The Estimation of Fertility Effects on Happiness: Even More Difficult than Usually Acknowledged

Abstract

There have been many studies of how the number of children in a family affects the parents’ or the children’s lives. One strand of this research focuses on the implications of fertility for the parents’ level of self-reported well-being or happiness. It is argued in this paper that an overall “happiness effect” is not very informative because of the presumably large variation in individuals’ perceived gains from having children. Furthermore, it is explained that such an effect would be difficult to estimate. Most importantly, the highly varying ideas about how a child will affect life quality are important for the decision about whether to have a child. Many of those who have few or no children have chosen this because they think their life will be best this way, and their happiness therefore tells us little about how happy their more fertile counterparts—who to a large extent have different views about the consequences of childbearing—would have been if they had few or no children. This estimation problem that arises when effects of a certain event (here childbearing) are heterogeneous, and the individuals who experience that event tend to be among those for whom the effects are particularly positive or negative, is acknowledged in the treatment effect literature. However, there is little consciousness about it in the fertility–happiness research. In addition, there is a more “standard” selection problem: factors with implications for childbearing desires, or for the chance of fulfilling these, may also affect or be linked to happiness for other reasons. Unfortunately, even the most advanced statistical approaches that have been used in this research area fail to handle all these problems, so reported results should be interpreted very cautiously.

Appendix: A More Formal Presentation of Main Ideas

Following standard terminology and ideas in the treatment effect literature (e.g. Angrist and Pischke 2009) and applying them to the fertility–happiness case, we can let h ⁰ _i be the potential happiness of a couple i if they do not have an additional child, while h ¹ _i is their potential happiness if they do. The “treatment effect” is the difference

$$ d_{i} = h_{i}^{0} - h_{i}^{1} . $$

What one can observe is h ⁰ _i if the couple does not have a child (i.e. if c _i  = 0) and h ¹ _i if they do (i.e. c _i  = 1).

A measure of potential interest is the “average treatment effect” ATE:

$$ E\left( {d_{i}} \right) = E\left( {h_{i}^{1} - h_{i}^{0}} \right). $$

In a situation where people are selected at random to have a child, this would be the same as the average happiness among those having a child minus the average happiness of those not having a child, i.e.

$$ {\text{ATE}} = E\left( {h_{i}^{1} |c_{i} = 1} \right) - E\left( {h_{i}^{0} |c_{i} = 0} \right) $$

However, with real non-experimental data, the relationship is typically more complex (see e.g. Xie et al. 2012):

$$ {\text{ATE}} = E\left( {h_{i}^{1} |c_{i} = 1} \right) - E\left( {h_{i}^{0} |c_{i} = 0} \right) - \left( {E\left( {h_{i}^{0} |c_{i} = 1} \right) - E\left( {h_{i}^{0} |c_{i} = 0} \right)} \right) - \left( {{\text{ATT}} - {\text{ATU}}} \right)q, $$

where ATT is the “average treatment effect among the treated” (i.e. among those having another child), ATU is the “average treatment effect among the untreated” (i.e. among those not having another child), and q is the proportion not having another child. Mathematically, ATT = E(h ¹ _i  − h ⁰ _i |c _i  = 1) and ATU = E(h ¹ _i  − h ⁰ _i |c _i  = 0). To simplify the discussion below, we can write the equation as

$$ {\text{ATE}} = {\text{Obsdiff}} + {\text{Term}}1 + {\text{Term}}2, $$

where Obsdiff = E(h ¹ _i |c _i  = 1) − E(h ⁰ _i |c _i  = 0), Term1 = − (E(h ⁰ _i |c _i  = 1) − E(h ⁰ _i |c _i  = 0)) and Term2 = − (ATT − ATU) q.

Term1 and Term2 have the following simple interpretations: if there are two groups in the population, both with the same potential happiness if they do not have a child, but with different effects of childbearing on happiness and (therefore) also different probabilities of having a child, Term1 = 0 and Term2 ≠ 0. If effects of childbearing instead are the same in the two groups, but one group has generally lower level of happiness and the probabilities of having a child differ, Term1 ≠ 0 and Term2 = 0.

In this paper, there is one example of a situation where ATE is equal to the difference in happiness between one- and two-child couples (Obsdiff), and some examples where this is not the case (Term1 or Term2 being non-zero). It is also discussed what kind of substantive mechanisms that give rise to such gaps between ATE and Obsdiff. In a simple regression analysis including only a child variable, the corresponding coefficient will be Obsdiff. In the second-last section of the paper, it is discussed whether addition of control variables or use of different types of models can give estimates closer to ATE.

1.1 Specification of Examples

In Example 1 in this paper, two groups are defined: group 1 (g = 1) and group2 (g = 2). They consist of 200 and 100 couples, respectively. The potential happiness outcomes for the two groups are defined as ${{h_{i}^{x}}_{\text{g}}} = 12 + 5x + \left( {g - 1} \right)\left( {5 - 10x} \right)$, where x = 1 in the potential situation where they have a child and x = 0 otherwise. The probabilities of having a child are assumed to be p ₁ = 0.50 if g = 1 and p ₂ = 0.50 if g = 2. Then,

$$ \begin{aligned} E\left( {h_{\text{i}}^{1} |c_{i} = 1} \right)& = \left( {E\left( {{{h_{i}^{1}}_{1}}|c_{i} = 1} \right)\cdot 200 \cdot p_{1} + E\left( {{{h_{i}^{1}}_{2}} |c_{i} = 1}\right) \cdot 100 \cdot p_{2}} \right)/\left( { 200 \cdot p_{1} +100 \cdot p_{2}} \right) \\ & = \left( {E\left({{{h_{i}^{1}}_{1}} |c_{i} = 1} \right) \cdot 200 + E\left({{{h_{i}^{1}}_{2}} |c_{i} = 1} \right) \cdot 100} \right)/300 \\& = \left( {17 \cdot 200 + 12 \cdot 100} \right)/300 = 15.33\\ \end{aligned} $$

when p ₁ = p ₂, regardless of whether they are 50 % as assumed or lower or higher.

Similarly, one can calculate E(h ⁰ _i |c _i  = 0), E(h ⁰ _i |c _i  = 1), ATE, ATT, etc., and it is not difficult to show that Obsdiff = ATE = ATT = ATU. (Term1 = 0 in spite of the two groups having different happiness if they remain one-child couples, because the chance of childbearing is the same in the two groups. Also Term2 = 0 as a result of the non-varying fertility.)

Example 2 is similar except that the probabilities of having a child are assumed to be different in the two groups: p ₁ = 0.90 and p ₂ = 0.30. Then, Term1 ≠ 0 (reflecting that the happiness associated with having one child is not the same for the two groups); ATT ≠ ATU, so that Term2 ≠ 0; and Obsdiff is not equal to ATE (and not equal to ATT or ATU either).

In Example 3, the happiness function is the same as in Examples 1 and 2 except that a term that differs between poor (r = 1) and non-poor (r = 0) is added:

$$ {{h_{i}^{x}}_{gr}} = 12 + 5x + \left( {g - 1}\right)\left( {5 - 10x} \right) - 6r $$

Furthermore, poverty is assumed to affect the chance of having a child through the chance of belonging to one of the two main groups g = 1 and g = 2: there are 70 poor and 230 non-poor; the poor have a chance of 20/70 of being in group g = 1, while the non-poor have a chance of 180/230 of being in group g = 1. The group-specific chances of having a child are as before p ₁ = 0.90 and p ₂ = 0.30. Thus, one may say that the difference compared to Example 2 is that the happiness levels in group 2 (averaged over poor/non-poor) are no longer reversed compared to those in group 1 (17 and 12 vs. 12 and 17), but generally lower because of a larger proportion of poor in group 2 than in group 1. It can be shown that Term2 is the same as in Example 2, but Term 1 is different. As in Example 2, Obsdiff ≠ ATE (and not equal to ATT or ATU either).

Example 4 is simpler in that poverty affects fertility (p) more directly, not via the distribution over the two main groups (which would symbolize differences in expected consequences of fertility and thus fertility desires). There is supposed to be only one group (g = 2 above, i.e. those less keen on childbearing), so happiness is given by

$$ {{h_{i}^{x}}_{r}} = 17 - 5x - 6r $$

The non-poor are assumed to have a 10 % chance of having a child and the poor a 50 % chance. Furthermore, 50 of the 200 are supposed to be poor. In this case, Term1 ≠ 0, Term2 = 0, and Obsdiff ≠ ATE (but ATE = ATT = ATU).