# 1 Introduction

This research question is not new. Mason (1997) discussed contemporary explanation models of the fertility transition and argued that there cannot be one explanation alone. More recent studies of Swedish circumstances were given by Dribe and Scalone (2014) and Bengtsson and Dribe (2014). One perhaps convincing theory is that the fertility decline was a necessary result of the mortality decline. For a detailed and initiated overview of the research field, see the aforementioned papers and also two thoughtful papers by Ní Bhrolcháin (Ní Bhrolcháin 1992, 2011).

# 2 The study area

Our data cover the time period 1821–1950, see Figure 4.1, and the geographical area defined by the Skellefteå and Umeå towns with surrounding rural parishes. Where in Sweden? See map in Figure 2.1.

The Skellefteå region was one big parish in the eighteenth century. It successively split up in smaller parishes.

The development of population size in the two regions is shown in Figure 2.2.

## 2.1 Skellefteå

At the outset of the study the region consisted of one large rural parish, Skellefteå. By 1900 three new parishes had been detached into separate units, but their populations are still included in the study. The region was large, both in terms of area and of population. With an area of about 1700 square miles, Skellefteå was considerably larger than most rural parishes in Sweden. It was considered a one-day journey to travel from the northern to the southern border, and a ride from the coast to the more remote and sparsely populated parts of the parish in the west could take even longer, especially in wintertime. The main part of the population was, however, concentrated in the coastal area and in river valleys. In the early 19th century the population size was around 6900, and it increased rapidly during the first half of the century. By 1850 it had reached to about 17000 and at the turn of the century it had further doubled. Despite the large increase in population, which was mainly the result of a high natural growth, the population density on the whole remained low (Alm Stenflo 1994).

Skellefteå was during the studied period a rural area with a mixed economy, based on animal husbandry, forestry and sidelines such as tar and saltpeter production. By the mid-19th century export of tar and lumber became an increasingly important part of the economy. The majority of the farmers in the region were smallholders and there were no large estates. Some small sawmills were established early in the century, but before 1900, industrialization had little impact on the local economy. In 1835, approximately 85 percent of the population made their living from farming. Although the distribution of economic resources was more uniform than in several other Swedish regions, the social stratification became more pronounced throughout the 19th century. The increasing proletarianization was mainly a consequence of rapid population growth. The number of farming households remained fairly stable, while the number of landless households increased. The socio-economic development was also influenced by two devastating subsistence crises in the region, in the 1830s and in the 1860s (Engberg 2005).

Infant mortality was comparatively low. Fertility was high, not only by Swedish standards, but also in European comparison and there are no indications of family planning. Total fertility fluctuated around five children per woman and, although fertility did decline during the nineteenth century, the actual fertility transition occurred late in the district (Alm Stenflo 1994; Ansley J. Coale and Watkins 1986). The rate of illegitimacy was low in comparison with many other parts of Northern Sweden, where frequent pre-nuptial conceptions and illegitimate births were common. The illegitimacy rate fluctuated between three and six per cent during the nineteenth century .

## 2.2 Umeå

The Umeå region is one of the newest in the collection of regions digitized at the Demographic Data Base, Umeå University. The covered time period is shorter than the one for the Skellefteå region in that registration starts around January 1, 1901.

Umeå is older and substantially larger than Skellefteå as a town: The birth year of the town Umeå is generally recognized as 1622. Umeå became early a center for administration and education, but Umeå University is not a fact before 1963.
Early in the twentieth century, two military regiments were placed in Umeå, and they remained there during the first half of the century. The population size as defined in our data sets was 19000 on January 1, 1900, 33000 on January 1, 1950 and 104000 when the end of the century.

The surrounding rural area is similar in demography and economy to the corresponding parts of the Skellefteå region.

# 3 Models

The statistical modeling is conveniently done in the framework of counting (birth) processes, but there are still choices to contemplate: Choice of time scale, and how to handle the dynamic aspects of the counting process. In most of the earlier attempts to investigate changes in stopping and spacing behavior over time, models for birth intervals have been employed , but in my opinion this approach is too indirect. Instead I argue for the counting process approach, where each married woman contributes an age interval (or a union of such intervals), starting at age of marriage or age 20, whichever comes last, and ends with the dissolution of marriage or age 50, whichever comes first. We also allow intervals to start and end with migration events. In the so defined interval, birth time (age) points are recorded.

An obvious alternative for studying marital fertility is to start the clock at the date of marriage, but it complicates things in some ways, one being that it will be difficult (but not impossible) to apply the Coale-Trussell model . Therefore, in the following, the basic time scale is age.

## 3.1 Fertility as a counting process

We define

$\begin{equation*} \{N_i(t), m_i \le t \le 50\}, \quad i \in \text{mother's ID} \end{equation*}$

as the number of births before and including age $$t$$ and after marriage at age $$m_i$$ for woman No.
$$i$$. By age we mean exact age, measured by a precision of a day, but with time unit year. Thus $$\{N_i, \, i = 1, 2, \ldots\}$$ are counting processes with jumps of size one (except for multiple births) at the age of the mother at deliveries. As an example, see Figure 3.1 for the marital fertility history of mother No. 233: She married at age 25, died at age 46, and in between she had seven births. See (Aalen, Borgan, and Gjessing 2008) for details on the counting process theory.

$$N(t)$$ is a right-continuous function, meaning essentially that at jump points, the value is the larger of the two possible values. $$N(t-)$$ denotes its left-continuous counterpart, that is, it is defined as the number of births up to but not including age $$t$$.

## 3.2 The Coale-Trussell (CT) model

We will assume a piecewise constant hazard model on the age span 20–50, with jumps every fifth year. We apply the restriction proposed by A. J. Coale (1971) and A. J. Coale and Trussell (1978):

$\begin{equation*} \lambda(t) = M n(t) e^{m v(t)}, \quad 20 \le t < 50, \end{equation*}$

where the piecewise constant functions $$n, v$$ are given by

     (20-25] (25-30] (30-35] (35-40] (40-45] (45-50]
n(t)    0.46   0.431   0.395   0.322   0.167   0.024
v(t)    0.00  -0.279  -0.667  -1.042  -1.414  -1.671

The parameter $$m$$ governs the form of the (cumulative) hazard function, while $$M$$ measures level. Figure 3.2 shows two cases, $$(m, M) = (0, 1)$$ and $$(m, M) = (1, 1.57)$$. These two cases generate the same expected number of births over a full reproductive period (20–50), that is, the same Total Marital Fertility Rate (TMFR). The first is an example of spacing, but no stopping, while the second is a case of stopping (and eventually spacing).