Multistate Analysis of Internal Migration, Life Table of Hungary by County of Residence

Multistate Analysis of Internal Migration, Life Table of Hungary by County of Residence

Miklós Faragó chief senior councillor Hungarian Central Statistical Office

E-mail: Miklos.Farago@ksh.hu

Using internal migration and mortality data and the methodology of multistate analysis, an age-specific regional life table was constructed for Hungary by county of residence that, similarly to the classic two- state (life-death) life tables, contains the survival and migration probabilities and also the expected duration of living of the random members of the population in the counties (given their county of origin, age, and sex). From these the expected migration numbers of the 2010 Hungarian population were calculated by pairs of counties. The study presents the description of the multistate life table method including an original estimation method for the transitional rates between states used for the calculation.

KEYWORDS: Multistate life table.

Multiregional demography.

Internal migration.

Markov-process.

M

ultistate demography is a generalisation of the classical mathematical demography: inspecting demographic processes of populations, the members of which can be characterized by several time-varying values of certain demographic and social- economic attributes such as residence, marital, health or labour market status. The values of these attributes (e.g. Somogy…; married, divorced…; active, pensioned…) are called states in general. Multistate demography is a generalisation of the classic two-state (life, death) mathematical demographic model not only in the sense of increasing the number of states but also because of allowing transitions in any direction between these states, therefore allowing the return to a state occupied earlier.

As the first application of the theory, Rogers [1975] has created a so-called

“multistate life table” for migration between 17 countries. The model also has handled mortality (e.g. the 18. state was death). The greatest virtue of the method was the introduction of matrix formalism using of which the examination of a multistate population has turned out to be quite simple. Afterwards, series of publications have come out in various topics such as marital status (Schoen–Nelson [1974], Schoen–Land [1979], Willekens et al. [1982], Keyfitz [1988]), labour market status (Hoem [1977]), Schoen–Woodrow [1980], Willekens [1980]), fertility (Suchindran–Namboodiri–West [1977], Lutz–Wolf [1986]), and international migration (DeWaard–Raymer [2012]). However, multistate analysis has spread slowly due to the lack of statistical data in sufficient detail and presumably to the relative difficulty of its methodology.

Multi-state models have a common assumption that the state changes of the population members are pairwise independent, and within a cohort are of the same probability e.g. cohorts are homogenous. This probability depends neither on the former states of population members nor on the elapsed time in their current state:

the transition of individuals between states is modelled by a finite-state, time- inhomogeneous Markov process (later it will be expounded). In an examination, obviously, it is a fundamental issue whether these criteria are confirmed by experience. For instance, in the study of marital status, the time elapsed since the beginning of the marriage influences certainly the likelihood of a divorce or remarriage, but the independence of the death of spouses is also questionable.

The first chapter of this study outlines the model, while the second presents the calculation results of the one applied to the Hungarian internal migration. In Section 3.1. the mathematical model is described in detail; in Section 3.2. the expected numbers of the 2010 Hungarian population evolving in consequence of internal migration and mortality are calculated by multiplying the values computed for one person in Section 3.1. with the corresponding 2010 baseline cohort numbers. Section

3.3. gives the formulae used for estimating the rates of migration. In Sections 3.2.

and 3.3. an original calculation method is included.

1. Multistate life table model – a short description

On the analogy of the theory of two-state “periodic life tables”, we create a model, called multistate life table, for the estimation of age-specific life expectancies spent in different states and for that of other indicators of a given population. This is a stochastic process what describes the transition of a randomly chosen person between various states. From the model, the so-called “table functions” (such as probabilities of being in various states; several life expectancies) are computed.

These are regarded applicable to the given population as the “model population”

hopefully behaves similarly to the “real population” when its “transition rates”

calculated from statistical data are inserted into the age-specific “real transition rates” (between pairs of states) that are parameters of the functions, assuming constant rates in the long term. (This is a fundamental presumption of the theory of life tables that is never met. The correct – dialectic – interpretation is that the results of calculation apply to the future but characterise the present.)

Figure 1. Graphs of life tables

a) b)

1

3 2

0 P_2,1( , )x h

1 lives

0

Figure 1. a) shows the graph of a “residence life table” where being in one of the states 1–3 means (e.g. permanent) residence in the region corresponding to the state and change in this state means migration. 0 state is death. Pi, j

( )

x,h is the probability that a person having his x^th birthday and staying in state i (e.g. living in region i) will be in j after time h. (Edges with zero probability for every x are omitted.) Pi, j

( )

x,h -s are determined from the statistical data of the population of the given time period. On the one hand, Pi, j

( )

x,h is a “summation“ of transition intensities at time points within time interval h and a piecewise (yearly) exponential function by our theory. On the

other hand, according to the classical theory, Pi, j

( )

x,h is piecewise linear. (The formulae of both theories are derived in Chapter 3.) In comparison, Figure 1. b) shows the graph of a classic two-state life table. Both tables have one source state and one target state. (Interestingly, life tables are called mortality tables in Hungarian.)

Among others, the following questions are answered (ages are exact whole numbers e.g. they refer to persons having birthday):

a) What is the probability that a person aged y in state i will (or will not) be in state j at the age of x

(

^{y x≤}

)

^?

b) What is the expected number of years that a person spends in state j between age x₁ and x₂

(

y x≤ 1<x2

)

? It is clear that, for instance, in the case of y x= ₁ and x₂" "= ∞, the sum of answers by j gives the total remaining life expectancy of a person whose initial state is i in age y.

The answers could be given by Monte Carlo simulation as follows. We place a person of age x into the box i of the corresponding figure, and, in accordance with the directed edges, we select the next state j with probability Pi, j

( )

x for his staying one year later. Repeating this step, we make him “walk” as long as he gets into state 0 (it occurs with probability 1). If we repeat this procedure often enough, the average time spent in the individual boxes between age x₁ and x₂ (the total time per the number of procedures) will approximate the answer of question “a) with arbitrary accuracy.

However, the table functions defined by questions a) and b) and others can also be given explicitly. Their computation method can be found in Section 3.1. of Chapter 3, while the rates needed to compute probabilities Pi, j

( )

x are estimated in Section 3.3.

2. Calculation results

In this study the Hungarian population living at the beginning of 2010 is examined by sex and counties. The source of data is the Hungarian Central Statistical Office. The period is the calendar year 2010. The baseline data are as follows:

population numbers at the beginning of 2010 and the numbers of death and migration registered during that year by age, sex and (pair of) usual place(s) of residence. A state is a county of usual residence that is the registered place of stay or in absence of this, the registered place of residence. Migration is the change in this place of residence e.g. the change of state. Due to lack of space, the tables and figures

henceforward refer exclusively to the male population of Hungary unless otherwise specified. For the same reason, we omit the detailed analysis of data, we only make some substantial observations and draw attention to some hidden links helping the further analysis.

It is important to bear in mind that the results are neither registered statistical data nor averages of them but are probabilities and expected population numbers, migration numbers and lengths of time; this is why we are using the attribute

“expected” in each occasion. The fundamental assumption is that the 2010 mortality and migration rates will not change in the future.

To orient readers, we refer to the corresponding formulae of Chapter 3 in the table and figure titles. The tables and figures account only for a tiny fraction of the results that can be computed. For example, question b) (see Chapter 1) alone defines over hundred thousand (20

×

20) tables of county level.

2.1. Multiregional life table of Hungary for males by current county of residence, 2010

It is worth making the title clear: not the Hungarian life tables by counties are discussed here. Those would mean individual life tables for each county (with two states: life, death) with the assumption that residents spend all their life in their county of birth and die according to the mortality rate of that. In this case, migration does not even occur.

Life tables yield two kinds of calculation results: probabilities and life expectancy for one person, answering the questions a) and b) in Chapter 1. The following results refer to special cases: y=0, (e.g. newborns) in question a) (see Table 1 and Figure 2) and

y x= 1 and x₂ = ∞ (e.g. the remaining life of y year-olds in question b) (see Table 2).

2.1.1. Survivors

Table 1 gives the probability in percentage that a 0-aged person staying in a county that corresponds to a row will be alive at age 60 in a county corresponding to a column. This also means the expected number of the survivors of 100 such newborns. Therefore, the column of row-sums contains the probabilities of being alive (anywhere) at age 60. Subtracting from a row sum the corresponding element of the main diagonal gives the probability of being alive “at home”, however, not necessarily staying there permanently (the model is able to handle remigration). (See Chapter 3.) The last column (e.g. the proportion of the entry of the main diagonal compared to the sum of the row) can be considered as a “retention indicator”. In

Table 1, the thin and thick cell/column borders mark the excessively high and low values (outliers). Without analysing the table in detail, one can expect heavy migration to Budapest and county Pest (and a high number of survivors) (see their columns with thicker border) and observe the phenomenon that Győr-Moson-Sopron is the only developed county – in addition to the undeveloped ones – that can expectedly retain its residents in a 60-year timeframe.

Table 1 The percentage of survivors aged 60 out of newborns

County 1. Budapest 2. Baranya 3. Bács-Kiskun 4. Békés 5. Borsod-Abaúj-Zemplén 6. Csongrád 7. Fejér 8. Győr-Moson-Sopron 9. Hajdú-Bihar 10. Heves 11. Komárom-Esztergom 12. Nógrád 13. Pest 14. Somogy 15. Szabolcs-Szatmár-Bereg 16. Jász-Nagykun-Szolnok 17. Tolna 18. Vas 19. Veszprém 20. Zala Total Of which those living at home

1. 25.1 1.4 2.6 1.4 2.5 1.7 3.2 2.2 2.0 1.6 1.8 1.2 20.5 1.8 2.2 2.1 0.9 1.0 1.9 1.2 78.4 32 2. 11.0 28.2 2.5 1.0 1.4 1.6 2.3 2.7 1.2 0.7 1.5 0.6 7.8 4.1 0.9 1.0 3.9 1.1 1.6 1.8 76.9 37 3. 12.1 2.1 26.9 1.6 1.6 5.2 2.8 1.9 1.3 0.9 1.3 0.8 10.2 1.3 1.2 2.1 1.5 0.7 1.5 0.9 78.2 34 4. 12.7 1.1 3.2 21.1 1.7 7.1 2.1 3.4 3.1 1.0 1.9 0.6 9.8 1.3 1.4 2.6 0.7 0.8 1.5 1.0 78.3 27 5. 13.7 0.8 1.7 1.1 24.3 1.3 2.0 2.5 3.6 2.8 1.9 0.9 10.1 1.1 3.0 1.7 0.6 0.8 1.3 0.8 76.0 32 6. 10.6 1.4 5.1 4.1 1.3 30.7 1.8 2.0 1.6 0.8 1.3 0.6 7.9 1.1 1.2 1.7 0.8 0.8 1.1 0.8 76.8 40 7. 13.9 1.8 2.7 1.1 1.9 1.5 21.2 2.9 1.5 1.0 2.9 0.8 11.4 2.2 1.3 1.5 2.3 1.2 3.8 1.5 78.3 27 8. 10.3 1.2 1.5 1.1 1.5 1.3 2.0 36.4 1.4 0.8 2.4 0.5 6.9 1.3 1.4 1.2 0.7 2.7 3.1 1.3 79.0 46 9. 11.2 0.8 1.5 1.8 3.4 1.4 1.9 2.4 29.7 1.4 1.4 0.7 8.5 1.0 4.5 2.1 0.5 0.8 1.1 0.8 76.7 39 10. 15.5 1.1 2.0 1.3 4.6 1.4 2.2 2.6 2.9 17.4 1.7 2.3 12.5 1.4 1.9 3.4 0.7 0.9 1.4 0.8 77.8 22 11. 12.7 1.4 1.9 1.4 2.0 1.5 3.8 5.4 1.7 1.0 25.1 0.8 10.0 1.5 1.3 1.6 0.8 1.2 2.2 1.1 78.3 32 12. 16.3 1.0 1.9 1.0 2.5 1.3 2.2 2.5 1.7 3.7 1.7 18.5 16.1 1.3 1.3 2.0 0.6 0.9 1.5 0.9 78.8 23 13. 19.6 1.2 3.1 1.4 2.1 1.6 2.9 2.0 1.8 1.6 1.8 1.4 28.3 1.5 1.9 2.4 0.9 0.9 1.7 1.0 79.2 36 14. 13.1 5.1 2.2 1.1 1.6 1.5 3.3 3.1 1.4 0.9 1.7 0.7 9.3 18.9 1.2 1.3 2.3 1.6 2.3 5.0 77.6 24 15. 13.7 0.8 1.7 1.1 3.8 1.4 2.0 2.1 6.4 1.2 1.4 0.7 10.3 0.9 24.4 1.6 0.6 0.7 1.2 0.8 76.7 32 16. 14.1 1.0 3.7 2.1 2.3 2.7 2.3 2.3 3.4 2.6 1.7 1.1 12.1 1.3 1.7 19.1 0.7 0.8 1.4 0.8 77.4 25 17. 12.3 7.2 3.9 0.9 1.4 2.0 4.9 2.1 1.3 0.9 1.5 0.6 8.9 3.3 1.1 1.3 19.7 1.1 2.1 1.5 78.0 25 18. 10.7 1.2 1.4 0.7 1.3 1.0 1.9 6.9 1.1 0.8 1.4 0.5 6.9 1.6 1.0 0.9 0.7 29.0 4.0 5.1 78.0 37 19. 12.5 1.5 2.1 1.2 1.6 1.3 4.2 6.8 1.1 0.9 2.1 0.7 9.0 1.8 1.1 1.2 1.2 3.0 22.1 2.9 78.1 28 20. 11.9 2.1 1.7 0.9 1.3 1.2 2.2 3.6 1.1 0.7 1.4 0.6 7.9 4.0 1.1 1.1 0.9 4.2 3.4 26.7 78.0 34

Note. Due to lack of space, here and in Table 2 the ordinal numbers in the first column correspond to the county names in the table header.

Figure 2. a) shows the probability in percentage that a person aged 0 in Budapest will be (alive) in a given county at age x. This also means the expected number of the survivors of 100 newborns in Budapest that will stay in that county after x years. Figure 2. b) shows the opposite direction. Similar pairs of graphs can be drawn for the remaining 19 counties.

As far as the migration of the residents of Budapest is concerned, Pest plays the most important role in both directions. (See Figure 2.) In Figure 2. a) the maximum points of the curves are over 40 years of age, but in 2. b) they are under 40. The figures can be interpreted as follows: “on the left side” of the maximum point, the expected in- migration can compensate mortality but cannot do it later when the in-migrated cohorts start expectedly declining in size. It indicates that people out-migrated from Budapest

“die later” at their new places of residence than those who in-migrate to the capital. That is, the place of residence at an early age is a crucial factor. In Figure 2. b) the gradients of the curves suddenly start to increase around the age of 18 and become permanent at a new level. It displays more intense expected in-migration to Budapest at the beginning of adulthood. However, these do not hold for the in-migration somewhere from Budapest where the expected intensity by age is nearly constant. (See Figure 2. a).)

Figure 2. The number of survivors as a function of age

a) The number of survivors living in the counties b) The number of survivors living in Budapest by age out of 100 newborns in Budapest by age out of 100 newborns in other counties

0 5 10 15 20 25 30

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85

Person

year of age

0 5 10 15 20 25 30

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85

Person

year of age

2.1.2. Life expectancies

Table 2 shows the expected years of life for a newborn male in a given county (corresponding to a row) that is spent – not necessarily permanently – in a county corresponding to a column.

The entries of the main diagonal show the expected duration spent at birthplace. The row sums give evidently the total life expectancy figures by counties of birth. These substantially differ from the classic life expectancies for counties, because the latter ones are calculated on the assumption that a person lives his whole life in one county – and

Pest county

Pest county

dies according to its mortality rates. Therefore, they apply only to such people (!). In Table 2 the dispersion of row sums is little compared to that of a classic table with life expectancies by counties: they are between 69.54% and 71.86%. The little dispersion by counties of birth means that a newborn migrating “through all the counties”, dying by the mortality rate of a given county, lives nearly for the same period wherever he was born. It should be interpreted as “the average migration of many thousand people”, since one person is not able to change more than one or two counties of residence in his lifetime.

Table 2 Expected lengths of remaining life by counties /19/

County 1. Budapest 2. Baranya 3. Bács-Kiskun 4. Békés 5. Borsod-Abaúj-Zemplén 6. Csongrád 7. Fejér 8. Győr-Moson-Sopron 9. Hajdú-Bihar 10. Heves 11. Komárom-Esztergom 12. Nógrád 13. Pest 14. Somogy 15. Szabolcs-Szatmár-Bereg 16. Jász-Nagykun-Szolnok 17. Tolna 18. Vas 19. Veszprém 20. Zala Total Of which the percentage of those living at home

At age 0

1. 34.7 0.9 1.5 0.9 1.6 1.0 2.1 1.3 1.2 1.0 1.2 0.7 15.7 1.1 1.4 1.3 0.5 0.6 1.2 0.7 70.5 49 2. 7.7 38.7 1.4 0.5 0.8 1.0 1.4 1.7 0.7 0.4 0.8 0.3 4.5 3.0 0.5 0.5 3.0 0.6 0.9 1.1 69.8 56 3. 8.6 1.4 37.4 1.0 1.0 3.6 1.8 1.1 0.7 0.6 0.8 0.4 6.3 0.8 0.7 1.4 1.1 0.4 0.9 0.5 70.6 53 4. 9.2 0.6 2.1 32.9 1.0 5.1 1.2 2.3 2.1 0.6 1.3 0.3 6.1 0.8 0.9 2.0 0.4 0.4 0.9 0.6 70.9 46 5. 9.6 0.4 0.9 0.7 36.6 0.8 1.2 1.6 2.5 2.2 1.1 0.6 5.9 0.6 2.0 1.0 0.3 0.4 0.7 0.5 69.5 53 6. 7.4 0.9 3.5 3.0 0.8 40.8 1.0 1.1 1.0 0.5 0.8 0.3 4.6 0.6 0.7 1.1 0.5 0.5 0.6 0.4 70.1 58 7. 9.9 1.1 1.7 0.6 1.2 0.9 33.1 1.7 0.9 0.6 2.1 0.4 7.1 1.5 0.8 0.8 1.7 0.6 2.8 0.9 70.8 47 8. 7.2 0.7 0.8 0.7 0.8 0.7 1.2 44.5 0.8 0.5 1.6 0.3 3.8 0.7 0.9 0.7 0.5 1.7 2.1 0.8 71.1 63 9. 7.8 0.4 0.8 1.4 2.4 0.8 1.2 1.5 39.7 0.9 0.8 0.4 5.0 0.5 3.1 1.5 0.3 0.4 0.6 0.4 70.0 57 10. 11.3 0.7 1.1 0.8 3.3 0.8 1.3 1.8 2.0 30.4 1.1 1.7 7.8 0.9 1.2 2.6 0.4 0.4 0.8 0.4 70.8 43 11. 9.0 0.9 1.1 0.9 1.1 1.0 2.6 3.6 1.1 0.6 36.5 0.5 6.3 1.0 0.8 1.0 0.5 0.6 1.4 0.6 71.0 51 12. 11.4 0.5 1.0 0.6 1.8 0.7 1.4 1.6 0.9 3.0 1.1 31.2 11.0 0.8 0.7 1.3 0.3 0.5 0.9 0.5 71.2 44 13. 16.3 0.7 2.0 0.9 1.3 1.0 1.9 1.2 1.1 1.0 1.2 0.9 35.9 0.9 1.2 1.6 0.5 0.5 1.1 0.6 71.9 50 14. 9.3 3.9 1.3 0.7 1.0 0.9 2.1 1.9 0.9 0.5 1.0 0.3 5.7 31.6 0.7 0.8 1.6 0.9 1.6 3.7 70.5 45 15. 9.7 0.4 0.9 0.6 2.8 0.9 1.2 1.3 4.7 0.7 0.9 0.4 6.2 0.5 36.1 1.0 0.3 0.4 0.7 0.4 70.2 51 16. 10.1 0.6 2.5 1.6 1.6 1.8 1.4 1.4 2.4 2.1 1.2 0.7 7.6 0.8 1.1 31.8 0.5 0.5 0.8 0.4 70.8 45 17. 8.7 5.5 2.7 0.5 0.8 1.3 3.9 1.2 0.8 0.5 0.9 0.3 5.2 2.3 0.7 0.8 31.9 0.7 1.4 0.9 71.1 45 18. 7.7 0.7 0.8 0.4 0.7 0.5 1.1 4.8 0.7 0.5 0.9 0.3 3.8 0.9 0.6 0.5 0.5 38.4 3.0 3.7 70.3 55 19. 8.9 0.9 1.2 0.7 1.0 0.8 3.0 4.8 0.7 0.5 1.5 0.4 5.5 1.1 0.6 0.7 0.8 2.1 33.6 2.0 70.8 47 20. 8.6 1.3 1.0 0.5 0.7 0.7 1.4 2.2 0.7 0.4 0.8 0.4 4.4 3.2 0.6 0.6 0.5 2.9 2.4 37.8 71.1 53

2.2. Expected internal migration of the male population at the beginning of 2010 Based on the methodology, the expected durations do not depend on the age distribution of the population but – in accordance with the classic life table theory – only on the transition (in our case on the migration and mortality) rates. Hence, the expected durations calculated for two populations having different age distributions but the same transition rate are equal.

In this section we leave the theoretical framework and compute (not per capita figures but) the expected number of those migrants of the whole population at the beginning of 2010 who remained alive. In doing so, we simply multiply the probabilities of survival and life expectancies per person in every cohort by the initial cohort sizes and then take their sum (see formulae /21/–/26/).

2.2.1. Survivors

This subsection contains the expected population numbers evolved from the 2010 population in consequence of internal migration and mortality, as a function of time.

That is, neither the expected number of births nor the external migration is taken into account. Therefore, after a possible initial growth due to the positive migration balance, the expected county populations of the country that is regarded as a closed system in this sense decline to zero.

Table 3 The number of survivors of the 2010 population as a function of time /22/

(1000 persons)

Year

1. Budapest 2. Baranya 3. Bács-Kiskun 4. Békés 5. Borsod-Abaúj-Zemplén 6. Csongrád 7. Fejér 8. Győr-Moson-Sopron 9. Hajdú-Bihar 10. Heves 11. Komárom-Esztergom 12. Nógrád 13. Pest 14. Somogy 15. Szabolcs-Szatmár-Bereg 16. Jász-Nagykun-Szolnok 17. Tolna 18. Vas 19. Veszprém 20. Zala

Initial 778 186 250 173 327 198 206 215 257 146 150 97 590 152 267 186 111 124 172 136 Stayers 10 521 136 182 122 235 148 145 166 193 99 109 67 428 103 194 129 78 91 122 99 40 137 40 53 32 65 47 39 57 62 24 31 16 136 24 56 33 20 26 32 28

80 4 1 1 1 1 1 1 2 1 0 1 0 4 0 1 1 0 0 1 1

(Continued on the next page.)

(Continuation.)

Year

1. Budapest 2. Baranya 3. Bács-Kiskun 4. Békés 5. Borsod-Abaúj-Zemplén 6. Csongrád 7. Fejér 8. Győr-Moson-Sopron 9. Hajdú-Bihar 10. Heves 11. Komárom-Esztergom 12. Nógrád 13. Pest 14. Somogy 15. Szabolcs-Szatmár-Bereg 16. Jász-Nagykun-Szolnok 17. Tolna 18. Vas 19. Veszprém 20. Zala

In-migrants 10 193 17 27 17 25 22 32 28 25 19 20 11 128 21 21 22 13 13 23 16 40 233 26 41 25 37 34 47 45 37 25 30 17 192 29 30 32 19 20 33 25

80 10 1 2 1 2 2 2 3 2 1 2 1 12 1 1 2 1 1 2 1

Total 10 140 22 32 24 44 21 32 21 30 24 20 16 100 25 37 29 17 14 26 17 40 175 33 47 35 69 33 45 33 47 34 30 23 130 34 58 42 24 21 37 24

80 9 2 3 2 4 2 2 2 3 2 2 1 7 2 3 2 1 1 2 1

Out-migrants 10 831 181 245 165 309 199 206 222 253 140 150 92 618 148 250 179 107 122 169 135 40 279 56 78 41 72 73 65 98 85 31 50 21 268 37 61 45 27 42 49 45

80 15 4 5 3 5 5 4 7 6 2 3 1 24 2 4 3 2 3 3 3

Figure 3. Survivors of the 2010 population of three counties as a function of time, with and without in-migration /22/,/23/

0 100 200 300 400 500 600 700 800 900 1 000

10 20 30 40 50 60 70 80

1000 persons

year

Budapest Budapest

Pest Pest

Borsod-Abaúj-Zemplén Borsod-Abaúj-Zemplén Broken: without in-migration

In Table 3 “initial” means the population at t = 0 e.g. at the beginning of 2010.

“Stayers” mean those members of the initial 2010 population who have not left or have remigrated to their county of origin by the time t year(s) later. The term “in- migrants” means those who have come from other counties and are still alive in the given county. “Total” is the sum of stayers and in-migrants e.g. the expected survivors of the 2010 population in the given county. The word “out-migrants”

means those who have left and are still alive in other counties at time t.

As shown in Figure 3, the expected net in-migration increases the population of Budapest and county Pest at a higher rate than that of county Borsod-Abaúj- Zemplén.

2.2.2. Expected numbers of migration

The expected numbers of migration are generally greater (but in no way less) than that of migrants, since one person may repeatedly in-migrate into a county. However, not only for this reason are the numbers of in- and out-migration greater in Table 3 than the corresponding values of Figure 4 but also because in-migration figures include solely those in-migrants who remained alive.

The expected migration numbers cumulated for t = 80 years are displayed in Figure 5 (divided by the county’s population number in Figure 5. b)).

Figure 4. The expected numbers of in- and out-migration of the 2010 population for t years, by counties /23/

a) In-migration b) Out-migration

0 100 200 300 400 500 600 700

10 20 30 40 50 60 70 80 1000 persons

year of age

0 100 200 300 400 500 600 700

10 20 30 40 50 60 70 80

1000 persons

year of age

In Figure 4 the expected in-migration numbers of Budapest and Pest (700 000 and 650 000 under 80 years of age) are higher by an order of magnitude than that of others but their out-migration numbers (500 000 and 650 000) are not much smaller either.

The same numbers of other counties fall between 50 000 and 120 000. County Fejér shows large but balanced, slightly positive expected in-migration. The three counties having lowest expected migration (in both directions) are Nógrád, Tolna and Vas.

Pest county Budapest

Pest county Budapest

Large in-migration (also in relative terms) is expected in counties Győr-Moson- Sopron, Zala and Pest, while large net out-migration (also in relative terms) is probable in counties Borsod-Abaúj-Zemplén, Szabolcs-Szatmár-Bereg and Nógrád.

Figure 5. The expected numbers of in- and out-migration of the 2010 population for t = 80 years, by counties /23/

a) In-migration b) Out-migration

0 100 200 300 400 500 600 700

0 100 200 300 400 500 600 700

Out-migration (1000 persons)

In-migration (1000 persons) Budapest

Pest

Fejér Borsod-

Abaúj- Zemplén

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Out-migration (2010 = 1.0)

In-migration (2010 = 1.0)

Budapest Pest Nógrád

Győr-Moson- Sopron Borsod-Abaúj-

Zemplén Szabolcs- Zala

Szatmár- Bereg

For correct interpretation of Figure 5. b), note that, for example, the expected migration of 83% from Budapest comes not only from the 2010 population of Budapest but also from those who are expected to in-migrate there in the meanwhile.

Figure 5. b) shows the phenomenon that often, even large and small expected per- capita out-migration numbers come with large and small expected per-capita in- migration numbers (for absolute numbers, this is natural [see Figure 5. a)]). Note that in Figure 5. a) the sum of x and y coordinates are equal e.g. the centroid of points falls on the main diagonal, since the sum of in- and out-migration figures by counties is obviously equal (a closed system is concerned).

3. Methodology

In Section 3.1. we describe the random movement of an individual between states by the theory of finite-state, time-inhomogeneous Markov processes. In the description, the age-specific transitional probabilities between states (from which all the table functions are derived and which are computed from transitional rates) play a central role. The transitional rates are assumed to be known, but their estimation

from statistical events – from age-specific population numbers and number of transitions – is included in Section 3.3. The estimation of transitional rates is an exciting field of demography.

3.1. The general model

In the following, matrices are denoted using bold normal letters; x, y, h, t, τ stand for real numbers, and i, j, k, n rates are assumed integers. Matrices with tilde refer to the examined population; the rest signifies table functions.

We denote the states by 0, 1,… n; 0 is for death; it is allowed to step from any state to another, except 0, from where there is no possibility to step over. Let ξ_t be a random variable, whose value is the state of a person in the model in time-point t.

We suppose that the ξ_t stochastic process is a Markov process, that is, it statisfies the Markov property:

Pr

(

ξt = jξu1 =i ,ξ1 u2 =i ,...,ξ2 u_n =in

)

=Pr

(

ξu_n =in

)

(

u1<u2< <... un <t

)

. /1/

We assume that the probabilities of events in the model depend on the time only through the age of the person e.g. do not depend on the time explicitly. Therefore, time is equivalent to age, and we denote the time by x instead of the accustomed t.

Kolmogorov's first equation for finite-state Markov processes is:

0

i, j n

x,x h k , j i,k

x h x,x h k

dp μ p

dh

+ + +

=

= ∑ ^{, pi, j}_x,x ^=δ_{i, j}

(

^{h>0, i, j=1 2, ,...,n}

)

^{, /2/}

where μ^{i, j}_x is the force of transition from the state i to j at age x with the meaning that for small dx-s, μ^{i, j}_x dx is the probability that a person aged x in state i will be in state j at age x dx+ . μ^i,_x⁰ is the force of mortality in state i at age x, where μ⁰_x^{, j} =0 for every j – this time. The p^{i, j}_{x,x h+} transition probability is the probability that a person aged x in state i will be in state j at age x h+ . Each equation of the differential equation system /2/ is in fact the total probability theorem with the restriction that the forces of transition do not depend on x explicitly (only the latter

x h+ ). In our case, in a state when one can stay, leave, or die, it holds:

0

1 ⁿ

j , j j ,k

x x

kk j

μ dx μ dx

=≠

⎡ ⎤

⎢ ⎥

= − ⎢ ⎥

⎢ ⎥

⎣ ⎦

∑

^and

1 n 1

i, j i

x,x h x,x h

k

p ₊ q ₊

= = −

∑ ^{, /3/}

where q_{x,x h}ⁱ ₊ is the probability of death of the persons aged x staying in state i within h years. Thus /2/ in matrix form is:

^d

(

^{x,x h )}

) (

^{x h}

) (

^{x,x h ,}

) ( )

^x,x

dh

+ = − + + =

P μ P P I, /4/

where

( )

1 1 2 1

0

2 1 2 2

0

1 2

0

n , j , ,n

x x x

j

, n , j ,n

x x x

j

n, n, n n, j

x x x

j

μ μ .... μ

μ μ .... μ

x

μ μ .... μ

=

=

=

⎡ − − ⎤

⎢ ⎥

⎢ ⎥

⎢ ⎥

− −

⎢ ⎥

= ⎢ ⎥

⎢ ⎥

⎢ ⎥

⎢− − ⎥

⎢ ⎥

⎣ ⎦

∑

∑

∑ μ

#

and

( )

1 1 1 2 1

2 1 2 2 2

1 2

, , ,n

x,x h x,x h x,x h

, , ,n

x,x h x,x h x,x h

n, n, n,n

x,x h x,x h x,x h

p p ... p

p p ... p

x,x h

p p ... p

+ + +

+ + +

+ + +

⎡ ⎤

⎢ ⎥

⎢ ⎥

+ = ⎢ ⎥

⎢ ⎥

⎢ ⎥

⎣ ⎦

P . /5/

In this way, the 0^th row and column are not included in the matrices, and the row sums of μ give the force of mortalities belonging to x in the corresponding state.

The row sums of P equal to 1 minus the probability of death.

An equivalent rewriting of /4/ is:

^P

(

^{x,x h+}

)

^{= −I}

_∫

₀^hμ

(

^x+τ

) (

^{P x,x+τ}

)

^{dτ. /6/}

If μ in /4/ is continuous on

[

^{x,x h+}

]

, and P is also continuous in its second variable, then (by the Bolzano–Weierstrass theorem) there exists an ^m

( )

^x,h

satisfying /7/ and /8/.

min0_{≤ ≤τ h}μ_{i, j}

(

x+τ

)

≤m_{i, j}

( )

x,h ≤max0_{≤ ≤τ h}μ_{i, j}

(

x+τ

) (

i, j=1 2, ,...,n

)

/7/

^P

(

^{x,x h+}

)

^{= −I m}

( )

^x,h

_∫

₀^hP

(

^x,x+τ

)

^{dτ /8/}

If we prescribe that ^μ

( )

^t is a constant on

[

^{0≤ ≤t h}

]

, that is, if

^μ

( )

^{t ≡m}

( )

^x,h

(

^{0≤ ≤t h}

)

^{, /9/}

then from /8/

^d

(

^{x,x t}

) ( ) (

^{x,h x,x t ,}

) ( )

^x,x

(

^{0 t h}

)

dt

+ = − + = ≤ ≤

P m P P I . /10/

The unique solution to the initial value problem /10/ is:

^P

(

^{x,x t+ =}

)

^exp⎡_⎣^−tm

( ) (

^{x,h ⎤}_⎦ ^{0≤ ≤t h}

)

^{. /11/}

However, if instead of /9/ it is ^P

(

^{x,x t+}

)

that is linear on

[ ]

^0,h – this is the assumption made in the classic theory – then from /8/ the known formula directly follows:

^P

(

^{x,x h+}

)

^=⎡_⎣^I+

(

^{h /2}

) ( )

^{m x,h ⎤ ⎡}_{⎦ ⎣}^{−1 I−}

(

^{h /2}

) ( )

^{m x,h ⎤}_⎦^{, /12/}

since replacing the left side of the identity:

_∫

₀^hP

(

^{x,x t+ =}

)

₂^h⎡_⎣^{I P+}

(

^{x,x h+}

)

^⎤_⎦ ^{0≤ ≤t h /13/}

expressing linearity by the right side of /8/, we get /12/. However, this gives proper values for ^P

(

^{x,x h+}

)

with h-s small enough (and also for small m^{i, j}_x,h-s), otherwise it results in such a large bias that the row sums will differ considerably from 1−qⁱ_{x,x h+} , perhaps negative probabilities can occur. (For an example, see Nour–Suchindran [1984].) This is why we choose the calculation method /9/–/11/ and define the matrix of transition probabilities by /11/ instead of the classical formula /12/. This method therefore prescribes the piecewise constancy of m in the model, instead of the piecewise linearity of P. Hence, ^m

( )

^x,h is a valid constant transient rate at each point of the interval

[

^{x,x h+}

]

. (See formula /9/.) Then by /11/, P is piecewise exponential in its second variable.

From the ^P

(

^{x,x h+}

)

-s other functions can be computed by h steps.¹ Since let

i, j

ly ,x the probability that a person aged y in state i will be in state (alive) j at age x.

1We note that during the computation the first six terms of the Taylor series

(x,x h+ )=e^{−hm( )x,h} =∑_i^∞₌0⎡⎣h ( )x,h⎤⎦ⁱ/ i!

P m has been sufficiently accurate.

Denote z the beginning of the last age interval and let z – x be divisible by h. Then by the total probability theorem, the survival matrices

{ }

^{li, jy ,x} can be created applying the following recursion to x-s of the formy kh+ , by h-year steps (where k is an integer):

ly ,x h₊ =l Py ,x

(

x,x h+

)

, l_{y ,y}=I

(

^{y x z h≤ ≤ −}

)

^{. /14/}

Denote L^{i, j}_{y ,x,h} the expected number of years (less than h) lived in state j between age x and x h+ by a person who is in state i at age y. Using /14/ and integrating based on /11/, we get:

^Ly ,x ,h=∫₀^hly ,x t₊ dt=∫₀^hP

(

x, x t+

)

^ly ,xdt=^{l m}y ,x

( )

x,h ⁻¹⎡⎣I−^P

(

x,x h+

)

⎤⎦ ^/15/

except the last half-open interval

[

^z,∞

)

, where we assume transition probabilities to be exponential, similarly to /11/, with constant force of transition:

^P

(

^{z,z t+ =}

)

^exp⎡_⎣^−tm

( ) (

^{z ⎤}_⎦ ^0≤t

)

^{. /16/}

It follows that (for convenience, using the index h for /18/):

^Ly ,z ,h=∫₀^∞P

(

z,z t+

)

^ly ,zdt=^m

( )

z ^−1ly ,z. ^/17/

Finally, the matrix of expected duration (or length of life) at age y above ^x

( )

^≥y

in the model is:

_{y ,x} ^z _{y ,k ,h}

k xstep h=

= ∑

e L , /18/

where the general entry e^{i, j}_{y ,x} of the matrix is the expected number of years that a person aged y in state i will live (not necessarily permanently) in state j above age x.

(Formally, e_{y ,x} also depends but not essentially on h just as a time step; hence we omit the index h.)

Moving on the estimation of the examined population: in the former formulae (starting from /7/) ^m

( )

^x,h is replaced by the transition rate ^m

( )

^x,h relating to the

members of the examined population aged between x and x h+ , and also, in /17/

( )

^z

m is replaced by ^m

( )

^z related the ones above age z. (The calculation of these rates can be found in Section 3.3.) The variables changed for this reason get tildes.

Hence

Ly ,x,h ≅m

( )

x,h ⁻¹⎡⎣I P−

(

x,x h+

)

⎤⎦ly,x and Ly,z ,h =m

( )

z ⁻¹ly ,z, /17’/

and the estimated expected length of life of an arbitrary member aged y of the examined population above age ^x

( )

^{≥y is:}

_{y ,x} ^z _{y ,k ,h}

k xstep h=

= ∑

e L . /18’/

The difference between the model of linear transition probability and the formulae mentioned formerly is that using the classic /12/ instead of /11/ and integrating in /15/ by /13/, we get – instead of /17/ – the common formula of life tables:

( )

( ) ( )

0 0

2 2 ,

h h

y,x,h y,x t y ,x

y,x y ,x y ,x h

dt x,x t dt

h h

x,x h

+

+

= = + =

⎡ ⎤

= ⎣ + + ⎦ = +

∫ ∫

L l P l

I P l l l 19/

in which replacing ^m

( )

^{x,h by m}

( )

^{x h,} (in the expression of P in /12/) we get:

^{Ly ,x,h ≅} ₂^h

(

^{ly ,x+ly ,x h+}

)

⁽

^{y x z h≤ ≤ −}

⁾

^{and Ly,z ,h =m}

^{( )}

^{z −1ly ,z. /19’/}

(For the last age interval, the exponential formula is used in the linear model, too.) The estimation of life expectancy is given by /18’/ again.

3.2. Expected migration of the 2010 population

In this section, we leave the theory of life tables that computes expected values for one person. Here we produce the expected numbers of survivors and those of internal migration for a population with known age distribution at the beginning of

the period. For this purpose, we multiply probabilities of survival and expected durations (which have already been computed) by cohort numbers, using again the assumption that the events concerning different population members take place independently and within a cohort with the same probability.

a) Denote R ys

( )

the number of population of completed age y (from now on y is an integer) at the beginning of the period (2010) in county s. Of them

R y ls

( )

y,x^{s, j} /20/

will be in state j at age x e.g. after x y− years. Therefore, /20/ gives the number of survivors (by age and county) in the population (by age and county) at the beginning of 2010.

The total expected number of survivors in county j after t years

( )

0

( )

0

( )

0 1 z t n

s, j j , j * j

s y,y t

y s

R y l λ t λ t

−

+ + +

≥ = = +

∑ ∑

^/21/

consists of those who were outside county j at the beginning of and also those who were (stayed or re-migrated) there during the period:

0

( ) ( )

0 1 z t n

* j s, j

s y ,y t

y s

s j

λ₊ t ⁻ R y l ₊

≥ =

≠

=

∑ ∑

^; 0

( ) ( )

0

j , j z t j , j

j y ,y t

y

λ₊ t ⁻ R y l ₊

≥

=

∑

^/22/

(where 0+ abbreviates the summation for y≥0 years of age).

b) The expected number of migration from county i to j within t years and the total expected number of in-migration is:

0

( ) ( ) ( )

0 0 1

t z τ n

i, j s,i

i, j s y ,y τ

τ y s

v ₊ t ⁻ P y τ R y l ₊

= = =

⎛ ⎞

=

∑ ∑

+ ⎜⎝

∑

⎟⎠

(

^{i≠ j}

)

^; 0

( )

0

( )

1

* j n i, j

ii j

v ₊ t v₊ t

=≠

=

∑

^{. /23/}

These are, therefore, not the expected numbers of migrants: one person can migrate to the same county several times. A short explanation for the first formula of /23/:

( )

1

n s,i

s y ,y τ

s

R y l ₊

∑= is the expected number of persons aged y at the beginning of 2010, being alive in county i after τ year. Multiplying it by the probability of transition from i to j in year τ, we get the expected number of transition from i to j

in year τ. In contrast with /21/ where the “time jump” from 0 to t can be managed in one step, here the “stream” has to be summarized from 0 to t by years.

The expected number of years lived in county j above age x by R ys

( )

persons of age y staying at the beginning of 2010 in county s is:

Ty,x^{s, j} =R y es

( )

y ,x^{s, j}. /24/

3.3. Calculating transition rates

The following methodology is a generalisation of that developed by the Human Mortality Database (Wilmot et al. [2007]) for death rates, since it calculates age specific transition rates for a population from the number of transition cases between states of finite number, happened during a given time interval (period). (In this article states mean staying in counties, numbers of transition mean numbers of migration.)

States are denoted by 0 1 2, , ,...,n; 0 refers to death, from each state one can step to any other, except for 0, from where one can step nowhere. Figure 6 shows a Lexis diagram (hereinafter called diagram) corresponding to state i. For every i>0, there is a similar diagram. The person belonging to the “skew line” in-migrated from state k to state i (to the plane of the diagram) at completed (integer) age x−1, at the beginning of the (calendar) year t−1 and later, in the middle of year t at age x migrated to j. The length of the line divided by 2 is the time spent in i (the diagonal of the square means exactly one year). For each member of the population, there is a lifeline which is a line segment divided into sub-segments in such a way that each of them is visible on exactly one diagram. The lifespan of a person begins with a line starting from x=0 (somewhere in the bottom left of the diagram that belongs to his birth state) and closes in a point labelled by 0 on the “last” figure diagram. (The easiest solution would be to illustrate the whole life span on one figure diagram in such a way that the sub-segments “spent” in individual states have colours corresponding to those states.)

Denote Vi,k^a

( )

x,t and Vk ,i^a

( )

x,t the number of migrants from state i to state k and that from k to i, respectively (in the lower triangle of the thick square in Figure 6).

Similarly, stand Vi,k^f

( )

x,t and Vk ,i^f

( )

x,t for out- and in-migrants in the upper triangle. The net out-migration from these two triangles is:

i^a

( )

i,k^a

( )

k ,i^a

( )

k i

V x,t V x,t V x,t

≠

=∑ − ^and i^f

( )

i,k^f

( )

k ,i^f

( )

k i

V x,t V x,t V x,t

≠

=∑ − ^{. /25/}

Figure 6. The Lexis diagram of state i

Definition: We call exposed to risk period belonging to calendar year t, age x and state i and denote by E x,ti

( )

the total time spent in state i by the members aged x of a population in year t (given in life-years), that is, the total length of life lines of the population inside the square

(

^i,x,t

)

^.

Definition: If t is a calendar year, x is an age and i, j are states, then the transition rate Mi, j

( )

x,t for a population is the number of out-migration

( )

^a

( )

^f

( )

i i,k i,k

V x,t =V x,t +V x,t from the square

(

^i,x,t

)

of the Lexis diagram divided by the expose to risk figure E x,ti

( )

. (Mi, j

( )

x,t is identical to mi, j

( )

x,h in 3.1. if

1

h= and t is the year of the period.)

Thus, E x,ti

( )

still needs to be determined. Let N x,ti

( )

be the number of those whose x^th birthday is in year t e.g. whose lifeline intersects the bottom line of the square

(

^i,x,t

)

. The intersection points mean birthdays. Assume that the distribution of these days is uniform, thus that is also the points on the bottom line.² If these people did not out-migrate from i in year t, then they would spend, on average, 1/2 year per person in the lower triangle. Similarly, if the N xi

(

+¹,t

)

number of persons having x+1^st birthday in year t were in state i from the beginning of the year (and were not out-migrate from it), and the distribution of their birthdays were uniform in

2 Instead of uniformity, it is sufficient that the distribution of the points on the top and bottom lines of the square are nearly the same.

( )

^{x t}

N_i ,

(

^{x t}

)

N_i +1,

( )

^{x t}

R_i , ^R_i

(

^x,t+1

)

+1 x

−1 x

x

−1

t t t+1

k

j