Estimation of migration from census data
Description of the methods
Estimating migration from census data is not technically complicated. Provided that the census(es) gather the appropriate information and are reasonably accurate it is possible to produce estimates of net immigration (i.e. immigration less emigration) of the foreignborn population (people born outside a particular country) and internal migration between (to and from) subnational regions of a country, over the period between two censuses.
To estimate net immigration of foreigners one essentially subtracts from the number of foreignborn people enumerated in a census, the number of foreigners expected to have survived since being enumerated in the previous census.
In a similar way, if the censuses record the subnational region of birth one can estimate net inmigration (i.e. net inmigration of those born outside the region less net outmigration of those born in the region) between subnational regions of a country. However, if the census asks of people where they were living at some prior point in time, say at the time of the previous census, one is able to estimate directly the number of surviving migrants (i.e. migrants still alive at the time of the latest census) into and out of each subnational region of the country since that prior point in time.
In order to estimate the number of migrants from the number of surviving migrants at the time of the second census one needs to add to these figures an estimate of the number of migrants who are expected to have died between moving and the time of the latest census.
If the latest census records other information such as year in which the migrant moved to the place at which the person was counted in the census, it is possible also to establish a trend of migration over time.
Migration is different from fertility and mortality both in that migrating is not final in the sense of a birth or death, but also that we are concerned not only with the population of origin, from which the migrant moved (which corresponds to a population exposed to the risk from which rates of migration akin to those of fertility and mortality can be calculated) but we also have a population to which the migrant moves, the destination population. Apart from this, in order to understand migration one is often interested in distinguishing between different types of migration (whether temporary or more permanent, whether circulatory or unidirectional, etc.). For these reasons there is a much wider range of measures and terminology associated with migration than there is with either fertility or mortality. It is not the purpose of this chapter to cover these issues and the interested reader is referred to the standard texts on the subject such as the UN Manual VI (UN Population Division 1970), Shryock and Siegel (1976), Siegel and Swanson (2004).
Data requirements and assumptions
Tabulations of data required
 To estimate net immigration of foreigners:
 the number of foreignborn females (males), in fiveyear age groups, and for an open age interval A+, at two points in time, typically two censuses
 For the deaths: either a suitable model life table or the numbers of nativeborn females (males), in fiveyear age groups, and for an open age interval A+, at two points in time, typically two censuses. Failing these, the central crude death rate for the population
 To estimate subnational regional net inmigration from place of birth data:
 the number of females (males) by subnational region and by subnational region of birth, in fiveyear age groups, and for an open age interval A+, at two points in time, typically two censuses
 For the deaths: either a suitable model life table, the numbers of nativeborn females (males), in fiveyear age groups, and for an open age interval A+, at two points in time, typically two censuses or numbers of deaths by region from the vital registration. Failing these, the central crude death rate for the population
 To estimate internal migration between subnational regions from place of residence at previous census data:
 The numbers of females (males) by subnational region and by subnational region at some prior date, typically that of the preceding census, in fiveyear age groups, and for an open age interval A+.
 If agespecific numbers are not available, aggregated data is still useful for estimating allage migration.
Important assumptions
 Estimating net immigration of foreigners:
 Censuses identify all foreignborn people accurately
 One is able to estimate the mortality of the foreignborn population accurately (either that the life table used is appropriate, or that the mortality is the same as that implied by the censuses for the nativeborn (locallyborn) national population)
 No return migration of locally born emigrants
 Estimating subnational regional net inmigration from place of birth data:
 Censuses count the population by subnational region accurately and identify the region of birth accurately
 One is able to estimate the mortality of people moving between two regions accurately (either that the life table used is appropriate, or that the mortality is the same as that implied by the censuses for the nativeborn national population).
 Estimating internal migration between subnational regions from data on place of residence at previous census:
 Latest census identifies correctly all people who have moved from one region to another since the prior date (e.g. previous census)
 One is able to estimate the mortality of people moving between two regions accurately (either that the life table used is appropriate, or that the mortality is the same as that implied by the censuses for the nativeborn national population). Since one is estimating in and outmigration separately (as opposed to net migration) this assumption is of less importance.
Preparatory work and preliminary investigations
Before applying this method, you should investigate the quality of the data in at least the following dimensions
 age structure of the population (by subnational region as appropriate); and
 relative completeness of the census counts (by subnational region as appropriate).
Caveats and warnings
Estimating migration using place of birth data from two censuses not only requires that the censuses count the population reasonably completely, but that the place of birth be accurately recorded. Often this is not the case, particularly when estimating immigration, where immigrants wish to hide the fact that they are foreign, but also in the case of internal migration where there may have been boundary changes or the respondent is ignorant about the place of birth of the person.
Estimating migration by asking questions of migrants is quite dependent on the census identifying completely all those who have migrated, as well as identifying the place from which moved correctly. To the extent that recent migrants are not yet established as residents of the region to which they have moved at the time of the census, they could be missed in the count.
Net migration, by definition, underestimates the flows of migrants into and out of a region or country. Thus, for example, people who moved into a region and then returned within the period being considered will result in zero net inmigration and yet moved twice.
Application of the method
A: Estimating net immigration of foreigners using place of birth data
This method produces estimates of the net immigration of foreigners using place of birth data. It is important to stress that this method does not take into account or measure the immigration of returning nativeborn people who left the country prior to the previous census and returned before the second census. Thus this method is not recommended for the measurement of immigration where significant return migration of nativeborn people (for example, after exile or forced migration of refugees) is in progress.
Step 1: Decide on survival factors
If data on the number of foreignborn people in the population are available by age group for each census then one needs to estimate the survival factors to be applied to the numbers of foreignborn in the first census to estimate the numbers surviving to the time of the second census. The user can choose between years of life lived in fiveyearly age groups (_{5}L_{x}) based on the standard from the General family of United Nations model life tables or one of any of the four families of Princeton model life tables or a model life table of a population experiencing an AIDS epidemic (Timæus 2007) which appear in the Models spreadsheet of the associated workbook. This spreadsheet also allows the user to input years of life lived in fiveyearly age groups of an alternative life table if there is reason to assume that the life table has a similar pattern of mortality to that of the population in question, or failing this, the survival factors can be derived from the proportion of each fiveyear age group of the nativeborn population surviving from the first to the second census (assumed to be n years apart, where n is a multiple of 5). Thus
$${\text{\hspace{0.17em}}}_{5}{S}_{x,n}{\text{,}}_{\infty}{S}_{An,n}\text{}$$
and
$$\text{}{S}_{B,n}\text{}$$
, the nyear survival factor for a group of people aged x to x + 5 at the previous census, An and older at the previous census, and born between censuses, respectively are estimated as follows:
$$\begin{array}{l}{\text{}}_{5}{S}_{x,n}=\frac{{}_{5}{L}_{x+n}}{{}_{5}{L}_{x}}\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{{\text{}}_{5}{N}_{x+n}^{nb}(t+n)}{{\text{}}_{5}{N}_{x}^{nb}(t)}\text{\hspace{0.17em}},\\ {\text{}}_{\infty}{S}_{An,n}=\frac{{T}_{A}}{{T}_{An}}\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{{\text{}}_{\infty}{N}_{A}^{nb}(t+n)}{{\text{}}_{\infty}{N}_{An}^{nb}(t)}\text{\hspace{0.17em}},\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\\ \text{}{S}_{B,n}=\frac{{}_{n}{L}_{0}}{n{l}_{0}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{{\text{}}_{n}{N}_{0}^{nb}(t+n)}{\text{}{B}^{nb}}\text{\hspace{0.17em}}.\end{array}$$
where the superscript nb represents ‘nativeborn’,
$${\text{}}_{5}{N}_{x}^{nb}(t)\text{\hspace{0.17em}}$$
represents the nativeborn population in the census at time t and B^{nb} represents the number of nativeborn births between time t and t + n.
If the data are not available in fiveyear age groups, the net number of immigrants can still be estimated in total provided we have an estimate of the crude death rate for the population (which might, in the absence of any evidence to the contrary, be assumed to be that of the nativeborn population).
Step 2: Estimate the number of deaths of the immigrants
If data on the number of foreignborn people in the population are available by age group for two censuses (n years apart) then one needs to estimate the number of deaths of foreignborn people (denoted by the superscript F) aged between x and x+5 at the first census (at time t),
$${\text{}}_{5}{D}_{x}^{F}\text{}$$
, aged An and older at the first census,
$${\text{}}_{\infty}{D}_{An}^{F}\text{}$$
, and those born between the censuses,
$$\text{}{D}_{B}^{F}\text{}$$
, as follows:
$$\begin{array}{l}\text{\hspace{0.17em}}{\text{}}_{5}{D}_{x}^{F}=\frac{1}{2}\left({\text{}}_{5}{N}_{x}^{F}(t)\cdot {\text{}}_{5}{S}_{x,n}+{\text{}}_{5}{N}_{x+n}^{F}(t+n)\right)\left(\frac{1}{{\text{}}_{5}{S}_{x,n}}1\right)\text{\hspace{0.17em}},\\ \text{\hspace{0.17em}}{\text{}}_{\infty}{D}_{An}^{F}=\frac{1}{2}\left({\text{}}_{\infty}{N}_{An}^{F}(t)\cdot {\text{}}_{\infty}{S}_{An,n}+{\text{}}_{\infty}{N}_{A}^{F}(t+n)\right)\left(\frac{1}{{\text{}}_{\infty}{S}_{An,n}}1\right)\text{\hspace{0.17em}},\\ \text{and}\text{\hspace{0.17em}}\text{}{D}_{B}^{F}=\frac{1}{2}\left({\text{}}_{n}{N}_{0}^{F}(t+n)\right)\left(\frac{1}{\text{}{S}_{B,n}}1\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{}.\end{array}$$
where
$${\text{\hspace{0.17em}}}_{5}{N}_{x}^{F}(t)\text{\hspace{0.17em}}$$
represents the number of foreignborn people according to the census at time t who were aged between x and x+5.
If data and/or survival factors are not available by age group then one can estimate the total number of deaths of the foreignborn people as follows:
$$\text{\hspace{0.17em}}{\text{}}_{\infty}{D}_{0}^{F}=\frac{n}{2}\left({\text{}}_{\infty}{N}_{0}^{F}(t)+{\text{}}_{\infty}{N}_{0}^{F}(t+n)\right){\text{\hspace{0.17em}}}_{\infty}{m}_{0}\text{\hspace{0.17em}}$$
where _{∞}m_{0} is an estimate of the crude mortality rate of the population in the country of the census.
However, if the age distribution of the foreignborn population is markedly different from that of the population in the country of the census, then this can produce a poor approximation to the true number of deaths.
Step 3: Estimate the net number of immigrants (of foreigners)
If data are available by age group for each census then agespecific net immigration can be estimated as follows:
$$\text{\hspace{0.17em}}\text{Net}{\text{}}_{5}{M}_{x}^{F}={\text{}}_{5}{N}_{x+n}^{F}(t+n){\text{}}_{\infty}{N}_{x}^{F}(t)+{\text{}}_{5}{D}_{x}^{F}\text{\hspace{0.17em}}$$
for x = 0, 5, … , A5n where
$$\text{\hspace{0.17em}}\text{Net}{\text{}}_{5}{M}_{x}^{F}\text{\hspace{0.17em}}$$
represents the net number of immigrants between times t and t+n who were aged between x and x + 5 at time t. For x > A  5  n
$$\text{\hspace{0.17em}}\text{Net}{\text{}}_{\infty}{M}_{An}^{F}={\text{}}_{\infty}{N}_{A}^{F}(t+n){\text{}}_{\infty}{N}_{An}^{F}(t)+{\text{}}_{\infty}{D}_{An}^{F}\text{\hspace{0.17em}}.$$
The net number of immigrants of those born between times t and t+n is estimated as follows:
$$\text{\hspace{0.17em}}\text{Net}\text{}{M}_{B}^{F}={\text{}}_{n}{N}_{0}^{F}(t+n)+\text{}{D}_{B}^{F}\text{\hspace{0.17em}}.$$
If data and/or survival factors are not available by age group then one would estimate of the total net number of immigrants as follows:
$$\text{\hspace{0.17em}}\text{Net}{\text{}}_{\infty}{M}_{0}^{F}={\text{}}_{\infty}{N}_{0}^{F}(t+n){\text{}}_{\infty}{N}_{0}^{F}(t)+{\text{}}_{\infty}{D}_{0}^{F}\text{\hspace{0.17em}}.$$
B: Estimating net internal migration between subnational regions from place of birth data
Net inmigration into a particular subnational region from other regions in the country can be estimated in exactly the same way as the international immigration, described above, by replacing the foreignborn population with the population born outside the region.
In addition, applying the same method to data on the change in the numbers of population born in (rather than outside) and living outside the region of interest allows us to estimate the net outmigration of those born in the region to other regions in the country. Subtracting this from the net inmigration of those born outside the region gives an estimate of the overall net inmigration into the region of interest.
If there is reason to suspect that there is a material difference in the mortality experienced by those born outside who moved into the region and those born in the region who moved out, and one has appropriate survival factors then one could apply different survival factors to each when estimating the net number of migrants. However, in practice it is likely that inaccuracies in the census data on place of residence at previous census are likely to outweigh any increase in accuracy achieved by using differential mortality.
C: Estimating internal migration between subnational regions from place of residence at previous survey
Net subnational interregional migration is estimated directly from the numbers of people in each region at the time of the census who moved since the previous census by place (e.g. region) they were in at a given prior date (e.g. at the time of the previous census). Confining the estimates to interregional flows the sum of the numbers of interregional inmigrants should be equal to the sum of interregional outmigrants; however, if the data include immigration to the subnational regions from outside the country one can extend the estimates of inmigration to include international immigration into each region.
Since one of the major areas of interest is the magnitude of interregional flows of the population, one is as interested in the total numbers of migrants between regions as one is in the age distributions of particular flows.
The number of migrants is derived from the number of surviving in and outmigrants as follows:
$${\text{\hspace{0.17em}}}_{5}{M}_{x}=\left({\text{}}_{5}{{I}^{\prime}}_{x}{\text{}}_{5}{{O}^{\prime}}_{x}+{\left({\text{}}_{5}{{I}^{\prime}}_{x}{\text{}}_{5}{{O}^{\prime}}_{x}\right)}_{x}/{\text{}}_{5}{S}_{x}\right)/2\text{\hspace{0.17em}},$$
where the superscript (’) represents numbers surviving and _{5}I’_{x} and _{5}O’_{x} respectively represent the number of surviving inmigrants into, and the surviving number outmigrants from, a particular region at the time of the second census who were aged between x and x+5 at the second census.
Worked example
This example uses data on the numbers of males in the population from the South African Census in 2001 and a ‘census replacement survey’, the Community Survey in 2007. (Although the survey was conducted approximately 5.35 years after the night of the census in 2001, it is assumed for the purposes of presentation here to have been exactly five years after the census in 2001.) The examples appear in the Migration_South Africa_males.xlsx workbook.
A: Estimating net immigration of foreigners using place of birth
Step 1: Decide on survival factors
The survival factors are shown in the fifth column of Table 1. The values are derived from (the years of life lived in each age group of) the alternative life table entered in the Models spreadsheet, for those aged 20 to 24 last birthday and those aged 80 and over at the time of the first census, and those born between the two censuses, as follows:
$$\begin{array}{l}{\text{}}_{5}{S}_{20,5}=\frac{{}_{5}{L}_{25}}{{}_{5}{L}_{20}}=\frac{\text{4}\text{.3382}}{\text{4}\text{.4975}}=0.96458\text{\hspace{0.17em}}\\ {\text{}}_{\infty}{S}_{80,5}=\frac{{T}_{85}}{{T}_{80}}=\frac{\text{0}\text{.75180}}{\text{1}\text{.19603}}=0.40912\text{\hspace{0.17em}}\\ \text{and}\text{\hspace{0.17em}}\text{}{S}_{B,5}=\frac{{}_{5}{L}_{0}}{5{l}_{0}}=\frac{\text{4}\text{.707549}}{\text{5}}=\mathrm{0.94151.}\text{\hspace{0.17em}}\end{array}$$
Table 1 Estimation of deaths of foreignborn and the net number of immigrants by age group, South Africa, 20012006
Age 
2001 
2006 
x 
_{5}S_{x} 
Age at 2^{nd} census 
D^{F} 
Net M 



B 
0.94151 



0 4 
8,963 
12,577 
0 
0.97896 
0 4 
391 
12,968 
5 9 
10,390 
13,724 
5 
0.99547 
5 9 
242 
5,003 
1014 
13,508 
13,998 
10 
0.99427 
1014 
55 
3,664 
1519 
27,835 
27,943 
15 
0.98602 
1519 
119 
14,555 
2024 
69,787 
59,493 
20 
0.96458 
2024 
616 
32,275 
2529 
87,381 
95,763 
25 
0.93161 
2529 
2,994 
28,970 
3034 
73,338 
100,450 
30 
0.90960 
3034 
6,675 
19,743 
3539 
66,663 
85,490 
35 
0.89780 
3539 
7,563 
19,715 
4044 
59,152 
75,684 
40 
0.89092 
4044 
7,701 
16,721 
4549 
45,184 
66,113 
45 
0.88633 
4549 
7,274 
14,234 
5054 
40,398 
55,913 
50 
0.87224 
5054 
6,154 
16,883 
5559 
30,640 
42,833 
55 
0.84731 
5559 
5,717 
8,153 
6064 
24,376 
34,433 
60 
0.80885 
6064 
5,442 
9,234 
6569 
17,895 
25,588 
65 
0.75468 
6569 
5,353 
6,564 
7074 
13,561 
18,989 
70 
0.66991 
7074 
5,281 
6,375 
7579 
10,238 
12,850 
75 
0.56388 
7579 
5,404 
4,693 
8084 
7,658 
7,461 
80+ 
0.40912 
8084 
5,118 
2,341 
85+ 
4,455 
5,305 


85+ 
7,410 
602 
Total 
611,423 
754,608 


Total 
79,509 
222,693 
Step 2: Estimate the number of deaths
Since we have data on the number of foreignborn people in the population by age group for each census we can estimate the number of deaths of foreignborn people which occurred in the period between the two censuses by age group using the numbers of foreigners in each census given in the second and third columns of Table 1. For those aged 20 to 24 last birthday and those aged 80 and over at the time of the first census, and those born between the two censuses, the calculations are as follows:
$$\begin{array}{l}\text{\hspace{0.17em}}{\text{}}_{5}{D}_{20}^{F}=\frac{1}{2}\left({\text{}}_{5}{N}_{20}^{F}(2001)\cdot {\text{}}_{5}{S}_{20,5}+{\text{}}_{5}{N}_{25}^{F}(2006)\right)\left(\frac{1}{{\text{}}_{5}{S}_{20,5}}1\right)\\ \text{}=\left(69787\cdot 0.96458+95763\right)\left(\frac{1}{0.96458}1\right)=2994\text{\hspace{0.17em}}\\ \text{\hspace{0.17em}}{\text{}}_{\infty}{D}_{80}^{F}=\frac{1}{2}\left({\text{}}_{\infty}{N}_{80}^{F}(2001)\cdot {\text{}}_{\infty}{S}_{80,5}+{\text{}}_{\infty}{N}_{85}^{F}(2006)\right)\left(\frac{1}{{\text{}}_{\infty}{S}_{80,5}}1\right)\\ \text{}=\left(\left(7658+4455\right)0.40912+5305\right)\left(\frac{1}{0.40912}1\right)=7410\text{\hspace{0.17em}}\\ \text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{}{D}_{B}^{F}=\frac{1}{2}\left({\text{}}_{5}{N}_{0}^{F}(2006)\right)\left(\frac{1}{\text{}{S}_{B,5}}1\right)=12577\left(\frac{1}{0.94151}1\right)=391\text{\hspace{0.17em}}.\end{array}$$
If data and/or survival factors were not available by age group then one could estimate the total number of deaths of the foreign born people as follows, given an estimate of the crude mortality rate in the population of 14 per 1,000:
$$\text{\hspace{0.17em}}{\text{}}_{\infty}{D}_{0}^{F}=\frac{5}{2}\left({\text{}}_{\infty}{N}_{0}^{F}(2001)+{\text{}}_{\infty}{N}_{0}^{F}(2006)\right){\text{\hspace{0.17em}}}_{\infty}{m}_{0}=\frac{5}{2}\left(611423+754608\right)\frac{14}{1000}=47811\text{\hspace{0.17em}}.$$
Step 3: Estimate the net number of immigrants (of foreigners)
Since data are available by age group for each census, agespecific net immigration of those born outside the country can be estimated as follows:If data and/or survival factors were not available by age group then one could estimate the total net number of immigrants as follows:
$$\begin{array}{l}\text{\hspace{0.17em}}\text{Net}{\text{}}_{5}{M}_{20}^{F}={\text{}}_{5}{N}_{25}^{F}(2006){\text{}}_{\infty}{N}_{20}^{F}(2001)+{\text{}}_{5}{D}_{20}^{F}=9576369787+2994=28970\text{\hspace{0.17em}}\\ \text{\hspace{0.17em}}\text{Net}{\text{}}_{\infty}{M}_{80}^{F}={\text{}}_{\infty}{N}_{85}^{F}(2006){\text{}}_{\infty}{N}_{80}^{F}(2001)+{\text{}}_{\infty}{D}_{80}^{F}=5305\left(7658+4455\right)+7410=602\text{\hspace{0.17em}}\\ \text{\hspace{0.17em}}\text{Net}\text{}{M}_{B}^{F}={\text{}}_{5}{N}_{0}^{F}(2006)+\text{}{D}_{B}^{F}=12577+391=12968\text{\hspace{0.17em}}.\end{array}$$
If data and/or survival factors were not available by age group then one could estimate the total net number of immigrants as follows:
$$\text{\hspace{0.17em}}\text{Net}{\text{}}_{\infty}{M}_{0}^{F}={\text{}}_{\infty}{N}_{0}^{F}(2006){\text{}}_{\infty}{N}_{0}^{F}(2001)+{\text{}}_{\infty}{D}_{0}^{F}=754608611423+47811=190996\text{\hspace{0.17em}}$$
B: Estimating subnational regional net inmigration using place of birth
The second and third column of Table 2 show the numbers of people living in the Western Cape province of South Africa who were born outside the province, as counted by the 2001 Census and the 2007 Community Survey, respectively. Although the same survival factors (column 5) have been used as were used in the example of Method A, this should not be the case if it was thought that the mortality experience of nativeborn and immigrants were very different. The final column of Table 2 gives the net numbers of migrants into the Western Cape who were born in provinces other than the Western Cape for the different age groups. Thus in total 213,911 people born outside the Western Cape moved to the Western Cape (after excluding those who moved out).
Table 2 Estimation of the net number of inmigrants of those born outside by age group, Western Cape, South Africa, 20012006
Age 
2001 
2006 
x 
_{5}S_{x} 
Age at 2^{nd} census 
D_{O} 
Net M (born out) 



B 
0.94151 



0 4 
16,443 
19,012 
0 
0.97896 
0 4 
591 
19,602 
5 9 
24,406 
28,743 
5 
0.99547 
5 9 
482 
12,782 
1014 
31,134 
30,792 
10 
0.99427 
1014 
125 
6,511 
1519 
44,478 
53,933 
15 
0.98602 
1519 
245 
23,043 
2024 
74,011 
82,526 
20 
0.96458 
2024 
896 
38,944 
2529 
80,187 
89,522 
25 
0.93161 
2529 
2,954 
18,466 
3034 
65,833 
90,783 
30 
0.90960 
3034 
6,074 
16,670 
3539 
56,393 
76,475 
35 
0.89780 
3539 
6,776 
17,417 
4044 
44,420 
59,692 
40 
0.89092 
4044 
6,268 
9,567 
4549 
32,862 
47,612 
45 
0.88633 
4549 
5,338 
8,529 
5054 
28,178 
37,969 
50 
0.87224 
5054 
4,303 
9,409 
5559 
19,983 
30,205 
55 
0.84731 
5559 
4,012 
6,039 
6064 
17,569 
25,593 
60 
0.80885 
6064 
3,832 
9,442 
6569 
11,216 
20,802 
65 
0.75468 
6569 
4,137 
7,371 
7074 
8,365 
12,612 
70 
0.66991 
7074 
3,426 
4,822 
7579 
5,919 
8,434 
75 
0.56388 
7579 
3,458 
3,528 
8084 
4,063 
5,061 
80+ 
0.40912 
8084 
3,248 
2,390 
85+ 
2,152 
2,183 


85+ 
3,413 
620 
Total 
567,613 
721,949 


Total 
59,576 
213,911 
The second and third columns of Table 3 present the numbers of people living in provinces other than the Western Cape who were born in the Western Cape, as counted by the 2001 census and the 2007 Community Survey, respectively. The net number of outmigrants of those born in the Western Cape (i.e. the number of people born in the Western Cape who moved out, less those who have returned) is given in column 8. The negative numbers mean that there was negative net outmigration (i.e. the number of those born in the Western Cape who moved to other provinces in the period was less than the number born in the Western Cape who were living outside who returned during the period). Thus the total of 19,017 means that the number of people born in the Western Cape, who returned to the Western Cape during the period having lived in another province until 2001 exceed those who were born in the Western Cape and moved to another province in the period by 19,017.
These estimates were derived using the same survival factors as were used for those born outside the Western Cape who moved into the province, but if there was reason to suppose that the mortality differed for those born in the Western Cape who moved out, then a different set of survival factors would be used to estimate the Net M (born in) numbers.
The overall net inmigration for the province is thus given in the final column of Table 3. Thus in total 232,928 more people moved into the Western Cape than left the Western Cape to live in another province.
In this example those born outside the province include those born outside the country and thus the overall net migration includes immigrants who settle in the province. Excluding the foreignborn from Table 2 would produce numbers of internal inmigrants net of internal outmigrants, and the sum of these numbers for all the provinces together would be zero.
Table 3 Estimation of the net number of outmigrants of those born inside by age group, Western Cape, South Africa, 20012006
Age 
2001 
2006 
x 
_{5}S_{x} 
Age at 2^{nd} census 
D_{I} 
Net M (born in) 
Overall Net M 



B 
0.94151 




0 4 
22,055 
11,747 
0 
0.97896 
0 4 
365 
12,112 
7,490 
5 9 
21,895 
12,509 
5 
0.99547 
5 9 
367 
9,180 
21,962 
1014 
21,382 
11,593 
10 
0.99427 
1014 
76 
10,226 
16,737 
1519 
18,265 
13,455 
15 
0.98602 
1519 
100 
7,827 
30,870 
2024 
14,645 
10,477 
20 
0.96458 
2024 
202 
7,587 
46,531 
2529 
13,501 
9,534 
25 
0.93161 
2529 
434 
4,676 
23,142 
3034 
13,118 
11,047 
30 
0.90960 
3034 
867 
1,587 
18,257 
3539 
12,121 
14,614 
35 
0.89780 
3539 
1,319 
2,815 
14,602 
4044 
11,725 
12,195 
40 
0.89092 
4044 
1,311 
1,384 
8,183 
4549 
10,335 
10,538 
45 
0.88633 
4549 
1,285 
98 
8,431 
5054 
9,211 
9,881 
50 
0.87224 
5054 
1,221 
768 
8,642 
5559 
7,264 
10,568 
55 
0.84731 
5559 
1,362 
2,720 
3,319 
6064 
6,691 
7,723 
60 
0.80885 
6064 
1,250 
1,710 
7,732 
6569 
4,643 
5,297 
65 
0.75468 
6569 
1,265 
128 
7,499 
7074 
3,954 
3,766 
70 
0.66991 
7074 
1,182 
304 
4,517 
7579 
2,331 
2,384 
75 
0.56388 
7579 
1,240 
330 
3,858 
8084 
1,402 
2,140 
80+ 
0.40912 
8084 
1,336 
1,145 
1,244 
85+ 
707 
555 


85+ 
1,024 
531 
89 
Total 
195,246 
160,023 


Total 
16,206 
19,017 
232,928 
C: Estimating internal migration between subnational regions from data on place of residence at previous census
Table 4 presents the results of the answers to the question about place (province in this example) of residence at the time of the 2001 Census given by those counted in each of the provinces in the 2007 Community Survey. (In actual fact the question asked whether the person was staying at the same place at the time of the prior census and if not, where they were staying at the time they moved to the place at which they were counted in the Community Survey. However, work by Dorrington and Moultrie (2009) shows that using these data and the year of movement to back project the population in order to estimate the numbers by province of residence at the time of the previous survey suggests that the assumption that there was only one move in the five years since the previous census was reasonably accurate.)
By far the largest numbers of migrants are those that moved within each of the provinces, however, these have been excluded from Table 4 because one is usually more interested in interprovincial migration than migration within a province.
Table 4 Interprovincial migration, South Africa, 20012006

Province where counted (destination) 


Previous residence (origin) 
WC 
EC 
NC 
FS 
KZ 
NW 
GT 
MP 
LM 
Total 
WC 

12,173 
4,060 
1,745 
3,221 
2,113 
16,400 
1,405 
874 
41,992 
EC 
52,239 

1,120 
7,187 
25,209 
14,430 
28,633 
4,693 
2,116 
135,626 
NC 
4,813 
1,942 

3,480 
908 
3,728 
4,956 
1,062 
357 
21,246 
FS 
2,943 
3,145 
2,546 

2,352 
12,733 
19,920 
4,293 
1,963 
49,896 
KZ 
6,762 
7,015 
631 
2,358 

3,573 
50,980 
8,886 
1,194 
81,399 
NW 
1,478 
907 
9,811 
5,555 
2,329 

47,633 
3,090 
4,337 
75,140 
GT 
24,891 
12,948 
3,962 
11,437 
18,145 
32,433 

18,598 
15,133 
137,547 
MP 
2,134 
1,317 
280 
1,724 
4,546 
5,767 
42,941 

8,628 
67,338 
LM 
2,754 
1,583 
255 
1,709 
2,209 
9,773 
81,394 
24,211 

123,889 
OSA 
21,221 
5,467 
1,209 
9,584 
10,933 
11,437 
51,873 
8,335 
9,286 
129,346 
DNK 
500 
3 
15 
124 
132 
78 
228 
89 
0 
1,170 
UNS 
1,058 
1,029 
107 
208 
875 
508 
3,558 
408 
633 
8,384 
Total 
120,794 
47,528 
23,996 
45,111 
70,860 
96,573 
348,516 
75,070 
44,524 
872,973 
WC = Western Cape, EC = Eastern Cape, NC = Northern Cape, FS = Free State, KZN = KwaZuluNatal, NW = North West, GT = Gauteng, MP = Mpumalanga, LM = Limpopo, OSA = Outside SA, DNT = Do not know, UNS = Unspecified 
In addition to the allage numbers in Table 4 (in actual fact these numbers exclude, as is often the case, migration of those born between the census and survey) one can also produce numbers of in and outmigration by age groups as shown in Table 5. For completeness these numbers include estimates of the number of migrants who were born since the previous census. However, relative to the other migrants these numbers look implausibly high, and the reason for this is discussed below.
The net number of migrants is estimated for those aged 2529 at the time of the Community Survey (i.e. were aged 2024 at the time of the 2001 census), for example, as follows:
$${\text{\hspace{0.17em}}}_{5}{M}_{x}=\left(\text{}20675\text{}5649+\left(\text{}20675\text{}5649\right)/\text{}0.96458\right)/2=15301\text{\hspace{0.17em}}.$$
Table 5 Estimation of the net number of inmigrants by age group, Western Cape, South Africa, 20012006
Age 
Surviving in migrants (I’) 
Surviving out migrants (O’) 
x 
_{5}S_{x} 
Net inmigrants 






0 4 
20,846 
11,747 
B 
0.94151 
9,381 
5 9 
6586 
3,554 
0 
0.97896 
3,065 
1014 
6685 
2,882 
5 
0.99547 
3,812 
1519 
10402 
3,967 
10 
0.99427 
6,454 
2024 
21266 
4,488 
15 
0.98602 
16,897 
2529 
20675 
5,649 
20 
0.96458 
15,301 
3034 
15584 
6,008 
25 
0.93161 
9,928 
3539 
10584 
5,098 
30 
0.90960 
5,758 
4044 
7264 
3,045 
35 
0.89780 
4,458 
4549 
4648 
2,714 
40 
0.89092 
2,053 
5054 
3095 
1,500 
45 
0.88633 
1,698 
5559 
3940 
935 
50 
0.87224 
3,225 
6064 
3776 
527 
55 
0.84731 
3,541 
6569 
3127 
818 
60 
0.80885 
2,582 
7074 
1540 
437 
65 
0.75468 
1,282 
7579 
561 
206 
70 
0.66991 
442 
8084 
797 
116 
75 
0.56388 
944 
85+ 
264 
47 
80+ 
0.40912 
374 
Total 
141,640 
53,739 


91,194 
Diagnostics, analysis and interpretation
Checks and validation
Perhaps the simplest check, on the reasonableness of the ‘shape’ (i.e. distribution of the numbers by age) of the estimates but not the level, is to see if it conforms to the standard shape (or a variation thereof). Rogers and Castro (1981a; 1981b) point out that the distribution of the number (or rate) of in and outmigrants tends to conform to standard patterns, with a peak in the young adult ages (usually associated with seeking employment), a second, usually less pronounced peak amongst very young children falling to a trough amongst young teenagers (the size depending on the extent to which it is families rather than individuals moving in the young to middle aged adults). Sometimes there is also a ‘hump’ (or trough) around retirement age if there is a strong flow of migrants moving to (or away from) the place to retire.
These patterns (not necessarily the same pattern) apply to in and outmigration flows separately, but not necessarily to net migration (which is the difference between the two flows) unless one flow (either the inmigration or the outmigration) is much greater than the other.
Figure 1 illustrates this using some of the estimates calculated above, expressed as proportions of the total number in each case (to allow them to be presented on a single figure). From this we can see that in broad terms (with the exception in some cases, where the proportion of migrants at the very young ages looks implausibly high) each conforms to the expected shape.
The net outmigrants of those born in the Western Cape (excluded from the figure for ease of illustration) does not conform to a standard model of migration, which could indicate these numbers are not very reliable, however, they are small relative to the inmigration of those born outside the province, and thus such a deviation may tolerated. In addition to this there are two other features to be noted from Figure 1. The first is that the outmigration from the Western Cape as estimated from data on place of residence at previous census, suggests that adult outmigrants peak at a somewhat older age (and possibly are likely to represent family rather than individual migration). The second is the fact that the net immigration into the country follows the standard shape which indicates that the flow into the country is much stronger than the return flow of those migrants.
[[wysiwyg_imageupload:230:]]
If the census asked place of birth and place of residence at the previous census then one can compare the two estimates of net inmigration into a specific subnational region. If they are similar this gives one some confidence in the results. In the case of the place of birth data for South Africa the net number of inmigrants into the Western Cape is 232,928 (Table 3) while the estimate from the data on place of residence at the time of the previous census data produced an estimate of 92,194 (Table 4), which suggests that one or both of these sets of data are suspect.
The most basic check of the estimates of migration is to project the population (of the country or the province) at the first census to the time of the second census making use of the estimates of the number of migrants and compare that with the census estimates from the second, more recent, census to see how well the two match, especially in the age range in which migration is concentrated. In the case of the net inmigration into the Western Cape, projecting the population forward from 2001 using the estimates derived from the change in the numbers by place of birth produced a much closer fit to the population in the 2029 year age range, suggesting that the data on place of birth are probably more complete than those on the place of residence at the date of the previous census. To some extent this is supported by a comparison of the change in the number of foreignborn in the country between the two censuses, 222,693 (Table 1) with the sum of the numbers who reported that they had moved from outside South Africa to one of the provinces since the previous census, 129,346 (Table 4).
Ideally, if one had independent estimates of the number of migrants one might compare those numbers against estimates using the above methods. Unfortunately, reliable independent estimates are rare. Although most countries try to record people entering and leaving the country, these data are often not reliable, particularly in developing countries with relative porous borders. And unless the country is extremely well regulated and maintains a complete and accurate register of the population, the only other way to measure internal migration is through migrationspecific surveys, which tend to be much more useful for understanding the type of migration (whether permanent, temporary, cyclical, etc.) than for producing reliable estimates of the number of migrants, given the often less structured situation that (particularly recent) migrants find themselves living in and an understandable reluctance to identify themselves as being migrants.
Interpretation
Considering the numbers of migrants estimated from the data on place of residence at the previous census given in Table 4 (and taking into account the suspicion that these probably underestimate the true migration), some 24% of the population changed province of residence in the 5 years between the 2001 Census and the Community Survey. Had we included the number who moved within, but did not change, province then between 7 and 15 per cent of the population moved in the 5‑year period.
The main provinces of destination are Gauteng (by a big margin) and Western Cape, which are predominantly urban and the wealthiest provinces. The main provinces of origin are Gauteng (inspection of the age distribution would show that this is mainly return migration of ‘retiring’ workers) Eastern Cape and Limpopo, which are poor, mainly rural provinces, from which people seeking work migrate to the urban areas.
It appears that migration is predominantly of individuals (seeking work) rather than of families.
Methodspecific issues with interpretation
Scanning errors
A particular feature of the data relying on province of birth is the apparently relatively high number of children born since the first census who have moved to another province. In all likelihood this is an artefact of the data capturing process. Scanning was used to capture the data from the questionnaires on which Western Cape was coded as a “1”, written in the appropriate space by hand. It appears that in a small percentage of cases the scanner might have had trouble distinguishing a handwritten “1” from a handwritten “7” (the code for Gauteng). The result of this is, for example, that some of the children coded as having been born outside the province in which they were counted, and thus appear to be migrants, but probably were not. Even though the percentage error in scanning is very small, the number of births can be large relative to the number migrants, and thus the error can produce noticeable errors. Since an increasing number of developing countries are using scanning to capture data, this sort of problem may be quite common.
Where scanning errors or other situations make it impossible to produce reliable estimates of the number of migrants of those born since the previous census one can use CWR from second census as follows:
$$\text{\hspace{0.17em}}{\text{Net}}_{5}{M}_{0}=\frac{1}{4}CW{R}_{0}\cdot \text{Net}{\text{}}_{30}{M}_{15}^{f}\text{\hspace{0.17em}}$$
for those born in the most recent five years, and
$$\text{\hspace{0.17em}}{\text{Net}}_{5}{M}_{5}=\frac{3}{4}CW{R}_{5}\cdot \text{Net}{\text{}}_{30}{M}_{20}^{f}\text{\hspace{0.17em}}$$
for those born in the five years before that if the censuses are 10 years apart, where CWR_{x} represents ratio of the number of children aged between x and x+5 to the number of women in the population aged between 15+x and 45+x in the population (regional or national) at the time of the second census, and
$${\text{}}_{30}{M}_{x}^{f}\text{}$$
represents the number of women migrants aged between x and x+30.
Applying this to the data for the Western Cape suggest that the number of migrants born since the previous census should be less than half the numbers being estimated from the data on place of birth.
Detailed description of method
Mathematical exposition
The indirect estimation of migration derives from the balance equation for two censuses n years apart, namely:
$$\begin{array}{l}{\text{\hspace{0.17em}}}_{5}{N}_{x+n}(t+n)\text{}={\text{}}_{5}{N}_{x}(t){\text{}}_{5}{D}_{x}+{\text{}}_{5}{{I}^{\prime}}_{x}{\text{}}_{5}{{O}^{\prime}}_{x}\\ \text{\hspace{0.17em}}\text{}\text{}={\text{}}_{5}{N}_{x}^{}(t){\text{}}_{5}{D}_{x}^{}+{\text{}}_{5}{{M}^{\prime}}_{x}\text{\hspace{0.17em}}\end{array}$$
where
$${\text{\hspace{0.17em}}}_{5}{{M}^{\prime}}_{x}=\text{}{\text{}}_{5}{{I}^{\prime}}_{x}{\text{}}_{5}{{O}^{\prime}}_{x}\text{\hspace{0.17em}}$$
is the net (i.e. in less out) number of inmigrants, aged x to x+5 at the time of the first census, surviving to the second census, and _{5}D_{x},_{ 5}I’_{x} and_{ 5}O’_{x}, represent the number of deaths, surviving inmigrants and outmigrants, aged x to x+5 at the time of the first census, who died or moved in the period between the censuses.
For those born after the first census the equation becomes:
$${\text{\hspace{0.17em}}}_{n}{N}_{0}^{}(t+n)=\text{}B\text{}{D}_{B}^{}+\text{}{{M}^{\prime}}_{B}$$
and those in the open age interval:
$${\text{\hspace{0.17em}}}_{\infty}{N}_{A}^{}(t+n)={\text{}}_{\infty}{N}_{An}^{}(t){\text{}}_{\infty}{D}_{An}^{}+{\text{}}_{\infty}{{M}^{\prime}}_{An}\text{\hspace{0.17em}}$$
where B represents the number of births in the population between the two censuses, D_{B} the number of deaths of those births in the period between the censuses and M’_{B} the net number of surviving migrants, born outside the country in the period between the two censuses, _{∞}D_{An} the number of deaths in the intercensal period aged An and older at the time of the first census, and _{∞}M’_{An} the net number of migrants aged An and older at the time of the first census.
Thus
$$\begin{array}{l}{\text{\hspace{0.17em}}}_{5}{{M}^{\prime}}_{x}={\text{}}_{5}{N}_{x+n}^{}(t+n){\text{}}_{5}{N}_{x}^{}(t)+{\text{}}_{5}{D}_{x}^{}\text{}\text{\hspace{0.17em}}\\ \text{\hspace{0.17em}}{{M}^{\prime}}_{B}={\text{}}_{n}{N}_{0}^{}(t+n)\text{}B+\text{}{D}_{B}^{}\text{\hspace{0.17em}}\\ {\text{\hspace{0.17em}}}_{\infty}{{M}^{\prime}}_{An}={\text{}}_{\infty}{N}_{A}^{}(t+n){\text{}}_{\infty}{N}_{An}^{}(t)+{\text{}}_{\infty}{D}_{An}^{}\text{}\text{\hspace{0.17em}}\end{array}$$
or alternatively
$$\begin{array}{l}{\text{\hspace{0.17em}}}_{5}{{M}^{\prime}}_{x}={\text{}}_{5}{N}_{x+n}^{}(t+n){\text{}}_{5}{N}_{x}^{}(t){\text{}}_{5}{S}_{x}\text{}\text{\hspace{0.17em}}\\ \text{\hspace{0.17em}}{{M}^{\prime}}_{B}={\text{}}_{n}{N}_{0}^{}(t+n)\text{}B{S}_{B}\text{\hspace{0.17em}}\\ {\text{\hspace{0.17em}}}_{\infty}{{M}^{\prime}}_{An}={\text{}}_{\infty}{N}_{A}^{}(t+n){\text{}}_{\infty}{N}_{An}^{}(t){\text{}}_{\infty}{S}_{An}\text{}\text{\hspace{0.17em}}\end{array}$$
where _{5}S_{x} , S_{B} and _{∞}S_{An} represent the proportion of the populations aged x to x+5 at the time of the first census, born between the censuses, and aged An and older at the time of the first census, respectively, surviving to the second census.
The net number of migrants can thus be estimated from the net number surviving to the second census as follows:
$$\begin{array}{l}{\text{\hspace{0.17em}}}_{5}{M}_{x}=\left({\text{}}_{5}{{M}^{\prime}}_{x}+{\text{}}_{5}{{M}^{\prime}}_{x}/{\text{}}_{5}{S}_{x}\right)/2={\text{\hspace{0.17em}}}_{5}{{M}^{\prime}}_{x}\frac{\left({\text{}}_{5}{S}_{x}+1\right)}{2{\text{}}_{5}{S}_{x}}\text{\hspace{0.17em}}\\ \text{\hspace{0.17em}}{M}_{B}=\text{\hspace{0.17em}}{{M}^{\prime}}_{B}\frac{\left(\text{}{S}_{B}+1\right)}{2\text{}{S}_{B}}\text{\hspace{0.17em}}\\ {\text{\hspace{0.17em}}}_{\infty}{M}_{An}={\text{\hspace{0.17em}}}_{\infty}{{M}^{\prime}}_{An}\frac{\left({\text{}}_{\infty}{S}_{An}+1\right)}{2{\text{}}_{\infty}{S}_{An}}\text{\hspace{0.17em}}.\end{array}$$
Unfortunately, since the net number of migrants is usually small relative to the size of the population, age misstatement or errors in either or both census counts can lead to very poor estimates being produced. Better estimates of the net number of immigrants into a country can be produced by confining one’s attention to the population of foreigners (defined as those born outside the country) and assuming that return migration of emigrants from the country of interest is insignificant. Thus one replaces each of the symbols above by equivalents specific to the foreignborn population in the country. Since it is unlikely that one has an accurate record of the number of the foreignborn deaths these need to be estimated in one of the following ways:
 Option 1 (Life table survival ratios): Applying rates from a suitable model life table, then
$${\text{\hspace{0.17em}}}_{5}{S}_{x}=\frac{{}_{5}{L}_{x+n}}{{}_{5}{L}_{x}},\text{}{S}_{B}=\frac{{}_{n}{L}_{0}}{n\cdot {l}_{0}}{\text{and}}_{\infty}{S}_{An}=\frac{{T}_{A}}{{T}_{An}}\text{\hspace{0.17em}}.$$
 Option 2 (Census survival ratios): Assuming that emigration of the nativeborn population is insignificant and that the proportions surviving are the same as those in the nativeborn population, then
$${\text{\hspace{0.17em}}}_{5}{S}_{x}=\frac{{}_{5}{N}_{x+n}^{nb}(t+n)}{{}_{5}{N}_{x}^{nb}(t)},\text{}{S}_{B}=\frac{{}_{n}{N}_{0}^{nb}}{{B}^{nb}}{\text{and}}_{\infty}{S}_{An}=\frac{{}_{\infty}{N}_{A}^{nb}(t+n)}{{}_{\infty}{N}_{An}^{nb}(t)}\text{\hspace{0.17em}},$$
where the superscript “nb” designates nativeborn.
 Option 3 (Vital registration): Where one has access to numbers of births and deaths from another source such as vital registration (which is only likely to be the case, if at all, with internal migration), one could work with deaths and births corresponding to the migrant population directly instead of survival ratios to estimate the net number of surviving inmigrants. Alternatively the net number of migrants can be derived as above by setting
$${\text{\hspace{0.17em}}}_{5}{S}_{x}=1\frac{{}_{5}{D}_{x}}{{}_{5}{N}_{x}(t)},\text{}{S}_{B}=\frac{{D}_{B}}{B}{\text{and}}_{\infty}{S}_{An}=\frac{{D}_{An}}{{\text{}}_{\infty}{N}_{An}(t)}\text{\hspace{0.17em}}$$
where the births and deaths are from the vital registration.
However, for most developing countries, particularly those in Africa, vital registration systems are too incomplete to be used in this way.
Internal migration
When it comes to internal migration one can estimate net inmigration (i.e. inmigration of those born outside the region less outmigration of those born outside the region who had previously moved into the region) into each subnational region of those born outside the region by making use of place of birth information to identify the change in numbers of those born outside the region, in the same way as described above. However, since one also has the place of residence of those born in the region who have moved out of the region since birth (but not emigrated) one can also estimate the net outmigration of those born in the region (i.e. outmigration of those born in the region less those born in the region who have returned after having previously moved out of the region) by applying the method described above to the population born in the region (as opposed to those born outside the region).
When estimating the survival of those born in the various regions the census survival ratios could have an advantage over the life table survival ratios in that any under or over count of the population by region, may well be matched by a similar distortion in the national population and hence in the survival ratios, thus resulting in a more accurate estimate of the number of migrants than would be produced by using life table survival ratios.
Apart from place of birth a census can ask of those who moved since the previous census (or some other suitable date) where they were at that census (or some other suitable date) which allows one to measure outmigration and hence (gross) inmigration separately for each subnational region.
If the census asks for the year when the migrant moved (or how long the person has been living in the place where counted in the second census) one can get a sense of the timing of migration, and estimate yearly migration rates. This is a complicated process and is not covered here, but the interested reader is referred to the paper by Dorrington and Moultrie (2009).
Working with total numbers only
If agespecific numbers are not available or the allocation to age is considered to be unreliable one can still produce estimates by age by estimating the total number of migrants as described below, and then apportioning this total to the age groups using either an age distribution for the same population at a different time (since the age distribution of migration flows tend be consistent over time, or (more likely) an appropriate standard model Rogers and Castro (1981a; 1981b).
$$\text{\hspace{0.17em}}\text{Net}{\text{}}_{\infty}{M}_{0}^{F}={\text{}}_{\infty}{N}_{0}^{F}(t+n){\text{}}_{\infty}{N}_{0}^{F}(t)+{\text{}}_{\infty}{D}_{0}^{F}\text{\hspace{0.17em}}$$
where
$$\text{\hspace{0.17em}}{\text{}}_{\infty}{D}_{0}^{F}=\frac{n}{2}\left({\text{}}_{\infty}{N}_{0}^{F}(t)+{\text{}}_{\infty}{N}_{0}^{F}(t+n)\right){\text{\hspace{0.17em}}}_{\infty}{m}_{0}\text{\hspace{0.17em}}$$
and _{∞}m_{0} is an estimate of the crude mortality rate of the population in the country of the census.
Limitations
The primary limitation of using censuses to estimate immigration and net inmigration is the quality of the census, in particular the extent of undercount of the censuses, in general but more significantly one relative to the other. However, even if the census undercount is low, the census might not identify all the migrants. In general recent migrants are often difficult to include in a census because they have yet to settle. More specifically, immigrants may not be keen to identify themselves as immigrants and either avoid being counted or do not admit to being foreignborn.
Apart from this, place of birth and/or place of residence at previous census, in the case of internal migrants, might be misreported due to boundary changes or ignorance (or even bias) on the part of the respondent.
The third drawback of census data is that it cannot be used to measure emigration from the country of the census. Emigration is particularly difficult to estimate for most countries, but one option is to apply the method for identifying net immigration of the foreigners described above to the censuses of the main countries of destination to which the emigrants move to estimate the change in the numbers of emigrants to those countries. Of course, this is only useful if the censuses of these countries identify the numbers of foreignborn by their countries of birth reasonably accurately.
Generally, statistics on immigrants and particularly emigrants that are collected at border posts provide quite poor estimates of the true numbers, unless the borders of the country are quite impenetrable and there are a few wellcontrolled ports of entry. Even then there may still be many ‘visitors’ who end up living in the country.
A final drawback occurs when working with data aggregated over all ages. In these cases one usually has to make use of the crude death rate for the population of the country of the census in order to estimate the number of deaths of the migrant population. However, since the distribution of the migrant population by age can differ from that of the population of the country of the census quite markedly, the estimated number of deaths can be quite inaccurate.
Extensions of the method
Some censuses ask additional questions which can be of use in interpreting the patterns of migration, if not improving the estimate of the level of migration. Most common of these is probably a question asking about when the migrant moved. These data allow one to estimate annual rates of migration, however, it possible that there could be a tendency for respondents to report moves as occurring more recently than is actually the case (Dorrington and Moultrie 2009).
Where a census asks, such as the recent censuses in South Africa, of those who moved since the previous census, where they moved from most recently and when they moved, and not where they were at the time of the previous census, it is possible to backproject the numbers of migrants by applying annual rates of migration between subnational regions to estimate the number by place at the time of the previous census (Dorrington and Moultrie 2009). However, in the case of South Africa, at least, it appears that the assumption the most migrants moved only once in the past five years, and thus that the place of residence before the most recent move is the same as the place at the time of the previous census, is quite reasonable (Dorrington and Moultrie 2009).
Where one has data on both the subnational region of birth and the place at the time of the previous census, one can crosstabulate the place of residence data by the place of birth and thus be able to classify recent migrants into primary, secondary and return migrants.
Further reading and references
For general background to the topic of migration, definition of terms and detail on the analysis and interpretation of the data on internal migration the interested reader is referred to the excellent UN Manual on topic, Manual VI (UN Population Division 1970). The textbook by Shryock and Siegel (1976) or its modern replacement by Siegel and Swanson (2004) also provides an introduction to the topic of migration and cover, in particular, the estimation of international migration.
Those interested in the estimation of annual migration rates and the backprojection of migration to estimate the numbers by place of residence at the time of the previous census from data on place of residence before the most recent move and year of move are referred to the paper by Dorrington and Moultrie (2009).
Dorrington RE and TA Moultrie. 2009. "Making use of the consistency of patterns to estimate agespecific rates of interprovincial migration in South Africa," Paper presented at Annual conference of the Population Association of America. Detroit, US, 30 April  2 May.
Rogers A and LJ Castro. 1981a. "Age patterns of migration: Causespecific profiles," in Rogers, A (ed). Advances in Multiregional Demography (RR81006). Laxenburg, Austria: International Institute for Applied Systems Analysis, pp. 125159. http://webarchive.iiasa.ac.at/Admin/PUB/Documents/RR81006.pdf
Rogers A and LJ Castro. 1981b. Model Migration Schedules (RR81030). Laxenburg, Austria: International Institute for Applied Systems Analysis. http://webarchive.iiasa.ac.at/Admin/PUB/Documents/RR81030.pdf
Shryock HS and JS Siegel. 1976. The Methods and Materials of Demography (Condensed Edition). San Diego: Academic Press.
Siegel JS and D Swanson. 2004. The Methods and Materials of Demography. Amsterdam: Elsevier.
Timæus IM. 2007. "Impact of HIV on mortality in Southern Africa: Evidence from demographic surveillance", in Caraël M and JR Glynn (eds) HIV, Resurgent Infections and Population Change in Africa. Springer, pp 229–243. doi: https://dx.doi.org/10.1007/9781402061745_12
UN Population Division. 1970. Manual VI: Methods of Measuring Internal Migration. New York: United Nations, Department of Economic and Social Affairs, ST/SOA/Series A/47. http://www.un.org/esa/population/techcoop/IntMig/manual6/manual6.html
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