Backwards Retropolation of Unknown Migration and Exploitation of the - - PDF document

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Backwards Retropolation of Unknown Migration and Exploitation of the Lee-Carter Model to Modelling Migration Profile

Ondrej Simpach1

• Abstract. During last two years, the European Union has experienced a significant

immigration wave. Knowing the age-and-sex specific migration profile is important for social policy planning and decision making. European countries are classified as a “transit” and as a “target”. The methods of demographic projections address the issue of migration only rarely. Net migration balance is often determined by expert guesses and is used to correct the final form of the population development. There are currently not sufficient methods of the migration projections. Therefore, the aim of the contribution is to present the methodology of the projection of the age-and-sex specific net-migration profile on the case study of the Czech Republic and other Visegrad Four Countries (Slovakia, Poland, Hungary) (as the transit countries which the migration wave only passes through) in Middle Europe. The procedure is based

• n Lexis approach and stochastic modelling by Lee-Carter model. The principle of

the method is based on the backward retropolation of the needed data and consequent application of the main component method. The projection horizon might vary depending on the available data. Results are largely universal and serve as the supplement to already used demographic models and expert guesses. Key words: immigration, Middle Europe, Visegrad Four Countries, Lexis diagram, Lee-Carter model, population projection JEL Classification: C22, C32, J11

1 Introduction

Current immigration development to some countries of the European Union (EU) does not have the parallel in the history. Hence, the modelling and extrapolation is difficult. There are countries in Europe which in relation to migration wave serve as target and countries through which the migrants are passing. Regardless if it is a country of transit or of destination, analysis always encounters the problem of a solid database (Šimpach, Pechrová, 2016). According to the Eurostat (2016) database it is clear, that many countries have started to consistently detect age-and-sex-specific numbers of immigrants only in year 2004. In the Visegrad Four Countries (Czech Republic, Slovakia, Poland, Hungary), which currently belong among the transit countries (i.e. immigrants assume that will only pass through the country to some western European country, where they have already their family or relatives) have been a good database gathered by the Eurostat since 2000. However, it is not sufficient to make quality analyses based

• n sophisticated quantitative methods.

1 University of Economics Prague, Faculty of Informatics and Statistics, Department of Statistics and probability.

Winston Churchill sq. 4, 130 67 Prague, Czech Republic. Email: ondrej.simpach@vse.cz, tel.: +420 737 665 461.

Simpach, O. (2017). Backwards Retropolation of Unknown Migration and Exploitation of the Lee-Carter Model to Modelling Migration Profile. In: 28th IUSSP International Population Conference, 29th October – 4th November 2017, Cape Town, South Africa, pp. 1–9.

2 Demographic processes of mortality and fertility have one common denominator that currently allows the analysis of these processes and makes possible to perform their modelling. It is the existence of a quality database and of the statistics can be divided by the gender and age group that are crucial for modelling of any demographic process, including migration. Many statistical offices (including the Czech Statistical Office – CZSO) lacks a long-term database of the number of immigrants by gender and by age at time t (Ix,t = immigrants x-years

• ld in time t) and the number of emigrants by sex and age at time t (Ex,t = emigrants x-years old

in time t). These statistics began to be collected in the Czech Republic only after 2000, in Slovakia, Poland and Hungary after 2004. Until then, there were examined only aggregate numbers of immigrants and emigrants in a given year, specific maximally by the gender. This is not sufficient to use the approaches that are currently commonly applied on other demographic processes. Therefore, the aim of this contribution is to elaborate the methodology

• f the projection of the age-and-sex migration on the case study of the Czech Republic and
• ther Visegrad Four Countries population. The method is based on stochastic modelling.

2 Literature and Motivation

Demographic analyses and population projections are very important. “On the basis of the data

• n the sex-and-age structure of the population it is possible to anticipate relatively well the

long-term development and foresee the future requirements, for instance, in the fields of education, the health service, social services, etc.”, (Fiala, Langhamrová and Langhamrová, 2009). Similarly, Lassilla, Valkonen and Alho (2014) demonstrate “that, although demographic forecasts are uncertain, they contain enough information to be useful in forward-looking policy rules”. However, the models are typically made under the assumption that future demographic development is deterministic. Immigration crisis can represent serious distortions to the models. Also, the longer is the projection horizon, the less probable is that the assumptions of the models would hold (Hyndman, Shang, 2009). Therefore, the stochastic approach must be considered. Despite that it enables more precise projections of population, its assumptions might not hold in turbulent development of the reality. Migration is a very important demographic component and it cannot be ignored in models

• r cannot be considered as unchangeable in time (or at least not in case of the larger areas.)

Currently the EU is contending with problems of immigration. The problem of possible datasets distortion should be discussed as many demographic models are not able to adequately explain the migration process. “Population migration involves the relocation of individuals, households

• r moving groups between geographical locations” (Vitanov and Vitanov, 2016). Some of the

population pressures might be relieved by immigration, but it must not be too high as “with growing disparities between the levels of material wealth in rich and poor countries, migration appears to be an attractive option for inhabitants of less developed countries”, (Rowlands, 1999). The increased migration from countries affected by wars and poor economic situation has recently raise many concerns. Our contribution deals with immigration to European countries which are considered to be the transit for the immigrants from third countries. Particularly we examine the migration to Czech Republic, Slovakia, Poland and Hungary.

3 Methodology and Data

One of the basic assumptions about the demographic processes is classical balance equation

   

F M t F M t F M t F M t F M t F M t

E I D B S S

/ / / / / 1 /

    



, (1) (where M = males, F = females) that is sex-specific, but not age-specific. The difference in the number of live born and death is natural increase (decrease) of population while the difference

Simpach, O. (2017). Backwards Retropolation of Unknown Migration and Exploitation of the Lee-Carter Model to Modelling Migration Profile. In: 28th IUSSP International Population Conference, 29th October – 4th November 2017, Cape Town, South Africa, pp. 1–9.

3 in numbers of immigrants and emigrants is mechanical increase (decrease) of population, denoted as

F M t F M t F M t

E I

/ / /

   . (2) Mechanical increase (decrease) of population has its interpretation as net migration balance, and the division of this balance by the average state of the population of particular sex in time t yields so-called net migration rate calculated as

F M t F M t F M t F M t F M t F M t

e i S E I

/ / / / / /

     . (3) (S ͞ t is in this case the mid-state number of population in time t that is assumed to be at 1st July

• f particular year). From above stated equations it is possible to define the cohort sex-specific

form of balance equation as

F M t x F M t x F M t x F M t x

D S S

/ , / , / 1 , / ,

   



. (4) Using Lexis diagram (see Figure 1),

Figure 1 Diagram for calculation of unknown number of net migrants by Lexis approach (Lexis, 1875). There is time on axis x and age on axis y. Marked area is divided to 2 × 1/8 and 2 × 3/8 = 8/8. Source: own elaboration based on Lexis (1875) approach.

knowing the initial, the middle and the final state of the demographic event and knowing completed ages, the Figure 1 is utilized where in the surface of the angles is 8 1 and 8 1 , 8 3 , 8 3     IV III II I . (5) Consequently, we express the new form of sex-specific cohort balance equation and include those angles from Lexis diagram as

Simpach, O. (2017). Backwards Retropolation of Unknown Migration and Exploitation of the Lee-Carter Model to Modelling Migration Profile. In: 28th IUSSP International Population Conference, 29th October – 4th November 2017, Cape Town, South Africa, pp. 1–9.

4                               

          F M t x IV F M t x F M t x III F M t x F M t x II F M t x F M t x I F M t x F M t x F M t x

D D D D S S

/ 1 , / 1 , / , 1 / , 1 / 1 , 1 / 1 , 1 / , / , / , / 1 , 1

8 1 8 3             , (6) while by simple rearrangement as

F M t x F M t x F M t x F M t x F M t x F M t x F M t x F M t x F M t x F M t x

D D D D S S

/ 1 , / 1 , / , 1 / , 1 / 1 , 1 / 1 , 1 / , / , / , / 1 , 1

8 1 8 1 8 1 8 1 8 3 8 3 8 3 8 3

         

             , (7) we gain a set of four equations with four unknowns in the form of

F M t x F M t x F M t x F M t x F M t x F M t x F M t x F M t x F M t x F M t x

D D D D S S

/ 1 , / 1 , / , 1 / , 1 / 1 , 1 / 1 , 1 / , / , / 1 , 1 / ,

8 1 8 1 8 1 8 1 8 3 8 3 8 3 8 3

         

             , (8a)

F M t x F M t x F M t x F M t x F M t x F M t x F M t x F M t x F M t x F M t x

D D D D S S

/ 1 , / 1 , / , 1 / , 1 / 1 , 1 / , / , / , / 1 , 1 / 1 , 1

8 1 8 1 8 1 8 1 8 3 8 3 8 3 8 3

         

             , (8b)

F M t x F M t x F M t x F M t x F M t x F M t x F M t x F M t x F M t x F M t x

D D D D S S

/ 1 , / 1 , / , 1 / 1 , 1 / 1 , 1 / , / , / , / 1 , 1 / , 1

8 1 8 1 8 1 8 3 8 3 8 3 8 3 8 1

         

             , (8c)

F M t x F M t x F M t x F M t x F M t x F M t x F M t x F M t x F M t x F M t x

D D D D S S

/ 1 , / , 1 / , 1 / 1 , 1 / 1 , 1 / , / , / , / 1 , 1 / 1 ,

8 1 8 1 8 1 8 3 8 3 8 3 8 3 8 1

         

             . (8d) After having the number of net migrants for each year t, each age x and each sex, the calculations are done according to the formula (3). This approach yields net migration rates, so- called net migration profiles, on which can be applied e.g. Lee-Carter method (Lee, Carter, 1992, Lee, Tuljapurkar, 1994), that is used for modelling of death rates and after certain modification also in the case of fertility rates. Because the empirical data matrix for the Czech Republic and the other Visegrad Four countries looks like in Figure 2 (male population is on the left side, female population on the right side), it is definitely better to use backwards reconstructed data, as shown in Figure 3. The results are complete, time series are longer and even through there is higher variability (mainly in advanced ages) the datasets are possible to be analysed. Migration data will be analysed only in the age range x = 0, 1, ..., 85+ of completed years and in the “Case study” section there will be presented results for the Czech Republic only.

Simpach, O. (2017). Backwards Retropolation of Unknown Migration and Exploitation of the Lee-Carter Model to Modelling Migration Profile. In: 28th IUSSP International Population Conference, 29th October – 4th November 2017, Cape Town, South Africa, pp. 1–9.

5

Figure 2a Empirical values of age-and-sex specific numbers of immigrants (only in 5-year age groups) in years 2004–2013 in the Czech Republic (perceived as transit). Source: Eurostat (2016). Figure 2b Empirical values of age-and-sex specific numbers of immigrants (only in 5-year age groups) in years 2004–2013 in Slovakia (perceived as transit). Source: Eurostat (2016). Figure 2c Empirical values of age-and-sex specific numbers of immigrants (only in 5-year age groups) in years 2004–2013 in Poland (perceived as transit). Source: Eurostat (2016).

Simpach, O. (2017). Backwards Retropolation of Unknown Migration and Exploitation of the Lee-Carter Model to Modelling Migration Profile. In: 28th IUSSP International Population Conference, 29th October – 4th November 2017, Cape Town, South Africa, pp. 1–9.

6

Figure 2d Empirical values of age-and-sex specific numbers of immigrants (only in 5-year age groups) in years 2004–2013 in Hungary (perceived as transit). Source: Eurostat (2016). Figure 3 Age-and-sex specific rates of net migration of males and females in the Czech Republic in years 1968–

• 2012. Rates are due to high variability in the highest age groups consider only for the age range 0–85+

completed years of life. Source: author’s calculations.

3 Case study

Values of age-and-sex-specific net migration rates in the Czech Republic will be decomposed according to the Lee-Carter method (Lee, Carter, 1992, or Lee, Tuljapurkar, 1994) as

F M t x F M t F M x F M x F M t x

k b a

/ , / / / / ,

     

, (9) where x = 0–85+, t = 1,2, …, T, parameters axM/F are age-and-sex-specific net migration profiles independent on time, bxM/F are supplementary age-and-sex-specific components determining how much the level of net migration in each age group changes when the indicator ktM/F changes, and finally ktM/F are time variant parameters – indexes of total net migration. εx,tM/F is residual term with the character of white noise, where E(εx) = 0, D(εx) = σ2, cov(εx ; εx’) = 0 and εx ≈ N

• distribution. Estimation of the parameters bxM/F and ktM/F is based on singular decomposition of

the matrix of age-and-sex specific net migration rates, as presented on the case of mortality by

Simpach, O. (2017). Backwards Retropolation of Unknown Migration and Exploitation of the Lee-Carter Model to Modelling Migration Profile. In: 28th IUSSP International Population Conference, 29th October – 4th November 2017, Cape Town, South Africa, pp. 1–9.

7 Bell, Monsell (1991), Lee, Carter (1992) or Hyndman, Ullah (2010). Age-and-sex specific net migration rates δt,xM/F in exact age x and time t create 86 × T dimensional matrix E BK A Δ   

T

. (10) Identification of Lee-Carter model is ensured by following constrains and 1

1 / 85 /

 

 

   T t F M t x F M x

k b . (11) Finally, an arithmetic average of age-and-sex specific net migration rates according to the gender is calculated as T a

T t F M t x F M x







1 / , /

 . (12) For the prediction of the future age specific net migration rates it is necessary to predict only the values of total indexes ktM/F. These calculations are mostly done (similarly as in the case of mortality and fertility) by ARIMA modelling. We estimated axM/F, bxM/F and ktM/F parameters and predicted the indexes ktM/F for both populations (males and females) up to year 2040. (This year was chosen based on the empirical verification of the model’s prediction power – there is not sufficient amount of information for longer extrapolation to the future, prediction will converge to unconditional mean value – to the constant). On the basis of estimates and forecast for each sex there are smoothed and consequently estimated the future values of δx,tM/F as

F M t F M x F M x F M t x

k b a

/ / / / ,

ˆ ˆ ˆ     . (13) The results (with added empirical data) for the Czech population of males are displayed in the Figure 4, where the perspective is rotated by 180° in comparison with Figure 2 and 3.

Figure 4 Empirical values of age specific net migration rates of males and females in the Czech Republic (years 1968–2012) with added predicted values by Lee-Carter model up to year 2040. Source: author’s calculations.

Simpach, O. (2017). Backwards Retropolation of Unknown Migration and Exploitation of the Lee-Carter Model to Modelling Migration Profile. In: 28th IUSSP International Population Conference, 29th October – 4th November 2017, Cape Town, South Africa, pp. 1–9.

8 The final form of the population projection is not corrected by predicted age-specific numbers

• f net immigrants, i.e. not by the components, but by the rates of age-and-sex specific net

migration (i.e. by the net migration profile). Taking into account the formula (3), from which the net migration balance can be calculated as Δx,tM/F = Ix,tM/F – Ex,tM/F , we yield

F M t x F M t x F M t x

S

/ , / , / ,

    . (14) Correction of the calculations by net migration of age-specific projection tables according to the sex is done as follows

 

t x t x t x t x

S S S

, , , ,

    , (15) which results in complex form of potential future age-and-sex structure of the population in horizon of h years. (In the case of this study it is for the Czech Republic up to 2040 in one-year age intervals). This correction can be added to the algorithm of matrix operator of cohort- component method elaborated by Keyfitz (1964) and Bogue, Anderton, Arriaga (1993).

4 Conclusions

The case study, which was presented in this contribution, can help the transit countries to calculate certain age-and-sex-specific assumption of net migration profile. With high probability this approach cannot be used on states which are the final destinations for the immigrants, because the resulting flow of migrants is currently unknown. Therefore, it will be necessary to rely rather on behavioural models and expert judgments published by relevant ministries of the countries concerned, the European institutions or other international

• rganizations. A certain option is also the combination of stochastic approach that was

introduced in this paper with expert guess and migration estimates, possibly by incorporating upper and lower limits. The expert guesses can be done for example based on the knowledge

• f migration quotas.

Acknowledgements The paper was supported from the Internal Grant Agency of University of Economics Prague

• no. 35/2017 “Demographic models in R Software”.

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