What a Surname Averages Out

Introduction

In an earlier post I argued that the two-generation mobility slope is not the object we want. If observed status \(y\) is a noisy reading of a latent family endowment \(e\),

\[ y_t = \lambda e_t + u_t, \qquad e_t = \rho\, e_{t-1} + v_t , \]

then the \(k\)-generation association is \(b_k = \lambda^2 \rho^k\). The loading \(\lambda\) is a toll paid twice: once to get from an ancestor’s title into his endowment, once to get from a descendant’s endowment back out into his title, no matter how many generations lie between. Persistence \(\rho\), the thing we actually care about, is paid once per generation. So \(b_1\) understates persistence, and the ratio \(b_2/b_1 = \rho\) recovers it because the toll cancels.

That trick buys identification with time depth: it needs three linked generations. There is a second route to the same parameter that needs only two, and buys it with the cross-section instead. That is the grouped surname estimator, and it is what Clark and Cummins are really doing.¹

The estimator

What do they do? Take every occupation-bearing individual and sort them into surname-by-birth-cohort cells. Give each cell the mean status of its members (in our data, HISCLASS, oriented so higher is better) and rank the cell means within each cohort, which purges the secular drift in the occupational structure. Then regress a surname’s rank in one cohort on the same surname’s rank in the previous cohort. The slope \(b\) is the persistence parameter, from \(b \approx 0\) (each cohort starts afresh) to \(b \approx 1\) (perfect inheritance).

Notice what is not required: nobody has to be linked to their father. The cells are measured from different people in each cohort, which is exactly why the method survives records where perhaps a tenth of individuals carry a legible occupation. Ours are Dutch genealogical lineages, cohorts born before 1800, cells with at least five occupation-bearing members, precision- and near-border-weighted so the slope reflects municipalities hugging the Mechelen boundary.

Why grouping identifies \(\rho\)

Average the measurement equation over the \(n\) members of a cell:

\[ \bar y_{g,t} = \lambda \bar e_{g,t} + \bar u_{g,t}, \qquad \mathrm{Var}(\bar u) = \frac{1-\lambda^2}{n}, \]

where \(g\) indexes the surname group and \(t\) the birth cohort, so \(\bar y_{g,t}\) is the mean observed status of the \(n\) occupation-bearing people called \(g\) who were born in window \(t\), and \(\bar e_{g,t}\) is the endowment those people share. The cell, not the individual, is now the unit: \(t\) advances one 50-year window at a time rather than one father–son link at a time, and it is over the \(n\) members within a cell that we are averaging. We normalise \(\mathrm{Var}(y) = \mathrm{Var}(e) = 1\) as before, which pins the noise variance: \(1 = \lambda^2 + \mathrm{Var}(u)\), so \(\mathrm{Var}(u) = 1 - \lambda^2\), and averaging \(n\) independent draws of it leaves \((1-\lambda^2)/n\). That is the only term the averaging touches. The endowment is shared by the members of the cell, so it does not shrink: \(\mathrm{Var}(\bar e_{g,t}) = 1\) still, and it is still an AR(1) in \(\rho\), so \(\mathrm{Cov}(\bar e_{g,t}, \bar e_{g,t-k}) = \rho^{k}\) exactly as at the individual level.

Now run the regression. The cell-level slope of \(\bar y_{g,t}\) on \(\bar y_{g,t-k}\) is a covariance over a variance. The covariance kills the noise, because the \(\bar u\)’s are uncorrelated across cohorts and with the endowment:

\[ \mathrm{Cov}(\bar y_{g,t},\, \bar y_{g,t-k}) = \mathrm{Cov}(\lambda \bar e_{g,t} + \bar u_{g,t},\; \lambda \bar e_{g,t-k} + \bar u_{g,t-k}) = \lambda^{2}\, \mathrm{Cov}(\bar e_{g,t}, \bar e_{g,t-k}) = \lambda^{2}\rho^{k} . \]

The denominator does not. The variance of an observed cell mean carries the surviving noise alongside the signal:

\[ \mathrm{Var}(\bar y_{g,t-k}) = \lambda^{2}\,\mathrm{Var}(\bar e_{g,t-k}) + \mathrm{Var}(\bar u_{g,t-k}) = \lambda^{2} + \frac{1-\lambda^{2}}{n} . \]

Divide one by the other:

\[ b^{\text{grouped}} \;=\; \frac{\mathrm{Cov}(\bar y_{g,t}, \bar y_{g,t-k})}{\mathrm{Var}(\bar y_{g,t-k})} \;=\; \frac{\lambda^{2}\rho^{k}}{\lambda^{2} + (1-\lambda^{2})/n} \;\xrightarrow[\;n \to \infty\;]{}\; \rho^{k} . \]

So the loading appears in both places, and the size of the cell decides how much of it cancels. The numerator is the same \(\lambda^2 \rho^k\) we had before, grouping buys you nothing there. What grouping does is shrink the denominator’s noise term toward zero, and once \((1-\lambda^2)/n\) is negligible the \(\lambda^2\) upstairs is divided by the \(\lambda^2\) downstairs and the toll is refunded. At \(n = 1\) nothing is refunded and we are back to \(\lambda^2 \rho^k\), the individual slope; the ratio \(\lambda^2 / (\lambda^2 + (1-\lambda^2)/n)\) is exactly the reliability of a cell mean as a measure of the endowment it proxies. As the cells fill up, the idiosyncratic slippage between endowment and job title averages away and that reliability climbs to one. The structural content is identical to the ratio test: both estimate \(\rho\) free of \(\lambda\); one does it by iterating the process forward, the other by averaging the noise down.²

Where the two routes disagree

This is the useful part. Suppose the Protestant and Catholic sides of the border differ only in \(\lambda\) — occupational titles simply track family standing more loosely on one side. Then grouping should shrink the measured gap: the reliability factor tends to one on both sides, so \(b^{\text{grouped}}_P / b^{\text{grouped}}_C \to 1\). A pure measurement story predicts that the better you group, the more the two sides converge.

In our case, they diverge. The individual father–son slopes are \(0.420\) Protestant against \(0.485\) Catholic, a ratio of \(0.87\). The grouped surname slopes are \(0.290\) against \(0.397\) before 1800, and \(0.347\) against \(0.595\) over the full genealogical window with larger cells, ratios of \(0.73\) and \(0.58\), moving away from one, with the full-window difference of \(0.248\) significant. The differential-persistence IV, interacting a surname’s lagged rank with the instrumented Protestant share, is negative throughout (\(-0.35\) to \(-0.90\) pre-1800).

Read through the model, that is a claim about \(\rho\): Catholic endowments regressed to the mean more slowly. Which sits awkwardly next to the ratio test, where \(b_2/b_1 = 0.69\) on both sides and pins the gap on \(\lambda\) instead.

Closing thoughts

I do not think that tension is resolvable with these data, and I would rather say so than pick the estimate I like. The ratio test is underpowered on the thin Catholic side. The grouped estimator has its own well-known wound: surnames correlate with place, trade and endogamy, so cell-level persistence can absorb group-level advantages that were never transmitted through the family at all — and there is no reason such contamination is symmetric across a confessional border.

What survives both routes is the sign. Whether the Protestant advantage sits in the loading or in the persistence, allocation or transmission, every estimator we have puts it on the same side of zero, in the nineteenth century and three centuries earlier.

Footnotes

1. Clark, G., & Cummins, N. (2015). Intergenerational wealth mobility in England, 1858–2012: Surnames and social mobility. The Economic Journal, 125(582).

2. Two liberties here. Surname cells pool kin, not clones, so the shared endowment is a fraction of each member’s; and a 50-year window is not one generation. Both make the mapping from \(b^{\text{grouped}}\) to \(\rho^k\) approximate rather than exact — which is why I read the estimator comparatively rather than as a point estimate of \(\rho\).