Beyond unidimensional poverty analysis using distributional copula models for mixed ordered-continuous outcomes
Maike Hohberg, Francesco Donat, Giampiero Marra, Thomas Kneib
BBeyond unidimensional poverty analysis using distributionalcopula models for mixed ordered-continuous outcomes
Maike Hohberg ∗ , Francesco Donat † , Giampiero Marra and Thomas Kneib Chair of Statistics, University of Goettingen, Germany Single Resolution Board, Brussels, Belgium Department of Statistical Science, University College London, UKSeptember 9, 2020
Abstract
Poverty is a multidimensional concept often comprising a monetary outcome and other welfaredimensions such as education, subjective well-being or health, that are measured on an ordinalscale. In applied research, multidimensional poverty is ubiquitously assessed by studyingeach poverty dimension independently in univariate regression models or by combining severalpoverty dimensions into a scalar index. This inhibits a thorough analysis of the potentiallyvarying interdependence between the poverty dimensions. We propose a multivariate copulageneralized additive model for location, scale and shape (copula GAMLSS or distributionalcopula model) to tackle this challenge. By relating the copula parameter to covariates, wespecifically examine if certain factors determine the dependence between poverty dimensions.Furthermore, specifying the full conditional bivariate distribution, allows us to derive severalfeatures such as poverty risks and dependence measures coherently from one model for differentindividuals. We demonstrate the approach by studying two important poverty dimensions:income and education. Since the level of education is measured on an ordinal scale whileincome is continuous, we extend the bivariate copula GAMLSS to the case of mixed ordered-continuous outcomes. The new model is integrated into the
GJRM package in R and appliedto data from Indonesia. Particular emphasis is given to the spatial variation of the income-education dependence and groups of individuals at risk of being simultaneously poor in botheducation and income dimensions. Although poverty is widely regarded a multidimensional phenomenon and poverty measures movingbeyond a single monetary dimension – such as the Multidimensional Poverty Index (MPI, Alkireet al., 2012) – have emerged, little progress has been made on analysing poverty as a multidimen-sional concept. To study poverty at the micro level, univariate linear regression is the standardtool of the empirical economist. Despite their widespread use, however, univariate models for ∗ Corresponding author: [email protected]. The author received funding from the Ministry for Scienceand Culture of Lower Saxony as a part of the project “Reducing Poverty Risks in Developing Countries” and theGerman Science Foundation within the research project KN 922/9-1. † This paper should not be reported as representing the views of the Single Resolution Board. The views expressedare those of the authors and do not necessarily reflect those of the Board. a r X i v : . [ s t a t . M E ] S e p overty analyses require either studying each poverty dimension separately in different equations,or using as response variable an index that subsumes all dimensions in a single number (e.g. Alkireand Fang, 2018). Both approaches neglect the interdependence between poverty dimensions andignore that the dependence itself should be part of the analysis. In fact the level of poverty andwell-being depends on the strength of the dependence (Duclos et al., 2006): for example, lowertail dependence can explain persisting poverty where performing low in one dimension is stronglyassociated with a low outcome in the other dimensions.To overcome such limitations, multivariate regression can be used to tackle multidimensionality inpoverty analyses. The relationship between two or more outcomes can also modeled using copulaswhich have been proven to be useful and flexible tools in this regard (see Nelsen, 2006, for anintroduction to copula theory). A second issue in poverty analysis concerns distributional aspects.Especially for program targeting and risk factor analysis, it is important that poverty studies movebeyond the simple mean effects. In fact, concepts like vulnerability to poverty – a forward-lookingmeasure of individuals’ exposure to poverty – look at both the location and scale of the targetdistribution. Previous studies on vulnerability to poverty used a step-wise procedure to explicitlymake the scale parameter dependent on covariates (see Günther and Harttgen, 2009; Calvo andDercon, 2013; Thi Nguyen et al., 2015; Zereyesus et al., 2017, for recent works). Another exampleis inequality, which has become growingly relevant for both the political agenda and for projectsimplemented in developing countries. The World Bank, for example, centers its shared prosperityinitiative around the goal to reduce inequality (World Bank, 2018). Hence, it is necessary to analysenot only effects on the mean but also on the other parameters characterising the distribution of theoutcomes of interest. Generalized additive models for location, scale, and shape (GAMLSS, Rigbyand Stasinopoulos, 2005) are able to capture the effects of covariates on the whole conditionaldistribution of a single poverty dimension.Both issues of multidimensionality and distributional aspects can be addressed with a combinationof GAMLSS and multivariate copula models, also referred to as copula GAMLSS. These modelsare implemented in the R package GJRM (Marra and Radice, 2019) and comprise a wide range ofpotential marginal distributions (continuous, binary, discrete) and copulas. A Bayesian version ofthis model class is implemented in the software
BayesX (Belitz et al., 2015) while Klein and Kneib(2016) provide the related literature. The advantage of embedding copula regression into GAMLSSis that each parameter of the marginals and the copula association parameter can be modeled todepend flexibly on covariates. This allows us to not only measure the strength of the dependence,which has been the focus of previous literature on interrelated poverty dimensions, but also toanalyse which factors related to household location and composition drive this dependence. Thislatter aspect has not been previously considered in poverty studies.When studying poverty, it often occurs that one dimension is reported in ordered categories whereasthe other is continuous. For example, two possible dimensions of interest could be income (mea-sured on the continuous scale) and the highest level of education, which is often assessed in orderedcategories such as “no schooling”, “elementary school”, “high school”, and “higher education”. Thisis a very relevant case, especially in economics and poverty research where several outcomes aremeasured on the ordinal scale (health, education, subjective well-being, etc.).The aim of this paper is twofold. First, to theoretically extend copula GAMLSS to a mixed ordered-continuous case. Second, to practically demonstrate how multidimensional poverty analysis canbenefit from flexible models that allow for covariate effects on the interdependence between thepoverty dimensions. 2or the theoretical part, we rely on the latent variable approach relating the ordered categoriesto an underlying continuous variable as in Donat and Marra (2018). In this way we can followthe approach developed in Marra and Radice (2017), which estimates the copula dependence andmarginal distribution parameters simultaneously within a penalized likelihood framework using atrust region algorithm. The new model is incorporated into the R package GJRM (Marra and Radice,2019).Regarding the application to multidimensional poverty, there is an extensive literature dealingwith the measurement of multidimensional poverty. Yet, the methods proposed for analyzingmultidimensional poverty, including its determinants and poverty profiles, are rather limited. Forexample, Alkire et al. (2004) suggest employing Generalized Linear Models using a single numberindex as the response variable. To demonstrate how a more comprehensive poverty analysis can beconducted by researchers, the empirical study in this paper applies copula GAMLSS in this context.Our application deals with two important poverty dimensions that are interrelated: income andeducation. In many developing countries, there is potentially a vicious cycle of poor education andlow income. This cycle is also called poverty trap and is a long-established concept in economics:capable children stay under-educated due to their parents’ restricted resources and hence remainpoor when grown-up (Barham et al., 1995). Understanding what determines the interdependencebetween poverty dimensions helps designing strategies to interrupt this cycle. To this end, wemodel the income-education dependency in Indonesia and draw an in-depth picture of monetaryand education poverty across the population. We address the following questions: 1) Whichfactors determine the distributions of household income per capita and individual education andtheir interdependence? 2) How does this dependence differ spatially across Indonesia? 3) What arethe probabilities of being poorly educated and income poor for different population’s sub-groups?We will answer these questions using a rich dataset from Indonesia which is made publicly availableby the RAND corporation (RAND, 2017).The dependencies between different poverty dimensions have been widely addressed in the eco-nomics literature during the last two decades. However the literature on using copulas to modelmultidimensional poverty is scarce and, to the best of our knowledge, restricted to the measurement of the strength of such dependence. Existing approaches do not place the model into a regressionframework and hence neither relate the copula association parameter nor the other parameterscharacterising the marginal distributions to covariates. For example, Quinn (2007) quantifies thedependence between income and an ordinal health measure in four industrial countries. Decancq(2014) uses copula models to measure dependence over time between income, health, and school-ing (all of them assumed to be continuous) in Russia. A similar approach was used by Perezand Prieto (2015) to study the dependence between income, material needs and work intensity inSpain. Kobus and Kurek (2018) analyse the distributions of health and education. In contrast toQuinn (2007) and this paper, that make use of a latent variable approach to represent the orderedcategories of education, Kobus and Kurek (2018) overcome the unidentifiability issue when usingcopulas with discrete marginals by concordance ordering. In a Bayesian context, Tan et al. (2018)re-construct the MPI using a one-factor copula model and data from East-Timur. These exam-ples emphasize once more the importance of extending copula GAMLSS also to the case of mixedordered-continuous outcomes when these models are applied to poverty analyses.The remainder of the paper is organised as follows: Section 2 introduces a bivariate copulaGAMLSS for mixed ordered and continuous outcomes. Section 3 presents the estimation pro-cedure. Finally, Section 4 studies poverty dimensions with copula GAMLSS using data from3ndonesia and discusses practical approaches to model selection. Section 5 concludes the paper.
The model considered in this paper deals with a pair of random variables, ( Y , Y ) (cid:48) , with support R × R , where ( R , (cid:22) ) is a totally ordered set under the ordering relation (cid:22) . The elements of R aredenoted by r and represent the levels of the categorical variable Y , namely R := { , . . . , r, . . . , R +1 } with R + 1 < ∞ . The variable Y is assumed to be continuous. In the case study of Section4, response Y will represent the income and Y the highest level of education attained by eachindividual surveyed.We are interested in building up a statistical model for the joint distribution of the response vari-ables ( Y , Y ) (cid:48) where their dependence structure is represented by means of a copula specification.The bivariate cumulative distribution function can then be written as F ( r, y ) = C ( F ( r ) , F ( y )) ∈ [0 , , (1)where the copula function is C : [0 , −→ [0 , , with F ( r ) := P ( Y (cid:22) r ) and F ( y ) := P ( Y ≤ y ) being the marginal distributions. A significant advantage of the copula representation is that itdecomposes the joint distribution into two marginals distributions, that may come from differentfamilies, and a copula function C that binds them together. The dependence structure of the twomarginals is captured by an association parameter γ that is specific to the copula employed asdescribed below.If both F and F are continuous, Sklar’s theorem ensures that the copula function is uniquelydetermined (Sklar, 1959). However, since Y is categorical in our case, the uniqueness of the copuladoes not apply directly. We address this limitation by representing the ordinal variable as a coarseversion of a latent continuous variable.Let Y ∗ ∈ R denote the unobserved (or latent) continuous variable that drives the decision for theobserved categories in R . This continuous latent variable can be modeled as Y ∗ = x (cid:48) β + (cid:15) , (cid:15) iid ∼ N (0 , , (2)where β is a vector of regression coefficients, x a vector of covariates, and (cid:15) the error term withdensity f ∗ and cumulative distribution function (CDF) F ∗ . Later on, the latent variable in (2)will be placed into the more sophisticated GAMLSS framework, but this model formulation withonly linear effects shall serve as a starting point. In line with McKelvey and Zavoina (1975), thefollowing observation rule linking the latent to the observed variable is applied: { Y = r } ⇐⇒ { θ r − < Y ∗ ≤ θ r } , r = 1 , . . . , R + 1 , (3)where θ r is a cut point on the latent continuum related to the level r of Y . We observe category r if the latent variable is between the cutoffs θ r − and θ r . There is a total of R + 2 cut points: −∞ = θ < θ < . . . < θ R +1 = ∞ . However, only R of them are estimable, namely { θ , . . . , θ R } .To guarantee the monotonicity of the cut points, we apply the transformation θ ∗ := θ and θ ∗ r := θ r − θ r − for any r > (Donat and Marra, 2017). This implies that θ r ≥ θ r − for any r ∈ R and θ r ∈ R . However, the equality θ r = θ r − can be problematic in practice because it resultsin estimated parameters at the boundary of the parameter space. This happens, for example,wherever a given level of Y has no observations in the sample (Haberman, 1980).From equation (2) and (3), we derive the cumulative link model P ( Y (cid:22) r ) = P ( Y ∗ ≤ θ r ) = P ( (cid:15) ≤ θ r − x (cid:48) β ) := F ∗ ( θ r − x (cid:48) β (cid:124) (cid:123)(cid:122) (cid:125) := η r ) , (4)where η r is the predictor associated with the ordinal categorical response in the model. It dependson the observed level r of y through cut point θ r . In Section 2.2 the predictor η r will be replacedwith a generalized additive form. With this information in hand, equation (1) can equivalently bewritten as F ( r, y ) = F ∗ ( η r , y ) = C ( F ( r ) , F ( y )) = C ( F ∗ ( η r ) , F ( y )) . (5)Since both marginals are now continuous, the applicability of Sklar’s theorem in ensured.Finally, deriving the analytical form of the density function f ∗ yields f ∗ ( η r , y ) = ∂ C ( F ∗ ( η r ) , F ( y )) f ( y ) ∂F ( y ) for r = 1 (cid:18) ∂ C ( F ∗ ( η r ) , F ( y )) ∂F ( y ) − ∂ C ( F ∗ ( η r − ) , F ( y )) ∂F ( y ) (cid:19) f ( y ) for < r ≤ R + 1 . This will form the basis for the derivation of the penalized log-likelihood function in Section 3.2.
The bivariate copula model is embedded into the distributional regression framework to modelflexibly both the dependence parameter and the marginal distributions. To this end, the responsevector y i = ( y ∗ i , y i ) (cid:48) , i = 1 , . . . , n, is assumed to follow a parametric distribution where potentiallyall parameters, except of the cut-points, are related to a regression predictor and consequently tocovariates. We write the joint conditional density as f ∗ ( ϑ i , . . . , ϑ Ki | ν i ) , where the vector ν i collects any covariates associated to the parameters ϑ ki , k = 1 , . . . , K of density f ∗ . Accordingly,the distributional parameter vector ϑ i = ( θ ∗ , . . . , θ ∗ R , ϑ i , . . . , ϑ Ki ) (cid:48) includes the transformed cut-points { θ ∗ r } , the location parameter of the first marginal distribution, all other distributionalparameters related to the second marginal distribution, and the copula parameter γ i . Subscript i attached to parameters is made explicit to stress their potential dependence on individual-levelcovariates. For the ordinal response, logit and probit link functions can be applied and the scaleparameter for density f is set to one in order to achieve identification as for a probit/logit model.The second marginal distribution can be selected from a wide range of options that are availablein GJRM and listed in Marra and Radice (2017). At the current stage, some of them are notimplemented for the mixed-ordinal case, but will be made available in the near future. In thispaper we only consider one-parameter copulas; some available options are summarized in Table 4(Appendix A) although rotated versions are also implemented in
GJRM . Since the copula parameter γ i is not directly comparable over different models, we relate it to the Kendall’s τ which can beused for interpreting the dependence. For optimisation and modelling purposes, an appropriate5ransformation of the copula parameter, γ ∗ i , is used in the estimation algorithm as highlighted inthe last column of Table 4 (Appendix).In the spirit of the GAMLSS approach, each distributional element in the parameter vector isrelated to an additive predictor via ϑ ki = h k ( η ϑ k i ) and η ϑ k i = g k ( ϑ ki ) , (6)where η ϑ k i ∈ R is the predictor belonging to distributional parameter ϑ ki , and h k = g − k is aresponse function mapping the real line into the domain of ϑ ki .For the ordinal response, η ri in equation (4) can now be represented as η µ ri = θ r − η µ i , where η µ i is a predictor as in (6). The predictor η ϑ k i takes the additive form η ϑ k i = J k (cid:88) j =1 s ϑ k j ( ν i ) , where functions s ϑ k j ( ν i ) , j = 1 , . . . , J k , can be chosen to model a range of different effects of (asubset) of explanatory variables ν i . In particular, • Linear effects are represented by setting s ϑ k j ( ν i ) = ν ϑ k ji β ϑ k j , where ν ϑ k ji is a singleton elementof ν i and β ϑ k j a regression coefficient to be estimated. For the second marginal, this alsoincludes an intercept with s ϑ k j ( ν i ) = β ϑ k to denote the overall level of the predictor whilefor the ordinal equation the intercept is already accounted for by the cut-point θ r . • For continuous covariates, nonlinear effects are achieved by including smooth functions s ϑ k j ( ν i ) represented by penalized regression splines. Ruppert et al. (2003) and Wood (2017)provide various definitions and options for computing basis functions and related penalties. • An underlying spatial pattern can be accounted for by specifying spatial information suchas geographical coordinates or administrative units in ν i . Smoothing penalties can accountfor the neighbourhood structure and ensure that effects are similar for adjacent regions. Rueand Held (2005) interpret this penalty as the assumption that the vector of spatial effects forall regions follows a Gaussian Markov random field. • If the data are clustered, random effects s ϑ k j ( ν i ) = β ϑ k jc i can be included with c i denoting thecluster the observations are grouped into. From the analytical expression of the bivariate density f given in Section 2.1, the model’s log-likelihood function is derived as (cid:96) ( β ) = n (cid:88) i =1 (cid:32)(cid:88) r ∈R { y i = r } (log { F . ( η ri , y i ) − F . ( η r − i , y i ) } ) + log { f ( y i ) } (cid:33) , (7)6here {·} is a Boolean operator that takes on value if condition {·} is verified, and otherwise.We define F . ( η ri , y i ) := ∂ C ( F ∗ ( η ri ) , F ( y i )) ∂F ( y i ) with F . ( η , − ,i , y i ) = 0 . The log-likelihood function is maximized with respect to the complete vector of regression coef-ficients β = ( θ ∗ , . . . , θ ∗ R , β ϑ , . . . , β ϑ K ) (cid:48) . Each vector of regression coefficients β ϑ k includes thecoefficients for one parameter ϑ k .Embedding the model into the distributional regression framework with highly flexible predictors,including regression spline components, typically requires penalization to avoid overfitting. Thepenalized log-likelihood (cid:96) p ( β ) with ridge-type penalty can be written as (cid:96) p ( β ) = (cid:96) ( β ) − β (cid:48) S λ β , (8)where S λ is a block diagonal matrix consisting of the penalties associated to each model parameter.For un-penalized parameters (like the cut points or categorical covariates) the corresponding blockof S λ is set to . Penalty matrices are associated with smoothing parameters λ = ( λ ϑ , . . . , λ ϑ K J K ) (cid:48) . Marra and Radice (2017) proposed maximizing the penalized likelihood in equation (8) using atrust region algorithm with integrated automatic selection of the smoothing parameters. As inRadice et al. (2016), Marra and Radice (2017) and Klein et al. (2019), the estimation proceeds intwo steps:
Step 1.
At iteration a , equation (8) is maximized for a given parameter vector β [ a ] holding λ [ a ] fixed at a vector of values. A trust region algorithm is applied as follows β [ a +1] = β [ a ] + argmin p : (cid:107) p (cid:107)≤ ∆ [ a ] ˘ (cid:96) p ( β [ a ] ) (cid:124) (cid:123)(cid:122) (cid:125) := p [ a +1] , (9) ˘ (cid:96) p ( β [ a ] ) := −{ (cid:96) p ( β [ a ] + p (cid:48) g p ( β [ a ] ) + 12 p (cid:48) H [ a ] p p } , where the Euclidean norm is denoted by (cid:107) · (cid:107) and ∆ [ a ] is the radius of the trust region. The radiusis adjusted in each iteration (see Geyer, 2015, for details). The gradient vector at iteration a isgiven by g [ a ] p = g [ a ] − S λ β [ a ] is and H [ a ] p = H [ a ] − S λ is the Hessian matrix - both penalized bymatrix S λ .The vector g ( β [ a ] ) consists of g [ a ] ( β [ a ] ) = (cid:32) ∂(cid:96) ( β ) ∂θ ∗ (cid:12)(cid:12)(cid:12)(cid:12) θ ∗ = θ ∗ [ a ]1 , . . . , ∂(cid:96) ( β ) ∂θ ∗ R (cid:12)(cid:12)(cid:12)(cid:12) θ ∗ R = θ ∗ [ a ] R , ∂(cid:96) ( β ) ∂ β ϑ (cid:12)(cid:12)(cid:12)(cid:12) β ϑ = β ϑ [ a ]1 , . . . , ∂(cid:96) ( β ) ∂ β ϑ K (cid:12)(cid:12)(cid:12)(cid:12) β ϑK = β ϑ [ a ] K (cid:33) (cid:48) and the elements of the Hessian matrix are H ( β [ a ] ) l,m = ∂ (cid:96) ( β ) ∂ β l ∂ β m (cid:48) (cid:12)(cid:12)(cid:12)(cid:12) β l = β l [ a ] , β m = β m [ a ] , l, m = ϑ , . . . , ϑ K . The second-order partial derivatives of the log-likelihood with respect to cut points θ ∗ , . . . , θ ∗ R (cid:96) p ( β [ a ] ) , and the solution p [ a +1] is chosen such that it falls within a trust regionwith centre β [ a ] and radius ∆ [ a ] . Step 2.
Holding the parameter vector value fixed at β [ a +1] , the following problem is solved λ [ a +1] = argmin λ (cid:107) M [ a +1] − A [ a +1] M [ a +1] (cid:107) − Kn + 2tr( A [ a +1] ) , (10)where, after defining I [ a +1] = − H [ a +1] , the key quantities are M [ a +1] = (cid:113) I ( β [ a +1] ) β [ a +1] + (cid:113) I ( β [ a +1] ) − g ( β [ a +1] ) , A [ a +1] = (cid:113) I ( β [ a +1] )( I ( β [ a +1] ) + S λ ) − (cid:113) I ( β [ a +1] ) , tr( A [ a +1] ) is the number of effective degrees of freedom (edf) of the penalized model while K is thenumber of penalized parameters in vector ϑ . The expression in (10) is solved using the methodproposed by Wood (2004). The gradient vector g and the Hessian H are obtained as a side productin step 1. Both are analytically derived in a modular fashion for each parameter, see Appendix Cfor details.Step 1 and 2 are iterated until they no longer improve the objective function, that is until thefollowing criterion is met: | (cid:96) ( β [ a +1] ) − (cid:96) ( β [ a ] ) | . | (cid:96) ( β [ a +1] ) | < e − . . To obtain the starting values for the marginals’ parameters and the cut-off value, a generalizedadditive model is fitted using gam() (Wood, 2017) or a GAMLSS using the gamlss() functionwithin the
GJRM package. A transformed Kendall’s τ between the responses is used as a startingvalue for the copula parameter. Further details on the trust region algorithm and smoothingparameter selection can be found in Appendix B while asymptotic considerations on the proposedmaximum penalized likelihood estimator are reported in Appendix D. At convergence, reliable point-wise confidence intervals are constructed based on Bayesian largesample approximation as in Wood (2017), for generalized additive models (GAM), i.e. ˆ β a ∼ N ( β , − H p (ˆ β ) − ) . The result for the Bayesian covariance matrix V β = − H − p is an alternative to the frequentistcovariance matrix V ˆ β = − H − p HH − p . For unpenalized models, the two matrices are equal.Applying the Bayesian framework to the GAM or copula GAMLSS context, follows the notion thatpenalisation in the estimation implicitly assumes certain prior beliefs about the model’s features(Wahba, 1978). In this view, a normal prior for the parameter vector β , i.e. f β ∝ exp( − / β (cid:48) S λ β ) means that wiggly models are less likely than smoother ones (Wood, 2006). Marra and Wood (2012)give a full justification for using the above approximation and show that V β gives close to across-the-function frequentist coverage probabilities since it includes bias and variance components in afrequentist sense, which is not the case for V ˆ β . 8o obtain intervals for non-linear functions of the model parameters (e.g. Kendall’s τ ), Radice et al.(2016) simulate from the posterior distribution of β and give examples of interval construction.They propose the following procedure: Step 1
Draw n sim random vectors ˜ β m , m = 1 . . . , n sim , from N (ˆ β , ˆ V β ) . Step 2
Calculate n sim realizations of the function under consideration, say R ( ˜ β m ) . Step 3
Calculate the ( ζ/ -th and (1 − ζ/ -th quantile of the realizations where ζ is typically setto 0.05. The confidence interval is then constructed as CI − ζ = [ R ( ˜ β m ) ζ/ , R ( ˜ β m )) − ζ/ ] A value of n sim equal to 100 typically produces reliable results although it can be increased if moreprecision is required. To evaluate the effectiveness and implementation of the proposed methodology, we conducted asimulation study with four scenarios that differ in terms of the continuous marginal distributionand the copula specification. All four scenarios are assessed using sample sizes of n = 1 , , , and , . The data generating process and detailed results can be found in Appendix E.Our approach is able to capture the effect of both linear and nonlinear covariates fairly welland performance improves significantly with increasing sample size. In addition to recoveringthe coefficients, we calculated the AIC in every simulation run for the bivariate model and thecorresponding independence model. The share of a runs in which the bivariate model had asmaller AIC was 1, providing evidence for the ability of our model to identify dependence betweenthe responses if this is indeed required by the data generating process. We refrain from detailledsimulations concerning the selection of marginal distributions and/or copulas since these havebeen considered before in the literature on copula GAMLSS, albeit for the case of two continuousmarginal distributions (e.g., Marra and Radice, 2017; Radice et al., 2016). To analyse poverty dimensions in a bivariate copula model and to identify 1) the determinantsof the income-education relation, 2) its spatial distribution and 3) groups at risk of being bothconsumption and education poor, we rely on the most recent wave (IFLS 5) of the IndonesianFamily Life Survey (IFLS). The IFLS is a publicly available, longitudinal survey on individual,household, and community level that is designed to study the health and socioeconomic situationof Indonesia’s population. The first wave was implemented in 1993 and covered individuals from7,224 households representing 83 percent of the population from 13 out of 27 Indonesian provinces(Strauss et al., 2016). The sample was drawn by stratifying the population on provinces andurban/rural areas before randomly selecting enumeration areas and households within the strata.Due to a large number of split-off households the sample grew up to 16,204 households interviewedin IFLS 5.In the IFLS, individuals of an IFLS-household older than 15 years were asked to fill in an “adultindividual book” containing questionnaires on subjects such as income, education, employment,9nd subjective health. We use the level of education as the ordinal response variable, and incomeas the continuous response. Education is proxied by the highest educational institution attendedand can take on five different levels: 1 "no schooling", 2 "primary school", 3 "middle school", 4"high school", 5 "tertiary education". In analyses for developing countries, income is often proxiedby expenditures for consumption. Expenditures are calculated at the household level and thendivided by the number of household members. Our income variable is thus precisely expendituresper capita. Due to different price levels in the provinces, we used the province specific minimumwage to adjust expenditures across provinces. Data on the individuals from the “adult individualbook” are extracted and merged with relevant information on the household head, such as genderand education, and complemented with information on the household’s location, such as provinceor whether the household lives in an urban area. We only included complete cases and individualsfrom the age of 18 as most of them already attained or are studying towards their highest educationlevel. The final dataset contains 32,884 individuals.
Applying flexible bivariate copula GAMLSS requires the researcher to decide on the specificationof multiple parameters, on the form of the continuous marginal distribution and of the copula.
Continuous marginal distribution
In line with, e.g. Klein et al. (2015) and Marra and Radice (2017), we propose to use normalizedquantile residuals for selecting the continuous marginal first. This allows us to assess graphicallythe appropriateness of the chosen distribution which could firstly be done using separate univariatemodels. A normalized quantile residual ˆ q mi for the second, i.e. the continuous marginal, is definedas: ˆ q i = Φ − { ˆ F ( y i ) } for i = 1 , . . . , n, where ˆ F ( · ) is the estimated marginal CDF for the continuous response component, and Φ − ( · ) isthe quantile function of a standard normal distribution. If ˆ F is close to the true distribution, ˆ q i approximately follows the standard normal distribution. Quantile residuals are fairly robust to thespecification of the distribution parameters’ specification (Klein et al., 2015; Marra and Radice,2017).We fit univariate models and select the model by inspecting the corresponding QQ-plots. Gooddistribution candidates for income and expenditure are generally the lognormal distribution, theSingh-Maddala distribution and the Dagum distribution (e.g. Kleiber and Kotz, 2003). Figure 1shows the QQ-plots for the univariate income model and all potential covariates using a lognormaland the Dagum distribution. Fitting the model with the Singh-Maddala distribution leads toconverge failure, which may signal an inappropriate choice of the marginal distribution. The QQ-plots suggest an appropriate fit for the lognormal distribution and it is hence used. Note thatonce the final bivariate model is built, the QQ-plot for the continuous margin are re-examined.However, we find that the plot (shown in Appendix F) looks almost identical to Figure 1 (leftpanel) indicating that a good fit for the continuous margin of the proposed copula model has beenobtained. 10 llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll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Dagum
Figure 1: Normal QQ-plots for the univariate income model and different distributions with 95%reference bands.
Variable selection
For the specification of the link function of the ordinal response, as well as variable selection for thebivariate model, and the choice of the copula function, the Akaike’s Information Criterion (AIC)and the Bayesian Information Criterion (BIC) can be used. These are defined asAIC := − (cid:96) (ˆ β ) + 2edf , BIC := − (cid:96) (ˆ β ) + log( n )edf , where (cid:96) (ˆ β ) is the log likelihood of the bivariate model evaluated at the penalized parameter estimateand edf = tr( ˆ A ) as defined in Section 3.2. Theoretical knowledge about the problem at handfacilitates the variable selection procedure by pre-selecting candidate predictors. Radice et al.(2016) also suggest to start with a model specification where all distributional and the associationparameter depend on all covariates. In case the algorithm does not converge, an instance that oftenindicates that the sample size is too small for the model’s complexity, they recommend trying outa series of more parsimonious specifications. To test smooth components for equality to zero, wehave adapted the results of Wood (2017) to the current context.To fit the bivariate model, we specify an equation for each distributional and the copula parameteras follows: We start with a set of variables selected according to economic reasoning. Note thatoften in income or expenditure equations, household size is used as a covariate in addition tonumber of children and elderly. However, we do not wish to separate the child effect in a “pure”child effect and children as additional household member effect. Moreover, the outcome variableis already adjusted for household size. Religion is included because it defines minority groups.While education on the individual level is part of the response vector, the level of education of thehousehold head is included as a control in the predictor for capita income. For the bivariate model,11e fit a full specification for the location parameter of each marginal distribution, i.e. µ educ and µ inc and perform variable selection using the AIC for the scale parameter of the second marginal, σ inc , and for the copula parameter, γ . More specifically, a backwards selection procedure isapplied for σ inc given a full specification for γ and then a second backwards selection is performedon γ given the reduced model for σ inc . This excludes only four variables for the scale predictorand two variables for the copula parameter specification, i.e. we arrive at: η µ educ = θ r − { s ( age ) + β µ · hhmarstat + β µ · hhmale + β µ · urban + β µ · num _ child + β µ · elderly + β µ · relig } η µ inc = β µ + s ( age ) + β µ · hhmarstat + β µ · hhmale + β µ · urban + β µ · num _ child + β µ · elderly + β µ · relig + β µ · hheduc + s ( prov ) η σ inc = β σ + s ( age ) + β σ · hhmarstat + β σ · num _ child + β σ · relig + s ( prov ) η γ = β γ + s ( age ) + β γ · hhmarstat + β γ · urban + β γ · num _ child + β γ · elderly + β γ · hheduc + s ( prov ) . Continuous variables enter the equations with smooth non-parameteric effects s() represented viathin plate regression splines with ten bases and second order derivative penalties. Spatial effectsof the provinces and their neighbourhood structure are modeled using Markov random fields. Wechoose to model the spatial effect at the province level since minimum wages are set at the provincelevel affecting individual wages and thus expenditure measure as well (Hohberg and Lay, 2015).
Ordinal model
The ordinal outcome education is fitted using an ordered model. Table 1 compares the AIC andBIC between a probit and a logit link of the bivariate model using the lognormal as the continuousmarginal and a Gaussian copula. Both AIC and BIC favor the logit model for the first marginal.Table 1: AIC and BIC of bivariate ordered-continuous model using the logit and probit links.AIC BIClogit 1,075,191 1,076,241probit 1,075,588 1,076,637
Choice of the copula
For the copula selection, a good starting point would be the use of a Gaussian copula and thenconsider all consistent alternatives depending on the direction of the dependence (Radice et al.,2016; Klein et al., 2019). Again, AIC and BIC can help choosing among several candidate copulas.Starting off with the Gaussian yields an average value for the copula parameter (with 95% confi-dence interval in brackets) of γ = 0 .
163 (0 . , . . Building on this finding, we test a range ofsuitable possible candidates. After checking convergence, we can only eliminate the un-rotated Joeand Clayton copula. The remaining candidates are compared using the AIC and BIC; see Table 2for the results. The AIC and BIC indicate that a Gaussian copula should be used for our model,and all copula models should be favoured over the independence model. Using the Gaussian copulasuggests that the dependence between per capita expenditures and education is symmetric withasymptotically independent extremes. 12able 2: AIC and BIC for different copula specifications.AIC BICGaussian 1,075,191 1,076,241F 1,075,233 1,076,295FGM 1,075,280 1,076,335PL 1,075,226 1,076,286AMH 1,075,298 1,076,359C0 1,075,448 1,076,508C180 1,075,379 1,076,380J0 1,075,514 1,076,462J180 1,075,516 1,076,573G0 1,075,351 1,076,360G180 1,075,306 1,076,341Independence 1,075,892 1,076,678Note: Abbreviations correspond to Frank, Farlie-Gumbel-Morgenstern, Plackett, Ali-Mikhail-Haq,Clayton, rotated Clayton (180 degrees), Joe, rotatedJoe (180 degrees), Gumbel, rotated Gumbel (180 de-grees), respectively. After deciding on the marginal distributions, the copula and covariates by comparing different can-didates, we check the final bivariate model. To this end, we use a multivariate generalization of thequantile residuals introduced in Section 4.2 that was proposed by (Kalliovirta, 2008). Multivariatequantile residuals for two continuous responses are defined as ˆ q i = (cid:32) ˆ q i ˆ q i (cid:33) = (cid:32) Φ − ( ˆ F ( y i ))Φ − ( ˆ F | ( y i | y i )) , (cid:33) where ˆ F | is the (estimated) conditional CDF of Y given Y . In our case, the first marginal isdiscrete such that we resort to randomized quantile residuals where uniformly distributed randomvariables on the interval corresponding to cumulated probabilities are plugged into Φ − ( · ) . If themodel is correctly specified, then ˆ q approximately follows a bivariate standard normal distribution.The contour plot for the bivariate model in Figure 2a shows the density of the quantile residuals ˆ q by means of a multivariate kernel density estimator. This estimated density is compared to thedensity of the standard normal distribution. The contour lines of both densities are close to eachother indicating a good fit of the bivariate copula model.In Figure 2b, the sum of the squared elements of the multivariate quantile residuals are considered.That is, ˆ q (cid:48) i ˆ q i = ˆ q i + ˆ q i , where ˆ q i is the multivariate quantile residual for the i -th individual. Since ˆ q i a ∼ N (0 , and ˆ q i a ∼ N (0 , , it follows that ˆ q (cid:48) i ˆ q i a ∼ χ (2) which is assessed in the QQ-plot inFigure 2b. The reference bands are obtained by repeatedly simulating from a χ (2) distribution.We draw n rep = 100 samples and compute the . and . quantiles for each observationsacross the sorted n rep samples. The plot supports our model choice.13 ll lll l l ll ll l l llll lll l lll lll lll ll l l lll l ll lll lll ll l lll lll ll lll llllll l l ll l llll ll l ll l lll ll l ll l lll llll lll l lll llll lll l ll l l lll ll llll lll ll l llll lll l ll llll lll l lll ll ll ll l lll ll lll l ll ll l ll l llll l ll ll l ll llll llll lll l ll llll ll l ll llll ll l ll lll l l lll lll ll ll l lll ll l lll l l lll l ll ll l lll ll ll ll l lll ll l ll lll ll lll l lll l ll l ll l ll lll lll lll lll lll lll l lll lll lll ll ll l ll llll l llll lll l l lllll l l ll lll ll lll ll ll ll ll ll l ll ll l l lll llllll lllll l lll ll lll ll l l ll ll lll ll ll l l ll l lll ll ll lll l ll ll lll ll lll lll ll l ll ll ll llllll l llll llll ll ll ll ll l l llll lll ll lllll ll l llll l lll ll ll l lll lll ll ll ll l l ll lll ll l llll ll l lll l lll llll ll l l lll ll l ll l ll ll l lll lll lll llll lll l lll lll ll lll ll ll ll l lll l lll ll ll lll l ll ll l l lll lll ll ll ll lll ll lll ll ll ll llll l lll ll 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lll lll llll ll lll lll ll l ll lll l ll lll lll ll l ll llll l l l ll ll llll l ll l l llll llll l ll l l lll ll l l lllll l l ll l ll ll l ll ll lll ll ll ll ll ll l lll ll l ll l lll ll llll l l lll ll l lll ll l lll ll ll l ll ll lll ll ll lllll llll lll l ll lll l lll ll ll ll lll ll l lllll ll ll l llll ll ll ll lll l lll ll l lll llll l lll l ll ll l l ll lll l l ll l ll ll l l lll l ll ll l l llll l ll l l lll lllllll ll l ll l lll l ll ll lll ll l l lll l ll llll l ll llll l lll l ll ll l llll ll lllll lll ll ll ll l ll lll ll l ll lll ll ll l ll lll ll ll ll lll ll ll ll llll l lll l llll llll l lll l lll ll ll llll ll ll ll l lll ll lll ll l llll l llll ll l l ll l l ll llll lll ll ll lll lll ll ll l l ll lll ll llll lll ll ll ll l lllll l ll l lllllll lll l ll ll l ll ll ll lll l l lll lll ll lll ll ll lll ll l ll ll l l llll l ll lll l lll ll l lll ll ll ll lll llll l lll ll llll l lll llll llll l ll llll ll lll ll ll lllll l ll l l l ll lll lll ll ll lllll lll ll l ll lll 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ll ll ll llll lll lll l lll ll ll l ll ll l ll ll ll ll ll llll ll llll l llll ll l l lll ll lll ll lll ll llll ll lll ll lll llll lll ll llll l l ll ll l l ll l ll ll llll ll ll ll llll lll l llll l l l lll ll l ll l l ll ll lll ll ll ll ll ll ll ll l ll ll ll lll l ll ll l ll ll lll l ll ll l l lll ll l lll ll llll lll l lll l ll ll l ll ll ll ll ll lllll ll lll l l ll l lll ll lll ll l ll lll ll l lll ll l lll lllll l lll ll lll ll lll ll lllll ll ll ll l lll llll ll lll l lll l llll ll l ll ll l l ll ll ll ll lll l lll l ll lll l l ll ll lll ll llll ll ll lll l ll ll l lll l lll lll lll lll lll llll ll l ll l l ll l l l lll lll ll lll ll l l ll l lll ll ll ll ll lll lll lll llll llll ll l l ll l l ll lllll ll ll lll l ll ll ll l ll l ll ll ll lll l lll ll lll ll l llll l lllll ll ll l lllll llll ll l ll ll llll llll lll ll llll l ll ll ll lll lll l ll ll llll l l lll llllll l l ll lll lllll lll ll ll lll lll lll l lll l ll l ll ll lll ll l l ll l ll l lll l ll ll ll ll l ll ll ll 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ll ll lll ll ll lll ll lll l ll l ll llll llll ll ll l ll l ll lll ll ll ll ll ll lll ll ll l lll l lll ll ll l ll lllll ll l llll lll ll lll ll l llll ll llll lllll ll l l lllll llll lll ll ll l llll l ll lll lll l lll ll ll l lll l ll l l ll ll l ll ll ll ll ll ll l ll ll lll lll l ll ll l ll ll l l ll ll l ll lll ll ll ll l lll lll l l lllll ll l l lll ll l ll lll ll l l ll ll ll l ll ll ll lll l l ll ll llllll lll ll l llll lll l lll lll l lll ll l l llll llll llll l l lll l ll ll ll ll lll ll ll l l l lll ll lll ll ll ll l llll ll ll lll ll lllllll ll ll lllll ll ll l lll lll ll ll llll ll l ll lllll lll ll llll lll ll lllll l l ll lll l llll lll l l ll l ll l ll l ll l lll ll l l lllll l ll ll lll ll ll lll ll l l lll l l l lll llll l ll lll ll l ll l l lllllll l ll ll ll lll l l ll lll lllll ll ll ll lllll llll lll ll ll ll ll l lll llll l l lll lll l l l ll lll ll ll ll lllll l l lll ll ll ll l lll ll l ll llll ll l l llll lll ll ll ll l lll lll llll lll ll ll llll lll l ll lll 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ll lll ll lllll ll l l l lll l llll ll ll l l llll llll ll ll llll l ll llll ll l ll l ll ll l l l llll ll ll lll ll ll ll l ll l llll ll ll ll ll lll lll ll ll ll ll ll l ll llll lll l ll lll ll lll l l l ll l l ll l llll lll ll l l ll ll ll lll ll ll ll l l l l l llll llll ll ll llll ll ll ll ll lll ll lll llll l lll l ll ll l llll l ll ll l ll llll lll l lll lll lll l llll lllll ll l l ll ll lll ll ll ll ll ll ll l l lll l ll llll ll ll ll lll l llll llll ll l ll l l llll ll lll l ll l ll l lll lll ll ll l lll l l ll ll l l llll l lll lll l l ll ll ll llllllll lll ll ll ll l lll lll lll ll lllll ll ll llll l lll ll l ll ll l ll ll ll lll l lll llll ll ll l lllll ll llllllll l llll l llll lll ll llll lll ll ll lllll ll l l ll ll l l ll l l ll ll llllll llll lll llll ll ll llll lll l l l ll ll ll lll lll l ll ll ll lllll lll l llll ll l ll l lll llll l l ll lllll ll lll l lll ll llll ll l ll l llll l ll l ll ll ll l ll l lll lll ll l ll ll ll l ll llll l l ll l lll l lll ll lll llll lllll ll l llll l ll lll ll lll l ll lll ll l lll ll lll lll llll ll ll l ll lll lll ll ll ll lll l ll lll l lll ll ll lll ll ll l llll ll ll l l lll ll ll ll l llll lllll lll l l ll l l llll ll lll ll llll ll l ll ll lll ll llll ll lll ll llll ll l ll l lll ll ll l lll ll ll ll lll ll l ll lll ll ll lll l llll l ll lll ll l ll lll l ll lll l llll l ll llll ll lll l llll lllll ll llll lll l l lll ll ll ll l ll ll ll l l l ll l l l lll llll lll l ll l lllll ll ll ll ll ll lll l l lll ll ll ll llll ll llllll l l l llll lll lll lll ll ll ll ll ll llll lll l ll ll ll ll ll lllll ll l ll ll lll l llll l l lll lll ll ll ll lll ll lll lll l l lll ll ll ll l l ll l ll lll ll lllll ll l lll l l ll ll lll ll lll llll lll lllll lll lll ll ll lll ll ll ll l lll l llllll lll lll l ll ll lll ll lll llll l ll ll l lll ll ll ll l lll lll l l ll lll ll ll l ll lll l l ll l ll llll ll lll ll lll ll l l lll ll ll l lllll ll llll lll ll llll l ll l ll lll ll l ll lll l ll lll lllll l ll l l lll ll lllll l lll lll ll ll l ll l ll l lll l ll l ll lll l l ll ll ll lll l l llll ll l lll lll l ll llll l l l lll l lll ll l l ll ll llll ll ll ll lll ll l ll l ll l ll l l llll ll llll ll l ll ll llll llll llllllll ll l ll lll ll l ll l ll llll ll l lll ll l l llll l lll ll ll ll l l ll ll ll l lllll l lllll ll ll l l lll l ll ll ll ll lll lll ll ll ll l ll ll l lll ll ll l lll l l lll l ll l l llll l lll l lll ll llllll ll llll ll l l ll l ll ll llll ll ll l ll lll ll lll lllll l l llllll ll l l l lll ll lll llll llll l l l llll ll lll ll ll lll ll llll ll llll ll ll ll lllll ll ll llll lll l l lll ll lllll ll llll ll lll l ll l ll ll lll lll l ll ll ll lll l lll lll lll ll lll ll l ll ll l lll l l ll ll ll l lll ll ll l llll l l ll ll l lll l ll ll l ll ll ll l ll ll l l ll lll l l lllll ll ll l ll lll ll l ll ll ll l ll llll l llll l lllll l ll l ll l ll ll l lll l lll lll ll ll llll l ll l ll l lll llll lll l lll ll l l l ll l ll lll ll l ll l llll lll lll l ll ll l l ll l l l lll l lll lll l ll llll lll llll l l ll ll ll lll lll ll ll ll ll ll ll lllll lll ll ll lll lll ll l lll ll ll lll ll l ll l lll lll llll ll lll l ll ll l ll ll lll ll lll l l lll lllll l l ll lll l lll ll ll lll l ll lll lll ll lll l ll lllll llllll ll llll ll l l l lll l l ll llll ll ll ll ll l ll ll ll ll ll ll ll lll ll l ll l ll ll l l lll lll ll lll lll lll ll ll llll ll ll l lll l lll lll ll ll ll ll ll ll l ll llllll l l l ll ll ll llll l ll ll l ll ll lll lll ll lll l lll ll lll ll ll ll l l ll l ll lll l ll llll l ll lll ll lll llll l lllll l lll ll lll l ll lll llll ll ll l l ll ll llll lll lll l lll ll l lll lll l lll l llll l ll llll l l llll lll l ll l ll l ll lll ll lll l llll l l lll l llll ll lll llll l l lll ll ll lll ll ll ll l l ll ll lll l lll ll ll lll l llll ll ll lll lll ll ll l ll ll l l lll llll llll ll lll l ll llll ll ll lllll l l ll ll lll l l lll lll ll ll lll ll ll ll l ll llll l lll ll ll lll l l l ll lll lll ll lll l ll ll ll l lll lll l lll llll l l lll l l llll l l 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lll l ll lll ll lll lll l llllll ll ll l l lll l ll llll l llll l ll lll ll ll l llll l l lllll l lll l ll ll l ll l l lll ll ll l llll ll l l lllll llllll ll ll llllll l l l ll l lll l l lllllll lll lll l ll llll lll ll ll llll ll l l ll lll lll l l ll l lllll l ll l lll l ll lll ll l lll lll lll ll l llll lll ll ll ll l l llll ll lll ll lll l lll ll l ll l lll ll ll l ll l lll ll lllll l ll ll ll ll llll ll l ll l lll l ll l ll lll ll l lllll ll l ll ll lll l ll ll l llll ll ll ll lll ll l ll l lll l ll lll ll lllll l ll lll ll ll lll ll lll ll ll l l lll ll lll lll lll ll ll l ll l lll l lll ll l ll lll lll l ll ll l lll ll l ll ll ll l llll l lll l l lll ll lll ll ll l ll l l ll l lll ll lll l ll ll ll ll ll llll l ll ll ll lll l lllllll ll ll llll llll l ll ll lll ll ll lll ll l ll llll ll lllll l ll ll lll lll l lll l lll l lll l l l lllll l lll ll ll l ll ll ll lll l l ll ll lll l l lll lll l l lll lllll l lll lllll l lll lll lll lll ll ll lll l ll ll ll l l ll l ll ll l l ll ll l lllll lll l ll llll llllll l ll ll ll llll lll lll l ll ll ll l l ll ll llll lll ll ll l ll ll llll l l lll l l ll lll lll ll ll ll llllll ll l l ll lll ll ll ll ll ll l llll l lll ll ll lll lll lll ll l ll l ll lll lll l lll l lll l ll llllll lll l ll ll l ll lll llll l ll ll ll ll ll l ll lll ll l ll l l ll l lll lll l lll l ll lll l lll lll llll l ll llll l lll llll l ll lll ll llll ll l lll llll lll lll l ll ll lll lll l l l llll ll llll ll ll l l ll lll l ll l l l ll lll lll ll l lll lll ll l ll ll ll lll l ll lll lll lll lllll lll lll lll lll ll ll ll lll lll ll lll lllll l ll ll l lllll ll l llll lllll lll ll ll llll ll l ll l l lll ll l lll l lll lllll llll ll ll l ll llll lll ll l ll l lllll ll l ll l lll l ll l ll lll ll ll l ll ll l l l ll ll ll l ll ll ll llllll lll ll lll ll llllll ll ll ll ll ll ll l ll ll lllll l l lllll l ll llll ll l lll ll ll ll lll ll l l ll l ll ll l ll ll ll l l ll ll llll ll llll lll ll llll llll lll l ll ll l lll ll ll ll l lllll ll lll l l ll lll lll lllll llll ll ll llll l lll l ll ll ll l lll ll ll lll lllll lll lll l lll ll llll ll ll ll lll l ll ll l llll ll l l lll l lll l llll l ll ll l l lll l lll ll l l ll ll ll ll ll ll ll l ll ll lllll l ll ll llll ll lll ll ll ll ll l lll l ll l l l llll l ll lll l l ll ll l llll lll l ll ll ll lllll lll llll ll ll ll l lllll lll lll ll l ll lll lll llll ll l lll ll lll ll ll l lll llll llll lll l lll ll ll l llll ll ll ll l lll l ll l lll ll ll l lll lll lll ll l ll l ll l ll l lll ll lll l ll lll l lll l lllll ll ll llll l lll l lll l lll l l lll ll l ll l llll llll lll ll ll lllllll lll lll l lll l ll ll lll lll ll lll lll l lll ll lll llll lll lll ll l ll l lll lll l ll ll l lll l ll l l lll llll ll ll lll l ll l ll ll ll l l l l lll ll llll ll ll ll l lll ll llll ll l llll llll l lll llll lll lll ll ll lllll lll l lll ll ll l ll lll ll ll l ll lll lll ll l lll l l llll lll ll l lll l ll lll l llll l l l lll l ll lll llll l lll ll lll lll ll lll l lll l lll ll ll lll l ll ll llll ll lll ll l llll l l ll ll ll llll ll ll ll llll l l l ll ll lll l lll ll l lll ll ll lll ll l ll l lllll ll ll llll lll l l l lll ll lll l lll ll ll ll l l l ll llll ll l lllll ll ll lllll ll ll ll lll ll lll l ll lll lll l l lll ll ll l ll ll ll lll lll l l ll l llllll ll lll llll ll l ll ll l lll l lll ll lll ll l llll ll lll lll llll l l ll l l ll ll ll ll lll ll ll l l l ll l lll ll lll llll l llllll ll l llll lll ll l lllll l ll ll lll l lll lll lll l llll ll ll ll ll l ll ll l lll l ll ll l ll l lllll l llll l lllll l ll lll l ll ll lll lll ll l l llll l ll lll ll l ll lll ll l ll l lll lll lll ll lll ll ll ll l llll ll lll ll l ll l l lll lll l ll ll l l ll ll ll ll lll ll l lll lllll l llll l l ll l lll l l lll llll llll ll ll l ll ll l lll lll l l lll ll lll ll ll l lll ll llll lll l lllll ll ll ll l ll lll llll l ll ll lll ll lll ll ll ll ll ll llll l ll ll ll lll lll ll ll l ll lll l lll l llll lll l l l ll lll lllll l l ll lll lll ll l ll lll lll lll ll lll l ll l ll ll ll l llll l ll llll ll l lll lll llll lll ll l lllll ll ll llll lll lll ll ll lllll l ll lll lll lllll ll lll l lll l lll lll lll lll ll l ll l llll l lll ll lll ll lll l ll llll ll l ll llll ll lll l ll lll lll lll ll llll lll ll ll ll lll ll l lllll llll ll ll lll l ll ll ll l l ll lll llll ll llll l lll l ll lll l ll l l lll ll llll ll l l lll l llllll ll l lll lll ll ll lll ll l llllll llll ll llll l ll l l ll lll l lll ll ll ll ll l ll ll ll ll l l l ll ll l ll ll ll ll l ll lllll l lllll ll ll ll l lll ll llll l ll llll llll l ll lll ll l ll l llll l ll lll ll l llll lll ll lll llll llllll l llll lll ll ll ll ll ll ll ll lll ll l ll l lll l llll ll ll lll l lll ll lll lll lll ll ll ll ll lll l ll l ll ll l llllll ll ll lll l ll lll ll l lll l l ll ll llllll llll ll l llll lll l l l ll ll ll ll l lllllll lll l l ll l lll lll l ll lll lll l ll lll ll l llll l l llll ll lll ll lllll lll ll lll l lll lll l ll ll llll l ll ll l l l ll l llll ll ll ll ll lll llll ll ll l l l l lll l l l ll lll ll l l ll l ll ll ll ll ll llll l llll l l ll l l ll l ll lll l lll llll lll l ll lll l l ll ll ll ll ll l ll lll l l lll llll lll l ll llll l ll ll ll ll ll l ll ll lllll ll ll ll lll lll ll ll ll l l ll ll ll lll l ll ll lll llll lll lll lll ll ll lll llll l ll lllllll ll lll ll l ll l l l ll l ll llll ll ll ll l lll l lll ll ll l ll lllll l lll ll ll lll lll lll l lll l lll lll l lll l ll l ll lll ll lll ll ll l lllll ll ll l l ll lll ll l llll llll ll ll l ll ll ll lll lll l lll l l ll ll ll ll l l ll lll l l l lll llll lll ll ll llll llll llll lll l ll l lllll lllll l l l ll ll ll l lll ll ll lll l l llll ll l llll lll ll l ll lll ll lll ll ll ll ll ll ll ll ll lll l lll lll l ll ll lllll ll l l ll lll ll l lll l lll ll l l ll ll ll l ll ll ll l l ll ll lllll l ll ll l l lll ll llll ll ll ll ll l lllll ll llll lll ll ll l l ll l l lll ll llll ll l ll ll l ll ll l ll lll ll l llllll l l ll ll ll lll l l lll l ll lllll ll lllll ll l ll llllll ll lll llll lll lll ll ll lll lllll lll llll ll l ll l l ll llll lllll ll ll l lll l ll l l lll l lll lll lll l ll ll ll ll ll ll lll ll ll l l ll lll l ll lll ll llll ll lll ll ll ll l ll l ll l l lll l l l ll l ll llll l ll lll lll lll ll lllll ll lll ll l lllll l lll lll lll ll lll l ll lll ll l lll ll l l ll lll ll l ll ll lll lllll l lll l ll ll l l lll ll llll ll lll ll ll l ll ll l l ll ll lll l l l l ll l ll lll l l lll ll l llll ll l ll l ll ll llll l lllll ll lllll l llll llll ll lll ll ll lllll l llll l ll llllll l llll lll ll l l ll llll l llll lllll l llll lll ll lll lllll lll llll ll llll lll l ll lll l l lll lll l ll ll lll lll lllll ll l ll ll lll l l llll l ll ll lll l lll lll l ll lll llll ll ll l ll lll llll ll ll lll ll ll lll l l l ll ll l ll lll ll l ll ll l ll l ll ll ll lll ll l lll llll ll l ll ll lll lll l llll ll lll ll l ll ll lll l llll ll lll l ll lll lll l lll l l l llll l lll l ll ll lll l lll ll l l l ll lllll l lll l l lll l ll ll llll l lll l ll ll lll lll l ll llll l lll ll lll ll l ll ll ll llll ll ll llll ll lll llll l llll llll lll ll lll llll l lll llll llll ll ll l ll ll ll l llll lll ll l lllll l l ll l lll l ll ll l lll lll l llll llll lll lll lll lll ll ll ll l lll ll lll l lll l ll ll l ll ll ll ll lll lll l ll lll l l ll ll lll lll l l ll l lllll lll l lll ll ll lll ll ll lll ll l ll lll llll lll l lll ll ll ll l l lll lll l ll l l ll lll l l l lllll ll l ll ll l l lll l ll l llll lll lll lll lll l lll llll l ll ll l ll l ll l ll l l ll l ll ll l l lll l lll l ll l lll l ll lll llll ll lll lll ll lll llll ll ll l ll ll ll llll ll ll l ll ll lllll l l lll llll ll l lll ll ll ll ll l llll ll ll lll ll ll lll lll lll ll l ll ll ll ll ll lll l l lllllll l l llll ll l llllll lll ll l lll ll lll l llll l ll ll llll l llll l ll l ll llll ll l llll l lll l lll lll l ll l lll ll ll l l ll ll lll ll l lll l lll ll ll ll l ll l lll l l ll ll ll llll l l ll lll ll lll lll lll l ll l lllll ll lll l lll ll ll l l lll l l lll ll lll lll llll llll ll lll lll lll lll ll ll lll lll ll l llllll ll ll lll ll lll ll lll ll l ll lll lll l lll lll lll lll ll ll lllll lll llll l l ll lll l llll l ll ll ll ll l l llll ll l ll ll ll llll lll l ll l ll lllll l llll l lllll ll ll ll lll l lll l lll lll lllll ll ll ll lll lll ll l ll lll llll ll l lllll lllll l ll l ll l lll lll l l l lll ll l lll ll l llll l lll ll lll llll llll ll llllll ll l ll lll lll ll l ll ll ll ll lll lll lll l ll l lll l llll ll ll l llll ll ll l l ll ll ll ll l ll ll lll l l ll lll l ll l lll lll ll l ll ll ll lll l ll lll ll lll ll ll lll l l l l ll ll l ll ll l lllll lll ll ll l l lll ll l lll ll lll llll l l ll l ll ll llll l llll ll ll lll ll l l ll ll lll lll ll ll ll lll l ll llll l llll ll ll lll lll l ll ll ll l ll lll ll l ll ll l ll ll ll l ll ll llll ll ll l lll lll l ll ll ll lll llll ll ll ll ll lll ll lll ll ll ll ll l ll l lll l ll ll lll lll ll l ll ll ll lll ll lll lll ll l ll l ll ll l l ll ll ll l lll llllll l ll ll l lll l llll lll ll llll ll l llll ll l lll lll l ll ll lll ll ll l ll lll lll l l ll l lll l lll ll l ll ll ll l llll l lll lll l l ll l llllll l ll ll llll lll ll lllll ll ll llll lll llll l lll l ll lll ll ll ll l ll l ll llll l l ll lll ll ll ll l llll ll l ll ll lll ll ll ll llll ll ll ll l lllll ll lll ll ll l lll ll lll ll l l l lll l lll l ll lllll l ll lll l lll l ll l lll ll lll l ll l ll llll ll lll lll l ll ll lll lll l lll ll llll ll l lllll ll ll lll l ll ll lll ll llll ll ll ll l l ll l ll l ll l l llll ll ll lll llllll lllll ll lll llll ll l ll ll lll ll l llll lll ll ll ll l ll lll lll llll l l ll ll ll l ll l lll l lll l lll l ll ll l lll ll lllll ll l ll ll ll l l lll l ll l ll llllll ll lll ll lll llll l lll llll lll ll ll ll l ll lll l ll ll l l ll ll lll l l ll l lllll lllll l lll ll l ll l ll llll ll ll l lll l ll lll lllll ll ll llll lllll l lll l ll llll l ll l llll ll ll l lll lll lll ll l llll ll lll ll ll l ll l ll ll llll ll lll l l ll l ll l l lll l ll llll ll l ll l ll l l ll ll ll l l ll ll llll ll l llll ll lllll l ll ll lll ll llll l ll lll l lll lll l ll ll ll lll ll l l ll l lll ll lll lll lll l lll l ll ll l ll l ll l ll ll ll l ll ll l ll l llll ll lll lll l ll llll ll l llll lll ll lll ll l llllll l lllll lll lll l ll l ll llll ll ll lll ll ll ll l lll ll llll ll l lll ll ll l l ll ll ll lll l l llll ll ll ll l ll ll ll ll ll llll llll ll ll ll ll ll ll l lll ll ll l ll llllll l ll lll l ll ll lll lll llll lll ll l ll l ll llll ll ll llllll ll lll ll ll lll ll l ll lll l l lll ll ll ll ll lll lll ll l lll ll l ll ll ll ll ll l ll ll lll l ll ll ll ll lll lll l lll l ll l lll l ll l lll ll l ll ll ll llll l l l ll ll lllllll l ll ll ll l llll ll l l l lll l lll lll ll l l ll ll ll ll lll llll l l lll ll l ll lll l l ll ll ll l ll lll lll ll l lll l lll ll ll ll lll lll l l ll lll ll ll ll ll l ll llll lll ll l l ll ll ll ll llll lllll ll ll ll l ll l ll l lll ll l ll l ll ll ll ll lllll llll l ll ll lll l lll l l ll ll l ll ll ll l llll llll ll ll llll ll l lll l l lll ll l lll ll lllll lll lll ll ll l ll ll ll ll l ll l ll l l llll l −4 −2 0 2 4 − − q q (a) Contour plot of multivariate quantile residuals.The red lines indicate the density of the quantileresiduals estimated by a multivariate kernel densityestimator. The blue circles are the contour lines ofthe density of the standard normal distribution withradius , and . l lll l lll lll ll lll l lll lllll ll lll l llllll llllll ll lll lll ll ll l ll lll lll ll lll lll lll llll l llll ll lll lll l llll lllll lll ll lll l lll lll llll ll lll ll llll ll llllll ll lll lll ll llllllll l ll l llllllll llll lll lllll lllll l lll ll lll lll ll ll llllll l lllllllll ll ll lll ll lll ll llll ll ll lll l lll ll lll l l ll lllll l ll ll ll llll l lll ll lll ll lllll lll ll lllll ll lll llll ll lll ll llll llll lll lll lllll llllll ll lll llll ll l lllllllll ll ll ll l lllllll lll lll l lllllllll lll llll ll llllll ll ll llll ll llllllllllll lll lll lllll lll ll l llll llllll ll lllll lllll lll lll l ll ll llll l lll lll l l lllll ll llll lllll lll lllllllll ll ll lllll ll ll llll l l l llll ll ll l lll ll lllll lllll lllllll llllllll llll lll llllll l llll l ll llll llll lllll l lllll lllll ll l lll llll ll llll l llll l lllll lllllll ll llll l l ll l l lllll lllll l ll ll llll l ll lll llllll l llll llll l llllllll lllll lll ll ll llllllll l lll l l llllllll lllll lllllll l l l ll lll lll ll l ll ll llllllll lll lll llll lll llllllll lllll lllllllll l llllll l ll ll lll llll llll lll ll ll l lll llllll ll lll llll llll ll lllllllll ll ll llll lllll lllll ll lll ll lll llll llllll llllll l llll lllllllll ll llll lll llll llll lll l lll lllll ll lll lll l lllllll lllll lllll ll llllll llllll lllll llll l lllll lll llll ll l lll l ll lllllll ll llll llllllll l ll llll ll ll lll ll l l lll ll llll ll ll ll lll lllll lll lll l lllll lll ll lll ll l ll lll l l llllllll l llll lll ll llllll ll llll l lll ll ll llll ll llllll ll llll lllllllll ll lllllll ll l ll ll lll lllll l lll ll llll l llll lllllll lllll l ll lll lll lllll lllllll llll l ll lll lllll llllll lll llll lllll ll ll lll lll lllll lllll l lll lll lllll ll lll lll llllll ll llllll ll ll llllllll llll ll ll llll lllllllllll lll lll llll lllllllllll l lllll llll ll lll lllll ll llll lll ll llll lllll llllll lll l llll l llll ll ll ll lllll lllllllll llll llllllllll ll lll lll lll ll llllllll ll l llll ll lllll l l ll lll ll lllll lll ll ll l lllll lll ll llll lll lll l llll lll ll llll ll lll lll ll lllll ll ll lll ll ll llll lll ll l ll llllll ll lll lll l ll ll ll l lllll lllllllll lllll llll ll ll lll l ll ll lll l lll ll ll llllll lll lllllll lll ll lll llll llll llll ll llllll lll lll llllll lllll l lll llll ll ll llll ll lll ll lll ll ll lll llll lllllllllllllllll ll l ll llll ll llll l llll llllll lllll lll l ll l ll llll l llll ll lll lllll llllll l lll l lllll llll ll lll lllll lllll llll llllllll llllllllll llll ll ll lll llll llllllll lll llll lllllll ll ll lllllllll lllll lll ll llllll llllll ll ll llll lll lllll ll ll llll ll ll ll l lllllllll ll lllll lllll llll llll lllll llllllll ll ll lll lll llll llll l lllll lll ll l ll l lll lll ll lllllll ll l ll lll l llllll ll l llllllll l lll ll ll lll ll ll ll ll lllllll lll llll ll ll llll llll lllll lll ll llll lll lllll l lll lll lll ll llll llll llllll ll llll ll ll l llllllll l lll lll lllllll lll ll llll lllll ll lllll llll l ll llll lllll ll llll lll lll llll lll ll ll lll ll lllll llll l lllllll l lllll llll llllllllll ll llll l ll ll llllll lllll lll llll l llll ll lllllll l lllllll l llll llll llllll llll ll ll lllllll lll ll ll lllll llll lll lll lllll l l ll ll lllllll l ll lll lll lll ll llll l lll llll ll lll llll lllll ll lll lllll llllll ll llllllll lll llll lll ll lll llll ll llll lllll llll l llll ll lll lllll llll ll lll lll l ll llll ll lll llll lllllllll ll lll ll l llll lllll l llll lllll ll llll llllll lllll lll ll ll lllll ll l lllllll ll l llll l llll llll ll lll lll lll llll llllll ll l ll lll lllll lll lll lll l l llllll ll lll lll l l llll l ll l lll ll llll ll lllll l ll llll lllll ll lll ll llll llll lll lll llll l ll ll ll ll ll l llll ll llll lllllll ll lllll ll l l lllllll lll ll ll lll lll lll l ll ll ll ll l lll ll lll llllllll lll llll ll ll ll ll 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ll lllll lll lllllll lllll lllll ll lll ll llll l lll l ll llllll llll ll ll ll ll ll llll ll l ll ll l llll llllll lll lll lll ll l ll llll lll lll lll l lllll ll lll lll l l ll lll llll ll l ll llll l lll ll l ll l lll lll lll llll ll lll l lll l l llll llll l ll l lll ll lll llll ll lll llll ll llll l ll l ll ll lll lll lll lllll ll lll lll ll llll llll l l llll ll lllll ll ll ll ll lllll ll lllll ll l ll llll ll l llllllllll llll l l ll lll lll lll l lllllll llll lll lll ll l llllllllll llllllllll llllllll ll ll ll lllllll llllll l llllll lllll ll ll ll ll llll llll lll l lllllllllllll l lll lll l lll l lllll ll l l lll lllll ll lllllll ll llllll llll l lll llllllll lll ll ll lllll ll lll ll ll llllll l ll lllll lllllll llll lllll ll llllll llllll ll lllll lll l lllll llllll lll lll lllllllll llll ll lllll lllll ll ll llllllll lllllll lllllllll l ll lllllll llllllll lll llll lllllllll lll llllllll lll ll llll llllllll lll lllll lllllll lll llll llllllll lll llllllll llllll lllllll llllll llll ll l llllll lll lllll llll llll llllllllllllllll llll llll l llllllll lllllllllllllllllllllllllllllllll llll lllll lll l llllll l l lll llllll llll llllll ll l l ll llllllll lll llllll ll lllll ll llll lll llll llll llllllll lll llllll llll ll lllll llllllllll lll lllllll llll lllll lll llllllllll ll ll llllllllll llllllll llll lllllll lllll llllllllll ll lllll ll llll lllllllll ll lll llllll lllllllllllll lll lllllllll llllll llllll ll ll l llll l lllllll l lll llll lllllll lll llll l llll llll llllll lll lll llllllllllllll ll lll lllllllll lllll lllllll ll llll lllll ll lllllll ll llll lllllll llllll l llllll lll llllllll lllllll l lllll l llll lll lll llll llllllllll lll l llll l llllllllll lllllll lll ll lllll lllll lll llllllllllll llllllllll lllllll lllll ll llllllll llll llllll lll ll lll lll lll lll ll lll llll llll llll lll ll ll ll lllll ll lll lllllll l lll ll llll lllllllll lllllll llll llllll lllll llll llll ll l lll ll llll lllllll l llllll l ll llll lllll llll ll lllll l lllll lllllllllll llll lll lll lll ll lll ll llllllll lll lllllll ll lllllllll llllllll ll llll lllllll lllllllll llll lll l l ll lll l llll ll lll lll lllll llllllllllll llllllll llllll ll llll lllll ll lllllllll llllll l lllllllll lll lllll lll ll ll lll ll ll ll llll llllllll llll ll lllllllll ll lllll ll lll lllll llllll l lllll lll lllllllll llll llllllll ll lllll lll ll lllll lll ll lll l lll llll l ll lll ll l llll ll ll ll l llllll llllll llllll l ll l lllllllllllll lllll ll lll ll llllllllll ll ll l lll l llll lllll llllllll ll lll lllllll llllllllllll lll ll l lll lllllllll llllllll lllllll llll ll lll llllllll l lllll ll ll lll llll ll llll l lll lll lll ll l lll lll llll lllllllll lll llll l llll ll l lll lll lllll llllllllll lllll ll ll ll l llll ll lll ll ll l llllll ll l ll ll llllllll llll ll llll ll lll llll ll lll lllll llllllllllll llllll lllll lll l lll ll ll lllll llll ll lllll lllllllll ll l llllllll lll ll llll ll ll ll lll lll lll llllll l l lllll ll lllll llllll ll llllllllll llllll llllll llll ll lll ll lll lllllll lllll llllll lllllllllll llll ll ll llll lll llllllll llll lll lll ll ll ll ll llll llll lll lll ll l llllll ll lll lll lll l lllllll ll ll lllllllllll llll lllll lll ll lll ll lll lllll lll ll lll llllll l l lllll llllllllll llll ll lll l ll ll llllll ll lll lll lllll l ll ll ll lll llllllll ll ll llll lll ll l ll lll lll llllll l ll llll ll ll llllll llllll llllll l ll lllll lllll l l ll lll lllll lll ll l ll ll ll l l ll lll l lll l l lllllllll lll lll lll l ll llll llll ll l ll l lll llll llll ll llllllll ll llll llll lll ll llll l ll ll lllll ll lll l ll lll llll ll lll ll ll llll ll lll l lllll ll lll ll lll llllll ll l lll ll llllll lllll l ll ll ll llllll llllll ll llll lll ll ll lllll lllll ll ll lll lll l ll lll lllll lll l lll ll ll l lll lllll lll l llllll ll ll ll lllll lll lll ll llll ll lll lll lllll ll ll l llllll ll lll llll ll lllll ll l lllllll l lll llll llll lll lllllll llll ll l llllllll lll ll lllll ll lllllll l l lll ll llll ll lllllllllll l ll lll ll l llllllll ll lll ll lll l llllll l ll ll lll ll lll ll ll lll ll lll l ll lll lll ll llll l lllll ll llll lllll lll lllll ll ll llll l ll ll llll ll llll lll ll lll ll lll ll ll llll ll l lll lll ll ll lllll llll ll ll l lll lll lll lll llll llllll ll lllllllll lllllll llllll l ll ll llll llll llllll l lllll ll ll lllll ll llll llllll ll l llll l lllll llllll l lll l l lllll lll lllll lll l lll l llll llllll lllll lllll llll l lll lllllll ll ll lll lllllllll llll l llll lll ll l llll lllll ll lll ll l ll lll llllll lllll ll lll ll llll ll lll llll lll lll llll lll llll ll l llll lllll l ll lllll lllllll llll lllll ll lll ll ll llllll lll lllll llll ll lll l lll ll ll l ll llll lllll lll ll lllll l lll llll ll llll ll llllll lll lll ll llllll ll llllll ll ll lllll llll llllll ll l lll l llll ll llll lll ll ll ll lll llll ll l ll llllll llllllll ll ll llll l ll lllll llll ll ll ll llll l ll llll l ll lllll lllll llll ll llllll ll llllll ll l llll llllll ll lll lll ll ll lllll ll lll l ll lll lllll ll lll llll lllll ll lllll ll llll ll ll lll ll lll llll ll lll l lll llll ll llllllll ll ll lll lllllll ll lll lllll ll lll ll ll ll ll l lll ll lll lllll ll ll ll llll lll lll lll l lll lllll l lll ll ll lllll lll llllllll lll ll lll lll llll lll llll llllll ll ll lll lll llllll lll l ll lll lllll ll lllll ll ll lll llll l llllllllll lll lll ll llll ll l llll ll lll ll lll ll llllllll llll llllllll ll ll lllll lll l l ll lllll ll llll lll lll llll llll llll llllll ll ll lll l lllll ll llll lll ll llll ll lllll ll lll ll ll ll ll llll l lllll lll ll ll lll lllll lll lll l llll ll ll llll ll ll l lll llll lll lll lll llllllll ll ll llll lll lll ll Theoretical Quantiles S a m p l e Q uan t il e s (b) QQ-plot depicting the sum of the squared ele-ments of the multivariate quantile residuals with 95%reference bands. Figure 2: Multivariate quantile residuals of the bivariate model.
This section demonstrates the results that poverty researchers can derive from fitting a copulaGAMLSS model for the joint analysis of two inter-related poverty dimensions. We summarisebriefly the covariate effects on the marginals before taking a closer look at the dependence structureand risk groups. This way, we are able to answer questions on how dependence and risk are affectedby a household’s location or composition.
Effects of covariates on the marginal distributions
The full list of effects on each parameter of the marginals and on the copula parameter is included inthe Appendix F. Note that the effects are subject to a ceteris paribus interpretation. For example,more schooling is associated with more income with tertiary education having the highest effect.Households with more children or elderly people living in the household have on average a lowerincome per capita and less education (except for the effect of one elderly on education which isnegative and not statistically significant). Urban households are associated with both better incomeand better education. A little surprising is that living in a household where the head is marriedis correlated with a reduction in income and education compared to not (yet) married households(Table 7 and 6). One possible explanation is that non-married households compared to marriedhouseholds include a larger share of young, single persons that do not need to share their incomeand have a comparatively higher level of education. Non-Muslim households are associated withhigher income per capita and more education although they represent a minority in Indonesia. Fora male household head, the effects are not as expected since it is negative for income but positive14or education. For the second parameter of the continuous margin, i.e. the scale parameter for theincome equation, the number of children and a marital status other than not married have negativeeffects while the effects of elderly and other religions compared to Islam are positive. All of thecovariates have a positive effect on the copula parameter though not all effects are significant.Age is modeled in a non-linear way and Figure 3 displays the smooth effects of age on each param-eter. Education attainments are lower for higher ages which can be explained by the educationexpansion that Indonesia has undergone since the 1970’s and younger individuals benefited from.For example, between 1974 and 1978 over 61,000 primary schools were built. In 1984, compulsoryeducation was set to six years which was extended to nine years in 1994 (Akita, 2017; Duflo, 2004).The effect of income on the location parameter is inverted u-shaped until the age of 60 with a peakaround the age of 40. After 80, the confidence intervals become very wide due to a lower number ofobservations in this age span and the effect is thus less clear. The effects on the scale parameter arearound zero. For the copula parameter, the age effect indicates that the dependence is decreasingfor individuals up to their mid 30ies and stays around zero afterwards until it decreases again afterthe age of 60.
20 40 60 80 100 − − − Logit, m age in years s m ( age )
20 40 60 80 100 − . . . . Lognormal, m age in years s m ( age )
20 40 60 80 100 − . − . . Lognormal, s age in years s s ( age )
20 40 60 80 100 − . − . . Copula parameter, g age in years s g ( age ) Figure 3: Estimated smooth functions of age and respective point-wise 95% confidence intervals.Figure 4 shows the effect of the underlying spatial pattern on the parameters µ and σ of thesecond response, and on the copula parameter γ . Households located in provinces of Java seemto have higher income per capita compared to the observed provinces in Sumatra, Borneo andSulawesi. Provinces with a negative effect on the location parameter have higher effects on thescale parameter except for Borneo whose scale effect is negative.15 Effect on −0.0914 0.07240 Effect on −0.0706 0.0352 Effect on
Figure 4: Spatial effects of the provinces on the distribution parameters of income and on thecopula parameter. 16 ependence structure
When focusing on the dependence, its structure can be represented via contour plots for specificcovariate combinations. For example, we focus here on the location of the household (urban/ruraland province) as they are the significant drivers of the copula parameter, while the remainingplots can be found in Appendix F. For comparison purposes, we include provinces with the highestfrequencies in the dataset but select only one of Java’s provinces. To compare the dependencestructure across different locations, we create an example of typical individual whose characteristics,other than the one under consideration, are set to their mean value or to their most frequentobservation. The only exception is the education of the household head which is set to the secondmost frequent observation. For a household head with "high school" degree the dependence is a bitmore pronounced we hence selected this level for demonstration purposes. This is the covariates’combination that we call an “example individual” henceforward.Figure 5a shows that the dependence is stronger for individuals in urban households comparedto rural households. One reason might be that average education levels are lower in rural areas(x-axis) while at the same time high paid job opportunities are restricted in a rural environment,resulting in more equal incomes compared to an urban environment. Figure 5b compares thedependence structure across selected provinces. The dependence seems weakest in the province ofNusa Tenggara Barat, which is one of the poorest provinces in Indonesia. It is surprising that theaverage per capita consumption (straight line) of Jakarta is about the same level than that one ofNusa Tenggara Barat. Most likely this is due to the high price level in Jakarta and the deflationmeasure applied which scaled down our expenditure measure maybe too drastically. Though thecopula coefficient for Jakarta is similar to Jawa Timur, the latter has more variation in incomesand the contour levels lie further apart. Considering all – and not just the selected provinces – thevalue of the copula parameter for the example individual ranges from 0.16 for Kalimantan Timurand Kalimantan Selatan up to 0.27 for Yogyakarta.A policy maker might be interested to know in which locations each individual in the dataset, andnot the example individual, have higher dependencies in order to efficiently design policy strategies.Although the lower panel of Figure 4 shows the effect of the provinces on the copula parameter, itmight be more helpful for interpretation to transform it into the Kendall’s τ , an association measurethat takes on values on [ − , . Each individual with his/her specific covariates’ combination isrelated to an individual-specific τ . One way to present the differences across provinces is to averagethe τ over all individuals in a particular province. This is shown in Figure 6. The KalimantanSelatan (South Borneo) is the province with the lowest average of Kendall’s τ with a value of0.0468 and Kepulauan Riau (Riau Islands, northwest of Borneo) has a value of 0.1467 which is thehighest average value that also indicates spatial heterogeneity in the strength of the dependence.The provinces of Sumatra seem to have higher dependence between income and education thanprovinces in Borneo or Sulawesi. Interestingly, for Java and its neighbouring smaller islands on theeast, the dependence seem to decrease from west to east. Joint probabilities
Other results we can derive from a copula GAMLSS models are joint probabilities for differentsub-groups. That is, we calculate the probability for the example individual of being poor in boththe education and income dimensions. As an example, we again focus here on household locationand additionally consider household composition. To define poverty, we classify individuals that17 ural latent education pe r c ap i t a i n c o m e −4 −2 0 2 4 + + + g = urban latent education pe r c ap i t a i n c o m e −4 −2 0 2 4 + + + g = (a) Contour plots for an example individual in an urban or rural household in theprovince of Jawa Timur. Jakarta Raya latent education pe r c ap i t a i n c o m e −4 −2 0 2 4 + + + g = Jawa Timur latent education pe r c ap i t a i n c o m e −4 −2 0 2 4 + + + g = Nusa Tenggara Barat latent education pe r c ap i t a i n c o m e −4 −2 0 2 4 + + + g = Sumatera Utara latent education pe r c ap i t a i n c o m e −4 −2 0 2 4 + + + g = (b) Contour plots for an example individual in an urban household in differentprovinces. Figure 5: Contour plots for (education, income)’ and a Gaussian copula by households’ location.Contour lines of densities are at levels from 0.00000005 to 0.00000025 in 0.00000001 steps. Thevertical straight lines represents the cut off values for the education categories, horizontal straightlines are the consumption average, and dashed horizontal line are at two standard deviations aroundthis average. 18 .0473 0.1473
Average per province
Sumatera Utara
LombokSumbawa
Nusa Tenggara Barat
JavaSumatra Borneo SulawesiBali
Jakarta RayaJawa Timur
Figure 6: Kendall’s τ for each individual averaged within provinces.have only primary or even less education as education poor and set a relative poverty line of60% of the median of unique values of per capita expenditures. Note that Indonesia also has anational absolute poverty line that is, however, based on a different expenditure measure thanthe one we constructed from the IFLS data. We thus decided to use a relative poverty line. Anindividual that is poor in both dimensions has expected values below each of these thresholds.One of the sub-groups is set to one and the probability of the other sub-group is compared tothis base category. Figure 7 shows that the probability for being poor in both dimensions is 2times higher for the example individual in a rural household compared to the same individualin an urban household. Compared to Jawa Timur the joint probabilities of Jakarta Raya, NusaTenggara Barat, and Sumatera Utara are about 8 times, 6 times, and 4 times higher, respectively.Not surprisingly, the risk of being poor in both dimensions increases with the number of childrenand elderly in the household. Vulnerability to poverty
The higher the dependence between education and income, the higher the chances that we misssome individuals at risk by only looking at the marginal distributions of each poverty dimension.To identify the individuals at risk, we calculate the probabilities of being poor in two dimensionsfor each individual in the dataset, first for the independence model and then for the copula model.A vulnerability threshold is arbitrarily defined at a probability of 0.1 for simplicity. All individualswith a probability of being poor above this threshold are declared as vulnerable. We then comparethe individuals that are identified as vulnerable by the copula and the independence models totheir actual poverty status and calculate the specificity or true positive rate. Results are displayedin Table 3. We find that the copula model has better specificity for two-dimensional poverty thanthe independence model although the difference is fairly small.19 ural urban
Joint probabilities for urban/rural r e l a t i v e r i sk . . . . . . Jakarta R. Jawa Timur NTB Sumatera Utara
Joint probabilities for different provinces r e l a t i v e r i sk Joint probabilities for different number of children r e l a t i v e r i sk Joint probabilities for different number of elderly r e l a t i v e r i sk . . . . . Figure 7: Relative joint poverty risks of an example individual differentiated by household locationand household composition. Baseline categories are urban, Jawa Timur, 1 child, and 1 elderly,respectively, and are set to one. The abbreviation NTB denotes the province of Nusa TenggaraBarat. 20able 3: Specificity of the copula and independence models. Individuals that are declared vulner-able are compared to their actual poverty status.copula independenceincome dimension 0.88 0.88both dimensions 0.66 0.63
Discussion of Results
The main advantage of applying copula GAMLSS in the precedent poverty analysis is that we areable to analyse the dependence between poverty dimensions in more detail than studies that arelimited to measuring the dependence. Even though the dependence is not very strong, once wecontrol for covariates in the marginals, we consider this an interesting outcome. We could furtheridentify heterogeneities in the strength of the dependence between income and education, andpoverty risk with respect to a household’s location. For example, an example individual in a ruralhousehold exhibit lower dependencies compared to the same individual in an urban household.One explanation might be that opportunities in rural areas are restricted and thus the individual’seducation level has a smaller influence on per capita expenditures.There is also a strong spatial heterogeneity between provinces. High dependencies can be foundin the northwest while the values decrease for the more central provinces. The explanation forlow dependence due to low opportunities could also apply to the province of Nusa Tenggara Baratwhich showed little average dependence and lower dependence for a specific example individualcompared to other provinces. Nusa Tenggara Bararat is one of the poorest provinces, with a lowGDP per capita and with agriculture and fishery being the most important industries. On the otherhand, provinces such as Jakarta and Kalimantan Timur, that have the highest per capita GDP outof all provinces in Indonesia, have higher probabilities of being poor in both dimensions comparedto most other provinces. These probabilities are calculated for an example individual that haveaverage characteristics and a high school degree. Possible reasons for the discrepancy betweenrich provinces and high relative poverty risks might be that in Jakarta income and consumptionare very unequally distributed and in East Kalimantan the high GDP is a result of high naturalresource exploitation which yields little benefit for the population (see for example Bhattacharyyaand Resosudarmo, 2015, who found that growth in the mining sector has had no effect on povertyand inequality in Indonesia).
Though poverty is conventionally regarded a multidimensional phenomenon, regression analysesin the poverty context either examine dimensions separately or use a scalar index such as theMultidimensional Poverty Index as an outcome. Both approaches neglect the dependence betweenpoverty dimensions. This paper presents an alternative model for an in-depth poverty study thatexplicitly analyses this dependence and its determinants. It is important to understand whatdrives the dependence between poverty dimensions since high dependencies can explain persistingpoverty.For this type of poverty analysis, we propose the use of bivariate copula GAMLSS which relateeach distributional parameter of the marginals and of the copula parameter to flexible covariateeffects. Since poverty analyses often include one monetary measure such as income or consumption21nd some ordinal measure such as education level or health status, we extended the class of copulaGAMLSS to incorporate ordinal data based on a latent variable approach. This extension has beenincorporated in the
GJRM R -package.We use data from Indonesia to show how copula GAMLSS can be applied to a poverty analysis. Themodel identified the number of elderly and children in the household, the education of the householdhead, and the household’s location as risk factors for low income or poor education or both, andthe probability of being poor in both dimensions. The gender of the household head, belongingto a minority religion and being widowed or separated compared to being married, has shown lessor the opposite influence than expected. We did not find evidence for strong tail dependenciesbetween education and expenditures after conditioning on the covariates in the marginals and inthe copula parameter.Focusing more on the household’s location, we find that an example individual in a rural householdexhibits lower dependencies compared to the same individual in an urban household, potentiallydue to restricted employment opportunities for highly educated individuals in rural areas. Thereis a strong spatial heterogeneity regarding poverty risk and the strength of the dependence. Highdependencies can be found in the northwest of Indonesia while central provinces have lower ones.For Jakarta and Kalimantan Timur we found a discrepancy between being rich in terms of GDPand exhibiting high relative poverty risks, potentially indicating unequally distributed economicgains.Thanks to the flexibility of our approach, the analysis of the dependence between poverty dimen-sions, and the various results that can be derived consistently by only estimating one model, weadvocate to include copula GAMLSS into the tool box of poverty researchers. Other applications inthe context of poverty may include analysing the drivers and spatial patterns of inter-generationalpoverty persistence or upward social mobility. On the methodological side, future research canbe directed at combining copula GAMLSS with experimental or quasi-experimental methods toevaluate Indonesia’s poverty policies on the micro level. Extensions beyond the bivariate case aredependent on the availability of a suitable copula. The implementation of these models subse-quently becomes more numerically and technically demanding and interpretation of the regressionresults will be challenging. 22 eferences
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World Devel-opment , 113–124. 25 ppendix A Considered copulas and corresponding (trans-formed) parameter Table 4: Families of some bivariate copula functions.
Name C ( u, v ) range of γ γ ∗ Kendall’s τ Gaussian Φ (Φ − ( u ) , Φ − ( v )) [ − , tanh − ( γ ) π arcsin γ Clayton ( u − γ + v − γ − − /γ (0 , ∞ ) log( γ − ε ) γγ +2 Frank − γ − log[1+( e − γu − e − γv − / ( e − γ − R \ { } γ − ε − γ (1 − D ( γ )) Gumbel exp (cid:8) − [( − log u ) γ + ( − log v ) γ ] /γ (cid:9) [1 , ∞ ) log( γ −
1) 1 − γ Joe − [(1 − u ) γ +(1 − v ) γ − (1 − u ) γ (1 − v ) γ ] /γ (1 , ∞ ) log( γ − − ε ) 1 + γ D ( γ ) FGM uv [1 + γ (1 − u )(1 − v )] [ − , tanh − ( γ ) γ AMH uv − γ (1 − u )(1 − v ) [ − , tanh − ( γ ) 1 − γ ( γ +(1 − γ ) log(1 − γ )) Plackett γ − u + v ) − √ [1+( γ − u + v )] − γ ( γ − uv γ − (0 , ∞ ) log( γ − ε ) - Note:
The association parameter is denoted by γ and u and v are the marginals F ( η k ) and F ( y ) , respectively. The abbre-viations FGM and AMH correspond to the Farlie-Gumbel-Morgenstern and the Ali-Mikhail-Haq copula, respectively. Function Φ (˙ , ˙; γ ) is the CDF of a bivariate standard normal distribution, while D ( γ ) = γ (cid:82) γ t exp( t ) − dt is the Debye function and D ( γ ) = (cid:82) t log( t )(1 − t ) − γ ) γ dt . The quantity ε denotes the machine smallest floating point multiplied by and isintroduced to force the transformed association parameters to lie in their respective supports throughout estimation. Appendix B Further details on the trust algorithm and smooth-ing parameter selection
B.1 General idea of the trust region algorithm
The trust region algorithm has the advantage of being more stable and faster compared to in-line search methods (Marra and Radice, 2017). Both line search and trust region methods use aquadratic model of the objective function and generate steps from one iterate to the next. Whileline search methods use the model to find first a search direction and suitable step lengths alongthis direction, trust region algorithms search the step that minimizes the objective function withina previously defined region around the current iterate such that both direction and step length areselected at the same time. If a function exhibits long plateaus and the current iterate β [ a ] is inthat region, line search methods may search the next step β [ a +1] far away from the current iterate.In this case, it is possible that the evaluation of the log likelihood will not be finite causing thealgorithm to fail. Before evaluating the objective function, trust region methods define a maximumdistance first based on the trust region. This has two advantages: first, the new iterate will not lietoo far away from the current one; second, in case of a non-definite evaluation of ˘ (cid:96) p , step p [ a +1] willbe rejected. If a candidate that minimizes the quadratic approximation of the objective functionand also lies in the trust region does not improve the function sufficiently or gives a non definiteevaluation of ˘ (cid:96) p , the trust region shrinks and the algorithm moves back to step 1. On the otherhand, if the improvement is large enough, the trust region expands in the next iteration. Detailson the trust region algorithm can be found in Nocedal and Wright (2006), Chapter 4.26 .2 Details on smoothing parameter selection Simultaneous optimization of β and λ causes overfitting. The reason is that the penalized log-likelihood will be highest at λ = . Instead, the smoothing parameters λ should be selected in away such that the function estimates are close to the true functions.Providing a full justification, Marra et al. (2017) propose an approach that yields the result inequation (10). They use the fact that, close to convergence, the trust region algorithm behaveslike a classic unconstrained algorithm (Radice et al., 2016). Considering the first order Taylorexpansion of g [ a +1] p and setting this to zero, they obtain = g [ a +1] p ≈ g [ a ] p + H [ a ] p ( β [ a +1] − β [ a ] ) , (11)where the quantities as those defined in the paper. The aim is now to find an expression for β [ a +1] that uses g [ a ] and H [ a ] as a whole. Marra et al. (2017) argue that this reduces the frequency ofsituations where H is not positive definite compared to when working with the n components thatmake it up. The situations where H turns out not to be positive definite can then be dealt with byperturbing H to positive definiteness (Wood, 2015). Recalling that I [ a ] = − H [ a ] , the expressionin (11) can be re-written as = g [ a ] p + ( − I [ a ] − S λ )( β [ a +1] − β [ a ] ) g [ a ] p = ( I [ a ] + S λ )( β [ a +1] − β [ a ] ) g [ a ] − S λ β [ a ] = ( I [ a ] + S λ )( β [ a +1] ) − ( I [ a ] + S λ )( β [ a ] )( I [ a ] + S λ ) β [ a +1] = g [ a ] − S λ β [ a ] + ( I [ a ] + S λ ) β [ a ] = g [ a ] + I [ a ] β [ a ] β [ a +1] = ( I [ a ] + S λ ) − (cid:112) I [ a ] ( (cid:112) I [ a ] β [ a ] + (cid:112) I [ a ] − g [ a ] ) . Thus, at convergence the parameter estimator takes the following form β [ a +1] = ( I [ a ] + S λ ) − (cid:112) I [ a ] M [ a ] , where M [ a ] := µ M [ a ] + (cid:15) [ a ] , µ M [ a ] := (cid:112) I [ a ] β [ a ] and (cid:15) [ a ] := (cid:112) I [ a ] − g [ a ] .From likelihood theory we have that (cid:15) ∼ N ( , I ) and M ∼ N ( µ M , I ) , with I being the identitymatrix and µ M := √ I β , where β denotes the true parameter vector. The predicted value vectorfor M is ˆ µ M = √ I ˆ β = AM , where A = √ I ( I + S λ ) − √ I . To obtain an estimate of λ thatsuppresses the complexity of smooth terms not supported by the observed data, ˆ µ M should beclose to µ M . Hence, employing the expected squared error, yields E ( (cid:107) µ M − ˆ µ M (cid:107) ) = E ( (cid:107) ( M − (cid:15) ) − AM (cid:107) )= E ( (cid:107) M − AM − (cid:15) (cid:107) )= E ( (cid:107) M − AM (cid:107) ) + E ( − (cid:15) (cid:48) (cid:15) − (cid:15) (cid:48) µ M + 2 (cid:15) (cid:48) Aµ M + 2 (cid:15) (cid:48) A(cid:15) )= E ( (cid:107) M − AM (cid:107) ) − Kn + 2 tr ( A ) . (12)The smoothing parameter is found by minimizing an estimate of the expectation in equation (12).For a given β [ a +1] we arrive then at equation (10).27 ppendix C Gradient and Hessian Throughout the derivations of the gradient vector and the Hessian matrix, we denote by β theregression coefficient associated to the fist ordinal equation, and by β := ( β µ , β σ ) (cid:48) the coeffi-cients for the location and scale parameters of the second continuous equation. Similarly β γ refersto the coefficients of the copula association parameter. C.1 Gradient vector
A general expression for the contribution of the i -th individual to the gradient vector is the following (cid:96) (cid:48) β i ( β ) := ∂(cid:96) i ( β ) ∂ β = (cid:88) r ∈R { y i = r } ∇ r F . ( η ri ) ∂ ∇ r F . ( η ri ) ∂ β + f ( y i ) − ∂f ( y i ) ∂ β , where ∇ r is the backward difference operator applied to r , in particular ∇ r F . ( η ri ) := F . ( η ri ) − F . ( η r − i ) . We next define the quantities F . ( η ri ) := ∂ C ( F ( η ri ) , F ( y i )) ∂F ( η ri ) ∂F ( y i ) with F . ( η ri ) := ∂ C ( F ( η ri ) , F ( y i )) ∂F ( y i ) . C.1.1 Derivatives with respect to the transformed cut points • h = 1 : (cid:96) (cid:48) θ ∗ i = 1 ∇ r F . ( η ri ) ∂θ ∂θ ∗ (cid:18) F . ( η ri ) ∂F ( η ri ) ∂η ri ∂η ri ∂θ − F . ( η r − i ) ∂F ( η r − i ) ∂η r − i ∂η r − i ∂θ (cid:19) = 1 ∇ r F . ( η ri ) (cid:0) F . ( η ri ) f ( η ri ) − { (cid:22) r } F . ( η r − i ) f ( η r − i ) (cid:1) • h = 2 , . . . , R : (cid:96) (cid:48) θ ∗ h i = 1 ∇ r F . ( η ri ) ∂θ h ∂θ ∗ h (cid:18) F . ( η ri ) ∂F ( η ri ) ∂η ri ∂η ri ∂θ h − F . ( η r − i ) ∂F ( η r − i ) ∂η r − i ∂η r − i ∂θ h (cid:19) = 2 ∇ r F . ( η ri ) (cid:0) { h (cid:22) r } F . ( η ri ) f ( η ri ) − { h +1 (cid:22) r } F . ( η r − i ) f ( η r − i ) (cid:1) θ ∗ h C.1.2 Derivatives with respect to β and β • β : (cid:96) (cid:48) β i = (cid:20) ∇ r F . ( η ri ) ∇ r (cid:18) ∂F ( η ri ) ∂η ri ∂F . ( η ri ) ∂F ( η ri ) (cid:19)(cid:21) ∂η ri ∂ β with ∂η ri ∂ β = ∂η r − i ∂ β • β ; ϑ k = { µ , σ } : (cid:96) (cid:48) β i = (cid:34) ∇ r F . ( η ri ) ∇ r (cid:18) ∂F . ( η ri ) ∂F ( y i ) (cid:19) ∂F ( y i ) ∂ϑ k + f ( y i ) − ∂f ( y i ) ∂ϑ k (cid:21) ∂ϑ k ∂η ϑ k i ∂η ϑ k i ∂ β ϑ k .1.3 Derivatives with respect to the copula association parameter (cid:96) (cid:48) β γ i = (cid:20) ∇ r F . ( η ri ) ∇ r (cid:18) F . ( η ri ) ∂γ (cid:19)(cid:21) ∂γ∂η γi ∂η γi ∂ β γ C.2 Hessian matrix
The general expression for the contribution of the i -th individual to the Hessian matrix is given by (cid:96) (cid:48)(cid:48) ββ (cid:48) ( β ) := ∂∂ β (cid:18) ∂(cid:96) i ( β ) ∂ β (cid:19) (cid:48) = (cid:88) r ∈R { y i = r } ∇ r F . ( η ri ) (cid:34) ∂ ∇ r F . ( η ri ) ∂ β ∂ β (cid:48) − ∇ r F . ( η ri ) (cid:18) ∂ ∇ r F . ( η ri ) ∂ β (cid:19) (cid:18) ∂ ∇ r F . ( η ri ) ∂ β (cid:19) (cid:48) (cid:35) + f ( y i ) − ∂ f ( y i ) ∂ β ∂ β (cid:48) − f ( y i ) − (cid:18) ∂f ( y i ) ∂ β (cid:19) (cid:18) ∂f ( y i ) ∂ β (cid:19) (cid:48) . C.2.1 Hessian components for the transformed cut points • ( θ ∗ ) : (cid:96) (cid:48)(cid:48) ( θ ∗ ) i = (cid:18) ∂θ ∂θ ∗ (cid:19) (cid:40) ∇ r F . ( η ri ) ∇ r (cid:32)(cid:18) ∂η ri ∂θ (cid:19) (cid:34) ∂ F . ( η ri ) ∂F ( η ri ) (cid:18) ∂F ( η ri ) ∂η ri (cid:19) + ∂F . ( η ri ) ∂F ( η ri ) ∂ F ( η ri ) ∂η ri (cid:21)(cid:19) (cid:18) ∇ r F . ( η ri ) (cid:19) (cid:20) ∇ r (cid:18) ∂η ri ∂θ ∂F . ( η ri ) ∂F ( η ri ) ∂F ( η ri ) ∂η ri (cid:19)(cid:21) (cid:41) • θ ∗ ( θ ∗ h ) (cid:48) ; h = 2 , . . . , R : (cid:96) (cid:48)(cid:48) θ ∗ ( θ ∗ h ) (cid:48) i = ∂θ ∂θ ∗ (cid:26) ∇ r F . ( η ri ) ∇ r (cid:32) ∂η ri ∂θ (cid:34) ∂ F . ( η ri ) ∂F ( η ri ) (cid:18) ∂F ( η ri ) ∂η ri (cid:19) + ∂F . ( η ri ) ∂F ( η ri ) ∂ F ( η ri ) ∂η ri (cid:35) (cid:18) ∂η ri ∂ θ h (cid:19) (cid:48) (cid:33) − (cid:18) ∇ r F . ( η ri ) (cid:19) ∇ r (cid:18) ∂η ri ∂θ ∂F . ( η ri ) ∂F ( η ri ) ∂F ( η ri ) ∂η ri (cid:19) ∇ r (cid:32) ∂F . ( η ri ) ∂F ( η ri ) ∂F ( η ri ) ∂η ri (cid:18) ∂η ri ∂ θ h (cid:19) (cid:48) (cid:33)(cid:41) (cid:18) ∂ θ (cid:48) h ∂ θ ∗ h (cid:19) (cid:48) θ ∗ β (cid:48) : (cid:96) (cid:48)(cid:48) θ ∗ β (cid:48) i = ∂θ ∂θ ∗ (cid:26) ∇ r F . ( η ri ) ∇ r (cid:32) ∂η ri ∂θ (cid:34) ∂ F . ( η ri ) ∂F ( η ri ) (cid:18) ∂F ( η ri ) ∂η ri (cid:19) + ∂F . ( η ri ) ∂F ( η ri ) ∂ F ( η ri ) ∂η ri (cid:35)(cid:33) − (cid:18) ∇ r F . ( η ri ) (cid:19) ∇ r (cid:18) ∂η ri ∂θ ∂F . ( η ri ) ∂F ( η ri ) ∂F ( η ri ) ∂η ri (cid:19) ∇ r (cid:18) ∂F . ( η ri ) ∂F ( η ri ) ∂F ( η ri ) ∂η ri (cid:19)(cid:27) (cid:18) ∂η ri ∂ β (cid:19) (cid:48) • θ ∗ β (cid:48) ; ϑ k = { µ , σ } : (cid:96) (cid:48)(cid:48) θ ∗ β (cid:48) i = ∂θ ∂θ ∗ (cid:40) ∇ r F . ( η ri ) ∇ r (cid:18) ∂η ri ∂θ ∂ F . ( η ri ) ∂F ( η ri ) ∂F ( y ,i ) ∂F ( η ri ) ∂η ri (cid:19) ∂F ( y i ) ∂η ϑ k i − (cid:18) ∇ r F . ( η ri ) (cid:19) ∇ r (cid:18) ∂η ri ∂θ ∂F . ( η ri ) ∂F ( η ri ) ∂F ( η ri ) ∂η ri (cid:19) ∂ ∇ k F . ( η ri ) ∂F ( y i ) ∂F ( y i ) ∂η ϑ k i (cid:41)(cid:32) ∂η ϑ k i ∂ β ϑ k (cid:33) (cid:48) • θ ∗ ( β γ ) (cid:48) : (cid:96) (cid:48)(cid:48) θ ∗ ( β γ ) (cid:48) i = ∂θ ∂θ ∗ (cid:26) ∇ r F . ( η ri ) ∇ r (cid:18) ∂η ri ∂θ ∂ F . ( η ri ) ∂F ( η ri ) ∂γ ∂F ( η ri ) ∂η ri (cid:19) − (cid:18) ∇ r F . ( η ri ) (cid:19) ∇ r (cid:18) ∂η ri ∂θ ∂F . ( η ri ) ∂F ( η ri ) ∂F ( η ri ) ∂η ri (cid:19) ∂ ∇ r F . ( η ri ) ∂γ (cid:41) ∂γ∂η γi (cid:18) ∂η γi ∂ β γ (cid:19) (cid:48) • θ ∗ ¯ h ( θ ∗ h ) (cid:48) ; ¯ h, h = 2 , . . . , R : (cid:96) (cid:48)(cid:48) θ ∗ ¯ h ( θ ∗ h ) (cid:48) i = 1 ∇ r F . ( η ri ) (cid:40) ∂ θ (cid:48) h ∂ θ ∗ h ∇ r (cid:32)(cid:34) ∂ F . ( η ri ) ∂F ( η ri ) (cid:18) ∂F ( η ri ) ∂η ri (cid:19) + ∂F . ( η ri ) ∂F ( η ri ) ∂ F ( η ri ) ∂η ri (cid:35) ∂η ri ∂ θ ¯ h (cid:18) ∂η ri ∂ θ h (cid:19) (cid:48) (cid:33)(cid:18) ∂ θ (cid:48) h ∂ θ ∗ h (cid:19) (cid:48) + ∇ r (cid:32) ∂F . ( η ri ) ∂F ( η ri ) ∂F ( η ri ) ∂η ri (cid:18) ∂η ri ∂ θ h (cid:19) (cid:48) (cid:33) ∂ θ h ∂ θ ∗ ¯ h ∂ ( θ ∗ h ) (cid:48) (cid:41) − (cid:18) ∇ r F . ( η ri ) (cid:19) ∇ r (cid:32) ∂F . ( η ri ) ∂F ( η ri ) ∂F ( η ri ) ∂η ri (cid:18) ∂η ri ∂ θ h (cid:19) (cid:48) (cid:33) (cid:18) ∂ θ (cid:48) h ∂ θ ∗ h (cid:19) (cid:48) ∂ θ (cid:48) h ∂ θ ∗ h ∇ r (cid:18) ∂F . ( η ri ) ∂F ( η ri ) ∂F ( η ri ) ∂η ri ∂η ri ∂ θ h (cid:19) where ∂ θ h ∂ θ ∗ ¯ h ∂ ( θ ∗ h ) (cid:48) = (cid:34) ∂ θ h ∂θ ∗ ¯ h ∂ ( θ ∗ h ) (cid:48) (cid:35) ¯ h =2 ,...,R ; h =1 ,...,R and ∂ θ h ∂θ ∗ ¯ h ∂θ ∗ h = (cid:40) if ¯ h = h o/w 30 θ ∗ h β (cid:48) ; h = 2 , . . . , R : (cid:96) (cid:48)(cid:48) θ ∗ h β (cid:48) i = ∂ θ (cid:48) h ∂ θ ∗ h (cid:26) ∇ r F . ( η ri ) ∇ r (cid:32) ∂η ri ∂ θ h (cid:34) ∂ F . ( η ri ) ∂F ( η ri ) (cid:18) ∂F ( η ri ) ∂η ri (cid:19) + ∂F . ( η ri ) ∂F ( η ri ) ∂ F ( η ri ) ∂η ri (cid:35)(cid:33) − (cid:18) ∇ r F . ( η ri ) (cid:19) ∇ r (cid:18) ∂η ri ∂ θ h ∂F . ( η ri ) ∂F ( η ri ) ∂F ( η ri ) ∂η ri (cid:19) ∇ r (cid:18) ∂F . ( η ri ) ∂F ( η ri ) ∂F ( η ri ) ∂η ri (cid:19)(cid:41)(cid:18) ∂η ri ∂ β (cid:19) (cid:48) • θ ∗ h β (cid:48) ; h = 2 , . . . , R ; ϑ k = { µ , σ } : (cid:96) (cid:48)(cid:48) θ ∗ h β (cid:48) i = ∂ θ (cid:48) h ∂ θ ∗ h (cid:26) ∇ r F . ( η ri ) ∇ r (cid:18) ∂η ri ∂ θ h ∂ F . ( η ri ) ∂F ( η ri ) ∂F ( y i ) ∂F ( η ri ) ∂η ri (cid:19) − (cid:18) ∇ r F . ( η ri ) (cid:19) ∇ r (cid:18) ∂η ri ∂ θ h ∂F . ( η ri ) ∂F ( η ri ) ∂F ( η ri ) ∂η ri (cid:19) ∇ r (cid:18) ∂F . ( η ri ) ∂F ( y i ) (cid:19)(cid:41) ∂F ( y i ) ∂ϑ k ∂ϑ k ∂η ϑ k i (cid:32) ∂η ϑ k i ∂ β ϑ k (cid:33) (cid:48) • θ ∗ h ( β γ ) (cid:48) ; h = 2 , . . . , R : (cid:96) (cid:48)(cid:48) θ ∗ h ( β γ ) (cid:48) i = ∂ θ (cid:48) h ∂ θ ∗ h (cid:26) ∇ r F . ( η ri ) ∇ r (cid:18) ∂η ri ∂ θ h ∂ F . ( η ri ) ∂F ( η ri ) ∂γ ∂F ( η ri ) ∂η ri (cid:19) − (cid:18) ∇ r F . ( η ri ) (cid:19) ∇ r (cid:18) ∂η ri ∂ θ h ∂F . ( η ri ) ∂F ( η ri ) ∂F ( η ri ) ∂η ri (cid:19) ∇ r (cid:18) ∂F . ( η ri ) ∂γ (cid:19)(cid:41) ∂F ( y i ) ∂γ ∂γ∂η γi (cid:18) ∂η γi ∂ β γ (cid:19) (cid:48) C.2.2 Hessian components for β • β β (cid:48) : (cid:96) (cid:48)(cid:48) β β (cid:48) i = ∂η ri ∂ β (cid:40) ∇ r F . ( η ri ) ∇ r (cid:32) ∂ F . ( η ri ) ∂F ( η ri ) (cid:18) ∂F ( η ri ) ∂η ri (cid:19) + ∂F . ( η ri ) ∂F ( η ri ) ∂ F ( η ri ) ∂η ri (cid:33) − (cid:18) ∇ r F . ( η ri ) (cid:19) (cid:34) ∇ r (cid:18) ∂F . ( η ri ) ∂F ( η ri ) ∂F ( η ri ) ∂η ri (cid:19)(cid:21) (cid:41) (cid:18) ∂η ri ∂ β (cid:19) (cid:48) • β β (cid:48) ; h = 2 , . . . , R ; ϑ k = { µ , σ } : (cid:96) (cid:48)(cid:48) β β (cid:48) i = ∂η ri ∂ β (cid:26) ∇ r F . ( η ri ) ∇ r (cid:18) ∂ F . ( η ri ) ∂F ( η ri ) ∂F ( y i ) ∂F ( η ri ) ∂η ri (cid:19) − (cid:18) ∇ r F . ( η ri ) (cid:19) ∇ r (cid:18) ∂F . ( η ri ) ∂F ( η ri ) ∂F ( η ri ) ∂η ri (cid:19) ∇ r (cid:18) ∂F . ( η ri ) ∂F ( y i ) (cid:19)(cid:41) ∂F ( y i ) ∂η ϑ k i (cid:32) ∂η ϑ k i ∂ β ϑ k (cid:33) (cid:48) β ( β γ ) (cid:48) : (cid:96) (cid:48)(cid:48) β β (cid:48) γ i = ∂η ri ∂ β (cid:26) ∇ r F . ( η ri ) ∇ r (cid:18) ∂ F . ( η ri ) ∂F ( η ri ) ∂γ ∂F ( η ri ) ∂η ri (cid:19) − (cid:18) ∇ r F . ( η ri ) (cid:19) ∇ r (cid:18) ∂F . ( η ri ) ∂F ( η ri ) ∂F ( η ri ) ∂η ri (cid:19) ∇ r (cid:18) ∂F . ( η ri ) ∂γ (cid:19)(cid:41) ∂γ∂η γi (cid:18) ∂η γi ∂ β γ (cid:19) (cid:48) C.2.3 Hessian components for β • β β (cid:48) ; ϑ ¯ k , ϑ k = { µ , σ } : (cid:96) (cid:48)(cid:48) β β (cid:48) i = ∂η ϑ ¯ k i ∂ β ϑ ¯ k (cid:40) ∇ r F . ( η ri ) (cid:32) ∂ ∇ r F . ( η ri ) ∂F ( y i ) ∂F ( y i ) ∂η ϑ ¯ k i ∂F ( y i ) ∂η ϑ k i + ∂ ∇ r F . ( η ri ) ∂F ( y i ) ∂ F ( y i ) ∂η ϑ ¯ k i ∂η ϑ k i (cid:33) − (cid:18) ∇ F . ( η ri (cid:19) (cid:32) ∂ ∇ r F . ( η ri ) ∂F ( y i ) ∂F ( y i ) ∂η ϑ ¯ k i (cid:33) (cid:32) ∂ ∇ r F . ( η ri ) ∂F ( y i ) ∂F ( y i ) ∂η ϑ k i (cid:33) + f ( y i ) − ∂ f ( y i ) ∂η ϑ ¯ k i ∂η ϑ k i − f ( y i ) − ∂f ( y i ) ∂η ϑ ¯ k i ∂f ( y i ) ∂η ϑ k i (cid:41) (cid:32) ∂η ϑ ¯ k i ∂ β ϑ k (cid:33) (cid:48) • β ( β γ ) (cid:48) ; ϑ k = { µ , σ } : (cid:96) (cid:48)(cid:48) β ( β γ ) (cid:48) i = ∂η ϑ k i ∂ β ϑ k (cid:40) ∇ r F . ( η ri ) (cid:32) ∂ ∇ r F . ( η ri ) ∂F ( y ,i ) ∂γ ∂F ( y i ) ∂η ϑ k i ∂γ∂η γi (cid:33) − (cid:18) ∇ r F . ( η ri ) (cid:19) (cid:32) ∂ ∇ r F . ( η ri ) ∂F ( y i ) ∂F ( y i ) ∂η ϑ k i (cid:33) (cid:18) ∂ ∇ r F . ( η ri ) ∂γ ∂γ∂η γi (cid:19)(cid:41) (cid:18) ∂η γi ∂ β γ (cid:19) (cid:48) C.2.4 Hessian components for the copula association parameter • β γ ( β γ ) (cid:48) : (cid:96) (cid:48)(cid:48) β γ ( β γ ) (cid:48) i = ∂η γi ∂ β γ (cid:40) ∇ r F . ( η ri ) (cid:32) ∂ ∇ r F . ( η ri ) ∂γ (cid:18) ∂γ∂η γi (cid:19) + ∂ ∇ r F . ( η ri ) ∂γ ∂ γ∂ ( η γi ) (cid:33) − (cid:18) ∇ r F . ( η ri ) (cid:19) (cid:18) ∂ ∇ r F . ( η ri ) ∂γ ∂γ∂η γi (cid:19) (cid:41) (cid:18) ∂η γi ∂ β γ (cid:19) (cid:48) Appendix D Remarks on asymptotic properties
Asymptotic results for the proposed estimator can be derived along the lines of Marra and Radice(2017) and Donat and Marra (2017), for instance. Consider the Maximum Penalized Likelihoodestimator ˆ β = arg max β (cid:96) p ( β ) , (cid:96) p is given in equation (8) and the parameter vector β comprisescoefficients for the transformed cut points, all distributional parameters, and the copula parameter,i.e. β = ( θ ∗ , . . . , θ ∗ R , β (cid:48) , β (cid:48) , β γ (cid:48) ) (cid:48) . The situation under consideration has a fixed number of splinebases such that the unknown smooth functions may not be exactly represented as linear combi-nations of given basis functions. However, as for example noted in Kauermann (2005), using alarge number of basis functions the approximation bias plays a minor role compared to estimationvariability.If L t is the likelihood of the true model, its Kullback-Leibler distance to likelihood L ( β ) is KL ( L t || L ( β )) = E ( (cid:96) t − (cid:96) ( β )) . Defining the minimizer of this distance as β = ( θ ∗ , . . . , θ ∗ c − , β (cid:48) , β (cid:48) , β γ (cid:48) ) (cid:48) leads to β = arg min β KL ( L t || L ( β )) . Thus, β minimizes the unpenalized log-likelihood (cid:96) ( · ) , that is E ( g ( β )) = 0 , with g being thegradient of (cid:96) ( · ) . The Hessian is denoted by H ( β ) . Both gradient and Hessian have penalizedversions: g p ( β ) = g ( β ) − S λ β , H p ( β ) = H ( β ) − S λ . We assume the following set of conditions related to gradient and Hessian:(A1) g ( β ) = O P ( n / ) (A2) E ( H ( β )) = O ( n ) (A3) H ( β ) − E ( H ( β )) = O P ( n / ) (A4) S λ = o ( n / ) The first three assumptions are standard conditions when showing consistency of the un-penalizedMLE and (A1) and (A3) mean that n g ( β ) and n H ( β ) converge in probability to their expectedvalues with rate n / . Kauermann (2005) gives an alternative formulation of (A4): λ ϑ k = o ( n / ) ,which means that the penalty sub-matrices of S λ corresponding to ϑ k are asymptotically boundedand the penalty term becomes less and less important for the fitting procedure as n → ∞ .Regarding consistency, one has to show that ˆ β − β = O P ( n − / ) with n → ∞ . (13) Proposition 1.
Let β be the "true" parameter vector as defined above, under conditions (A1) -(A4) the penalized ML estimator ˆ β satisfies ˆ β − β = ( − E ( H ( β )) + S λ ) − ( g ( β ) − S λ β )( I + o P (1)) implying convergence in probability at rate n − / and hence consistency of ˆ β . roof. Using the Taylor expansion of g p (ˆ β ) at point β , yields: g p (ˆ β ) = g p ( β ) + H p ( β )(ˆ β − β ) + . . . since g p (ˆ β ) = 0 ⇐⇒ ˆ β − β = − H p ( β ) − g p ( β )= − H p ( β ) − ( g ( β ) − S λ β ) We decompose H p ( β ) and obtain: H p ( β ) = H ( β ) − E ( H ( β )) − ( − E ( H ( β )) + S λ ) . Upon defining R = H ( β ) − E ( H ( β )) as a stochastic error term and the penalized Fisher infor-mation matrix F = − E ( H ( β )) + S λ ) , we calculate the Taylor expansion of f ( R ) = H − p ( β ) at f ( ) . The auxiliary function f ( · ) = ( · − F ( λ )) − takes a matrix as input. We obtain: H − p ( β ) = − F ( λ ) − − F ( λ ) − R ( F ( λ )) − ) (cid:48) + . . . = − F ( λ ) − ( I + RF ( λ ) − + . . . )= − F ( λ ) − ( I + O P ( n − / )) with (A2)-(A4) ⇐⇒ ˆ β − β = ( − E ( H ( β )) + S λ ) − ( g ( β ) − S λ β )( I + o P (1)) , which proves the stated proposition and hence consistency as in equation (13).The argumentation above is in line with maximum likelihood theory and is also adopted by Kauer-mann (2005) and Kauermann et al. (2009) to derive asymptotic results on penalized spline smooth-ing. With Proposition 1, we can also derive the bias and covariance matrix of ˆ β , i.e.bias: E (ˆ β ) − β = − F ( λ ) − S λ β ( I + o P (1)) with E ( g ( β )) = 0 covariance: Cov (ˆ β ) = − F ( λ ) − E ( Hβ F ( λ ) − ( I + o P (1)) with Cov ( g ( β )) = − E ( H ( β )) . Taking these considerations together with (A2) and (A4), we can characterise the asymptotic orderof bias and covariance matrix as o ( n − / ) and O ( n − ) , respectively.To guarantee an asymptotically normal behaviour of the score, we need an additional assumption:(A5) ∀ β s ∈ β : ∂ (cid:96) ( β ) /∂β s exists and is bounded in the neighbourhood of β s , that is | ∂ (cid:96) ( β ) /∂β s | ≤ M ( ν ) , with E ( M ( ν ) | β s ) < ∞ , for all ν ∈ R . Furthermore, ≤ I ( β s ) < ∞ .As we assume the observations to be independent, g ( β ) and H ( β ) consist of sums of i.i.d.random variables. Therefore, ( − E ( H ( β ))) − / g ( β ) d −→ N ( , I ) , which, together with Proposition (1), implies asymptotic normality of ˆ β . As in Radice et al. (2016)note, the normal approximation is not accurate in case the copula parameter is bounded and thesample size is small. 34 ppendix E Simulations To assess the method’s effectiveness, we report the results of four simulation exercises. The firstscenario corresponds to our application setting in terms of marginals and type of copula, i.e. usinga normal copula and the lognormal distribution as the continuous marginal. We then change eitherthe second marginal to a gamma distribution (scenario 2) or the copula to a Joe copula (scenario3), or both (scenario 4).In all scenarios, the response vector consists of an ordinal outcome with 3 levels and a continuousvariable which follows either a lognormal (scenario 1 & 3) or a gamma (scenario 2 & 4) distribution.The five covariates, x , x , x , ν , ν are uniformly distributed on the [ − , interval and two ofthem, ν and ν , enter the model in a non-linear fashion. Each parameter’s additive predictor isconstructed in a way that they are roughly symmetrical around zero with most values in the [ − , interval. The copula is Gaussian in scenario 1 and Joe copula in scenario 2.The i -th predictors are constructed as follows: η µ i = θ ri − ( β µ x i + s µ ( ν i ) + s µ ( ν i )) η µ i = β µ + β µ x i + β µ x i + s µ ( ν i ) η σ i = β σ + β σ x i η γi = β γ + s γ ( ν i ) , where s , s , s are the three different smooth functions below: s ( ν ) = ν ∗ sin(3 ∗ ν ) s ( ν ) = sin(2 ∗ ν ) + 0 . ∗ νs ( ν ) = 3 ∗ ν ∗ cos( ν ) . For each scenario we run three different cases that differ in number of observations. We considercase 1 with n = 1 , , case 2 with n = 3 , and case 3 with n = 10 , . For each iteration,we store the coefficients of the linear effects and estimated smooth functions. There were alsoiterations that gave warning messages about the convergence of the model. As these warningsoften indicates that the model is too complicated for the number of observation, convergence failsmore frequently for the cases with n = 1 , . Sometimes, even if these warnings occur, the fitmight be reasonable. Users are nonetheless advised to check the model carefully whenever warningmessages are returned. We decided to drop iterations with warnings and move on to the next onesuntil 100 simulation runs without warnings were obtained. The number of iterations displayingwarning messages are reported in Table 5. The number of those warnings decreases drastically asthe sample size increases. For scenario 1, no iterations had to be excluded due to non-convergence,while for scenarios 3 and 4 more iterations returned warning messages. Although the number ofparameters is equal in all four scenarios, based on the number of repetitions, it seems that scenarios1 and 2 using a Gaussian copula are relatively easier to be fitted than scenarios 3 and 4 whichemploy a Joe copula.Note that the aim of this simulation is to check the implementation and the ability to estimatereliably the model’s coefficients. How well GJRM() selects the correct copula specification via theAIC and BIC is considered elsewhere (e.g., Marra and Radice, 2017; Radice et al., 2016).35 l ll X (a) coefficients for x of µ l X (b) coefficients for x of µ l −0.22−0.20−0.18 1000 3000 10000n X (c) coefficients for x of µ l ll X (d) coefficients for x of σ Figure 8: Simulation results for linear effects in scenario 1. The boxplots represent the estimatedlinear coefficients in N = 100 iterations. The true values of the coefficients are denoted by theblack diamond symbols.Figure 8 shows the boxplots of all estimated coefficients for the linear effects in scenario 1. Theestimator captures the effect fairly well and the performance improves with the sample size.Figure 9 exemplarily shows the estimated smooth functions against the true ones for scenario 1and n = 1 , . The procedure is able to recover the smooth functions satisfactorily although someof the curves are wigglier or too smooth compared to the true functions. This effect vanishes inthe other cases as the sample size grows. Note that for the location parameter of the continuousmarginal, the smooth effects seem to be easier to estimate as all curves are very close to the truefunctions. If we increase the sample size, the fit of the smooth functions improves significantlyespecially at the local minima and maxima (plots are available upon request).Note that some simulation iterations for the case of n = 1 , were problematic in that the smoothfunctions were not estimated adequately. Their number was rather small and we excluded them.Table 5 gives an overview of how many iterations were excluded.The results for scenarios 2, 3 and 4 are similar to the first one and shown in Figures 10, 11 and 12,respectively. Again, the estimation of the smooth terms improves considerably as the sample sizeincreases but we only show the more difficult case of n = 1 , here to demonstrate that resultsare still acceptable also at relatively small sample sizes.36 s ( Z ) (a) s µ ( ν ) −2−101 −2 −1 0 1 2Z2 s ( Z ) (b) s µ ( ν ) −2−1012 −2 −1 0 1 2Z1 s ( Z ) (c) s µ ( ν ) −2.50.02.55.0 −2 −1 0 1 2Z2 s ( Z ) (d) s γ ( ν ) Figure 9: Simulation results for non-linear effects in scenario 1. The pink solid lines represent thetrue functions.scenario case rejected by algorithm manually deletedscenario 1 n = 1,000 0 3n = 3,000 0 0n = 10,000 0 0scenario 2 n = 1,000 1 1n = 3,000 1 0n = 10,000 1 0scenario 3 (J0) n = 1,000 62 2n = 3,000 3 0n = 10,000 0 0scenario 4 n = 1,000 77 1n = 3,000 3 0n = 10,000 2 0Table 5: Number of repeated iterations due to warning messages and number of manually deletediterations that were extreme outliers. 37 l l X (a) coefficients for x of µ ll ll X (b) coefficients for x of µ lll −0.22−0.20−0.18 1000 3000 10000n X (c) coefficients for x of µ ll X (d) coefficients for x of σ −101 −2 −1 0 1 2Z1 s ( Z ) (e) s µ ( ν ) −2−1012 −2 −1 0 1 2Z2 s ( Z ) (f) s µ ( ν ) −2−1012 −2 −1 0 1 2Z1 s ( Z ) (g) s µ ( ν ) −2.50.02.55.0 −2 −1 0 1 2Z2 s ( Z ) (h) s γ ( ν ) Figure 10: Simulation results for linear and non-linear effects (case n = 1000 ) in scenario 2. Theboxplots represent the estimated linear coefficients in N = 100 iterations. The true values of thecoefficients are denoted by the black diamond symbols. The pink solid lines represent the truefunctions of the non-linear effects. 38 llll l X (a) coefficients for x of µ l X (b) coefficients for x of µ l ll −0.22−0.21−0.20−0.19−0.18 1000 3000 10000n X (c) coefficients for x of µ l X (d) coefficients for x of σ −101 −2 −1 0 1 2Z1 s ( Z ) (e) s µ ( ν ) −2−101 −2 −1 0 1 2Z2 s ( Z ) (f) s µ ( ν ) −2−1012 −2 −1 0 1 2Z1 s ( Z ) (g) s µ ( ν ) −10−50 −2 −1 0 1 2Z2 s ( Z ) (h) s γ ( ν ) Figure 11: Simulation results for linear and non-linear effects (case n = 1000 ) in scenario 3. Theboxplots represent the estimated linear coefficients in N = 100 iterations. The true values of thecoefficients are denoted by the black diamond symbols. The pink solid lines represent the truefunctions of the non-linear effects. 39 l X (a) coefficients for x of µ ll ll X (b) coefficients for x of µ lll l −0.23−0.22−0.21−0.20−0.19−0.18−0.17 1000 3000 10000n X (c) coefficients for x of µ X (d) coefficients for x of σ −101 −2 −1 0 1 2Z1 s ( Z ) (e) s µ ( ν ) −2−101 −2 −1 0 1 2Z2 s ( Z ) (f) s µ ( ν ) −2−1012 −2 −1 0 1 2Z1 s ( Z ) (g) s µ ( ν ) −5.0−2.50.02.5 −2 −1 0 1 2Z2 s ( Z ) (h) s γ ( ν ) Figure 12: Simulation results for linear and non-linear effects (case n = 1000 ) in scenario 4. Theboxplots represent the estimated linear coefficients in N = 100 iterations. The true values of thecoefficients are denoted by the black diamond symbols. The pink solid lines represent the truefunctions of the non-linear effects. 40 ppendix F Additional material for the application case Histogram and Density Estimate of Residuals
Quantile Residuals D en s i t y −4 −2 0 2 4 . . . . . ll ll ll lll l llll lll ll l ll lll l ll l lll l ll lllll l ll lll llll llll l lll ll ll lll ll ll llll l lll ll ll ll llll l lll l llll lll l llllllllll l ll lll ll l lll ll lll lll l llll lll lll l lllll l l l l lllllll l llll llll lllll l lllll ll lllllll llll lll ll lll l ll lll llll ll lllllll l llll llll ll l llll ll l ll ll llll lll ll lll llll llllll llllllll l l llllllllll l ll l l lll lllllll ll lllll llll l llll l llll lll lll lll lll llll ll llllllll llll ll lll lllllll llll l lll ll ll ll ll l lll lll llllll lll lll ll ll lllll lll llllll ll llllllllll lll ll lllllllll l llllll llllllllllll lll llll ll llll llll ll llll lll l lll l llll lllll l lll llll lllllll ll lllllll llll llllll lll llll l lllllllll ll ll lllll lllll l llll llll llll lll l llllll llll llllllll llllllllll llllll lllllllllllllllll llll llll llll ll llll lllll llllll ll ll llllllllllll lllllllllllll ll llll llllllll lll ll lllllllll lllllll llll ll lll ll lllllll llllll ll llll llll lllllllll ll lll ll lll ll lll lllllll llllllll llllll llllllllllllll lllllllllll llllll llllll lll llll llllllll lllllllllllll lllll llll llll llllll lllll l lll llll l llll lllllll lllllll l ll lll ll llllll llllll llll llllll lll llll llllll l lll lllll l llll lllllllll ll l lllll ll lll lll llll l lllllllll lllllllll lll ll llll llll ll lllllll lllllll ll lllllllll ll lll l llllllllllll l l llllllllllll ll lllll lllll llllllll l lllllllllll llllllll llll l lll llllll lllllllll ll l ll llll lll llll lllllll ll llll ll lllllllll lll llllllllllllllllll ll lllll ll ll ll llll lllllllll lll lll llll llll l ll lllll llllllllll l lllllllll lll lllll llll llllll lll l lllll ll llllll lll lll llllll lllll lllll llll lllllll llllllllll lllll ll lllllllllllll lll lll l ll lllll l ll l lllllllllll lllllllllllll llll ll llll llll lll l lll llllll lll ll ll ll lll llll ll lllll l lll l lllll llllll llllll llllllll llllll l ll lllll llll l l ll llll llll l lll lll l ll lllllll lllll lll ll lllll lllll ll lll llllll lll lll ll ll ll lll ll l llll ll ll ll llllll llll ll ll lll l lll lll lll lll lll llll llll llll lll ll lll ll lll ll ll l lllll lll l l ll lll ll ll l lll l lll llll lllll ll ll lllllllllllll lll lll l llll lll ll ll llll lll ll llll ll l llll l l lll ll l ll l ll lll lll lll llll l ll lll ll l lll llllll l l lll lllll l llll ll lllll llllll lll l ll lll lll ll ll llll l l ll ll ll ll ll ll ll ll llllll lllll ll ll llll l lllll ll l l lllll ll llll ll ll ll llll l ll lllll llll llll ll l l lll llll l llll llllll lll lll l llll ll lll ll llllll ll ll ll llll l lll ll lll ll lll ll l ll ll lll ll ll lll l ll l ll ll ll lll l ll lll ll ll llll lll ll ll lll ll ll l ll l lllll ll l ll l l lll ll ll ll l llll ll llll lll ll ll lll lll l ll ll lll ll ll l llll llll ll lll ll lll l l lll l lllll l ll lll ll llll lllll l lllll ll l lll ll ll ll ll ll ll lll ll ll l l lll ll lll l lll ll ll lllll l ll lll ll ll ll lll lll l lll llll l 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ll llllll lll lll ll ll l llll ll lll lll lll ll ll ll ll lll ll llll ll l llll l lll ll llll ll llll l llll ll l lll ll ll ll l ll ll l l ll lll l lll lll lll ll lll llll l l lll ll ll lll ll ll lll lll l ll lll ll l lll lllllll l l l ll ll ll lll l lll lll lllllll lll l lllll lll ll lll l ll llll l lll l lll llllll lll llll ll lll l llll l ll ll ll l llll l lll ll llll lll ll l ll ll llll ll l ll llll ll l lll ll llllllllll llll lllll l llll l ll lll l lll llllll llll llll lll ll l llll llll l ll lll lll lll llll l ll llll l llll l lll ll l llllll l lll llll l l ll llll l ll llllll lll lllll l lll llll lll llllll ll lll lll llll lll ll llll l l l lll ll lllll ll lll ll ll llll ll llll llll lll llll l llllll ll lll llll lllll llll ll llll lllllllll ll ll l llllll ll lllllll l llllll llll lll l ll lllll lll llll lll llll lll llll lllll ll llllll lll l llll lll l llll ll llll lll l lll ll llll lllllllllllll ll l ll lll llll lllll llllll llll lllllll l ll lll llll llll ll lllllllllll 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lllllll ll ll llll l ll ll lllll lll ll lllll l lllll llll l ll lllllll ll lllll lllll lllll l lll llll lll lll ll l lll lllllll ll llll llll ll ll ll ll llll ll llllllllllllllll llll ll l ll ll lllll lll lll ll llll lll lll ll ll ll ll ll ll l ll lllll ll l lllll l llll l lllllll ll lll ll l lll ll ll lll ll ll l l ll ll lll l ll l lllll lll l lll lllll llll ll lll llllll ll ll ll lll l lll l llll l l ll ll l ll lll lll llll lll llll l ll l lll ll ll ll llll lll ll l ll lll l l lll ll l ll ll lllll lll ll llll llll ll ll l lll ll llll ll l lll ll llll ll llll lll ll llll l lll ll ll lll lll lll ll ll l l lll lll l lll lll llllll l ll l lll lllllllll lll lll llll lllll llllllll l ll lll l lll ll lll ll l llll ll ll ll lll l ll lll lll llll ll llll lllll lll lll ll l llll lllll lllll lll ll l ll lllll ll lllllllll ll lllll lll llll lll lll lllllllllllllll ll lll l llll llll lll ll l ll llll l lll ll ll llll l lll llll lllll llll l lllll llllll lll lllll l l ll ll ll ll l ll l lllll 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lll llllll l llllll lllll lll ll lllll lll llllll ll l llllllll lllllllll lllllll lllllll llll ll llllll ll lllllll lll llllll llll ll ll lllllllll llllll llll ll lllllll llllll lllllllllll llll lll llll ll llllll llllll ll lllll llll lll lllll l llllllllll lll llllll lllllllll llllllll lllllll lllllllllll ll llll ll llll llll ll lll lll lll ll lllllllll llll llll llllllllll llll ll lll llllllll ll llll llllllllllll lllll llllllll llll llll llll lllllllllll ll l llllllll lll llllll llll lll lllll lllllllll lll lllllll ll llllllll lllllll lllll lll lllllll ll llll ll lll llll llllll lllllllllll ll llll ll l llllll l lllll llllll llllll llllll lllllllll ll l lllll lllll lllll ll llllllllll ll lllll llll ll ll lllllllllll ll lll lll lllll lll lllll llll l lllll lllllll l llll l llllll lllll lll lll ll llll llllll lll llll ll llllll llll l lllllll llll lll llll lll lll lll lllllllllll lllllllll l ll ll lllllllll l llll lllll l llllll lllll ll l llll llllll ll lll ll llll llll lll l l lllllllllllllllll ll ll ll l lll lll lllll lll llll llll ll ll llllllll ll lllll llll llll lllllll llllllll llll l lllll l ll l ll l ll lllll l ll llllllll lll lllll llll llll lll lll lllllllll lllllllllllllll l llllll ll llll llllll ll llllll lll llllllll l ll ll llll lll ll ll ll ll lllll llll ll llll ll ll lll ll l lllll ll l lll l lll l lll lllll lllll llllll l ll lllll ll llll l llll lll ll ll l ll llllll l lllllll l lll ll lll lll l llll lll lll llll lll llll ll ll ll llll l lllll l lll lll ll ll llll llll l ll l lll ll ll l ll lll lll ll ll ll ll l l l llll lll lll lll l l llll l lllll ll llll ll ll ll l l llll l lllll lll ll lll l lll lll ll l ll lll lll lllll l ll ll ll ll l ll lll l l llllll lll l ll ll ll lll lll ll lllllll llll l lll ll llll l ll l lll ll lll l lll ll lllll lll l lll ll ll lll llll l llll ll lll ll lll ll ll l l lll lll l ll ll l ll ll l ll lll l lll ll ll ll l llll llll l ll l lll ll lll lll ll l lll l llllll ll ll ll ll lll l ll lll lll l llllll ll ll ll ll ll l lll ll llll ll l lllll ll l lll ll l ll l lllll lllll llllll ll l ll lllll lllll ll l lllll ll lll ll llll lll lll lll lll l ll lll ll ll lllll l ll lll l lll lllll llll lll ll ll llll lll lll l lll ll ll ll llll l lll llll ll lll llll l lllll l lll ll l l llllll lllll l l lll ll ll ll ll lll ll ll ll l l ll ll ll llll ll ll ll l lll llll lll ll lllll lll ll l lllll ll lll ll lll llll lll ll lll llll ll l ll lll lll ll ll l lllll l lllll l ll ll ll l llll lll ll lllll l ll l lll lll llll ll lll l ll ll lll l lll ll ll lll llll l l llllll ll lll l ll lllll lll l lll l ll lll llllll lll ll ll l ll llll l llll lll l ll llll lllll llll lllll ll l ll lllll lll l l l llllll ll llll lll ll ll lll ll lll l lllll lll ll l lll llll lllll ll l lll lll l llll lll ll ll llll l lll ll ll lll ll l lll lll ll llll lll lll l lll lllll lllll lll l ll ll lll l llll l lllll lll ll lll llll ll lllll l ll lllll lll lll l ll ll lll l llll lllll ll l lll l ll l lllll ll ll lll lllll l ll ll lll lll lll ll llll lll ll l l ll l lllll lll lll ll l ll ll l l llll lll ll lll ll llll l lll llll ll lll llll lll l ll lll ll l ll lll ll l llll lll ll lll lll ll ll llllllllll l ll l ll l l llll l ll llll ll lll lll l lll lll ll ll ll l lllll ll ll ll lll l lll llll lll ll ll l lllll ll ll lll l ll ll llll lll ll l ll ll l lll ll llll l llll llll ll ll l llll ll ll ll llll ll ll l ll ll ll lll lllll ll lll l ll lll llll l lll l ll lll ll llll ll ll l lll l lll lll llllll l ll lll l llll l ll lll ll lll l l ll l ll llll l llll ll llll lllll llll ll l lll ll l l llllll ll llll ll ll ll ll lll llll llll lll llll ll lll lll ll l ll ll l ll ll llll l lll llll ll lll l lll l lllll ll llll lll l l lllll ll lllll l lll l lllll l ll lll lll l ll lll l ll l llll ll lllll llll llll llll llll ll ll lll llll ll l ll ll ll lll ll l l llll ll lll ll l lll lll llllll lll lll l lll ll ll llll l lll ll ll lllll lll l lll l lllll l ll lll l lll ll ll ll ll ll llll llll ll l ll llllll l lll l ll lll l llll lll ll ll ll ll lll l llll lll llll lll lll lll l ll l −4 −2 0 2 4 − − Normal Q−Q Plot
Theoretical Quantiles S a m p l e Q uan t il e s Figure 13: Histogram and normal Q-Q plots for the log-normal continuous margin of the finalbivariate copula model. 41 children latent education pe r c ap i t a i n c o m e −4 −2 0 2 4 + + + + g = latent education pe r c ap i t a i n c o m e −4 −2 0 2 4 + + + + g = latent education pe r c ap i t a i n c o m e −4 −2 0 2 4 + + + + g = latent education pe r c ap i t a i n c o m e −4 −2 0 2 4 + + + + g = latent education pe r c ap i t a i n c o m e −4 −2 0 2 4 + + + + g = latent education pe r c ap i t a i n c o m e −4 −2 0 2 4 + + + g = latent education pe r c ap i t a i n c o m e −4 −2 0 2 4 + + + g = latent education pe r c ap i t a i n c o m e −4 −2 0 2 4 + + + g = Figure 14: Contour plots for (education, income)’ and a Gaussian copula by different numbers ofchildren and elderly living in the same household. Contour lines of densities are at levels from0.00000005 to 0.00000025 in 0.00000001 steps. The vertical straight lines represents the cut offvalues for the education categories, horizontal straight lines are the consumption average, anddashed horizontal line are at two standard deviations around this average.42able 6: Effects on educationEstimate Std. Error z value Pr( > | z | )cutoff 1 -3.170 0.071 -44.348 0.000cutoff 2 -0.201 0.008 -24.059 0.000cutoff 3 0.761 0.006 129.930 0.000cutoff 4 2.618 0.006 413.481 0.000marital status of household head: married -0.460 0.063 -7.272 0.000marital status of household head: separated -0.637 0.089 -7.128 0.000marital status of household head: widowed -0.448 0.073 -6.151 0.000household head is male 0.185 0.042 4.353 0.000urban dummy 1.042 0.022 48.383 0.000number of children: 1 -0.166 0.028 -5.969 0.000number of children: 2 -0.131 0.030 -4.366 0.000number of children: 3 -0.168 0.043 -3.936 0.000number of children: 4-7 -0.332 0.067 -4.949 0.000number of elderly: 1 -0.003 0.031 -0.111 0.912number of elderly: 2 or 3 0.191 0.059 3.250 0.001religion: Christian 1.002 0.047 21.175 0.000religion: Hinduism and other -0.028 0.048 -0.576 0.565 Note:
Base categories for marital status is “not yet married”, for number of children “no children”, for number ofelderly “no elderly”, and for religion “Islam”.
Table 7: Effects on µ of the income distributionEstimate Std. Error z value Pr( > | z | )Intercept 14.430 0.031 465.128 0.0000marital status of household head: married -0.142 0.021 -6.674 0.0000marital status of household head: separated -0.140 0.029 -4.834 0.0000marital status of household head: widowed -0.145 0.024 -5.997 0.0000household head is male -0.060 0.013 -4.754 0.0000education of household head: primary 0.127 0.022 5.734 0.0000education of household head: middle school 0.245 0.023 10.713 0.0000education of household head: high school 0.388 0.023 16.944 0.0000education of household head: tertiary education 0.663 0.026 25.048 0.0000urban dummy 0.120 0.007 17.852 0.0000number of children: 1 -0.259 0.008 -32.190 0.0000number of children: 2 -0.414 0.009 -47.637 0.0000number of children: 3 -0.559 0.012 -45.353 0.0000number of children: 4-7 -0.738 0.019 -39.319 0.0000number of elderly: 1 -0.189 0.009 -21.082 0.0000number of elderly: 2 or 3 -0.312 0.017 -18.660 0.0000religion: Christian 0.059 0.016 3.818 0.0001religion: Hinduism and other 0.166 0.026 6.441 0.0000 Note:
Base categories for marital status is “not yet married”, for education “no schooling”, for number of children“no children”, for number of elderly “no elderly”, and for religion “Islam”. σ of the income distributionEstimate Std. Error z value Pr( > | z | )Intercept -0.426 0.022 -19.394 0.000marital status of household head: married -0.176 0.023 -7.617 0.000marital status of household head: separated -0.075 0.033 -2.262 0.024marital status of household head: widowed -0.116 0.026 -4.458 0.000number of children: 1 -0.065 0.011 -6.132 0.000number of children: 2 -0.081 0.011 -7.021 0.000number of children: 3 -0.066 0.017 -4.000 0.000number of children: 4-7 -0.076 0.026 -2.958 0.003religion: Christian 0.046 0.019 2.394 0.017religion: Hinduism and other 0.035 0.026 1.310 0.190 Note:
Base categories for marital status is “not yet married”, for number of children “no children”, for number ofelderly “no elderly”, and for religion “Islam”.
Table 9: Effects on the copula parameterEstimate Std. Error z value Pr( > | z | )Intercept -0.179 0.045 -3.998 0.000marital status of household head: married 0.143 0.034 4.219 0.000marital status of household head: separated 0.170 0.048 3.517 0.000marital status of household head: widowed 0.177 0.039 4.543 0.000education of household head: primary 0.061 0.033 1.877 0.060education of household head: middle school 0.104 0.036 2.906 0.004education of household head: high school 0.161 0.033 4.853 0.000education of household head: tertiary education 0.120 0.036 3.337 0.001urban dummy 0.101 0.013 7.816 0.000number of children: 1 0.018 0.016 1.128 0.259number of children: 2 0.041 0.017 2.411 0.016number of children: 3 0.084 0.025 3.391 0.001number of children: 4-7 0.063 0.036 1.746 0.081number of elderly: 1 0.075 0.017 4.402 0.000number of elderly: 2 or 3 0.113 0.030 3.748 0.000 Note:
Base categories for marital status is “not yet married”, for education “no schooling”, for number of children“no children”, and for number of elderly “no elderly”. ppendix G Software We incorporated the models proposed this paper into the
GJRM package (Marra and Radice, 2019)in R (R Core Team, 2019). A mixed ordered-continuous model is called by setting the option ordinal = TRUE . The main fitting function, gjrm() , is very easy to use in that its syntax followsthose of linear models, generalized linear models, or generalized additive models. The function CopulaCLM() is called internally to fit this specific model. An example of model specification isgiven below. eq.educ <- educ_att ~ s(age) + as.factor(hhmarstat) + as.factor(hhmale) +as.factor(urban) + as.factor(num_child) +as.factor(elderly) + as.factor(relig)eq.mu <- pce.defl ~ s(age) + as.factor(hhmarstat) + as.factor(hhmale) +as.factor(urban) + as.factor(num_child) +as.factor(elderly) + as.factor(relig) +as.factor(hheduc) + s(prov, bs = "mrf", xt = xt1, k = 15)eq.si <- ~ s(age) + as.factor(hhmarstat) +as.factor(num_child) + as.factor(elderly) +as.factor(relig) + s(prov, bs = "mrf", xt = xt1, k = 15)eq.theta <- ~ s(age) + as.factor(hhmarstat) +as.factor(urban) + as.factor(num_child) +as.factor(elderly) + as.factor(hheduc) +s(prov, bs = "mrf", xt = xt1, k = 15)form.list <- list(eq.educ, eq.mu, eq.si, eq.theta)mod.edu <- gjrm(form.list, data = na.omit(df), ordinal = TRUE,Model = "B", BivD = "N", margins = c("logit", "LN"),drop.unused.levels = FALSE, gamlssfit = TRUE)
45e user first specifies the four equations for the model’s parameters of the marginal distributionsand of the copula, which are stored in the list form.list . Continuous variables enter the modelspecifications via smooth effects s() represented (by default) via thin-plate regression splines (ar-gument bs = "tp" ) with ten basis function and second order derivative penalties. Spatial effectsof the provinces are modeled using Markov random fields with neighbourhood structure xt1 and15 knots (argument bs = "mrf" ). Argument
Model = "B" specifies that a bivariate model willbe estimated, margins = c("logit", "LN") gives the marginal distributions and
BivD = "N" specifies the Gaussian copula. The argument ordinal must be set to
TRUE in order to fit amixed ordered-continuous model and the ordinal outcome educ_att must be numeric. The op-tional argument gamlssfit = TRUE uses starting values obtained from a univariate gamlss and drop.unused.levels = FALSE is needed because not all of the provinces specified via the Markovrandom fields have observations in the data frame df . Functions summary() , plot() , AIC() and
BIC() can employed in the usual manner. It is advisable to use post.check() after fitting themodel to produce plots of normalized quantile residuals. More details, options, and the availablechoices for the marginal distributions and copula functions can be found in the documentation ofthe