MD Bruno Etain
email: bruno.etain@inserm.fr

Mohamed Lajnef

Frank Bellivier

Flavie Mathieu

Aurélie Raust

Barbara Cochet

Sébastien Gard

Katia M ' Bailara

Jean-Pierre Kahn

Katia M'bailara

Orly Elgrabli

Renaud Cohen

Stéphane Jamain

Eduard Vieta

Marion Leboyer

Henry Chantal

Clinical expression of bipolar disorder type I as a function of age and polarity at onset: convergent findings in

Keywords: bipolar disorder, onset, polarity, age at onset, clinical expression, course

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Bipolar disorder type I (BD1) is highly heterogeneous in terms of its clinical expression, course, comorbid psychiatric disorders, comorbid medical conditions and treatment response [START_REF] Goodwin | Manic-Depressive Illness: Bipolar Disorders and Recurrent Depression 2nd Revised edition[END_REF] . The factors underlying this variability are poorly understood. Nevertheless, recent studies have suggested that some onset characteristics, such as age and polarity at the first mood episode, may be associated with different patterns of clinical expression, course and comorbid conditions [START_REF] Leboyer | Age at onset in bipolar affective disorders: a review[END_REF][START_REF] Colom | The road to DSM-V. Bipolar disorder episode and course specifiers[END_REF][START_REF] Tohen | The International Society for Bipolar Disorders (ISBD) Task Force report on the nomenclature of course and outcome in bipolar disorders[END_REF] . This may have major implications for the development of preventive strategies, clinical management and drug prescription.

Age at onset (i.e. age at the first major mood episode) is probably the associated factor most widely studied to date (for review see [START_REF] Leboyer | Age at onset in bipolar affective disorders: a review[END_REF]. A threshold at about 21 years of age, defining an early-onset subgroup of the disorder, has consistently been identified in seven independent bipolar I samples from Europe and the USA [START_REF] Bellivier | Admixture analysis of age at onset in bipolar I affective disorder[END_REF][START_REF] Bellivier | Age at onset in bipolar I affective disorder: further evidence for three subgroups[END_REF][START_REF] Hamshere | Age-at-onset in bipolar-I disorder: Mixture analysis of 1369 cases identifies three distinct clinical sub-groups[END_REF][START_REF] Lin | Clinical correlates and familial aggregation of age at onset in bipolar disorder[END_REF][START_REF] Manchia | Age at onset in Sardinian bipolar I patients: evidence for three subgroups[END_REF][START_REF] Severino | Association study in a Sardinian sample between bipolar disorder and the nuclear receptor REV-ERBalpha gene, a critical component of the circadian clock system[END_REF][START_REF] Tozzi | Admixture analysis of age at onset in bipolar disorder[END_REF] . Early-onset BD1 has been associated with a particular pattern of clinical expression (psychotic features during episodes), a more severe course (rapid cycling) and a higher frequency of comorbid conditions (suicidal behavior, alcohol/substance misuse and anxiety disorders) (for review see [START_REF] Leboyer | Age at onset in bipolar affective disorders: a review[END_REF] ).

The polarity of the first episode (i.e. manic or depressive) may also be a relevant factor associated with the course of the disorder. It has been suggested that depressive onset is the most frequent [START_REF] Daban | Clinical correlates of first-episode polarity in bipolar disorder[END_REF][START_REF] Perugi | Polarity of the first episode, clinical characteristics, and course of manic depressive illness: a systematic retrospective investigation of 320 bipolar I patients[END_REF][START_REF] Forty | Polarity at illness onset in bipolar I disorder and clinical course of illness[END_REF][START_REF] Perlis | Revisiting depressive-prone bipolar disorder: polarity of initial mood episode and disease course among bipolar I systematic treatment enhancement program for bipolar disorder participants[END_REF][START_REF] Chaudhury | Does first episode polarity predict risk for suicide attempt in bipolar disorder?[END_REF] and is associated with suicide attempts [START_REF] Daban | Clinical correlates of first-episode polarity in bipolar disorder[END_REF][START_REF] Perugi | Polarity of the first episode, clinical characteristics, and course of manic depressive illness: a systematic retrospective investigation of 320 bipolar I patients[END_REF][START_REF] Forty | Polarity at illness onset in bipolar I disorder and clinical course of illness[END_REF][START_REF] Perlis | Revisiting depressive-prone bipolar disorder: polarity of initial mood episode and disease course among bipolar I systematic treatment enhancement program for bipolar disorder participants[END_REF][START_REF] Chaudhury | Does first episode polarity predict risk for suicide attempt in bipolar disorder?[END_REF][START_REF] Azorin | Risk factors associated with lifetime suicide attempts in bipolar I patients: findings from a French National Cohort[END_REF] , a rapid cycling course [START_REF] Perugi | Polarity of the first episode, clinical characteristics, and course of manic depressive illness: a systematic retrospective investigation of 320 bipolar I patients[END_REF][START_REF] Forty | Polarity at illness onset in bipolar I disorder and clinical course of illness[END_REF] , comorbid anxiety disorders [START_REF] Perlis | Revisiting depressive-prone bipolar disorder: polarity of initial mood episode and disease course among bipolar I systematic treatment enhancement program for bipolar disorder participants[END_REF][START_REF] Colom | Clinical and therapeutic implications of predominant polarity in bipolar disorder[END_REF] and axis II comorbid conditions [START_REF] Daban | Clinical correlates of first-episode polarity in bipolar disorder[END_REF] . A manic onset of the disease has been associated with lifetime psychotic symptoms [START_REF] Daban | Clinical correlates of first-episode polarity in bipolar disorder[END_REF][START_REF] Perugi | Polarity of the first episode, clinical characteristics, and course of manic depressive illness: a systematic retrospective investigation of 320 bipolar I patients[END_REF][START_REF] Chaudhury | Does first episode polarity predict risk for suicide attempt in bipolar disorder?[END_REF] , comorbid alcohol misuse [START_REF] Chaudhury | Does first episode polarity predict risk for suicide attempt in bipolar disorder?[END_REF] and substance misuse preceding onset [START_REF] Daban | Clinical correlates of first-episode polarity in bipolar disorder[END_REF] . Polarity at onset is also of particular importance because it has been shown to correlate with the predominant polarity of the disease 12-16, 18, 19 .

These results have led some authors to suggest that both polarity and age at onset should be added as specifiers in the future DSM-V, given their relevance to both the course and outcome of the disease [START_REF] Colom | The road to DSM-V. Bipolar disorder episode and course specifiers[END_REF] . Polarity and age at onset have essentially been studied independently, but depressive onset has been associated with an earlier age at onset 14-16, 20, 21 .

Thus, to avoid a confounding bias due to these two factors being studied independently, we investigated the relationship between both polarity and age at onset and the clinical, progressive and comorbid expression of BD1 in two well characterized independent samples of BD1 patients from France and the USA.

Methods

Sample of patients (France)

Adult patients meeting DSM-IV criteria [START_REF]American Psychiatric Association. Diagnostic and Statistical Manual of Mental Disorders, Fourth Edition[END_REF] for BD1 were recruited from three universityaffiliated psychiatry departments and interviewed by trained psychiatrists using the French version of the Diagnostic Interview for Genetic Studies (DIGS) [START_REF] Nurnberger | Diagnostic interview for genetic studies. Rationale, unique features, and training. NIMH Genetics Initiative[END_REF][START_REF] Preisig | Diagnostic interview for genetic studies (DIGS): inter-rater and test-retest reliability of the French version[END_REF] . Patients were of French origin and were euthymic at inclusion (i.e. having MADRS and Mania Rating Scale scores of no more than five) [START_REF] Bech | The mania rating scale: scale construction and inter-observer agreement[END_REF][START_REF] Montgomery | A new depression scale designed to be sensitive to change[END_REF] . Written informed consent was obtained from participants. This study was approved by the institutional review board.

Sample of patients (USA)

Clinical data were extracted from the Bipolar Disorder Phenome Database [START_REF] Potash | The bipolar disorder phenome database: a resource for genetic studies[END_REF] . We selected only BD1 patients, based on the best-estimate diagnosis reported in the database. A description of the construction of the database, the review of interview items, extraction from the original dataset and quality control of the database has been reported elsewhere [START_REF] Potash | The bipolar disorder phenome database: a resource for genetic studies[END_REF] . Briefly, bipolar patients were ascertained for genetic linkage studies. Diagnosis was determined with the Schedule for Affective Disorders and Schizophrenia-Lifetime Version [START_REF] Endicott | A diagnostic interview. The schedule for affective disorders and schizophrenia[END_REF] or the DIGS [START_REF] Nurnberger | Diagnostic interview for genetic studies. Rationale, unique features, and training. NIMH Genetics Initiative[END_REF] .

Definition of age at onset

In the French sample, age at onset (AAO) was defined as the age at which the first mood episode occurred (depressive, manic, hypomanic or mixed), according to DSM-IV criteria. It was determined by reviewing case notes and information from the DIGS.

In the American sample, AAO was defined, using the DIGS or the SADS, as the age at which the first DSM-IV criteria manic or depressive episode occurred, on the basis of the fields 'age at first major depression' and 'age at first mania', with the youngest age recorded taken as the age at onset. Hypomanic and mixed onsets were not taken into account because they were not recorded in the database.

Age at onset was classified into two groups: early age at onset (before 21 years) and later age at onset (onset at or after the age of 21 years). This threshold was selected on the basis of the replicated results of seven independent admixture analyses in European and American bipolar type I samples [START_REF] Bellivier | Admixture analysis of age at onset in bipolar I affective disorder[END_REF][START_REF] Bellivier | Age at onset in bipolar I affective disorder: further evidence for three subgroups[END_REF][START_REF] Hamshere | Age-at-onset in bipolar-I disorder: Mixture analysis of 1369 cases identifies three distinct clinical sub-groups[END_REF][START_REF] Lin | Clinical correlates and familial aggregation of age at onset in bipolar disorder[END_REF][START_REF] Manchia | Age at onset in Sardinian bipolar I patients: evidence for three subgroups[END_REF][START_REF] Severino | Association study in a Sardinian sample between bipolar disorder and the nuclear receptor REV-ERBalpha gene, a critical component of the circadian clock system[END_REF][START_REF] Tozzi | Admixture analysis of age at onset in bipolar disorder[END_REF] .

Definition of polarity at onset

In the French sample, two types of onset polarity were defined: depressive onset when the first mood episode met DSM-IV criteria for a major depressive episode, and manic onset when the first mood episode met DSM-IV criteria for a manic, hypomanic or mixed episode.

In the American sample, polarity at onset was determined by comparing the reported age at the first major depressive episode and the age at the first manic episode according to DSM-IV criteria. Subjects for whom the first major depressive episode and the first manic episode occurred at the same age were not included in the analysis. In the American sample, hypomanic and mixed onsets were not taken into account in the definition of polarity at onset because they were not recorded in the database.

Dependent variables

Several clinical categorical and continuous variables were extracted from the two databases (France and USA). Categorical variables (suicidal behavior, mixed episodes, rapid cycling, psychotic symptoms, alcohol/drug misuse, comorbid anxiety disorders) were categorized into lifetime presence versus absence. The term 'misuse' defined the abuse of or dependence on substances. For continuous variables, a density measure was obtained by dividing the continuous variable (e.g. number of events) by the duration of the disease (age at interview minus age at onset).

Statistical analysis

We investigated the association of demographic and clinical characteristics with both polarity and age at onset. A logarithmic transformation of continuous variables (density of events per year of disease duration) was performed to achieve the normality assumed for parametric procedures. Generalized regression models adjusting for sex were then used to examine relationships between continuous variables and polarity and age at onset as independent variables. Associations between categorical dependent variables and independent variables (polarity and age at onset) were tested by logistic regression with adjustment for sex. As the analysis was exploratory, a p-value of 0.05 or less was considered significant. For variables showing association in both samples, sensitivity (Sen) and specificity (Spe) were calculated (reported as the area under the curve (AUC) or a percentage). Statistical analyses were performed with SAS software, version 9.1.

Results

Characteristics of the samples

We studied 480 BD1 patients from France and 714 from the USA. The characteristics of the two samples differed significantly for numerous variables (table 1); all differences remained significant when duration was used as a covariate (data not shown).

Depressive onset was associated with an earlier age at onset in the American sample (² = 20.97, p<0.001), whereas no such association was observed in the French sample (² = 0.56, p=0.45).

Polarity at onset

Most BD1 patients experienced a depressive onset, and a depressive onset was much more frequent among American patients than among French patients (71% versus 57.9%, respectively; ²=21.82, p<0.001).

Several associations were observed in both samples (see table 2). A depressive onset was associated with a higher density of depressive episodes (France: p=0.02 AUC=0.68; USA: p<0.001 AUC=0.61), a lifetime presence of suicidal behavior (France: OR=1.48 95% CI [1.01-2.16] Sen=47% Spe=63%; USA: OR=1.52 95% CI [1.06-2.17] Sen=44% Spe=70%) and lifetime alcohol misuse (France: OR=2.08 95% CI [1.28-3.36] Sen=26% Spe=85%; USA: OR=1.46 95% CI [1.03-2.08] Sen=42% Spe=66%). A manic onset was associated with a higher density of manic episodes (p<0.001 in both samples, AUC France =0.59 and AUC USA =0.63).

In the French sample, 61.3% of the mood episodes of bipolar patients who had experienced a depressive onset were major depressive episodes, whereas 60.1% of mood episodes in bipolar patients who had experienced a manic or hypomanic onset were manic or hypomanic. Similar percentages were observed for the American sample (61.3% and 57.8%, respectively). Thus, in both samples, polarity at onset (but not age at onset) was associated with the subsequent predominant polarity of the disease. Indeed, polarity at onset was the only variable found to be associated with this percentage (France: beta=-0.21, 95% CI [-0.26, -0.16], p<0.001; USA: beta=-0.19, 95% CI [-0.23, -0.15], p<0.001). Some associations were observed in only one sample. A depressive onset was associated with rapid cycling in the French sample only, and with panic disorder and anxiety disorders in the US sample only. A manic onset was associated with a higher density of hospitalizations in the US sample only.

Age at onset

A large proportion of patients belonged to the early-onset subgroup in both the French and American samples (42 % in France versus 68 % in the USA; p<0.001). Several associations were observed in both samples (see table 2). An early age at onset was associated with suicidal behavior (France: OR=2. Some associations were observed in only one sample. An earlier onset was associated with a lower density of hospitalizations in the French sample only. An earlier onset was associated with a higher density of major episode (both depressive and manic), rapid-cycling, alcohol misuse, panic disorder and anxiety disorders in the US sample only.

Discussion

In two large independent samples of BD1 patients, we found that a depressive onset was the most frequent initial presentation of the disease and that polarity at onset was strongly correlated with the predominant polarity. We suggest that age and polarity at onset are associated with particular patterns of clinical expression, disease course and comorbid conditions. Indeed, a depressive onset was associated with suicidal behavior and alcohol misuse and earlier onset was associated with suicidal behavior and cannabis and cocaine/opiate misuse. These findings were obtained in two independent samples with different clinical presentations, recruitment procedures, geographic origins, level of familial/genetic loading, age at onset and duration of illness. Furthermore, slightly different criteria were used to define age at onset and polarity at onset in the two samples. Nevertheless, despite these major differences, we found consistent associations in the two samples. These findings suggest that both age and polarity at onset should be systematically investigated in BD1 patients and could be considered specifiers of the course of the disease, as previously proposed [START_REF] Colom | The road to DSM-V. Bipolar disorder episode and course specifiers[END_REF] .

In both samples, a major depressive episode was the most frequent mode of onset of BD1, as previously suggested [START_REF] Daban | Clinical correlates of first-episode polarity in bipolar disorder[END_REF][START_REF] Perugi | Polarity of the first episode, clinical characteristics, and course of manic depressive illness: a systematic retrospective investigation of 320 bipolar I patients[END_REF][START_REF] Forty | Polarity at illness onset in bipolar I disorder and clinical course of illness[END_REF] . Therefore, most BD1 patients will initially be thought to be suffering from a major depressive episode (single or recurrent) before the occurrence of the first manic, hypomanic or mixed episode. This may account for the high estimated rate of misdiagnosis (about 70%) [START_REF] Bowden | Strategies to reduce misdiagnosis of bipolar depression[END_REF][START_REF] Perlis | Misdiagnosis of bipolar disorder[END_REF][START_REF] Keck | Clinical and economic effects of unrecognized or inadequately treated bipolar disorder[END_REF][START_REF] Fajutrao | A systematic review of the evidence of the burden of bipolar disorder in Europe[END_REF] and the long delay to treatment in many patients [START_REF] Hirschfeld | Recognition and diagnosis of bipolar disorder[END_REF] . The awareness of psychiatrists and general practitioners should be increased, to encourage them to explore in detail all possible indicators of a progression towards bipolar disorders when treating a patient suffering from a major depressive episode (particularly when recurrent).

Careful screening for certain characteristics of major depressive episodes that have been identified as potential indicators of bipolar disease, and for undiagnosed hypomanic episodes and a family history of BD should immediately trigger suspicion of possible progression towards bipolar disorders [START_REF] Ghaemi | Cade's disease" and beyond: misdiagnosis, antidepressant use, and a proposed definition for bipolar spectrum disorder[END_REF] .

Our second major finding for these two independent samples was the strong correlation between the polarity at onset of the disorder and subsequent predominant polarity, consistent with previous reports 12-15, 18, 35 . Polarity at onset is thought to be familial [START_REF] Kassem | Familiality of polarity at illness onset in bipolar affective disorder[END_REF] and underpinned by genetic and/or shared environmental factors, accounting for the notion that polarity at onset reflects a more stable trait (i.e predominant polarity). This may have many implications for treatment. Indeed, the drug treatment strategy to be used could be anticipated early in the course of the disorder on the basis of polarity at onset. Some drugs may be significantly more effective at preventing manic relapses, whereas others may be better at preventing depressive symptoms [START_REF] Beynon | Pharmacological interventions for the prevention of relapse in bipolar disorder: a systematic review of controlled trials[END_REF] . Some adjustment of the serum concentrations of mood regulators may also be required, as a function of polarity patterns. Lithium concentrations in the lower part of the therapeutic range may be sufficient for the optimal prevention of depressive episodes, whereas higher lithium concentrations, within the therapeutic range, may be required for optimal protection against manic/mixed episodes [START_REF] Severus | Is the polarity of relapse/recurrence in bipolar-I disorder patients related to serum lithium levels? Results from an empirical study[END_REF][START_REF] Kleindienst | Are serum lithium levels related to the polarity of recurrence in bipolar disorders? Evidence from a multicenter trial[END_REF][START_REF] Kleindienst | Is polarity of recurrence related to serum lithium level in patients with bipolar disorder?[END_REF] .

In both samples, depressive onset was associated with suicidal behavior and alcohol misuse, as reported in previous studies [START_REF] Daban | Clinical correlates of first-episode polarity in bipolar disorder[END_REF][START_REF] Perugi | Polarity of the first episode, clinical characteristics, and course of manic depressive illness: a systematic retrospective investigation of 320 bipolar I patients[END_REF][START_REF] Forty | Polarity at illness onset in bipolar I disorder and clinical course of illness[END_REF][START_REF] Perlis | Revisiting depressive-prone bipolar disorder: polarity of initial mood episode and disease course among bipolar I systematic treatment enhancement program for bipolar disorder participants[END_REF][START_REF] Chaudhury | Does first episode polarity predict risk for suicide attempt in bipolar disorder?[END_REF][START_REF] Azorin | Risk factors associated with lifetime suicide attempts in bipolar I patients: findings from a French National Cohort[END_REF] . An early onset was associated with suicidal behavior and the misuse of cannabis or cocaine/opiates, as previously reported [START_REF] Leboyer | Age at onset in bipolar affective disorders: a review[END_REF][START_REF] Hamshere | Age-at-onset in bipolar-I disorder: Mixture analysis of 1369 cases identifies three distinct clinical sub-groups[END_REF][START_REF] Lin | Clinical correlates and familial aggregation of age at onset in bipolar disorder[END_REF] . These findings suggest that careful and precise screening for substance use disorders and risks, and closer clinical monitoring could help to prevent poor outcome (due to alcohol misuse and suicide attempts, in particular) in early onset mood disorders.

This study was subject to several limitations. First, the determination of onset characteristics was based on retrospective assessment and may therefore be subject to recall bias. Second, the definition of polarity and age at onset slightly differed between samples, particularly in terms of the frequency of depressive polarity and early onset. Misclassifications may have occurred for some patients in the American sample, for which hypomanic and mixed-onset episodes were not recorded because they may be difficult to identify retrospectively [START_REF] Perugi | Polarity of the first episode, clinical characteristics, and course of manic depressive illness: a systematic retrospective investigation of 320 bipolar I patients[END_REF][START_REF] Perlis | Revisiting depressive-prone bipolar disorder: polarity of initial mood episode and disease course among bipolar I systematic treatment enhancement program for bipolar disorder participants[END_REF] . Third, some associations were observed only in the American sample and this lack of replication might be due to insufficient power, due to the smaller size of the French sample. Finally, the samples differed in terms of clinical presentation. This may be due to differences in recruitment procedures, geographic origins, level of genetic loading, age at onset and duration of illness.

As previously reported [START_REF] Post | Incidence of childhood-onset bipolar illness in the USA and Europe[END_REF][START_REF] Baldessarini | Onset-age of bipolar disorders at six international sites[END_REF] , BD1 patients from the USA have an earlier onset than patients from France. Indeed, the French sample was a mixture of sporadic and familial cases (60% versus 40%), whereas all the American patients had at least one affected sibling; the American sample may therefore have been enriched in early-onset cases 2 . This may appear to be a limitation of the study. However, despite differences between the samples in the frequencies of depressive and/or earlier onset, several results were consistent between samples. This suggests that the principal associations observed between onset characteristics and subsequent course were robust.

The aim of this study was to determine whether onset characteristics were associated with the subsequent clinical expression of the disorder. The sensitivity and specificity of onset characteristics for predicting clinical variables appeared to be low. Indeed, the sensitivity of age at onset for predicting clinical characteristics ranged from 1 to 35% in the French sample and from 7 to 30% in the American sample. For polarity at onset, sensitivity was higher (26-47 %), with an AUC between 0.59 and 0.68, and specificity was acceptable (63-85%). Given the sensitivity and specificity of each parameter, this suggests that many other factors may affect outcome, including duration of untreated period, the appropriateness of treatment and compliance with treatment. This issue could be addressed, at least in part, by making use of information about current psychotropic treatments and previous treatment sequences, but this information was not available in this study. Moreover, given the differences in sensitivity between the two factors considered, polarity at onset seems to have the better potential as a specifier, further investigations being required for age at onset.

Conclusion

Polarity at onset is associated with subsequent predominant polarity during the course of BD1. Suicidal behavior and substance misuse are associated with both onset characteristics (i.e. a depressive onset and an earlier age at onset). These findings are of particular interest for the identification of more homogeneous subgroups of patients for clinical and research studies, the definition of focused prevention strategies for comorbid conditions and selection of the optimal and best targeted drugs for treatment. Our findings and those of previous studies provide evidence that onset polarity should be considered as a relevant specifier in future BD1 classifications, as proposed for the DSM-V 3 . Age at onset, given it relatively low sensitivity for predicting clinical variables, requires further investigation to determine its potential relevance as a specifier.

Acknowledgments (France)

This work was supported by INSERM, Assistance Publique -Hôpitaux de Paris, RTRS Santé Mentale (Fondation Fondamental), Agence Nationale pour la Recherche (ANR), Fondation pour la Recherche sur le Cerveau (FRC) and National Alliance for Research on Schizophrenia and Depression (NARSAD). We thank E. Abadie, C. Bulach and M.J. Pereira Gomes for technical assistance. We also thank J.R. Richard. We thank bipolar patients for their participation. We thank the NIMH and J.B. Potash for providing access to the Bipolar Phenome Database.

Acknowledgment for Bipolar

Disorder Biomaterials and Clinical Data (USA) Data and biomaterials were collected as part of 10 projects from the National Institute of Mental Health (NIMH) Bipolar Disorder Genetics Initiative. From 1999-2003, the principal investigators and co-investigators were: Indiana University, Indianapolis, IN, R01 MH59545, John Nurnberger, M.D., Ph.D., Marvin J. Miller, M.D., Elizabeth S. Bowman, M.D., N. Leela Rau, M.D., P. Ryan Moe, M.D., Nalini Samavedy, M.D., Rif El-Mallakh, M.D. (at University of Louisville), Husseini Manji, M.D. (at Wayne State University), Debra A. Glitz, M.D. (at Wayne State University), Eric T. Meyer, M.S., Carrie Smiley, R.N., Tatiana Foroud, Ph.D., Leah Flury, M.S., Danielle M. Dick, Ph.D., Howard Edenberg, Ph.D.; Washington University, St. Louis, MO, R01 MH059534, John Rice, Ph.D, Theodore Reich, M.D., Allison Goate, Ph.D., Laura Bierut, M.D.; Johns Hopkins University, Baltimore, MD, R01 MH59533, Melvin McInnis M.D. , J. Raymond DePaulo, Jr., M.D., Dean F. MacKinnon, M.D., Francis M. Mondimore, M.D., James B. Potash, M.D., Peter P. Zandi, Ph.D, Dimitrios Avramopoulos, and Jennifer Payne; University of Pennsylvania, PA, R01 MH59553, Wade Berrettini M.D.,Ph.D.; University of California at Irvine, CA, R01 MH60068, William Byerley M.D., and Mark Vawter M.D.; University of Iowa, IA, R01 MH059548, William Coryell M.D. , and Raymond Crowe M.D.; University of Chicago, IL, R01 MH59535, Elliot Gershon, M.D., Judith Badner Ph.D., Francis McMahon M.D., Chunyu Liu Ph.D., Alan Sanders M.D., Maria Caserta, Steven Dinwiddie M.D., Tu Nguyen, Donna Harakal; University of California at San Diego, CA, R01 MH59567, John Kelsoe, M.D., Rebecca McKinney, B.A.; Rush University, IL, R01 MH059556, William Scheftner M.D., Howard M. Kravitz, D.O., M.P.H., Diana Marta, B.S., Annette Vaughn-Brown, MSN, RN, and Laurie Bederow, MA; NIMH Intramural Research Program, Bethesda, MD, 1Z01MH002810-01, Francis J. McMahon, M.D., Layla Kassem, PsyD, Sevilla Detera-Wadleigh, Ph.D, Lisa Austin, Ph.D, Dennis L. Murphy, M.D. In addition, families were provided by Dr. Carlos Pato at the University of Southern California and his staff. This work was sponsored by NIMH grants MH52618 and MH058693. Genotyping services were provided by the Center for Inherited Disease Research (CIDR). CIDR is fully funded through a federal contract from the National Institutes of Health to The Johns Hopkins University, Contract Number N01-HG-65403. Data and biomaterials were collected and supported by NIMH grant R01 MH59602 (to Miron Baron, M.D.) and by funds from the Columbia Genome Center and the New York State Office of Mental Health. The main contributors to this work were Miron Baron, M.D. (Principal Investigator), Jean Endicott, Ph.D. (Co-Principal Investigator), Jo Ellen Loth, M.S.W., John Nee, Ph.D, Richard Blumenthal, Ph.D., Lawrence Sharpe, M.D., Barbara Lilliston, M.S.W., Melissa Smith, M.A., and Kristine Trautman, M.S.W., all from Columbia University Department of Psychiatry, New York, NY, USA. A small subset of the sample was collected in Israel in collaboration with Bernard Lerer, M.D. and Kyra Kanyas, M.S. from the Hadassah -Hebrew University Medical Center, Jerusalem, Israel. We are grateful to the patients and their family members for their cooperation and support, and to the treatment facilities and other organizations that collaborated with us in identifying families. Most importantly, we thank the families who participated in and contributed to these studies.


Alexandru Minea
email: alexandru.minea@u-clermont1.fr.

Antoine Parent
email: antoine.parent@univ-nancy2.fr.

Bd F Mitterrand

Patrick Plane

Catherine Araujo Bonjean

Is High Public Debt Always Harmful to Economic Growth? Reinhart and Rogoff and some complex nonlinearities

Keywords: public debt, economic growth, nonlinear effects, cliometrics. JEL Codes: H63, E62, N10

Les commentaires et analyses développés n'engagent que leurs auteurs qui restent seuls responsables des erreurs et insuffisances.

I. Introduction

In their already-famous article "Growth in a Time of Debt", published by The American Economic Review in 2010 (AER 100(2), pp. 573-78), Carmen Reinhart and Kenneth Rogoff analyze the relation between economic growth and public debt. 1 Drawing upon a new public debt database, Reinhart & Rogoff (2010, hereafter RR) exhibit the presence of debt-to-GDP ratios that they assess in terms of threshold. Regarding advanced economies, RR show that post-WW2 economic growth is dramatically declining on average once the public debt-to-GDP ratio is above the 90% threshold. Policy recommendations by official institutions, including the OECD, the EU Commission or the French Report on Public Finance (April 2010), seem to have adopted this threshold as an ultimate guide for policy making, by taking for granted the "Reinhart and Rogoff's debt intolerance ratio" and transforming it, rather mechanically, into a "natural" target to shift public spending.

The goal of the present manuscript is to contribute to the current strand of literature on public debt and economic growth,2 by exploring the relevance of the 90% threshold emphasized by RR. Using up-to-date econometric techniques, which allow dealing properly with complex nonlinearities on panel data, we find the existence of an endogenouslyestimated threshold, at roughly 115%, in the public debt to economic growth relation. Below this threshold, a debt increase damages growth; however, this negative effect is declining as public debt is increasing. Above this threshold, the link between public debt and economic growth changes sign.

In light of these estimations, we reproduce the stylized facts emphasized by RR. On the one hand, we show that, similarly to RR, average economic growth is lower for countries with debt levels between 90 and 115%, compared to countries with debt levels between 60 and 90%. On the other hand, we show that average economic growth is higher for countries with public debt above 115%, compared to countries with debt levels between 90 and 115%, and more importantly, that average economic growth is not statistically different for the former group compared to countries with debt levels between 60 and 90%. Although one should reasonably refrain from concluding that governments should adopt loose fiscal policies, leading to high public debt levels, to foster economic growth, this latter result provides a new perspective on the "debt intolerance ratio" emphasized by RR. Indeed, an increasing public debt path is not generating bottomless pit growth losses, but is bordered by a high public debt regime in which economic growth may increase, emphasizing that additional evidence is needed before suggesting policy recommendations regarding growth effects of fiscal policy in such high debt regimes, which may be subject to complex nonlinearities.

Section 2 emphasizes the RR result, section 3 presents the econometric method, our results and discusses some policy implications and robustness, and section 4 concludes.

II. Public debt and economic growth: A discussion of Reinhart & Rogoff (2010)

The result we aim at discussing regards the correlation between public debt and economic growth. Using a new database, RR distinguish between four regimes, namely advanced economies with low public debt (below 30% in ratio of GDP), medium-low public debt (between 30% and 60% in ratio of GDP), medium-high public debt (between 60% and 90% of GDP), and high public debt (above 90% in ratio of GDP). According to their Figure 2 entitled "Government Debt, Growth andInflation: Selected Advanced Economies, 1946 -2009" 3 (page 575), over the period 1946-2009 the average economic growth rate is dramatically lower in countries presenting a debt ratio above 90%, compared to the other countries in the sample.

However, this result is somehow puzzling. Indeed, their Figure 2 shows that compared to the sharp contraction of economic growth (around 3 percentage points higher for mediumhigh debt with respect to high debt countries), the difference between the median growth rates is considerably lower (around 1 percentage point). This observation could suggest a typical outlier problem, and is the starting point of our analysis.

To investigate the robustness of the findings of RR, we begin by reproducing their Figure 2. The data we are using cover the same time period (1945-2009) and come from two sources. First, using real GDP data from Maddison (2007), we compute the real economic growth rate. Second, we use the very recent Ali Abbas et al. (2010) database on public debt.

The reason for using this database is double. On the one hand, we aim at investigating if RR findings still hold when using a different database on public debt. 4 On the other hand, we aim at exploring the robustness of our results from a historical perspective, namely for the 1880-2009 period. To ease comparison, we focus on the same sample of twenty advanced economies.

Figure 1. GDP growth and the debt-to-GDP ratio 5

Similar to RR, we find that a debt-to-GDP ratio over 90% is reducing average economic growth. However, contrary to RR, the contraction in economic growth is much less obvious, as suggested by simple equality tests which fail to support the presence of a statistically significant difference between the average growth rates of the last two regimes. 6 Consequently, one should be cautious in deriving economic policy implications from this chart alone, before exploring more in detail the properties of the data.

III. Public Debt and Economic Growth: An Econometric Reappraisal

The previous section emphasized the difficulties of establishing a clear-cut relation between public debt and economic growth, on the basis of simple statistics. In addition, the results that could be derived from Figure 1 suffer from major econometric shortcomings, including (i) the specification of exogenous thresholds in the public debt-to-GDP ratio, (ii) the absence of econometric tests for the relevance of the regimes, and (iii) the presence of brutal transitions 4 Our efforts for obtaining the database used by RR were, up until now, unfortunately unsuccessful. In addition, the database we use is an official database from the IMF and is, to the best of our knowledge, the most complete data collection on public debt. 5 Figure 1 is based on the same countries as RR. The number of observations in our sample (in RR sample) in the four regimes is 307 (443), 452 (442), 246 (199) and 129 (96), for a total of 1134 (1180) observations. 6 Under the null hypothesis of an identical economic growth rate in the last two regimes, the Anova and Welch F-tests equal 1.305 and 0.836 respectively, and the associated p-values are 0.25 and 0.36 respectively. According to RR, public debt is found to reduce economic growth above the 90% threshold, while this relation seems to be less obvious and not significant in our Figure 1. In addition, even RR cast doubts on the shape of the effect of public debt on economic growth below the 90% level. Consequently, it is difficult to infer a clear-cut shape of the public debteconomic growth link, and particularly regarding the presence or the absence of nonlinearities. To overcome these shortcomings, we first present the results of the identification procedure, which, as usual with nonlinearities, tests their merely existence; provided the presence of nonlinearities, this phase also establishes the number of transition functions and thresholds of the general model (1). Based on these findings, we estimate in the second phase the PSTR model.

( ) ( ) ( )                      - - + = Γ + Γ + + = - = = ∏ ∑ 1 1 1 2 1 exp 1 , ; ;. k H h k h it k k h k it k it K k it k it k it i it B B B B B B B Y γ γ ε β β α , ( 1

The effect of public debt on economic growth

Table 1 presents the identification procedure of the PSTR model (1).

7 Appendix B presents an extensive discussion of the PSTR technique. The tests are based on the linearized form of model ( 1) (see Appendix B). We considered up to 3 = H thresholds, as suggested by Gonzalez et al. (2005). Bolded values signal the strongest rejection of the null hypothesis (namely, a linear panel). p-values are reproduced in brackets. Since this is a sequential procedure, the significance level (1% or 5%) is reduced after each sequence by a constant factor 5 . 0 = τ (see Gonzalez et al., 2005).

The first column of Table 1 depicts the values of the tests for the first transition function with up to three thresholds, as suggested by Gonzalez et al. (2005). The low p-values confirm the existence of strong nonlinearities in the effect of public debt on economic growth. Moreover, both LM and F tests show that the strongest rejection of the null hypothesis criterion (see Gonzalez et al., 2005) is consistent with a first transition function with 3 = H thresholds. Thus, we block the number of thresholds to three in this first transition function, and search next for a second transition function. Irrespective of the number of thresholds considered, the rejection of the null hypothesis arises for p-values largely lower than for the first transition function, and in particular lower than half of the significance level considered in the first sequence (namely, 1% or 5%, see Gonzalez et al., 2005). Consequently, the PSTR model that comes out from the identification procedure consists of one transition function with three thresholds, namely

- =               - - + = = = = = Γ + Γ + - = ∏ h h it it it it it it B B B B B B B B Y γ γ ε .
(2)

The results of the PSTR regression (2) were estimated following the algorithm presented in Appendix B, with standard errors (corrected for heteroskedasticity) reproduced in brackets, and the stars *** denoting the significance at the 1% level. In line with the results emphasized in the identification procedure, the presence of a nonlinear relation between public debt and economic growth is confirmed by the strong significance of the 2

β coefficient.
Let us explore in the following the effect of the public debt ratio on growth, namely

( ) ( ) ( ) (
)

1 3 3 2 2 1 3 2 1 2 2 2 1 2 3 1 . B B B B B B B B B B B B dB dY it it it + + + + + - Γ - Γ + Γ + = γ β β β .(3)
According to (3), the influence of the level of public debt on economic growth transits through a rather complex nonlinear way. To have a better look at these effects, we present in The results depicted in Figure 2 confirm some of the findings of RR, and equally go beyond them. Indeed, according to our estimations, an increase in the debt ratio decreases economic growth (the derivative is negative). This result, which is in accordance with our Figure 1 illustrating a negative correlation between economic growth and debt, extends the analysis of RR who fail to derive a clear-cut relation between public debt and growth for debt ratios below 90%. Besides, although economic growth declines as public debt increases, its reduction progressively vanishes, contrary to the conclusions of RR who support a dramatic reduction in economic growth when public debt is above 90%.

Moreover, the effect of high (above 90%) public debt on growth displays much more complex nonlinearities that stressed by RR. Indeed, an increase of public debt above the 90% ratio reduces economic growth, as emphasized by RR. But in addition we depict the existence of a threshold, around a public debt-to-GDP ratio of 115%, above which the correlation between public debt and growth changes sign. 8 Consequently, contrary to RR, countries with public debt ratios above 90% not only do not see their economic growth plunging, but they could even experience an increase in their growth, once public debt is above the endogenously-estimated threshold of 115%. 9

Figure 3 appends our Figure 1 above in light of these results, by accounting for the presence of a threshold at around 115% for the public debt ratio (for simplicity, we focus on public debt ratios above 60%).

Figure 3. GDP growth and the debt-to-GDP ratio: accounting for threshold effects

The first two charts, reproduced from our Figure 1, recall that average growth is lower in countries with a public debt ratio above 90%, compared to countries with a debt ratio between 60 and 90%. 10 The last two charts develop the analysis by accounting for the presence of threshold effects for a debt ratio of 115%; thus, we distinguish between high 8 Compared to RR, where the 90% debt threshold brutally distinguishes between countries with high, respectively low growth rates, the transition between negative and positive effects of debt on growth occurs smoothly, around the 115% debt threshold. 9 Besides, the magnitude of positive effect of public debt on economic growth is rather important, and may be even stronger (in absolute value) than the negative effect illustrated for low public debt ratios. 10 However, remind that equality tests show that there is statistically no difference between average growth rates for the two regimes.

0% 1% 2% 3% 4% 60%-90% > 90% 90%-115% > 115%
Average Median public debt countries below this threshold (namely, with a debt ratio between 90 and 115%)

and respectively above it, and compute average and median economic growth rates.

In the spirit of RR, we find that compared to countries with a debt ratio between 60 and 90%, countries with a debt ratio between 90 and 115% experience a decline in their average economic growth rates. Although this decline is statistically significant,11 notice that the economic growth contraction is much less pronounced than acknowledged by RR. In addition, contrary to RR, we find that countries with a public debt ratio above 115% present an average growth rate which is higher compared to the average growth rate of countries with a public debt ratio between 90 and 115%. More important, the growth rate of countries with a debt ratio above 115% is not found to significantly decline compared to the growth rate of countries with a debt ratio between 60 and 90% (the values of Anova and Welch mean equality F-tests equal 0.169 and 0.050 respectively, and the associated p-values are 0.68 and 0.82 respectively).

Economic policy implications and robustness

Our findings reveal a different perspective in terms of economic policy recommendations. Indeed, according to RR, countries should be cautious regarding their fiscal policy, since loose fiscal policies, raising the public debt level, may drive them towards high debt ratio regimes (characterized by a public debt above 90%), associated with dramatically lower economic growth. Our results do not go against this general idea, since we find that increasing public debt is detrimental to economic growth, when public debt is relatively low. However, we also find that the negative effect of public debt on growth is declining, and thus the growth contraction might be less important that acknowledged by RR.

In addition, we even show that raising public debt can even increase economic growth, in a context of high debt levels, namely when the public debt-to-GDP ratio is above a threshold level estimated at around 115%.

It is important to observe that we are not making the case for the use of loose fiscal policies as an engine to promote economic growth, when public debt reaches high levels.

Instead, we aim at showing that an increasing public debt path is not generating bottomless pit growth losses, but is bordered by a regime in which economic growth and debt may be positively correlated. According to our analysis, countries in this regime, characterized by a public debt-to-GDP ratio above 115%, present on average growth rates that are comparable to the growth rates of countries with significantly lower public debt ratios, namely between 60 and 90%.

An immediate question regarding our findings concerns this last regime, namely countries with high public debt ratios (above 115%) and high economic growth. Among them, we can enumerate Belgium, Canada, New Zealand or United Kingdom, and the periods concerned are the decade just after the WW2 and, to a lesser extent, the last decade of the 20 th century. However, one may argue that these events are singular, in the sense that they are not likely to have been observed in other time periods; in this case, the positive association between high public debt and growth may not be more but a simple artifact conditional to the considered time sample. To tackle this critique, we aim at investigating if our results still hold when considering a historical perspective. To this end, we perform our analysis on the same sample of countries, but on the period 1880-2009, namely the double of the time length considered previously (1945-2009).

The results of the identification procedure performed on the general PSTR model (1), presented in Appendix A, support the presence of one transition function with one threshold.

The results of the estimation of this model are the following

( ) ( ) ( ) ( ) [ ] 1 1
- - + = = = Γ + Γ + - = B B B B B B Y it it it it it it γ γ ε , ( 4
)
and the associated effect of public debt on economic growth is illustrated in Figure 4. this positive correlation can be also explained by historical events that occurred during the 1880-1945 period, and in particular by the "Roaring Twenties". During this period, many countries from our sample, including Belgium, France, Italy or UK, despite presenting elevated (above our estimated threshold) public debt ratios, experienced remarkable high economic growth rates.

Consequently, our analysis for the 1880-2009 period unveils two important facts. On the one hand, we present an additional perspective on the outstanding period of the Roaring Twenties, contributing to the fascinating existing literature on this topic (see, for example, Schumpeter, 1946). On the other hand, we confirm that the positive association between debt and growth is not singular to the recent period (1945-2009), but finds support from a historical perspective. 12 12 Besides, Appendix A confirms that economic growth is yet again statistically identical for countries with a debt ratio between 60 and 90%, compared to countries with a debt ratio above the threshold of 130%.

-

IV. Conclusion

This manuscript discusses one of the main results recently emphasized by the very influential contribution of Reinhart & Rogoff (2010), namely the substantial economic growth decline observed for advanced economies with a debt ratio above 90%. Using the new IMF database on public debt, we show that, although higher public debt reduces growth for debt ratios below the threshold considered by RR (namely 90%), this negative effect is declining as debt increases. Moreover, we extend RR findings by emphasizing the presence of nonlinearities in the effect of debt on growth for debt ratios above 90%. On the one hand, debt still reduces growth for countries with a debt-to-GDP ratio below a threshold estimated at around 115%, as acknowledged by RR. On the other hand, we reveal that economic growth and public debt are positively associated for debt ratios above 115%; for countries in this regime, simple equality tests support that average economic growth is not significantly different from the average growth rate of countries with a public debt ratio between 60 and 90%. In addition, this latter finding is confirmed when adopting a historical perspective, namely when considering the same panel of countries but for the 1880-2009 period.

The implications of our results should be considered with caution. In particular, one should refrain from recommending the adoption of loose fiscal policies as an engine to foster economic growth in a context of high public debt levels. In turn, our results extend the conclusions of RR, by showing that an increasing public debt path is not generating bottomless pit growth losses, but is bordered by a regime is which economic growth can be positively correlated with high public debt levels.

Consequently, the present paper, revealing nonlinear effects of high public debt levels on economic growth, emphasizes that further analysis regarding the mechanisms underlying these nonlinearities is needed before drawing economic policy recommendations, all the more in a high-debt context as the one currently experienced by an important number of countries worldwide.

Simple equality tests show that (i) average growth declines for countries with a debt ratio between 90 and 130% (the Anova and Welch mean equality F-tests equal 5.917 and 5.385, and the associated p-values equal 0.01 and 0.02 respectively), and (ii) countries with high debt ratios, namely above 130%, do not experience significantly different growth rates compared to countries with a debt ratio between 60 and 90% (the Anova and Welch mean equality Ftests equal 0.100 and 0.045, and the associated p-values equal 0.74 and 0.83 respectively).

These findings extend and confirm our conclusions for the 1945-2009 period.

presence of a brutal transition between regimes, involving that either the elasticity fully depends on the transition variable or is completely independent of it. This is the case in the Figure 1 of RR, according to which the correlation between public debt and economic growth is significantly different for countries with a public debt ratio on the left and respectively the right hand side of the 90% threshold. However, such structural differences are hard to justify for countries with fairly close public debt-to-GDP levels. Second, even if we develop (A1) to the presence of multiple thresholds, the obtained PTR model would still allow for a reduced number of regimes. For example, a two-threshold PTR model yields three different regimes (i.e. three different elasticities), which may be considered as rather limited when one deals with panel data with (i) an important time dimension (in this paper, 65 and up until 130 years), and (ii) important heterogeneity among countries (since the large time period we consider may cover different stages of economic development). To overcome these shortcomings in an adequate manner, we focus in the following on the PSTR technique.

Keeping the same notations as in (A1), let us assume the following PSTR model (the model described by equation ( 1) in the main text)

( ) ( ) ( )                      - - + = Γ + Γ + + = - = = ∏ ∑ 1 1 1 2 1 exp 1 , ; ;. k H h k h it k k h k it k it K k it k it k it i it B B B B B B B Y γ γ ε β β α , (A2) with k Γ ( K k , 1 =
) transition functions depending on the level of the public debt-to-GDP ratio Gonzalez et al. (2005). Irrespective of the number of thresholds, the value of Γ is close to one when public debt approaches its maximum ( max B ), while its value for low levels of the public debt ratio (around the minimum min B ) depends on number of thresholds. In addition, increasing the number of thresholds impacts on the monotonicity of the transition function. Finally, even in the presence of a unique threshold, the model is still much more general than a simple PTR model (provided that the smoothing parameter is low enough, as this is the case in all simulations of Figure A

above), since it virtually generates an infinity of regimes, namely an infinity of different values for the effect of public debt on economic growth, depending on the public debt ratio in every point of time and for every country.

In search of nonlinearities: the identification procedure

According to RR, the relation between public debt and economic growth is nonlinear, since an increase of the public debt ratio over the 90% threshold reduces considerably growth.

However, as usual with nonlinearities, the first operation should consists in testing their

H

. The algebra for a model with three thresholds is easily computable and available upon request. 16 Denoting by 0 S and 1 S the sum of squared residuals under the null hypothesis (linear panel) and the alternative hypothesis (PSTR model) respectively, the tests are computed as

(

)

1

Figure 2 a

 2 Figure 2 a graphical representation of (3), based on results from regression (2).

Figure 2 .

 2 Figure 2. The effect of the public debt-to-GDP ratio on economic growth

Figure 4 .

 4 Figure 4. Public debt and economic growth in a historical perspective(1880-2009)

(

  B ), the threshold ( B ), and the smoothing parameter ( 0 > k γ ) to be discussed below. In this general formulation, we consider that each transition function k the transition function, and of the PSTR model, crucially depend upon the (non-negative) transition parameter γ . Indeed, when 0 = γ , the transition function is constant (it equals 0.5), and the PSTR model collapses to a linear model (since the dB dY / elasticity does not depend on the level of the public debt ratio B anymore). In the other extreme case ( ∞ → γ ), the transition function takes only two independent values, and the PSTR model collapses to a PTR model with brutal transition between regimes. 14 Moreover, 14 Consequently, the PTR model may be viewed as a special case of a PSTR model, namely when PSTR model with smooth transition between regimes (the lower the smoothing parameter, the smoother the transition), as depicted by the left-hand chart of the Figure A below.

Figure A .

 A Figure A. Different smooth transition functions

  In a PTR setup, such tests suffer from a nuisance problem (seeHansen, 1999), which however may be overcome by an adapted bootstrapping procedure. Fortunately, such complications may be avoided in a PSTR model; given that the transition function ( ) . Γ satisfies the necessary condition of continuity and presents sufficient conditions for derivability, a simple way to check for the existence of nonlinear effects is to compute the first-order Taylor-linearization of the transition function around the smoothing parameter 0 → γ . For simplicity, consider a simple one-transition function one-threshold model , the PSTR model (A3) collapses to a simple second-order polynomial nonlinear panel model. Moreover, since * 2 β is a multiple of γ , we can test the absence of nonlinearities in the PSTR model in the presence of two thresholds, as the parameters associated respectively with 2 B and 3 B in the linearized model are also multiples of γ . 15 To check for the existence of such nonlinearities, we use the two tests developed by the related literature, namely the Lagrange Multiplier-based (LM) test and its Fisher (F) version. 16To identify the structure of the PSTR model, we use the following cascade procedure.In the first phase, we consider a PSTR model with one transition function, depending on up to three thresholds. Based on the linearized form of the assumed PSTR model (A1) (or (1

Table 1 .

 1 Identification of the PSTR model (1): nonlinearities in the public debt ratio

			First transition	Second transition function
			Function	(First transition function has three thresholds)
	1 threshold	LM Test	44.6 (2.38e-011)	2.57 (0.109)
		F Test	45.6 (2.33e-011)	2.52 (0.113)
	2 thresholds	LM Test	50.1 (1.34e-011)	5.29 (0.071)
		F Test	25.7 (1.25e-011)	2.60 (0.075)
	3 thresholds	LM Test	59.5 (7.53e-013)	8.84 (0.032)
		F Test	20.5 (6.17e-013)	2.90 (0.034)

To better focus our analysis, we leave for further investigation the issue of economic growth and inflation, equally discussed by[START_REF] Reinhart | Growth in a Time of Debt[END_REF].

For a recent survey on this literature, seeParent (2012).

The twenty advanced economies included are: Australia, Austria, Belgium, Canada, Denmark, Finland, France, Germany, Greece, Ireland, Italy, Japan, Netherlands, New Zealand, Norway, Portugal, Spain, Sweden, the United Kingdom and the United States.

The values of Anova and Welch mean equality F-tests equal 4.568 and 6.250 respectively, and the associated p-values are 0.03 and 0.01 respectively.

As the number of thresholds increases, the estimation of the PTR model is technically more difficult (for convergence issues, for example).

Appendix A: Public debt and economic growth in a historical perspective (1880-2009) The tests are based on the linearized form of model (1) (see Appendix B). We considered up to 3 = H thresholds, as suggested by Gonzalez et al. (2005). Bolded values signal the strongest rejection of the null hypothesis (namely, a linear panel). p-values are reproduced in brackets. Since this is a sequential procedure, the significance level (1% or 5%) is reduced after each sequence by a constant factor 5 . 0 = τ (see Gonzalez et al., 2005).

The above identification procedure leads to the estimation (4), and Figure 4 shows that this estimation is consistent with the presence of a debt threshold at around 130%. Consequently, the following chart appends Figure 3

General considerations

The Panel Smooth Threshold Regression (PSTR) model, recently coined by Gonzalez et al. (2005), can be seen as an upgrading of two existing techniques. On the one hand, as a generalization to panel data of thresholds with smooth transition used in time series (see the seminal paper on STAR models by Chan & Tong, 1986). On the other hand, as a generalization to smooth transitions of panel threshold models (PTR) with brutal transitions.

In the following, we present the PSTR method in the light of the latter view.

PTR models were introduced by Hansen (1999), in an attempt for providing a tool for estimating threshold effects on panel data. Indeed, recall that RR illustrate the presence of a threshold at a debt-to-GDP ratio of 90%, above which growth declines dramatically.

Consequently, assuming a panel model of dimension . However, Hansen (1999) developed a procedure to endogenously estimate the value of the threshold B , as well as for testing its relevance (namely, the absence of nonlinearities described by the null hypothesis 2 1 0 :

H

) through a pertinent bootstrapping analysis. Besides, a sequential identification procedure can be implemented to extend the PTR model to multiple thresholds. 13 Despite its novelty compared to standard polynomial approaches for modeling nonlinearities, the PTR model suffers however of two important shortcomings. First, the main text), we compute the above-mentioned linearity tests, for each number of thresholds.

Following Gonzalez et al. (2005), the number of thresholds for the first transition function is chosen according to the strongest rejection of the null hypothesis (linear model). Once the number of thresholds for the first transition function identified, we check, in phase two, for the presence of a second transition function again with up to three thresholds. The procedure stops when the null hypothesis of linear effects can no longer be rejected. Consequently, contrary to traditional polynomial forms used to account for nonlinearities, this identification procedure puts no ex-ante constraint on the functional form of the nonlinear relations. Indeed, given the possibility of virtually combining multiple transition functions, with as much as three thresholds for each one of them, provides sufficient place for obtaining dB dY / elasticities that can depend in a rather complex manner on the level of the public debt-to-GDP ratio.

The estimation of a PSTR model

At the end of the identification procedure, the number of transition functions, as well as the number of thresholds for each transition function, is known. Next, we proceed to the estimation of the PSTR model (i.e. the threshold(s), the transition parameter(s), and the slopes), in a three-step procedure. For convergence issues, the first step consists of eliminating fixed effects, an operation which, contrary to a traditional linear panel model, can lead to some complications. 17 Since the quality of the initial values for the parameters γ and h B is crucial for the convergence of the estimation, the second step focuses on this problem.

Among several available methods, we use a "grid search" technique, which consists in generating combinations of vectors ( )

Supplementary references for Appendix B

-Chan, K., Tong, H. (1986), "On estimating thresholds in autoregressive models", Journal of Time Series Analysis 7, 179-190.

- Hansen, B., (1999), "Threshold effects in non-dynamic panels: Estimation, testing, and inference", Journal of Econometrics 93, 345-368.


Sandra Rousseau

Christian Jutten

Marco Congedo

The error-related potential and BCIs

The error-related potential is an event-related potential triggered by errors. Recently it has been the subject of many attentions notably for its possible use in BCI systems. Since it is linked to error occurrence, it could be used in the design of control loop to build more robust systems. In this paper we studied the characteristics of the error potential and present how it could be used for BCI systems improvement.

Introduction

The error-related potential (ErrP) is an event-related potential (ERP) which is generated when a subject commits or observes the commitment of an error. It was first reported in 1991 [START_REF] Falkenstein | Effects of crossmodal divided attention on late ERP components.II. Error processing in choice reaction tasks[END_REF]. Lately several authors have become interested in its integration in a control loop for BCI systems. The integration of the ErrP in BCIs involves two main operations: its single trial detection and the use of this information to modify the system on-line. As any ERP the ErrP can easily be seen by summing up several trials (its signal to noise ratio is very low). The detection of the ErrP is a crucial point for its integration in BCIs, thus learning its characteristics to design optimal filtering method is a key point for its integration in BCIs. The paper is divided into three parts. First we present the ErrP and what makes it interesting for BCIs. Then we present an experiment we designed to study the characteristics of the ErrP and the results we obtained. Eventually we make propositions on how this ErrP could be integrated in BCIs.

The error-related potential

There exists different types of ErrPs characterized by the way the error is reported and depending on the agent committing the error. The most known ErrP is the response ErrP (ErrPr) [START_REF] Falkenstein | Effects of crossmodal divided attention on late ERP components.II. Error processing in choice reaction tasks[END_REF]. It appears when subjects perform an incorrect action instantly detected (mostly studied in reaction time tasks). The second major type is the feedback ErrP (ErrPf) [START_REF] Miltner | Event-related brain potentials following incorrect feedback in a time-estimation task: Evidence for a generic neural system for error detection[END_REF]. It appears when the subject gains access to the outcome of an action through an external feedback (mostly studied in gambling tasks). More recently, in [START_REF] Ferrez | Error-related EEG potentials in brain-computer interfaces[END_REF] the authors reported an ErrP when a subject interacts with an interface that does not respond in the expected way. It is called the interaction ErrP (ErrPi) (mostly studied in BCI-like context). All these potentials share some common characteristics. They are mainly characterized by a negative deflection with a fronto-central localization (electrodes FCz and Cz) (see figure 1) which latency depends on the type of ErrP observed. Grand averaged error-potential. Data were averaged over 10 subjects. 72 trials were used for each subject with a mean error rate of 18%. Data were bandpass filtered between 1-12Hz.Blue line corresponds to the error condition, red dashed line to the correct condition.

The ErrP and the BCIs

ErrPs are much studied today in the framework of BCI. Indeed they provide an opportunity to check if the machine has selected the command sought by the user. Such an information could be used in a control loop for current BCI protocols. Some authors, as in [START_REF] Ferrez | You are wrong!-automatic detection of interaction errors from brain waves[END_REF] and [START_REF] Bollon | EEG error-related potentials detection with a Bayesian filter[END_REF], already studied the on line detection of this potential and its possible integration on BCI devices. In the case of a BCI experiment the ErrPi would be the potential to study. In a BCI context the user aims at controlling a device by means of cerebral orders. EEG signals are classified and turned into commands. A misclassification results into an incorrect action of the device. Thus the subject observes an error of the interface (generating the ErrPi). In some cases, the error could come directly from the subject which would use the wrong cerebral order or change his mind. Then the corresponding error would be the ErrPf. Thus two different types of ErrPs could co-exist and their disentanglement might be difficult. However when looking at the temporal shape of both potentials it seems that they share the same characteristics (some authors even make no distinction between both). Thus one can hope that both could be detected by the same classifier.

Presentation of an error-potential experiment

We studied the ErrPf in the case of a memory game, with no monetary gain or loss. The experiment involved two sessions, each consisted of six blocks of six trials, for a total of 72 trials. Stimuli were presented on a computer screen in front of the subjects. Nine square boxes were arranged in circle. Each trial consisted in a random sequence of two to nine digits appearing sequentially in random positions. Subjects had to memorize positions of all digits. At the end of the sequence the target digit was displayed and subjects had to click on the box where it had appeared. If the answer was correct, the box background turned into green, otherwise it turned into red. Subjects were then asked if the result matched their expectation, giving four classes of outcomes (expected corrects, unexpected corrects, expected errors, unexpected errors). A random rest period was allowed before starting the next trial. The number of digits to memorize was fixed within blocks (between two and nine) and adaptively increased or decreased of one digit at each block, according to subject performance. The temporal order of each trial is illustrated in Figure 2. 22 healthy volunteer subjects participated. Due to the presence of artifacts, four subjects were excluded from analysis. The age of participants ranged from 20 to 30 with a mean (standard deviation) of 24(2.5). The mean error rate (standard deviation) was equal to 18(4.6)% of the trials. Analysis was conducted to determine the differenciation between errors and corrects (expectancy was not taken into account in this analysis). Two types of analysis were conducted: a temporal and a time frequency analysis. Temporal analysis was applied to the EEG signal filtered between 2-20Hz. Time-Frequency analysis was conducted using fieldtrip software ( [START_REF] Oostenveld | Fieldtrip: open source software for advanced analysis of meg, eeg, and invasive electrophysiological data[END_REF]). Frequency analysis was performed using ERD/ERS (event-related desynchronization / event-related synchronization) measures ( [START_REF] Pfurtscheller | Event-related eeg/meg synchronization and desynchronization: basic principles[END_REF]), in order to mitigate subject and inter-trials variability. This measure is defined as follows:

ERD/ERS(s, f, t) = T F R(s, f, t) -T F Rb(s, f, tj) T F Rb(s, f, tj)
where TFR(s,f,t) is a time-frequency representation averaged over trials of our signal at sensor s, frequency bin f and time t and TFRb is the TFR of the baseline (baseline was chosen to be [-1s : 0s] pre-stimulus) averaged over time. Here TFRs were calculated using a hanning sliding window over the 2-32Hz band using a 1Hz step, on times [-0,5s 1,2s] using a time step of 0,05s. Frequency dependent time window were used so as to reduce the spectral leakage. TFRs were calculated for each subject and each trial and then averaged. Statistical differences between error and correct trials were assessed using a within subjects non-parametric cluster based randomization test. 1000 permutations were performed. A cluster based multiple comparison correction was applied: at each permutation only the t values which p values are higher than a given threshold (set here to 0.05) are kept. Clusters are then formed by means of time, spatial and frequency adjacency. For each cluster the statistic is defined as the sum of the t value contained in this cluster. The maximum cluster statistics is kept as the test statistic for this permutation. The global significance threshold was set to p value ≤ 0.05. This method allows not to take into account outliers and to assess for global effect.

Statistical tests on the temporal signal allowed the identification of three components: a significant positivity for errors at time [300ms 400ms] at electrode Cz (p ≤ 0.01), a significant negativity for errors at time [450ms 520ms] at clustered electrodes Fz,FCz,Cz (p ≤ 0.01) and a significant positivity for errors at time [600ms 700ms] at clustered electrodes Fz,FCz (p = 0.025) (see figure 3). For the frequential domain, TFRs showed a post-stimulus synchronization in the theta band in both conditions. Statistical tests showed a significantly higher synchronization for errors in the band 5-8Hz for the time [350ms 600ms] over the electrode Fz and FCz (p = 0.015)(see figure 4). The integration of the ErrP in BCIs starts by its single trial detection. Several authors have already studied this point leading to varying results going from 64% (60% for errors and 64% for corrects) [START_REF] Seno | Online detection of p300 and error potentials in a bci speller[END_REF] of good detection up to 88% [START_REF] Jrad | Spatio-temporal feature extraction and classification of event-related potentials. 7e Plateforme[END_REF]. The characterization of the ErrP allows us to better know its components. This is of great use for the design of specific spatial and frequential filters which could greatly improve the detection rate. Indeed most authors only used the temporal evolution of the ErrP, our results show that some frequential components might also exist and could be used to improve results. Once an ErrP is detected the question that arises is how it can be used to improve a BCI. Different approaches have already been proposed. To assess the performance of a BCI the commonly used measure is the information transfer rate (in bits per trial (BpT)) which takes into account both the accuracy and the speed of the system. The first method to be proposed was the cancellation of the erroneous action. In [START_REF] Ferrez | You are wrong!-automatic detection of interaction errors from brain waves[END_REF] the author obtained an average gain on the BpT of 72% for a 2-class paradigm and 28% for a 3-class paradigm. In [START_REF] Visconti | Automatic recognition of error potentials in a P300-based brain-computer interface[END_REF] the author observed that this improvement depended on the initial performance of the BCI. Another possible integration has been proposed in the case of a P300 speller in [START_REF] Combaz | Error-related potential recorded by eeg in the context of a p300 mind speller brain-computer interface[END_REF]. The author proposed to select the second best letter when an error was detected, the total number of mistypes was reduced for every subject. Another possible integration could be to improve the classifier. For example in the case of a SVM (support vector machine) classifier [START_REF] Jrad | sw-svm: sensor weighting support vector machines for eeg-based brain-computer interfaces[END_REF] the regularization parameter which determines the trade off between the regularized term and the empirical error could be adapted in the case of an error detection. For future works we plan to select different methods and to test them first in a theoretical framework for different BCI initial BpT and different ErrP classification rates, in order to determine what strategy is the best and in which conditions. In a second time we will test the selected strategies on a real BCI experiment in order to see if the theoretical results match the real one and how the subject responds to this error correction.

Conclusions

In this paper we have discussed the integration of the ErrP in BCI. First we have shown temporal and frequential results on the characterization of the ErrP in the aim of improving its single trial detection detection which is the first step toward its integration in BCI systems. Then we have discussed how this late could be integrated in a BCI system by presenting already existing results and proposing future works to test several methods and compare them. Future works will focus on studying the impact of error (and correct) expectation on the different characteristics of the ErrP and its possible impact on BCI integration.

Fig. 1 :

 1 Fig. 1: Temporal evolution of the ErrPf at electrode Cz

Fig. 2 :

 2 Fig. 2: Temporal order

Fig. 3 :Fig. 4 :

 34 Fig. 3: Temporal evolution of the ErrPf averaged over electrodes Fz,FCz,Cz and the significant time windows.Grand averaged on 18 subjects with 72 trials (mean error rate of 18%). Data were bandpass filtered between 2-20Hz. Blue line corresponds to the ERP on error condition, red dashed line to correct condition, yellow bars to the statistically significant time windows.


Déborah Idier

Jeremy Rohmer

Thomas Bulteau

Etienne Delvallée

Alexandre Magnan

An inverse methodology for coastal risk management

Keywords: coastal flooding, inverse approach, level of acceptability, modelling, statistics

come

For the last decade, extensive work has been done on the estimation/analysis of coastal flooding hazard/risk. One approach to assess the risk is to use a "direct" method, hazard-based, starting from offshore conditions scenarios (combination of wave, tide and surge) where each scenario is characterized by a return period. This allows estimating the induced hazards (flooding), and finally the risk. Such an approach is based upon three main hypotheses. Firstly, it makes the assumption that the probability of risk (or hazard) is equal to the probability of the offshore forcing conditions: this is not true because risk R is generally defined as R=Pf.D (D: expected damages) where Pf is the probability of flooding resulting from the failure of the pathways which is different from that related to offshore conditions. Secondly, it can lead to neglect forcing conditions that are not dramatic individually, but that can lead to major events when taking together. Coastal defense failures are good examples of such a mistake, as shown during Katrina (USA) event. Thirdly, it leads to the design of answers that are not always the most relevant in terms of risk management. Indeed, many other drivers than just environmental ones are involved in decision-making, territorial planning and societal relationships to hazards.

An alternative approach, risk-based, could consist in inversing the usual direct approach and to start from a level of risk which is judged acceptable by the society (decision-makers, stakeholders and populations), to end up with the exceedance probability associated to this risk level. Such an approach would allow identifying all the forcing conditions (and their occurrence probability or return period) inducing a threat for critical assets of the territory.

In the present paper, we first describe very briefly this new methodology of coastal risk assessment. Then, assuming we already know the maximal acceptable risk level, a first application to a real case, located on the French Mediterranean coast, is presented. The advantages of the inverse methodology for decision-making support are discussed before the conclusion outlying the key issues for future research directions is drawn.

Basically, risk can be defined as the potential adverse consequence that may or may not be realized in the future. Hence it involves two main components: extent of losses (i.e. damages) and probability of occurrence. A widely-used tool for displaying risk is the probability / damage graph (e.g., [START_REF] Ballard | Guest editorial: societal risk-progress since Farmer[END_REF]. On this basis, the critical limit between what is acceptable or not can be defined. Note that in what follows, we use both the terms "maximum acceptable" and "intolerable" to qualify the limit between the two areas. This limit can either be small damage / high probability, or high damage / low probability, and both could be "intolerable" for the society. A complete presentation of the different risk classes within this space (probability -impact) has been done by WBGU (2000). The risk analysis should be adapted to the level of what can be accepted by the society. For instance, if we consider the example of a nuclear plant, the risk of a nuclear plant accident (extreme event) can be categorized within the risk class "sword of Damocles" in the classification of WBGU (2000), such that the damage level is very high, whereas the probability of occurrence is very low. In the case of flooding, [START_REF] Haimes | Risk of Extreme Events and the Fallacy of the Expected Value[END_REF] found that people are "often more concerned with low probability catastrophic events than with more frequently occurring but less severe accidents". Then, as stated by [START_REF] Renn | Concepts of Risk: An Interdisciplinary Review -Part 1: Disciplinary Risk Concepts[END_REF], the investigation of the risk perception of these rare but possible catastrophic events shows that the probability plays hardly any role. In other cases, for smaller impact events, society accepts the later to occur up to a critical occurrence probability (or return period), above which it cannot be tolerated [START_REF] Renn | Concepts of Risk: An Interdisciplinary Review -Part 1: Disciplinary Risk Concepts[END_REF]. Furthermore, it should be noted that even if there are some differences, one of the most consistent findings across cultures and within cultures is that usually the same concepts (e.g. : dreadfulness, unknown risks, etc.) emerge, but receiving different priorities [START_REF] Kasperson | Six Propositions on Public Participation and Their Relevance for Risk Communication[END_REF], leading to different level of risk acceptance. For these reasons, the present approach should be based on the acceptable risk, allowing to deal with both extreme events and small damage events. The main steps of the methods are presented in the paper: identification of the acceptable risk threshold based on a global approach of the territory vulnerability; estimation (through inversion of the flooding assessment model) of the offshore conditions leading to the acceptable risk threshold; estimation of the exceedance probability of the associated combinations, and thus to the maximal acceptable risk; feedback to coastal management.

To investigate the feasibility of the inverse methodology, a first application to a real case, located on the French Mediterranean coast, is presented and analyzed. For this case, the level of acceptable risk is assumed to be known and to correspond to the water level threshold at the coast (Rc). Then, a modeling approach is used to find out which set of combinations of forcing offshore conditions lead to a water level overcoming the critical threshold Rc. Finally, the occurrence probability of these combinations is computed (using the JOIN-SEA program, [START_REF] Hawkes | The joint probability of waves and water levels in coastal engineering design[END_REF], HR-Wallingford, 2000).

Moreover, the paper outlines the main advantages of such an inverse approach, as well as the future challenges within this research.