Confidence sets for dynamic poverty indexes
CConfidence sets for dynamic poverty indexes
Guglielmo D’Amico a, ∗ , Riccardo De Blasis b , Philippe Regnault c a Universit`a ”G. DAnnunzio”, Chieti, Italy b Universit`a Politecnica delle Marche, Ancona, Italy c Universit´e de Reims Champagne-Ardenne, Reims, France
Abstract
In this study, we extend the research on the dynamic poverty indexes, namely the dynamicHeadcount ratio, the dynamic income-gap ratio, the dynamic Gini and the dynamic Sen,proposed in D’Amico and Regnault (2018). The contribution is twofold. First, we extend thecomputation of the dynamic Gini index, thus the Sen index accordingly, with the inclusionof the inequality within each class of poverty where people are classified according to theirincome. Second, for each poverty index, we establish a central limit theorem that gives usthe possibility to determine the confidence sets. An application to the Italian income datafrom 1998 to 2012 confirms the effectiveness of the considered approach and the possibilityto determine the evolution of poverty and inequality in real economies.
Keywords:
Markov process, Population dynamic, Nonparametric estimation, Micro-data
JEL:
I32, P46
1. Introduction
Economists and econometricians have dedicated a lot of efforts to the investigation ofpoverty. The literature is vast and comprehends theoretically oriented contributions as wellas applied researches. The advancement to more powerful indexes of poverty is always ofinterest and it aims at capturing specific peculiarities of the phenomenon that were ignored byprevious indicators (see, e.g., Hirsch et al., 2020). In particular, multidimensional measures ofpoverty are relevant in this context, with the inclusion of non-monetary sources of deprivationwhich affect the well-being of individuals and households such as disability, exposure to ∗ Corresponding author
Email address: [email protected] (Guglielmo D’Amico) a r X i v : . [ ec on . E M ] J un nvironmental hazards and limited availability of healthcare services (see, e.g., Park andNam, 2020; Annoni et al., 2015; Parodi and Sciulli, 2008). This research stream has hisorigin in the seventies of the last century when the stately contribution by Professor Senappeared (Sen, 1976). Since then, continuous improvements and generalisations have beenmade (see, e.g., Takayama, 1979; Shorrocks, 1995; Foster, 2009).Almost at the same time, it was recognised that poverty is not a static notion andthat this characteristic should be investigated in relation to time. The main idea is tostatistically assess frequencies of poverty condition and their variations through time. Thesestatistical properties of poverty mobility were determined and confirmed in several studies(see, e.g., Bane and Ellwood, 1986; Duncan et al., 1993; Whelan et al., 2000). In general,understanding the features of poverty over time needs the adoption of a stochastic modelof income evolution. Therefore, Markov chain models were frequently used. Some examplesare available in McCall (1971), Breen and Moisio (2004), Cappellari and Jenkins (2004),Formby et al. (2004), and Langheheine and Pol (2016). In addition, poverty rates andtransition probabilities have been estimated in relation to noisy data by Lee et al. (2017).In contrast to previous research, the idea of advancing dynamic indexes of poverty andincome inequality is relatively new and limited to a few contributions. Precisely, at authorsknowing, two approaches can be identified. They share the same starting idea, namely theextension of static indicators into a dynamic framework, but then move away in relation tomethods and results to answer different questions. On the one hand, Ewald and Yor (2015)consider a sequence of distributions parametrised in time and they look for conditions underwhich the corresponding sequence of the indicator (in the specific case, the Gini index)increases over time. On the other hand, starting from the work by D’Amico and Di Biase(2010), D’Amico et al. (2012) and D’Amico et al. (2014) built up an economic system byadvancing a set of assumptions on the time evolution of income for every agent member ofthe economic system. This approach was also adopted in D’Amico and Regnault (2018) inrelation to dynamic measures of poverty where the dynamic indexes were evaluated bothfor finite and infinite size economic system. Specifically, the infinite size system (i.e., aneconomy with an infinite number of agents) revealed to be particularly interesting. In fact,for each index, using probabilistic arguments (i.e., strong law of large numbers), it is possible2o determine a deterministic function (of the parameters of the model) to which the indexof the real economy converges to.In this paper, we move further steps into this direction. First, we extend the computationof the dynamic Gini index including the inequality within each class of poverty where peopleare classified according to their income. This extension impacts also the Sen index that isa function of the Gini index. Second, for each poverty index, we establish a central limittheorem that gives us the possibility to determine confidence sets, i.e., bounds that at afixed probability level express the goodness of the approximations based on the strong lawof large numbers. These results derive from the specific assumptions defining the modelwhich are based on the probabilistic equality of the incomes of people belonging to thesame income class and that may migrate in time from one class to another according to acontinuous time Markov process. Finally, we present an application of the aforementionedprobabilistic approximations on the Italian income data provided by the Italian CentralBank from 1998 to 2012 which contains information about family net disposable incomesand household members. The results of the application suggests the effectiveness of theconsidered approach and confirm the possibility to apply it for the determination of theevolution of poverty and inequality in real economies.The remainder of the paper is organised as follows. Section 2 sets out the assumptions thatdefine the model and presents the main theoretical results of the paper including probabilisticapproximations of the indexes and their confidence sets. Section 3 illustrates the result ofthe application to real data and demonstrates the adequacy of the proposed approach tothe investigation of the evolution in time of dynamic indexes of poverty in real economicsystems. Section 4 summarises our contribution and results. All proofs are deferred to theAppendix.
2. The stochastic model and confidence sets for dynamic poverty indexes
In this section we present the mathematical model. First, we introduce the stochasticmodel of income evolution and the dynamic version of four poverty indexes in the case ofinfinite size economic systems. Then, we derive the confidence sets for the poverty indexes.They are obtained by proving the central limit theorem for the stochastic processes expressing3he dynamic poverty indexes.Following D’Amico and Regnault (2018), we consider an economic system composed ofa set H of N individuals. The income produced by each economic agent evolves randomlyin time and can be described through a stochastic process Y = ( Y h ( t )) t ∈ R + , where h denotesthe h-th individual in the economic system and t is the time variable. For our purposes weclassify individuals according to their income in one of three exhaustive and exclusive incomeclasses denoted by a random process C h ( t ) such that: C h ( t ) := C if Y h ( t ) ≤ y ep , C if y ep < Y h ( t ) ≤ y p , C if Y h ( t ) > y p , where y p and y ep are the poverty and extreme poverty threshold rates, respectively. Clearly,the possibility to extend the model to multiple richness classes is straightforward.In the remainder of the paper, the simplifying notation { , , } will be used to denotethe set {C , C , C } .The following assumptions advanced in D’Amico and Regnault (2018) define the model: A1 : the number N of individuals in the economic system is finite and constant in time; A2 : the income rate processes ( Y h ) h ∈H are independent and hence the class allocationprocesses ( C h ) h ∈H ; A3 : the processes ( C h ) h ∈H are identically distributed ergodic Markov processes taking valuesin the set {C , C , C } with infinitesimal generator matrix Λ ; A4 : For any time t ∈ R and any individual h ∈ H , the conditional distribution of the income Y h ( t ) knowing that C h ( t ) = C i , with C i ∈ E , does not depend on past income values, noron t or h . We denote it as F i and we assume that it possesses finite first and second ordermoments. In symbol, D ( Y h ( t ) | σ t − ( Y h ) , C h ( t ) = C i ) = D ( Y h ( t ) | C h ( t ) = C i ) =: F i ( · ) , ∀ t ∈ R , ∀ h ∈ H , (1)where σ t − ( Y h ) := lim s → t − σ s ( Y h ) is the sigma-algebra generated by the income process ofagent h up to time t but excluding it. 4ow, according to D’Amico and Regnault (2018) we present the stochastic extension ofthe poverty indexes. To this end we denote by P ( t ) = { h ∈ H : Y h ( t ) ≤ y p } , the set of poor agents at time t and by n ( t ) = { n ( t ) , n ( t ) , n ( t ) } , t ∈ R + , (2)the multivariate counting process denoting the composition of the income classes in time.Precisely, n i ( t ) is the number of individuals allocated in class C i at time t . Definition 1.
The Dynamic Headcount ratio, The Dynamic Income-gap ratio, the DynamicGini and the Dynamic Sen Index are defined as follows: H ( t ) := n ( t ) + n ( t ) N , (3) I ( t ) := 1 − (cid:80) h ∈P ( t ) Y h ( t ) y p ( n ( t ) + n ( t )) , (4) G ( t ) := (cid:80) h ∈P ( t ) (cid:80) l ∈P ( t ) | Y h ( t ) − Y l ( t ) | n ( t ) + n ( t ))( (cid:80) h ∈P ( t ) Y h ( t )) , (5) S ( t ) = H ( t ) · [ I ( t ) + (1 − I ( t )) · G ( t )] . (6)Although the previous indexes share the same functional form with their static counter-parts, they are of different nature being stochastic processes due to the randomness of thecounting process n ( t ) and of the incomes { Y h ( · ) } h ∈P ( t ) .This set of assumptions defines an economic system that describes the evolution of peopleaccording to their income. The study of this system is very complex and since the numberof involved individuals N is very large, it requires a big computational effort. An alternativestrategy has been implemented in D’Amico and Regnault (2018) where assumption A1 isrelaxed in favour of a new assumption: A1 (cid:48) : (large-size population) the number of individuals N in the economy is large enough tobe considered as infinity. 5his new hypothesis allows us to use stochastic approximations based on limit theorems.Next proposition is the first results of this strategy: Proposition 2.
Under assumptions A1 (cid:48) - A4 , we have that: H ( t ) a.s. −→ H ∞ ( t ) = H ( µ, Λ , t ) := µ (cid:48) ( P . ( t ) + P . ( t )) , (7) where µ (cid:48) is the transpose of the initial distribution and P . ( t ) and P . ( t ) are the first andsecond column of P ( t ) = exp( t Λ ) , respectively.Similarly, the Dynamic Income-gap ratio I ( t ) , the Dynamic Gini index G ( t ) and theDynamic Sen index S ( t ) converge almost surely to: I ∞ ( t ) := 1 − y y p µ (cid:48) P . ( t ) H ∞ ( t ) − y y p µ (cid:48) P . ( t ) H ∞ ( t ) , (8) G ∞ ( t ) := ( µ (cid:48) P . ( t )) z +2( y − y )( µ (cid:48) P . ( t ))( µ (cid:48) P . ( t ))+( µ (cid:48) P . ( t )) z H ∞ ( t )( y µ (cid:48) P . ( t )+ y µ (cid:48) P . ( t )) (9) S ∞ ( t ) := H ∞ ( t ) · [ I ∞ ( t ) + (1 − I ∞ ( t )) · G ∞ ( t )] , (10) where y = ( y , y ) is the vector of mean incomes per poor classes and z := (cid:90) y ep (cid:90) y ep | y − x | dF ( y ) dF ( x ) ,z := (cid:90) y p y ep (cid:90) y p y ep | y − x | dF ( y ) dF ( x ) . Proof:
See Appendix.
Remark 3.
Proposition 2 was already demonstrated in D’Amico and Regnault (2018). How-ever, with respect to the Gini index, and in turn to the Sen index, the proof was limited to thecase of equivalence of the incomes of people belonging to the same class while the hypothesesof the model advance only the equivalence of their probability distributions. The proof weprovide in this paper overcomes this limitation with the addition of the inequality within eachclass.
The next step forward in global understanding of the time evolution of the dynamicpoverty indexes is the assessment of specific central limit theorems for each index and the6ubsequent derivation of the confidence sets. The confidence sets are centred to the asymp-totic values obtained in Proposition 2 and have amplitudes proportional to their variancescomputed in next proposition. This finding represents the main result. However, before itsexposition, we anticipate an auxiliary Lemma which is a useful tool for obtaining the proofof our main result concerning the central limit theorems for the considered dynamic indexesof poverty.
Lemma 4.
For any h ∈ H and for any t ∈ R let F ( t ; x ) := P [ Y h ( t ) { C h ( t ) ∈{ C ,C }} ≤ x ] .Then, F ( t ; x ) = F ( x ) µ (cid:48) P . ( t ) + F ( x ) µ (cid:48) P . ( t ) + µ (cid:48) P . ( t ) , (11) and accordingly, for any integer r ≥ E [( Y h ( t ) { C h ( t ) ∈{ C ,C }} ) r ] = y ( r )1 µ (cid:48) P . ( t ) + y ( r )2 µ (cid:48) P . ( t ) , (12) where y ( r )1 := E [( Y ) r ] and y ( r )2 := E [( Y ) r ] .Moreover, if Θ t := E [ | Y h ( t ) { C h ( t ) ∈{ C ,C }} − Y l ( t ) { C l ( t ) ∈{ C ,C }} | ] , then Θ t = ( µ (cid:48) P . ( t )) (cid:90) y ep (cid:90) y ep | y − x | dF ( x ) dF ( y )+ 2( y − y )( µ (cid:48) P . ( t ))( µ (cid:48) P . ( t ))+ ( µ (cid:48) P . ( t )) (cid:90) y p y ep (cid:90) y p y ep | y − x | dF ( x ) dF ( y ) . (13) Finally, using the notation σ ( t ) + Θ t := (cid:90) + ∞−∞ (cid:20)(cid:90) + ∞−∞ | y − x | dF ( t ; x ) (cid:21) dF ( t ; y ) , (14)7 e find that σ ( t ) + Θ t = ( µ (cid:48) P . ( t )) (cid:90) y ep (cid:20)(cid:90) y ep | y − x | dF ( x ) (cid:21) dF ( y )+ ( µ (cid:48) P . ( t )) ( µ (cid:48) P . ( t ))( y (2)1 − y y + y )+ ( µ (cid:48) P . ( t )) ( µ (cid:48) P . ( t ))( y (2)2 − y y + y )+ ( µ (cid:48) P . ( t )) (cid:90) y p y ep (cid:34)(cid:90) y p y ep | y − x | dF ( x ) (cid:35) dF ( y ) . (15) Proof:
See Appendix.Let us introduce basic notations for expectation and variance of the random variable Y h ( t ) { C h ( t ) ∈{ C ,C }} : x ( t ) := E [ Y h ( t ) { C h ( t ) ∈{ C ,C }} ] , σ ( t ) := V [ Y h ( t ) { C h ( t ) ∈{ C ,C }} ] (16) Proposition 5.
Under assumptions A1 (cid:48) - A4 , we have the following convergences in law: i ) √ N (cid:0) H N ( t ) − H ∞ ( t ) (cid:1) L −→ N (cid:0) , H ∞ ( t )(1 − H ∞ ( t ) (cid:1) ; (17) ii ) √ N (cid:18) − I ( t ) − x ( t ) y p H ∞ ( t ) (cid:19) L −−−−→ N → + ∞ N (cid:18) , σ ( t ) y p H ∞ (cid:19) ; (18) iii ) √ N ( G N ( t ) − G ∞ ( t )) L −→ N (cid:18) , σ ( t )4 H ∞ ( x ( t )) (cid:19) ; (19) where σ ( t ) = (cid:90) + ∞−∞ (cid:20)(cid:90) + ∞−∞ | y − x | dF ( t ; x ) (cid:21) dF ( t ; y ) − Θ t , (20) and Θ t is given in formula (13). iv ) √ N ( S N ( t ) − S ∞ ( t )) L −→ N (cid:16) , (1 − H ∞ ( t )) S ∞ ( t ) H ∞ ( t ) (cid:17) . (21) Proof:
See Appendix.Proposition 5 tells us that the dynamic poverty indexes, properly centralised and nor-malised, converge in distribution to normal random variables. Accordingly, it is possible toobtain their confidence sets, e.g., from formula (17) we can easily determine ∀ a, b ∈ R an8stimate of the probability P ( a ≤ H N ( t ) ≤ b ). Indeed, once we denote by Z the standardnormal distribution, the following approximation holds according to Proposition 5: P ( a ≤ H N ( t ) ≤ b ) ≈ P a − H ∞ ( t ) (cid:113) H ∞ ( t )(1 − H ∞ ( t )) N ≤ Z ≤ b − H ∞ ( t ) (cid:113) H ∞ ( t )(1 − H ∞ ( t )) N = Φ b − H ∞ ( t ) (cid:113) H ∞ ( t )(1 − H ∞ ( t )) N − Φ a − H ∞ ( t ) (cid:113) H ∞ ( t )(1 − H ∞ ( t )) N . (22)A similar argument can be used to construct confidence sets for the other indexes. Herewe just focus on the Sen index which is the most powerful index among those considered inthis paper.Let us denote by E ( t ) the asymptotic variance in formula (21), i.e. E ( t ) := (1 − H ∞ ( t )) S ∞ ( t ) H ∞ ( t ) . Then, ∀ a, b ∈ R the following approximation can be established: P ( a ≤ S N ( t ) ≤ b ) = P ( a − S ∞ ( t )) ≤ S N ( t ) − S ∞ ( t )) ≤ b − S ∞ ( t )))= P (cid:32) a − S ∞ ( t )) E√ N ≤ S N ( t ) − S ∞ ( t )) E√ N ≤ b − S ∞ ( t )) E√ N (cid:33) = P (cid:32) a − S ∞ ( t )) E√ N ≤ Z ≤ b − S ∞ ( t )) E√ N (cid:33) = Φ (cid:32) b − S ∞ ( t )) E√ N (cid:33) − Φ (cid:32) a − S ∞ ( t )) E√ N (cid:33) . (23)The construction of confidence sets is of crucial importance for the application of themodel since it shows the accuracy of the infinite size economic systems approximation to thereal system.
3. Empirical application
We test the model on the Italian income data provided by the Italian central bank, Bancad’Italia. The historical database is based on the survey of Italian households budgets from9 ount mean std min 25% 50% 75% max
Table 1: Summary statistics of the net disposable income in euro of Italian households from 1998 to 2012by number of family components. Data sourced from the Italian households budgets survey from Bancad’Italia.
Table 2: Italian poverty thresholds in euro for the years 1998-2012. Data sourced from the Italian NationalInstitute of Statistics (ISTAT) ount % mean std min 25% 50% 75% max C C C Table 3: Summary statistics of the standardised income in euro of Italian households from 1998 to 2012 bypoverty class, where C is the class of extreme poor households and C is the class of poor households. For the application of the model with two poverty classes, we need to define an extrapoverty class. We identify a class for extremely poor households by setting its threshold, y ep ,at 60% of the poverty threshold, y p . Moreover, to make the incomes comparable betweenthe number of household components and to account for the inflation during the years, westandardise the net disposable incomes. The standardisation by the components is performedeach year using the income for 1-component household as base income. Conversely, theinflation adjustment is performed setting the first year income as base income. Consequently,the poverty threshold y p is represented by the 1998 1-component value, i.e. 5479.50, and theextreme poverty threshold y ep is 60% of previous threshold, i.e. 3287.70.In addition, to work on a clean sample, we require that all households show an incomeeach year, thus we exclude households with any missing income, reducing the number ofhouseholds to 914, and the number of observed incomes to 7312. Table 3 reports the summarystatistics for the standardised income divided by classes.Now, considering that we do not observe the household incomes continuously but everytwo years, we can estimate the generator matrix using the periodic sampling of class alloca-tion processes described in D’Amico and Regnault (2018). According to this methodologyand with 914 independent trajectories of the class allocation processes D , D (1998) , D (2000) , . . . , D (2012) , ... D (1998) , D (2000) . . . , D (2012) , where D i ( t ) denotes the income class occupied by household i at year t .11e first estimate the transition probability matrix ˆ P = ˆ p ij withˆ p ij = K ij K i , where K ij = (cid:88) k =1 6 (cid:88) t =0 { D k (2 t +1998)= i,D k (2 t +2000)= j } , is the number of transitions from class i to class j , and K i = (cid:88) j =1 K ij , is the total number of times households have been allocated to class i .Then, the maximum likelihood estimator ˆΛ of the generator matrix Λ satisfies the relationˆ P = exp ( η ˆ Λ ); accordingly, it can be obtained as the logarithm matrix,ˆ Λ = log ( ˆ P ) η , (24)where η is the period of observation, i.e. 2 years in our application. It should be remarkedthat the maximum likelihood estimator of Λ under this observational scheme is not guaran-teed to exist or to be unique (see e.g. Bladt and Srensen (2005), Regnault (2012)) but asproved in D’Amico and Regnault (2018), estimator (24) exists and is unique whenever thetransition probability matrix is irreducible with positive eigenvalues.The estimated transition probability matrix isˆ P = .
37 0 .
38 0 . .
11 0 .
38 0 . .
01 0 .
03 0 . , (25)which can be readily recognised as an irreducible stochastic matrix. This matrix has positiveeigenvalues: x = 1; x = 1200 (cid:0)
71 + √ (cid:1) ; x = 1200 (cid:0) − √ (cid:1) . Λ = − .
59 0 .
58 0 . . − .
59 0 . .
00 0 . − . . (26)Finally, given the estimated initial distribution in the year 1998,ˆ µ (cid:48) = (0 . , . , . , (27)and the average income for the poverty classes as reported in Table 3, we calculate thefour indexes, i.e. Headcount ratio, Income gap ratio, Gini Index, and Sen index, and theirrespective confidence intervals at 95% level of significance. Figure 1 shows the four indexesestimated from the model against the observed index. In all cases, the computed indexesfollow the trajectories of the observed indexes. Also, it is important to notice that theobserved indexes fall within the 95% confidence intervals with an extra small variabilityfor the Gini index, in which the observed value in year 2010 which goes above the upperconfidence limit. In general, the plots show that the model has a very good power in capturingthe dynamic of the observed indexes.Figure 1 also indicates that all the indexes show a decreasing path in time. This meansthat the different aspects of poverty represented by them are moving towards better economicconditions of the given households which include a reduction of the percentage of poor, alower mean short-fall of people below the poverty line and a reduction of disparities amongthe poor. However, it is relevant to remark that at year 2012 all the indexes are very closeto their stationary levels. This implies that a further decrease of poverty must necessarilybe accompanied by a reinforcement of poverty containment policies or by the adoption ofnew ones because, if left in current conditions, the economic system cannot evolve towardsa lower level of poverty.As a robustness test for the model, we now proceed to estimate the generator matrixusing a reduced set of data with only three years of observation, i.e. from 1998 to 2002. Thenumber of households and poverty thresholds remains the same. However, the number of13 igure 1: Dynamic indexes of poverty for Italian households income from 1998 to 2012 computed withparameters given in (25) and (27). P = .
32 0 .
41 0 . .
12 0 .
37 0 . .
01 0 .
02 0 . , ˆ Λ = − .
70 0 .
69 0 . . − .
63 0 . .
00 0 . − . , (28)and the average income for the poverty classes become y = 2046 .
57 and y = 4430 .
4. Conclusion
The analysis of the literature on poverty has demonstrated the importance of a dynamicapproach to the determination of poverty and inequality. With the advancements proposedin this paper we aim at giving an additional tool to help the definition of the policies for thepoverty in the real economies.In this study, we first proposed an extension of the dynamic Gini index, and consequentlythe Sen index, with the inclusion of the inequality within each class of poverty where peopleare classified according to their income. Then, we established the central limit theoremfor each poverty index for the determination of their confidence sets. An application to theItalian income data from 1998 to 2012 confirmed the effectiveness of the considered approachand demonstrated that the model has a very good power in capturing the dynamic of theobserved indexes.This study leaves some open possibilities for further research. For example, the extensionto more complex dynamics or the relaxation of some of the model’s assumptions. On the15 igure 2: Dynamic indexes of poverty for Italian households income from 1998 to 2012 computed withparameters estimated using only the initial three years of data, from 1998 to 2004. The vertical linesrepresent the separation between observed data on the left and forecast on the right.
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Appendix A. Mathematical Proofs
Proof of Proposition 2
As already explained in Remark 3, we prove only the result concerning the dynamic Giniindex that here is more general than that presented in D’Amico and Regnault (2018) whereit is also possible to find the proof for the remaining indexes.20rom the definition of the dynamic Gini index it follows that G ( t ) = (cid:80) h ∈P ( t ) (cid:80) l ∈P ( t ) | Y h ( t ) − Y l ( t ) | N · N n ( t ) + n ( t ))( (cid:80) h ∈P ( t ) Y h ( t )) . (A.1)We first determine the value to which the first factor of (A.1) converges to. Since the incomeclasses are mutually exclusive we obtain1 N (cid:88) h ∈P ( t ) (cid:88) l ∈P ( t ) | Y h ( t ) − Y l ( t ) | = 1 N (cid:40) N (cid:88) h =1 N (cid:88) l =1 { C h ( t )=2 ,C l ( t )=1 } | Y h ( t ) − Y l ( t ) | + N (cid:88) h =1 N (cid:88) l =1 { C h ( t )=1 ,C l ( t )=2 } | Y h ( t ) − Y l ( t ) | + N (cid:88) h =1 N (cid:88) l =1 { C h ( t )= C l ( t )=1 }| Y h ( t ) − Y l ( t ) | + N (cid:88) h =1 N (cid:88) l =1 { C h ( t )= C l ( t )=2 } | Y h ( t ) − Y l ( t ) | (cid:41) . (A.2)Analysing the four addends separately and observing that by construction Y h ( t ) > Y l ( t )for C h ( t ) = 2 and C l ( t ) = 1, we obtain:1 N N (cid:88) h =1 N (cid:88) l =1 { C h ( t )=2 ,C l ( t )=1 } | Y h ( t ) − Y l ( t ) | = 1 N N (cid:88) h =1 N (cid:88) l =1 { C h ( t )=2 ,C l ( t )=1 } ( Y h ( t ) − Y l ( t ))= (cid:80) Nh =1 (cid:80) Nl =1 { C h ( t )=2 } { C l ( t )=1 } Y h ( t ) − (cid:80) Nh =1 (cid:80) Nl =1 { C h ( t )=2 } { C l ( t )=1 } Y l ( t ) N = (cid:80) Nh =1 { C h ( t )=2 } Y h ( t ) (cid:80) Nl =1 { C l ( t )=1 } N − (cid:80) Nh =1 { C h ( t )=2 } (cid:80) Nl =1 { C l ( t )=1 } Y l ( t ) N = n C ( t ) N · (cid:80) Nh =1 { C h ( t )=2 } Y h ( t ) N − n C ( t ) N · (cid:80) Nl =1 { C l ( t )=1 } Y l ( t ) N (A.3)Now observe that from the strong law of large numbers for i ∈ { , } it holds n C ( t ) N a.s. −−→ E [ { C h ( t )= C i } ] = P [ C h ( t ) = C i ] = µ (cid:48) P .i ( t ) (A.4)21 further application of the strong law of large numbers guarantees that (cid:80) Nl =1 { C i ( t )=1 } Y i ( t ) N a.s. −−→ E [ { C i ( t )=1 } Y i ( t )]= E [ { C i ( t )=1 } E [ Y i ( t ) | σ ( C ( s ) , . . . , C N ( s ) , s ≤ t )]]= E [ y i { C i ( t )= i } ] = y i P [ C h ( t ) = C i ] = y i µ (cid:48) P .i ( t ) . (A.5)A substitution of (A.4) and (A.5) in (A.3) gives1 N N (cid:88) h =1 N (cid:88) l =1 { C h ( t )=2 ,C l ( t )=1 } | Y h ( t ) − Y l ( t ) | a.s. −−→ µ (cid:48) P . ( t ) y µ (cid:48) P . ( t ) − ( µ (cid:48) P . ( t )) y µ (cid:48) P . ( t ) = ( y − y )( µ (cid:48) P . ( t ))( µ (cid:48) P . ( t )) . (A.6)Similarly, it can be proved that1 N N (cid:88) h =1 N (cid:88) l =1 { C h ( t )=1 ,C l ( t )=2 } | Y h ( t ) − Y l ( t ) | a.s. −−→ ( y − y )( µ (cid:48) P . ( t ))( µ (cid:48) P . ( t )) . Now, let us analyse the third addend of formula ( A. Z h,l ;1 ( t ) = { C h ( t )= C l ( t )=1 } | Y h ( t ) − Y l ( t ) | and consequently consider the followingrepresentation: 1 N N (cid:88) h =1 N (cid:88) l =1 { C h ( t )= C l ( t )=1 } | Y h ( t ) − Y l ( t ) | = 1 N (cid:88) h ∈ C (cid:88) l ∈ C Z h,l ;1 ( t ) , As h and l belong to P ( t ), we have that { Z h,l ;1 ( t ) } h,l ∈P ( t ) represents an array of ( n C ( t )) − n C ( t ) i.i.d. random variables with mean z , i.e. z := (cid:90) y ep (cid:90) y ep | y − x | dF ( y ) dF ( x ) . N n C ( t ) (cid:88) h =1 n C ( t ) (cid:88) l =1 Z h,l ;1 ( t ) = ( n C ( t ))( n C ( t ) − N n C ( t ))( n C ( t ) − (cid:88) h,l ∈ C Z h,l ;1 ( t ) a.s. −−→ ( µ (cid:48) P . ( t )) · z . The same reasoning can be applied to the fourth addend of (A.2):1 N n C ( t ) (cid:88) h =1 n C ( t ) (cid:88) l =1 Z h,l ;2 ( t ) a.s. −−→ ( µ (cid:48) P . ( t )) · z , where z := (cid:90) y p y ep (cid:90) y p y ep | y − x | dF ( y ) dF ( x ) . Therefore, the first factor of equation (A.1) converges almost surely to( µ (cid:48) P . ( t )) (cid:90) y ep (cid:90) y ep | y − x | dF ( y ) dF ( x ) + 2( y − y )( µ (cid:48) P . ( t ))( µ (cid:48) P . ( t ))+ ( µ (cid:48) P . ( t )) (cid:90) y p y ep (cid:90) y p y ep | y − x | dF ( y ) dF ( x ) . Analysing the second factor we obtain:2( n C ( t ) + n C ( t )) (cid:80) h ∈P ( t ) Y h ( t ) N = 2 n C ( t ) + n C ( t ) N (cid:80) h ∈P ( t ) Y h ( t ) N .
Moreover we have the following convergence:2 n C ( t ) + n C ( t ) N a.s. −−→ E [ { C i ( t ) ∈{ C ,C }} ] = 2 µ (cid:48) ( P . ( t ) + P . ( t )) = 2 H ∞ ( t ) , and in virtue of (A.5) we have (cid:80) h ∈P ( t ) Y h ( t ) N = (cid:80) Ni =1 { C i ( t )= C } Y i ( t ) + (cid:80) Ni =1 { C i ( t )= C } Y i ( t ) N a.s. −−→ y µ (cid:48) P . ( t ) + y µ (cid:48) P . ( t ) . (A.7)23herefore, for N → + ∞ , G N ( t ) a.s. −−→ G ∞ ( t ) := ( µ (cid:48) P . ( t )) z + 2( y − y )( µ (cid:48) P . ( t ))( µ (cid:48) P . ( t )) + ( µ (cid:48) P . ( t )) z H ∞ ( t )( y µ (cid:48) P . ( t ) + y µ (cid:48) P . ( t )) . Proof of Lemma 4
In general, because the random variables Y i are non-negative, ∀ a < F i ( a ) = 0 andaccordingly it results F ( t ; x ) := P [ Y h ( t ) { C h ( t ) ∈{ C ,C }} ≤ x ]= (cid:88) k =1 P [ Y h ( t ) { C h ( t )= { C ,C }} ≤ x | C h ( t ) = C k ] · P [ C h ( t ) = C k ]= F ( x ) µ (cid:48) P . ( t ) + F ( x ) µ (cid:48) P . ( t ) + 1 · µ (cid:48) P . ( t ) , where the last equality follows from assumption A4 . The computation of the r-th momentcan be now accomplished by using the cdf F ( t ; x ). Indeed, for any integer r ≥ E [( Y h ( t ) { C h ( t ) ∈{ C ,C }} ) r ] = r (cid:90) ∞ x r − P [ Y h ( t ) { C h ( t ) ∈{ C ,C }} > x ] dx = r (cid:90) ∞ x r − (1 − F ( t ; x )) dx. (A.8)Now observe that 1 − u (cid:48) P . ( t ) = µ (cid:48) P . ( t ) + µ (cid:48) P . ( t ) , thus1 − F ( t ; x ) = µ (cid:48) P . ( t )(1 − F ( x )) + µ (cid:48) P . ( t )(1 − F ( x )) . Now by substitution of the latter expression in (A.8) we have: E [( Y h ( t ) { C h ( t ) ∈{ C ,C }} ) r ] = r (cid:90) ∞ x r − (cid:0) µ (cid:48) P . ( t )(1 − F ( x )) + µ (cid:48) P . ( t )(1 − F ( x )) (cid:1) dx = rµ (cid:48) P . ( t ) (cid:90) ∞ x r − (1 − F ( x )) dx + rµ (cid:48) P . ( t ) (cid:90) ∞ x r − (1 − F ( x )) dx = y ( r )1 µ (cid:48) P . ( t ) + y ( r )2 µ (cid:48) P . ( t ) , (A.9)where y ( r )1 := E [( Y ) r ] and y ( r )2 := E [( Y ) r ].Next point is to prove formula (13). The expected value that defines Θ t can be evaluated24y computing the following double integralΘ( t ) = (cid:90) + ∞ (cid:18)(cid:90) + ∞ | y − x | dF ( x ) (cid:19) dF ( y ) . In order to compute it, we first observe that ∀ t ∈ R dF ( t ; x ) = , if x < dF ( x ) µ (cid:48) P . ( t ) , if 0 < x < y ep dF ( x ) µ (cid:48) P . ( t ) , if y ep < x < y p , if x > y p . Thus, we haveΘ( t ) = (cid:90) y ep (cid:18)(cid:90) y ep | y − x | dF ( x ) (cid:19) dF ( y ) + (cid:90) y p y ep (cid:32)(cid:90) y p y ep | y − x | dF ( x ) (cid:33) dF ( y )+ (cid:90) y ep (cid:32)(cid:90) y p y ep ( − y + x ) dF ( x ) (cid:33) dF ( y ) + (cid:90) y p y ep (cid:18)(cid:90) y ep ( y − x ) dF ( x ) (cid:19) dF ( y ) . Now, we separately proceed to compute previous integrals: (cid:90) y ep (cid:18)(cid:90) y ep | y − x | dF ( x ) (cid:19) dF ( y ) = (cid:90) y ep (cid:90) y ep | y − x | µ (cid:48) P . ( t ) dF ( x ) µ (cid:48) P . ( t ) dF ( y )= ( µ (cid:48) P . ( t )) (cid:90) y ep (cid:90) y ep | y − x | dF ( x ) dF ( y ) . (cid:90) y p y ep (cid:32)(cid:90) y p y ep | y − x | dF ( x ) (cid:33) dF ( y ) = (cid:90) y p y ep (cid:90) y p y ep | y − x | µ (cid:48) P . ( t ) dF ( x ) µ (cid:48) P . ( t ) dF ( y )= ( µ (cid:48) P . ( t )) (cid:90) y p y ep (cid:90) y p y ep | y − x | dF ( x ) dF ( y ) . y ep (cid:32)(cid:90) y p y ep ( − y + x ) dF ( x ) (cid:33) dF ( y ) = (cid:90) y ep (cid:90) y p y ep − ydF ( x ) dF ( y ) + (cid:90) y ep (cid:90) y p y ep xdF ( x ) dF ( y )= − (cid:90) y ep y (cid:32)(cid:90) y p y ep µ (cid:48) P . ( t ) dF ( x ) (cid:33) µ (cid:48) P . ( t ) dF ( y ) + (cid:90) y ep (cid:32)(cid:90) y p y ep xµ (cid:48) P . ( t ) dF ( x ) (cid:33) µ (cid:48) P . ( t ) dF ( y )= − ( µ (cid:48) P . ( t ))( µ (cid:48) P . ( t )) (cid:90) y ep ydF ( y ) + ( µ (cid:48) P . ( t ))( µ (cid:48) P . ( t )) (cid:90) y ep y dF ( y )= ( µ (cid:48) P . ( t ))( µ (cid:48) P . ( t ))[ − y + y ] = ( y − y )( µ (cid:48) P . ( t ))( µ (cid:48) P . ( t )) . Similarly, it is possible to prove that (cid:90) y p y ep (cid:18)(cid:90) y ep ( y − x ) dF ( x ) (cid:19) dF ( y ) = ( y − y )( µ (cid:48) P . ( t ))( µ (cid:48) P . ( t )) . Therefore, Θ( t ) = ( µ (cid:48) P . ( t )) (cid:90) y ep (cid:90) y ep | y − x | dF ( x ) dF ( y )+ 2( y − y )( µ (cid:48) P . ( t ))( µ (cid:48) P . ( t ))+ ( µ (cid:48) P . ( t )) (cid:90) y p y ep (cid:90) y p y ep | y − x | dF ( x ) dF ( y ) . (A.10)It remains to prove formula (15). For the application of this results, it remains to computethe variance σ ( t ) in formula (20). In order to reach this objective we decompose the integralaccording to the values of dF ( t ; · ). It results (cid:90) + ∞ (cid:20)(cid:90) + ∞ | y − x | dF ( t ; x ) (cid:21) dF ( t ; y )= (cid:90) y ep (cid:20)(cid:90) y ep | y − x | dF ( t ; x ) (cid:21) dF ( t ; y ) + (cid:90) y p y ep (cid:34)(cid:90) y p y ep | y − x | dF ( t ; x ) (cid:35) dF ( t ; y )+ (cid:90) y ep (cid:34)(cid:90) y p y ep ( − y + x ) dF ( t ; x ) (cid:35) dF ( t ; y ) + (cid:90) y p y ep (cid:20)(cid:90) y ep ( y − x ) dF ( t ; x ) (cid:21) dF ( t ; y ) . Now, we separately proceed to compute previous integrals: (cid:90) y ep (cid:20)(cid:90) y ep | y − x | dF ( t ; x ) (cid:21) dF ( t ; y ) = (cid:90) y ep (cid:20)(cid:90) y ep | y − x | µ (cid:48) P . ( t ) dF ( x ) (cid:21) µ (cid:48) P . ( t ) dF ( y )= ( µ (cid:48) P . ( t )) (cid:90) y ep (cid:20)(cid:90) y ep | y − x | dF ( x ) (cid:21) dF ( y ) , (cid:90) y p y ep (cid:34)(cid:90) y p y ep | y − x | dF ( t ; x ) (cid:35) dF ( t ; y ) = (cid:90) y p y ep (cid:34)(cid:90) y p y ep | y − x | µ (cid:48) P . ( t ) dF ( x ) (cid:35) µ (cid:48) P . ( t ) dF ( y )= ( µ (cid:48) P . ( t )) (cid:90) y p y ep (cid:34)(cid:90) y p y ep | y − x | dF ( x ) (cid:35) dF ( y ) . Furthermore we have (cid:90) y ep (cid:34)(cid:90) y p y ep ( − y + x ) dF ( t ; x ) (cid:35) dF ( t ; y )= (cid:90) y ep (cid:34)(cid:90) y p y ep ( − y + x ) µ (cid:48) P . ( t ) dF ( x ) (cid:35) µ (cid:48) P . ( t ) dF ( y )= ( µ (cid:48) P . ( t )) ( µ (cid:48) P . ( t )) (cid:90) y ep (cid:34) − y (cid:90) y p y ep dF ( x ) + (cid:90) y p y ep xdF ( x ) (cid:35) dF ( y )= ( µ (cid:48) P . ( t )) ( µ (cid:48) P . ( t )) (cid:90) y ep [ − y + y ] dF ( y )= ( µ (cid:48) P . ( t )) ( µ (cid:48) P . ( t )) (cid:90) y ep (cid:2) y − yy + y (cid:3) dF ( y )= ( µ (cid:48) P . ( t )) ( µ (cid:48) P . ( t )) (cid:26)(cid:90) y ep y dF ( y ) − y (cid:90) y ep ydF ( y ) + (cid:90) y ep y dF ( y ) (cid:27) = ( µ (cid:48) P . ( t )) ( µ (cid:48) P . ( t )) (cid:110) y (2)1 − y y + y (cid:111) . Similarly, it is possible to prove that (cid:90) y p y ep (cid:20)(cid:90) y ep ( y − x ) dF ( t ; x ) (cid:21) dF ( t ; y ) = ( µ (cid:48) P . ( t )) ( µ (cid:48) P . ( t )) (cid:110) y (2)2 − y y + y (cid:111) . Then by substitution we get formula (15).
Proof of Proposition 5i) Dynamic Headcount Ratio
The random variable expressing the Headcount ratio can be expressed as a sum of i.i.d.27andom variables, H N ( t ) = n C ( t ) + n C ( t ) N = (cid:80) Ni =1 { C i ( t ) ∈{ C ,C }} N , where { C i ( t ) ∈{ C ,C } = , P ( C i ( t ) ∈ { C , C } )0 , − P ( C i ( t ) ∈ { C , C } ) . It is simple to observe that E [ { C i ( t ) ∈{ C ,C }} ] = P ( C i ( t ) ∈ { C , C } ) = H ∞ ( t ) ,V ( { C i ( t ) ∈{ C ,C }} ) = H ∞ ( t )(1 − H ∞ ( t )) . Then, as a direct application of the central limit theorem for i.i.d. random variable we canconclude that Z N := (cid:80) Ni =1 { Ci ( t ) ∈{ C ,C }} N − H ( t ) (cid:113) H ∞ ( t )(1 − H ∞ ( t )) N L −−−−→ N → + ∞ N (0 , , or equivalently, √ N (cid:32) H N ( t ) − H ∞ ( t ) (cid:112) H ∞ ( t )(1 − H ∞ ( t )) (cid:33) L −→ N (0 , , and in turn √ N (cid:0) H N ( t ) − H ∞ ( t ) (cid:1) L −→ N (cid:18) , H ∞ ( t )(1 − H ∞ ( t )) (cid:19) . ii) Dynamic Income Gap Ratio From the definition of the Income gap ratio we have that1 − I ( t ) = (cid:80) h ∈P ( t ) Y h ( t ) y p ( n ( t ) + n ( t )) = (cid:80) Nh =1 Y h ( t ) { Ch ( t ) ∈{ C ,C }} Ny p ( n ( t )+ n ( t )) N . Note that the denominator y p n ( t ) + n ( t ) N = y p H N ( t ) a.s. −−−−→ N → + ∞ y p H ∞ ( t ) ,
28s argued above in the proof of Proposition 2. The numerator expresses the sample meanof the sample (cid:0) { C ( t ) ∈{ C ,C }} Y ( t ) , . . . , { C N ( t ) ∈{ C ,C }} Y N ( t ) (cid:1) . According to Lemma 4, eachelement of this random sample has E [ Y h ( t ) { C h ( t ) ∈{ C ,C }} ] = y µ (cid:48) P . ( t ) + y µ (cid:48) P . ( t ) := ¯ x ( t ) (A.11) V [ Y h ( t ) { C h ( t ) ∈{ C ,C }} ] = E [ Y h ( t ) { C h ( t ) ∈{ C ,C }} ] − (¯ x ( t )) = [ y (2)1 µ (cid:48) P . ( t ) + y (2)2 µ (cid:48) P . ( t )] − [ y µ (cid:48) P . ( t ) + y µ (cid:48) P . ( t )] =: σ ( t ) . (A.12)Then, from the CLT for i.i.d. random variable we get √ N (cid:32) (cid:80) Nh =1 Y h ( t ) { C h ( t ) ∈{ C ,C }} N − ¯ x ( t ) (cid:33) L −−−−→ N → + ∞ Z σ ( t ) ∼ N (0 , σ ( t )) . Now from Slutsky’s theorem (see e.g. Vaart (1998)) we can deduce that the random vector √ N (cid:32) (cid:80) Nh =1 Y h ( t ) { C h ( t ) ∈{ C ,C }} N − ¯ x ( t ) (cid:33) y p n C ( t ) + n C ( t ) N L −→ Z σ ( t ) y p H ∞ ( t ) , In addition, consider the function f : R → R defined as f ( x, y ) = xy for y (cid:54) = 00 for y = 0 , againfrom Slutsky’s theorems we could deduce that f √ N (cid:32) (cid:80) Nh =1 Y h ( t ) { C h ( t ) ∈{ C ,C }} N − ¯ x ( t ) (cid:33) y p n C ( t ) + n C ( t ) N = √ N (cid:18) (cid:80) Nh =1 Y h ( t ) { Ch ( t ) ∈{ C ,C }} N − ¯ x ( t ) (cid:19) y p n C ( t )+ n C ( t ) N L −→ f Z σ ( t ) y p H ∞ ( t ) = Z σ ( t ) y p H ∞ ( t ) ∼ N (cid:16) , σ ( t )( y p H ∞ ( t )) (cid:17) if P Z σ ( t ) y p H ∞ ( t ) ∈ C ( f ) = 1, being C ( f ) the continuity set of f . In our case, the29unction f is discontinuous at every point belonging to the set { ( x, y ) ∈ R : y = 0 } butwe can observe that the probability distribution of the limiting random vector Z σ ( t ) y p H ∞ ( t ) assigns mass zero to this set, i.e. P Z σ ( t ) y p H ∞ ( t ) ∈ { ( x, y ) ∈ R : y = 0 } = 0 , In this way we proved that √ N (cid:18) (cid:80) Nh =1 Y h ( t ) { Ch ( t ) ∈{ C ,C }} N − x ( t ) (cid:19) y p n C ( t )+ n C ( t ) N L −−−−→ N → + ∞ N (cid:16) , σ ( t )( y p H ∞ ( t )) (cid:17) . Simple algebra and the application of the convergence H N ( t ) a.s. −−→ H ∞ ( t ) as N → + ∞ produces the following result: √ N (cid:18) − I ( t ) − x ( t ) y p H ∞ ( t ) (cid:19) L −−−−→ N → + ∞ N (cid:16) , σ ( t ) y p H ∞ ( t ) (cid:17) . iii) Dynamic Gini index among the poor We first observe that (cid:88) h ∈P ( t ) (cid:88) l ∈P ( t ) | Y h ( t ) − Y l ( t ) | = 2 (cid:88) ≤ l 31e find that equation (A.16) is equivalent to √ N (cid:18) G N ( t ) − Θ t H ∞ ( t )¯ x ( t ) (cid:19) L −→ N (0 , A ( t )) , and therefore √ N ( G N ( t ) − G ∞ ( t )) L −→ N (0 , A ( t )) . iv) Dynamic Sen Index The Dynamic Sen Index is defined according to S ( t ) = H ( t )[ I ( t ) + (1 − I ( t )) G ( t )]. Sincewe proved that I ( t ) a.s. −−→ I ∞ ( t ) and G ( t ) a.s. −−→ G ∞ ( t ), then from the continuous mappingtheorem (see, e.g. Vaart (1998)) we can write[ I ( t ) + (1 − I ( t )) G ( t )] a.s. −−→ [ I ∞ ( t ) + (1 − I ∞ ( t )) G ∞ ( t )] ∈ R . Furthermore, from point i ) of Proposition 5 we know that √ N ( H N ( t ) − H ∞ ( t )) L −−−−→ N → + ∞ N (cid:0) , H ∞ ( t )(1 − H ∞ ( t )) (cid:1) , and then if we consider the function f : R → R defined as f ( x, y ) = xy due to the continuityof f we have f √ N ( H N ( t ) − H ∞ ( t )) I ( t ) + (1 − I ( t )) G ( t ) L −→ f Z σ H ( t ) I ∞ ( t ) + (1 − I ∞ ( t )) G ∞ ( t ) = Z σ H ( t ) · (cid:0) I ∞ ( t )+(1 − I ∞ ( t )) G ∞ ( t ) (cid:1) , where Z σ H ( t ) ∼ N (0 , H ∞ ( t )(1 − H ∞ ( t )).Thus, we have that √ N ( H N ( t ) − H ∞ ( t ))[ I N ( t )+(1 − I N ( t )) G N ( t )] L −−−−→ N → + ∞ N (cid:16) , σ H ( t )( I ∞ ( t )+(1 − I ∞ ( t )) G ∞ ( t )) (cid:17) Simple algebraic manipulations give √ N ( S N ( t ) − H ∞ ( t )[ I N ( t ) + (1 − I N ( t )) G N ( t )]) L −→ N (cid:16) , (1 − H ∞ ( t )) S ∞ ( t ) H ∞ ( t ) (cid:17) , I N ( t ) + (1 − I N ( t )) G N ( t )] a.s. −−→ [ I ∞ ( t ) + (1 − I ∞ ( t )) G ∞ ( t )], we can conclude that √ N ( S N ( t ) − S ∞ ( t )) L −→ N (cid:16) , (1 − H ∞ ( t )) S ∞ ( t ) H ∞ ( t ) (cid:17) ..