Appendix Appendix F Common Support and Renormalization
A practical issue in the decomposition is that the set of observed occupation $\times$ seniority $\times$ industry cells changes over time. Some cells are observed in both the 2021 baseline and period $t$, while others appear in only one of the two periods. This support non-overlap can mechanically affect the decomposition, especially in the three-fold specification, where part of the change may be absorbed by the interaction term.
To address this issue, we define the period-specific common support between 2021 and period $t$ as
$$ S_{t}=\{c:w_{c,2021}>0\text{ and }w_{ct}>0\}. $$
Thus, $S_{t}$ contains only cells that are observed in both the baseline and the current period.
After restricting attention to $S_{t}$, we renormalize posting shares within the common-support sample. Specifically, for each $c\in S_{t}$, define
$$ \tilde{w}^{(t)}_{ct}=\frac{w_{ct}}{\sum_{j\in S_{t}}w_{jt}},\qquad\tilde{w}^{(t)}_{c,2021}=\frac{w_{c,2021}}{\sum_{j\in S_{t}}w_{j,2021}}. $$
These renormalized weights sum to one within the common support in each period:
$$ \sum_{c\in S_{t}}\tilde{w}^{(t)}_{ct}=1,\qquad\sum_{c\in S_{t}}\tilde{w}^{(t)}_{c,2021}=1. $$
Using these weights, the common-support aggregate exposure in period $t$ is
$$ \bar{E}^{CS,\;renorm}_{t}=\sum_{c\in S_{t}}\tilde{w}^{(t)}_{ct}E_{ct}, $$
and the corresponding baseline object is
$$ \bar{E}^{CS,\;renorm}_{2021}(t)=\sum_{c\in S_{t}}\tilde{w}^{(t)}_{c,2021}E_{c,2021}. $$
Renormalization is useful because, without it, a decomposition on the common support would still reflect not only changes among persistent cells, but also changes in how much total posting mass lies on the common support. By renormalizing, we isolate the change among cells that are observed in both periods.
The renormalized common-support object is not, in general, identical to the raw aggregate exposure. We therefore report the difference between the raw aggregate and the renormalized common-support aggregate as a summary measure of support non-overlap:
$$ \text{Gap}_{t}=\bar{E}_{t}-\bar{E}^{CS,\;renorm}_{t}. $$
A similar comparison can be made for the baseline period using $\bar{E}^{CS,\;renorm}_{2021}(t)$.
We also report two simple overlap diagnostics:
$$ m^{cur}_{t}=\sum_{c\in S_{t}}w_{ct},\qquad m^{base}_{t}=\sum_{c\in S_{t}}w_{c,2021}. $$
Here, $m^{cur}_{t}$ is the share of current-period posting mass that lies on the common support, and $m^{base}_{t}$ is the corresponding share for the 2021 baseline. These measures provide a transparent summary of how much of the posting distribution is comparable across the two periods.
F.1 Diagnostic results for the renormalized common-support decomposition
Table F3 reports diagnostic results for the renormalized common-support decomposition used in the main text. For each quarter, the table reports the raw total change in aggregate exposure, the corresponding renormalized common-support total, and the residual difference between the two, together with the composition, within-cell, and interaction components of the renormalized decomposition. The residual remains small in magnitude throughout, though it becomes somewhat more visible around the mid-2023 turning point. The numerical reconstruction gap is negligible in all quarters, confirming that the renormalized total is exactly accounted for by the three-fold decomposition.
| Quarter | Raw total | Renorm. total | Residual |
|---|---|---|---|
| 2022Q1 | 0.0169 | 0.0170 | -0.0001 |
| 2022Q2 | 0.0130 | 0.0130 | 0.0000 |
| 2022Q3 | 0.0071 | 0.0071 | -0.0001 |
| 2022Q4 | 0.0032 | 0.0033 | -0.0000 |
| 2023Q1 | 0.0028 | 0.0030 | -0.0002 |
| 2023Q2 | -0.0020 | -0.0017 | -0.0003 |
| 2023Q3 | -0.0183 | -0.0179 | -0.0005 |
| 2023Q4 | -0.0181 | -0.0177 | -0.0004 |
| 2024Q1 | -0.0091 | -0.0089 | -0.0002 |
| 2024Q2 | -0.0086 | -0.0083 | -0.0003 |
| 2024Q3 | -0.0062 | -0.0060 | -0.0002 |
| 2024Q4 | -0.0074 | -0.0071 | -0.0002 |
| 2025Q1 | -0.0099 | -0.0097 | -0.0002 |
| 2025Q2 | -0.0200 | -0.0196 | -0.0004 |
- Notes: Raw total is the change in aggregate exposure relative to the 2021 baseline in the original data. Renorm. total is the corresponding change computed on the period-specific common-support sample after renormalizing posting shares. Residual is the difference between the raw total and the renormalized total. By construction, the renormalized total is exactly reconstructed by the sum of the composition, within-cell, and interaction effects; the numerical reconstruction gap is negligible throughout.
Table F3: Diagnostic Decomposition Results under Renormalized Common Support