Marriage and the Intergenerational Mobility of Women

Introduction

One of the persistent headaches in the historical mobility literature is that women are very hard to track. The standard approach links people across censuses by name, but women changed their surnames at marriage, so the moment a woman marries she effectively disappears from the record. As a result, almost everything we know about long-run intergenerational mobility in the 19th century is really about men (fathers and sons), and we have had to guess at what was happening to daughters.

A recent paper by Eriksson, Niemesh, Rashid and Craig (NBER WP 34821) takes this problem head on, and I found it a nice example of how a clever data source plus a small structural model can let you measure something you cannot directly observe. The question they ask is how women’s economic mobility evolved in the US between 1850 and 1920, and what role the marriage market played in it. Their answer is that women’s mobility improved substantially over this period, well before married women had any real access to the labour market, and that most of this improvement came not from labour-market opportunities but from a decline in assortative mating: the weakening of the tendency to marry within one’s own socioeconomic group.

I want to walk through how they get there, because the interesting part is the inference problem: a married woman’s own economic status is essentially never recorded, so how do you say anything about who she “matched” with?

Getting women into the data

The trick is the data source. The authors digitised over 1.2 million Massachusetts marriage certificates (1850–1914), and the crucial feature of these registers is that they record a woman’s birth surname. That single piece of information is what makes it possible to follow a woman backwards to her childhood household, and hence to her father, despite the name change at marriage.

The linking is done in two steps. First, couples in the marriage register are matched forward to a later adult census; then each spouse is matched backward to their own childhood census to recover their father’s occupation. Because the marriage register also lists the parents’ names, this backward step is much more reliable than ordinary census-to-census linking. Insisting that both spouses link to their fathers is demanding, and it leaves about 38,760 couples, a 3% match rate, but those are couples for which we observe the husband’s status, the husband’s father, and the wife’s father all at once. That triplet is what the whole exercise rests on.

Economic status throughout is an occupational wealth score: property values from the 1870 census, averaged by occupation (and region and immigrant status), with everyone then ranked into percentiles. So the regressions below are all rank-rank regressions of the now-familiar kind.

Women’s mobility improved: the puzzle

The headline mobility numbers are rank-rank slopes, estimated separately for four twenty-year marriage cohorts. A higher slope means more persistence, i.e. less mobility. The men’s slope is the usual son-on-father regression; the women’s slope regresses the husband’s rank on the wife’s father’s rank, the “father-in-law correlation”, which is the best one can do when the wife’s own status is unobserved.

Cohort	Men	Women
1850–1870	0.38	0.37
1860–1880	0.38	0.37
1880–1900	0.27	0.33
1900–1920	0.21	0.30

Both sexes start at the same high level of persistence around 0.37. Both become more mobile, but men improve much faster (a 46% drop in the slope) than women (18%), so a gender gap opens up by the end of the century. The estimation is careful in ways worth flagging: the father’s rank is instrumented with a second observation of his status from a nearby census, following Ward (2023), which removes the attenuation bias from measurement error that plagued earlier work and is why these persistence estimates are higher than the older literature’s.

So far this looks like a clean story about improving mobility. The puzzle appears when you try to measure the thing that is supposed to be driving it. A natural proxy for assortative mating is the correlation between the two fathers-in-law (how similar in status were the families that married into each other). And that proxy goes the wrong way: it rises by about 31% over the period, from roughly 0.32 to 0.42. Taken at face value, that says sorting was getting stronger even as women’s mobility improved, which makes no sense. Resolving that contradiction is what the structural model is for.

Why you need a model at all

The object we actually care about is the correlation in economic status between husband and wife,

\[\rho = E[X^h_i \cdot X^w_i],\]

which is the textbook definition of assortative mating. The problem is stark: the wife’s status \(X^w_i\) is never observed. We only ever see her father’s status. So \(\rho\) has to be backed out indirectly, and the question is what observable quantities pin it down.

The authors borrow the framework of Espín-Sánchez, Ferrie and Vickers (2023)¹. Throughout, every status variable is a percentile rank rescaled to have mean zero and variance one, so the expectation of a product of two ranks is just the correlation between them. We have three things we can measure from the data. Let \(X^h\) be the husband’s rank, \(X^f\) his father’s, and \(X^{fl}\) his father-in-law’s (the wife’s father):

\[b^h_f = E[X^h X^f], \qquad b^h_{fl} = E[X^h X^{fl}], \qquad b^f_{fl} = E[X^f X^{fl}].\]

These are men’s mobility, women’s measured mobility, and the father-on-father-in-law proxy respectively. The model writes each child’s status as inherited from their two parents,

\[X^h_i = \beta_f X^f_i + \beta_m X^m_i + e^h_i, \tag{H}\]

\[X^w_i = \beta_{fl} X^{fl}_i + \beta_{ml} X^{ml}_i + e^w_i, \tag{W}\]

where \(\beta_f\) and \(\beta_m\) are the inheritance from the husband’s father and mother, and \(e^h, e^w\) are idiosyncratic terms uncorrelated with the parents’ status. Equation (W) is the same object for the wife. It has to be there, because without an expression for \(X^w\) there is nothing to substitute for the unobserved wife’s status in the definition of \(\rho\).

The identifying assumption, which the paper is upfront about, is symmetry: inheritance works the same way for sons and daughters, and the same regardless of whether status flows through the father or the mother, so \(\beta_{fl} = \beta_f\) and \(\beta_{ml} = \beta_m\). There is no way around an assumption like this in a period with no data on women’s own outcomes. What symmetry buys you is that the wife’s two unknown inheritance parameters in (W) collapse onto the same \(\beta_f, \beta_m\) from (H), turning what would be six unknowns into three: \(\beta_f\), \(\beta_m\), and \(\rho\).

Two further facts about how status sorts across families close the system. First, a husband’s own parents are themselves a married couple, so the correlation between them is the same assortative mating parameter we are after, \(E[X^f X^m] = \rho\). Second, the two families joined by a marriage are linked only through that marriage, so every cross-family pair of parents shares the same correlation as the two fathers, \(E[X^m X^{fl}] = E[X^f X^{ml}] = E[X^m X^{ml}] = b^f_{fl}\).

Now do the substitutions. Each one takes an observable, replaces the husband’s (or wife’s) rank using (H) and (W), and uses the facts above.

Take the husband’s mobility \(b^h_f\) and substitute (H) for \(X^h\):

\[b^h_f = E[X^h X^f] = \beta_f \underbrace{E[(X^f)^2]}_{=\,1} + \beta_m \underbrace{E[X^m X^f]}_{=\,\rho} + \underbrace{E[e^h X^f]}_{=\,0} = \beta_f + \rho \beta_m. \tag{1}\]

Take the women’s measured mobility \(b^h_{fl}\), again substituting (H) for \(X^h\):

\[b^h_{fl} = E[X^h X^{fl}] = \beta_f \underbrace{E[X^f X^{fl}]}_{=\,b^f_{fl}} + \beta_m \underbrace{E[X^m X^{fl}]}_{=\,b^f_{fl}} = b^f_{fl}(\beta_f + \beta_m). \tag{2}\]

Finally take \(\rho\) itself and substitute (H) for \(X^h\) and (W) for \(X^w\). Symmetry turns the wife’s coefficients into \(\beta_f, \beta_m\), the error terms drop out, and each of the four surviving cross-family expectations equals \(b^f_{fl}\):

\[\rho = E[X^h X^w] = b^f_{fl}\left(\beta_f^2 + 2\beta_f\beta_m + \beta_m^2\right) = b^f_{fl}(\beta_f + \beta_m)^2. \tag{3}\]

That is the system: three equations, (1) to (3), in the three unknowns \(\beta_f, \beta_m, \rho\).

It solves in one pass. Equation (2) gives the sum of inheritance parameters directly, as the ratio of two observable slopes,

\[\beta_f + \beta_m = \frac{b^h_{fl}}{b^f_{fl}}.\]

Substituting that into (3) gives the assortative mating parameter,

\[\rho = b^f_{fl}\left(\frac{b^h_{fl}}{b^f_{fl}}\right)^2 = \frac{(b^h_{fl})^2}{b^f_{fl}}.\]

I find this a clean result. The unobservable spousal correlation turns out to equal the square of the women’s mobility slope divided by the father-on-father-in-law slope, both of which we can estimate. The two inheritance parameters then come apart from equation (1). We now know the sum \(\beta_f + \beta_m\) and the value of \(\rho\), so (1) reads \(b^h_f = \beta_f + \rho\beta_m\) with only \(\beta_f\) and \(\beta_m\) left unknown. That is two linear equations in two unknowns,

\[\beta_f + \beta_m = \frac{b^h_{fl}}{b^f_{fl}}, \qquad \beta_f + \rho\beta_m = b^h_f,\]

and subtracting the second from the first gives \(\beta_m = \left(\frac{b^h_{fl}}{b^f_{fl}} - b^h_f\right)\big/(1-\rho)\), with \(\beta_f\) the remainder.

Resolving the puzzle

The same algebra also explains why the naive proxy misleads. Rearranging the last equation,

\[b^f_{fl} = \frac{\rho}{(\beta_f + \beta_m)^2}.\]

The father-on-father-in-law correlation is not \(\rho\); it is \(\rho\) inflated by how strongly status is inherited. So if inheritance weakens, \(b^f_{fl}\) can rise even while \(\rho\) falls. And that is exactly what happens in the data. Plugging the estimated slopes through the formula, the structural \(\rho\) falls from about 0.34 in the 1850–70 cohort to 0.13 by 1900–20, a decline of roughly 61%, at the same time that the raw father-on-father-in-law proxy rises by a third. The reconciling object is the inheritance sum \(\beta_f + \beta_m\), which falls from about 0.97 to 0.57 over the same window. Sorting really was falling; the proxy just pointed the wrong way because inheritance was loosening underneath it. Any study using \(b^f_{fl}\) as a stand-in for assortative mating would have concluded the opposite of the truth.

How much does this matter for women’s mobility? You can answer that by counterfactual: take a later cohort’s inheritance parameters but feed in the strong sorting \(\rho \approx 0.34\) of the mid-century cohort. Doing this for 1900–1920 raises the implied women’s persistence slope from the observed 0.23 to about 0.59, i.e. women’s mobility would have been far worse (more than double the persistence) had sorting stayed at its mid-century strength. The decline in assortative mating was doing most of the work in improving women’s mobility.

Why did sorting fall?

The last question is what was actually weakening the link between a woman’s family background and her husband’s status. The authors decompose the change in \(\rho\) across demographic groups, and the dominant factor is immigration. Holding the share of immigrant-parent couples fixed at early-cohort levels accounts for roughly half of the decline in \(\rho\); holding the wealth gap between immigrant and native families fixed accounts for another third or so. As immigrant families became both more numerous and economically closer to native families, the marriage market mixed across origins that had previously stayed separate, and sorting on family status fell.

Internal migration and urbanisation, by contrast, explain almost none of the aggregate decline; rural and urban, stayers and movers all saw similar drops in sorting, so there is no group-specific mechanism hiding there.

Closing thoughts

What I like about this paper is that the central message, that for 19th-century American women the marriage market, not the labour market, was the main channel of economic mobility, only becomes visible once you stop treating the father-on-father-in-law correlation as if it were the spousal correlation. The structural model is doing real work: it disentangles the unobservable \(\rho\) from the observable proxy by embedding both in one inheritance framework, and the payoff is the compact estimating equation \(\rho = (b^h_{fl})^2 / b^f_{fl}\). The identifying symmetry assumption is strong, and worth keeping in mind when reading the magnitudes, but it is hard to see what else one could do given that women’s own economic outcomes simply were not recorded.

I hope this has been a useful summary. If you have any remarks or spot anything I got wrong, let me know, and thanks for reading.

Footnotes

1. Espín-Sánchez, J. A., Ferrie, J. P., & Vickers, C. (2023). Women and the Econometrics of Family Trees (No. w31598). NBER.