Decay IV: Vig's “annuity model”
21. marts 2015
In this blog post, we will examine Vig’s annuity model, described by Nyborg in the Decay article.
In basic textbooks of demography, the expression for the geometric increase of a population is usually given as a projection to time t:
where $x$ is a fractional of increase per unit time; and $t$ counts the units of time, in this case, years. If the rate $x$ is compounded instantaneously, this function limited by the natural exponential.
In finance theory, a similar equation describes the growth of capital subjected to a fixed interest rate, which likely inspired the economist Vig to term his population model an “annuity model”.
In the Decay article, Nyborg presents the “annuity model” as follows:
where $i_\text{fo}$ is given as the increase in the number of citizens of foreign origin and $i_\text{np}$ is the increase in naturalized people. In the Decay article, Nyborg defines $b$ as “official UN birth rates” and $d$ as “the total common mortality rate for DK”.
There are, however, several problems with this formula.

Division by one (1.000) is not correct, what is meant is probably division by one thousand. However, Statistics Denmark reports crude birth rates per one thousand population, and death rates per ten thousand, so these numbers should be used to arrive at the correct units of $b$ and $d$.

The expression is mathematically malformed, as it will only compute the “Status count”, i.e. the population size, at 1980.

The expression given in $\eqref{eq:annuity}$ is not what is calculated by the spreadsheet, the formula actually used in the spreadsheet will be given below.
Equation $\eqref{eq:geomincr}$ can be reformulated:
highlighting its iterative character. The rate of natural increase $x$ is defined:
where $b$ is the crude birth rate and $d$ is the mortality rate (crude death rate). Thus, with the knowledge of the natural rate of increase $x$, and the population size $N_t$ it is possible to estimate the population size of the following year.
With this in mind, we reformulate the annuity formula $\eqref{eq:annuity}$ reflecting how it is employed in the spreadsheet :
From the rate of natural increase $x$, the birth excess $f_{t}$ for the year $t$, in a population exhibiting geometric growth, can be calculated using:
so $\eqref{eq:annuity_corrected}$ becomes:
which is the actual “annuity model” as it is used in the retrocorrection step.
Vig and Statistics Denmark
Vig does not believe the population statistics published by Statistics Denmark reflects the truth. He believes that the political elite in Denmark is involved in a conspiracy to conceal the true number of immigrants for the Danish population, and therefore, the numbers published by Statistics Denmark can not be trusted. There is abundant documentation that this idea was fostered in “Den Danske Forening” in the 1990’s and that it led Vig to begin his population projections, where he attempted to predict the “true” number of immigrants and the associated costs for society.
Vig also (falsely) believes that children born to foreign nationals automatically obtain Danish citizenship, and that they therefore somehow “escape” what he believes is the correct categorization. For this reason, Vig thinks he needs to use population statistics where foreign nationals are listed by citizenship; another reason for this is so he can add naturalized foreign citizens back to the category where they belong, the reason for this being that Vig believes naturalized foreigners should not be counted in the ethnically pure Danish stock.
In reality, they are not, but these false premises have been annexed and reproduced by Nyborg in the Decay article:
Children born to foreign citizens/citizens of foreign origin in DK are counted as Danish citizens and so are the children born to naturalized citizens after naturalization.
In reality, Statistics Denmark counts every single person living in the country, including their country of origin and their citizenship. Every child born to immigrants are categorized according to the mother’s country of origin, unless one of the parents is a Danish citizen, in which case marital status and the gender of the Danish parent is taken into account.
The retroprojection analysis
Vig’s analysis consists of two steps:

A retrocorrection, which goes back to the starting year 1979, and progresses for 31 years by applying the annuity $\eqref{eq:annuity_corrected}$ yearbyyear until 2010. Yearbyyear SBdata for immigration, death rate and naturalizations, are used to compute the size of each foreign subpopulation living in Denmark. Constant values for crude birth rates for all subpopulations is used. Naturalized foreigners are added back ethnic group they originally came from. The size of the ethnic Danish population is computed by subracting the total number of foreigners from the total population. In Decay, Nyborg refers to this group as the “residual danish” population.

A projection, that continues the population profile resulting from the retrocorrection. The future numbers for immigration, emigration, birth and death rates, naturalizations are naturally unknown, and are set to constant values. This step starts in 2011 and continues for 63 years until year 2072.
Both steps make use of the annuity model from the Decay article, only the data that goes into it is varying.
The population related data used by Vig is available from Statistics Denmark’s public webservice Statbank Denmark (SB) from 1979 and onward.
It is not clear exactly what Vig means by the “increases in the number of citizens of foreign origin” $i_\text{fo}$. If $i_\text{fo}$ means net immigration, the value of $i_\text{fo}$ can be determined from the SB tables on immigration (VAN21A, VAN211, VAN21AAR) and emigration (VAN22A, VAN222, VAN22AR). However in Vig’s spreadsheet the increase in number of foreign nationals $i_\text{fo}$ in a subpopulation is calculated by subtraction of population counts:
Numbers for $P_t$ is given in tables BEF2A, KRBEF2A and FOLK1, and numbers for $i_\text{np}$ can be found in the table DKSTAT. The value of $P_{1979}$, the initial population size per subpopulation, can be found in table BEF2A. Population count is available from SB stratified by country of origin, but Vig has used the population count stratified by citizenship, apparently because he wants to add naturalized citizens back to their original subpopulation.
Thus the annuity model ignores the actual number of citizens counted by Statistics Denmark but uses their data to compute increase in the size of subpopulations $i_\text{fo}$.
For the the immigrant groups, Vig uses a set of crude birthrates of the country of origin for an arbitrary year which is held constant for 93 years, while the Danish mortality rates (Statistics Denmark Table HISB3) are used for all immigrant groups. In general, the mortality in the countries that have high birth rates is higher than the Danish, thus Vig chooses the highest value of $b$, and the lowest value of $d$. The effect of this choice is to inflate the rate of natural increase of the immigrant populations.
The assumption that immigrant groups maintain the birth rates of the country of origin is clearly wrong and contrary to facts. Statistics Denmark has detailed data on fertilities of immigrants and descendants of immigrants which shows a rapid convergence to the fertilities of the indigenous population.
When Vig has computed the population sizes of the 235 national groups represented in Denmark the total population of foreigners is computed thus:
where $N_t^i$ is calculated according to $\eqref{eq:annuity_corrected}$. Finally the ethnically pure Danish “residual” population is calculated using:
where $T_t$ is the total population according to Statistics Denmark (tables BEF2A, BEF707 & FOLK1). Thus, the total population count will always be in agreement with Statistics Denmark, but the distribution of the subpopulations will be grossly different.
The annuity model contains a systematic error
When Statistics Denmark receives the quarterly population data from the Civil Registration (CPR), the previous quarter’s increase in a certain immigrant subpopulation will consist of: Number of newborns plus number of immigrants minus number of emigrants minus number of deceased. Similarly, the yearly increase in a immigrant subpopulation by citizenship is given by:
but $P_t  P_{t1}$ is Vig’s $i_\text{fo}$, so we equation $\eqref{eq:annuity_corrected}$ can be rewritten:
where $b_t$ is the birth excess, $m_t$ is net migration and $i_\text{np}$ the number of naturalized in year $t$.
However, in this equation, $f_t$ and $b_t$ refer to the same quantity, namely the birth excess. Thus, in Vig’s annuity model, the birth excess is counted twice: Once from the rate of natural increase $x_t$ (equation $\eqref{eq:ft}$) and a second time from Statistics Denmarks’s population statistics (equation $\eqref{eq2}$). The number of naturalized persons $i_\text{np}$ cancels out of the equation.