Why more firms do not always mean more precise estimates of economic shocks

More firms do not automatically mean more information about an economy-wide shock. That is the central message of a theoretical preprint by Mahmut Ablay, Mengshan Xu and Endong Wang, which compares two ways economists estimate how businesses with different characteristics respond to the same disturbance.

The paper, When Does N Buy Precision? Heterogeneous Responses to Aggregate Shocks in Micro Panels, finds that a correctly specified dynamic model can gain precision from both the number of firms and the number of periods observed. A more flexible alternative often remains constrained by the number of periods. The distinction matters for researchers using large administrative datasets with comparatively short histories.

Two approaches, different information limits

The authors compare panel local projections, or LPs, with panel vector autoregressions, or VARs. Both study differences in responses—for example, whether firms with different leverage respond differently to a monetary policy shock.

An LP estimates a separate regression for each future horizon, relating an outcome to the shock interacted with a firm's pre-shock characteristics. A VAR instead estimates a system describing how outcomes evolve from one period to the next, then repeatedly applies that system to construct longer-horizon responses.

The analysis removes persistent firm-specific differences and common period-specific effects. For the VAR, it also uses a split-panel jackknife: estimates from the full time sample and its two halves are combined to remove the leading bias associated with estimating dynamic models with firm effects.

Under the paper's assumptions and a linear exposure function, VAR estimation error shrinks at the rate 1/sqrt(NT), where N is the number of units and T the number of periods. At positive horizons, LP error generally shrinks at 1/sqrt(T). These are large-sample theoretical results, not measured improvements in a completed empirical study.

Why shared shocks remain a bottleneck

The difference comes from what remains unexplained in each regression. In the correctly specified VAR, the one-period error is idiosyncratic: it can be averaged away across both firms and periods.

An LP looking further ahead also encounters aggregate shocks that arrive after the initial disturbance. Firms differ in their exposure to those later shocks. Period effects remove the component common to all firms, but not these differing exposures.

Adding firms measures the cross-sectional pattern more precisely; it does not create additional realizations of the shared shock. Averaging across time is therefore what reduces the leading uncertainty.

There is an important exception at impact, before later shocks enter the projection error. The paper defines the bias-corrected impact LP estimator to coincide with the corresponding VAR estimator, allowing both to attain the faster rate under the stated conditions. The slower positive-horizon LP result also requires a nonvanishing aggregate uncertainty component.

Flexibility can come without a slower leading rate

The authors extend the comparison beyond linear exposure. They approximate an unknown smooth relationship between firm characteristics and shock exposure using a series of basis functions, while retaining the model's common linear propagation dynamics.

For the VAR, learning that relationship introduces the usual nonparametric cost: its integrated error rate depends on both smoothness and the number of characteristics. Where the required growth conditions hold, the rate becomes (NT)^(-r/(2r+m)), with r describing smoothness and m the state dimension.

LPs also incur approximation and coefficient-estimation errors. But with sufficient smoothness and enough firms relative to periods, those costs can be smaller than their existing time-series uncertainty. Their leading integrated error rate then remains 1/sqrt(T). That does not establish equal variances, or automatically guarantee the same behavior for estimates at particular characteristic values.

Precision depends on getting the dynamics right

The VAR advantage is conditional. The main results assume stationary processes, shocks independent of the aggregate past, and independent firm-level innovations. The faster linear VAR result also requires N to grow more slowly than T squared.

Omitted lags or variables, and delayed shock effects excluded from the fitted recursion, can bias VAR responses. Under the alternatives analyzed, LPs remain consistent. An unobserved aggregate shock with heterogeneous exposure can also eliminate the VAR's faster rate without making it inconsistent.

The authors propose a Hausman-type test comparing LP and VAR estimates. A sufficiently large discrepancy provides evidence against the maintained dynamic model. The test has theoretical power against the specified alternatives when their response estimates diverge; non-rejection is not a guarantee that every assumption holds.

The empirical application remains in progress. The paper describes linking Turkish administrative records into a firm-month panel spanning January 2009 to April 2025, but reports no completed response estimates. Its present contribution is methodological—not evidence about which Turkish firms monetary policy affects most.