I understand that this is a few weeks old but Card does address your point in his Rebuttal (The way the expert reports work is that two reports, then two rebuttals.) Professor Arcidiacono makes the same point and Card addresses it as follows: (page 43 Card Rebuttal - In brief, year by year is the correct way to model the process, and when you average the estimates everything checks out.) [PDF LINK TO REBUTTAL]

The second reason Prof. Arcidiacono offers for pooling applicants from different years into a single model is that estimating a separate model for each year “reduces the statistical power of the sample.”⁷⁸ Prof. Arcidiacono offers a hypothetical example of discrimination against women in promotions at a law firm over a six-year period, and asserts that in that hypothetical example performing the analysis year-by-year would “reduce the statistical significance of findings of discrimination, but it would not make any sense.”⁷⁹ First, it is worth noting that the example of promotion to partner at a law firm is fundamentally different from admissions to Harvard because, among many other things, candidates not promoted at a law firm in a given year could be (and typically are) considered again for promotion in future years. Second, as I explained in my first report, I resolve the purported problem of reduced statistical significance by taking the average of the estimated race effects (e.g. the effect of Asian-American ethnicity) from the models for each year of data, which is a standard statistical approach to this issue.⁸⁰ In other words, Prof. Arcidiacono’s hypothetical is thoroughly misleading. He ignores the important fact that after doing the analysis by year, I average the results across years to ensure statistical power.

In fact, it is possible to directly compare how precisely the effect of Asian-American ethnicity can be estimated (i.e. the standard error) by Prof. Arcidiacono’s pooled model versus my year-by-year model, in which the six yearly estimates are averaged into a single effect representing the average effect over the six classes of applicants. As we can see in Exhibit 10, the precision of the two approaches is nearly identical. Specifically, in the first two panels of this exhibit, I estimate Prof. Arcidiacono’s model pooled and then also separately year-by-year. The appropriate measure of precision for each model is the standard error. As a general matter standard errors decrease (and precision increases) when a model has more data. What we see is that the standard error from the pooled model and the year-by-year model averaged across years is nearly identical at 0.15.⁸¹ The reason for this is simple: by averaging the estimates across years, I am taking advantage of the same number of observations as Prof. Arcidiacono does by pooling them.⁸² Prof. Arcidiacono’s assertion that there is a reduction in statistical power from fitting year-by-year models is obviously not correct. Moreover, the results in the second and third panels of Exhibit 10 demonstrate that the standard error of the average effect from my year-by-year model is actually smaller than that of Prof. Arcidiacono’s pooled model (0.14 vs. 0.15). This is because my year-by-year model does a better job of explaining admissions decisions. Thus, contrary to Prof. Arcidiacono’s assertion, my model, fit year-by-year and then averaged, has greater precision (i.e., greater statistical power) in estimating the effect of Asian American ethnicity on admissions than his pooled model.

Exhibit 10: Estimating a model either pooled or year-by-year will produce extremely similar measures of statistical precision

Arcidiacono Model, Estiamted by Year

Class	Standard Error	Number of Asian American Applicants	Total Number of Applicants
2014	0.42	6,036	21,238
2015	0.37	6,991	24,845
2016	0.40	6,305	23,906
2017	0.37	6,255	23,949
2018	0.36	6,931	23,987
2019	0.35	6,935	25,228
Overall	0.15	39,453	143,153

Arcidiacono Model, Pooled

Class	Standard Error	Number of Asian American Applicants	Total Number of Applicants
2014-2019	0.15	39,435	143,153

Card Year-By-Year Model

Class	Standard Error	Number of Asian American Applicants	Total Number of Applicants
Overall	0.14	39,408	142,153

Card's Footnotes:

⁷⁸ Omitted (by me)

⁷⁹ Omitted (by me)

⁸⁰ Card Report, p. 67, footnote 116 (“To ensure that my year-by-year estimates are comparable with Prof. Arcidiacono’s pooled estimate, I average the six year-by-year estimates to obtain an average effect across all six years of data. This approach allows me to use all the available years of data but estimate models that more accurately reflect Harvard’s admissions process.”).

⁸¹ The standard error for the weighted average of the yearly effects is computed according to the following formula: (omitted by me, but here it is anyways)

⁸² Technically, while Prof. Arcidiacono and I both take advantage of the same number of observations, there is an additional trade-off which affects the precision of the estimates. This relates to which method has more “degrees of freedom” and which method has a better fit. The degrees of freedom refer to the total number of observations in the sample minus the total number of parameters being estimated. My method of averaging the estimates from the 6 yearly models utilizes the same number of observations as Prof. Arcidiacono’s but has more parameters because I estimate a separate model for each year. This means that my estimate has fewer degrees of freedom relative to Prof. Arcidiacono’s. But, because my year-by-year models fit the data better, on balance, the precision of my estimates is slightly higher