The Gap Has Two Margins

The linked application records reveal a puzzle about the conversion to paid employment: a forecast can miss whether employer entry happens, and it can miss the scale reached after entry. Separating those two margins makes the evidence easier to interpret. It shows why a large population of applications does not translate directly into a large population of employers, and why the average gap can point in a different direction from the gap among businesses that succeed in entering.

Entry errors: who crosses the employer threshold?

The first question is binary in form, even though the process behind it is not: did the applicant expect to become an employer, and did the linked record later show positive employment within the relevant horizon? The study defines expected entry from a positive expected number of employees. Realized entry occurs when the linked business records positive employment in the matched one-year window. These two indicators create four groups, each describing a different relationship between a plan and an outcome.

The largest group is the first one. Most applications in the linked sample neither report positive expected employment nor become employers within the window. That concentration at zero is important for interpreting every average that follows. It does not mean that 84.1 percent of applicants lacked a serious idea, nor does it prove that all of them abandoned the business. Some likely intended nonemployer activity; some may have delayed hiring; some may never have operated. What the data establish is narrower: the study does not observe positive paid employment for them within its employer-entry definition.

The two off-diagonal groups reveal the directional error. In 8.9 percent of applications, the applicant expected employer entry but did not realize it in the window. In 3.6 percent, the outcome went the other way: employer entry occurred even though the application reported no expected employees. The mean entry-expectation error is therefore the first share minus the second, 0.089 - 0.036 = 0.053, or 5.3 percentage points. If the question is whether the prediction was wrong in either direction, the two off-diagonal groups add to 12.5 percent.

These are different summaries. The 5.3-point figure says that expected employer entry exceeds unexpected employer entry in the aggregate. The 12.5-percent figure says how often the forecast and the observed entry state disagree, regardless of direction. Neither is a universal probability that any future application will fail. Both depend on the population and the study’s comparison window.

The timing pattern reinforces the same conclusion. Expected cumulative entry rises rapidly after application and reaches about 12.5 percent by the sixth quarter, after which applicants no longer anticipate a first wage payment in the measure used for the comparison. Realized entry accumulates more slowly and continues after that point. Some late entries narrow the gap, but the early difference persists. Applicants are therefore not merely describing a pool of employers that already exists; they are making a forecast about when an employer will materialize, and that forecast is systematically early or optimistic relative to the observed path.

A useful way to read the four groups is to treat the application as a classification signal rather than a certificate. A positive expectation selects a subset that is more likely to enter than someone who reports zero, but the selected group still contains many non-entrants. A zero expectation lowers the predicted chance of entry, but it does not make entry impossible. This is the logic of a noisy classifier: it can separate groups on average while producing false positives and false negatives.

That logic matters for public interpretation. Suppose an analyst counts every application with a positive hiring expectation as an impending employer. The 8.9 percent expected-but-not-observed group shows why that count is too high even before the analyst asks how many people each business will hire. Suppose instead the analyst counts only applicants who correctly predict entry. The 3.6 percent unexpected-entry group shows why a positive plan is not the only route into paid employment. A policy that allocates attention or support from applications alone will necessarily work with incomplete information.

The zero-heavy distribution also explains why application totals and employer totals can look dramatically different without either data source being defective. Applications measure a decision to take an administrative step and report a plan. Employer records measure a later state that requires actual paid employment. The space between them is an observed mismatch that may reflect timing, changed plans, nonemployer choices, or other factors the entry table does not identify. It does show that the conversion is not automatic.

The distinction between “not observed” and “did not exist” is especially important here. A four-quarter outcome window cannot establish that a business never became an employer. A late entrant may fall outside the window; a nonemployer business may continue successfully without hiring; and a plan may be postponed rather than abandoned. The paper’s evidence supports a statement about employer entry during the measured period, not a permanent verdict on the applicant or the business.

The classification can be represented as a path from the forecast to the realized state:

The lower part of the diagram introduces the second margin without treating it as a footnote. Only the groups with realized employer entry can be compared directly on realized employment as an entrant outcome. The entry decision is the gate; scale is the outcome conditional on passing through that gate. Keeping the gate and the scale separate is what turns a confusing average into an interpretable result.

Scale errors: how much employment follows entry?

The intensive margin asks how the expected maximum number of employees compares with the maximum quarterly employment actually observed. Across all applications, applicants expect 0.530 employees on average and realize 0.388, for an average overprediction of 0.142 employees. That overall figure is not the average experience of a typical entrant. It is an average across the entire application population, including the large group with zero expected and realized employment.

The group breakdown shows where the positive average comes from. Applicants who expected employer entry but did not enter overpredicted by 4.087 employees on average: their expected employment was 4.087 and their realized employment was zero. Applicants who did not expect entry but nevertheless entered underpredicted by 4.462: their expected employment was zero and their realized employment was 4.462. These two groups contribute errors in opposite directions, just as they did at the entry margin. Their shares are not equal, however, and the positive-error group is larger, so the entry failures contribute more to the aggregate overprediction.

The group that both expected and achieved entry produces the apparent reversal. Its average expected employment was 4.779, while average realized employment was 6.683. The mean error—expected minus realized—was therefore -1.903, an underprediction. If we looked only at these successful entrants, we could conclude that applicants tend to be too conservative. If we looked at the full application population, we would conclude that applicants overpredict. Both statements are supported because they refer to different populations selected by different conditions.

Selection helps explain the reversal. To be in the group that actually enters, a business must realize enough value and employment to cross the observed threshold during the window. Among people who predicted entry, those whose outcomes receive a favorable shock are more likely to appear in the successful group. A group selected on positive realized outcomes will tend to look better than the broader set of people who made the same forecast. The average forecast among those successful entrants can consequently sit below their realized employment even while the initial applicant pool, dominated by failures and zeros, overestimates.

The successful group is not homogeneous either. Within the 3.4 percent that both expected and achieved entry, 47 percent underpredicted by 6.237 employees, 24 percent overpredicted by 3.872, and 29 percent forecast employment exactly. The exact-forecast subgroup had relatively small expected and realized employment, around two employees on average. The broad average of -1.903 therefore conceals at least three kinds of experience: businesses that grew beyond the plan, businesses whose plan overshot, and small operations whose scale was easier to anticipate.

This is why the two margins should not be compressed into a single “forecast accuracy” score. Consider two applications that both predict four employees. One never becomes an employer; its scale error is four employees because realized employment is zero. The other becomes an employer and reaches eight employees; its error is negative four because the plan understated the realized maximum. The numerical distance from expectation may be the same, but the business states and practical implications are different. One is an entry problem; the other is a scale-surprise problem.

The evidence also shows that expectations do contain information about the ordering of outcomes. The correlation between expected and realized employment is 0.30 across all applications and 0.32 among applications with positive expected employment. Higher expectations are associated with higher realized employment, but the correlations are far from one. An analyst using expectations to rank applications might obtain useful information. An analyst treating the expected number as a reliable headcount would make a much stronger claim than the correlation supports.

Correlation and calibration answer different questions. Correlation asks whether applications with higher expectations tend to have higher realized employment. Calibration asks whether the level of the expectation matches the level of the outcome without systematic bias. A forecast can preserve the ranking—one applicant predicts more than another and later hires more—while being too high for both. It can also have a positive average relationship while producing large errors for particular groups. The reported correlations establish predictive association, not accurate one-for-one forecasting.

The variance decomposition adds another perspective. For all applications, the variance of the employment error is 12.69, and the variance of expected employment accounts for 75 percent of that error variance, compared with 68 percent for realized employment; the covariance contribution is -43 percent. Among applications with positive expected employment, the error variance rises to 69.37, the expectation component accounts for 88 percent, the realized-employment component for 58 percent, and the correlation is 0.32. These shares are parts of an algebraic decomposition and do not mean that expectations cause the error. They do indicate that variation in what applicants forecast is central to the variation in the eventual gap.

The measurement design places limits on the interpretation of the scale numbers. Expected and realized maximum employment are winsorized at 100 employees, affecting fewer than one percent of observations. Realized employment is measured quarterly, while an applicant may imagine a monthly or even more frequent path. The researchers examine conservative upper bounds for finer measurement frequencies and find that the main conclusions remain because employment within firms is highly persistent across quarters. That robustness supports the comparison, but it does not turn a quarterly maximum into a complete account of a business’s workforce.

Nor should these group averages be read as effects of optimism, planning, or legal form. They are conditional sample comparisons. The fact that an applicant predicted four employees does not mean the prediction caused a later hiring decision, and the fact that a successful entrant underpredicted does not prove that conservative forecasts produce growth. The data reveal how expectations and outcomes line up. They do not, at this stage, identify whether the gap arose from information, belief formation, demand, financing, execution, or a shock after filing.

The central empirical lesson is therefore more precise than “entrepreneurs are too optimistic.” At the extensive margin, expected employer entry exceeds realized entry because expected entrants often do not cross the paid-employment threshold. At the intensive margin, businesses that do cross the threshold can realize more employment than they expected, while the full applicant population still shows aggregate overprediction. Expectations are useful enough to carry information about outcomes, but noisy enough that their level cannot be read as a promised workforce.

That combination creates the next analytical problem. If the forecast is neither empty nor accurate, what is inside it? The answer may include information about business quality, but also heterogeneous priors, selection into application, and beliefs that do not track outcomes one-for-one. Understanding that mechanism is necessary before assigning the gap to a single trait such as optimism.

References

• Expectations versus Reality in Business Formation