Amemiya A Ronald Takeshi

John F Gallant

Cheng Geweke

Peter Hsiao

Arnold Robinson

Zellner

Aït-Sahalia Yacine

H Badi

M W Baltagi

Marcus J Brandt

Songnian Chambers

Manfred Chen

Miguel A Deistler

Jean-Marie Delgado

Sylvia Dufour

Eric Fruhwirth- -Schnatter

John C Ghysels

Javier Ham

Han Hidalgo

Yongmiao Hong

Bo E Hong

Maxwell L Honoré

Yuichi King

G M Kitamura

Chung-Ming Koop

Naoto Kuan

Kajal Kunitomo

Q Lahiri

Tong Li

Oliver Li

James G Linton

Robert Mackinnon

Rosa L Mcculloch

Franz C Matzkin

Dale J Palm

Nicholas Poirier

B M Polson

Ingmar Pötscher

Peter C Prucha

Eric Reiss

Frank Renault

Robin Schorfheide

Fallaw Sickles

G J Sowell

Herman Van Den Berg

Dijk Van

H Quang

Edward Vuong

Tom Vytlacil

Andrew Wansbeek

Weiss

Sonia Bhalotra
email: s.bhalotra@bristol.ac.uk

Arthur Van Soest

Julie Davanzo

Helene Turon

Birth Spacing, Fertility and Neonatal Mortality in India: Dynamics, Frailty and Fecundity ♣

Keywords: fertility, birth spacing, neonatal mortality, health, dynamic panel data models, siblings, India I12, J13, C33

Using microdata on 30,000 child births in India and dynamic panel data models, we analyze causal effects of birth spacing on subsequent neonatal mortality and of mortality on subsequent birth intervals, controlling for unobserved heterogeneity. Right censoring is accounted for by jointly estimating a fertility equation, identified by using data on sterilization. We find evidence of frailty, fecundity, and causal effects in both directions.

Birth intervals explain only a limited share of the correlation between neonatal mortality of successive children in a family. We predict that for every neonatal death, 0.37 additional children are born, of whom 0.30 survive.

Interest in the determinants of child mortality has recently surged, with the inclusion of targets for child mortality amongst the Millennium Development Goals [START_REF] Lawn | Neonatal Survival 1-4 million neonatal deaths: When? Where? Why?[END_REF], UNDP 2003), and short birth-spacing and high fertility are widely regarded as among the most important causes of early childhood death. However, reproductive behaviour is endogenous to mortality and both are influenced by characteristics and choices of families, some of which are difficult to observe. For these reasons, there is limited evidence of the true causal associations of these variables.

In developing countries, 30% of deaths are of children under five, compared to less than 1% in rich countries [START_REF] Cutler | The determinants of mortality[END_REF]). Almost half of child deaths are in the neonatal period, the first month of life, when the tie between mortality and fertility is closest [START_REF] Sathar | The effect of birth spacing on childhood mortality in Pakistan[END_REF]. About 4 million neonates died in 2000, 99% of them in developing countries, and 27% in India. The proportion of neonatal in under-5 deaths has increased, since interventions like immunization, control of acute respiratory infection, or oral rehydration have had more of an effect on post-neonatal death [START_REF] Lawn | Neonatal Survival 1-4 million neonatal deaths: When? Where? Why?[END_REF]. It is thus important to focus attention on the causes of neonatal death.

Despite a long-standing interest of economists and demographers in the relation between childhood mortality and reproductive behaviour, the literature is scarce in a complete micro-data analysis of all inter-relations of these variables [START_REF] Wolpin | Determinants and consequences of the mortality and health of infants and children[END_REF]. The main contribution of this paper is to use panel data based on retrospective fertility histories to estimate causal effects of birth interval length on subsequent neonatal mortality risk and of neonatal mortality on subsequent birth interval length, controlling for unobserved heterogeneity in both processes (referred to as frailty and fecundity, respectively). It also Understanding the way in which biological and behavioural factors shape the familylevel relation between reproductive behaviour and childhood mortality is crucial to understanding the demographic transition that has historically preceded economic growth [START_REF] Kalemli-Ozcan | Does mortality decline promote economic growth[END_REF], and the endogenous processes by which societies evolve past the Malthusian spectre [START_REF] Galor | Population, technology, and growth: From Malthusian stagnation to the demographic transition and beyond[END_REF]. Time series analyses of historical data for today's industrialized countries suggest that declining mortality stimulated fertility decline (e.g. [START_REF] Ben-Porath | Fertility response to child mortality: Micro data from Israel[END_REF][START_REF] Eckstein | A quantitative analysis of Swedish fertility dynamics, 1751-1990[END_REF], and a similar tendency can be seen in recent data for developing countries (e.g. [START_REF] Nyarko | Child loss and fertility behaviour in Ghana[END_REF]. Cross-sectional studies using household survey data have emphasized the reverse direction of causation, namely that high fertility, associated with close birth spacing or an early start to childbearing, causes an increase in childhood mortality (e.g. Cleland and [START_REF] Sathar | The effect of birth spacing on childhood mortality in Pakistan[END_REF].

In families with multiple children, there is a recursive bi-causal relation of these variables. The death of a child is often followed by a shorter interval to the next birth, which may be explained either by volitional replacement (e.g. [START_REF] Olsen | Cross-sectional methods for estimating the replacement of infant deaths[END_REF] or by the fact that the mother stops breastfeeding, enabling her to conceive the next child sooner than otherwise (e.g. [START_REF] Chen | A prospective study of birth interval dynamics in Rural Bangladesh[END_REF]. A short birth interval, in turn, increases the mortality risk death trap, creating persistence in death risk within families. This mechanism operates by the endogenous shortening of intervening birth intervals. Of course a birth interval is only observed if the mother has another birth, and this fertility decision is also influenced by whether her previous birth survived or not. While these relationships have each been studied, their interactions have rarely been studied jointly, and unobserved heterogeneity, another potential source of correlation of death risks within a family, is often ignored.

The analysis in this paper provides estimates, using survey data from India, of a dynamic panel data model that describes the complete process of child survival and birth spacing (and thus fertility), allowing for endowment heterogeneity, input endogeneity, right-censoring and accounting for the initial conditions problem. We find evidence that childhood mortality risk is influenced by the pattern of childbearing, that is, by the timing and spacing of births, and that birth-spacing and fertility are, in turn, a function of realized mortality. We find a replacement effect of 0.37, in line with the few available estimates in the literature. The results suggest that the full impact of family planning interventions extends to reducing mortality and that mortality-reducing interventions like provision of piped water also affect birth spacing and fertility. Our finding of causal effects of sibling mortality on both mortality and reproductive behaviour implies that interventions that reduce mortality or lengthen birth intervals will have multiplier effects.

The paper is organized as follows. Section 2 summarizes related research. Section 3 describes the data. The econometric model is presented in section 4 and estimation and simulation results are reported in section 5. Section 6 summarizes and concludes. Previous demographic research provides estimates of some of the main effects analyzed in this paper, although not in a unified framework: for example, see [START_REF] Curtis | Birth interval and family effects on postneonatal mortality in Brazil[END_REF], [START_REF] Madise | Determinants of infant mortality in Malawi: An analysis to control for death clustering within families[END_REF], [START_REF] Hobcraft | Demographic determinants of infant and early child mortality: A comparative analysis[END_REF], and [START_REF] Whitworth | Birth spacing, sibling rivalry and child mortality in India[END_REF] for analysis of the effects of birth-spacing on mortality, [START_REF] Olsen | Cross-sectional methods for estimating the replacement of infant deaths[END_REF], [START_REF] Zenger | Siblings' neonatal mortality risks and birth spacing in Bangladesh[END_REF] or [START_REF] Frankenberg | The relationship between infant and child mortality and subsequent fertility in Indonesia, 1971-1991[END_REF] for analysis of the effects of mortality on birth-spacing. The limitation of these studies is that their estimates cannot be given a causal interpretation (see [START_REF] Moffitt | Causal analysis in population research: An economist's perspective[END_REF].

There is limited previous research in economics in this area. [START_REF] Bhargava | Family planning, gender differences and infant mortality: Evidence from Uttar Pradesh, India[END_REF] estimates a single-equation probit model of infant mortality in India, and argues that endogeneity of birth spacing is taken care of by controlling for the survival status of older siblings, which is instrumented using household possessions and number of previous births. [START_REF] Maitra | Birth spacing and child survival: Comparative evidence from India and Pakistan[END_REF] estimate a simultaneous hazards model of birth-spacing and child mortality, relying upon similarly strong identifying assumptions. Rosenzweig and Schultz (1983b) estimate a model of infant mortality in which birth-spacing is instrumented using household incomes and local prices. However, as discussed in Rosenzweig andWolpin (1988, 1995), the implied exclusion restrictions typically do not hold. Rosenzweig andWolpin (1988, 1995) instead use sibling differences to eliminate the mother-specific endowment. To allow for differences across siblings in frailty, they instrument inputs into the health of the index child using inputs relating to older siblings and parental characteristics. The econometric strategy in our paper is similar in that it relies upon natural information restrictions associated with the sequencing of births.

A c c e p t e d m a n u s c r i p t 5

Estimates of the effects of childhood mortality on subsequent birth spacing and fertility have mostly relied on the implausible assumption that parents have no influence on the survival chances of their offspring [START_REF] Ben-Porath | Fertility response to child mortality: Micro data from Israel[END_REF][START_REF] Wolpin | Determinants and consequences of the mortality and health of infants and children[END_REF][START_REF] Cigno | Fertility decisions when infant survival is endogenous[END_REF], although there are exceptions [START_REF] Olsen | Estimating the effect of child mortality on the number of births[END_REF][START_REF] Olsen | Cross-sectional methods for estimating the replacement of infant deaths[END_REF][START_REF] Olsen | The impact of exogenous child mortality on fertility: A waiting time regression with exogenous regressors[END_REF]. Our approach is different, in that we use a dynamic panel data framework and provide estimates of causal effects in both directions, between mortality and reproductive behaviour, and for both birth spacing and total fertility. The recent demographic literature has highlighted the widespread phenomenon of sibling death clustering, emphasizing the role of unobserved heterogeneity, estimated using multi-level models that incorporate a random effect at the mother-level [START_REF] Guo | Use of sibling data to estimate family mortality effects in Guatemala[END_REF][START_REF] Zenger | Siblings' neonatal mortality risks and birth spacing in Bangladesh[END_REF][START_REF] Curtis | Birth interval and family effects on postneonatal mortality in Brazil[END_REF][START_REF] Sastry | A nested frailty model for survival data, with an application to the study of child survival in Northeast Brazil[END_REF][START_REF] Whitworth | Birth spacing, sibling rivalry and child mortality in India[END_REF]. Arulampalam and Bhalotra (2006a,b) contribute to this introducing state dependence in mortality. They identify state dependence effects in 13 of 15 Indian states using a singleequation model for mortality. Otherwise, there is little previous research on statedependence effects in analysis of sibling data, although sibling correlations in outcomes have been widely studied (e.g. [START_REF] Solon | A longitudinal analysis of sibling correlations in economic-status[END_REF].

Data & Descriptive Statistics

The data are from the second round of the National Family Health Survey of India (NFHS-II) which recorded complete fertility histories for ever-married women aged 15-49 in 1998-99, including the time and incidence of child deaths. 1 Mothers constitute the crosssectional dimension of the data. As mothers are observed repeatedly, in relation to every birth, birth-order creates the time dimension of the (unbalanced) panel. We use data for 1 For details on sampling strategy and context, see IIPS and ORC Macro (2000). Strictly, neonatal death refers to death in the first four weeks of life. We include deaths up to a month to allow for age-heaping. The birth interval is the interval between reported dates of birth, rather than the inter-conception interval. As a result, measured birth intervals will be shorter on account of premature births [START_REF] Gribble | Birth intervals, gestational age and low birth weight: are the relationships confounded?[END_REF]) and longer on account of miscarriage or stillbirth [START_REF] Madise | Determinants of infant mortality in Malawi: An analysis to control for death clustering within families[END_REF]. The first problem is dealt with by removing intervals shorter than 9 months; the second is harder to address. Ignoring miscarriage and stillbirth may lead to under-estimation of the mortality-raising effect of short birth intervals if women who have these problems also tend to produce weaker live births, since then falsely long intervals will be associated with higher mortality. However, this bias may be expected to be small once we control for mother-specific frailty and fecundity.

A c c

Means and standard deviations of all variables used in the analysis are in Appendix Although contraceptive prevalence is increasing, contributing to the fertility decline witnessed in India since the mid-1980s, it seems to have had little impact on neonatal mortality [START_REF] James | Neonatal mortality in India: Emerging paradoxes[END_REF]. This is because contraception is used primarily to limit fertility rather than to control early childbearing and lengthen birth intervals. At the time of the survey, women were asked what their current contraceptive method was. In the state of UP, 65.8% were using no method, and 19.6% reported female sterilization, which is the predominant form of contraception, as in other parts of India. In section 4 we will argue that information on sterilization helps identify the fertility equation.

Figure 1 is a non-parametric regression of neonatal death on the log of the preceding birth interval. The curve declines monotonically. At short birth intervals, the probability of neonatal death is highest, and the gains from an additional month's spacing are largest.

Figure 2 plots the kernel density function of the birth interval separately for whether or not the previous child in the family survived its first month. The birth interval distribution for the case where the preceding child has died clearly lies to the left of the other, with median birth intervals of 23 months after a neonatal death and 27 months otherwise (the means are 24.3 and 31.2 months). The raw data thus suggest the patterns that, as argued in section 1, contribute to "death traps" at the family level: short birth intervals raise subsequent mortality risk (Fig. 1), and mortality results in a shorter subsequent birth interval (Fig. 2).

A c c e p t e d m a n u s c r i p t 8

In the sample of second and higher-order children, the average probability of neonatal death is 6.28%. In the sub-sample in which the previous sibling survived, this probability is 5.20%, but amongst those whose previous sibling died, it is a remarkable 18.80%. Thus the death of a preceding sibling is associated with an increase in mortality risk of 13.6%-points.

This can be explained by both unobserved heterogeneity and genuine state dependence, and state dependence can, in turn, be explained by short birth-spacing or other mechanisms. Our analysis will disentangle these three explanations.

The Model

The model has a recursive dynamic structure: the risk of neonatal mortality depends upon mortality of the previous sibling and on the preceding birth interval, while the birth interval depends upon survival of the preceding sibling. Similarly, the probability of continuing fertility depends on previous mortality and parity. Identification of the main causal effects rests on exploiting the natural sequencing of the birth and mortality processes, avoiding the need for exclusion restrictions. Amongst other covariates in the model are maternal age at birth of the child, and the year of birth of the child. Together with birth order, these are endogenous because they depend upon the entire history of birth intervals and maternal age at first birth. The model accounts for this endogeneity. A limitation of our approach is that it does not readily extend to analysis of infant or under-5 mortality since, for e.g., infant mortality may occur after the next birth. As discussed in section 1, this is not a strong restriction since neonatal mortality is particularly closely tied to reproductive behaviour.

The mortality equation can be regarded as a health production function in which the

A c c e p t e d m a n u s c r i p t

9 birth interval is an endogenous input, as in Rosenzweig and Schultz (1983a,b). The birth spacing equation is an input equation, but it also describes an outcome that depends upon tastes and technology. These two equations are estimated jointly with an equation for continued fertility that accounts for right-censoring of birth intervals, and an equation for mortality risk of the first-born child, that addresses the initial conditions problem.

The estimation allows for endowments (persistent mother-specific traits), unobservable by the econometrician but potentially known to the mother, and for the agency of the parent in influencing outcomes. The health endowment is referred to as frailty. We also incorporate inter-family unobserved heterogeneity in the birth spacing and fertility equations (for convenience both of these terms are henceforth referred to as fecundity), and allow this to be correlated with frailty since, for e.g., women who are more careful about contraception may also better maintain the health of their children. Ignoring unobserved heterogeneity would bias estimates of the dynamics of each process [START_REF] Heckman | The incidental parameters problem and the problem of initial conditions in estimating a discrete time-discrete data stochastic process[END_REF][START_REF] Hyslop | State dependence, serial correlation and heterogeneity in inter-temporal labor force participation of married women[END_REF]) and of the causal effect of each variable on the other [START_REF] Alessie | Ownership of stocks and mutual funds: A panel data analysis[END_REF]).

The econometric model is an extension of the univariate model of [START_REF] Heckman | The incidental parameters problem and the problem of initial conditions in estimating a discrete time-discrete data stochastic process[END_REF] and is broadly similar to the bivariate model of [START_REF] Alessie | Ownership of stocks and mutual funds: A panel data analysis[END_REF]. Our approach to dealing with right-censoring is new, exploiting data on sterilization. The way in which the initial conditions problem is addressed is also novel. To take account of the sampling design, we use random effects at the community (cluster) level. The model is estimated by simulated maximum likelihood.

Let n i be the number of live births of mother i at the time of the survey. Let M ij be an indicator variable with value 1 if child j in family i suffers neonatal death, and 0 otherwise.

B ij is the log of the length of the interval between the birth of child j-1 and child j in family

Neonatal Mortality

For child j (j=2,…,n i ) in family i (i=1,2,…, N), the equation for neonatal mortality is

M ij * =g( x i , x i1 , x ij , M i1 ,… M i,j-1 , B i2 ,…, B ij ; θ m ) + α mi + u mij ;
(1)

M ij =1 if M ij * >0 and M ij =0 if M ij * <0
To explain the assumptions needed for consistent estimation, it is initially written in a general form. α mi is mother specific unobserved heterogeneity, reflecting the child's health endowment or "frailty", which may derive from genetic sources (e.g. maternal propensities to low birth weight and prematurity), environmental factors, or child-care behaviours (e.g., [START_REF] Sastry | A nested frailty model for survival data, with an application to the study of child survival in Northeast Brazil[END_REF]Rosenzweig and Schultz 1983a,b). As emphasized in [START_REF] Rosenzweig | Heterogeneity, intrafamily distribution and child health[END_REF], the fact that endogenous inputs like breastfeeding are not explicitly incorporated implies that the estimated mother-effect will also reflect inter-family heterogeneity in preferences. The error term u mij varies over mothers as well as children. It is revealed at the birth of child j and assumed not to influence parental inputs to child j in the one month of life during which parental choices can influence neonatal mortality risk. However, we allow u mij-1 to influence parental inputs into child j through past mortality in the family, M ij-1 .

The vectors x i , x i1 , and x ij are exogenous explanatory variables, partitioned into variables that vary over children (x ij , j=2,…,n), are specific to the first child (x i1 ), or do not vary over children (x i ). The vector of unknown parameters is denoted by θ m . The variables We assume that x i , x i1 , and x ij are independent of α mi and u mij . Mean independence of (x i , x i1 ) and α mi is the usual assumption in a random effects model, needed for identification; the conditional mean of α mi given x i and x i1 is subsumed in g. In x i , we A potential drawback of random effects models as compared with fixed effects models is the assumption that the "time-varying" (in this context, varying across siblings) regressors x ij are assumed to be independent of the individual effects α mi . In our case, however, the only variables included in x ij are child gender and birth-order, which will be uncorrelated with mother-level frailty, so the independence assumption seems plausible.

M i1 ,… M i,j-1 , B i2

Birth Spacing

The log length of the birth interval is modeled in a similar way as mortality:

B ij =h( x i , x i1 , x i,j-1 , M i1 ,… M i,j-1 , B i2 ,…, B ij-1 ; θ b ) + α bi + u bij ;
(2)

The family-specific effect, α bi , is referred to as "fecundity" though it will include not only biological fecundity but also, e.g., variation in preferences for family planning or desired fertility. A causal effect of mortality of child j-1 on the birth interval to child j is allowed through M i,j-1 . We include x i,j-1 since gender of child (j-1) may affect the interval to the birth of child j. The function h is specified as a linear combination of x i , x i,j-1 , M i,j-1 , and the calendar year and age of the mother at the time of the birth of child j-1 and their squares, as in section 4.1. Biomedical and demographic research give no argument for a causal effect of B ij-1 on B ij , conditional on α bi , so we do not allow for this. 4 Assumptions on familyspecific effects and errors u bij are similar to those for equation (1). We assume that x i , x i1 , and x ij are independent of α bi and u bij and that u bij is independent of the past.

A c c e p t e d m a n u s c r i p t 13

We allow for correlation between the unobserved heterogeneity terms α bi and α mi in equations ( 1) and ( 2). This allows an alternative, non-causal explanation for the correlation between birth interval lengths and mortality in the raw data and for the potential endogeneity of the preceding birth interval in equation ( 1), which may be correlated with frailty, α mi . For example, parents with weak endowments may choose shorter birth intervals to meet their target number of children in a given time. Similarly, our model allows M ij-1 in equation ( 2) to be correlated with family-level fecundity, α bi .

The distribution of the family effects (α mi , α bi ) is assumed to be bivariate normal with mean zero, variances σ m 2 , σ b 2 , and covariance σ m σ b ρ α . The child-specific error terms u mij and u bij are assumed to be independent of α mi and α fi and normally distributed with mean zero. Without loss of generality, the variance of u mij is set to 1.

Right-Censoring

Inclusion of the birth spacing equation, (2), in the model demands a correction for right-censoring because some mothers will not have completed their fertility at the time of the survey. To account for this, we model the probability that mother i will have another child after the birth of child j, as follows:

F ij * =f( x i , x i1 , x ij , M i1 ,… M i,j-1 , B i2 ,…, B ij ; θ f ) + α fi + u fij ;
(3)

F ij =1 if F ij * >0 and F ij =0 if F ij * <0
We specify f as a linear combination of x i , the calendar year and age of the mother at the time of the birth of child j-1 and their squares (functions of x i1 and B i2 ,…, B ij-1 ), dummies for the presence of boys and girls in the family that did not suffer neonatal death, and the total numbers of boys and girls in the family who survived the neonatal period (functions of The data contain information on whether a mother is sterilized at the time of the survey, which helps to estimate the parameters of the model more efficiently. For sterilized mothers (19.6% of the sample), the complete birth process is observed. Of the remaining mothers, some will have another child after the survey date, and others will not.

Sterilization is an incomplete indicator of whether the mother will have another child; it is an implicit dependent variable in our model but not a variable of interest as such, as it is the decision to have another child or not that is modelled here. To identify equation (3) with data on sterilization, we assume that women who have decided to have no more children get sterilized with a fixed probability κ (a nuisance parameter).

To be precise: If mother i has more than j children, we know she has given birth to another child after child j, and the likelihood will incorporate the probability that F ij =1. If the mother reports that she has had exactly j children and was sterilized after the birth of the j-th child, then the likelihood will incorporate the probability that F ij =0 and the probability κ. If, at the time of the survey, the mother had j children but was not (yet) sterilized, then it A c c e p t e d m a n u s c r i p t 15 is unclear whether child j is the last child or not -the birth interval after the birth of child j may extend beyond the time of the survey. The probability that this will happen, given that there will be another birth and given unobserved heterogeneity components, follows from

(2) and is given by

Φ([T-{h( x i , x i1 , x i,j-1 , M i1 ,… M i,j-1 , B i2 ,…, B ij-1 ; θ b )+ α bi }]/σ),
where T is the length of the time interval elapsed between the birth of child j and the time of the survey, and σ is the standard deviation of the error term in (2). In this case, the likelihood (conditional on unobserved heterogeneity) contains a factor that accounts for the fact that we do not observe whether or not there will be another birth after birth j. 5 The usual approach to right-censoring is to assume that the same process continues but is not observed after the time of the survey (e.g. Wooldridge, 2002, Chapter 20). This approach does not work well here since the fertility process is necessarily finite and ended well before the time of the survey for many women in the sample. 6 In the absence of information on sterilization, an alternative would be to assume that fertility stops at a given age (e.g. 40) for all mothers, or to estimate equation (3) without the sterilization information. In the latter case, the fertility equation would only be indirectly identified and the estimates are likely to be much less precise.

The Initial Conditions Problem

5 An appendix with likelihood details is available at http://www.efm.bris.ac.uk/www/ecsrb/bhalotra.htm 6 Initial experimentation with our data showed that the usual procedure produces a poor fit, being unable to explain why so many women suddenly completely stop having children.

A c c e p t e d m a n u s c r i p t

16 "Lagged" mortality, M ij-1 , is endogenous in equation ( 1) by virtue of being correlated with frailty, α mi . This creates the initial conditions problem common in this type of model (e.g. [START_REF] Heckman | The incidental parameters problem and the problem of initial conditions in estimating a discrete time-discrete data stochastic process[END_REF]). This is addressed by formulating a separate equation for the mortality risk of the first-born child of every mother:

M i1 * = g 1 ( x i , x i1 ; θ m,1 ) + λ m α mi + λ b α bi + λ f α fi + u mi1 ; (4) M i1 =1 if M i1 * >0 and M i1 =0 if M i1 * <0
In most applications of this type of models (e.g., [START_REF] Hyslop | State dependence, serial correlation and heterogeneity in inter-temporal labor force participation of married women[END_REF] the true process is ongoing and the first observation is generated in the same way as later observations. Heckman et al.

( 1985) is an exception. They model birth spacing and observe the process from its natural start, the start of menarche. Here, similarly, we observe the birth and mortality processes from their beginning for each mother, and the first child is a genuine starting point of that process. This makes Heckman's approach quite natural compared to, for example, the alternative approach of [START_REF] Wooldridge | A framework for estimating dynamic, unobserved effects panel data models with possible feedback to future explanatory variables[END_REF].

We will work with a linear specification of g 1 , in line with the specification of (1). It is likely that M i1 will be correlated with α mi and we also allow it to be correlated with the other family specific effects α bi and α fi . The error term u mi1 is assumed to be standard normal and independent of the other error terms in the model, of the individual effects, and of the exogenous regressors x ij and x i . θ m1 , λ m , λ b and λ f are auxiliary parameters. Equation ( 4) is a flexible function of the exogenous variables. We do not impose restrictions on the relation of the parameters in (4) to those in (1).

A c c e p t e d m a n u s c r i p t 4.5 Community Effects and Estimation

The data are collected in 333 geographical clusters ("communities") with, on average, 24.4 mothers per cluster. To allow for the possibility that mothers (and children) within a cluster share unobservable traits (for example, sanitation or social norms), we need to include a cluster-level term in the equation error. 7 As the large number of clusters makes it infeasible to use cluster dummies, we incorporate random cluster effects in equations ( 1) and ( 2) and (3) in the same way as the mother-specific effects, with similar assumptions. 8 A linear combination of the cluster effects in (1), ( 2) and ( 3) is added to equation ( 4), with three additional auxiliary parameters as coefficients. For identification, it is assumed that the cluster effects are independent of mother-specific effects.

The complete model can be estimated by maximum likelihood, including the nuisance parameters of the initial conditions equation, and the fertility equation. 9

Conditional on the random effects, the likelihood contribution of a given mother can be written as a product of univariate normal probabilities and densities over all births of a mother, and the likelihood for a given cluster can be written as the product over all mothers in that cluster. The actual likelihood contribution is the expected value of the conditional likelihood contribution, taking the expectation over all (unobserved) random effects (three 7 The data has some information on community characteristics at the time of the survey. We did not use this since it may not reflect community characteristics at the time of birth.

8 That is, trivariate normal with arbitrary covariance structure to be estimated, independent of exogenous variables and error terms.

9 An explicit specification of the likelihood function is in the online Appendix referred to in footnote 5.

A c c e p t e d m a n u s c r i p t

in the model without cluster effects, six in the model with cluster effects). This is a three or six-dimensional integral, which could in principle be approximated numerically using, for example, the Gauss-Hermite-quadrature.

In this paper, we instead use (smooth) simulated ML, drawing multivariate errors from N(0,I 3 ), transformed into draws of the random effects using the parameters of the random effects distribution. The conditional likelihood contribution is averaged over R independent draws. If R→∞ with the number of clusters, this gives a consistent estimator; if draws are independent across households and R/√N→∞, the estimator is asymptotically equivalent to exact ML (see, e.g., [START_REF] Hajivassiliou | Classical estimation methods for LDV models using simulation[END_REF]. We use Halton draws, which give more accurate results for given R than independent random draws [START_REF] Train | Discrete Choice Methods With Simulation[END_REF].

The results we present use R=100. Using R=50 gives very similar results.

Results

This section presents the results of the complete "benchmark" model (Table 1)

Neonatal Mortality

The left hand panel of Table 1 reports the estimates of the equation for neonatal mortality. In the discussion, we focus on the implied marginal effects for the second child, assuming that the first child survived the first month of life, and setting all family characteristics to benchmark values (boy, Hindu, not of a backward caste, maternal and paternal education zero average date of birth (1986.8), age of mother at birth (21.3 years), and previous log birth interval (3.32). The estimated probability of neonatal mortality for this benchmark child is 4.84%.

A c c e p t e d m a n u s c r i p t

19 A 10% increase in the length of the preceding birth interval reduces the probability of death by about 0.45 percentage-points in the benchmark case (and the marginal effect is similar for higher birth-orders). Some studies have found that the deleterious effects of short birth intervals are enhanced if the previous sibling has survived (e.g. Zenger 1993, Cleland and[START_REF] Sathar | The effect of birth spacing on childhood mortality in Pakistan[END_REF]. We therefore included an interaction of lagged mortality with the birth interval. This was insignificant-a result that contrasts with [START_REF] Whitworth | Birth spacing, sibling rivalry and child mortality in India[END_REF]. Our results suggest that maternal depletion is more important than sibling competition in explaining the mortality-increasing effects of short birth intervals. Maternal depletion will be especially pronounced amongst poor women who need longer to replenish stocks of nutrients like calcium and iron that are needed to support a healthy pregnancy.

Neonatal mortality of the previous sibling makes neonatal death significantly more likely for the index child, even with the birth interval held constant. For the benchmark second child, the estimated difference is 4.16 percentage-points. Similar effects are found for the third and later children. 10 This suggests that, for neonatal mortality, learning effects (a mother is better able to avoid a further child death once she has experienced one) or reduced competition for scarce resources are dominated by state dependence mechanisms that create a positive association of sibling deaths and do not operate via birth spacing. We hypothesize that the loss of a child may create psychological effects that the mother may not have recovered from by the time she conceives her next child, as a result of which there may be physiological effects that make this child more vulnerable both in the womb and 10 A similar positive effect of lagged mortality on current mortality is reported in [START_REF] Whitworth | Birth spacing, sibling rivalry and child mortality in India[END_REF], who do not interpret this finding. They use earlier data for all India and control for unobserved heterogeneity and birth intervals, but take birth intervals to be exogenous.

Birth Spacing

Estimates of the birth spacing equation are in the second panel of Table 1. Since the dependent variable is in logs, the interpretation of the parameters is in terms of percentage changes in the expected length of the birth interval. Note that all covariates in this model refer to the preceding child (i.e. the child born at the start of the birth interval). Neonatal death of the previous child reduces the subsequent birth interval by about 21%, consistent with replacement behaviour (e.g. [START_REF] Ben-Porath | Fertility response to child mortality: Micro data from Israel[END_REF]. Feeding this into equation ( 1), we find that the effect of M ij-1 operating via B ij results in an increase in M ij of about 1.06%-points.

Since the direct effect of M ij-1 on B ij in equation ( 1) is 4.16%-points, total state dependence increases the risk of death by 5.2%-points. Thus genuine state dependence accounts for 37% of the clustering of sibling deaths ("raw persistence"), which was 14%-points on average (section 2). The residual 63% is (observed and unobserved) heterogeneity.

The gender of the last-born child is significant and its sign is consistent with sonpreference. If the last birth was a girl, the expected birth interval is about 3% shorter than if it was a boy. There is a significant hump shaped trend in birth spacing, with a maximum in

Fertility Equation

Table 3 presents estimates of the probability of having another child after each birth.

Of particular interest are the family composition variables, which indicate son-preference.

The probability of continued fertility is decreasing in the number of surviving children, but more than three times as rapidly in the number of surviving boys than in the number of surviving girls. Similar results have been reported for other countries in Asia and North

Africa (e.g. Rahman andDaVanzo 1993, Nyarko et al. 2003). 12 Fertility is hump-shaped in time, with a maximum at about 1981. The quadratic in mother's age is decreasing over the age range, until age 47. Fertility falls with the level of education of both mother and father, with mother's education having larger effects.

Muslims show a higher tendency to continue fertility. Mothers in backward castes other than scheduled castes and tribes have lower fertility than others, not in the raw data, but after conditioning upon covariates.

Unobserved Heterogeneity

11 It is well-known that breastfeeding lengthens birth intervals [START_REF] Habicht | The Contraceptive Role of Breastfeeding[END_REF], and its effects are, in this study, subsumed in the birth interval effect. Direct investigation of the role of breastfeeding is difficult because of incomplete data and since breastfeeding is a choice variable.

A c c e p t e d m a n u s c r i p t 23

We find that the covariance structure of the mother and community-specific effects is sensitive to specification choices (like the number of draws in simulated ML), but the covariance structure of the sum of these terms is not. We therefore focus on the latter (see Table 2). There is significant evidence of both effects in the mortality and birth interval equations, but only community-specific effects are significant in the fertility equation.

Overall, the heterogeneity terms are statistically significant but small compared to the idiosyncratic errors. Compared to the idiosyncratic noise term (with variance 1), the two heterogeneity terms in the mortality equation capture about 15% of the total unsystematic variation in M ij * (0.173/(1+0.173)). More than half of this is heterogeneity across communities. In the birth interval equation, the idiosyncratic noise term has estimated variance 0.206, and total heterogeneity is about 11% of the total unsystematic variation.

The estimated covariance between the total unobserved heterogeneity terms in the birth spacing and mortality equations is virtually zero, implying a correlation coefficient of -0.004.

The heterogeneity terms in the fertility equation explain about 15% of the unsystematic variation in F ij * , but this estimate is not very accurate. We find a large negative correlation between the total heterogeneity terms in the fertility and birth interval equations of -0.73. This suggests that mothers who desire many children tend to use shorter birth intervals to achieve this, other (observed) explanatory variables constant. This is consistent with replacement behaviour in, for example, the target fertility model (see [START_REF] Wolpin | Determinants and consequences of the mortality and health of infants and children[END_REF]. On the other hand, the small correlations between unobserved heterogeneity in the mortality equation and both the birth interval and the fertility equations suggest that hoarding does not play much of a role: there is hardly any evidence that mothers who

Robustness of the state dependence effect and other specification checks

A challenging finding is the strong positive effect of lagged mortality on mortality in equation ( 1), even after unobserved heterogeneity and the length of the preceding birth interval are controlled for. There are three potential sources of positive correlation between lagged mortality and current mortality: unobserved heterogeneity, the causal mechanism operating through the birth interval, and other sources of state dependence. We would probably overestimate the importance of the last one if we underestimated the importance of the first two because of model misspecification. We therefore investigated model extensions generalizing either the specification of unobserved heterogeneity, or the mechanism through which birth intervals affect mortality.

In the first category, we considered auto-correlated error terms, which can be seen as unobserved heterogeneity that changes gradually over time, and a more general distribution of unobserved heterogeneity. In the second category, we considered correlation between errors in birth interval and mortality equations (which may be due to measurement error in reported birth intervals), other functional forms of the relationship between birth intervals and neonatal mortality, and interactions of the birth interval with socio-demographics.

Finally, we also looked at interactions between lagged mortality and socio-demographics in order to see whether the positive effect of lagged mortality can be attributed to specific ). The upshot is that the coefficient on lagged mortality is robust to all these specification checks.

We also investigated robustness of the main coefficients of interest (those on the lagged endogenous variables) to several simplifications of the model. The results are generally as expected. For example, failing to control for unobserved heterogeneity or not controlling for the birth interval leads to overestimation of state dependence in equation ( 1).

See the online Appendix for details.

Simulations

Table 3 shows the results of simulations performed with the benchmark model, to Births as a result of replacement are about 2.6% of all births, that is, 0.37 births for every neonatal death. Because replacement increases mortality, the replacement effect on the number of surviving children is smaller, about 0.30 surviving replacement children for every death. These estimates, although they refer to neonates only, are in line with existing estimates, which lie between 0.2 and 0.5 (Schultz 1997:384-385). In particular, Olsen (1988) finds a replacement effect of 0.35 using Malaysian data, and shows that replacement is greatest for children who die soon after birth.

In addition to suppressing the effects of mortality on birth interval and fertility, the next simulation (col. 4) also suppresses the effect of previous mortality on index child mortality (M ij-1 on M ij ), so that all effects of previous mortality are eliminated. Mainly because of suppressing the replacement effects, the average length of birth intervals increases by 0.50 months and the total number of children born falls by 2.6%. Neonatal mortality falls by about 0.66 %-points, because of suppressing state dependence and the longer birth intervals. Since the negative fertility effect dominates the positive mortality effect on surviving children, the total number of surviving children falls.

In column 5, we further eliminate hoarding. High frailty mothers are now not expected, ex ante, to have shorter birth intervals or higher probabilities of having another child. Thus all behavioural and non-behavioural relations between mortality and birth spacing/fertility are eliminated. The total reduction in neonatal mortality compared to the benchmark is 10.9%, most of which is due to eliminating state dependence. On average, birth intervals are 2.6% longer, mainly because the replacement effect is eliminated. The

Summary and Conclusions

The main findings are as follows. The difference in neonatal mortality according to whether or not the preceding sibling died in the neonatal period is 14 %-points. This is enormous, given that the average risk of neonatal death in our sample is 7.4 percent. We estimate that genuine state dependence accounts for 37% of this, the remaining 63% being explained by inter-family heterogeneity. The analysis confirms that endogenously determined birth spacing is a mechanism generating state dependence in mortality, but it explains only about a fourth of total state dependence. Identification of the mechanisms driving the remaining state dependence is an important avenue for further research, not addressed in the demographic literature. We suggest maternal depression as a possibility.

We find direct evidence of replacement behaviour: a child death results in a shortening of the interval to the next birth, and also increases the probability of a next birth.

Our model simulations imply that, accounting for direct and indirect effects, 37 in 100 children who die during the neonatal period are replaced by new births. Of these, about 30 survive. There is no evidence that frailty is correlated with fecundity. This suggests that couples do not practice hoarding, i.e., we find no evidence that women who know that their children are at relatively large risk of neonatal death anticipate this by reducing the length of their birth intervals ex ante.

As a measure of the importance of allowing for the joint determination of death risk and reproductive behaviour, we estimated the effects on the main outcomes of eliminating

  the effects of expected mortality (hoarding) and realized mortality (replacement) on fertility. Third, we model the mortality dynamics within families, estimating both the extent to which observed persistence in death risk is explained by state dependence, and the contribution of endogenously determined birth-spacing to state dependence effects. Other contributions are methodological, relating to the way in which we deal with right-censoring of birth intervals and with the initial conditions problem that arises in dynamic models with unobserved heterogeneity.

  next child in the family, possibly because the mother has not recuperated from the previous birth (e.g. DaVanzo and Pebley 1993). Thus vulnerable families are caught in a

  Thus B ij refers to the interval closed by the birth of child j. As it is the preceding birth interval for child j, it is, by definition, predetermined with respect to M ij . The rest of this section describes the equations and the model in detail.

  education levels of the mother and father, and caste and religion dummies. In x i1 we also include calendar year and age of mother at first birth.

  i1 ,… M i,j , and B i2 ,…, B ij-1 ). The variables are gender specific to allow for son-preference, of which there is considerable evidence for UP (e.g.[START_REF] Drèze | Uttar Pradesh: The burden of inertia[END_REF]. Endogeneity of the sibship variables is taken care of in the same way as in the other equations -they are functions of lagged dependent variables. Moreover, confounding unobserved factors are controlled for by allowing arbitrary correlations of α fi with α mi and α bi , assuming joint trivariate normality with arbitrary covariance matrix and independence of exogenous variables. We make similar assumptions on u fij as on the other error terms: normality, independence of individual effects and error terms for other birth-orders or other equations, and independence of exogenous variables.

  1980. That birth intervals got shorter in the last 20 years might be because rising nutritional standards for mothers allow them to support shorter intervals. Alternatively, the rise in women's labour force participation may have encouraged clustering of fertility. Birth spacing is hump-shaped in maternal age, with a maximum at about 29 years. For the average mother, birth intervals increase until the sixth child is born. Parental education and caste have no significant effect. Birth intervals of Muslim families are 7.6% shorter than for families. Other things equal, birth-order exhibits a non-monotonic pattern, with the shortest birth intervals preceding the birth of the fourth child. 11

  to have relatively high mortality risk react ex ante by having persistently shorter birth intervals.

  groups. The changes in the estimates are discussed in detail in the online Appendix (see

  investigate the effects of neonatal mortality on birth intervals and fertility, and the importance of state dependence and hoarding. Column 1 is the benchmark simulation where all mechanisms at work in the estimated model are active (see the online Appendix for details). Other columns present percentage deviations from the benchmark for scenarios in which some behavioural or non-behavioural mechanisms are "switched off."Switching off the effect of previous mortality on the birth interval (col. 2) increases average birth interval length by 1.7%. Since longer birth intervals depress subsequent mortality, neonatal mortality falls by almost 3.2%. The longer birth intervals also imply that women have their children later and, through the negative effect of mother's age on the decision to have another child, this reduces the number of births by 0.72%. Due to the fall in mortality, the reduction in the number of surviving children is smaller (0.47%).The next simulation(col. 3) shows what happens if mortality affects neither birth intervals, nor the probability of having another child. Mainly because of longer birth falls by 4.4%. Comparing these results with the benchmark, the total size of the replacement effect can be estimated: total simulated mortality is 7.08% of births.

  children born falls by 3.3%; the number surviving falls by less, because of the reduced mortality probability.

Figure

  Figure 1: Nonparametric (lowess) relation of (predicted) neonatal mortality and the preceding birth interval

  Figure 2: Density of log birth interval by survival status of preceding sibling

e p t e d m a n u s c r i p t 6

  Uttar Pradesh (UP), the largest Indian state, which, in the year 2000, contained 17.1% of the country's population (approximately 165 million people). It has social and demographic indicators that put it well below the Indian average[START_REF] Drèze | Indian Indian Development: Selected Regional Perspectives[END_REF].

	After dropping
	mothers with at least one multiple birth, the sample contains 29,747 live births of 7286
	mothers, that occurred between 1963 and 1999. 2

Table 1 .

 1 The incidence of neonatal death over the sample period in UP was 7.39%,

	compared with an all-India average of 5.21%. Previous research on developing country data
	suggests that preceding birth intervals less than 24, and especially 18, months raise
	mortality risk. In our sample, 17.5% of birth intervals are shorter than 18 months, 18.3%
	2 Elimination of multiple births is in line with the demographic literature on mortality. Children of a
	multiple birth face hugely higher odds of dying, other things equal.

A c c e p t e d m a n u s c r i p t are

  18-23 months long. The mean number of births per mother is 4.04, the median is 4, and the maximum is 14. The mean age of mothers at first birth is 18.4, and the mean age of

	mothers at (any) birth is 22.2. As many as 28.3% of all live births are to mothers under 19
	and 14.3% to mothers under 18.

  , and include quadratic terms in the year of birth of the child, and in the age of the mother at birth of the index child, both of which are functions of x i1 and B i2 ,…, B ij . 3 Since the age of the mother at birth of child j depends upon her age at birth of child j-1 and the length of the intervening birth interval, B ij , it is clear from recursivity of the model that maternal age at birth of j can be expressed as a function of maternal age at first birth (in x i1 ) and the history of birth intervals up until that date (B i2 ,.. B ij ). Thus, by allowing for endogeneity of birth intervals and conditioning on x i1 , we also allow for endogeneity of maternal age. Since our data include births that

,…, B ij are realized at or before the birth of child j.

A c c e p t e d m a n u s c r i p t 11

We will specify g as a linear function of x i , x ij , M i,j-1 , B ij occurred across a span of about 30 years, a quadratic in the year of birth of the child is included to capture technological change in health production. This is also a function of the birth year of the first child and previous birth intervals.

We expect a negative effect of B ij on M ij , consistent with the hypotheses of maternal depletion (section 1) and competition amongst closely spaced siblings

[START_REF] Sathar | The effect of birth spacing on childhood mortality in Pakistan[END_REF]

. The effect of lagged mortality, M i,j-1 , on M ij can be negative if learning or siblingcompetition effects dominate, or positive if there is a strong role for factors like maternal depression. The first-order Markov assumption implicit in our specification of g is justified by the nature of the mechanisms driving state dependence (that is, a causal effect of M ij-1 on M ij ): see

[START_REF] Zenger | Siblings' neonatal mortality risks and birth spacing in Bangladesh[END_REF]

.

  28all behavioural and non-behavioural relations between the mortality process and the birth interval and fertility processes. The predicted reduction in neonatal mortality is 10.9%, most of which is due to eliminating state dependence. On average, birth intervals are 2.6% longer, mainly because the replacement effect is eliminated. The total number of children born falls by 3.3%, while the total number of children surviving the neonatal period falls by 2.5% (because of the reduced mortality probability).Our estimates of fertility behaviour are consistent with son-preference. These are probably the first estimates of son-preference in fertility that allow for endogeneity of mortality. We find that mortality falls with maternal age for most of range of the sample, indicating benefits to interventions that delay first birth (and lengthen birth intervals). The literature is scarce in estimates of maternal age effects on mortality that account for its endogeneity. Maternal education decreases both mortality and fertility, but has no effect on birth spacing. Paternal education depresses the probability of another birth but has no significant effect on the other endogenous variables. There are strong religion effects: 29 highest neonatal death rate in India. Extension of the analysis to consider other Indian states or other developing countries will lend important insight into the mechanisms driving the key relationships analysed here. Many countries in sub-Saharan Africa have higher rates of neonatal mortality and persistently higher fertility than in India. It would be interesting to investigate this "demographic trap". And this is easily done. The Indian survey used in this paper is one of about 70 Demographic and Health Surveys conducted across the developing world (see www.measuredhs.com). The methods used in this paper are therefore immediately applicable to a vast array of countries with different profiles of the structural processes.

	A c c e p t e d Muslims exhibit higher fertility, shorter birth intervals and, yet, lower mortality. In contrast, m a n u s c r i p t fixed effects associated with caste are weak. Conditional upon all covariates, we estimated a trend reduction in mortality of 0.16%-points p.a. during 1963-99, which is about 3.3% of the benchmark probability. Fertility decline in India seems to have set in from 1981. Despite this, birth intervals have got shorter since about then. A c c e p t e d m a n u s c r i p t
	Future work could extend the framework to analyze infant or under-5 mortality.
	This creates the additional complication that mortality events and births can take place in
	overlapping time periods, requiring a different modelling approach. These results are for
	one Indian state, albeit a state with the largest population (166 million in 2001) and the

Table 1 : Model Parameter Estimates

 1 Notes: s.e. denotes standard error. * indicates that the parameter is significant at the two-sided 5% level.

		Neonatal Mortality	Log Birth Interval	Prob(Further Birth)
		Parameter	s.e	Parameter	s.e	Parameter	s.e
	lagged mortality	0.320*	0.068	-0.237*	0.017
	log birth interval	-0.447*	0.050
	Religion:
	Muslim	-0.197*	0.065	-0.076*	0.014	0.379*	0.045
	other	-0.043	0.338	-0.098	0.070	-0.404	0.242
	Caste:
	scheduled caste	0.098	0.136	0.025	0.034	0.101	0.088
	scheduled tribe	0.100	0.054	0.001	0.013	-0.044	0.039
	other backward caste	-0.061	0.052	-0.005	0.012	-0.158*	0.038
	caste missing	0.063	0.099	0.001	0.024	-0.038	0.074
	Maternal education:
	incomplete primary	-0.029	0.094	0.011	0.024	-0.115	0.070
	complete primary	-0.197*	0.093	0.034	0.021	-0.136*	0.059
	incomplete secondary	-0.093	0.099	0.013	0.024	-0.266*	0.067
	secondary & higher	-0.297*	0.144	0.034	0.022	-0.544*	0.062
	Paternal education:
	incomplete primary	-0.004	0.083	0.013	0.020	0.116*	0.057
	complete primary	-0.109	0.076	0.002	0.015	-0.035	0.049
	incomplete secondary	-0.103	0.059	0.004	0.015	-0.098*	0.042
	complete secondary	-0.134*	0.067	-0.001	0.016	-0.198*	0.048
	higher than secondary	-0.031	0.066	0.010	0.016	-0.228*	0.047
	Gender:
	female	-0.043	0.038	-0.028*	0.008
	Birth year of child:
	year of birth of child/10	-0.028	0.62	0.415*	0.121	14.462*	0.850
	(year/10) squared	-0.008	0.036	-0.026*	0.007	-0.895*	0.051
	Maternal age:
	maternal age at birth/10	-0.817*	0.341	0.308*	0.072	-1.195*	0.190
	(age/10) squared	0.143*	0.063	-0.054*	0.015	0.126*	0.033
	Child birth-order:
	birth-order	0.044	0.049	-0.025*	0.008
	square of birth-order	-0.001	0.004	0.003*	0.001
	Surviving children:
	1 if no boys					0.206*	0.049
	1 if no girls					0.147*	0.041
	number of boys					-0.249*	0.025
	number of girls					-0.072*	0.020
	Constant	1.633	2.706	1.402*	0.509	-53.872*	3.420
	sigma error			0.454*	0.002

Table 2 . Unobserved Heterogeneity: Mother plus community level effects

 2

	Covariance matrix
		Mortality Birth interval	Fertility
	Mortality	0.173
	Birth interval	-0.000	0.025
	Fertility	0.003	-0.049	0.181
	Correlation matrix
		Mortality Birth interval	Fertility
	Mortality	1.000
	Birth interval	-0.004	1.000
	Fertility	0.015	-0.725	1.000
	Notes:

See section 5.4 of the text Table 3. Simulations

  Column 1 presents sample averages of the simulated outcomes for the benchmark model. Columns 2-5 show percentage deviations from the benchmark that arise when selected mechanisms are "switched off" as follows: 2: No effect of mortality on birth interval. 3: No effect of mortality on birth interval or probability of having another child. 4: No effect of mortality on birth interval, probability of having another child, or next child's mortality. 5: No effect of mortality on birth interval, probability of having another child, or next child's mortality; no hoarding (i.e. unobserved heterogeneity in the mortality equation is not correlated with unobserved heterogeneity in the equations for birth spacing and fertility).

		(1)	(2)	(3)	(4)	(5)
	Neonatal mortality (%)	7.404	-3.17	-4.40	-8.92	-10.86
	Birth interval (months)	30.586	1.70	1.66	1.66	2.63
	Number of births	4.125	-0.72	-2.59	-2.59	-3.34
	(fertility)
	Number of survivors	3.819	-0.47	-2.25	-1.90	-2.50
	Notes:

[START_REF] Heckman | New evidence on the timing and spacing of births[END_REF] show, for a sample of (married) Swedish mothers, that there is no state dependence in the birth spacing process once controls for unobserved heterogeneity are introduced.

A c c e p t e d m

a n u s c r i p t 20 after birth. Several studies report negative effects of depression on pregnancy outcomes [START_REF] Steer | Self-reported depression and negative pregnancy outcomes[END_REF]) and early childhood mortality [START_REF] Chung | Maternal depressive symptoms and infant health practices among low-income women[END_REF][START_REF] Drewett | Failure to thrive in the term and preterm infants of mothers depressed in the postnatal period: A population-based birth cohort study[END_REF]). [START_REF] Rahman | Impact of maternal depression on infant nutritional status and illness[END_REF] find that maternal depression in the prenatal and postnatal periods is a risk factor for malnutrition and illness in infants in Pakistan. Mental health is increasingly recognised as an important health problem that is often neglected in poor countries [START_REF] Who | The World Health Report. Mental health: New Understanding[END_REF]. Overall, depression is a plausible causal mechanism, but further research is merited

to investigate this and other potential pathways.

Conditional on the other covariates, gender and birth-order are insignificant. Over the period 1963-99, neonatal mortality exhibits a trend reduction of 0.16 percentage-points per year. It is U-shaped in mother's age at birth, a familiar pattern in developing country data.

The minimum is at about 29 years of age. On average, mothers are much younger than this when giving birth to their second child (21.3 years). Mortality risk is decreasing in both maternal and paternal education, with larger effects of maternal education. We find no significant differences between castes. A striking result, that deserves further investigation, is that children of Muslim families (who, on average, have higher fertility, shorter birthspacing and lower socio-economic status) are significantly less likely to die in the first month than Hindu children, with an estimated difference of about 1.7 %-points.

Estimates of the "reduced form" probit equation for mortality of first-born children (equation 4) are available in the online Appendix (see footnote 5). The female dummy is now negative and significant at the two-sided 10% level, consistent with the fact that girls are born with a survival advantage, and with research that shows that discrimination against girls is smallest for first-borns (DasGupta 1990). Other effects are broadly similar to those obtained for second and higher-order births.

A c c e p t e d m a n u s c r i p t

Appendix


Franc ¸ois Chaplais
email: francois.chaplais@ensmp.fr

Nicolas Petit
email: nicolas.petit@ensmp.fr

INVERSION IN INDIRECT OPTIMAL CONTROL

Keywords: Optimal control, indirect methods, adjoint, inversion, nonlinear control theory

In this paper we explain how to use inversion (as defined in nonlinear control theory) for indirect optimal control. Given the relative degree r, it is possible to recover r adjoint states and thus to simplify the problem. Explicit proof is given and relies on the triangular structure of the underlying normal form. An example from the literature is treated in the last section.

Introduction

Geometric tools have long been used in control theory for feedback linearization [START_REF] Isidori | Nonlinear Control Systems[END_REF][START_REF] Nijmeijer | Nonlinear Dynamical Control Systems[END_REF]. The induced change of variables lead to straightforward resolution of inverse problem, i.e. computation of required inputs for a prescribed behavior of outputs. Optimization of the obtainable trajectories is an important topic, especially for applications, but generally requires numerical solvers. Inversion has lately been used in numerical optimal control and the numerical importance of the relative degree of the output chosen when casting the optimal control problem into a numerical collocation scheme was emphasized in [START_REF] Petit | Inversion based constrained trajectory optimization[END_REF]. It yields substantial reduction in required CPU time [START_REF] Milam | A new computational approach to real-time trajectory generation for constrained mechanical systems[END_REF]. The bestcase scenario is full feedback linearization (flatness) as considered in [START_REF] Faiz | Differentially flat systems with inequality constraints: An approach to realtime feasible trajectory generation[END_REF].

Given the system dynamics and an optimal cost, it was shown how to take advantage of the geometric structure of the dynamics to reduce the dimensionality of a numerical collocation scheme. In general collocation methods, coefficients are used to approximate with basis functions both states and inputs [START_REF] Hargraves | Direct trajectory optimization using nonlinear programming and collocation[END_REF]. While it was known since [START_REF] Seywald | Trajectory optimization based on differential inclusion[END_REF] that it is numerically efficient to eliminate the control, it was emphasized in [START_REF] Petit | Inversion based constrained trajectory optimization[END_REF] that it is possible to reduce the problem further. Choosing outputs with maximum relative degrees is the key to efficient variable elimination that makes the required number of coefficients smaller. When combined to a nonlinear programming solver, this can induce drastic speed-ups in numerical solving [START_REF] Milam | Real-time constrained trajectory generation applied to a flight control experiment[END_REF].

In this paper we focus on indirect methods (i.e. adjoint methods) as opposed to the previously discussed direct collocation schemes. We show that in this framework it is also possible to take advantage of the geometric structure of the dynamics. More precisely we show that given a SISO system with a ndimensional state, the relative degree of the system also plays a role in its dual dynamics. We derive an equivalent rewriting of the dynamics of the two-point boundary value problem where we eliminate as many variables as we can. In the case of full feedback linearisability (flatness) the induced equation is a 2n-degree differential equation in the linearizing output.

The article organizes as follows. In section 1 we detail the problem we address (SISO system with generic terminal and integral cost). In section 2 we explain how to recover the geometric structure of the adjoint equation. Further we detail how to take boundary conditions into account for two special cases: terminal cost with or without terminal constraints. In section 3 we treat a numerical example from the literature.

Problem settings

Consider an optimal control problem with dynamics

dx dt = f (x) + g(x)u , u ∈ R, x ∈ R n (1)
where all vector fields and functions are real-analytic.

It is desired to find a trajectory of (1), i.e. [0, t f ] t → (x, u)(t) ∈ R n+1 , that minimizes the performance index

min φ(x(t f )) + t f 0 L(x(t), u(t))dt (2)
where L is a smooth nonlinear function A (SISO) single input single output system dx dt

= f (x) + g(x)u y = h(x) (3)
is said to have relative degree (see for instance [START_REF] Isidori | Nonlinear Control Systems[END_REF])

r at point x 0 if L g L k f h(x) = 0
, in a neighborhood of x 0 and for all k < r-1, and

L g L r-1 f h(x 0 ) = 0 where L f h(x) = n i=1 ∂h ∂xi f i (x)
is the derivative of h along f . Roughly speaking, r is the number of times one has to differentiate y before u appears.

An important result (see [START_REF] Isidori | Nonlinear Control Systems[END_REF]) is as follows. Suppose the system (3) has relative degree r at x 0 . Then r ≤ n. Set

φ 1 (x) = h(x) φ 2 (x) = L f h(x) . . . φ r (x) = L r-1 f h(x)
If r is strictly less than n, it is always possible to find nr more functions φ r+1 (x), ..., φ n (x) such that the mapping

φ(x) =    φ 1 (x) . . . φ n (x)   
has a Jacobian matrix which is nonsingular at x 0 and therefore qualifies as a local coordinates transformation in a neighborhood of x 0 . The value at x 0 of these additional functions can be fixed arbitrarily. Moreover, it is always possible to choose φ r+1 (x), ..., φ n (x) in such a way that L g φ i (x) = 0, for all r + 1 ≤ i ≤ n and all x around x 0 . The implication of this result is that there exists a change of coordinates

x → (z 1 , z 2 , ..., z r , η) ≡ (ζ, η) ∈ R n
such that the systems equations may be written under the following normal form

dz i dt = z i+1 for i = 1 . . . r -1 (4)
dz r dt = b(ζ, η) + a(ζ, η)u (5)
dη dt = q(ζ, η) (6)
where

a(ζ, η) is nonzero for all (ζ, η) in a neighborhood of (ζ, η) = φ(x 0 ).
The inverse change of coordinates gives

x(t) = A(ζ(t), η(t)) (7)
u(t) = B(ζ(t), dz r dt , η(t)) (8)
In the coordinates (ζ, η), (2) is rewritten as

min φ(A(ζ(t f ), η(t f ))) + t f 0 L(A(ζ(t), η(t)), u(t))dt (9)
with dynamics (4,5,6) (when one does not substitute u using ( 8)).

Hamiltonian computations

The Hamiltonian

Because of the triangular structure of the previous normal form, the Hamiltonian of cost (9) with dynamics (4,5,6) is

H(ζ, η, u, λ, µ) = L(A(ζ, η), u) + i=r-1 i=1 λ i z i+1 +λ r [b(ζ, η) + a(ζ, η)u] + µ T q(ζ, η)

Adjoint equations

The necessary conditions of Pontryaguin on the Hamiltonian imply

dλ 1 dt = - ∂L ∂A ∂A ∂z 1 -λ r ∂b ∂z 1 + ∂a ∂z 1 u -µ T ∂q ∂z 1 ( 10
)
dλ i dt = - ∂L ∂A ∂A ∂z i -λ i-1 -λ r ∂b ∂z i + ∂a ∂z i u -µ T ∂q ∂z i for i = 2 . . . r (11)
dµ i dt = - ∂L ∂A ∂A ∂η i -λ r ∂b ∂η i + ∂a ∂η i u -µ T ∂q ∂η i (12)
on the optimal trajectory.

Optimality conditions

Since there is no constraint on the control u, ∂H ∂u is necessary zero along the optimal trajectory. Observe that this partial derivative does not involve the adjoint states (λ 1 , . . . , λ r-1 , µ), i. e. , there is a function G 0 (z 1 , . . . , z r , u, η) such that

∂H ∂u = G 0 (z 1 , . . . , z r , u, η) + λ r a(ζ, η) = 0 (13)
along the optimal trajectory. Inserting ( 8) into (13) shows that there exists F 0 such that

∂H ∂u = F 0 (z 1 , . . . , z r , dz r dt , η, λ r ) = 0 (14)
along the optimal trajectory.

Its derivatives are thus also zero along the trajectory. Their structure is given by the following lemma

Lemma 1. For i > r, let z i (t) = d r-i z r dt r-i = d i-1 z 1 dt i-1 . For i = 1 . . . r -1, the i th time derivative of ∂H
∂u along the optimal trajectory has the structure

d i dt i ∂H ∂u = F i (z 1 , . . . , z r+i+1 , η, λ r-i , . . . , λ r , µ) (15)
where F i is affine with respect to (λ r-i , . . . , λ r , µ) and satisfies

∂F i ∂λ r-i = (-1) i a(z 1 , ..., z r , η) (16)
Proof. Equation (13) gives the desired result for i = 0. The proof proceeds then by induction. Let us assume that (15) holds for i < r -1. Taking one more derivative of (15) and using (8) yields equation (17). This last function is of the form (15).

If F i is affine with respect to (λ r-i , . . . , λ r , µ), then the previous computations show that F i+1 is affine with respect to (λ r-i-1 , . . . , λ r , µ) (H is also affine with respect to the adjoint states). Observe that dλ 1 dt does not appear in the computations since ri > 1. Also, λ r-i-1 only appears once (for j = ri) and the derivative of the right handside expression with respect to it is -∂Fi ∂λ r-i , which proves (16) for i + 1.

d i+1 dt i+1 ∂H ∂u = r+i+1 j=1 ∂F i ∂z j z j+1 + ∂F i ∂η q + r j=r-i ∂F i ∂λ j - ∂L ∂A ∂A ∂z j -λ j-1 -λ r ∂b ∂z j + ∂a ∂z j u -µ T ∂q ∂z j - ∂F i ∂µ (z 1 , . . . , z r+i+1 , η) ∂H ∂η T (ζ, η, λ r , µ) = r+i+1 j=1 ∂F i ∂z j z j+1 + ∂F i ∂η q + j=r j=r-i ∂F i ∂λ j - ∂L ∂A ∂A ∂z j -λ j-1 -λ r ∂b ∂z j + ∂a ∂z j B(z 1 , . . . , z r+1 , η) -µ T ∂q ∂z i - ∂F i ∂µ ∂H ∂η T ≡ F i+1 (z 1 , . . . , z r+i+2 , η, λ r-i-1 , . . . , λ r , µ) (17)
The adjoint states (λ 1 , ..., λ r ) corresponding to the invertible part of the normal form (or upper part) can be eliminated from (15) along the optimal trajectory.

Lemma 2. For i = 0 . . . r -1, there exists a function W i such that

λ r-i = W i (z 1 , . . . , z r+i+1 , η, µ) (18)
along the optimal trajectory at times where a(z 1 , . . . , z r ) is not zero. For concision we shall note λ = W (z 1 , ..., z 2r , η, µ).

Proof. Gathering the conditions H u = 0 and d i dt i

∂H ∂u

= 0 for i = 1, ..., r -1, one get the following system of r equations

F 0 (z 1 , . . . , d r+1 z 1 dt r+1 , η, λ r ) = 0 F 1 (z 1 , . . . , d r+2 z dt r+2 , η, λ r-1 , λ r , µ) = 0 . . . F r-1 (z 1 , . . . , d 2r z dt 2r , η, λ 1 , . . . , λ r , µ) = 0
The system (F i = 0) is linear and triangular with respect to the λ i . Up to the sign, the diagonal term is ∂F0 ∂λ r = a(z 1 , . . . , z r ). Hence the system (F i = 0) has a unique solution with respect to the λ i . For i = 0, equation ( 18) is easily drawn from (13). Consider now F i = 0 for i > 0. From (15) we see that λ r-i is a function of (z 1 , . . . , z r+i+1 , λ r-i+1 , . . . , λ r ); for j < i, we assume recursively that λ r-j is a function W j of (z 1 , . . . , z r+j+1 ), with r + j + 1 < r + i + 1. Hence λ r-i is a function W i of (z 1 , . . . , z r+j+1 ).

For i = r -1, we have

λ 1 (t) = W r-1 (z 1 (t), . . . , z 2r (t), η, µ) (19)
Differentiating this equation and using the adjoint equation on λ 1 leads to

Proposition (Main result).

The optimal output z 1 satisfies the following differential equation of order (2r):

i=2r i=1 ∂W r-1 ∂z i z i+1 + ∂W r-1 ∂η q - ∂W r-1 ∂µ ∂H ∂η T = -µ T ∂q ∂z 1 - ∂L ∂A ∂A ∂z 1 -W 0 (z 1 , . . . , z r+1 , η) ∂b ∂z 1 + ∂a ∂z 1 B(z 1 , . . . , z r+1 , η) (20
)

with z i (t) = d i-1
z1 dt i-1 . Conversely, assume that z 1 satisfies (20); let η a solution of ( 6) and µ a solution of [START_REF] Ross | Pseudospectral methods for optimal motion planning of differentially flat systems[END_REF][START_REF] Seywald | Trajectory optimization based on differential inclusion[END_REF] and ∂H/∂u(ζ, η, u, λ, µ) = 0, then ζ satisfies [START_REF] Hargraves | Direct trajectory optimization using nonlinear programming and collocation[END_REF][START_REF] Isidori | Nonlinear Control Systems[END_REF], λ = Λ and µ satisfies (12).

dµ i dt = - ∂H ∂η i - ∂L ∂A ∂A ∂η i -W 0 (z 1 , . . . , z r+1 , η) ∂b ∂η i + ∂a ∂η i u -µ T ∂q ∂η i (21) with u = z r+1 -b(z 1 , . . . , z r ) a(z 1 , . . . , z r ) (22) Define Λ = W (z 1 , . . . z 2r , η, µ) and ζ = (z 1 , . . . z r ). If λ = (λ 1 , . . . λ r ) satisfies
Proof. The left handside is the time derivative of (19). The right handside is the insertion of (8) and of (18) for i = 0 into [START_REF] Ross | Pseudospectral methods for optimal motion planning of differentially flat systems[END_REF].

The converse part essentially shows that λ can be drawn from z 1 and its derivatives and, of course η and µ. It is closely related to the direct part. From the definition (22) of u, it follows that ζ satisfies [START_REF] Hargraves | Direct trajectory optimization using nonlinear programming and collocation[END_REF][START_REF] Isidori | Nonlinear Control Systems[END_REF]. Since ∂H/∂u = 0, then λ r satisfies (13), and hence, λ r = W 0 (z 1 , . . . , z r+1 , η) = Λ r . Together with (21), it follows that µ satisfies (12). Since ∂H/∂u = 0, it follows from lemmas 1 and 2 that λ = W (z 1 , . . . , z 2r , η, µ) = Λ.

In other words the 1+2(n-r) unknowns to be found are z 1 , η, and µ. Then the Λ as defined above are appropriate candidates for the adjoint states corresponding to (z 1 , ...z r ) (the only adjoint states that are missing).

Endpoint conditions

With the proposed terminal cost φ(x(t f )) = φ(A(z 1 , ..., z r , η)), the terminal conditions on the adjoint states are then

λ i (t f ) = ∂ ∂z i φ(A(z 1 (t f ), ..., z r (t f ), η(t f ))), for i = 1, ..., r
That can be rewritten in terms of conditions upon the derivatives of the flat output

W i (z 1 (t f ), . . ., z r+i+1 (t f ), η(t f ), µ(t f )) = ∂ ∂z r-i φ(A(z 1 (t f ), ..., z r (t f ), η(t f ))), for i = 1, ..., r

Endpoint equality constraints

Following Bryson and Ho's [START_REF] Bryson | Dynamic optimization[END_REF] presentation of optimal control problems we explain how to treat a special case of practical interest. Our proposed approach applies in this situation where extra adjoint states appear.

Let us consider the original problem (1), (2) with the additional constraint

ψ(x(t f )) = 0. ( 23
)
Where ψ is a smooth function from R n → R q , 0 < q < n.

Introducing q extra multipliers π ∈ R q we get the terminal conditions

λ i (t f ) = ∂ ∂z i φ(A(z 1 (t f ), ..., z r (t f ), η(t f ))) + π T ∂ ∂z i ψ(A(z 1 (t f ), ..., z r (t f ), η(t f ))) (24)
for i = 1, ..., r along with the q terminal constraints (23). Under the assumption that one can extract a non singular q ×q matrix out of the Jacobian of ψ (which is a classical assumption to make (23) solvable), it is possible to isolate an invertible set of q equations in the q unknows π from the set of equations (24). This means that these multipliers also write in terms of z 1 and its successive derivatives, η and µ. Let (i 1 , ..., i q ) be a set of indexes such that the q × q matrix extracted from the Jacobian of ψ is invertible, let M q denote this matrix extracted from ∂ ∂z ψ(A(z 1 (t f ), ..., z r (t f ), η(t f )))

This matrix depends on z 1 (t f ), ..., z r (t f ), η(t f ). Let W denote the vector (W i k ), for k = 1, ..., q and z the vector (z i k ), for k = 1, ..., q. Then one can solve

π = M -1 q W (z 1 (t f ), ..., z r (t f ), η(t f ), µ(t f )) T - ∂ ∂ z φ(A(z 1 (t f ), ..., z r (t f ), η(t f )))

Example

To illustrate the previous result we derive here the optimality conditions for the forced van der Pol Problem.

The dynamics of this system discussed in [START_REF] Milam | A new computational approach to real-time trajectory generation for constrained mechanical systems[END_REF] and [START_REF] Ross | Pseudospectral methods for optimal motion planning of differentially flat systems[END_REF] is given by

ẋ1 = x 2 ẋ2 = -x 1 + (1 -x 2 1 )x 2 + u
It was used in the last two cited references as a benchmark.

The optimal control problem is to minimize the quadratic cost function

J = 5 0 (x 2 1 (s) + x 2 2 (s) + u 2 (s))ds
subject to the dynamics given above and the endpoint constraints

x 1 (0) = 1, x 2 (0) = 0, x 2 (5) -x 1 (5) = 1
The dynamics is flat and z 1 = x 1 is a flat output (ideal case). Following the procedure presented in the previous section we get the Hamiltonian

H = 1 2 (x 2 1 + x 2 2 + u 2 ) + λ 1 x 2 + λ 2 (-x 1 + (1 -x 2 1 )x 2 + u)
The adjoint equations are

λ1 = -H x1 = -(x 1 -2λ 2 x 1 x 2 ) (25) λ2 = -H x2 = - x 2 + λ 1 + λ 2 (1 -x 2 1 )
Solving the two equations H u = 0 and dH u dt = 0 for the adjoint states we get

λ 1 =x (3) 1 + ẋ1 + 2x 1 ẋ2 1 -(1 -x 2 1 )ẍ 1 -ẋ1 - -x (2) 1 -x 1 + (1 -x 2 1 ) ẋ1 (1 -x 2 1 ) (26) λ 2 = -x (2) 1 -x 1 + (1 -x 2 1 ) ẋ1 (27)
Now one just has to substitute the expression for λ 1 and λ 2 into the differential equation (25) to get the fourth order differential equation to be satisfied by the flat output on the optimal trajectory x (4)

1 + 2x 3 1 + 6x 1 ẋ1 ẍ1 + 2x 1 ẋ1 -(1 -x 2 1 )ẍ 1 (1 -x 2 1 ) + 2x 1 + ẍ1 = 0
Finally one gets the boundary conditions

x 1 (0) = 1 ẋ1 (0) = 0 ẋ1 (5) -x 1 (5) = 0 x (3) 1 (5) + ẋ1 (5) + 2x 1 (5) ẋ2 1 (5) -(1 -x 2 1 (5))ẍ 1 (5) -ẋ1 (5) -(-x (2) 1 (5) -x 1 (5) + (1 -x 2 1 (5)) ẋ1 (5))(1 -x 2 1 (5)) = x (2) 1 (5) + x 1 (5) -(1 -x 2 1 (5)) ẋ1 (5)
the last relation being just the endpoint condition λ 1 (5) + λ 2 (5) = 0 expressed in terms of the flat output. Solving this two point boundary value problem can be done with any appropriate numerical solver, e.g. Matlab bvp4c.

Results give solutions comparable to the one obtained in [START_REF] Milam | A new computational approach to real-time trajectory generation for constrained mechanical systems[END_REF] and [START_REF] Ross | Pseudospectral methods for optimal motion planning of differentially flat systems[END_REF], the obtained cost is 1.68588.

The adjoint states λ 1 and λ 2 also writes in terms of the flat output along the optimal trajectory as given by ( 26) and ( 27).

Conclusion

In summary, what we have shown is the following: if the system has relative degree r, then r of the adjoint states can be explicitly recovered. To do so one just has to rewrite the problem in its normal form and then write that H u and its first r -1 time derivatives are zero along the optimal trajectories.

As stated in the main result of the paper, finding an optimal trajectory is equivalent to finding time functions z 1 (t) ∈ R, η(t) ∈ R n-r and µ(t) ∈ R n-r that satisfy (20), ( 6) and (21).

We have fewer unknowns than the classical n states and n adjoint states. We are now left with 1 + 2(nr) unknowns. The ideal case is (as for collocation methods) the fully invertible case (flat case) where r = n, as treated in the example in the previous section. In any case we believe this structure could be exploited in implementation of shooting techniques (see [START_REF] Bonnans | The shooting algorithm for optimal control problems: A review of some theoretical and numerical aspects[END_REF] for an overview of shooting methods) as it involves fewer unknowns. Numerical relevance of this approach is currently under investigation.