As laid out in Grembi et al. (2016), there are three identifying assumptions underlying the diff-in-disc estimator. First, all potential outcomes should be continuous across the respective RDj thresholds. Second, the confounding seasonal effects need to be
constant over time. Third, there should be no interaction between the transition to a different registration regime at RDj and the seasonal trends. The first two
assumptions are sufficient for a causal interpretation of the diff-in-disc estimator, while the third allows for a more general estimand (Grembi et al., 2016). I am not able to able to test the last assumption in the present setting and focus instead on
38These common shocks are imperfectly captured by a combination of the γ coefficient and the
providing evidence for the first two assumptions.
As an indirect test of whether potential outcomes are likely to vary smoothly at the respective RDj cutoffs (assumption one), I consider whether pretreatment
characteristics are continuous across the same thresholds. Specifically, I use four household characteristics as measured in 1900, when the individuals in my sample were still children or teenagers: (i) an indicator for whether the family lived in an urban area, (ii) the number of siblings residing in the household, (iii) the age of a person’s father, and (iv) the logarithm of the father’s occupation income score. This exercise is implemented using regression (1), but replaces the outcome variable Yi
with one of these four household characteristics. A β coefficient that is close to zero would suggest that cohorts born on either side of RDj have similar backgrounds on
average. This may hint that their potential outcomes vary smoothly as well. I present the regression results in Table 3.5. Panels A and B look at the older and younger cohorts respectively. The first four columns use a bandwidth of six months while the next four columns use a three-month bandwidth. Consider the γ coefficients – these are often statistically significant, indicating that there are seasonal patterns in childhood household characteristics. The direction of these effects, however, is not always consistent across bandwidths and there is substantial variation in their magnitudes. The more important β estimates, on the other hand, are usually not statistically different from zero, particularly with a bandwidth of three months.39
This indirectly suggests that assumption one of the diff-in-disc estimator may be satisfied. It also allays concerns that any estimated effects may be driven by the particularly low match rates for the April and May cohorts as discussed above.
Can one find evidence that the confounding seasonal effects are constant over
39One might be concerned that, when using the age of one’s father as the outcome variable, the
β coefficients are imprecisely estimated rather than true zeros. This is unlikely to pose a problem because even if these β estimates had been significant, their magnitudes are small compared to the standard deviation of over eight years for any combination of age group and bandwidth.
time, as required by the second assumption underlying the diff-in-disc estimator? I use the “before” and “after” windows to verify this. The idea is as follows: if seasonal patterns in economic status are similar across birth years, then there should not be large differences in these patterns between the “before” and “after” windows. This, in turn, would give one confidence that the seasonal trends estimated with these windows can be extrapolated to the “effective” window. To operationalize this test, a modified version of regression (1) is used:
Yi = α + Af ter
+ p(Ri) + p(Ri) · Af ter
+ γAf ter· 1[Ri ≥ 0] + βAf ter · 1[Ri ≥ 0] · Af ter + vi
(2)
where most of the notation is as before. A small or insignificant βAf ter would consti-
tute evidence for stable seasonal effects. Table 3.6 presents the results of this exercise for the four measures of economic status. Following the layout of Table 3.5, Panel A of Table 3.6 looks at the older men, Panel B focuses on the younger men, columns (1) to (4) use a bandwidth of six months, and columns (5) to (8) use a three-month bandwidth. Consider first the γAf ter coefficients – these are typically significant when
using a six-month bandwidth but not with a more narrow bandwidth, suggesting that seasonal effects may be less important closer to the thresholds. This does not imply that seasonal trends do not matter at all with a three-month bandwidth. In the main specification that will be used for the baseline analysis below, the “before” and “after” windows are pooled together to estimate the seasonal effects, which improves precision and reveals the presence of some statistically significant seasonal patterns even with a bandwidth of three months. As for the βAf ter estimates, these are usually
not statistically significant regardless of the bandwidth, consistent with the assump- tion of time-invariant seasonal effects. This stability may also hint that the shocks
common to a particular birth month cohort may be reasonably similar across different birth years, apart from the WWI military service shock.40 If true, this would weaken
the case for clustering standard errors by cohort.