Théo Guyard

Clément Cédric Herzet

Clément Elvira

Elvira Screen

Relax

Ayşe-Nur Arslan

Screen & Relax for Sparse Support Identification

Keywords: Sparse problems, screening tests, relaxing tests

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Sparse representations arise in many application fields such as machine learning and signal processing. They usually result from the resolution of some convex optimization problem of the form x ∈ arg min x f (x) + λg(x), where f (•) is a loss function, g(•) is a sparsity-inducing regularization [START_REF] Shen | Structured sparsity promoting functions[END_REF] and λ > 0 is a tradeoff parameter. The latter problem can be addressed with first-order optimization methods such as Proximal Gradient (PG), Coordinate Descent (CD) or Alternating Direction Method of Multipliers (ADMM). A popular strategy to exploit the problem structure is to enhance these algorithms with screening tests to detect the position of zeros in x . This allows shrinking the problem dimensionality and saving computational resources during the resolution [START_REF] Xiang | Screening tests for lasso problems[END_REF]. We propose a complementary strategy, dubbed relaxing, to detect the position of non-zeros in x . We then leverage this knowledge to perform second-order iterations on the latter subset of entries, thus expecting improvements in the convergence rate. The figure shows the improvements allowed by screening and relaxing on two largescale datasets, namely bctgca and cancer, drawn from the HDRM database. We set f (•) as a Leastsquares and Logistic loss, respectively, and g(•) as an Elastic-net regularization. For each dataset, we solve the corresponding problem for 20 logarithmically spaced values of λ to span different working regimes and we report the overall solution time. We observe an acceleration factor up to three orders of magnitude when implementing both screening and relaxing tests compared to a vanilla implementation of PG, CD and ADMM, which assesses the interest for the presented "Screen & Relax" strategy.

cancer

  Without scr. or rlx. With scr. only With rlx. only With scr. and rlx.


Théo Guyard

Gilles Monnoyer

Cédric Herzet

Clément Elvira

Ayşe-Nur Arslan

Safe Peeling for L0-Penalized Problems

Keywords: Sparse problems, 0 -norm, mixed-integer programming, bound tightening

In many application fields such as machine learning, signal processing or inverse problems, it is of paramount interest to solve optimization problems of the form

min x∈R n f (x) + λ x 0 (1)
where f (•) is some loss function, • 0 is the so-called 0 -norm that promotes sparsity in the optimizers by counting the number of non-zeros in its argument and λ > 0 is an hyperparameter [START_REF] Bertsimas | Best subset selection via a modern optimization lens[END_REF]. This problem can be solved effectively via Branch-and-Bound (BnB) algorithms by adding a bound constraint l ≤ x ≤ u to reformulate the 0 -norm using a binary variable through Big-M relations [START_REF] Ben Mhenni | Global optimization for sparse solution of least squares problems[END_REF]. In practice, the choice of the values in the bounds l and u must respect two conflicting imperatives. On the one hand, the additional constraint must not modify the solutions of the initial problem. Hence, the interval [l, u] ⊆ R n should be chosen wide enough to contain all the optimizers of (1). On the other hand, the quality of the relaxations constructed across the BnB tree degrades with the spread of [l, u]. Hence, setting a too-wide interval is likely to increase the number of nodes explored during the BnB algorithm and penalize the solution time.

We propose a new strategy, dubbed peeling, to address this issue. By leveraging a duality property linking consecutive nodes, we locally tighten the interval [l, u] which results in a strengthening of the relaxations constructed. This procedure is carried out in a safe manner, i.e., we guarantee the validity of the BnB procedure and the optimality of the solutions obtained. Peeling can be seen as a generalization of the screening methodology recently introduced for 0penalized problems [START_REF] Atamturk | Safe screening rules for L0-regression from perspective relaxations[END_REF]. Through different numerical experiments, we show that peeling allows significant gains in the number of nodes explored and in the solution time.


Emma Naulin

Marine Lombard

Pascal Retailleau

Elsa Van Elslande

Luc Neuville

Géraldine Masson

Enantioselective and Regiodivergent Synthesis of Dihydro-1,2-oxazines from Triene-Carbamates via Chiral Phosphoric Acid-Catalysis

Conjugated trienes are fascinating building blocks for the rapid construction of complex polycyclic compounds. However, limited success has been achieved due to the challenging regioselectivity control. Herein, we report an enantio-and diastereoselective process allowing to regioselectively control the functionalization of NH-triene-carbamates. Synthesis of chiral cis-3,6dihydro-2H-1,2-oxazines is achieved by a chiral phosphoric acid catalyzed Nitroso-Diels-Alder cycloaddition involving ((1E,3E,5E)hexa-1,3,5-trien-1-yl)carbamates. Moreover, a modular access to three different regioisomers with excellent diastereoselectivities and high to excellent enantioselectivities is obtained by careful choice of the reaction conditions. Utility of each regioisomeric cycloadduct is highlighted by a variety of synthetic transformations. 16 (R)-3h DCM 50 9:0:1 49 94

a General conditions: 1a (0.05 mmol), 2a (0.075 mmol) and 3 (5 mol%) in solvent (0.05 M), 24 h at -40 °C. b Total yield of isolated pure product 5+6+7 after preparative TLC chromatography. c Determined on the crude based on 1 H NMR. d Determined by HPLC analysis on a chiral stationary phase vs racemic. e Performed at rt.

INTRODUCTION

Linear conjugated trienes are important structural motifs found in many natural products and biologically active compounds, 1 and they have many applications in materials science. 2 Consequently, the development of efficient protocols for their synthesis has attracted great attention over the recent decades. 3 Moreover, tThe presence of three double bonds that can be individually functionalized makes them a unique platform for constructing complex molecules. 4 For instance, linear conjugated trienes have successfully been employed in intramolecular Diels-Alder (DA) reactions 5 for synthesizing natural products. 6 By contrast, the use of such non-symmetrical polarized linear trienes in intermolecular and hetero-DA reactions has been rarely investigated, presumably owing to difficult regiocontrol stemming from the presence of two reactive dienes in the same molecule. 7 Furthermore, an unsymmetrical dienophile can react from the distal or proximal mode. [START_REF] Boger | Regioselectivity of the Intermolecular Diels-Alder Reaction of Acyl Nitroso Compounds (C-Nitrosocarbonyl Compounds) and Nitrosoformates (O-Nitrosocarbonyl Compounds). Preparation of Functionalized cis-. Δ.6-1-octalones[END_REF] As a result, up to four regioisomers can be formed, rendering such DA reactions highly challenging (Scheme 1a). Therefore, specific trienes [START_REF] Leach | Transition States and Mechanisms of the Hetero-Diels-Alder Reactions of Hyponitrous Acid, Nitrosoalkanes, Nitrosoarenes, and Nitrosocarbonyl Compounds[END_REF] with a defined geometry (E vs Z), or steric or electronic bias have been mainly used for minimizing the problem of regioselectivity in these reactions. 10 Additionally, only a few enantioselective DA reactions with trienes have been studied to date. [START_REF] Yu | Chiral Gold Complex Catalyzed Cycloisomerization/Regio-and Enantioselective Nitroso-Diels-Alder Reaction of 1,6-Diyne Esters with Nitrosobenzenes[END_REF] In 2011, Chen and Jørgensen et al. reported the first regioselective, diastereoselective, and enantioselective DA reaction of dienals with dienophiles derived from oxindoles and cyanoacetate, catalyzed by a secondary amine through linear conjugated trienamine formation (Scheme 1b). 12 Later, Jørgensen et al. achieved an asymmetric DA reaction between α,β-unsaturated enals and activated chromones through a similar catalytic strategy, affording complex tetrahydroxanthones with high enantioselectivity. 13a In both reports, only terminal dienes reacted in the cycloadditions. To the best of our knowledge, no example of enantioselective cycloaddition involving trienes with internal double bonds as reactants has been reported. In this context and despite the key achievements described above, 14 there is still room for developing new enantioselective DA reactions with linear trienes, paving the way for the production of regiodivergent 15 cycloadducts.

In view of the aforementioned current state, coupled with our recent success in the development of various cycloaddition reactions with diene-carbamates [16][17][START_REF]For the first examples using dienecarbamates in cycloadditions[END_REF] we explored the feasibility of conducting challenging regioselective and enantioselective DA reactions using 1,3,5-triene-carbamates. Up to now, the linear NH-1,3,5-triene-1-carbamates (RNHCOOR') have found limited use despite their potential to undergo diverse cycloadditions around of each unsaturation for the synthesis of advanced compounds. 19 A single example of a DA reaction involving a linear conjugated triene-tertiary-carbamate, namely, N-((1E,3E)-4-phenylbuta-1,3-dien-1-yl)pyrrolidin-2-one, was reported by Maio; the carbamate was found to undergo cycloaddition with maleic anhydride under thermal conditions. [START_REF] Golliher | The synthesis and use of γ-chloro-enamides for the subsequent construction of novel enamide-containing small molecules[END_REF] Interestingly, an opposite regioselectivity was observed when compared with Jørgensen's works 12,13a and only the internal diene reacted to afford the racemic endo-product (Scheme 1c). Motivated from this precedent along with our previous work on diene-controlled regiodivergent (4+2) cycloaddition, [START_REF] Ma | Chiral Phosphoric Acid-Catalyzed Enantioselective Formal [4+2] Cycloaddition between Dienecarbamates and 2-Benzothioazolimines[END_REF] we envisioned that the regioselectivity might be influenced by the nature of the substituent on the triene. Additionally, since the regio-and enantioselectivity can be also strongly influenced by the choice of the dienophile and chiral catalyst, they must be carefully chosen to develop regio-and enantioselective cycloaddition with triene-carbamates. In light of this and guided by our successful work in asymmetric nitroso-DA reaction (Scheme 1d), 17a the nitrosoarenes ,22 and chiral phosphoric acid [START_REF] Akiyama | Selected recent reviews on chiral phosphoric acid catalysis[END_REF] were selected as dienophiles and catalyst, respectively. Herein, we report our findings on a substrate/catalyst-controlled highly enantioselective chiral phosphoric acid-catalyzed nitroso-DA reaction that allows regioselective and regiodivergent production of three out of four possible dihydro-1,2-oxazines from simple linear NH-triene-carbamates (Scheme 1e).

Scheme 1. Trienes functionalization through cycloadditions.

RESULTS AND DISCUSSION

Reaction Optimization and Scope

To begin our investigation, we choose to study the reaction between ((1E,3E,5E)-hepta-1,3,5-trien-1-yl)carbamate 24 (1a) and 1-methyl-4-nitrosobenzene (2a) in presence of chiral phosphoric acid catalyst (Table 1). Performing the reaction with 5 mol% of BINOL-derived phosphoric acid catalyst 3a in dichloromethane (DCM) at room temperature led to the formation of three out of four regioisomeric adducts with an overall yield of 56% in a 6:2:2 ratio (entry 2). Each regioisomer was isolated as a single cis-diastereomer. The major product was found to be the internal distal cycloadduct 6aa, which exhibited an interesting enantiomeric excess of 65%. Terminal adduct 5aa did not benefit of any enantio-enrichment (0% ee), while only residual induction was observed for compound 4aa (<5% ee). Based on our previous work, we wondered whether the two racemic cycloadducts would arise from uncatalyzed process. Performing the reaction in absence of catalyst delivered isomers 5aa and 6aa in 19% and 37% yield, respectively (entry 1). This result supports our hypothesis that the thermal reaction competes with the catalyzed one. Fortunately, this uncatalyzed process could be significantly reduced by lowering the temperature (-15, -40 °C, entries 3-4), thereby improving the regioisomeric distribution (6:0:4 ratio for 4aa/5aa/6aa) but also the enantioselectivity of both compound 4aa and 6aa. A further drop in the temperature to -60 °C was not satisfactory as the reaction became slow and occurred presumably to a significant extend during the work up (entry 5). It is interesting to note that the regioisomers were found to be easily separable by column chromatography. Yields were essentially maintained over a range of solvents, but a strong impact on the regioselectivity was observed (entries 6-9); indeed, while DCM, ethyl acetate and acetonitrile furnished 6aa as the dominating isomer, compound 4aa was the main adduct when using aromatic solvents (toluene and trifluoromethylbenzene). DCM remained however the best choice when considering enantioselectivity for either 4aa or 6aa. Chiral phosphoric acid screening subsequently highlighted the importance of the 3,3' (or 6,6')-substituent of phosphoric acid scaffold (entries 10-16). While a bulky 2,4,6-triisopropylphenyl residue (catalyst 3a) favored the formation of 6aa over 4aa (ratio of 6:4), using less sterically demanding 3,3' aromatic side chains (3b-e and 3g-h), clearly favored compound 4aa (entry 4 vs entries 10-13 and 15-16, ratio of 4aa vs 6aa up to 95:5). In addition, extended aromatic side arms (1-naphtyl or 9-anthracenyl) on BINOL, H8-BINOL or SPINOL backbones, substantially increased the enantioselectivity of 4aa (entries 11-13 and 15-16). Overall catalyst 3d, bearing 9-anthracenyl residues, furnished the best result taking into account yield, ratio and enantioselectivity (4aa: 70% yield, ratio of 6aa/4aa: 5:95, 97% ee, entry 12).

With optimized conditions toward the enantioselective synthesis of regiodefined terminal distal cis-cycloadduct in hands (Table 1, entry 12), the scope of the transformation was examined with a set of diverse nitrosoarenes 2 (Scheme 2). Enantioenriched terminal distal (TD) cycloadduct could be isolated in each case (4ab-4ag) but the nature of the nitrosoarenes exerted a strong influence on results. Electro-rich para-substituted nitrosoarenes (OMe, Me) behaved well in the reaction furnishing terminal cis-cycloadducts with good yields and excellent enantioselectivities (4aa, 4ab, >95% ee). Surprisingly, 2 with p-MeO group, the minor isomer formed is terminal distal instead of expected internal distal (ID). Increasing the catalyst loading to 10 mol% helped in reaching near perfect regioselectivity in the case of 4ab. Overall, regioselectivity was less pronounced (ratio TD/ID between 8:2 and 6:4) for the neutral or electrodeficient nitrosobenzene, nevertheless furnishing high enantioselectivities (4ac and 4ad, 90% ee for both). Use of meta-substituted or para/meta-doubly-substituted nitrosoarenes was also tolerated. In these cases, yield of isolated terminal cis-cycloadducts remained decent and enantioselectivity was high (4ae-4ag, 91% ee). 25 Robustness of the conditions was also demonstrated by performing a large-scale synthesis of compound 4ab (0.5 mmol), thereby furnishing results in line with the optimization data.

Variation of the triene-carbamates was also investigated in the transformation. In addition to benzyl carbamate (NHCbz), tertbutyl carbamate protection (NHBoc) was well tolerated and the corresponding pure TD product 4bb was obtained uneventfully in similar yield and enantioselectivity compared to the N-Cbz protected 4ab. Pleasingly, 1,3,5-trien-1-yl carbamate with sterically demanding or functionalized -substituted alkyl side chains (isopropyl or 3-(benzyloxy)propyl) were also competent partners to build 3,6-dihydro-2H-1,2-oxazin-3-yl)vinyl)carbamates 4eb and 4cb with medium yields but excellent enantioselectivity (96% ee for both). The enantioenriched triene-carbamate 1f (prepared from (S)-citronellal) participated in the reaction with equal efficiency. It is worth mentioning that the enantioselective process was hampered neither by a remote chiral center nor by the presence of an unsaturation in the side chain. a For conditions see Table 1, entry 12 (0.05 mmol scale). b Yield referred to pure TD adduct. c TD/ID Ratio was determined on the crude based on 1 H NMR. d The absolute configuration of 4ab was determined to be 3S and 6S by X-ray analysis (cf Supporting information). e Performed with 10 mol% of 3d. f 0.5 mmol scale and performed with 10 mol% of 3d. g For 6 h of reaction time. h For 2 h of reaction time.

The presence of an alkyl side chain at  position of the triene seems to be important to effectively building enantioenriched terminal distal cycloadduct under such conditions. Indeed, reaction performed with unsubstituted benzyl ((1E,3E)-hexa-1,3,5-trien-1-yl)carbamate [START_REF] Ma | Chiral Phosphoric Acid-Catalyzed Enantioselective Formal [4+2] Cycloaddition between Dienecarbamates and 2-Benzothioazolimines[END_REF] or benzyl ((1E,3E,5E)-6-phenylhexa-1,3,5-trien-1-yl)carbamate (1g) led to poor results (not shown). Interestingly, inspection of the crude reaction performed with 1g, revealed that the main isomer was not in favor of the terminal cis-cycloadduct but in favor of the internal one (albeit with low ee). Form this, we hypothesized that alternative conditions could provide an entry to the selective formation of enantioenriched internal adducts. We rapidly turned our attention to catalyst 3a that favored the internal cis-cycloadduct in our first set of screening (Table 1). Pleasingly, phenylhexatrienyl carbamates 1g furnished a 4:1 ratio in favor of the internal distal vs terminal distal cis-cycloadduct when the reaction was performed with such catalyst under otherwise identical conditions (Table 2, entry 1). As such, compound 6ga was isolated in 42% yield and with 60% ee. The modest enantioselectivity arose probably from thermal background reaction found to deliver racemic 6ga as the sole adduct (entry 2). Enantioselectivity could not be improved using either a higher loading of catalyst 3a, a catalyst bearing less sterically hindered 3,3' substituent, nor by changing the solvent (entry 3 and 5-7). As previously observed, decreasing the temperature to -60 °C was also not possible because of solubility issues. Fortunately, conditions were found to furnish good results (67% yield of internal distal cycloadduct, 84% ee, 95:5 regioisomeric ratio) when using nitrosobenzene (2c) instead of nitrosotoluene (2a) in the reaction (entry 4). a General conditions: 1a (0.05 mmol), 2a (0.075 mmol) and 3 (5 mol%) in solvent (0.05 M). b Total yield of isolated pure product 4+6, after preparative TLC chromatography. c Determined on the crude based on 1 H NMR. d Determined by HPLC analysis on a chiral stationary phase vs racemic. e Performed at rt. f 2c (0.125 mmol). g With 10 mol% of 3a.

The influence of substitution on the nitrosobenzene was further investigated and engaging a set of diverse nitrosoarenes revealed that electronic properties exerted a strong effect on the outcome of the reaction (Scheme 3). Contrary to p-methyl-nitrosobenzene (2a), nitrosobenzene incorporating a phenyl, a chloro, an iodo or an ester group at the para position not only furnished the corresponding cis-3,6-dihydro-2H-1,2-oxazine 6gi-6gk in good yield (62-74%), but pleasingly also with an appreciable enantioselective induction (79 to 84% ee) and excellent regioselectivity. Strong withdrawing groups such as nitro or trifluoromethyl group exerted a clear beneficial effect on the enantioselective process allowing to isolate pure cis-ID regioisomers 6gl and 6gm in 94% ee, each. In addition, results were maintained by scaling the synthesis on a 0.5 mmol scale (6gm, 80%, 9:1; 96% ee). The reaction was not limited to the use of para-substituted nitrosobenzene and meta-substituted 1-chloro-or 1-nitro-3-nitrosobenzene proved themselves to be competent partners. In such cases, internal distal cis-cycloadduct 6gn and 6go were isolated in good yield and high enantioselectivity (85 and 90% ee, respectively). Interestingly, while the reaction with para-nitrosotoluene (2a) was modest in terms of enantioselectivity (6ga, 60% ee, entry 1,Table 2), use of disubstituted nitrosobenzene bearing a para donating (methyl) and a meta withdrawing group (such as nitro) furnished a significantly higher enantioselective induction (6gp, 86% ee). 6-Phenylhexa-1,3,5-triene protected with benzylthiocarbamate or Boc group uneventfully delivered product 6im and 6hm, respectively, in high yield and enantioselectivity. Electronically or sterically biased 6-arylhexa-1,3,5-trien-1-yl) carbamates were also suitable partners in the reaction as exemplified with the synthesis of compound 6jm-6lm, obtained in good yields (70 to 78% yield) and with excellent enantioselectivities (> 91% ee). It should be noted that under the described reaction conditions, 6-aryl hexa-1,3,5-trien-1yl) carbamates led systematically to the internal distal cycloadduct with high regioselectivity ranging from 9:1 to 99:1.

Scheme 3. Scope of enantioselective synthesis of ID cycloadducts 6. a,b,c

a For conditions see Table 2, entry 4 (0.05 mmol scale). b Yield referred to pure ID adduct. c ID/TD Ratio was determined on the crude based on 1H NMR. d The absolute configuration of 6gm was determined to be 3R and 6R by X-ray analysis (cf Supporting information). e 0.5 mmol scale.

Since electron poor nitrosoarenes were particularly effective in building the internal distal cycloadducts 6, we reasoned that nitrosopyridines could be suitable partners too. 26 Gratifying when engaged under the same reaction conditions, a clean reaction took place allowing cis-compound 7gq to be isolated in a good yield and high enantioselectivity (Scheme 4, 85% yield, 96% ee). Following a close inspection, the structure of compound 7gq was established as the internal proximal cycload-duct. A decisive role of the catalyst in this alternative regiochemical cyclisation mode was unveiled, since in the absence of catalysis, i.e., under thermal conditions, only the internal distal adduct 6 was observed (Not Shown).

Scheme 4. Scope of enantioselective synthesis of IP cycloadducts 7. a,b,c

a For conditions see Table 2, entry 4 (0.05 mmol scale). b Yield referred to pure IP adduct. c IP/ID Ratio was determined on the crude based on 1 H NMR. d 0.5 mmol scale. e The absolute configuration of 7jq was determined to be 3R and 6R by X-ray analysis (cf Supporting information).

The complementary regiochemistry was further explored using various nitrosopyridines. Substitution at the 3, 4, 5 or 6-position of the nitrosopyridines was possible leading to cis-cycloadducts 7gr-7gu in yields ranging between 67% and 78%. Enantioselectivities were generally high (96% ee) even if a slide decrease was observed when the substituent was at the 3-position (7gs, 76% ee). Position of the substituent also influenced the regioselectivity that was less pronounced when the substituent was positioned closer to the nitrogen (position 6, 7gu, ratio 7/6: 7:3) compared to remote substitutions (position 3-5, ratio 7/6: up to 99:1). Effective synthesis of internal proximal ciscycloadducts could also be achieved from diverse 6-arylhexa-1,3,5-trien-1-yl) carbamate, incorporating donating or withdrawing groups (OMe, Cl) at their aryl unit or bearing various carbamates protection (tert-butoxy or thio-carbamate). A variable regioselectivity that however remained high in each case allowed isolating the pure regioisomer 4jq-4kq in high yields and excellent enantioselectivity (>90% ee).

Mechanistic Studies

Control experiments were conducted to gain some mechanistic insights. A trapping experiment involving ethanol as an external nucleophile was performed (see the Supporting Information). In this experiment, intermediates could not be identified and the reaction uneventfully afforded the expected terminal distal 3,6-dihydro-2H-1,2-oxazine in 68% yield and 93% ee (Scheme 5a). Although not a definitive proof, this result suggests that this reaction follows the generally accepted one-step, asynchronous concerted mechanism, which is characteristic of hetero-DA reactions. [START_REF] Leach | Transition States and Mechanisms of the Hetero-Diels-Alder Reactions of Hyponitrous Acid, Nitrosoalkanes, Nitrosoarenes, and Nitrosocarbonyl Compounds[END_REF] Uncatalyzed (thermal) and catalyzed reactions differ in enantioinduction and regiochemical outcomes, particularly when targeting terminal distal and internal proximal regions of the carbamate molecule. Involvement of the H-bond between the Lewis basic site of phosphoric acid and the Lewis acidic site of the triene-carbamate (RCO2N-H) could be responsible for such differences in the regiochemistry. We therefore probed this further using N-methylated methyl-or phenyl-triene-carbamate (8 and 9, respectively) as a reactant under the three regiochemical conditions (Scheme 5). While oxazines did form in each case, we noticed that the N-methylated trienes were far less reactive compared to their parent NH trienes. [START_REF]For reduced nucleophilicity of tertiary enamides, see: (a) Carbery, D. R. Enamides: valuable organic substrates[END_REF] . Moreover, regio-distribution of cycloadducts as well as enantioselectivities were profoundly affected. An enantioenriched terminal distal adduct 10 (21% yield, 80% ee) and a racemic terminal proximal adduct 11 (21% yield) were isolated from the reaction that involved Nmethyl hepta-1,3,5-trien-1-yl carbamate 8 and 1-methyl-4-nitrosobenzene (2a) (Scheme 5b). Interestingly, the absolute configuration of compound 10 was found to be opposite to that observed for the one produced independently via a sequential reaction entailing first the cycloaddition of 1-methoxy-4-nitrosobenzene to the NH carbamate in the presence of the same (S)-3d catalyst and a subsequent methylation step (using NaH and MeI, see the Supporting Information). The reaction between Nmethylated 6-phenylhexatrienyl carbamate 9 and 1-nitroso-4-(trifluoromethyl)benzene (2m) predominantly produced a mixture of modestly enantioenriched terminal regioisomers (12 and 13) and the expected internal adduct 14, independently prepared (see the Supporting Information), was essentially absent from the crude mixture (Scheme 5c). Finally, the reaction involving nitrosopyridine 2q led to the formation of a fully racemic mixture of compounds 15 and 16 (Scheme 5d). Given that the regioisomeric distribution and the enantio-induction observed in these cases are clearly different from those observed in the reaction involving the NH precursors, these three control experiments highlighted a strong involvement of the acidic NH function in the reaction mechanism. The relative and absolute configuration of the internal distal adduct (6) and the internal proximal adduct (7) was unambiguously assigned to be (3R,6R) based on the results obtained from the X-ray single crystal structure analysis of enantioenriched 6gm and 7jq obtained using (R)-3a as the catalyst. While direct X-Ray analysis of the terminal distal adduct (4) was not achieved because of the lack of crystalline material, the configuration was determined via postreaction transformation (vide infra)-considering that the catalyst (S)-(3d) was used-and assigned to be (3S,6S).

Based on these results and considering that with complementary Lewis base/Brønsted acid properties, 16b,17,21,23,29 as previously demonstrated, formation of a hydrogen bond or, in other words, protonation between the nitrosoarene and the acidic catalyst can be anticipated, 17a we can propose that each individual regioselective and enantioselective product formation reaction could involve the formation of a highly organized ternary complex containing the catalyst, NH-triene-carbamate, and nitrosoarene, which upon dual activation would form the product (Scheme 6). To facilitate the comprehensibility of proposed transition states (TS), catalyst derived from (R)-BINOL 3a and 3d was chosen. In the general case, interaction between the catalyst and nitrosoarene would occur at the basic nitrogen center of the nitroso residue, as seen in the transition state (TS)-A and TS-B. However, nitrosopyridine could provide an alternative binding mode, in which the azaphilic phosphoric acid 17,[START_REF] Akiyama | Selected recent reviews on chiral phosphoric acid catalysis[END_REF] would interact with the nitrogen of the pyridine and be further stabilized by the lone pairs of oxygen (TS-C). Such a chelation has been proposed in the case of copper-catalyzed nitroso-DA reactions. 22i To add to this, the formation of an additional bond between the acidic NH of triene-carbamate and the Lewis-basic phosphoryl oxygen would result in a ternary complex in which the two consecutive double bonds in the triene need to adopt an s-cis conformation to allow the cycloaddition to proceed. On the one hand, in a narrow cavity, such as the one created as a result of catalyst 3a bearing the sterically demanding triisopropylbenzene side arms, the nitrosoarene will be in close proximity with the internal reaction sites of the 1,4-s-cis triene-carbamate (TS-B and TS-C). On the other hand, the less crowded and π-extended anthracenyl group present in catalyst 3d could favor a 3,6-s-cis conformation of the triene and allow the nitrosoarene to reach the terminal reaction site as seen in TS-A.

Scheme 5. Control experiments.

studies are required to identify the precise origin of the enantioselectivity in such reactions, but in close analogy to our computed results arrived in the context of the enantioselective nitroso-DA reaction with 1,3-diene-1-carbamates, 22i three plausible transition states TS-A, TS-B, and TS-C can be proposed. The (R)-catalyst would impose a fully defined arrangement of the chiral faces, presenting the Si face of the triene-carbamate to the nitrosoarene. Following a dual activation of both reactants, an asynchronous concerted exo-cyclization 30 through these well-organized transition states would effectively lead to the formation of the product with observed (3R,6R) enantioselectivity and diastereoselectivity based on the (R) chirality in the catalyst. Scheme 6. Proposed transition state models.

Applications and Derivatizations

To show the practicability of these protocols, 0.5 mmol-scale synthesis of 4ab, 6gm and 7gr were carried out and they were isolated in similar yields with same level of regio-enantio-and diastereoselectivity (Scheme 2, 3 and 4). To further demonstrate the value of individual regioselective processes, post-transformations were conducted using a representative example of each type of adduct (Scheme 7). Selective manipulation of the enecarbamate residue present in terminal distal cycloadducts was investigated first. The initial plan was to convert the enecarbamate into an α-carbamoylsulfide derivative. 31 However, surprisingly, when compound 4ab was subjected to the catalytic action of para-toluenesulfonic acid in the presence of an excess of thiophenol in DCM, an interesting chiral 2,3,4,7-tetrahydrooxepine derivative (17) was produced in 34% yield (Scheme 7a). [START_REF] Snyder | Recent developments in the synthesis of oxepines[END_REF][START_REF]Exploration of this unusual transformation is currently pursued[END_REF] Enantioselectivity was nearly fully preserved during the transformation, as is evident from the 95% ee for 4ab and the 94% ee measured for compound 17. In addition, the (2S,3R,4R,7S) absolute configuration observed via X-ray analysis allowed the absolute stereochemistry of compound 4ab to be tracked and assigned as (3S,6S). It was also possible to convert the internal distal adduct 6gm into 3,6-dihydro-2H-1,2-oxazine-3-carbonitrile (18) using boron trifluoride diethyl etherate Lewis acid along with trimethylsilyl cyanide (Scheme 7b). [START_REF] Dumoulin | Enantioselective Three-Component Amination of Enecarbamates Enables the Synthesis of Structurally Complex Small Molecules[END_REF] In this transformation, only marginal diastereocontrol was observed relative to the newly formed center (7:3 mixture of diastereoisomers); however, enantioselectivity was essentially pre-served throughout the transformation and each enantiomer presented a high degree of enantioenrichment (90% ee). Hydrogenation of internal proximal styryl-substituted 3,6-dihydro-2H-1,2-oxazin-6-yl carbamate (7gr) was also investigated (Scheme 7c). A clean reduction of both unsaturated bonds occurred to afford 1,2-oxazinane (19) in 66% yield and 92% ee in the presence of a catalytic amount of PtO2 (5 mol%) in EtOAc at room temperature under 1-atm hydrogen. Overall, these transformations demonstrate the value of the cycloadducts in building enantioenriched products with complex architectures. Scheme 7. Stereospecific derivatizations of TD, ID and IP cycloadducts.

CONCLUSIONS

We have explored the regioselective and enantioselective phosphoric acid-catalyzed nitroso-DA reaction using linear conjugated triene-carbamates. We have discovered regioselective divergent cycloaddition pathways that are dependent on the conditions and the terminal substitution of the triene carbamate. Three out of four possible regioisomers can be conveniently constructed with excellent enantioselectivities and regioselectivities. Each regio-cycloadduct was amenable to further transformations and could be converted into enantioenriched products with complex architectures.

ASSOCIATED CONTENT Supporting Information. The Supporting Information is available free of charge on the ACS Publications website at DOI: X-ray crystallographic data (CIF), experimental procedures and additional optimizations studies, computational details, charts of 1 Hand 13 C-NMR spectra, chiral and racemic HPLC traces (PDF). Accession Codes CCDC 2260833, CCDC 2260839 and CCDC 2285002 contains the supplementary crystallographic data for this paper. These data can be obtained free of charge via www.ccdc.cam.ac.uk/data_request/cif, or by emailing data_request@ccdc.cam.ac.uk, or by contacting The Cambridge Crystallographic Data Centre, 12 Union Road, Cambridge CB2 1EZ, UK; fax: +44 1223 336033

Scheme 2 .

 2 Scheme 2. Scope of enantioselective synthesis of TD cycloadducts 4. a,b,c

Table 1 . Optimization of catalytic enantioselective nitroso- DA reaction involving triene-carbamate 1a. a

 1

	entry Cat. Solvent Yield (%)	Ratio	6aa ee	4aa ee
			(4+5+6) b	4/5/6 c	(%) d	(%) d
	1 e None DCM	56	0:7:3	-	-
	2 e (R)-3a DCM	55	2:2:6	65	<5
	3 f (R)-3a DCM	54	4:0:6	84	<5
	4 (R)-3a DCM	55	4:0:6	92	39
	5 g (R)-3a DCM	60	1:4:5	85	10
	6 (R)-3a	Tol	50	7:1:2	60	15
	7 (R)-3a EtOAc	50	2:2:6	60	ND
	8 h (R)-3a MeCN	45	4:1:5	84	24
	9 i (R)-3a PhCF3	50	7:0:2	75	<5
	10 (R)-3b DCM	50	9:0:1	69	32
	11 (R)-3c DCM	55	9:0:1	78	74
	12 (S)-3d DCM	70	95:0: 5	-	97
	13 (R)-3e DCM	55	4:0:1	72	87
	14 j (R)-3f DCM	40	1:1:0	-	40
	15 (R)-3g DCM	55	95:0: 5	-	86

Table 2 . Optimization of catalytic enantioselective nitroso Diels-Alder involving triene-carbamate 1g. a

 2

	entry	Cat.	Solvent	Yield (%)	Ratio	6 ee (%) d
				(4+6) b	4/6 c
	1	(R)-3a	DCM	47	1:4	60
	2 e	None	DCM	71	0:1	-
	3	(R)-3b	DCM	50	1:1	-
	4 f	(R)-3a	DCM	67	5:95	84
	5 f,g	(R)-3a	DCM	60	5:95	82
	6 f	(R)-3a Toluene	30	1:9	46
	7 f	(R)-3a	CHCl3	33	1:4	38

ACKNOWLEDGMENT

ML and EN thank Paris-Saclay University for the doctoral fellowship. We thank the UPSaclay and CNRS for financial support of this work.

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AUTHOR INFORMATION

Notes

The authors declare no competing financial interests.


Rui Li
email: rui.li@orange.com

Bertrand Decocq
email: decocq.bertrand@orange.com

Yiping Fang
email: yiping.fang@centralesupelec.fr

Zhiguo Zeng
email: zhiguo.zeng@centralesupelec.fr

Anne Barros
email: anne.barros@centralesupelec.fr

Reliability Challenges of 5G and Beyond Networks Applications in High-Speed Trains

Keywords: Mobile network, 5G and Beyond, Reliability, High-speed trains, Discrete event simulation

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Introduction

For more than 20 years, ground-to-train communication has relied on the GSM-R system based on 2G. The International Union of Railway (UIC) decides to launch a new system, Future Railway Mobile Communication System (FRMCS), to replace it. As pointed out by UIC (2020), the goal is to usher in 5G for rail networks. GSM-R, often reinforced with redundancy in the application, has been, so far, one of the most reliable systems [START_REF] He | High-speed railway communications: From gsm-r to lte-r[END_REF]). Although GSM-R is still a universal solution for the communication between the train and control center, there are many reasons to upgrade this system, such as the end of the GSM-R system life-cycle and the need to improve the quality of service and quality of experience [START_REF] Masur | Lte/sae -the future railway mobile radio system: Long-term vision on railway mobile radio technologies[END_REF]).

5G and Beyond is undoubtedly the most advanced telecommunication system that will enhance the quality of railway services. The 5G New Radio (5G NR) extends to a higher spectrum band [START_REF] Niu | A survey of millimeter wave communications (mmwave) for 5g: opportunities and challenges[END_REF]), enabling a higher data transfer rate. The 5G Core will be fully virtualized [START_REF] Bonati | Open, programmable, and virtualized 5g networks: State-of-the-art and the road ahead[END_REF]), providing a flexible and tailored network to train services.

Nevertheless, just as GSM needs to be upgraded with further enhancements specific to the requirements to become GSM-R, 5G and Beyond networks need to be carefully implemented and designed to adjust to the specific requirements of railroad operation.

According to 3GPP (2022), seamless communication is crucial for train control service as it conveys important signals guaranteeing the operation of trains. Onboard, seamless communication is also required to provide high-quality services.

However, communication in the high-speed railway scenario faces many challenges. As discussed by [START_REF] Fan | 5g high mobility wireless communications: Challenges and solutions[END_REF], most of these challenges could be grouped under four categories: accurate channel estimation, advanced signal processing, optimized network deployment, and effective mobility management. Since this work addresses reliability-related issues, we focus mainly on network deployment and mobility management. The failure of the network facility is one of the main reasons a train loses its communication service since it would need to connect to different base stations during its movement. The faster a train moves, the faster it needs to change the anchoring base stations, thus the more network elements it uses during a given time. In network management, HandOver (HO) procedure can be another crucial reliability challenge. As 5G and Beyond networks introduce a high spectrum band, the dense small-cell (Al-Falahy and Alani ( 2017)) layout increases HO frequency for high mobility end-users. HO signaling procedure reliability becomes thus more important for providing a seamless connection to high-speed trains.

Some works have addressed the 5G reliability problem, considering low-mobility or nonmobility users [START_REF] Farooq | Continuous time markov chain based reliability analysis for future cellular networks[END_REF]; [START_REF] Qu | Reliability-aware service chaining in carriergrade softwarized networks[END_REF]; [START_REF] Thiruvasagam | A reliability-aware, delay guaranteed, and resource efficient placement of service function chains in softwarized 5g networks[END_REF]). Some works have investigated the HO process management under high mobility and sought to find a better way to avoid wrong HO, failed HO, or missed HO [START_REF] Song | Handover scheme for 5g c/u plane split heterogeneous network in high-speed railway[END_REF][START_REF] El Banna | Handover scheme for 5g communications on high speed trains[END_REF]; [START_REF] Sönmez | Handover management for nextgeneration wireless networks: A brief overview[END_REF]; [START_REF] Tanveer | An overview of reinforcement learning algorithms for handover management in 5g ultra-dense small cell networks[END_REF]). Nevertheless, little attention has been paid to the impact of network infrastructure failure and HO procedure failure on the reliability and availability of high-speed train communication service.

This paper aims to take up the challenges of 5G and Beyond reliability analysis in high-speed train applications. We developed a 5G and Beyond network element model and a moving train model. Combined together, these two models reflect the real communication-related problems a train could encounter during its mission. The reliability and availability of 5G and Beyond network and train telecommunication service are estimated by carrying out discrete event simulations. The main contributions of this work are the following:

• Main challenges in high-speed train communication are discussed The paper has been organized in the following way. We briefly introduce the high-speed train service problem in section 2. In section 3, we present the 5G and Beyond network model and the train model. A high-speed train mission scenario is presented, and the simulation results are given in section 4. Finally, section 5 concludes the work with some remarks and outlines future works.

Problem statement

We consider a generic 5G and Beyond network composed of the Radio Access Network (RAN) and the Core Network (CN). The network architecture is presented in Figure 1. RAN, which transmits, receives, converts and processes the signal, comprises a set of gNB base stations, and each is composed of Radio Units (RUs), Distributed Units (DUs) and Central Units (CUs). The CN, consisting of different Virtual Network Functions (VNFs), that take charge of aggregation, authentication, service control, etc., is divided into the User Plane (UP) with User Plane Function (UPF), and the Control Plane (CP), including VNFs such as Access Management Function (AMF), Session Management Function (SMF), Data Management (UDM), Authentication Server Function (AUSF), etc. As an end-user, a train will connect to the RU with the best signal that covers the area it passes via a 5G NR air interface. Once the train is registered to the network, it will request a Protocol Data Unit (PDU) session to start an endto-end UP connectivity between the UE and Data Network (DN). This connectivity is supported by User Plane, that is, RU, DU, CU-UP, UPF, and the links between them.

The main problem addressed in this work is the reliability and availability-related challenges of communication services applied to high-speed trains. More precisely, a train is considered connected to the internet if the user is registered to the network and it has initiated a PDU session and the whole user plane allocated by the PDU session is reachable and available to the train. We distinguish in the paper two kinds of connection failure: the failure related to User Plane failure and the failure related to reachability.

User Plane failure

When a train starts to travel on the railway, we assume that it is already registered to the 5G and Beyond network. While the train is running, failures from different parts of the network will impact the communication service in different ways:

• If the gNB facility (including RU, DU, and CU-UP) fails, the train directly loses the connection to DN. There are two possible solutions to reconnect to the DN. If there is another available gNB covering the train, then the train will try to re-establish the connection via this available gNB by a re-establishment procedure. Otherwise, the train becomes unconnected and untraceable. Communication service is stopped. The train will wait until the gNB is repaired or until it enters an available gNB coverage area. • If the UP in CN fails, i.e., UPF-UP fails, the end-to-end communication service is interrupted, yet the train is still connected to the gNB. The communication service resumes after the recovery of CN UP.

The Re-establishment procedure (3GPP ( 2021)) is simplified by considering the call flow involving only the RU, DU, CU, AMF, and UPF.

Reachability failure

Since the train is in high mobility, the RU to which it connects can only serve a specific area, as shown in the radio layout example in Figure 2. To guarantee a seamless connection, the train regularly changes the connected RU by HO process at the overlapping covered by multiple RUs. There are different types of HO regarding the implementation and layout of 5G (3GPP ( 2021)). In the scope of this work, we consider two of them:

• Inter gNB-DU and Intra gNB-CU Handover: In this HO procedure, the new and old gNB-DUs are connected to the same CU. The signaling message will not necessarily be sent to CN. This procedure will involve messaging over the source and target RUs, DUs, and their CU. • Inter gNB-CU Handover: In this HO procedure, the signaling will involve messaging over the source and target gNBs (including RUs, DUs, CU), AMF, and UPF.

If the HO procedure fails, the train stays connected to the previous RU. When the RU is no longer reachable to the train, the train will be disconnected from the network and need to reestablish the connection to resume the communication service.

Availability and reliability

To analyze the reliability challenges, the reliability-related terms should be well defined.

For the considered network, we define the availability and reliability from both network and highspeed train communication service perspectives:

• We define network availability as the percentage value of the amount of time the network operator can provide end-to-end service and response to CP signaling messages everywhere by using the 5G and Beyond network deployed in a considered area, divided by the total considered time. • We define network reliability as the ability of the 5G and Beyond network to provide endto-end connection and response to CP signaling messages everywhere in a considered area. We measure network reliability using the Mean Time To Failure (MTTF) of the considered network system. • We define train network communication service availability as the percentage value of the amount of time the end-to-end communication service is delivered, divided by the amount of time the train network communication service is expected to be delivered. • We define network communication service reliability as the ability of the communication service to perform as required for a given time in- terval under given conditions. We describe network communication reliability using MTTF of the train communication service.

Discrete event simulation model

We separate the considered system into two parts: the network facility, "Telecommunication network", and the service user, "high-speed train". A telecommunication network is a set of network functions composed of virtualized applications and physical resources. The train, whose position is known at a given moment, will consume the service the reachable network functions provide.

5G network model

The 5G and Beyond network comprises different elements, such as DU, CU, and AMF in Figure. 1. We assume they all have similar behavior as shown in Figure . 3. They all start from a working state (W) and may fall into a failed state (F) due to software and hardware reasons. This failure will be detected and identified (N).

Finally, it will be either fixed automatically in the case of software and application issues or repaired manually (R). When the element is not in the state (W), all end-users relying on this element fail to use the element, leading to a service connection or a signaling procedure (re-establishment or HO) failure.

Train model

From an end-user's perspective, the train is always in a moving situation. We divide the train's mission into a series of rounds. Each round is represented by Figure . 4. A round starts from the state where the train is initially connected to i th RU.

If the train runs into a Single RU area, it will stay at the connected state unless the connection fails (some of the network elements it uses are in states (F)). If the failure is due to UPF-UP, the train can return to the connected state when UPF-UP is repaired. If the gNB fails, the train will try to re-establish the connection to i th RU if the failed gNB is repaired, and the train then goes back to the connected state. If the train fails to re-establish the connection, it will remain disconnected until a successful re-establishment to j th RU when entering an overlapping zone, where j ̸ = i.

If the train runs into an overlapping area, it can request HO when a better signal is found. If the HO procedure succeeds, the train will connect to j th RU, where j ̸ = i. If the HO procedure fails, the train will retry HO until the train runs outside of the i th RU covering zone. Then the train will re-establish the connection instead of requiring HO. In this area, the connection is also at risk of facility failure. As the train runs in an overlapping area, another RU always exists. Should i th RU fails, it would immediately try to re-establish the connection to the other RU, j th RU, where j ̸ = i.

Both the HO and re-establishment processes change the state of a train by generating a call flow. The re-establishment process changes a train from a non-connected state to a connected state. The HO process allows a connected train to be handed over to another available RU. The train remains connected throughout the HO process.

Interactions between two models

The two models work together in the simulations. When a train starts either a HO or a reestablishment process, it informs the correspond- ing network elements that they will be needed or no longer be needed by the train. When a network element changes its state from (W) to (F), for instance, it will inform the train of the failure. If the train is already connected to the network, it will be disconnected and request a re-establishment process.

Simulation and results

We implement the proposed models in section 3 with the SimPy environment. We consider a railway line of 100 km with locally distributed RAN and one aggregated CN. The gNBs in RAN consist of co-located RUs and DUs at the edge data center and one aggregated CU at the gNB level data center. RUs are assumed to be purely physical equipment and are equally spaced alongside this 100 km line. First RU is at the starting point of the railway, and the last RU is at the endpoint. The RUs in this study can cover an area with a radius of 5 km using the spectrum it can provide. The failure process of the network system is given in Table 1, according to the data provided by the network service suppliers. The composition of our envisioned 5G and Beyond network is given in Table 2. Throughout the simulation, one train runs every hour from the start to the end of the line at a fixed speed of 200 km/h. All network links in this study are assumed ultra-reliable.

Unreliable Radio Unit

In the first scenario, we simplified the network elements to better explain the different perspectives from the network and the train. We consider that only RUs will fail in the network, and the rest of the system is highly reliable. We investigate how the density of radio installations may impact the network and service communication reliability.

From the network operator's perspective, the network availability and reliability are strictly defined by considering the capability to provide endto-end connection and signaling message response at every position (including both single RU zones and overlapping zones) in the considered area. From the train's perspective, the system we consider is changing between a single RU system and an overlapping system dynamically as it travels.

We simulate the trains traveling through the railway for 100 000 hours (about 11 years) and estimate the availability and the MTTF of train network communication service. Via Monte-Carlo simulation, we compared the impact of different numbers of RUs, varying from 12 to more than 20. Figure . 5 and 6 show the availability and reliability metric MTTF for network and service.

A direct computation of the series system helps us validate this result.

Obviously, neither availability nor reliability from these two perspectives is the same. For operators, when the number of RUs is below 20, some parts of the railway are always covered by a single RU. The more RU installation is dense, the larger the number of these single RU zones. The network availability and MTTF thus decrease with the number of RUs. However, if when the number of RUs is more than 20, there is a sudden jump. In fact, the RU setup is considered fully redundant everywhere, covered by at least two RUs (this redundant layout, in reality, is often not affordable for a network operator). The network service availability obtains nine nines (99.9999999%), and the MTTF is largely improved.

For train service, it only considers the RUs it can connect to at its position. A failed RU far from where the train is would not impact end-toend service delivery for the train. At the overlapping zone, the re-establishment procedure helps the train to resume the connection if one of the RU in the overlapping zone fails. Therefore, the more RUs installed, the less time it spends in a single RU area, and the more service can be guaranteed by at least two RUs in the overlapping area. Then the train communication service availability increases with the density of RU installation. With more than 20 RUs in the railway, the communication service availability reaches even 11 nines. However, the Radio Unit is expensive, and it is hard to do maintenance as they are often distributed. With a limited budget, one of the possible solutions could be deploying RUs according to geographical information of the train route and upgrading the existing 3G/4G facility.

Random failures

In the second scenario, we remove the assumption of high reliability on the rest of the network. All elements in gNBs and the CN can fail. Then the system becomes more complex. Still, we compare different Radio Unit densities alongside the railway. The simulation time is 100 000 hours to generate enough failure in the system.

For the network operators, the system is considered available when all network elements work as initially expected to provide end-to-end service, re-establishment request, and HO request anywhere in the considered railway network. The time to fail is the time from when at least one network element fails to when all the failed network elements are repaired.

For the high-speed train, the service is considered available when its connection is established, and all the UP functions it uses work. HO procedure provides seamless connection as it induces no service interruption and thus enhances service reliability. On the other hand, the re-establishment procedure helps an end-user reconnect to the network from either UP or HO failure. Re-establishment can not maintain a connection and always comes with a service interruption. Therefore, unlike HO, the re-establishment procedure can only enhance service availability but does not contribute to service reliability.

The estimated reliability and availability for the network and service from the simulation are shown in Table 3. Similar to the previous scenario, while we increase the number of RUs, the network availability and reliability decrease. However, for communication service, there are more failures during a train's mission, especially minor failures when the number of RUs increases. The reestablishment procedure can guarantee availability since the overlapping area gets larger. Nevertheless, as the number of failures still increases, the MTTF gets shorter, resulting in less reliable communication service. A possible solution for enhancing reliability could be adding redundant items, which may be energy-consuming and expensive for train and network operators.

Conclusion

This paper discussed the reliability of 5G and Beyond network applications on high-speed trains from two different angles. Service operators often focus on the overall system availability and reliability to provide end-to-end connection and signaling requests for the end-users everywhere in the network. In comparison, a high-mobility end-user focuses only on local issues. That is why high-speed train service has a different estimation of reliability and availability than the telecommunication network itself. We also modeled both the 5G and Beyond network and the high-speed train to simulate how high-speed train interacts with the network by reestablishment and HO procedures. The discrete event simulation helps us understand the differ-ent perspectives of network operators and service users on reliability and availability. The result also shows how they change with the density of the Radio Unit facility alongside the railway.

Our assumptions on the radio interface are ideal. Many aspects, such as weather conditions and moving speed, can cause other types of failures during the re-establishment and HO procedures. The failure rates of the system are assumed to be constant. When considering aging systems, degradation models should be applied. However, our current work has already provided valuable information on the reliability challenges of 5G and Beyond networks for high-speed train services.

The continuation of this work will focus on building an analytical model of the complex network system to validate our proposed approach and compare the performance with the discreteevent simulation. Further cooperation with railway companies will help refine the model by including additional information, such as railway geographical coordinates and train schedules, which will add more value to the approach.

Fig. 1 .

 1 Fig. 1. 5G network architecture.

Fig. 3 .

 3 Fig. 3. 5G and Beyond network element model.

Fig. 4 .

 4 Fig. 4. High-speed train model.

Fig. 5 .

 5 Fig. 5. Number of RUs' impact on network and service availability.

Table 1 .

 1 Failure processes of network system.

	Item	MTTF	repair time
	RU	50 years	1 hour
		constant failure rate fixed repair time
	Virtual	52 days	10 s
	application constant failure rate U (0, 10) continuous
	(container)		uniform distributions
	Server	1 year	1 hour
		constant failure rate fixed repair time

Table 2 .

 2 Components of network system.

	Items Instances	Description
	RU	Variable	Physical equipment
	DU	1 for 1 RU	1 app and 1 server
	CU	1 pair for 8 DUs	2 apps and redundant servers
	UPF 1 in total	2 apps and redundant servers
	AMF 1 in total	1 app and redundant servers

Table 3 .

 3 Performance with random failures

	Number Network Network Service	Service
	of RUs availability MTTF availability MTTF
			(hours)		(hours)
	12	99.86058%	55	99.99456% 359
	13	99.84895%	52	99.99512% 344
	14	99.83789%	50	99.99571% 333
	15	99.82612%	48	99.99628% 319
	16	99.81485%	46	99.99686% 308
	17	99.80219%	44	99.99742% 298
	18	99.79151%	42	99.99801% 288
	19	99.78031%	41	99.99859% 279
	20	99.76875%	39	99.99917% 270


John P Sincerely

Conley

Editor. John P Conley

Sy-Hoa Ho

Idir Hafrad
email: i.hafrad@yahoo.fr

Viet Dung Tran

Asymmetric exchange rate pass-through in Vietnam

Keywords: Exchange rate pass-through, asymmetric exchange rate, ARDL models, NARDL models JEL: C22, E41, F31, F33, F41

AcessEcon's editorial software was conceived in 1999 and deployed in 2000. The Economics Bulletin remains the oldest entirely electronic journal in Economics, as well as the oldest open access journal. Patches and improvements have been made continuously in the intervening years. The need for backward compatibility led us to continue to build on the existing foundation for as long as possible. This became increasingly difficult, expensive, and ultimately, infeasible.

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At this point, we estimate(!) that we will launch towards the end of the quarter 1 or beginning of quarter 2 of 2023. Our plan is to start publishing papers in the order they were accepted, in back issues, with 2022 dates.

Introduction

Assuring price stability through maintaining a moderate inflation rate is one of the most important missions of central banks in the world. Since the Bretton Woods agreement, economists has focused on the impact of exchange rates variation on consumer price indexes. The policy regime and central bank credibility are important factors for the exchange rate pass-through (thereafter, ERPT).

As an emerging market, Vietnam can be considered an interesting case study in macroeconomic stability, well-controlled inflation, and regulated monetary policy.

Vietnam is amongst the fastest-growing countries in the world, at 6.5% annual rate over the 2000-2018 period. Besides the high economic growth, the inflation is changing through different stages. It surged to the 2-digit inflation in the crisis period of 2008 and 2011, and it was around 6% for the whole period from 2000 to 2018. Faced by high inflation pressure, the government sets stability price targeting and macroeconomic stability as a priority target. The State Bank of Vietnam (SBVthe central bank of Vietnam) used many monetary policy tools in order to control inflation.

The SBV has geared the exchange rate regime toward more flexibility and in line with exchange foreign markets. USD/VND exchange rate follows a crawling peg exchange rate. The SBV sets the reference rate to orient exchange foreign market, and the exchange rate will move in a band setting by the central bank. Before 2015, the reference rate was the average of latest price trading. After 2015, the reference rate is calculated based on a basket of eight currencies in order-oriented exchange rate for the market. The band is small (±1%, ±2%) in the exchange rate pressure period, and the band is ±3% in the recent period [START_REF] Ho | Symmetric and asymmetric effects of exchange rates on money demand: empirical evidence from vietnam[END_REF].

There are several studies of ERPT in Vietnam, however the evidence is contrasting. [START_REF] Vo | Exchange rate pass-through and its implications for inflation in vietnam[END_REF] and [START_REF] Nguyen | Exchange rate pass-through into inflation in vietnam: an assessment using vector autoregression approach[END_REF] pointed out a low-end ERPT, whereas [START_REF] Pham | Exchange rate pass through into consumer prices in vietnam[END_REF] found a high ERPT. The empirical literature has paid little attention to the issue of asymmetries in ERPT, despite the importance of this assumption for monetary authorities, and a similar study for Vietnam is still missing.

Contextualizing on these unsettled debates on the ERPT, this study contributes to the existing literature by focusing on the scale of the exchange rate pass-through into domestic prices in Vietnam during a long period from 2000Q4 to 2018Q2. Most of the existing studies on ERPT use the linear approaches, whereas this study refers to a novel approach -the Non-Linear Autoregressive Distributed Lags (NARDL) estimator, developed by [START_REF] Shin | Modelling asymmetric cointegration and dynamic multipliers in a nonlinear ardl framework[END_REF], which accounts for the nonlinear and asymmetric aspects of ERPT. This methodology allows in the one hand, to distinguish the transmissions of positive and negative changes of exchange rate into domestic prices, which is not the case with linear models. In the other hand, it allows for analyzing the transmissions of exchange rate in both long run and short run. In addition, to the best of our knowledge, this is the first study on ERPT of Vietnam that takes into account the role of foreign competitor's price. By investigating the evidence on ERPT in a small but open and vibrant emerging country like Vietnam as a comparison to the previous literature which mostly concentrate on other developed countries, we believe this study is of interest to scholars and investors, as well as has profound implications for monetary authorities.

The remainder of this paper is organized as follows. Section 2 briefly reviews existing literature on the subject. Section 3 introduces the data employed and sets out a novel methodology to estimate ERPT in Vietnam. Section 4 discusses the results. Finally, section 5 draws the conclusion.

Literature review

There has been an extensive literature on exchange rate pass-through providing mixed evidence in different economies. For instance, [START_REF] Feinberg | The interaction of foreign exchange and market power effects on german domestic prices[END_REF] documents ERPT to domestic producer prices in the US and Germany to be higher in less concentrated industries. [START_REF] Goldberg | Goods Prices and Exchange Rates: What Have We Learned[END_REF] find ERPT to import prices are smaller in more segmented industries where firms are able to engage in third degree price discrimination. [START_REF] Mccarthy | Pass-through of exchange rates and import prices to domestic inflation in some industrialized economies[END_REF] shows that ERPT into consumer price index and producer price index in industrialized countries is small due to some reasons such as less volatile of exchange rate, stability of GDP growth, and less competitiveness. [START_REF] Campa | Exchange rate pass-through into import prices[END_REF] examine ERPT into import prices using quarterly data of 23 OECD countries from 1975 to 2003 and find compelling evidence of partial pass-through in the short run, especially within manufacturing industries. In the long run, producer-currency pricing is more prevalent for many types of imported goods. Countries with higher rates of exchange rate volatility have higher pass-through elasticities, although macroeconomic variables have played a minor role in the evolution of pass-through elasticities over time.

However, ERPT in the emerging market is higher than ERPT in advanced countries [START_REF] Webber | Dynamic and long run responses of import prices to the exchange rate in the asia-pacific[END_REF][START_REF] Mihaljek | A note on the pass-through from exchange rate and foreign price changes to inflation in selected emerging market economies[END_REF][START_REF] Ghosh | Exchange rate pass through, macro fundamentals and regime choice in latin america[END_REF]. Some studies of ERPT in ASEAN5 countries (Malaysia, Thailand, Singapore, Philippines, and Indonesia) were investigated by [START_REF] Cortinhas | Exchange rate pass-through in asean: implications for the prospects of monetary integration in the region[END_REF]; [START_REF] Siregar | Effectiveness and commitment to inflation targeting policy: Evidence from Indonesia and Thailand[END_REF][START_REF] Vo | Exchange rate pass-through in asean countries: An application of the SVAR model[END_REF]. However, only a few empirical studies have examined the exchange rate pass-through in Vietnam and the majority of those papers have used a vector autoregressive model (VAR). [START_REF] Nguyen | The impact of real exchange rate on output and inflation in vietnam: a var approach[END_REF] use monthly data from 1992M1 to 2005M4 and a set of variables including industrial output, domestic price, real bilateral exchange rate, money supply, trade deficit, and US interest rate. They found that the exchange rate has a small impact on inflation. [START_REF] Vo | Exchange rate pass-through and its implications for inflation in vietnam[END_REF] uses monthly data over only 7 years from 2001M1 to 2007M2 and a set of variables including consumer price index, oil prices, nominal effective exchange rate (NEER), output gap, import price, and money supply. He also found that the ERPT is very small (0.08) and the impact of exchange rate to consumer price is removed after 15 months. [START_REF] Bhattacharya | Inflation dynamics and monetary policy transmission in Vietnam and emerging asia[END_REF] tries to figure out what are the main drivers of inflation in Vietnam from 2004Q1 to 2012Q2, by using growth rates of CPI, output, credit to the economy, nominal effective exchange rate, and the nominal interest rate. The results suggest that the NEER and the foreign inflation are the most important variables explaining the inflation [START_REF] Bhattacharya | Inflation dynamics and monetary policy transmission in Vietnam and emerging asia[END_REF]. [START_REF] Pham | Exchange rate pass through into consumer prices in vietnam[END_REF] studied the same issue based on the period of 2002Q1-2016Q3 and applied a structural VAR model by including growth rate of output (GDP), consumer price (in headline level and its components (disaggregated level of CPI)), money supply, NEER, and interest rate (only this variable in level). He concluded that the correlation between exchange rate pass-through and the average of inflation is above 0.87 after 4 quarters. Some monetary policy variables such as credit to the economy, money supply or interest rate are less explanatory (less 5% or 10%) or have no clear relationship to inflation [START_REF] Imf | What drives inflation in vietnam? a regional approach, vietnam: Selected issues[END_REF][START_REF] Camen | Monetary policy in vietnam: the case of a transition country[END_REF][START_REF] Le | Var analysis of the monetary transmission mechanism in vietnam[END_REF]. Therefore, we will not include money supply, credit, and the interest rate in our model.

The shortcomings of these studies can be summarized as follows: First, all models have used the linear VAR model. These studies only analyze the reaction of a shock to domestic price through Impulse Response Function and do not interpret estimated coefficients. Second, based on linear models, the elasticity of exchange rate on inflation rate is the same for different variations of exchange rate (appreciation or depreciation).

However, several theoretical arguments point out that the ERPT to consumer price could be asymmetric due to market share, capacity constraints, and other rigidities (quantity or export prices) [START_REF] Marston | Pricing to market in japanese manufacturing[END_REF][START_REF] Knetter | Is export price adjustment asymmetric? evaluating the market share and marketing bottlenecks hypotheses[END_REF]Pollard and Coughlin, 2003).

The asymmetry in ERPT has been investigated in [START_REF] Delatte | Asymmetric exchange rate pass-through: Evidence from major countries[END_REF]; [START_REF] Bussière | Exchange rate pass-through to trade prices: The role of nonlinearities and asymmetries*[END_REF][START_REF] El Bejaoui | Asymmetric effects of exchange rate variations: An empirical analysis for four advanced countries[END_REF][START_REF] Brun-Aguerre | Heads i win; tails you lose: asymmetry in exchange rate pass-through into import prices[END_REF]. [START_REF] Delatte | Asymmetric exchange rate pass-through: Evidence from major countries[END_REF] and El Bejaoui ( 2013) investigated an asymmetric non-linear ARDL to assess the ERPT to consumer price index/trade price for Japan, Germany, the United Kingdom, and the United States. [START_REF] Bussière | Exchange rate pass-through to trade prices: The role of nonlinearities and asymmetries*[END_REF] analyzed the relationship between trade prices and exchange rate movements in the G7 economies. Brun-Aguerre et al. ( 2017) investigated asymmetry ERPT into import prices of 33 emerging and developed countries from 1980Q1 to 2010Q4.

In order to contribute to fill these gaps, we apply a NARDL model to reassess ERPT in Vietnam. Our main contributions are (i) to create a foreign competitor price index capturing the impact of foreign price on domestic price; to the best of our knowledge, this paper is the first one that capture the foreign competitor price for ERPT in Vietnam; (ii) to show the different magnitudes of appreciation and depreciation of the exchange rate to consumer price; (iii) to perform the ERPT in both short and long runs; and (iv) to generate dynamic multipliers showing new long-run equilibrium from an initial long-run equilibrium for the consumer price with the positive and negative shocks of the exchange rate.

Model

As we will discuss the potential variables explaining the ERPT in this section, we follow [START_REF] Delatte | Asymmetric exchange rate pass-through: Evidence from major countries[END_REF] to estimate ERPT to consumer price index as follows:

𝐶𝑃𝐼 𝑡 = 𝛼 0 + 𝛼 1 𝑁𝐸𝐸𝑅 𝑡 + 𝛼 2 𝑂𝑖𝑙 𝑡 + 𝛼 3 𝐹𝐶𝑃 𝑡 + 𝜖 𝑡 (1)
where: CPI is consumer price index; NEER is the nominal effective exchange rate;

Oil is the Brent oil price; FCP is the foreign competitor price; and 𝜖 𝑡 is the error term. We used the NEER instead of the bilateral nominal exchange rate because it takes into account (any) shifts throughout the sample period with Vietnam's respective trading partners, unlike the nominal bilateral exchange rate. We computed NEER as the product of the weighted average of the bilateral nominal exchange rate between Vietnam and its top 20 major trading partners1 as follows:

𝑁𝐸𝐸𝑅 𝑡 = 100 * ∏ 𝑆 𝑖𝑡 𝑊 𝑖 20 𝑖=1
where 𝑆 𝑖𝑡 is the nominal exchange rate quoted by a number of local currency unit (VND) relative to 1 foreign currency unit of the trading partner i. 𝑊 𝑖 is their weights, and is normalized (equal to 1) by the shares of the trade share (sum of export plus import) of the 20 trading partners. 𝑆 𝑖𝑡 , 𝑊 𝑖 are normalized in the base year 2000Q4. An increase (decrease) in the NEER implies a depreciation (an appreciation) of the home currency (VND). We expect a positive sign.

Foreign competitor price (FCP) is calculated as the weighted average of foreign price and is derived using the definition of the real effective exchange rate (see [START_REF] Brun-Aguerre | Heads i win; tails you lose: asymmetry in exchange rate pass-through into import prices[END_REF]; López-Villavicencio and Mignon ( 2017)). The specification of FCP is very important in our model. A better indicator to use should be the export price of Vietnam's trading partner weighted by the share of import from these countries [START_REF] Mihaljek | A note on the pass-through from exchange rate and foreign price changes to inflation in selected emerging market economies[END_REF]. However, these data are not available for the main trading partners of Vietnam, especially Asian countries. As a result, we use foreign competitor price. We compute trade-weighted foreign competitor price based on Lopez and Mignon ( 2007 . We expect a positive sign.

We hence use world oil price (Oil) as a proxy for supply shocks to capture effects of external demand pressure on the real side of the economy. We expect a positive sign.

To capture the short-run effect in equation ( 1), we transpose this equation into an ARDL model as follows:

Δ 𝐶𝑃𝐼 𝑡 = 𝑐 1 + 𝜌 𝑗 𝐶𝑃𝐼 𝑡-1 + 𝜌 𝑐 𝑁𝐸𝐸𝑅 𝑡-1 + 𝜌 𝑑 𝑂𝑖𝑙 𝑡-1 + 𝜌 𝑒 𝐹𝐶𝑃 𝑡-1 + ∑ 𝜑 𝑖 Δ 𝐶𝑃𝐼 𝑡-𝑖 𝑝 𝑖=1 + ∑(𝜋 1𝑗 Δ 𝑁𝐸𝐸𝑅 𝑡-𝑗 + 𝜋 2𝑗 Δ 𝑂𝑖𝑙 𝑡-𝑗 + 𝑞 𝑗=1 𝜋 3𝑗 Δ𝐹𝐶𝑃 𝑡-𝑗 + 𝜋 4𝑗 𝐺𝑎𝑝 𝑡-𝑗 ) + 𝑣 𝑡 (2)
Where Δ is the first difference of each variable. We include the output gap (Gap) in the short run (see de Brouwer and Ericsson (1998); Delatte and López-Villavicencio ( 2012)) . Gap is output gap that is computed from GDP in constant price by using Hodrick-Prescott filter. The output gap is used to capture the domestic demand pressure and to take into account the business cycles. When the actual GDP is above the potential GDP it reflects a positive output gap, which means that the economy is growing over its long-run capacity (or over the full employment capacity), leading to an increase in inflation. Hence, we expect a positive sign. [START_REF] Shin | Modelling asymmetric cointegration and dynamic multipliers in a nonlinear ardl framework[END_REF] introduced the decomposition of the exchange rate into its positive and negative partial sums as follows:

𝑁𝐸𝐸𝑅 𝑡 + = ∑ Δ 𝑁𝐸𝐸𝑅 𝑗 + = ∑ max(Δ 𝑁𝐸𝐸𝑅 𝑗 , 0) ; 𝑡 𝑗=1 𝑡 𝑗=1
(3)

𝑁𝐸𝐸𝑅 𝑡 -= ∑ Δ 𝑁𝐸𝐸𝑅 𝑗 -= ∑ min(Δ 𝑁𝐸𝐸𝑅 𝑗 , 0) ; 𝑡 𝑗=1 𝑡 𝑗=1
where 𝑁𝐸𝐸𝑅 𝑡 + 𝑎𝑛𝑑 𝑁𝐸𝐸𝑅 𝑡 -are the partial sums of positive and negative changes of the (log) nominal effective exchange rate, respectively, indicating the depreciation (superscripts +), and appreciation of VND (superscripts -), respectively. Therefore, equation ( 2) changes from symmetry effect to asymmetry effect as follows:

Δ 𝐶𝑃𝑖 𝑡 = 𝑐 + 𝜌 𝑗 𝐶𝑃𝐼 𝑡-1 + 𝜌 𝑐 + 𝑁𝐸𝐸𝑅 𝑡 + + 𝜌 𝑐 -𝑁𝐸𝐸𝑅 𝑡 -+ 𝜌 𝑑 𝑂𝑖𝑙 𝑡-1 + 𝜌 𝑒 𝐹𝐶𝑃 𝑡-1 + ∑ 𝜑 𝑖 Δ 𝐶𝑃𝐼 𝑡-𝑖 𝑝 𝑖=1 + ∑ (𝜋 1𝑗 + 𝑁𝐸𝐸𝑅 𝑡 + + 𝜋 1𝑗 -𝑁𝐸𝐸𝑅 𝑡 -+ 𝜋 2𝑗 Δ 𝑂𝑖𝑙 𝑡-𝑗 + 𝑞 𝑗=1 𝜋 3𝑗 Δ𝐹𝐶𝑃 𝑡-𝑗 + 𝜋 4𝑗 𝐺𝑎𝑝 𝑡-𝑗 ) + 𝑣 𝑡 (4)
We use bound tests in order to ascertain the existence of cointegration. assuming that all variables are integrated of order zero, I(0), whereas the upper critical values are calculated upon the hypothesis that all variables are integrated of order one, I(1).

If 𝑡 𝐵𝐷𝑀 and 𝐹 𝑃𝑆𝑆 are higher than the upper bound critical value, we may conclude on the existence of cointegration. . If we reject the null hypothesis of symmetry, this implies that the model allows for asymmetric effects in both long and short runs. Otherwise, if we cannot reject the null hypothesis of symmetry, this means that the model should only allow for symmetric effects (i.e., linear ARDL).

When the null hypothesis of symmetry is rejected, we can compute the asymmetric dynamic multipliers of the positive and negative changes in the exchange rate as follows:

𝑚 ℎ + = ∑ j=0 h 𝜕𝐶𝑃𝐼 𝑡+𝑗 𝜕𝑁𝐸𝐸𝑅 𝑡+𝑗 + ; 𝑚 ℎ -= ∑ j=0 h 𝜕𝐶𝑃𝐼 𝑡+𝑗 𝜕𝑁𝐸𝐸𝑅 𝑡+𝑗 - (5)
where ℎ → ∞, 𝑚 ℎ + → 𝐿 𝑛𝑒𝑒𝑟 + ; 𝑚 ℎ -→ 𝐿 𝑛𝑒𝑒𝑟 -. The dynamic multipliers could capture the positive and negative shocks of the exchange rate on the consumer prices from an initial equilibrium to the new equilibrium [START_REF] Shin | Modelling asymmetric cointegration and dynamic multipliers in a nonlinear ardl framework[END_REF].

Data and Results

Data

In this paper, we use quarterly data from 2000Q4-2018Q2 2 . The GDP (constant price, 2010) is from General Statistics Office of Vietnam. Other variables are from Datastream Refinitiv. All variables are in logarithm and seasonally adjusted except the output gap.

Results

In this section, we check whether all variables are either stationary I(0) or nonstationary I(1), as shown in Table 1. We then estimate an NARDL model of ERPT as shown in Table 2 and test for the existence of cointegration among all variables. Finally, we implement diagnostic and symmetry tests, as shown in Table 3.

First, we use the Augmented Dickey Fuller test based on the Schwarz information criterion to test unit roots, as presented in Table 1. According to Table 1, except for the variable Gap that is stationary in level, all variables are stationary in the first difference.

[Insert 1 here] Second, we estimate the asymmetry exchange rate pass-through in Equation ( 4) by using a general-to-specific approach to find the optimal lag. Estimations are displayed in Table 2, in which Panels A and C show the short-run and long-run estimations, respectively, whereas Panel B presents the cointegration vector estimations.

Panel A in Table 2 shows the short-run estimated coefficients. The elasticity (passthrough) of positive exchange rate 𝑁𝐸𝐸𝑅 𝑡 + is weak in the short run. In fact, the coefficient is around 0.13 and significant at the 5% level. These results are in line with [START_REF] Jongwanich | Determinants of producer price versus consumer price inflation in emerging asia[END_REF]. This result reflects that an increase of 1% of depreciation of the nominal effective exchange rate leads to an increase of 0.13% of domestic inflation in the short run. However, the effect of appreciation of NEER on domestic inflation is omitted in the short run. This can be explained by the permanent depreciation of VND for a long period.

The response of domestic inflation to movements in the output gap is statistically significant in the short run, and remains entirely coherent with economic theory. The coefficient is around 0.0018, which means that a 1% rise of the output gap might result in an increase of domestic inflation by 0.18%. This finding suggests that output gap drives 2

The choice of this period is based on the availability of GDP in constant price (2010).

inflation in the short run, and the government should be cautious about GDP growth rate target.

The elasticity of oil price (-0.025) is negative and statistically significant. This result may be surprising but is consistent with the objectives of the Vietnamese Petroleum Price Stabilization Fund (PPSF). The aim of the PPSF is to act as a shock absorber to maintain the stability of domestic oil price in the event of high oil price volatility. Thus, the PPSF allows a slow and gradual adjustment of consumer prices, which makes domestic prices (CPI) less responsive to variations of oil price. Consequently, the slight negative effect of oil price, in the short run, on domestic consumer price seems obvious.

Panel C presents the long-run estimated coefficients. The results show that, for all the long-run pass-through coefficients are as expected and statistically significant except for the negative change in NEER (i.e. 𝐿 𝑛𝑒𝑒𝑟 -

). The exchange rate depreciation pass-through on consumer price is 0.861 and significant at the 1% level. This result indicates that a 1% increase of depreciation of the nominal exchange rate will lead to an increase of 0.86% of domestic inflation. The high exchange rate depreciation passthrough level is consistent with [START_REF] Pham | Exchange rate pass through into consumer prices in vietnam[END_REF] who found an ERPT of the same magnitude (0.84). Our finding is similar to ERPT of emerging markets [START_REF] Mihaljek | A note on the pass-through from exchange rate and foreign price changes to inflation in selected emerging market economies[END_REF][START_REF] Mccarthy | Pass-through of exchange rates and import prices to domestic inflation in some industrialized economies[END_REF]. The possible explanation is related to the exchange rate regime. Indeed, the exchange rate regime and the credibility of the central bank play an important role in the transmission of exchange rate variations. In fact, Vietnam follows crawling peg exchange rate regime suggesting that a change in exchange rate is usually interpreted as permanent so economic agents adjust their prices. As a result, the pass-through should be more pronounced (closed to full pass-through).

The elasticity of appreciation exchange rates appears to be weaker at around -0.396 but is not statistically significant. One possible explanation is when there is an appreciation of VND (i.e., depreciation of foreign currency), exporting firms do not change exports.

Therefore, the transmission of exchange rate to domestic price is less responsive.

[Insert Table 2 here] Most interestingly, the long-run coefficient of foreign competitor price is 0.794 and significant at the 1% level. This means that a 1% rise of the price level of the major partners of Vietnam will lead to an increase of 0.794% of domestic inflation. The coefficient of foreign competitor price seems to be high, but it is coherent with economic features of Vietnam, and in line with theory. The argument is related to country size: As a small economy, Vietnam is considered as a price taker and not as a price maker. Therefore, the imports price of Vietnam is set by the world market, and this variable plays an important role in the movement of the domestic price.

The variable who has less impact on consumer price index is oil price. Indeed, oil price long-run coefficient is around 0.068, which means that a 10% rise of oil price will be translated into a rise of 0.68% of domestic inflation. As we explained above, the intervention of PPSF will absorb shocks in world oil price in order to maintain the stability of domestic oil price. Therefore, the impact of oil price on inflation is small.

Another important finding of our study is that depreciation of the exchange rate is more passed through to domestic prices than appreciation, in both short run and long run. This result is consistent with Delatte and Lopez (2012) and [START_REF] Brun-Aguerre | Heads i win; tails you lose: asymmetry in exchange rate pass-through into import prices[END_REF]. We attribute this finding to the fact that Vietnam is a price taker in the world markets. Therefore, following an appreciation, importer companies are less willing to decrease their prices in order to increase their mark-up (profits). On the other hand, following depreciation, importer companies tend to transmit (pass) shocks to domestic prices by increasing the prices to limit the reduction of their mark-up.

Third, diagnostic tests are applied to the estimations, including cointegration test, symmetry test, serial correlation in the residuals, heteroskedasticity tests, and CUSUM test.

The results are shown in Table 3. In Table 3, we use the F-statistic upper bound critical value in [START_REF] Narayan | The saving and investment nexus for China: evidence from cointegration tests[END_REF] and the t-statistic one in [START_REF] Pesaran | Bounds testing approaches to the analysis of level relationships[END_REF] for the cointegration test (bound tests). Both 𝐹 𝑃𝑆𝑆 and 𝑡 𝐵𝐷𝑀 statistics exceed the upper critical values, confirming the existence of cointegration. We check the existence of the asymmetry effect of exchange rate in long and short runs by using Wald test. The long-run symmetry test is rejected at the 1% level, indicating that the model allows the asymmetry effect of exchange in the long run. Only the positive change of NEER lag one (Δ 𝑁𝐸𝐸𝑅 𝑡-1 + ) remains significant at the 5% level in the short run (Table 2), implying that asymmetry effects of the exchange rate also apply in the short run. To sum up, exchange rate pass-through appears to be asymmetric in both long and short runs. This means that the linear ARDL is misspecified. Serial correlation and heteroskedasticity tests cannot reject the null hypothesis, indicating that our estimation is not spurious. We test structural break by CUMSUM test (cumulative sum of residuals)3 . The coefficients for the specification describing the dynamics of CPI seem to be stable over the sample period.4

[Insert Table 3 here] The dynamic multipliers up to 40 quarters are presented in Figure 1 based on Equation (5). They show the new long-run equilibrium for the consumer price following the positive and negative shocks of the exchange rate from an initial long-run equilibrium.

It is clear that the positive component of the exchange rate 𝑁𝐸𝐸𝑅 𝑡 + has a stronger impact than negative one 𝑁𝐸𝐸𝑅 𝑡 -in both short and long run effects. Indeed, depreciations of the exchange rate 𝑁𝐸𝐸𝑅 𝑡 -tend to reach their long-run value within a few quarters. In contrast, appreciations tend to be less pronounced and less persistent.

[Insert Figure 1 here]

Conclusion

In this paper, we have tried to measure the exchange rate pass-through on consumer price for Vietnam by using non-linear ARDL model over the quarterly period 2000Q4-2018Q2. We investigated the possibility of asymmetry in the reaction of domestic prices to variations of exchange rate. Our main results can be summarized as follows. First, the exchange rate still has an impact on domestic prices and depreciations are more strongly transmitted to CPI than appreciations, in both short and long runs. However, the passthrough is higher in the long run than in the short run. This finding is consistent with [START_REF] Delatte | Asymmetric exchange rate pass-through: Evidence from major countries[END_REF] and [START_REF] Brun-Aguerre | Heads i win; tails you lose: asymmetry in exchange rate pass-through into import prices[END_REF]. Second, the foreign competitors price plays an important role for domestic price movement. Finally, the passthrough is high in Vietnam. This could be explained by macroeconomic conditions in Vietnam. Among Tiger countries, Vietnam has one of the highest levels of inflation. Indeed, from 2000 to 2018, the annual average of inflation in Tiger countries is around 4.39% against 6.59% for Vietnam. This finding is in line with [START_REF] Taylor | Low inflation, pass-through, and the pricing power of firms[END_REF] and [START_REF] Frankel | Slow pass through around the world: A new import for developing countries? CID Working Papers 116[END_REF] who argued that the decline in pass-through is due to a low inflation environment and is also influenced by the environment of transitory exchange rate fluctuations. In the policy setting, these results suggest that movements of exchange rate and foreign competitors price are important indicators for policymakers and the State Bank of Vietnam in order to ensure price stability.

  ) and Brun-Aguerre et al. (2017), which is derived from the real effective exchange rate equation (𝑅𝐸𝐸𝑅 = 𝑁𝐸𝐸𝑅 * 𝑃 * /𝑃) as 𝐹𝐶𝑃 = 𝑃 * = 𝑅𝐸𝐸𝑅 * 𝑃 𝑁𝐸𝐸𝑅 where P is Vietnamese consumer price index; REER is calculated by using NEER adjusted by relative price between Vietnam (P) and its trading partners (𝑃 𝑖 ) as follows: 𝑅𝐸𝐸𝑅 𝑡 = 100 * ∏ (𝑆 𝑖𝑡 * 𝑃/𝑃 𝑖 ) 𝑊 𝑖 20 𝑖=1

  [START_REF] Banerjee | Error-correction mechanism tests for cointegration in a singleequation framework[END_REF] proposed 𝑡 𝐵𝐷𝑀 statistic to test the null hypothesis of 𝜌 𝑗 = 0.[START_REF] Pesaran | Bounds testing approaches to the analysis of level relationships[END_REF] proposed 𝐹 𝑃𝑆𝑆 statistic to be applied to the ARDL model.[START_REF] Shin | Modelling asymmetric cointegration and dynamic multipliers in a nonlinear ardl framework[END_REF] demonstrated that we can use 𝐹_𝑃𝑆𝑆 for the Nonlinear Autoregressive Distributed Lag (NARDL) model. The null hypothesis of ρ j = 𝜌 𝑐 + = 𝜌 𝑐 -= ρ d = ρ e = 0 is tested against the alternative hypothesis of coefficients different to zero. The lower bound critical values are computed

The.

  ERPT is computed by the positive and negative long-run coefficients of the exchange rate, which are defined as follows: long-run coefficients of oil price and foreign competitor price are computed by 𝐿 𝑜𝑖𝑙 = -𝜌 𝑑 /𝜌 𝑗 ; 𝐿 𝐹𝐶𝑃 = -𝜌 𝑒 /𝜌 𝑗 . The short-run coefficients are those in the first difference, 𝜋 𝑖𝑗 (𝑖 = 1, … ,4). We expect a positive sign for 𝐿 𝑛𝑒𝑒𝑟 + and a negative sign for the 𝐿 𝑛𝑒𝑒𝑟 -. The long-run symmetry can be tested by the Wald test of the null hypothesis of L To test the existence of short-run symmetry, we use the Wald test to test the null ∑

Figure

  Figure 1. Dynamic multipliers

Table 1 .

 1 Unit root testNotes: * and *** means the significant at the 10% and 1% level. 𝐿 𝑜𝑖𝑙 , 𝐿 𝐹𝐶𝑃 are long-run coefficients of depreciation, appreciation of exchange rate, oil price and foreign competitor price, respectively.

				Level		First
		𝑁𝐸𝐸𝑅 + 𝑁𝐸𝐸𝑅 -	-1.729 -0.128		-4.581*** -9.128***
	Oil			-1.989		-6.314***
	FCP			-0.106		-4.766***
	Gap			-2.868*
	Table 2. NARDL model estimation
	Panel A: Short-run coefficients
	Variable/Lag order	0	1	2	3	4
		Δ 𝐶𝑃𝐼		0.329***		-0.212**
					(0.072)		(0.087)
		Δ 𝑁𝐸𝐸𝑅 +		0.137**
		Δ 𝑁𝐸𝐸𝑅 -		0.055
		Δ𝐹𝐶𝑃	2.406***			1.311**
				(0.288)			(0.495)
		ΔOil				-0.025***
							(0.008)
		Gap				0.0018***
							(0.0006)
	Panel B: Cointegration vector (ECM form)
		𝑁𝐸𝐸𝑅 𝑡-1 +	𝑁𝐸𝐸𝑅 𝑡-1 -	𝑂𝑖𝑙 𝑡-1	𝐹𝐶𝑃 𝑡-1
		0.084**	-0.038	0.0067	0.077**
		(0.033)	(0.048)	(0.004)	(0.03)
	Panel C: Long-run coefficients
		𝐿 𝑛𝑒𝑒𝑟 +	𝐿 𝑛𝑒𝑒𝑟 -	𝐿 𝑜𝑖𝑙	𝐿 𝐹𝐶𝑃
		0.861***	-0.396	0.068*	0.794***
		(0.151)	(0.432)	(0.036)	(0.028)
	𝐿 𝑛𝑒𝑒𝑟 +	, 𝐿 𝑛𝑒𝑒𝑟 +	,

Notes: We go general-to-specific to find the best model by setting p=q=4 for quarterly data.

Table 3 .

 3 Symmetry test, cointegration tests, and diagnostic tests

	ECM	𝐴𝑑𝑗 -𝑅 2	AIC	𝑡 𝐵𝐷𝑀	𝐹 𝑃𝑆𝑆	LR-symmetry test Serial correlation Heteroskedasticity test
	-0.097** 0.854	-8.740	-2.62***	5.493***	18.472***	1.593 [0.191]	1.04 [0.427]
	Notes: The F-upper critical values for the case restricted intercept and no trend in case N=70,
	K=5 are 4.717 (1%): 3.650 (5%) and 3.161 (10%) (see Narayan, 2005); t-upper critical values
	for -4.44(10%); -3.83 (5%); -3.49(1%). P-value of Serial Correlation (Breusch-Godfrey) and
	Heteroskedasticity test (Harvey) are shown in [].

Vietnam's main trading partners are Japan, Singapore, China, Korea, US, Thailand, Hong Kong, Germany, Malaysia, France, Indonesia, UK, Netherlands, Russia, Philippines, Switzerland, Italy, Belgium, India, and Canada. The trade weight of these countries was approximately 80% of the total trade of Vietnam in

We thank an anonymous referee for this remark.

The result is available upon request.