An Nguyen

Stéphane Goutte
email: stephane.goutte@uvsq.fr

Earlier Preschool Education for Better Development? Evidence from Vietnam

Keywords: JEL Classification: J24, O15, I26 Early childhood education, Preschool education, Double Machine Learning

Using longitudinal Young Lives survey in Vietnam, this paper analyzes the effect of preschool starting age on skills, educational attainment, and health of poor children. Implementing the double machine learning estimator, we find short and medium-term positive effects of starting preschool younger on child cognitive skills but little evidence on child non-cognitive skills. Starting preschool younger significantly increases the highest grade achieved and creates a behavioral response of increasing subsequent parental investment in education. In contrast, we uncover the short-term negative impact of starting preschool younger with measures of child health, namely weight-for-age and BMI-for-age. The findings imply that expanding access to preschool and improving the quality of preschool education are important to achieve comprehensive child development.

Introduction

Does it matter at what age disadvantaged children attend preschool? Although an accumulating body of evidence shows the beneficial effects of preschool education on disadvantaged children [START_REF] Berlinski | Giving children a better start: Preschool attendance and school-age profiles[END_REF][START_REF] Figlio | 1. the behavioral consequences of prekindergarten participation for disadvantaged youth[END_REF]1 Fredriksson et al., 2010;[START_REF] Felfe | Does preschool boost the development of minority children?: the case of roma children[END_REF], none of the studies reviewed appears to assess directly the effect of preschool starting age on child outcomes.

Early childhood experiences have persisted and profound impact on brain development-affecting the accumulation of life-relevant skills and, ultimately, well-being [START_REF] Robert | Mothering style and methylation[END_REF][START_REF] Eric I Knudsen | Economic, neurobiological, and behavioral perspectives on building america's future workforce[END_REF]. Adverse early childhood environments raise serious concerns about the accumulation of these skills and eventually the life prospects of disadvantaged children. Recent studies prove the beneficial effect of high-quality early childhood education programs on the development of disadvantaged children by enriching their learning and nurturing environments. Theoretical evidence proposed by [START_REF] Cunha | Estimating the technology of cognitive and noncognitive skill formation[END_REF] also suggests that early childhood education can have a long-term impact on skills by directly improving current skills, which will beget later skills (i.e., self-productive) and increasing the productivity of later investment (i.e., dynamic complementarity). Based on these shreds of evidence, we might expect that early preschool enrollment would benefit disadvantaged children by preventing the negative effect of early life conditions and shaping the skills that promote their later life outcomes.

The question is specifically relevant to a country such as Vietnam, which already has universal preschool take-up for children one year before primary school. The proven benefit of early preschool enrollment will motivate countries like Vietnam to promote or even to require children from disadvantaged households, local poor areas, and ethnic minorities to enter preschool early to tackle poverty and inequality in the long run.

We address this question by analyzing the effect of preschool starting age on the development of poor children in Vietnam. In exploring this relationship, the contribution of our study is three folds. First, instead of focusing on pure preschool enrollment as most studies do, we directly tackle the effect of preschool starting age on a wide range of outcomes, including cognitive skills, non-cognitive skills, health, and educational attainment over a 10-year development period of children. This broad coverage of outcomes over a long period makes it possible to have a complete picture of the impact until adolescence.

Second, we also contribute to the literature by assessing the potential pathway through which early preschool enrollment improves child outcomes suggested by theory. Understanding the underlying mechanisms can help design and improve the early childhood education policy for disadvantaged children to benefit these children and the entire society. Finally, the paper contributes to the causal machine learning literature by applying the double machine learning estimator [START_REF] Chernozhukov | Double/debiased machine learning for treatment and structural parameters[END_REF] empirically for average treatment effects. The method depending on conditional independence assumption such as propensity score matching requires identifying a set of control covariates enter the analysis and the functional form. Many times the functional form is not clear, and a large number of control variables is needed. The recently introduced framework of double machine learning [START_REF] Chernozhukov | Double/debiased machine learning for treatment and structural parameters[END_REF] allows the use of many control variables and their flexible functional form in an objective and datadriven way to make the conditional independence assumption (CIA) plausible.

We use the Young Lives Survey, which aims to document child poverty in Vietnam. It yields a wealth of measurement of child skills and information of the household, community, and parental investments in children. The highly dimensional data makes use of the conditional independence assumption likely plausible by providing an enormous set of control variables. The treatment variable is the preschool starting age of children. Outcomes include cognitive skills measured by mathematics and language; non-cognitive skills captured by self-efficacy, self-esteem, parent-child relations, and peer relations; and health captured by height-for-age, weight-for-age, and BMI-for-age z-scores. We also look at other outcomes comprising education attainment and nutritional statuses. These measures cover a wide range of aspects of child development that can be positively affected by early preschool enrollment. Finally, to analyze a potential channel through which early preschool enrollment affects child development, we assess the impact on parental investment in education proxied by the number of hours of extra classes.

Exploiting high-dimensional data and the advance of machine learning, we reach the following findings. First, we uncover short-and medium-term favorable effects of early preschool enrollment on child cognitive skills and educational attainment but little evidence of the impact on non-cognitive skills. Specifically, at 15, a one-year younger preschool starting age raises child language skills by 0.091 standard deviations and increases the highest grade achieved by 0.047 year. We find that the favorable impacts could be, at least in part, explained by parental investment behavior (proxied by the number of hours of extra classes children attended) as early preschool enrollment is associated with an increase in the number of hours of extra classes in all ages observed in the survey. Second, we analyze the impact on child health using anthropometric scores. We uncover the unfavorable short-term effect of early preschool enrollment on weight-forage and BMI-for-age z-score. A one-year earlier preschool enrollment decreases weight-for-age z-score and BMI-for-age z-score by 0.056 and 0.092 standard deviation at age 8, respectively. This finding was unexpected and suggests that the quantity of nutritious value of preschool feeding might be less than home feeding. Interestingly, there is a negative association between preschool starting age and the probability of being thin at age 15, suggesting a link between health outcomes and educational attainment.

Although previous literature proves that the positive effect of preschool enrollment is higher for disadvantaged children, our study casts doubt on the qual-ity of school feeding for poor children in Vietnam. While preschool in Vietnam might enrich the cognitive learnings of poor children compared to childcare alternatives or home environments, we argue that improving the quality of school meals and non-cognitive skill stimulation is essential to achieve comprehensive child development.

The paper unfolds as follows. Section 2 presents the relevant literature.

Section 3 describes the data. Section 4 outlines the methodology. Section 5 presents our empirical results, and section 6 concludes.

Literature Review

Whereas the literature has provided evidence on the effectiveness of universal preschool education on outcomes of disadvantaged children, they only look at the effect of pure preschool attendance without considering variation in age when children enter preschool. A dearth of non-cognitive skills, long-term outcomes, and developing contexts also limits this current stock of evidence. There is little understanding about the medium-or long-run effects of preschool attendance for health, non-cognitive skills, as well as parental investment. Even though research has remained mainly confined to developed countries, the mechanisms by which it has generated the impacts have not been fully discovered. Analyses of preschool education in developing country contexts are still very limited and inconclusive. Assessing the role of preschool education on disadvantaged child development is especially important in these contexts, where expanding access to preschool provision has been one of the main education policies to addressing inequality, breaking the cycle of poverty, and improving outcomes later in life.

Within the context of developing countries, the recent study of [START_REF] Najeeb Shafiq | Are there longterm benefits from early childhood education in low-and middle-income countries? Education Policy Analysis Archives[END_REF] attempts to assess the relationship of participation in early childhood education (ECE) and various long-term outcomes in 12 low-and middle-income countries. They find positive and statistically significant associations between ECE participation and post-ECE educational attainment but fewer cases of positive associations between ECE and long-term socio-emotional outcomes.

They also find mixed evidence of ECE and labor market outcomes, with positive associations for skill use but weak associations with earnings. Other studies of the impact of preschool education on long-term outcomes mainly have been confined in developed countries and on outputs such as school enrollment or test scores. [START_REF] Peter Fredriksson | Do pre-school interventions further the integration of immigrants? evidence from sweden[END_REF] analyze the effects of preschool attendance on test scores at age 13 for four cohorts of children born in Sweden. Controlling for general cohort-specific trends, they find that preschool attendance significantly reduces the language score disparity between children of immigrants and those of native-born parents but does not affect the gap in scores on another test or academic secondary school degree completion. [START_REF] Berlinski | Giving children a better start: Preschool attendance and school-age profiles[END_REF] using data on children aged 7-15 in Uruguay and controlling for family fixed effects, find that preschool attendance is positively associated with school attendance and level of education with substantial effect for children with low-educated parents or living outside the capital city. Exploiting variation in child care prices as an instrument for preschool participation, [START_REF] Black | Care or cash? the effect of child care subsidies on student performance[END_REF] uncover a positive effect government-subsidized preschool on children's future national exam grades at age 16 in Norway, with the most significant impacts for children from lowincome families. [START_REF] Hazarika | The effect of early childhood developmental program attendance on future school enrollment in rural north india[END_REF] study effects on adolescents in India using presence in the village of a center or a school offering preschool classes as an instrumental variable. They find that preschool participation increases school enrollment and also speeds grade progression conditional on enrollment. [START_REF] Drange | Kindergarten for all: Long run effects of a universal intervention[END_REF] using a difference-in-difference approach to study the longrun effect on schooling of mandating kindergarten at age 5-6 in Norway, find no effect both overall, across sub-samples, and over the grading distribution. Dhuey (2011) using similar strategy, detects the negative effect of kindergarten expansions on grade retention among Hispanic children, non-English speakers, children of immigrants, and children from low socioeconomic status households in the US.

The impact of universal preschool education on measurements of child skills (e.g. cognitive or non-cognitive capabilities) has only been assessed mainly short-term outcomes in developed countries, especially in the US. Using the birthday cut-off for enrollment to construct RDD, [START_REF] Jr | The effects of universal pre-k on cognitive development[END_REF][START_REF] William T Gormley | Preschool programs can boost school readiness[END_REF] find positive effects of Oklahoma's universal pre-kindergarten program on children's literacy and math, as well as socio-emotional development, with the largest benefits for disadvantaged children. [START_REF] Fitzpatrick | Starting school at four: The effect of universal prekindergarten on children's academic achievement[END_REF] uses a difference-in-difference method to show that Georgia's universal prekindergarten program increased the reading and math scores of disadvantaged children and those in small towns or rural areas while reducing the probability of grade retention. Employing the presence or absence of a public pre-kindergarten program in the student's local zoned elementary school as an instrument, [START_REF] Figlio | 1. the behavioral consequences of prekindergarten participation for disadvantaged youth[END_REF] document that attending public pre-kindergarten school in Florida reduces behaviour problems and suspension or grade retention rates in short-term with the most significant benefits from children from families with low levels of education or disadvantaged neighborhoods. Applying the same approach with individual distance to the nearest preschool facility as intrumental variable, Felfe and Huber (2016) found significant short-term gains in terms of children's literacy. Preschool attendance also increases the prevalence of vaccinations but does not affect other observed health outcomes for Roma children in Europe.

When assessing the impact of preschool education, the fundamental identification problem is that preschool attendance is likely to be endogenous. The current literature uses different approaches to tackle this problem. The rigorous research by far used the quasi-experimental method, their methodological strategies, yet have some crucial drawbacks. For example, several research exploiting regional and time variations by using the difference-in-difference framework [START_REF] Fitzpatrick | Starting school at four: The effect of universal prekindergarten on children's academic achievement[END_REF][START_REF] Drange | Kindergarten for all: Long run effects of a universal intervention[END_REF] is inherent to a crucial drawback that if there are any differences in trends of unobserved characteristics between treatment and control group, the difference-in-difference estimates capture the differences in trend rather than the effect of the programs. Other literature [START_REF] Jr | The effects of universal pre-k on cognitive development[END_REF] exploited the nature of the data in the US to shape the regression discontinuity design on age eligibility. This strategy faces the problem of an insufficient number of children to identify the impact in the cutoff's vicinity and of selective attrition from the sample [START_REF] Elango | Early Childhood Education[END_REF].

Our paper aims to address these concerns and contribute to the literature in three aspects. First, instead of focusing on pure preschool enrollment, we focus on the impact of preschool starting age on a broad set of outcomes representing the development of poor children over ten years. This applies to a country like Vietnam where already has universal preschool enrollment among five-yearold children. Second, we assess the relationship between preschool starting age and child development using a new empirical approach compared to the studies mentioned previously. In particular, we exploit the double machine learning approach dealing with a high-dimensional set of control variables while prior studies adopt the quasi-experiment method as the identification strategy.

The richness of our longitudinal data allows us to construct numerous potential control variables. These variables controlled in flexible functional forms make the conditional independent assumption workable. Finally, we contribute to the literature by testing a potential channel through which preschool starting age improves child outcomes.

3 Data, Treatment Variable and Outcomes

Data

The data used here are the Young Lives Survey of Younger Cohort, which aims to document child poverty. The Young Lives survey uses quantitative and qualitative data collection methods, tracking children 2,000 children in five rounds in 2002,2006,2009,2013,2016, which correspond to the age of 1, 5, 8, 12, 15 of the Younger Cohort. The Young Lives data is intended to gather information on diversified life aspects of children, their caregivers, and families.

The survey has collected comprehensive information on child characteristics, families, caregiver/parent characteristics and resources, their preferences and feelings, as well as school and community factors. Moreover, the survey provides specific information for measuring both children cognitive and non-cognitive skills, allowing us to study children development.

Although the Young Lives household survey sampling is not designed to represent the Vietnamese population, it considers the diversity of children in the country regarding various attributes and experiences. These surveys use a sentinel-site sampling design, including a complex consultative process with a pro-poor selection rule based on each commune's poverty ranking. Overall, the data consists of 20 purposely selected sites representing diversity and pro-poor bias in five provinces: Lao Cai, Hung Yen, Da Nang, Phu Yen, and Ben Tre.

Each province contains four sites, and each site comprises one or two communes, in total, with 36 communes. At the site level, children have selected randomly in 2001 such that the data represents the birth cohort at each site.

Preschool starting age

There are three types of institutions for early childhood education in Vietnam, depending on the child's age. From three months to three years old, children can attend nurseries/creches. They go to kindergartens from three to six years old. Preschools in Vietnam incorporate the functions of these two schools and welcome infants from the age of 3 months up to the age of 6 years [START_REF] Thao | Early childhood education in vietnam, history, and development[END_REF].

From the age of 3, preschool functions as an educational facility that promotes developmental learning and growth and prepares children for primary education.

As a result, this study only considers preschool starting age from three years old. This exclusion also makes preschool in Vietnam similar to that of other countries.

The Young Lives Survey provides us with a wide variety of child characteristics. The primary explanatory variable is the child's preschool starting age, collected in the second round when the child is five years old. We exclude children who did not attend preschool or started preschool before the age of three years, accounting for 11% and 0.7% of the total number of young cohort children in the survey, respectively. -bi) , i = 1, 2, 3, ..., n P is the probability that a child j with ability θ j get the right item i. b i is the item difficulty. n is the number of items in the test. θ j is child's ability parameter.

P (θ j , b i ) = e (θj -bi) 1 + e (θj
We apply marginal maximum likelihood estimation to estimate item parameters and person parameters (Rasch score) and then construct θ score for each child.

The non-cognitive skills are measured by self-efficacy, self-esteem, parentchild relations, and peer relations in round 4 and 5 [START_REF] Ngo | Skill formation and transition to productive livelihood in Vietnam[END_REF]. We use factor analysis [START_REF] Gie | A beginner's guide to factor analysis: Focusing on exploratory factor analysis[END_REF] to compute the non-cognitive skills based on respondent's degree of agreement or disagreement with some statements. Agreement ranges from "strong agreement" to "strongly agreement" with a 4-point scale; all the items are recorded to reflect positive statements. The number of items in the questionnaire related to self-efficacy, self-esteem, parent-child relations, and peer relations is ten, eight, nine, and eight, respectively. The simple math-ematical model for factor analysis is:

X j = a j1 F + e j , j = 1, 2, ..., p p denotes the number of items in the group (X 1 , X 2 . . . , X p ). The factor loading a j1 is the factor loading of j th variable on the factor F , and e j is a residual variate. We assume that a group of items only represents one underlying non-cognitive skill dimension called factor. Since the data can violate the assumption of multivariate normality, we use the Principal Axis Factor for factor extraction and oblique rotation for rotation method, as suggested by Yong, & Pearce, 2013. We run four factor analysis for four measures of non-cognitive skills. Depending on the result, we only choose significant items having the value of factor loading greater than 0.4. We end up with nine, six, eight, and eight items in round 3 and six, six, eight, and seven items in round 4 to construct final measures of self-efficacy, self-esteem, parent-child relations, and peer relations, respectively. We also assess the impact on educational attainment using the highest grade achieved.

We explore one potential mechanism through which preschool starting age can affect child development by assessing its impact on parental investment in education proxied by the number of hours of extra classes. As suggested by theoretical evidence, early investment can promote subsequent investment and increase its return.

Child health

We measure child health with child's anthropometry, including height-for-age, weight-for-age and BMI-for-age z-score. In the Young Lives data, z-scores were estimated using World Health Organization (WHO) reference tables and software. All anthropometric scores adjusted by child's age and sex. Since the WHO reference tables apply only to children of certain ages, the weight-for-age z score is computed for the Younger Cohort only up in Round 3. We also consider the nutritional statuses built up based on the child's anthropometry. The survey used the WHO table (2007) to construct Stunting, Underweight and Thinness based on height-for-age, weight-for-age and BMI-for-age z-scores, respectively.

We observe three dummy variables taking the value of one if the corresponding z-score is below the tables' cut-off of -2, and zero otherwise. The summary statistics of all outcomes variables are shown in table appendix A 4 Empirical Framework

Identification

We estimate the effect of preschool starting age on child skills, long-term health and educational attainment. Our empirical strategy is motivated by a new double machine learning estimator for average treatment effects and the availability of longitudinal rich data of poor children in Vietnam. The double machine learning method and the advance of machine learning in solving prediction problems provide a promising way to draw causal inferences from observational data.

The fundamental problem of confounding is that preschool attendance is likely endogenous since the preschool investment decision of parents reflects several factors such as their taste, resources, and information about the current evolution of their child skills that may be correlated with child skills, education and health.

To eliminate confounding, we rely on the existence of a vector of covariates X i such that the following two assumptions are fulfilled:

1. Conditional independence: The treatment status D i is as good as randomly assigned and not correlated with potential outcome

Y t i conditional on covariates X i Y t i ⊥ ⊥ D i |X i 2.
Common support: Any unit needs to have a non-zero probability to receive each of the treatments:

P [D i = t|X i = x] > 0
In this case, the CIA requires observing all variables that influence the parent's decision to send children to preschool and the outcomes of interest simultaneously.

Following [START_REF] Cunha | Estimating the technology of cognitive and noncognitive skill formation[END_REF], inherited traits and prenatal investments affect skills at birth which are correlated with investments in early childhood and child outcomes. We control this concern by considering the child's characteristics, including child's age, child's sex, ethnic minority, number of brothers and sisters in the household, and child health using child's anthropometry at age 1, parental background and parental education. Moreover, household and community environments have an impact on both child investment and child development. We consider the household environment by controlling for household socioeconomic characteristics and the sentinel site environment by controlling for sentinel site id.

Parent assesses the cost of preschool depending on parental labour supply and other child care options. A working caregiver is more likely to search for other support. We control for the caregiver's working activity and the support caregiver can receive inside and outside of the household. The support can be childcare directly, in particular from household members, not household members or creche. She can also receive general support from groups for which she is a member or any other sources for which she can look. All these supports represent social capital that can have an impact on the relative cost of preschool education.

Preschool investment decisions of parents also reflect parental taste, parental altruism towards the child and parental expectation. We include the caregiver's attitude and perception and child care history to control for these factors.

All the control variables are measured in the first round except for the caregiver's attitude and perception. We assume that the caregiver's attitude and perception are not updated until the child goes to primary school. In total, we extract 50 potential control variables from the Young Lives survey.

Estimation

Double machine learning approach [START_REF] Chernozhukov | Double/debiased machine learning for treatment and structural parameters[END_REF] estimates outcome and treatment by machine learning algorithms and combines the estimates to derive average treatment effect estimator. The data-adaptive machine learning method provides small-mistake estimates of nuisance parameters to de-bias estimates of the average treatment effect.

The equations models outcome and treatment as follows:

Y i = θ 0 D i + g 0 (X i ) + e i , E[e i |D i , X i ] = 0 ( 1
)
D i = m 0 (X i ) + u i , E[u i |X i ] = 0 (2)
The confounding factors X i affect the treatment variable

D i via the propensity score, m 0 (X i ) := E[D i |X i ],
and the outcome variable via the function g 0 (X i ).

e i and u i are stochastic errors. Given the conditional independence assumption, θ 0 is interpreted as a structural or causal parameter that can be inferred from equation [START_REF] Goutte | Highest grade achieved 5[END_REF]. The double/debiased ML estimator θ0 solves the following condition: ) . Then we use this estimation to construct the estimation θ0 as the solution to the equation:

E[ψ(W, θ 0 ; η 0 ] = 0 (3) Where ψ(W ; θ, η) = (Y -Dθ -g(x)(D -m(x)) is score function, W = (Y, D, X),
1 N K k=1 i∈I k ψ(W i ; θ0 , η0,k ) = 0 (4)
The asymptotic variance of θ0 is estimated by:

σ2 = Ĵ-2 0 1 N K k=1 i∈I k [ψ(W i ; θ0 , η0,k )] 2 Ĵ0 = 1 N K k=1 i∈I k ψ a (W i ; η0,k )
where

ψ a (W ; η) = -(D -m(X))(D -m(X))
The specific sample splitting creates an additional source of variation, having impact on estimation results asymptotically in finite samples [START_REF] Chernozhukov | Double/debiased machine learning for treatment and structural parameters[END_REF]. They suggest further adaptation by re-partitioning the data multiple times and then aggregate it by the median of the resulting estimates as the final average treatment effect. The estimated standard errors are also corrected to capture the spread of the estimates.

We implement the double machine learning approach of Chernozhukov et al.

(2018) applying the DoubleML package (c, 2021) in R. We use eXtreme Gradient

Boosting -XGBoost by [START_REF] Chen | Xgboost: A scalable tree boosting system[END_REF] to fit the nuisance parameters.

XGBoost is a gradient boosted decision-tree-based ensemble Machine Learning algorithm that focuses on computational speed and model performance 1 . This model does not require the form of nuisance parameters to be specified and automatically handle missing data, implicitly conduct feature selection and characteristics. XGBoost is fast and dominates structured or tabular datasets on classification and regression predictive modelling problems 2 . We fit XGBoost to estimate each nuisance function with parameters chosen by internal tunning using five-fold cross-validation.

Results

We report estimates obtained using five-fold cross-fitting. All results are based on taking 100 different sample splits. The results are summarized across the sample splits using the median method.

1 Gradient boosting is an approach where new models are created that predict the residuals or errors of prior models and then added together to make the final prediction. It is called gradient boosting because it uses a gradient descent algorithm to minimize the loss when adding new models.

2 Since its introduction, this algorithm has been credited with winning numerous Kaggle competitions 5.1 Impact on skills, educational attainment and parental investment Panel A and B of table 2 shows the results for the effect of preschool starting age on child cognitive skills. All skill variables are standardized to have zero mean and variance one. The double machine learning estimates show a negative effect of preschool starting age on mathematics and language skills at age of 8. The estimates imply that starting preschool one year younger increases the mathematics score by about 0.09 standard deviations and language score by about 0.06 standard deviations. At age of 12 and 15, attending preschool one year earlier is only significantly associated with an increase in child's language skills, in which the estimates are almost 0.07 and 0.09 standard deviations, respectively. In particular, the magnitude of these estimates tends to increase over time. Under the light of the technology of skill formation proposed by [START_REF] Cunha | Estimating the technology of cognitive and noncognitive skill formation[END_REF], these findings support the idea of self-productivity for language skills in which "skills beget skills". However, we do not find enough evidence of self-productivity for mathematics skills.

We find little evidence on the effect on non-cognitive skills. As shown in panel B of table 2, early preschool enrollment is only significantly associated with selfesteem at age 12, but the effect disappears when children reach age 15. The estimates of other dimensions of non-cognitive skills turn out not statistically significant. Overall, panel C reports the impact on educational attainment. Our estimates suggest that starting preschool one year younger raises the highest grade achieved by 0.034 year at age 8 and 0.47 year at age 15.

Our estimated impacts of starting preschool younger converge on the prior research about the impact of preschool participation reviewed earlier. In particular, Shafiq et al. (2018) show positive statistically significant and positive associations between early childhood education participation and long-term cognitive skills and educational attainment in Vietnam. However, using different measurements of non-cognitive skills, they find mixed evidence on socio-emotional skills with both negative and positive impacts. Their data consist of urban adults from 20 to 64 years old in 2014, so this would not have been the same preschool program and underlying population with our study. However, an implication of the findings is the possibility that preschool education in Vietnam might focus on cognitive stimulation rather than enrich the learning of non-cognitive skills.

We explore an important channel through which preschool starting age affect child skills and child education. According to [START_REF] Cunha | Estimating the technology of cognitive and noncognitive skill formation[END_REF], early investment increases the stock of future skills that promote the productivity of future investment. We test the hypothesis that early preschool enrollment boosts the future parental investment in education, explaining the effect of preschool starting age on child's outcome. Estimates of preschool starting age on parental investment in education are reported in panel C of table 2. We use the number of hours of extra classes as a proxy for parental investment in education. Attending preschool earlier is accompanied by short-and medium-term increases in the number of hours children attend extra classes during an average week, and indeed, the magnitude of effect increases over time. Starting preschool one year younger increases the number of extra-class hours on an average week by 0.576, 0.582 and 0.673 at age 8, 12, and 15 years, respectively. The significant short and medium-term effects on parental investment in education can partly explain the positive effect of early preschool enrollment on educational attainment.

Impact on Health

We present the estimates of the effects of preschool starting age on child health in • I am confident that I could deal efficiently with unexpected events.

• I am confident that I could deal efficiently with unexpected events.

• It is easy for me to stick to my aims and accomplish my goals.

• I can always manage to solve difficult problems if I try hard enough.

• I can remain calm when facing difficulties because I can rely on my coping abilities.

• It is easy for me to stick to my aims and accomplish my goals.

• I can usually handle whatever comes my way.

• I can remain calm when facing difficulties because I can rely on my coping abilities.

• Thanks to my resourcefulness, I know how to handle unforeseen situations.

• I can usually handle whatever comes my way.

Round 3 Round 4

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  θ 0 is the parameter of interest, and η = (g, m) denotes nuisance functions with population value η 0 = (g 0 , m 0 ). The score ψ satisfies the Neyman orthogonality condition ∂ η Eψ(W ; θ 0 , η)| η=η0 = 0 which ensures that the condition (3) used to identify and estimate θ 0 is insensitive to small pertubations of the nuisance function η around η 0 and to the replacement of η 0 with η0 for high-quality inference. Therefore, the estimation θ0 of θ 0 is constructed base on the score ψ which replace the nuisance functions η 0 with η0 produced by data-adaptive machine learning method and on the moment conditions (3). This estimator is semi-parametric efficient, under weak conditions, due to an extra step of sample splitting. Since using one sample to fit the nuisance parameter, obtain prediction and construct the estimate of θ 0 causes overfitting and thus bias, sample splitting would help to avoid overfitting and thus reduce bias.[START_REF] Chernozhukov | Double/debiased machine learning for treatment and structural parameters[END_REF] propose randomly split the sample to k-folds equal partitions (1, 2, . . . , N ). For each set I k , let I c k denote all observation indices that are not in I k . The nuisance parameters are estimated in the auxiliary sample I c k : η0,k = η0,k (W i ) i∈I c k

••

  friends easily • I make friends easily • I am popular with kids of my own age • I am popular with kids of my own age • Most other kids like me Most I have lots of friends • I have lots of friends • I am easy to like • I am easy to like • I get along with other kids easily

Table 1

 1 reports the summary statistics of preschool starting age. The average years of preschool starting age are approximately 3.72 years old. Children start preschool latest at 5.75 years old.

	Statistic	N	Mean St. Dev. Min Pctl(25) Pctl(75) Max
	Preschool	1,763 3.723 0.757	3.000 3.000	4.167	5.750
	starting age
	Table 1. Summary statistics of preschool starting age variable

The Rash model is a one-parameter of the Item Response Theory model using logistic regression analyses where item responses are predicted by the difference between the item difficulty and the child's ability. The equation for the Rasch model is:

Table 2 .

 2 Panel A presents results on height-for-age, weight-for-age, and BMIfor-age score. Panel B shows the impacts on nutritional statuses. We find a Note: * p<0.1; * * p<0.05; * * * p<0.01. Outcome variables of Panel A and B are scaled to have mean 0 and variance 1. Estimated effects and standard errors from partially linear regression effect model based on orthogonal estimating equations. Results are based on 100 sample splits and the standard errors adjusted for variation across splits are provided in parentheses. Note: * p<0.1; * * p<0.05; * * * p<0.01. Outcome variables of Panel A are scaled to have mean 0 and variance 1. Estimated effects and standard errors from partially linear regression effect model based on orthogonal estimating equations. Results are based on 100 sample splits and the standard errors adjusted for variation across splits are provided in parentheses.

	(1)	(2)	(3)
	Age 8	Age 12	Age 15

Table 3 .

 3 Estimated effect of preschool starting age on child health. underweight. Although starting preschool younger does appear to benefit child skills and education, the downsides for child health imply that the quantity and quality of school feeding remain low for poor children. Nonetheless, we find a positive effect on the probability of the child being thin when children reach the age of 15 years. Incorporating with the prior results, this might be suggestive of a link between health and education attainment.The positive effect of younger preschool starting age provides a signal that "earlier is better" for the development of poor children. However, preschool education in Vietnam might only enrich cognitive learning without nurturing environments for non-cognitive skills and health. Our findings suggest that improving the quality of preschool education is as important as, if not more than, expanding access to preschool to achieve sustainable development goals.This study confines to poor children in Vietnam. Therefore its results cannot be generalized for the whole society. Further research needs to be done to assess the impact across different groups of children, cohorts, and countries.

	Variable	Mean	N	Mean	St. Dev.
	Weight-for-age z-score at age 1	Continuous 1,758 -0.956 1.088
	Father's age	Discrete	1,704 30.006 5.861
	Father's education	Discrete	1,697 6.478	4.112
	Mother's age	Discrete	1,759 27.092 5.680
	Mother's education	Discrete	1,749 5.914	4.106
	Average working days per week of	Discrete	1,648 1.257	0.538
	caregiver
	Age order of siblings in the house-	Categorical 1,761
	hold
	Index child is the eldest		5
	Index child is the youngest		925
	Index child has no siblings in the		831
	household
	Number of sisters in the household	Discrete	1,763 0.437	0.762
	Number of brothers in the household Discrete	1,763 0.332	0.516
	Number of males aged 0-5	Discrete	1,763 0.138	0.365
	Number of males aged 6-12	Discrete	1,763 0.208	0.460
	Number of males aged 13-17	Discrete	1,763 0.079	0.292
	Number of males aged 18-60	Discrete	1,763 1.268	0.694
	Number of males aged 61+	Discrete	1,763 0.111	0.316
	Number of females aged 0-5	Discrete	1,763 0.163	0.387
	Number of females aged 6-12	Discrete	1,763 0.268	0.571
	Number of females aged 13-17	Discrete	1,763 0.101	0.357
	Number of females aged 18-60	Discrete	1,763 1.353	0.720

short-term negative effect of starting preschool younger on weight-for-age and BMI-for-age z-score. Starting one year younger is associated with a decrease in weight-for-age and BMI-for-age z-score by 0.073 and 0.089 standard deviations. Given gender, BMI-for-age and weight-for-age reflect current and longterm health status, respectively [START_REF] Le | Shedding light on maternal education and child health in developing countries[END_REF]. It can therefore be suggestive that preschool starting age can harm the current health of young children. We do not find enough evidence for the long-term impact of preschool starting age on health. We also find no evidence on nutritional statuses except for being thin at age 15 years. Incorporated with the effect of education, this can support the link between educational attainment and health outcomes.

Conclusion

While most studies have shown the favorable effect of preschool enrollment on child outcomes, no study, to the best of our knowledge, investigate the impact of preschool starting age. We apply the double machine learning approach given the conditional independence assumption to investigate the effect of preschool starting age on development of poor children in Vietnam. Our results suggest that early preschool enrollment has significantly positive short-and mediumterm effects on child cognitive skills and educational attainment. However, we find little evidence of the effect on non-cognitive skills. We find that the favorable effect of early preschool enrollment can be, at least, partly explained by an increase in parental investment in education. Specifically, earlier preschool enrollment promotes the number of hours of extra classes in all rounds of the survey.

We uncover the unfavorable short-term effect of early preschool starting age on two measures of child health: BMI-for-age and weight-for-age, while we find no effect on height-for-age, the probability of the child being stunted, and Appendix A Summary statistics of outcome variables [START_REF] Goutte | Highest grade achieved 5[END_REF] (


J Martin

F Doumenc

Condensation-induced self-patterning of a thin clayey layer

We show through laboratory experiments that the self-patterning of a thin clayey layer can be triggered by condensation. The natural sediment used in the experiments was a highly polydisperse granular paste with smectite clay in the fine fraction. Under certain physicochemical conditions, condensation induces the solid-to-liquid transition of the sediment layer, resulting in sediment flow and the formation of band structures. These results suggest a physical mechanism for the formation of patterns commonly observed on the humid walls of underground cavities, referred to as vermiculations.

Introduction. -It has been well known for more than a century that the sedimentary material lying on the humid walls of underground cavities can self-organize to form a wide variety of patterns, called vermiculations (see for instance Adesso et al [1] for illustrative examples). Many attempts have been made to explain the formation of these patterns (see reviews in refs. [2,[START_REF] Hoerlé | Reports on "Vermiculations in Lascaux cave[END_REF]), but a physical mechanism supported by strong evidence remains to be found. Understanding this natural process is, however, of utmost importance for the conservation of painted caves, where valuable prehistoric paintings can be damaged by vermiculation crises [4][5][6].

The presence of water on cave walls is a necessary but not sufficient condition to obtain vermiculations [2]. When the sediment is saturated with water, it forms a granular paste that behaves as a yield stress fluid [7]. The yield stress, due to colloidal interactions, depends on the chemical composition of the water film present on the wall, through the pH, the ionic strength, or the nature of the ions in solution [START_REF] Larson | The structure and rheology of complex fluids[END_REF]. These physicochemical parameters may vary in time and space with the origin of the water (percolation, flooding or condensation) [6], or microbiological activity [START_REF] Barton | [END_REF]. We show through laboratory experiments that, under certain conditions, condensation reduces the sediment yield stress to zero, resulting in sediment flow and pattern formation.

Material characterization. -In our experiments, we used natural sediment collected in Maillol Cave (Montignac, France) in a limestone massif. The particle size distribution of this material is highly polydisperse. It spreads from 1 µm to 1 mm, with a significant fraction of fine particles (16% in volume under 10 µm). The mineralogical composition of Maillol Cave sediment was determined by X-ray diffraction. It consists of high quantities of quartz (silica) and calcite (calcium carbonate, CaCO 3 ), with lower amounts of many other minerals such as metallic oxides, hydroxides, and various clay minerals (phyllosilicates) with a predominance of smectites (80% of the fine fraction) [7]. Before use, the sediment was dried in ambient air and then smoothly crushed with a mortar. All particles with a diameter above 200 µm (4% volume fraction) were removed by dry sieving. A granular paste was finally obtained by mixing the dried powder with a given amount of water previously saturated with CaCO 3 . The initial volume fraction of solid in the sediment paste was set to 0.440 ± 0.012 for all experiments. The corresponding yield stress was 2300 ± 400 Pa (as a comparison, the gravity-induced shear stress on a vertical 100 µm-thick sediment layer is of the order of 1 Pa).

The rheological behavior of Maillol Cave sediment was recently studied by Freydier et al [6,7]. They showed that the yield stress of this material could go to zero if it was first soaked in an aqueous solution containing a salt with a monovalent cation (e.g., NaCl) and then placed in deionized (DI) water. This solid-to-liquid transition of the sediment was not observed if the aqueous solution used in the first step of the process contained a salt with a multivalent cation (e.g., CaCl 2 ), or if the sediment sample was directly soaked in DI water. This striking behavior is characteristic of smectites. Indeed, the faces of smectite platelets carry permanent negative charges. They are balanced by the cations of the diffuse double layers, whose valence strongly affects the smectite phase diagram. In a smectite paste with monovalent counterions (e.g., Na + ), interparticle forces are attractive for salt concentrations greater than the critical coagulation concentration (CCC), typically of the order of 10 mmol/L. In a smectite paste with multivalent counterions (e.g., Ca 2+ ), interparticle forces are always attractive [10][11][12]. This accounts for the experimental results of Freydier et al [6,7]. During the first soaking in a NaCl solution, Na + was substituted for Ca 2+ in the diffuse double layers, but the sediment yield stress was still significant because the NaCl concentration was greater than the CCC. Interparticle interactions turned to repulsion in the subsequent soaking in DI water, which resulted in the gel-sol transition of smectite, and the collapse of the sediment yield stress. This rheological behavior reveals that particles of various compositions and sizes included in the sediment are bound together by a smectite gel, which is responsible for the material yield stress.

Experimental setup. -An experimental setup was designed to demonstrate that the sediment loss of cohesion could induce self-patterning under cave environmental conditions. It enables the exposure of a sediment thin layer to a flow at specific physicochemical conditions followed by condensation at a controlled rate (see fig. 1 for a sketch of the setup, and the Supplementary Material for more details).

A thin layer of sediment paste was coated on the rectangular surface of the limestone rock (80×55 mm). The same rock sample was used for all experiments (after each experiment, the rock was cleaned with tap water and rinsed with DI water previously saturated with CaCO 3 ). The coating was done using a film applicator with a 200 µm-gap (Elcometer, ref. 3540). However, because of the waviness of the rock surface (measured with a comparator), the actual local gap between the rock surface and the applicator could vary from 100 to 200 µm. We can thus assume that the mean thickness of the sediment layer was approximately 150 µm p-2 with spatial fluctuations lower than 50 µm. The reproducibility of the deposit thickness was tested by weighing the rock before and after coating. The mass measurement of 7 deposits yielded a mean thickness of 163 µm and a standard deviation of 15 µm.

The coated stone was then inserted in a thermostatically controlled enclosure at 15.05 • C, which could be tilted at an angle α from the horizontal position (0 ≤ α ≤ 85 • ). This enclosure was closed with transparent double glazing to allow observation of the sediment layer. Humid air precooled at the enclosure temperature (15.05 • C) permanently flowed above the sediment layer with a mass flow rate of 0.215 g/s, CO 2 partial pressure (pCO 2 ) of 35 Pa and a dew point of 14.92 • C. Two temperature sensors were inserted into the limestone rock, which rested on two Peltier modules (small heat pumps in the form of 40 × 40 mm square plates). The rock thickness was 20 mm and the temperature sensors were located 9 mm below the upper surface of the rock. This device allowed us to control the rock temperature independently of its surroundings.

In the first step of each experiment, the rock temperature was set to 15.00 • C to prevent condensation. The tilt angle α was set to 30 • . An aqueous solution containing 20 mmol/L of NaCl and saturated with CaCO 3 at the same pCO 2 as the air flow ([Ca 2+ ] 0.6 mmol/L, pH 8.2, ionic strength 22 mmol/L) was injected with a syringe pump at the top of the setup at a constant flow rate of 100 mL/h. The solution flowed above the sediment layer by gravity drainage (fig. 1a). From lubrication theory, we estimate that the height of the liquid film was approximately a few tens of microns, resulting in a viscous stress on the sediment layer of less than 1 Pa. Note that the solution had been previously saturated with CaCO 3 to avoid the dissolution of sediment calcite during this step. After 3 hours, the syringe pump providing the flow of the aqueous solution was stopped.

In the second step, immediately after stopping the syringe pump, the set-point of the rock temperature was decreased below the dew point of the air to trigger water condensation at the surface of the sediment layer. After a transient regime due to the thermal inertia of the rock, the condensation rate was expected to be proportional to ∆T , the temperature drop between the air dew point and the rock temperature (see fig. 1b). The time τ th required to achieve 90% of the temperature drop was systematically measured in all experiments. τ th globally increased with ∆T , in the range from 2 min to 17 min. Assuming 8 × 10 -7 m 2 /s for the limestone diffusivity, the thermal diffusion time between the temperature sensors and the sediment layer was approximately 2 min. The temperature sensors and the rock upper surface thus reached the steady state approximately at the same time. We also determined the time for the drainage of the liquid film after stopping the syringe pump. The sediment layer was illuminated with a laser sheet. After stopping the syringe pump, the image of the laser sheet reflected on a white screen showed that the transition from specular reflection (presence of a liquid film) to diffuse reflection (due to sediment roughness) was achieved within a time of approximately 3 min. This drainage time does not exceed τ th , the duration of the thermal regime. τ th is thus used to estimate the uncertainty on the initial time. If modified, the tilt angle α was changed after a waiting time of 10 min to allow prior drainage of the aqueous solution.

To characterize the setup, the coated rock was weighed before and after being exposed to condensation at a given ∆T , before self-patterning began. These experiments provided the absorption rate q, defined as the volumetric flow rate of condensed water absorbed by a unit surface of sediment layer. We checked that no liquid flowed out from the rock during the condensation stage prior to pattern formation (we only noticed a few droplets under the rock after the end of the experiments performed at the highest temperature drop, i.e., ∆T 11 K). The absorption rate q is thus expected to approximately equal the condensation rate, which accounts for the scaling q ∝ ∆T observed in fig. 5a.

Results. -Figure 2 displays the initial state and the final pattern for α = 85 • and ∆T = 3.3 K. Video 1 in the Supplementary Material shows the same experiment. The origin of time is taken at the beginning of the second step (i.e., when the set-point of the p-3 rock temperature was decreased). No modification of the sediment layer is observed during the first step, i.e, during the flow of the NaCl-CaCO 3 solution (this remark applies to all experiments presented in this paper). Self-patterning takes place during the second step. Sediment motion begins approximately at times t 1 128 min and t 2 188 min at 2 cm from bottom and top edges, respectively. For all experiments, we define the onset time as t sp = (t 1 + t 2 )/2. In the specific case of fig. 2, t sp 158 min. The whole surface is covered with band structures at time t 270 min, and no significant motion is observed after t 290 min.

We see in fig. 3 that the same kind of pattern develops for a smaller tilt angle (α = 30 • ) and a wide range of temperature drops, from ∆T 0.3 K to ∆T 11 K (movie 2 in the Supplementary Material shows the full patterning process for the case ∆T = 3.3 K). This highlights the robustness of the underlying mechanism leading to the accumulation of particles in the band structures. Increasing ∆T enhances the contrast of the final pattern, with no obvious effect on its morphology. diffuse double layers. Ca-smectite, which is always in a gel state, becomes a Na-smectite, which is expected to disperse at low enough ionic strength. As no motion is observed in the sediment layer, one infers that the ionic strength of the aqueous solution is high enough to maintain the Na-smectite in a gel state. This is consistent with the rheological measurements displayed in table 1. After a 24 hours long soaking in the NaCl-CaCO 3 solution (same as the one injected in the first step), the sediment yield stress decreased by an order of magnitude but was still of the order of a few hundreds of Pascals, considerably above the stresses induced by gravity and by the flow of the aqueous solution (∼ 1 Pa).

We turn now to the second step of the experiment. When the injection of the solution is stopped, the liquid film above the sediment layer is drained away by gravity, but the sediment pores are still filled with the injected NaCl-CaCO 3 solution. In the following condensation step, however, the pore liquid is diluted by the intake of condensed water. This causes a decrease of the ionic strength resulting in the gel-sol transition of the smectites, the collapse of the sediment yield stress, and finally the flow of the sediment layer due to gravity (no motion is observed for α = 0, which proves the role of gravity, see fig. 4b). At this point, the sediment behaves as a suspension of macroscopic particles. Indeed, particles greater than 10 µm represent 84% of the particle size distribution, and these particles are no longer bound by the smectite gel. Hence, the mechanism proposed by Buchanan et al [13] to describe the formation of band structures for noninteracting particles is likely to apply. When the sediment flows, its thickness decreases. Due to the confinement by the free surface, some isolated particles are pinned by solid friction with the substrate. The subsequent accumulation of particles in band structures results from the jamming induced by convergent streamlines around the pinned particle and by further accumulation of particles in the jammed region acting as a filter [14].

Table 1: solid volume fraction ϕ s and yield stress τ c of sediment samples in the initial state and after a 24 hours long soaking in an aqueous solution of 20 mmol/L NaCl saturated with CaCO 3 (1-mm thick sample containing 2 cm 3 of sediment soaked in 250 mL of solution, see ref. [6] for a detailed description of the rheological measurements). , respectively. The uncertainty on V w is derived from the uncertainties on q and t sp .

In the following, we investigate in more detail the onset of the self-patterning process triggered by the solid-to-liquid transition of the sediment layer, prior to which it cannot flow. Let V w = q t sp be the volume of condensed water per unit of surface at the onset of selfpatterning. It results from the scalings q ∝ ∆T (fig. 5a) and t sp ∝ ∆T -1 (fig. 5b) that V w is almost constant (fig. 5c). Averaging all of the experimental results yields V w 180 ± 60 µm. It is worth noticing that this constant value of V w was obtained for onset times t sp varying over a decade (from 100 min to 2000 min, see fig. 5b), suggesting negligible kinetic effects over the experiment time scales.

It is possible to ascertain that the constant value of V w is consistent with the rheological behavior of the sediment. As the onset of self-patterning is reached at a fixed dilution factor of the pore solution, the concentration [NaCl] sp at the onset is constant and it is given by the relation

[NaCl] sp [NaCl] h (1 -ϕ s ) h (1 -ϕ s ) + V w , (1)
where the NaCl concentration [NaCl], the deposit height h and the solid volume fraction ϕ s should be assessed at the end of step 1. However, the sediment swelling during step 1 can be neglected (table 1 shows that the solid volume fraction decreased by only 0.051 after 24 hours soaking), so that h and ϕ s can be approximated by their initial values. With [NaCl] 20 mmol/L, h 150 µm and ϕ s 0.44, eq. ( 1) yields [NaCl] sp 6 mmol/L. As NaCl corresponds to the dominant species in the solution, [NaCl] sp is an estimation of the ionic strength at the onset. Its constant value is therefore in accordance with a corresponding gel-sol transition of the Na-smectites. This interpretation has been tested with the help of rheological characterization of sediment samples successively soaked in two NaCl aqueous solutions saturated with CaCO 3 . The first soaking was 24 h long in a bath at a fixed concentration [NaCl] = 20 mmol/L (same condition as in table 1), whereas the second soaking was shorter (3 h ± 30 min, chosen for experimental convenience) at bath concentrations varied from 0 to 20 mmol/L. We see in fig. 6 that the yield stress after these two soakings collapses from approximately 300 Pa to 0 when the NaCl concentration of the second soaking is decreased from 10 mmol/L to a value ranging from 3 to 2 mmol/L. The agreement with the value [NaCl] sp 6 mmol/L predicted by eq. ( 1) is quite satisfying if we consider that both configurations, condensation and soaking, are quite different (e.g., there is ion exchanges across the sample surface in soaking experiments, not during condensation). This result confirms that the self-patterning onset is controlled by the yield stress collapse due to the gel-sol transition of the Na-smectite gel. Conclusion. -These laboratory experiments demonstrate that the self-patterning of a clayey layer can be triggered by water condensation. Transposed to the natural cave environment, our experimental two-step process might correspond to a first time period in which the concentration in monovalent cations increases (due, for instance, to evaporation or microbiological activity), followed by a condensation event during a second time period. Other scenarios are also possible, since the rheology of smectites can be strongly modified by the pH [15], or the presence of multivalent anions [16][17][18]. In general, any change in the physicochemical conditions leading to the solid-to-liquid transition of a granular paste layer should result in the formation of new patterns or the evolution of existing patterns.

Fig. 1 :

 1 Fig. 1: Experimental protocol. (a) First step: aqueous solution flowing film; (b) Second step: condensation. The sediment layer coated on the rock is shown in yellow.

Fig. 2 :

 2 Fig. 2: (a) Initial state and (b) final pattern for α = 85 • and ∆T = 3.3 K (image size: 80 × 55 mm).

Fig. 3 : 4 Fig. 4 :

 344 Fig. 3: Effect of the temperature drop ∆T on the final pattern for α = 30 • . (a) ∆T = 0.31 K; (b) ∆T = 1.1 K; (c) ∆T = 3.3 K; (d) ∆T = 11.0 K (image size: 80 × 55 mm).

5 Fig. 5 :

 55 Fig. 5: Effect of the temperature drop ∆T on (a) the absorption rate q; (b) the onset time t sp ; (c) The corresponding volume of condensed water per unit of surface V w = q t sp . The error bars on ∆T show the difference between the temperatures measured by both sensors located in the rock. The error bars on q stem from the uncertainties on the measurement of the mass variation of the sediment layer due to condensation and the time during which it was exposed to condensation. The upper and lower error bars on t sp are |t 1 -t 2 |/2 and (τ th + |t 1 -t 2 |/2), respectively. The uncertainty on V w is derived from the uncertainties on q and t sp .

Fig. 6 :

 6 Fig.6: sediment yield stress τ c after two successive soakings in NaCl solutions saturated with CaCO 3 . NaCl concentration of the first soaking was fixed to 20 mmol/L. τ c is plotted as a function of the NaCl concentration of the second bath. Black arrows correspond to concentrations for which no yield stress was detected (1-mm thick sample containing 2 cm 3 of sediment soaked in 250 mL of solution, see ref.[6] for a detailed description of the rheological measurements).

The experimental setup used in this study was built in the framework of the project "Vermiculations in Lascaux Cave" funded by the French Ministry of Culture (Direction des Affaires Culturelles de Nouvelle-Aquitaine). We thank P.-Y. Jeannin from SISKA for sharing his knowledge about karst systems through many useful discussions. We gratefully thank A. Aubertin, L. Auffray, J. Amarni, R. Pidoux, and C. Manquest (FAST Laboratory) for technical support. We thank D. Calmels, C. Quantin, G. Monvoisin, O. Dufaure, and S. Miska (GEOPS Laboratory) for chemical analysis and geological characterization of our samples and M. Rossi (IJLRA) for useful discussions.