1 = [λI T + Q] -1 Q.
This yields that (2.13)

Q[λI T + Q] -1 Z = 0.
Moreover, the following identity holds.

(2.14)

(I T + λP ′ P ) -1 = ZZ ′ + Q[λI T + Q] -1 .
Since the matrix

( (1 + α 2 β 2 )I T + α 1 P ′ P ) is positive definite, the unique solution ŷ := y((α 1 , α 2 , β), (x, z)) to Problem (2.5) is ŷ = a(I T +λP ′ P ) -1 w, or, using (2.14), (2.15) ŷ = aZZ ′ w + aQ[bI T + Q] -1 w,
where, we have set a := 1/(1+α 2 β 2 ), b := α 1 /(1+α 2 β 2 ) and w := x+α 2 βz. Equation (2.15) defines the descriptive filter that associates a trend y to the time series (x, z) through w, depending on the smoothing parameters α 1 , α 2 , the disturbance operator P and the parameter β.

A criterion to find the best smoothing parameters

Following Schlicht (Schlicht [15], Theorem 1), a way to estimate the smoothing parameters α 1 and α 2 is to let the optimal solution ŷ in (2.15) be the best predictor of any trend y given the time series (x, z), i.e. (2.16) ŷ ≈ E[ y| (x, z)].

In other words, the conditional expectation E[ y| (x, z)] which is usually not smooth, is 'equal' to the smooth trend ŷ.

In the macroeconomic literature dealing with the HPMV filter such as Laxton and Tetlow [START_REF] Laxton | A simple multivariate filter for the measurement of potentiel output[END_REF], Richardson et al. [START_REF] Richardson | The concept, policy use of structural unemployment: estimating a time varying Nairu across 21 OECD countries[END_REF], Boone [START_REF] Boone | Signal extraction for nonstationary time series[END_REF] and Chagny and Lemoine [START_REF] Chagny | Ecart de production dans la zone Euro: Une estimation par le filtre de Hodrick-Prescott multivarie[END_REF], it is suggested that the best smoothing parameters are the noise-to-signal ratios α e 1 = σ 2 u /σ 2 v and α e 2 = σ 2 u /σ 2 ξ . Applying Schlicht's criterion to this filter we will show in Proposition 2.1 below, that there is a whole family of best smoothing parameters to which belong α e 1 and α e 2 .

Since ŷ is a (linear) function of w, adapting Schlicht's criterion, the best smoothing parameters of the HPMV filter, given a realization of (x, z), are those positive parameters α * 1 and α * 2 for which (2.17)

ŷ -Z γ = E[ y| w] -Zγ,
where, γ is the maximum likelihood estimator of γ which we determine using observations from w = x + α 2 βz:

γ = arg min γ (w -E(w)) ′ Σ -1 w (w -E(w)) Since, E(w) = Zγ/a and Σ w = σ 2 v a 2 [cI T + Q], we get, (2.18) γ = aZ ′ w.
In view of (2.10), we have

(2.19) E[ y| w] = Zγ + aQ[cI T + Q] -1 (w -Zγ),
where, c :=

σ 2 u +α 2 2 β 2 σ 2 ξ (1+α 2 β 2 ) 2 σ 2 v . From (2.15) and (2.19) we get ŷ -E[ y| w] = Z(aZ ′ w -γ) + aQ{[bI T + Q] -1 -[cI T + Q] -1 (w -Zγ)}. (2.20)
Hence, by Eq. (2.17), the best smoothing parameters are those positive

α * 1 and α * 2 for which b = c, which yields α * 1 = σ 2 u +α * 2 2 β 2 σ 2 ξ (1+α * 2 β 2 )σ 2 v , or, α * 1 /α e 1 = 1 + α * 2 β 2 (α * 2 /α e 2 -α * 1 /α e 1 )
, where,

α e 1 = σ 2 u /σ 2 v and α e 2 = σ 2 u /σ 2 ξ . Furthermore, (2.21) ŷ * -E[ y| w] = Z(aZ ′ w -γ),
where, ŷ * := y((α * 1 , α * 2 ), (x, z)). We have proved the following proposition Proposition 2.1. The best smoothing parameters of the HPMV filter according to the criterion (2.17) are those positive parameters α * 1 and α * 2 for which

(2.22) α * 1 /α e 1 = 1 + α * 2 β 2 (α * 2 /α e 2 -α * 1 /α e 1 ) , where, α e 1 = σ 2 u /σ 2 v and α e 2 = σ 2 u /σ 2 ξ . In particular, (1) α * 1 = α e 1 if and only if α * 2 = α e 2 . ( 2
) If α * 1 /α e 1 < 1, then necessarily α * 2 /α e 2 < α * 1 /α e 1 . (3) If α * 1 /α e 1 > 1, then necessarily α * 2 /α e 2 > α * 1 /α e 1 .

Optimality of the best smoothing parameters

In the main result of this section, Theorem 2.2, we will show that the best smoothing parameters α * 1 and α * 2 given in Proposition 2.1 minimize the gap between the descriptive HPMV filter ŷ:=y((α 1 , α 2 ), (x, z)) and the best predictor of the trend y given w, E[y|w], i.e. (α * 1 , α * 2 ) = arg min α 1 >0,α 2 >0 ∥ŷ -E[y|w]∥, and give an explicit formula for the expected value of the optimal quadratic gap. In particular, it follows that the noise-to-signal ratios α e 1 and α e 2 are optimal.

Theorem 2.2. We have

(2.23) (α * 1 , α * 2 ) = arg min α 1 >0,α 2 >0 ∥ŷ -E[ y| w]∥, Moreover, the optimal gap ŷ * -E[ y| w] = Z(aZ ′ w -γ) is a centered Gaus- sian vector with covariance matrix cov (Z (aZ ′ w -γ)) = ρZZ ′ . where, ŷ * := y((α * 1 , α * 2 ), (x, z)) and ρ := (σ 2 u + α * 2 2 β 2 σ 2 ξ )/(1 + α * 2 β 2 ) 2 . In particular, E[∥ŷ * -E[ y| x]∥ 2 ] = E[∥Z(γ -Z ′ x)∥ 2 ] = ρ trace(ZZ ′ ).
Proof. Recall that, by (2.20), we have ŷ -

E[ y| w] = Z(aZ ′ w -γ) + aQ{[bI T +Q] -1 -[cI T +Q] -1 }. Now, since Z ′ Q = 0, we also have (Z(aZ ′ w - γ)) ′ aQ{[bI T + Q] -1 -[cI T + Q] -1 } = 0. Hence, ∥ŷ -E[ y| x]∥ 2 = ∥Z(aZ ′ w - γ)∥ 2 + ∥aQ{[bI T + Q] -1 -[cI T + Q] -1 }x∥ 2 . This yields, using (2.21), ∥ŷ - E[ y| x]∥ ≥ ∥ŷ * -E[ y| x]∥ = ∥Z(γ -Z ′ x)∥,
for all positive α 1 and α 2 . Thus α * 1 and α * 2 are optimal. The rest of the proofs is straightforward.

Estimation of the noise-to-signal ratios

In this section we propose consistent estimators of the noise-to-signal ratios α e 1 = σ 2 u /σ 2 v and α e 2 = σ 2 u /σ 2 ξ , and the parameter β as ratios of explicit unbiased consistent estimators of the variances σ 2 u , σ 2 v and σ 2 ξ . To this end, we follow Dermoune et al. [START_REF] Dermoune | A consistent estimator of the smoothing parameter in the Hodrick-Prescott filter[END_REF], Section 3.2, to estimate σ 2 u , σ 2 v , using the fact that (3.1)

P x = v + P u ∼ N (0, σ 2 v I T -2 + σ 2 u P P ′ ).
We get the following unbiased consistant estimates of σ 2 u and σ 2 v .

(3.2)

σ 2 u = - 1 4(T -3) T -3 ∑ j=1 P x(j)P x(j + 1). and (3.3) σ 2 v = 1 T -2 T -2 ∑ j=1 P x(j) 2 + 3 2(T -3) T -3 ∑ j=1 P x(j)P x(j + 1).
Similarily, to estimate σ 2 ξ , we use the fact that the time series

(3.4) P z ∼ N (0, β 2 σ 2 v I T -2 + σ 2 ξ P P ′ ).
is Gaussian and stationary, with the following covariance matrix (V (i, j)) ij :

V (i, j) = β 2 σ 2 v δ j i + σ 2 ξ (P P ′ ) ij = r |i-j| , where, (3.5
) r k =            β 2 σ 2 v + 6σ 2 ξ , if k = 0; -4σ 2 ξ , if k = 1; σ 2 ξ , if k = 2; 0, otherwise,
whose unbiased consistent estimator is explicitly given by rk

= 1 (T -2) -k T -2-k ∑ j=1 P z(j)P z(j + k), k = 0, 1 , 2.
From that we derive the following consistent unbiased estimator of σ 2

( 3 2 + (T -3) ∑ T -2 j=1 P x(j) 2 (T -2) 4
∑ T -3 j=1 P x(j)P x(j + 1)

) -1 , (3.7) α e 2 = ∑ T -3 j=1 P x(j)P x(j + 1) ∑ T -3 j=1 P z(j)P z(j + 1) , (3.8)
As shown in Figure 1, we get a quite good fit of the trend by our method of the considered US real GNP time series. We note, in particular, that at the end of the sample our estimator is performant. The behavior of the HP(1600) filter under the period 2008Q4-2009Q1 is dramatic.

Calibration of the output gap

Figure 2 shows estimates of the output gap corresponding to the classical HP filter with α = 1600, while Figure 3 displays estimates of the output gap obtained using our estimates of (α e 1 , α e 2 , β). It appears clearly that our output gap estimator does significantly better in predicting important turning points of inflation and is more compatible and correlated with the inflation rate.

Conclusion

The HPMV filter is an alternative way of estimating unobserved variables.This method stems from the use of the standard HP filter augmented by relevant economic information. Intuitively, the estimation based on the HPMV filter is not only a simple average going through the observed series, but also allows for input from economic relationships. According to Basistha and Nelson [START_REF] Basistha | New measures of the ouput gap based on the forwardlooking new Keynesian Phillips curve[END_REF] in the same spirit, we can use the Keynesian Phillips curve to estimate the smoothing parameters or another added economic condition. We can also use the state-space model as in Harvey et al. [START_REF] Harvey | Reading in Unobserved Components Models[END_REF] to estimate the parameters by maximum likelihood method. In this paper we suggest an optimality criterion for the smoothing parameters and explicit consistent estimators of the HPMV filter. We show that this criterion admits a whole set of optimal smoothing parameters to which belong the widely used noise-to-signal ratios and devices a rule to fix the choice of these parameters by estimating the noise-to-signal ratios which in turn yields a new performant method to estimate the potential output and the output gap.

v

  and ξ are independent Gaussian vectors with mean zero and respective covariance matrices σ 2 u I T , σ 2 v I T -2 and σ 2 ξ I T , where, I T and I T -2 denote the T × T and (T -2) × (T -2) identity matrices, respectively:

Theorem 3 . 1 .

 31 j)P z(j + 1). Finally, in view of (3.5), combining (3.2), (3.3) and (3.6), we get the following consistent estimators of α e 1 , α e 2 and β. The following statistics α e 1 = -1

13ESTIMATIONFigure 1 :

 1 Figure 1: Potential Output Estimation.

Figure 2 :

 2 Figure 2: Potential Output Estimation.

∑ T -3 j=1 P x(j)P x(j + 1)

, based on the time series of observations P x and P z, are consistent estimators of the smoothing parameters α e 1 , α e 2 and the parameter β.

Simulation and calibration

Numerical examples

To illustrate the performance of the estimators αe 1 , αe 2 and β, we give a numerical example based on simulations drawn from the distributions of P x and P z for different values of the time horizon T , and specified values of α e 1 , α e 2 and β. The estimator αe 1 of α e 1 is exactly the one obtained in Dermoune et al. [START_REF] Dermoune | A consistent estimator of the smoothing parameter in the Hodrick-Prescott filter[END_REF] for the standard HP filter, thus suffering from the same deficiency in the sense that it may not be good enough to estimate values of α e 1 larger than 1. Therefore, we restrict our simulation exercise, reported in the numerical example below, to the case α e 1 ≤ 1. In Tables 1, 2 and3

Calibration

The data used are from U.S. Department of Commerce, Bureau of Economic Analysis.

• The quarterly US real GDP (output) under the period January In the sequel, we first estimate the potential output, y, with ŷ and then the output gap, u, which is the difference between the actual output, x, and potential output, ŷ.

Calibration of the potential output

In Table 4, we collect the estimated values of the statistics ,α e 1 , α e 2 and β based on our estimators