A27 methods and calculation notes
This is a comparative evidence synthesis. No new empirical estimate, meta-analysis, participant-level replication, college-return estimate or wage forecast is presented.
Different estimands
Cruces et al. randomize assigned AI access for one self-contained business task. Let H and L denote their pre-existing education categories and T assigned assistant access. The main interaction is [E(score | H,T=1) − E(score | L,T=1)] − [E(score | H,T=0) − E(score | L,T=0)]. This is a difference in access effects between education groups among analysed completers. It is not the causal effect of schooling. The score is scaled by the low-education control group's task-score standard deviation. The immediate related follow-up uses its own control-group standard deviation.
Contractor and Reyes randomize AI permission during a learning period for undergraduates. The reported fraction-correct contrasts are intention-to-treat effects of assignment on intended-unaided knowledge tests. Actual AI use, mode of use and study-time choices are endogenous; no treatment-on-the-treated or strategy estimate is used in the article. Their test had five questions immediately and ten about a week later. The percentages are percentage-point changes in the fraction correct, not a fraction of knowledge retained.
Deming analyses historical US worker panels and occupational sorting. His analysis motivates examining access to jobs and wage growth; it does not identify an AI-era response or isolate a signalling effect. The three studies do not estimate a common treatment, population or outcome, so we do not pool them or rank their effect sizes.
Mechanical reconstruction
Numbers are transcribed from Cruces Table 1 (PDF p. 31/printed p. 30) and Contractor/Reyes Table 4 (PDF p. 38); integrity numbers are from the latter's Table 7 (PDF p. 41). Approximate two-sided 95% intervals use estimate ± 1.96 × reported standard error, without access to unrounded regressions or microdata.
| Quantity | Calculation from printed table | Result / interpretation |
|---|---|---|
| Cruces main gap fraction compressed | (0.548 − 0.139) / 0.548 |
0.74635, described as about three quarters; no ratio interval available. |
| Cruces main interaction interval | −0.408 ± 1.96 × 0.122 |
−0.64712 to −0.16888 SD. The printed gap subtraction is −0.409 because of rounding. |
| Cruces treated-gap interval | 0.139 ± 1.96 × 0.087 |
−0.03152 to 0.30952 SD; crosses zero. |
| Cruces follow-up interaction interval | −0.100 ± 1.96 × 0.118 |
−0.33128 to 0.13128 of the follow-up control SD. |
| Contractor/Reyes immediate ITT interval | 100 × (0.067 ± 1.96 × 0.032) |
+0.428 to +12.972 percentage points. |
| Contractor/Reyes later ITT interval | 100 × (0.051 ± 1.96 × 0.023) |
+0.592 to +9.608 percentage points. |
Scroll the table sideways to read all columns.
We do not divide the follow-up gap by the task gap, divide the later student effect by the immediate effect, compare standard-deviation magnitudes across studies, or infer equality from a nonsignificant result. Required covariance information for a confidence interval on either ratio is absent from the printed table, and their denominators/outcomes differ. The confidence intervals are arithmetic approximations to printed standard errors, not a new uncertainty model.
Conceptual routes to earnings
The framework is unestimated. Current task output is q = F(h, a, z), where h is human capability, a effective AI assistance and z task requirements. Cruces varies assigned access to a; its education categories are neither randomly assigned nor exact observations of h.
Future capability can be written h(next) = (1 − δ)h + L(e, a, j), where δ is depreciation, e effort and j the learning setting or job. The sign of the AI effect on L is not imposed. Contractor/Reyes tests later topic knowledge under one fixed-time setting. Whether a person uses AI to augment or automate work is a choice, so use-pattern comparisons are not randomized learning mechanisms.
For a snapshot of observed group labour earnings, let D denote qualification status, A the AI environment, j a mutually exclusive employment state (including the relevant bundle for multiple-job workers), and Y the current rate of labour earnings. The accounting identity is E[Y | D,A] = Σ_j P(j | D,A) × E[Y | j,D,A]. Include nonemployment as a category with zero current labour earnings; this is not a claim that someone currently without work earned nothing earlier in the year. It shows two margins: allocation across jobs and earnings within jobs. It does not identify the causal return to acquiring D. AI could alter output, its observability, employer screening, task prices, worker supply, demand and wage setting separately. Assisted output could affect earnings while unaided capability remains unchanged. Conversely, capability could rise without a pay or access change. Net educational return also requires fees, foregone earnings, completion probability, timing and discounting. No pass-through coefficient or net-return parameter is calibrated from these studies.