Emna Trabelsi
email: emna.trabelsi2007@yahoo.fr.

What effects exert Economic Globalization and Central Bank Transparency on inflation of OECD countries? An Application of LSDVC Estimator on a dynamic Panel Model

Keywords: JEL Classifications: C23, E58 central bank transparency, economic globalization, inflation, inflation persistence, LSDVC, GMM

). (iii) Although our findings are generally confirmed by a wide set of robustness tests, some caveats associated with the length of time period and the characteristics of inflation pattern observed under the second estimation period (2006)(2007)(2008)(2009)(2010)(2011)(2012)(2013)(2014)(2015) should be taken into account. The estimations were run using a bias corrected Least Square Dummy variable (hereafter, LSDVC), developed by Bruno (2005) for short dynamic panels with fixed effects, and extended to accommodate unbalanced data. Alternative results using Generalized Method of Moments (hereafter GMM) estimators: (Arellano and Bond, 1991, hereafter AB; Blundell and Bond, 1998, hereafter BB) are also provided. (iv) Overall, we find that LSDVC estimator outperforms GMM estimators as it exhibits a satisfactory fit of our hypotheses.

Introduction

There's a general line of thought that central bank transparency helps improve the efficiency of monetary policy. "The reasoning is that if the monetary policy is transparent, economic agents can easily realize that the policy decisions are really aimed at achieving low and stable inflation" (Cihák, 2007, p.491). This paper revisits the question of whether further increases of transparency may generate adverse consequences.

Some theoretical opposing views regarding potential benefits of transparency argue that excess transparency exposes the central bank to external hurdles [START_REF] Amato | Communication and Monetary Policy[END_REF][START_REF] Morris | Social Value of Public Information[END_REF][START_REF] Kool | Central Bank Transparency and the Crowding Out of Private Information in Financial Markets[END_REF]. The influential paper by [START_REF] Morris | Social Value of Public Information[END_REF] found proponents like [START_REF] James | Optimal Policy Intervention and the Social Value of Public Information[END_REF] who brought a robust argument on the undesirable effect of increasing transparency (measured by the precision of public information) on welfare, taking into account the central bank's policy action. Some other views emerge in relation of the minutes1 and votes publishing or whether decisions should be accompanied by press conferences. In a recent statement made on 27 th January 2015, Ignazio Angeloni, the member of the Supervisory Board of the European Central Bank (ECB), identifies the disadvantages of disclosing the votes of the individual decision-making board members. Yet the success of the monetary policy depends on the understanding of central bank decisions and actions by the markets, which in return helps steer interest rates. Nevertheless, some central banks would refrain from "forward guidance" policy adoption2 . In this context, Andersson and Hoffman (2010) argue that announcing the future interest rate path tracks may neither improve the predictability of monetary policy, nor does anchor long term inflation expectations if the central bank has already a clear defined price stability objective and a high degree of transparency. Some authors [START_REF] Douglas | Bank Runs, Deposit Insurance, and Liquidity[END_REF][START_REF] Cukierman | The Limits of Transparency[END_REF] have made much more persuasive arguments against increasing transparency in the case of 'financial stability-related communication' or the use of models of bank runs3 . Whatever the case, the debate around the benefits and costs related to particular forms of communication practices didn't prevent a remarkable increase in the degree of transparency of the central banks around the world. Trends toward greater transparency seem not even knocked off course by the global financial crisis of 2007-2008. The question of further information disclosure is especially appealing for central banks with a high degree of credibility like OECD countries. In order to examine the effect of central bank transparency on inflation, we use the most recent index updated by [START_REF] Dincer | Central Bank transparency and independence: Updates and new measures[END_REF] which is available on an annual basis under the period 1998-2010.

We also focus on another issue that is as topical as central bank transparency. That is, we contribute to the empirical literature on the relationship between economic globalization and inflation. Nowadays, "it has been argued that globalization has had-or is likely to havesignificant implications for the conduct and effectiveness of monetary policy in controlling inflation" [START_REF] Papademos | Globalisation and central bank policies[END_REF]. Its link to inflation forms, then, an important objective of this paper. Both topics have been subject to empirical analysis separately in the literature. To our knowledge, there has not been a study that combines both issues. One exception is the contribution of [START_REF] Binici | Trade openness, market competition, and inflation: Some sectoral evidence from OECD countries[END_REF] who included central bank independence in addition to economic globalization into the same regression. However, their results didn't lead to a significant effect of central bank independence on inflation and conclude that it should be dropped from the regression [START_REF] Binici | Trade openness, market competition, and inflation: Some sectoral evidence from OECD countries[END_REF]. We fill this gap by including the aforementioned multidimensional index of transparency4 constructed according to the methodology of [START_REF] Geraats | Central bank transparency[END_REF] 5 . We believe our analysis is important because it suggests new evidence as of the optimal level of transparency in the case of our sample which appears higher than the threshold found previously by [START_REF] Van Der Cruijsen | Optimal Central Bank Transparency[END_REF]. Moreover, we show that economic globalization matters for monetary policy. Particularly, the composite sub-index constructed by [START_REF] Dreher | Measuring Globalisation -Gauging its Consequences[END_REF] seems to be a perfect proxy. It is also interesting to see how sensitive our core results are, when we subject the specification to a wide series of robustness tests.

The rest of the paper is structured as follows: Sec. 2 presents the methodology, explains the data used, and describes how well transparency is related to inflation persistence, thereby providing new insights with respect to the robustness of previous related research. Sec. 3 presents the results and Sec. 4 concludes.

Empirical analysis 2.1. Data

We collect data for 34 OECD countries under the period 1998-2010. The choice of the sample is restricted by the availability of data on certain variables. The dependent variable is given by the actual inflation rate as projected by the World Economic Outlook (WEO). We use a set of control variables among those which potentially affect inflation, as well as inflation persistence (see Table 1 for details and source of data). Whatever the case, we expect economic globalization (for instance, measured by the composite sub-index of [START_REF] Dreher | Measuring Globalisation -Gauging its Consequences[END_REF] to contribute to a lower inflation. The transparency index6 of Dincer and Eichengreen (2014)7 affects inflation through a quadratic form in the lines of van der Cruijsen et al. (2010)8 , suggesting the presence of a threshold effect of central bank transparency. See details about the definition of those variables in Appendix C.

We include a set of controls, such as the output gap which is linked to inflation positively through a Philips curve (see Table 2 for the expected signs according to existing literature). We use the nominal interest rate as the short-term monetary instrument of central banks, especially in inflation targeting framework9 . We also introduce the logarithm of GDP per capita in our regression which serves as a general proxy of the level of development [START_REF] Romer | Openness and Inflation: Theory and Evidence[END_REF]. Another set of controls is tested in a second econometric application (see Sec. 3.3).

Table 2 Determinants of inflation and inflation persistence and their expected signs

Determinants of inflation Determinants of inflation persistence

Institutional determinants

Transparency (-), Institutions (-) Transparency (-), Transparency 2 (+),

Macroeconomic determinants

Inflation Targeting(-), Output gap (+), logarithm of GDP per capita growth (-), Short-term interest rate (+) Inflation Targeting(-),

Financial determinants

Banking_crisis dummy (+)

External determinants openness (+/-), economic globalization (-), Degree of competition (-), exports as a share of GDP (-)

Degree of Competition (-), Technology Readiness (-), Customer Relations (+), Labor Share (+), Wage Flexibility (-)

Note:

The expected signs are between ( )

Model's specification

The empirical literature on measuring inflation persistence adopts usually univariate time series approaches (e.g [START_REF] Pivetta | The persistence of inflation in the United States[END_REF][START_REF] Stephen | Has the inflation process changed?[END_REF]. Inflation is a persistent process. Typically, the measure is given by the sum of the autoregressive coefficients extracted from the following equation

it j it K j j i it            1 Where   K j j 1
 is interpreted as the measure10 of the persistence. In this paper, we rely on an original and modified definition of inflation persistence proposed by [START_REF] Van Der Cruijsen | Optimal Central Bank Transparency[END_REF]. Therefore, our general estimated econometric model consists of this equation

it it Q p it p it it it it it it it i it Y T T ecoglob X                                 1 1 4 2 1 4 1 3 1 2 1 0 i ,...,t , t ,....,n , i 2 1 2 1   (1)
Where it  stands for the yearly inflation rate, expressed as the percentage increase of Consumer Price Index (CPI), it X is the set of control variables that determine inflation rate, ecoglobit is the component of KOF globalization index, it T is the transparency score and it Y is the set of potential variables that affect the inflation persistence. From (1) we can deduct the measure of persistence denoted (P) as follows

it it Q p p it it it it it i it P Y T T ecoglob X                                              1 4 2 4 3 2 1 1 0 i ,...,t , t ,....,n , i 2 1 2 1   (2)
First, we test the assumption under which inflation depends negatively on economic globalization. The corresponding hypothesis is given by: H01:

0 1   .
The coefficient of the squared term (2) is designed to capture non linearity effect. We can derive the influence (or effect) of transparency on inflation persistence through the following expression

2 4 3 it it T T B     (3)
A U shape form consists of including a quadratic term in a linear model as shown in (3). It requires that-in the words of Lind and Mehlum (2010, p.111): "the slope of the curve is negative at the start and positive at the end of a reasonably chosen interval" [ min T , max T ]. Therefore, there will be an estimated threshold up to which, increased transparency reduces inflation persistence and above which, the impact turns to be the opposite (see Fig. 1). T ˆ is the extreme point= the estimated value of transparency score at which the persistence of inflation is minimized. It is the value up to which increasing transparency is always beneficial and beyond which further increasing turns to be undesirable.

Throughout the forthcoming estimations, we test for the U-shape11 hypothesis using the corresponding test of Lind and Mehlum (2010) and applying "utest" Stata module. H02:

max min T T 4 3 4 3 2 0 2        
Lastly, we solve for the first-order condition of P with respect to T implies the extreme point:

4 3 2    T ˆ

Estimation method

We estimate a dynamic panel model with fixed effects. The inclusion of fixed effects in our model is suitable when a specific set of 𝑛 individuals is not randomly selected from some larger population. Since the sample data come specifically from OECD countries, the fixed effects model is more appropriate for the analysis. Furthermore, dynamic panels share the common feature of endogeneity of the lagged dependent variable. [START_REF] Arellano | Some tests of specification for panel data: Monte Carlo evidence and an application to employment equations[END_REF] GMM-DIFF and Blundell and Bond (1998) GMM-SYS solve this problem. But, Blundell and Bond's estimator (1998) is more appropriate than the one proposed by [START_REF] Arellano | Some tests of specification for panel data: Monte Carlo evidence and an application to employment equations[END_REF] when it comes to deal with high persistent response variable.

By looking to the dynamic model described in (1), three issues emerge:  Our cross-sectional dimension of our panel is small; so that 𝑛 consistent GMM estimators may be affected by potentially severe sample bias and could lead to a poor fit.

 The unbalanced nature of our panel doesn't permit to correct the within estimator by applying the bias approximation formulae derived in [START_REF] Kiviet | On bias, inconsistency, and efficiency of various estimators in dynamic panel data models[END_REF], [START_REF] Bun | Bias-corrected estimation in dynamic panel data models[END_REF] and [START_REF] Bun | On the diminishing returns of higher-order terms in asymptotic expansions of bias[END_REF], which are only valid for balanced panels. Thereby, our estimation strategy employs a bias corrected LSDV estimator as proposed by [START_REF] Bruno | Estimation and inference in dynamic unbalanced panel-data models with a small number of individuals[END_REF].

 In the case of missing values (which is also the feature of our sample), [START_REF] Flannery | Estimating Dynamic Panel Models in Corporate Finance[END_REF] show that LSDVC12 is the most accurate estimator applied on corporate finance data.

A comparison between GMM estimators and LSDVC is made available at the end of Sec. 3.2. Briefly, we describe the characteristics13 of all three types of estimators as follows (see Table 3 for more details):

 AB [START_REF] Arellano | Some tests of specification for panel data: Monte Carlo evidence and an application to employment equations[END_REF] difference GMM derive a consistent GMM estimator for datasets with many panels and few periods. The Stata command 'xtabond' implements this estimator, which applies one lag of the exogenous variables as the instrument set.

 BB [START_REF] Blundell | Initial conditions and moment restrictions in dynamic panel data models[END_REF] system GMM uses additional moment conditions. The estimator is designed for datasets with large panels and few periods. The Stata command 'xtdpdsys' implements this estimator.

 LSDVC [START_REF] Bruno | Estimation and inference in dynamic unbalanced panel-data models with a small number of individuals[END_REF] Least Square Dummy Variable corrected of the biased Fixed Effect (FE) model. It extends the bias approximation formula in [START_REF] Bun | On the diminishing returns of higher-order terms in asymptotic expansions of bias[END_REF] to accommodate short unbalanced panels with a strictly exogenous selection rule. The estimator is implemented through the Stata command 'xtlsdvc'. Its performance is evaluated via Monte Carlo experiments. [START_REF] Bruno | Estimation and inference in dynamic unbalanced panel-data models with a small number of individuals[END_REF] implements bias-corrected estimator for a small number of individuals and short temporal dimension (n=29, t=9). [START_REF] Bruno | The short-and long-run impacts of financial crises on youth unemployment in OECD countries[END_REF] applied the estimator for n=27, t=29. [START_REF] Buddelmeyer | Fixed Effects Bias in Panel Data Estimators[END_REF] showed that LSDVC is the preferred estimator for n=20, t=5 in dynamic panels with a medium size on the one-lagged dependent variable.

In our case, the LSDVC estimator is initialized by a dynamic panel estimate (GMM-DIFF) and then relies on a recursive correction of the bias of the fixed effects estimator.

Results

Impact of economic globalization on inflation: Sub-index of Dreher et al. (2008)

Economic globalization may refer to "the process of integration into global markets facilitated by reductions in transaction costs. Accordingly, economic globalization constitutes a threat of international economic competition and dependence on foreign markets." (Marshall and Fisher, 2015). An interest in the research of the relationship between economic globalization and inflation has increasingly grown. Earlier theoretical and empirical models, however, didn't reach a consensus on the globalization-inflation linkage. At least, Allardo (2007) showed that globalization decreases in Poland. Other authors arrive at ambiguous effect of globalization driven by other factors such as higher energy prices that could cause the relationship to alter [START_REF] Sylvester | Impact of Globalization on inflation[END_REF]. [START_REF] Binici | Trade openness, market competition, and inflation: Some sectoral evidence from OECD countries[END_REF] showed that economic globalization is negatively and significantly related to inflation. Economic globalization is a complex concept and it is difficult to find adequate measures. The KOF14 globalization index is still worth look and usage. Its economic dimension measures the actual flows of trade, foreign direct investment and portfolio investment, as well as the restrictions applying to these flaws 15 . Hence, we think it useful as a composite indicator. Fig. 2 depicts the evolution of economic globalization between 1998 and 2010 in the case of our sample of countries. Ireland jumps to the top of OECD list in 2010 with a value of 94.96, followed by Luxembourg and Netherlands with respective scores 94.09 and 91.70. When it comes to the change rate, Poland ranks the first with a positive change of 42.2% between 1998 and 2010, followed by Slovak Republic (34.1%). To first evaluate H01, we examine the impact of economic globalization on inflation. Our empirical results are shown in Table 4 and Table 5 and support an overwhelmingly negative relationship, either with inclusion of transparency in the specification or without inclusion of that variable. The index is also robust to the introduction of trade openness 16 . The remaining control variable are significant and the associated coefficients apprise the expected signs. 16 The correlation between economic globalization and openness is 0.63 (see Table A.1). Bootstrapped standard errors using 50 iterations are between () [START_REF] Bruno | Estimation and inference in dynamic unbalanced panel-data models with a small number of individuals[END_REF]. *, **, *** imply statistical significance at 10, 5, and 1%, respectively.

7,9%

Impact of central bank transparency on inflation: Overall index of Dincer and Eichengreen (2014)

We turn, now, to evaluate the second hypothesis H02. Table 5 shows the results of the estimation of our regression. Clearly, the coefficients associated with the quadratic form are highly significant. In fact, transparency enters with a negative and significant coefficient and transparency squared enters positively and significantly. A large number of previous researches attempt to test non monotone relationship, but hardly any of these used adequate formal procedures to test the U shape. Lind and Mehlum (2010) developed a nice test to detect such a non-monotone relationship. The results are given in the last lines of Table 5 and show a significant intermediate degree of transparency. Based on the quadratic form's coefficient signs and their significance, the findings confirm again a U-shaped relationship between transparency and the inflation persistence. Overall, LSDVC estimators exhibit a satisfactory fit of our hypothesis, and an optimal (intermediate) transparency is pronounced. The hypothesis H01 is again accepted since economic globalization mirrors the same magnitude and the statistical significance. All control variables have their expected signs. Particularly, the output gap and the nominal interest rate come out with significant coefficients. All other effects, when they do exist, are an order smaller of size and/or significance. IT and Lngdp turn to be insignificant. A possible reason might be that transparency is picking up the effect of these variables (Table A.1 shows a correlation of 0.3 and 0.49, respectively). Indeed, inflation targeting significance is restored when we drop transparency from the regression and it affects both inflation rate and inflation persistence (see columns (1), ( 4) and ( 5) of Table 4). The same applies to the logarithm of GDP per capita (see columns (2) and(3) of Table 4). We subject our results to a number of robustness checks. First, we introduce lagged values of transparency17 and the corresponding results are reported in Table 6. Again, the U test indicates a strong presence of an intermediate transparency level. The control variables follow almost the same pattern as in Table 5. Second, a comparison of results by using pure GMM estimators (AB and BB) is made available in Table 7 and suggests that LSDVC estimator exhibits very satisfactory observations 18 . It is important to note that our variants of the baseline regression in (1) do not include the financial crisis 2007-2008 which could result in structural breaks biasing (potentially) our estimators. So, another (separate) robustness task is performed on a regression including a crisis variable and/or controlling for time dummies to capture OECD common changes in inflation and the explanatory variables. We focus on banking_crisis since the global recession that spreads around the world has affected banking systems. The corresponding dates are picked from [START_REF] Laeven | Systemic banking crises Database: An update[END_REF]. Surprisingly, we don't find evidence that the crisis dummy disturbs the inflation path under the period 1998-2010. Turning to time dummies, only year 2 (1999), year 3 (2000) and year 13 (2010) display significant slopes (see Table A.2). Finally, we replicate the results of Tables 4, 5 and 6 and initialize LSDVC estimator by BB (GMM-SYS) instead of AB (GMM-DIFF). The observations confirm our previous findings and are available in An important feature of our first-step results consists of the finding of a high optimal score level of transparency (in some cases, it attains almost 13.6) that varies according to the set of controls included in the specification. That observation is in sharp contrast to that of van der Cruijsen et al. ( 2010) whose estimations showed an optimal level of 7.5. We think that our results are more in accordance with the actual data. Not only, is the level of 7.5 already exceeded by the advanced economies, but also the central banks around the world continue to increase their transparency practices (Table 8). The Sveriges Riksbank (the bank of Sweden), for example, is on the brink of maximum transparency with a score of 14.5 (since 2003), making it at the top of the most transparent central banks. The lowest score of the group in 2010 is Mexico, which rated 6 points. Central Bank of Turkey figures initially among the least transparent central banks, but increased its score from 2002 with a value of 8.519 . Overall, there was a substantial upward trend of transparency. [START_REF] Horvàth | Peer Effects in central banking[END_REF] explained the main forces that underpinned this trend through social interaction regression. He identified 'peer effects' and other external factors that play as equally important role as domestic factors of central bank transparency. We remark that some emerging OECD countries (i.e. Chile, Estonia and Mexico) may still benefit from further transparency increasing as the actual degree still doesn't' attain the estimated optimum found in our results (Table 9).

 (1) (2) (3) (4) (5) (6) 0  k 0  k 0  k 0  k 0  k 0  k 1  it  0.
 (1) (2) (3) (1) (2) (3) (1) (2) (3) 1  k 1  k 1  k 2  k 2  k 2  k 3  k 3  k 3  k 1  it  0.
 LSDVC_Bruno GMM_AB GMM_BB 0  k 0  k 0  k 0  k 0  k 0  k 0  k 0  k 0  k 1  it  0.

Results from another set of controls

Our previous results suggest somehow a causal negative relationship between inflation and economic globalization (H01), as well as an optimal level of transparency highlighted by a statistical support of a quadratic relationship hypothesis between transparency and inflation persistence (H02). However, it is interesting to see if the co-movement between inflation and economic globalization plus the existence of an optimal threshold of transparency are not just a coincidental phenomenon -though a series of robustness checks is already made in Sec. 3.2 and confirms both hypotheses -or are related to the choice of the accompanied controls and to the inflation path observed under the estimation period (1998-2010)20 . Our point is also motivated by the call of an effective comparison of our results with those of van der Cruijsen et al. ( 2010), hence we need to replicate our econometric exercise by using their set of controls, correcting for competition, institutions, labor share, technology readiness and customer relations (see Table 1 for the source of these data and their detailed definitions in Appendix C). A note of caution has to be in order here before proceeding to the analysis: 2010), we exploit two indicators for labor share: the first control is for "production process sophistication" labeled as LS. We expect that upward trend of inflation persistence might be induced by higher labor costs of services (see [START_REF] Lawless | Understanding the dynamics of labor shares and inflation[END_REF]Lünnemann and Matha, 2010). The second control is for "flexibility of wage determination" labeled as WF (Global Competitiveness Report, World Economic Forum). An increase in wage flexibility raises low inflation. Economic literature exploring the link between market competition and inflation suggest that higher competition leads to lower inflation (see for a related literature, [START_REF] Neiss | The markup and inflation: evidence in OECD countries[END_REF][START_REF] Cavelaars | Does Competition Enhancement Have Permanent Inflation Effects?[END_REF]Przybyla and Roma, 2005). Besides our variable of interest "economic globalization" denoted ecoglob, we test for the effect of globalization through the lens of increased competition. Hence, we use two further proxies: the first one controls for "the intensity of competition in the local market" and labeled as DoC, the second one is exgdp_ the total exports as a share of GDP. By doing so, we select the one that performs the best in explaining inflation dynamics. We also assume that inefficient institutions contribute to higher inflation persistence. Hence, we introduce the quality of public and private institutions Inst_pub and Inst_priv, both are taken from the Global Competitiveness Report, World Economic Forum. Lastly, we control for technological readiness (TR) and for customer relations which is proxied by the "degree of customer orientations" (CR). While, TR (Global Competitiveness Report, World Economic Forum) should exhibit a negative effect on inflation persistence, inflation is positively related to customer relations indicator (CR). We note that many of those indicators affect inflation persistence but could impact inflation series, as well. Because inflation targeting (IT) is timeinvariant in the second sub-sample (2006)(2007)(2008)(2009)(2010)(2011)(2012)(2013)(2014)(2015), we dropped that dummy variable from the regression and keep transparency practices as the second mechanism of interest in controlling inflation stability by central banks. We run various combinations of variables and we finally retain the variants of the specification described in (1) that contain the most of significant explanatory variables (see Table C.2→Table C.4 of Appendix C). We include the above indicators such that concerns about multicollinearity are the least warranted (see Table C.1).

Most of the controls have their expected signs. They have similar size and significance across all variants. Clearly, Technology readiness (TR) exhibits a highly significant negative impact on inflation persistence. The indicator of "production process sophistication" (LS) does effect inflation persistence positively as to be expected but doesn't impact inflation in level. Moreover, LS outperforms WF in explaining inflation persistence. The latter has the negative expected sign but is never significant. Although a high correlation is depicted between LS and TR, these indicators remain significant even when included solely in the regression. Our second econometric exercise apprises us of a change in correlation dynamics between economic globalization and inflation: ecoglob has now an insignificant impact on inflation or has a significant effect but exhibits the opposite sign (Table C.2). Interestingly, exgdp appears to be the most efficient competition representative candidate, followed by "the intensity of competition in local markets" indicator (DoC) which exhibits a negative estimate and keeps the same magnitude in all specifications, but is rarely significant. When estimating specifications including both exgdp and ecoglob variables, exgdp retains the same coefficient's size, sign and significance. This result is not surprising if we refer to the low correlation of 0.0048 between these two proxies (see Table C.1). "Customer relations" impacts significantly both inflation rate and inflation persistence when enters the specification jointly with exgdp, with a positive coefficient of 1.319 and 0.023, respectively (regressions not reported). Lastly, we don't find evidence for the quality of institutions (Inst) in explaining inflation dynamics. Turning to central bank transparency, we observe a negative and significant impact on inflation persistence as well as on inflation series though weaker in the last case, therefore favoring maximum transparency. When the regression jointly comprises economic globalization and transparency, ecoglob loses its significance and H01 is rejected.

The associated results are reported in Table C.3 and Table C.4. Note that adding this variable -Tto the regression compresses the sample size to 22 countries and the estimation period becomes 2006-2010. Transparency scores range between 6 (moderate) and 14.5 (high).

Notwithstanding, some data limitations might play a role in explaining the findings. By re-estimating the model using pooled OLS, we accept H02 if we include standard determinants such as he output gap and the logarithm of GDP per capita 21 , the latter is to correct for the level of economic development and if we consider a quadratic relationship between transparency and inflation rate rather than inflation persistence. The second part of overall results' explanation is related to inflation pattern under the estimation period. In fact, three puzzles evolve from inflation dynamics: first, many countries (especially advanced ones) experienced high inflation during 2007-2011 crisis-period. Second, we characterize a sudden fall in inflation rate from 2012 to the middle of 2014. Third, we observe a deflation between 2014 and 2015. [START_REF] Friedrich | Global Inflation Dynamics in the Post-Crisis Period: What Explains the Twin Puzzle?[END_REF] nicely examined the first two puzzles, called "twin puzzles" in order to understand inflation behavior since the global financial crisis, in an unrelated work 22 . Indeed, time dummies and/or banking_crisis variable, when added to the regression, turn to be relatively significant (see Table C.3, Table C.4 and Table 10 23 ), featuring inflation fluctuations (the switch from high to low) following the crisis period (2007)(2008). 22 Some factors such as the variation of economic slacks measured by the output gap contribute to explain the inflation dynamics [START_REF] Friedrich | Global Inflation Dynamics in the Post-Crisis Period: What Explains the Twin Puzzle?[END_REF]. 23 We didn't find a significant effect of banking_crisis dummy in OLS model. 𝑡{2006, 2007, 2008, 2009, 2010}.

Note:

Conclusion

In this paper, we gave a new evidence on the presence of an intermediate transparency at the optimum on the one hand and examine the relationship between economic globalization and inflation on the other hand. Classical methods would be restricted on the findings of a significantly negative coefficient on transparency term and a significant positive estimate on the quadratic term and take them as evidence supporting the U shape curve. Lind and Mehlum (2010) argue that earlier conventional approaches -although intuitive-could yield misleading results. We revisited the hypothesis of an intermediate optimal central bank transparency by introducing both technical and economic differences in our specification. Particularly, we have used the most recent monetary transparency index developed by [START_REF] Dincer | Central Bank transparency and independence: Updates and new measures[END_REF]. We found that the hypothesis of U-shaped relationship was strongly depicted in the case of central banks considered in our sample. Indeed, the test results overwhelmingly reject the combined null hypothesis of an inverted-U or monotone relationship in favor of a Ushaped linkage between central bank transparency and inflation persistence by using an appropriate test of [START_REF] Thori | With or Without U? The Appropriate Test for a U-Shaped Relationship[END_REF]. The results are robust for using lagged values of transparency. We find, also that LSDVC is the most suitable estimator in fitting the hypothesis of a quadratic relationship between transparency and inflation persistence compared to GMM estimators 24In the light of our results and the related previous research, we conclude some worth noting points:

 The index of transparency employed is subjective to the issue of quality. Yet, the existing measures focus on the quantity of disclosure, while the concept also encompasses features like accuracy, truthfulness, and information relevance.  The pros and cons of providing too much or too little information rely on the country specific media and communication culture of each central bank. Both aspects have to be taken into account when design the appropriate tools and channels for explaining the monetary policy.  Trends in transparency practices of each central bank are determined in an important part by other central banks' experiences and lessons (Called 'peer effects' as identified by Horvàth, 2014).  There has been a skeptical view as of the effective negative economic globalizationinflation relationship. In the words of Amtenbrink (2011, p.31): "globalization is considered to have a positive effect on inflation in industrial countries by inducing downward pressure on prices inter alia through the opening of labour markets, better allocation of (financial) resources and increased competition.." "…globalization may also induce upward pressure on prices as demand namely for energy and raw materials in emerging economies has grown notably". As our sample is dominated by advanced economies 25 , we found that economic globalization did not alter the central banks' tasks in pursuing their price stability primary objective, given the first set of controls under the estimation period (1998)(1999)(2000)(2001)(2002)(2003)(2004)(2005)(2006)(2007)(2008)(2009)(2010). Precisely, inflation is a decreasing function of the subindex of [START_REF] Dreher | Measuring Globalisation -Gauging its Consequences[END_REF].

We examined the extent to which the essence of our multiple estimations in the first econometric task (under the period 1998-2010), prevails if we use a set of controls in the lines of [START_REF] Van Der Cruijsen | Optimal Central Bank Transparency[END_REF]. Summarizing:

 Shortened data plus the unusual inflation path under the second estimation period (2006)(2007)(2008)(2009)(2010)(2011)(2012)(2013)(2014)(2015) engender an inconclusive impact of economic globalization (ecoglob) on inflation. At the same time, globalization affects inflation dynamics through enhanced competition channel. In contrast to economic globalization, we found that the exports as a share of GDP turns to be the leading candidate, hence outperforming the intensity of local competition (DoC). This means that some aspects of globalization might prevail others depending on the characteristics of inflation pattern.  Favoring instead maximum transparency doesn't oppose our primary results because we already observe a high optimal level of transparency (in some cases, it attains 13.6, see Table 5).  Future work should unravel potential sources explaining inflation path starting 2006 onwards.

As a further econometric exercise, we suggest testing for the existence of an intermediate optimal level of transparency by using a dynamic panel threshold model estimated in the lines of [START_REF] Seo | Dynamic Panels with Threshold Effect and Endogeneity[END_REF] and by instrumenting the transparency variable26 . Lastly, a panoramic debate regarding the effect on inflation persistence of some particular transparency practices pursued by the central bank (i.e. publishing the minutes, forward guidance..) must be the object of more discussion (theoretical and empirical) in forthcoming research. For instance, [START_REF] Gaus | Adaptive learning, Heterogeneous expectations and forward guidance[END_REF] shows through a Cagan-type model that the announcement of forward guidance may help decrease inflation persistence. ). Bootstrapped standard errors using 50 iterations are between () [START_REF] Bruno | Estimation and inference in dynamic unbalanced panel-data models with a small number of individuals[END_REF]. inf: inflation rate, L.inf: lagged inflation rate, ecoglob: economic globalization, inflagxT : lagged inflation x transparency, inflagxT2: lagged inflation x squared transparency, outgap: output gap, N: the total number of observations.

( , 2007, 2008, 2009, 2010, 2011, 2012, 2013, 2014, 2015}. N: the total number of observations. N: the total number of observations. 𝑡{2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013, 2014 𝑡{2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013, 2014 Bootstrapped standard errors using 50 iterations are between () [START_REF] Bruno | Estimation and inference in dynamic unbalanced panel-data models with a small number of individuals[END_REF]. inf: inflation rate, L.inf: lagged inflation rate, ecoglob: economic globalization, exgdp: exports as a share of GDP, CR: customer relations, DoC: degree of competition, T: transparency score, inflagxT: lagged inflation x Transparency, inflagxTR: lagged inflation x Technology readiness, inflagxLS: lagged inflation x Labor share, banking_cr~s: banking crisis dummy, year_k: dummy variable taking the value 1 at year t, 𝑡{2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013, 2014

inf inf inf inf inf inf inf inf inf (1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

Fig 1

 1 Fig 1 Example of the illustration of U shape

Fig 2

 2 Fig 2 Evolution of economic globalization between 1998 and 2010 within OECD countries AUS: Australia, BEL: Belgium, CHL: Chile, DNK: Denmark: FIN: Finland, DEU: Germany, HUN: Hungary, IRL: Ireland, ITA: Italy, KOR: Korea, MEX: Mexico, NZL: New Zealand, POL: Poland, SVK: Slovak Republic, ESP: Spain, CHE: Switzerland, GBR: United Kingdom. Data source: http://globalization.kof.ethz.ch/

  : Denmark, EST: Estonia, MEX: Mexico, NZL: New Zealand, POL: Poland, SVK: Slovak Republic, SWE: Sweden, TUR: Turkey, UK: United Kingdom.

Table 1

 1 Description of variables and data source

	Variables	Description	Source
	Dependent variable Inflation (  )	Consumer price index	IMF, World Economic Outlook Database
		(annual % increase)	http://www.imf.org/external/ns/cs.aspx?id=28
	Variables of interest
	ecoglob	Economic
		globalization: a
		component of KOF
		index of globalization

Table 1

 1 Description of variables and data source (continued)Countries considered in our sampleAustralia, Austria, Belgium, Canada, Chile, Czech Republic, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Iceland, Ireland, Israel, Italy, Japan, Korea, Luxembourg, Mexico, Netherlands, New Zealand, Norway, Poland, Portugal, Slovak Republic, Slovenia, Spain, Sweden, Switzerland, Turkey, United Kingdom, United States

	Estimation period: 1998-2010 / 2006-2015
	(a)

Table 3

 3 The efficiency of the estimators according to different situations

	Estimator Unobserved	Dynamic panel	Second order serial	Unbalanced	Endogenous
		heterogeneity	data	correlation	data	variables
	AB	yes	yes	no	yes	yes
	BB	yes	yes	no	yes	yes
	LSDVC yes	yes	yes	yes	no
	Source: Flannery and Hankins (2013)

Table 4

 4 Impact of economic globalization on inflation: Initial results without including central bank transparency

	150							50,0%
					42,2%
						34,1%		40,0%
		28,2%			30,0%	30,0%
	100
		18,0%	15,3%			20,0%
	50	3,2% -2,7% -4,4%	-1,1% 6,1% -0,3% 0,5% 0,8% 4,7%	6,5% -1,0% 8,6% -4,5% 5,5% 5,7% -3,2% 8,1% -1,0% -8,5% 2,0%	5,0%	-10,3% 1,4% 0,9% 2,4% 0,3% -6,2%	0,0% 10,0%
								-10,0%
	0							-20,0%

AUS BEL CHL DNK FIN DEU HUN IRL ITA KOR MEX NZL POL SVK ESP CHE GBR 1998 2010 Change rate Note: Bias correction initialized by Arellano-Bond estimator. Bias approximation is accurate up to O(1/n 𝑡 2 ).

Table 5

 5 Impact of economic globalization and Central Bank transparency on inflation: Dincer and Eichengreen (2014) data

	Dependent variable : it

Table 6

 6 Impact of economic globalization and Central Bank transparency on inflation: Dincer and Eichengreen (2014) data_ Lagged values

	Dependent variable:

it

Table 7

 7 Alternative results using GMM estimators and comparison with LSDVC estimator

	Dependent variable:

it

  Table B.1, Table B.2 and Table B.3 of Appendix B.

Table 8

 8 Descriptive statistics of central bank transparency: 1998-2010

		1998	1999	2000	2001	2002	2003	2004	2005	2006	2007	2008	2009	2010
	Min	3
		(POL,
		TUR)

Table 9

 9 Emerging OECD countries: Transparency index of[START_REF] Dincer | Central Bank transparency and independence: Updates and new measures[END_REF] in 2010

	Country	Chile Czech	Estonia Hungary Mexico Poland Slovak	Turkey
			Republic					Republic
	Transparency	8.5	12	6.5	13.5	6	9	n.a	10
	index
	Note: n.a=not available

Table 10

 10 Impact of central bank transparency on inflation: Pooled OLS with time dummies(2006)(2007)(2008)(2009)(2010)

	Dependent variable: it 	(1)
	T	it		-1.291* (0.733)
		2		0.059*
	T
		it		(0.034)
	outgap	0.291***
			it	(0.061)
	Lngdp	-1.205**
			it	(0.460)
	year	2	0.812 (0.555)
	year 3	1.344*** (0.556)
	year	4	1.048 (0.661)
	year 5	1.727*** (0.629)
		2		0.449
	R
		adj
	Ramsey overidentification test	2.00
	[p-value]	[0.12]
	Heteroscedasticity test	1.92
	[p-value]	[0.16]
	Utest		1.58*
	[p-value]	[0.06]
	Extreme point	10.997

21

Table

10

indicates that both indicators are significant. The overidentification test rejects the hypothesis of omitted variables. The adjusted coefficient of determination of 0.45 shows a moderate fit of OLS model.

  𝑦𝑒𝑎𝑟 𝑘 : Dummy variable taking the value 1 at year t and 0 otherwise,

	T :
	it
	transparency score of Dincer and Eichengreen
	(2014).

*, **, *** imply statistical significance at 10, 5, and 1%, respectively. it  : inflation rate, it outgap : output gap, it Lngdp : logarithm of GDP per capita.

Table A . 2

 A Impact of economic globalization and central bank transparency on inflation including time dummiesNote: Bias correction initialized by Arellano-Bond estimator. Bias approximation is accurate up to O(1/n 𝑡 2

Table B . 1

 B Impact of economic globalization on inflation: Initial results without including central bank transparencyNote: Bias correction initialized by Blundell-Bond estimator. Bias approximation is accurate up to O(1/n 𝑡 2 ). Bootstrapped standard errors using 50 iterations are between ()[START_REF] Bruno | Estimation and inference in dynamic unbalanced panel-data models with a small number of individuals[END_REF]. inf: inflation rate, L.inf: lagged inflation rate, ecoglob: economic globalization, openness : trade openness, IT : inflation targeting dummy, inflagxIT: lagged inflation x IT, outgap: output gap, lngdp : logarithm of GDP per capita, int_rate : interest rate. N: the total number of observations.

		(1)	(2)	(3)	(4)	(5)
		inf	inf	inf	inf	inf
	L.inf	0.362***	0.335***	0.398***	0.222***	0.219***
		(0.052)	(0.051)	(0.049)	(0.067)	(0.046)
	outgap	0.162***	0.150***	0.156***	0.102***
		(0.034)	(0.035)	(0.034)	(0.037)
	ecoglob	-0.182***	-0.197***	-0.174***	-0.157***
		(0.030)	(0.035)	(0.029)	(0.034)
	IT	-0.981*
		(0.581)
	openness		0.0141			0.0122*
			(0.012)			(0.007)
	lngdp		-1.184***		-0.508*
			(0.291)		(0.269)
	inflagxIT			-0.245**		-0.207***
				(0.109)		(0.056)
	int_rate				0.167***	0.284***
					(0.054)	(0.033)
	N	285	285	285	278	417
	Standard errors in parentheses
	* p<0.1, ** p<0.05, *** p<0.01

Table B . 2

 B Impact of economic globalization and Central Bank transparency on inflation:[START_REF] Dincer | Central Bank transparency and independence: Updates and new measures[END_REF] data Note: Bias correction initialized by Blundell -Bond estimator. Bias approximation is accurate up to O(1/n 𝑡 2 ). Bootstrapped standard errors using 50 iterations are between ()[START_REF] Bruno | Estimation and inference in dynamic unbalanced panel-data models with a small number of individuals[END_REF]. inf: inflation rate, L.inf: lagged inflation rate, ecoglob: economic globalization, openness : trade openness, IT : inflation targeting dummy, inflagxT: lagged inflation x Transparency, inflagxT2: lagged inflation x squared Transparency, inflagxIT: lagged inflation x IT, outgap: output gap, lngdp : logarithm of GDP per capita, int_rate : interest rate. N: the total number of observations.

		(1)	(2)	(3)	(4)	(5)	(6)
	Utest	1.82**	1.19	1.64**	1.39*	2.81***	3.02***
	[p-value]	[0.03]	[0.12]	[0.05]	[0.08]	[0.00]	[0.

00] Extreme point 13.078 13.946 10.006 12.299 10.249 10.207

		(1)	(2)	(3)	(4)	(5)	(6)
		inf	inf	inf	inf	inf	inf
	L.inf	0.734***	0.583***	0.563***	0.623***	0.746***	0.795***
		(0.081)	(0.091)	(0.095)	(0.089)	(0.062)	(0.060)
	inflagxT	-0.112***	-0.0814***	-0.0993***	-0.0865***	-0.154***	-0.163***
		(0.026)	(0.029)	(0.035)	(0.027)	(0.028)	(0.028)
	inflagxT2	0.00430*	0.00292	0.00496*	0.00352	0.00751***	0.00799***
		(0.002)	(0.002)	(0.003)	(0.002)	(0.003)	(0.003)
	outgap	0.169***	0.167***	0.166***	0.164**
		(0.063)	(0.063)	(0.064)	(0.064)
	ecoglob	-0.164***	-0.162***	-0.163***	-0.158***
		(0.047)	(0.051)	(0.052)	(0.054)
	IT		-1.004	-0.784		-0.154
			(0.810)	(0.839)		(0.537)
	lngdp			-0.681
				(0.459)
	inflagxIT				-0.184
					(0.194)
	openness					0.0295**	0.0292**
						(0.014)	(0.014)
	int_rate					0.322***	0.319***
						(0.052)	(0.051)
	N	136	136	136	136	244	244
	Standard errors in parentheses
	* p<0.1, ** p<0.05, *** p<0.01

Table B . 3

 B Impact of economic globalization and Central Bank transparency on inflation:[START_REF] Dincer | Central Bank transparency and independence: Updates and new measures[END_REF] data_ Lagged values Note: Bias correction initialized by Blundell-Bond estimator. Bias approximation is accurate up to O(1/n 𝑡 2 ). Bootstrapped standard errors using 50 iterations are between ()[START_REF] Bruno | Estimation and inference in dynamic unbalanced panel-data models with a small number of individuals[END_REF]. inf: inflation rate, L.inf: lagged inflation rate, ecoglob: economic globalization, openness : trade openness, IT : inflation targeting dummy, inf-Tlag_k: lagged inflation x lagged Transparency at t-k, inf-2lag_k:: lagged inflation x squared Transparency at time t-k, inflagxIT: lagged inflation x IT, outgap: output gap, lngdp : logarithm of GDP per capita, int_rate : interest rate. N: the total number of observations.

		𝑘 = 1 𝑘 = 1 𝑘 = 1 𝑘 = 2 𝑘 = 2 𝑘 = 2 𝑘 = 3 𝑘 = 3	𝑘 = 3
		(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)
	Utest	3.72***	1.20	1.44*	2.39***	1.18	1.73**
	[p-value]	[0.00]	[0.11]	[0.07]	[0.01]	[0.12]	[0.

04] 1.57* [0.06] 3.37*** [0.00] 2.83*** [0.00] Extreme point 9.333 5.696 9.804 9.461 15.649 12.453 8.040 10.058 10.076

  Impact of economic globalization on inflation without including central bank transparnencyNote: Bias correction initialized by Arellano-Bond estimator. Bias approximation is accurate up to O(1/n 𝑡 2 ). Bootstrapped standard errors using 50 iterations are between ()[START_REF] Bruno | Estimation and inference in dynamic unbalanced panel-data models with a small number of individuals[END_REF]. inf: inflation rate, L.inf: lagged inflation rate, ecoglob: economic globalization, exgdp: exports as a share of GDP, DoC: degree of competition, Inst: quality of private and public institutions, CR: customer relation, LS: labor share,

	 Econometric results
				Table C. 1 Correlation matrix
		(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)
		inf	inf	inf	inf	inf	inf	inf	inf	inf
	L.inf	0.872***	0.104	0.661***	0.859***	0.872***	0.948***	0.361***	1.119***	1.055***
		(0.048)	(0.083)	(0.067)	(0.047)	(0.075)	(0.077)	(0.066)	(0.037)	(0.042)
	inf~Tlag_1	-0.193***	0.0432	-0.101***
		(0.027)	(0.036)	(0.037)
	inf~2lag_1	0.0104*** -0.00379	0.00514
		(0.003)	(0.003)	(0.004)
	int_rate	0.294***			0.249***		0.0314	0.336***
		(0.057)			(0.063)		(0.080)	(0.074)
	openness	0.0240*	0.0106	-0.0199	0.0329**	0.0357**	0.0237	0.0301**	0.0234	0.0270*
		(0.013)	(0.021)	(0.019)	(0.013)	(0.015)	(0.016)	(0.013)	(0.015)	(0.015)
	outgap		0.153***	0.0925*		0.139***	0.129***		0.170***	0.166***
			(0.055)	(0.051)		(0.045)	(0.047)		(0.041)	(0.045)
	ecoglob		-0.182***			-0.145*** -0.0813*		-0.157***	-0.152***
			(0.051)			(0.033)	(0.044)		(0.034)	(0.035)
	IT		-1.694*			-0.847			0.322
			(0.890)			(0.773)			(1.053)
	inflagxIT			-0.319*			-0.211			-0.0578
				(0.191)			(0.163)			(0.184)
	inf~Tlag_2				-0.181***	-0.123***	-0.141***
					(0.035)	(0.038)	(0.037)
	inf~2lag_2				0.00958**	0.00394	0.00566*
					(0.004)	(0.003)	(0.003)
	inf~Tlag_3 Table C. 2						(0.035) -0.0959***	(0.037) -0.222***	(0.041) -0.203***
	inf~2lag_3							0.00597	0.0110***	0.0101***
								(0.004)	(0.003)	(0.003)
	N	266	147	147	248	137	132	230	127	127
	Standard errors in parentheses
	* p<0.1, ** p<0.05, *** p<0.01

inflagxWF: lagged inflation x wage flexibility, inflagxTR: lagged inflation x Technology readiness, inflagxLS: lagged inflation x Labor share, banking_-s: banking crisis dummy, year_k: dummy variable taking the value 1 at year t, 𝑡{2006

Table C . 3

 C Impact of Central bank transparency on inflation and inflation persistence

										banking crisis
	dummy,	year_k:	dummy	variable	taking	the	value	1	at	year	t,

Note: Bias correction initialized by Arellano-Bond estimator. Bias approximation is accurate up to O(1/n 𝑡 2 ). Bootstrapped standard errors using 50 iterations are between ()

[START_REF] Bruno | Estimation and inference in dynamic unbalanced panel-data models with a small number of individuals[END_REF]

. inf: inflation rate, L.inf: lagged inflation rate, exgdp: exports as a share of GDP, CR: customer relations, DoC: degree of competition, T: transparency score, inflagxT: lagged inflation x Transparency, inflagxTR: lagged inflation x Technology readiness, inflagxLS: lagged inflation x Labor share, banking_~s:

  , 2015}. N: the total number of observations.

		(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
		inf	inf	inf	inf	inf	inf	inf	inf	inf	inf
	L.inf	0.896***	0.484***	0.391***	0.129	0.232'	0.750***	0.875***	0.862***	1.127***	0.912***
		(0.141)	(0.178)	(0.179)	(0.175)	(0.181)	(0.148)	(0.153)	(0.148)	(0.116)	(0.146)
	exgdp	-0.170'	-0.215*** -0.156'		-0.189*		-0.207*** -0.208*** -0.230***
		(0.107)	(0.101)	(0.095)		(0.103)		(0.092)	(0.091)	(0.090)
	T	-1.272*	-1.133'	-1.252*** -1.611*** -1.126*	-1.810***
		(0.691)	(0.691)	(0.611)	(0.559)	(0.663)	(0.637)
	inflagxTR	-0.533*** -0.526*** -0.548*** -0.481*** -0.532*** -0.487*** -0.698*** -0.692*** -0.697*** -0.670***
		(0.246)	(0.207)	(0.224)	(0.203)	(0.213)	(0.225)	(0.175)	(0.175)	(0.165)	(0.174)
	inflagxLS	0.341'	0.394*	0.439***	0.409***	0.450***	0.312'	0.723***	0.709***	0.677***	0.739***
		(0.243)	(0.207)	(0.223)	(0.199)	(0.210)	(0.221)	(0.217)	(0.220)	(0.206)	(0.213)
	banking_~s		2.435*							2.189*
			(1.469)							(1.224)
	year2		1.106'		0.631	1.225		1.367'	1.202'	1.159'	0.811
			(0.848)		(0.796)	(1.004)		(0.893)	(0.855)	(0.770)	(0.714)
	year3		1.603***	1.647***	1.968***	2.401***		2.620***	2.512***	1.783***	2.201***
			(0.692)	(0.625)	(0.715)	(0.838)		(0.745)	(0.691)	(0.655)	(0.684)
	year5		-0.595		-0.181					-0.684	-0.244
			(0.876)		(0.779)					(0.756)	(0.738)
	DoC				-2.562	-2.307	-1.583	-1.593			-1.420
					(2.166)	(2.239)	(2.457)	(1.977)			(1.975)
	year4					0.615		0.697	0.670
						(0.903)		(0.802)	(0.798)
	inflagxT							-0.129*	-0.124*	-0.131*** -0.164***
								(0.070)	(0.071)	(0.066)	(0.066)
	N	80	80	80	80	80	80	80	80	80	80
	Standard errors in parentheses
	' p<0.2, * p<0.1, ** p<0.05, *** p<0.05

Table C . 3

 C Impact of Central bank transparency on inflation (continued)Note: Bias correction initialized by Arellano-Bond estimator. Bias approximation is accurate up to O(1/n 𝑡 2 ). Bootstrapped standard errors using 50 iterations are between ()[START_REF] Bruno | Estimation and inference in dynamic unbalanced panel-data models with a small number of individuals[END_REF]. inf: inflation rate, L.inf: lagged inflation rate, exgdp: exports as a share of GDP, CR: customer relations, DoC: degree of competition, T: transparency score, inflagxTR: lagged inflation x Technology readiness, inflagxLS: lagged inflation x Labor share, outgap: output gap, banking_~s: banking crisis dummy, year_k:

	dummy	variable	taking	the	value	1	at	year	t,

  , 2015}. N: the total number of observations.

		(1)	(2)	(3)	(4)	(5)	(6)
		inf	inf	inf	inf	inf	inf
	L.inf	1.098***	1.480***	1.044***	0.484***	0.391**	0.401**
		(0.185)	(0.149)	(0.156)	(0.178)	(0.179)	(0.180)
	outgap	0.409***	0.400***	0.371***
		(0.066)	(0.063)	(0.053)
	DoC	-2.516*
		(1.313)
	T	-0.484	-0.382	-0.185	-1.133'	-1.252**	-1.249**
		(0.426)	(0.444)	(0.439)	(0.691)	(0.611)	(0.591)
	inflagxTR	-0.301**	-0.472***	-0.350**	-0.526**	-0.548**	-0.550**
		(0.144)	(0.140)	(0.147)	(0.207)	(0.224)	(0.223)
	inflagxLS	0.001	0.124	0.104	0.394*	0.439**	0.438**
		(0.162)	(0.157)	(0.164)	(0.207)	(0.223)	(0.223)
	year2	-1.681**	-2.250***		1.106'
		(0.773)	(0.722)		(0.848)
	year3	-0.335	-0.881'		1.603**	1.647***	1.645***
		(0.523)	(0.592)		(0.692)	(0.625)	(0.619)
	year5	-0.629	-1.160*		-0.595
		(0.572)	(0.644)		(0.876)
	exgdp		-0.195**	-0.184***	-0.215**	-0.156'	-0.157*
			(0.078)	(0.058)	(0.101)	(0.095)	(0.094)
	banking_~s			2.024**	2.435*
				(0.804)	(1.469)
	year4			0.987*
				(0.528)
	CR						-0.022
							(2.617)
	N	48	48	48	80	80	80
	Standard errors in parentheses
	' p<0.2, * p<0.1, ** p<0.05, *** p<0.01

Table C . 4

 C Impact of Economic globalization and Central bank transparency on inflation and inflation persistenceNote: Bias correction initialized by Arellano-Bond estimator. Bias approximation is accurate up to O(1/n 𝑡 2 ).

  , 2015}. N: the total number of observations.

		(1)	(2)	(3)	(4)
		inf	inf	inf	inf
	L.inf	0.310*	0.490***	1.129***	0.858***
		(0.182)	(0.177)	(0.119)	(0.149)
	ecoglob	0.0806	0.0584	0.130	0.142
		(0.355)	(0.343)	(0.288)	(0.310)
	exgdp	-0.201*	-0.219**	-0.237***	-0.214**
		(0.106)	(0.103)	(0.091)	(0.091)
	T	-0.855	-1.121'
		(0.669)	(0.702)
	inflagxTR	-0.530**	-0.522**	-0.690***	-0.682***
		(0.220)	(0.213)	(0.169)	(0.180)
	inflagxLS	0.427**	0.388*	0.671***	0.702***
		(0.218)	(0.213)	(0.207)	(0.221)
	year2	0.976	0.998	0.899	0.952
		(1.166)	(0.871)	(0.825)	(1.034)
	year3	2.304***	1.582**	1.734***	2.500***
		(0.833)	(0.677)	(0.637)	(0.685)
	year4	0.634			0.707
		(0.923)			(0.784)
	banking_cr~s		2.449'	2.211*
			(1.511)	(1.227)
	year5		-0.616	-0.725
			(0.874)	(0.743)
	inflagxT			-0.133**	-0.127*
				(0.067)	(0.072)
	N	80	80	80	80
	Standard errors in parentheses
	' p<0.2, * p<0.1, ** p<0.05, *** p<0.01

The ECB resisted for years to the release of minutes. An act that makes it accused for failure to be transparent. Recently, the ECB decided to revise its decision in favor of a move to publish the minutes of its meetings.

For more discussions about the issue, see, for example,[START_REF] Kool | How effective is forward guidance[END_REF] and[START_REF] Geraats | Monetary Policy Transparency[END_REF].

See Born et al. (2011) for more details.

The index was originally compiled by[START_REF] Eijffinger | How Transparent Are Central Banks?[END_REF] for 9 major central banks in[START_REF] Blundell | Initial conditions and moment restrictions in dynamic panel data models[END_REF] and 2002[START_REF] Dincer | Central Bank Transparency: Causes, Consequences and Updates[END_REF] build their index on that of[START_REF] Eijffinger | How Transparent Are Central Banks?[END_REF] for 100 central banks under the period 1998-2006. Siklos (2011) updated the index until 2009. Finally, Dincer and Eichengreen (2014) extended the score until 2010.

[START_REF] Geraats | Central bank transparency[END_REF] distinguishes five dimensions: economic (data, models, forecasts), political (objectives, numeric targets, institution), procedural (strategy, minutes, and votes), policy (decision explanation, inclination) and operational (control errors, transmission shocks).

We also tested the effect of central bank independence index augmented by[START_REF] Dincer | Central Bank transparency and independence: Updates and new measures[END_REF] based on data of[START_REF] Cukierman | Measuring the Independence of Central Banks and Its Effect on Policy Outcomes[END_REF], but we didn't find any significant impact on inflation.

The index comprises 15 components; each component is assigned 0, ½ or 1 points. The overall score value ranges between 0 and 15.

In a first step, we consider a set of control variables that are different from that used in van der Cruijsen et al.(2010). The comparison between our results and those of van der Cruijsen et al. (2010) is made in a second step.

We are grateful to an anonymous referee for suggesting the inclusion of the interest rate in the regression.

[START_REF] Fuhrer | Inflation persistence" Handbook of Monetary Economics[END_REF] enumerated a battery of measures that attempt to capture the persistence of inflation. We recommend the paper for an overview.

Other (non-parametric) techniques to test U-shape are also suggested in literature. See, for example,Imbs and Wascziarg (2003) and[START_REF] Kostyshak | Non-parametric testing of U-shaped relationship[END_REF].

The LSDVC estimator becomes a widely applied methodology for short dynamic panels, devoted to many applications (e.g[START_REF] Flannery | Estimating Dynamic Panel Models in Corporate Finance[END_REF] Bogliacino et al., 2013).

Those definitions are extracted from Stata manual. Check http://www.stata.com/manuals13/xtxtabond.pdf ; http://www.stata.com/manuals13/xtxtdpdsys.pdf and Bruno (2005).

The KOF index encompasses three aspects: economic, social, and political.

The definition is taken from: http://www.mrglobalization.com/globalisation/292-measuring-globalization. See[START_REF] Dreher | Measuring Globalisation -Gauging its Consequences[END_REF] for details.

We considered lags k=1, 2, 3.

The control variables show almost the same pattern for all three estimators. Note that Sargan test for overidentification in GMM estimators is not applicable in case of robust standard errors.

The score is still superior to 7.5.

We thank an anonymous referee for noting such an interesting point.

Some differences are related to the coefficient's size of the Lngdp variable which becomes high in magnitude and statistically significant.

There are 8 emerging OECD countries in our sample.

For instance, van der Cruijsen et al. (2010) claimed that it is difficult to find reliable instruments for central bank transparency. However, one could follow[START_REF] Dincer | Central Bank Transparency: Causes, Consequences and Updates[END_REF] and use the fitted value of transparency on a constant and a rule of law.


Zhiming Guo

Shibin Wang

Linan Li

Hongwei Ji

Zhiyong Wang

Songbao Cai

Inflation of Stressed Cylindrical Tubes: An Experimental Study

Keywords: Inflation, Cylindrical tube, Critical and propagation pressure, Profile, Localization

The inflation of an initially stressed cylindrical shell provides a good illustration of the phenomenon of the initiation and propagation of an instability, which shares the same mathematical and mechanical features with a variety of other strain localization phenomena in engineering structures and materials. The high speed CCD camera and digital image processing system were used to measure the 3D shape of the inflated cylindrical tube. The localized bulge of a cylindrical tube with closed ends forms when the internal pressure reaches a critical value P cr . As more air is filled into the tube, the pressure drops but the radius at the centre of the bulge will increase until it reaches a maximum value r max . With continued inflation, the pressure stays at a constant value P p . The purpose of this study is to investigate the critical and propagation pressures in the tubes and the profile outside when the shells under axial tension and internal pressure were inflating. We focus on the influence of the axial tension on the critical pressure. In this paper the problem is explored through experimental efforts. A series of experiments were conducted on commercially available natural rubber latex tubes involving different geometries and initial axial tensions, which were regarded as isotropic, homogeneous, incompressible and hyper-elastic materials.

INTRODUCTION

Hyper-elastic tube structures are widely used in modern engineering and everyday life. Because of some features such as that they are collapsible, low weight and inflatable, they have promising applications in extra-terrestrial use and widely application prospect in the field of spaceflight. Many tissues and organs also consist of hyper-elastic tube structures which form the special bulge under the internal pressure and complex load. Localized bulging of an inflated hyper-elastic membrane tube shares the same mathematical and mechanical features with a variety of other strain localization phenomena in engineering structures and materials. The localized bulge mentioned above would cause harm to engineering safety and functions of biological tissue.

However, an inflation of a hollow rubber-like stressed cylindrical shell provides a good illustration of the phenomenon of initiation and propagation of an initially localized instability. The process of inflation is known as localization can be described as follows. When a cylindrical membrane tube with closed ends is inflated by an internal pressure, it firstly expands in the radial and axially in a uniform fashion. As more air is filled into the tube, the cylindrical configuration becomes unstable, and a local bulge appears somewhere along the length of the tube with the internal pressure drops. Because of the dropping pressure, the part of the tube away from the bulge experience unloading. In real engineering and life, the process shares the same features as a family of other problems, such as propagating buckles in long metal tubes under external pressure [START_REF] Kyriakides | Experimental determination of the propagation pressure of circular pipes[END_REF] . propagating necks in some polymeric materials when pulled in tension [START_REF] Hutchinson | Neck propagation[END_REF] and stress-induced phase transformations [START_REF] Ericksen | Equilibrium of bars[END_REF] .

Inflation of membrane tube and associated problem of instability is a classical subject that has been studied in many books and papers. The mechanisms for the initiation and propagation of bulges in an inflated tube were made more transparent through the analysis of Chater and Hutchinson [START_REF] Chater | On the propagation of bulges and buckles[END_REF] and experiments and numerical studies of Kyriakides and Chang [START_REF] Kyriakides | On the inflation of a long elastic tube in the presence of axial load[END_REF][START_REF] Kyriakides | The initiation and propagation of a localized instability in an inflated elastic tube[END_REF] . Fu et al. [START_REF] Fu | Post-bifurcation analysis of a thin-walled hype-elastic tube under inflation[END_REF] recalculated the solid line in the Figure 9 of Kyriakides and Chang [START_REF] Kyriakides | On the inflation of a long elastic tube in the presence of axial load[END_REF] and found almost perfect agreement between the theoretical limiting pressure and the experimental initiation pressure. Yin [START_REF] Yin | Non-uniform inflation of a cylindrical elastic membrane and direct determination of the strain energy function[END_REF] gave a detailed characterization of the propagation state (the kinked state). Additional experimental results are given by Pamplona et al. and Goncalves et al. [START_REF] Pamplona | Finite deformation of cylindrical membrane under internal pressure[END_REF][START_REF] Goncalves | Finite deformations of an initially stressed cylindrical shell under internal pressure[END_REF] . In this paper, a series of experiments are carried out to observe the process of bulging. The high speed CCD camera and digital image processing system were used to measure the 3D shape of the inflated cylindrical tube.

The problem of inflation is explored through experimental efforts and the rest of this paper proceeds as follows. After describe the specimen and experimental setup in the next section, we will present the method and show the results of measurement. In the final section, we make a summary of our findings.

EXPERIMENTAL SETUP AND SPECIMEN

The experimental setup used in measuring critical pressure and 3D shape of deformation is shown schematically in Fig. 1. The device had the functions of clamping steadily and providing internal pressure continuously. The initial axial load was applied by hanging a calibrated weight which was placed on a guide pulley. The friction of the pulley was small. The tube was inflated with compressed air from the left end. A pressure gage and a dynamic pressure sensor were used to monitor the pressure timely in the system. The pressure value could be read on the dial, and the output of the pressure sensor was recorded in computer using a data acquisition card at the same time. The test section was clamped the connection and sealing element at the ends of the tube. The rate of inflation was controlled manually through a throttle. A pressure maintaining valve was used to ensure a constant supply of air, moreover, a high speed camera system was used to monitor and record the deformed configurations of the tube during inflation.

The experiments were conducted on commercially available nature latex tubes as shown in Fig. 2. The tube had a nominal outside diameter of 7.5 mm and wall thickness of 1.5 mm and the length ranged between about 30 tube diameters. The tolerance of the diameter and wall thickness was ±0.2 mm. These natural rubber latex tubes were involving different geometries, which were regarded as isotropic, homogeneous, incompressible and hyper-elastic materials. In preparation for an experiment, the test specimens hand-picked were cyclically loaded axially to achieve the required material stability.

EXPERIMENTAL METHOD AND RESULTS

When the air is pumped into the tube at a slow rate, the internal pressure monitored by pressure sensor is first increased to a critical value P cr gradually, then it drops suddenly to a constant value P p . After inflating uniform, a bulge was formed rapidly along the length of the tube and the radius at the centre of the bulge will increase until it reached a maximum r max . The 3D shape of the inflated tube is recorded at the rate of 2000 fps. In order to record more clearly, the two ends should be fixed steadily but not be free. All the experiments are carried out at the temperature of 28 ℃.

The inflating process and pressure-time history of the tube under axial tension

The apparatus mentioned above fills the tube with air, measures and controls the internal pressure. Fig. 3 shows a series of photographs of the cylindrical tube subjected to increasing volume of air. The corresponding internal pressure shown in Fig. 4 can be observed in the dial of the pressure gage and collected into the computer by the data acquisition unit. In Fig. 3 (a) the tube is stretched. As the air is filled, the tube increases uniformly in Fig. 3 (b) until a maximum value r max is reached, after which there is a localization of deformation in Fig. 3(c), which is taken just prior to bulge initiation as shown in Fig. 3 (d). Fig. 3 (e-f) show the approximately axi-symmetric bulge spreads in both radial and axial directions. In Fig. 4 the curve section ①, ② and ③ corresponds to Fig. 3 (a-b), (c-d) and (e-f) respectively. In order to get a better look at the edge outside, a light box is used behind the tube.

The measurement of critical and propagation pressure at different sizes of axial loading

As the air is injected, the localized bulge forms while the internal pressure reaches a critical value P cr . With continued inflation, the pressure drops and finally stays at a constant value P p . Fig. 5 presents the pressure-time history recorded in one elastic tube. For the range of axial tension considered, when the cylindrical shells are initially under a set of different axial tensions, it is shown in Fig. 5 that axial tension lowers the initiation pressure observably but the influence of axial loading to propagation pressure is not obvious in the general trend. This is probably due to that the boundary condition has larger effect on the instability but smaller effect on the stable propagation of the localized bulge.

Measured profiles of the initiation and propagating bulge in an inflating tube

The profile of initiation and propagation of a bulge is measured using the high speed camera and digital image processing system. Fig. 6 shows a picture get through edge extraction, which explains the process involves forming of a localized bulge and unloading of the non-expansion section. This phenomenon is occur in a very short time about 30ms.

As shown in Fig. 6 the red line and yellow line represent the shapes before and after the inflation, after uniform expansion, the pressure jump causes a sudden expansion at the point O and a reduction in the diameter of undisturbed section in the green line. In the next moment, the bulge continues to increase in both radial and axial directions.

Fig. 6. The profile of the expansion and unloading section when the bulge is formed Fig. 7 presents the propagation of the bulge from right to left while the tube radius reaches a maximum value r max . The small and large diameter sections of the bulged tube remained the same, and the diameter of small section is the same with the unloading part in size. The velocity of propagation depend largely on the rate of inflation. The region which connects the small and large parts moves from location of bulging to both ends and keeps original shape. Fig. 7. The profiles of a propagating bulge in an inflated tube Fig. 8 shows a set of real detailed profiles of initiation of a bulge recorded by the camera mentioned above. We can observe the rapid expansion of the bulge, which is not a constant speed. The position of the bulge may differ from one specimen to another. This is probably due to the effect of initial material imperfection. More research will be carried out and relevant results will be reported in separate papers.

① ② ③ ④ ⑤ ⑥ ⑦ ⑧ ⑨ ⑩
Fig. 8 The initiation process of a bulge recorded by the high speed camera

4. CONCLUSION

In this paper the internal critical and propagation pressure and corresponding deformed configurations of an inflated tube are measured. As observed in the experiments, when the air is injected, the whole cylindrical shell deformed symmetrically and the pressure inside increases continuously until a maximum pressure P cr was reached. At this moment, the pressure drops suddenly and a bulge is initiated at a certain position along the tube and then propagated steadily, quasi state propagation of the bulge is constant.

Axial tension lowers the initiation pressure observably but the influence of axial loading to propagation pressure is not obvious in the general trend. The expansion of the bulge in an inflated tube occurs in a very short time, and the velocity is not a constant. In the paper, a detailed process of inflating of a cylindrical shell is studied experimental, meanwhile the pressure inside and shape of the tubes are recorded.

Fig. 1 .

 1 Fig. 1. Experimental setup Fig. 2. Nature latex tube

Fig. 3 .Fig. 4 .

 34 Fig. 3. A set of configurations in the stretched tube inflation experiment

Fig. 5 .

 5 Fig. 5. Measured critical and propagation pressure at different values of axial tension

Acknowledgement

This work presented was supported by the National Science Foundation (NSFC, Grant Nos. 11372212, 11072174) and the National Key Basic Research Program of China (973 Program, No. 2012CB937500).


Jiu-Sheng Ren

Dynamical response of hyper-elastic cylindrical shells under periodic load

Keywords: hyper-elastic cylindrical shells, nonlinear differential equation, periodic oscillation, quasi-periodic oscillation, critical load

Dynamical response of hyper-elastic cylindrical shells under periodic load

Introduction

In recent years, hyper-elastic materials, such as rubber, synthetic elastomers and polymeric materials, have been used in a broader and broader range of engineering fields due to their unique and non-replaceable properties. Therefore, the nonlinear problems, such as the instability of hyper-elastic materials and structures, have attracted much attention in the world as they play a fundamental role on the failure of materials [START_REF] Fu | Nonlinear elasticity[END_REF][START_REF] Beatty | Topics in finite elasticity[END_REF][START_REF] Gent | Elastic instability in rubber[J][END_REF] . The well-known examples are the inflation of spherical hyper-elastic balloons [START_REF] Gent | Elastic instability of inflated rubber shells[END_REF][START_REF] Needleman | Inflation of spherical rubber balloons[END_REF] and the deformation of the hyper-elastic spherical shells or cylindrical shells [START_REF] Haughton | On the incremental equations in non-linear elasticity-II: bifurcation of pressurized spherical shells[J][END_REF][START_REF] Haughton | Bifurcation of inflated circular cylinders of elastic material under axial loading-II: exact theory for thick-walled tubes[J][END_REF][START_REF] Jiusheng | Instability of incompressible thermo-hyperelastic tubes[J][END_REF] . Most of the literatures on such mechanics deal with elasto-statics which have been extensively studied. For example, when a cylindrical shell is inflated, it maintains a uniform inflation at the initial stage until a certain maximum of pressure is attained. After the maximum, the shell will undergo a non-uniform inflation. One part of the shell is highly stretched as a bubble while the remainder becomes lightly stretched. However, many physical problems are inherently dynamic, so a stability analysis of elasto-dynamics is important. As an example, intracranial aneurysms are typically subject to periodic or nearly periodic internal pressures [START_REF] Shah | Finite strain elastodynamics of intracranial aneurysms[J][END_REF] . Because of its property of large deformation, high elasticity and the application of hyper-elastic theory, the mathematical model for hyper-elastic dynamics is an ignition-boundary value problem with high nonlinear differential equations. Therefore, the solution of the dynamical problem is more difficult than the quasi-static problem. The system is autonomous when the forcing internal pressure is constant. Techniques for studying such autonomous systems are well-known, and they have been extensively studied [START_REF] Guo | Free and forced finite amplitude oscillations of an elastic thick-walled hollow sphere made of incompressible material[END_REF][START_REF] Calderer | The dynamical behavior of nonlinear elastic spherical shells[END_REF] . For example, when an incompressible cylindrical shell is inflated by a suddenly applied constant internal pressure, it will undergo a nonlinearly periodic oscillation. However, dynamical cases of a time varying internal pressure have not been well studied. The case of some kinds of biological soft tissues and rubber spherical membranes surrounded by a fluid under a periodic internal loading was studied by Haslach and Humphrey [START_REF] Haslach | Dynamics of biological soft tissue and rubber: internally pressurized spherical membranes surrounded by a fluid[END_REF] . A periodic orbit in a phase space exists near a static equilibrium, and a jump from one periodic orbit to another is possible for rubber models.

The purpose of the present paper is to further investigate the dynamical response and the destruction of incompressible hyper-elastic cylindrical shells under a periodic or constant internal pressure within a framework of finite elasto-dynamics. At first, the instability problem and the destruction of incompressible hyper-elastic cylindrical shells under a statically uniform internal pressure are examined within a framework of finite elasto-statics. The solution of the static equilibrium problem is the fixed point for the dynamic response of the corresponding autonomous system under a suddenly applied constant internal pressure. Then the dynamical response and the destruction of cylindrical shells under the suddenly applied constant internal pressure or the periodic internal pressure are examined within the framework of finite elastodynamics. The second order differential relationship between the deformation of the internal boundary of the shell and the internal pressure is obtained from the basic formulations. The displacement response curves, the phase portrait and the Poincaré maps are given out by the numerical computation through the Runge-Kutta integrator for the transformed first order differential equations. The dynamical response along with the destruction of the shell is discussed with these results following the usual dynamics. There exists a critical value for the shell under a suddenly applied constant internal pressure or a periodic internal pressure. When the pressure is less than the critical value, the shell will undergo a nonlinear periodic oscillation or quasi-periodic oscillation. When the pressure is larger than the critical value, the shell will be ultimately destroyed with time.

Formulations

Consider the finite deformation dynamics and the destruction for an incompressible hyperelastic cylindrical shell with an undeformed internal radius A and an outer radius B. Assume it is set into motion by a periodic internal pressure p(t) = p 0 + p 1 sin(ωt) at the initial time t 0 . A point (r, θ, z) at time t is assumed to occupy the point (R, Θ, Z) in the undeformed state. Then the deformation function of the shell is given as

r = r(R, t), θ = Θ, z = Z (A ≤ R ≤ B).
(

) 1
From the incompressibility condition of the material, we have

r = r(R, t) = (R 2 -A 2 ) + a 2 1 2 , ( 2
)
where a = r (A, t) is the deformed internal surface. The principal stretches are

λ 1 = ∂r(R, t) ∂R = R r , λ 2 = r(R, t) R = r R , λ 3 = 1. (3)
The strain energy function of the shell is given as the neo-Hookean material with the strain energy function as follows:

W = μ 2 (I 1 -3). (4)
Here, the first invariant I 1 = (λ 2 1 +λ 2 2 +λ 2 3 ), μ is the shear modulus for infinitesimal deformations. The corresponding non-zero principal components of the Cauchy stress tensor are

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ τ rr (r, t) = λ 1 ∂W ∂λ 1 -p(r, t) = μλ 2 1 -p(r, t), τ θθ (r, t) = λ 2 ∂W ∂λ 2 -p(r, t) = μλ 2 2 -p(r, t), (5)
where p(r, t) is the hydrostatic pressure to be determined. The motion equation with the absence of body forces is

dτ rr dr + 1 r (τ rr -τ θθ ) = ρr (t ≥ 0), ( 6
)
where ρ is the constant mass density of the material. The boundary conditions of the shell are

τ rr (a, t) = -p(t), τ rr (b, t) = 0. (7)
Here, b = r(B, t) is the deformed outer surface.

The initial stress-free conditions for the shell at time t = 0 are

r(R, 0) = R, ṙ(R, 0) = 0. ( 8
)

The governing differential equations

From the incompressible condition (2),

r(t) = 1 r 1 - a 2 r 2 ȧ2 + aä . ( 9
)
Substituting stresses ( 5) and ( 9) into the motion equation ( 6), we have

d dr μλ 2 1 -p(r, t) + 1 r μ λ 2 1 -λ 2 2 = ρ r 1 - a 2 r 2 ȧ2 + aä . ( 10
)
Integrating it with respect to r, then

μλ 2 1 -p(r, t) + p(a, t) + r a μ λ 2 1 -λ 2 2 ds s = ρ ȧ2 a 2 -r 2 + 2r 2 ln r -2r 2 ln a 2r 2 + ρaä(ln r -ln a). ( 11
)
Introducing it into (5), we have

τ rr (r, t) = ρ ȧ2 a 2 -r 2 + 2r 2 ln r -2r 2 ln a 2r 2 + ρaä(ln r -ln a) - r a μ λ 2 1 -λ 2 2 ds s -p(a, t). ( 12
)
Introducing it into (7), we have p(a, t) = p(t), and

p(t) =ρ ȧ2 a 2 -b 2 + 2b 2 ln b -2b 2 ln a 2b 2 + ρä(ln b -ln a) - b a μ(λ 2 1 -λ 2 2 ) ds s . ( 13
)
By defining new invariants

a A = x(t), B 2 A 2 -1 = δ, we have b 2 A 2 = x 2 (t) + δ, b 2 B 2 = x 2 + δ δ + 1 , b a = 1 + δ x 2 . Let ξ = r R , then ξ = 1 - a 2 -A 2 r 2 -1 2 , dr r = 1 1 -ξ 2 dξ ξ .
Equation ( 13) may be rewritten as

p(t) = ρA 2 ẋ2 -δ 2(x 2 + δ) + ln 1 + δ x 2 + ρA 2 xẍ ln 1 + δ x 2 - q x 2 +δ δ+1 x μ ξ -2 -ξ 2 ξ (1 -ξ 2 ) dξ = ρA 2 ẋ2 -δ 2(x 2 + δ) + ln 1 + δ x 2 + ρA 2 xẍ ln 1 + δ x 2 + 1 2 1 + δ δ + x 2 - 1 2x 2 -ln x + 1 2 ln δ + x 2 1 + δ . ( 14
)
Here,

a = xA, b = A √ x 2 + δ. The initial conditions are x(0) = 1, ẋ(0) = 0. ( 15
)
By putting x 1 = x and x 2 = ẋ, a system of first order equations are obtained from (14):

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ẋ1 = x 2 , ẋ2 = 1 x 1 ln 1 + δ x 2 1 1 ρA 2 - 1 2 1 + δ δ + x 2 1 + 1 2x 2 1 + ln x 1 - 1 2 ln δ + x 2 1 1 + δ + 1 x 1 ln 1 + δ x 2 1 p(t) ρA 2 -x 2 2 -δ 2(x 2 1 + δ) + ln 1 + δ x 2 1 . ( 16
)
The corresponding initial conditions are

x 1 (0) = 1, x 2 (0) = 0. ( 17
)

Static equilibrium solutions

The solution of the static equilibrium problem is the fixed point for the dynamic response of the corresponding autonomous system under a constant internal pressure. The property of the fixed point is related to the property of the dynamical solution and the motion of the shell. The equilibrium pressure function is the core of the static equilibrium problem. Let the right side of the motion equation ( 6) equal zero, the equilibrium equation for the cylindrical shell under a statically internal uniform pressure p 0 is obtained. Following a similar analogy as above, we have

p 0 = v(A) v(B) μ(v -3 -v) v 2 -1 dv. ( 18
)
Here,

v = v (R) = r (R) R = 1 + a 2 -A 2 R 2 1 2 , v(B) = 1 + a 2 -A 2 B 2 1 32 , v(A) = a A = x
describe the motion of the internal boundary of the shell. Numerical results of (18) for the shell with different thickness are shown in Fig. 1. Material constants are taken as μ = 2.63 MPa, ρ = 950 kg • m -3 . As shown in Fig. 1, the deformation of the shell increases with the pressure, and there exists a horizontal asymptote. When the pressure is close to the one corresponding to the horizontal asymptote, the deformation of the shell may have a quick increase, which means the destruction of the shell. Therefore, this pressure may be taken as the critical value p cr for the shell under the internal uniform pressure p 0 . Also, the critical value p cr increases with the thickness of the shell. For example, the values are 0.28 MPa, 1.45 MPa and 4.12 MPa for the shell with thickness δ = 0.234, 2.0, 4.0, respectively. Therefore, it is more difficult to destroy a thicker shell.

Dynamic response under a constant internal pressure

The system is autonomous when the forcing internal pressure is constant. Letting p(t) = p 0 and integrating (14) with respect to x, we obtain

x 2 ln 1 + δ x 2 ẋ2 - 2 ρA 2 x 1 x 1 2 1 + δ δ + x 2 - 1 2x 2 -ln x + 1 2 ln δ + x 2 1 + δ dx - p 0 ρA 2 (x 2 -1) = 0. ( 19
)
From the theory of vibrations, the motion x (t) is periodic if and only if the curve of x vs. ẋ = V in the phase diagram is closed and owns a finite period T = dx V . For a given load p 0 , the period motion x (t) will occur if there is a root x > 0 for (19) when V = 0. Letting

V = ẋ = 0 in (19) leads to 2 x 1 x 1 2 1 + δ δ + x 2 - 1 2x 2 -ln x + 1 2 ln δ + x 2 1 + δ dx - p 0 ρA 2 (x 2 -1) = 0. ( 20
)
For a given load p 0 , if there is a root x > 0 for (20), then it is the maximum radius of the internal surface of the shell in the oscillation process and denoted by x max . The curves between x max and p 0 for the shell with different thickness are shown in Fig. 2. Then the curves computed from (19) between the velocity V = ẋ and x are shown in Fig. 3 corresponding to different values of x max . It is seen that there exists a critical value p cr for the pressure of the shell (for example, when δ = 2, p cr = 1.204 MPa). When p 0 < p cr , x max increases with the pressure, and the corresponding phase curves in the phase plane are closed curves. Thus the shell undergoes a nonlinearly periodic oscillation. However, when p 0 ≥ p cr , the phase curves in the phase plane are not closed. So the shell will be destroyed ultimately with time. At the same time, the critical value p cr for the shell increases with the thickness of the shell, i.e., it is easier to destroy a shell with a thinner thickness.

The system is autonomous when the forcing internal pressure is constant. The fixed point for the system is (x 1 , x 2 ) = (x s , 0) . Here, x s is the corresponding static equilibrium of the system (the static deformation under the same load). The property of the fixed point is related to the property of the dynamical solution and the motion of the shell. The type of the fixed point should be determined from the Jacobian at the fixed point as usual. The Jacobian at the fixed point of equation ( 16) is

J = J 11 J 12 J 21 J 22 (xs,0) = 0 1 J 21 0 , ( 21
)
where,

J 21 = 1 x s ln 1 + δ x 2 s 2μ ρA 2 - 1 x 3 s - 2x s δ + x 2 s + 1 x s + (1 + δ) x s (δ + x 2 s ) 2 + 2δ x 4 s 1 + δ x 2 s ln 1 + δ x 2 s p 0 ρA 2 + μ ρA 2 1 2x 2 s - 1 + δ 2 (δ + x 2 s ) + ln x s -ln δ + x 2 s 1 + δ + 1 x 2 s ln 1 + δ x 2 s 2p 0 ρA 2 + 2μ ρA 2 1 2x 2 s - 1 + δ 2 (δ + x 2 s ) + ln x s -ln δ + x 2 s 1 + δ . ( 22
)
The trace of the Jacobian is tr J = J 11 + J 22 = 0, and the determinant of the Jacobian is D = det |J| = -J 21 . Numerical results for the determinant of the Jacobian for the shell in the case of δ = 2 from (22) are shown in Table 1. As usual, if D < 0, the fixed point is a saddle and it is unstable. If tr J = 0, D > 0, the fixed point is a center and it is stable. It is shown that if p 0 < p cr = 1.204 MPa (x s < 1.9), the fixed point is a center; and if p 0 ≥ p cr = 1.204 MPa, the fixed point is an unstable saddle in the case of δ = 2. And as usual, each center is surrounded by a homoclinic or heteroclinic orbit. As an example, for the shell in the case of δ = 2 under the constant pressure p 0 = 1.04 MPa, (x s , 0) = (1.6, 0) is the center of the closed curve (homoclinic orbit) with (x (0) , 0) = (1.0, 0) and (x max , 0) = (2.4, 0) in the phase space shown as the solid curve in Fig. 3.

Dynamic response under a periodic internal pressure

In practice, rubber hoses or biological structures must endure periodic or near periodic forcing. The response to the periodic forcing must be examined for its stability and for the destruction of the structure. Therefore, the dynamic response and the destruction of the shell for a periodic forcing, such as p (t) = p 1 + p 2 sin(ωt), are examined.

In the case of p (t) = p 1 + p 2 sin(3πt), numerical results, such as the displacement response curves, the phase portraits and the Poincaré maps, are given by numerical computation through the Runge-Kutta integrator of the first order differential equations ( 16). One of the displacement response curves is shown in Fig. As usual, for the internal periodic forcing p(t) = p 1 + p 2 sin(ωt), the pressure is p m = p 1 + p 2 . It is found that there exists a critical value p m cr for the mean pressure for a given load amplitude p 2 and a given frequency ω. When the mean pressure p m < p m cr , the displacement response curves, the phase portrait and the Poincaré maps as shown above may be obtained. It is shown by the figures that the motion of the shell presents nonlinear quasi-periodic oscillations. However, when the mean pressure p m ≥ p m cr , the figures as shown above cannot be obtained. That is to say the shell may be destroyed ultimately with time.

At the same time, the effect of the load amplitude p 2 and the frequency ω on the critical mean pressures p m cr may be ignored. For example, the value is always 1.4 MPa for the shell in the case of δ = 2.

Conclusions

Dynamical response and the destruction of internally periodic pressurized or constant pressurized incompressible hyper-elastic cylindrical shells are examined within the framework of finite elasto-dynamics. There exists a critical value for the pressure of the shell under static equilibrium. When the pressure approaches this critical value, the shell expands quickly and will be destroyed. The solution of the static equilibrium problem is the fixed point for the dynamical response of the corresponding system under a suddenly applied constant load. The property of the fixed point is related to the property of the dynamical solution and the motion of the shell. There also exists a critical value for the pressure of the shell under a suddenly applied constant internal pressure. When the pressure is less than this critical value, the fixed point is a center surrounded by a homoclinic orbit, and the shell will undergo nonlinear periodic oscillation. But when the pressure is larger than the critical value, the fixed point is an unstable saddle, the phase portrait is a non-closed curve, and the shell will be destroyed ultimately with time. There also exists a critical value for the pressure of the shell under a periodic internal pressure by numerical computation of the first order differential equations. When the pressure is less than this critical value, the displacement response curves, the phase portrait and the

Fig. 1

 1 Fig.1The equilibrium pressure function

Fig. 3

 3 Fig. 2 xmax vs. p0 curves

4 .Fig. 4 Fig. 5 Fig. 6 Fig. 7

 44567 Fig. 4 Displacement response curve with δ = 2, p1 = 0.3 MPa and p2 = 0.1 MPa

Fig. 8 Fig. 9 Fig. 10 Fig. 11

 891011 Fig. 8 Phase portrait with δ = 2, p1 = 1.4 MPa and p2 = 0.1 MPa

Table 1

 1 Determinant of the Jacobian (δ = 2)

	xs	1.1	1.2	1.3	1.4	1.5	1.6	1.7	1.8	1.9
	D/10 6	6.6	3.46	1.51	0.78	0.66	0.48	0.42	0.19	-0.16

Poincaré maps may be given, and the shell will undergo a nonlinear quasi-periodic oscillation. But when the pressure is larger than this critical value, the figures cannot be obtained, and the shell will be destroyed ultimately with time. The critical value demanded for the destruction of the shell under a periodic internal pressure or a constant pressure is less than that under the static equilibrium, and the value under the constant pressure is the lowest.