Estimating Economic Responses When the Records Have Gaps

Economists often follow many firms, households or countries through time to ask how an outcome changes after an economic event. But records rarely arrive as a complete grid. Firms enter and leave databases, report at different frequencies or have gaps in individual variables. At a given horizon, the outcome, current predictor and earlier controls may never all be available together.

In a November 2026 preprint, economist Yubo Tao proposes a way to estimate certain relationships from such incomplete panels. The method does not make missingness harmless: it relies on information in other, fully observed series and on restrictions about the data that cannot be guaranteed simply by having a large sample.

Why pairs can be enough

The paper studies panel local projections, which estimate a relationship separately at each future horizon. For example, a researcher might estimate how investment differs between firms with high and low exposure to geopolitical risk six months after a change in that risk.

A conventional regression uses complete cases: dates on which the outcome and every required predictor are observed together. Tao points out that the equations defining the regression coefficient can instead be expressed in terms of products of pairs of variables. Those pairs need not all be observed on the same dates.

Imagine a predictor and outcome are missing every third date, while the regression needs observations from three adjacent dates. No date supplies a complete regression case. Yet each required pair can still be observed repeatedly. Under the paper's assumptions, those pairs can help recover the projection for the intended period, even though an ordinary complete-case regression cannot be computed.

Restoring more than an average

The proposed estimator uses a factor model: it represents each series as a unit-specific baseline, a response to shared time-varying factors, and a remaining unit-specific component. Fully observed auxiliary series must contain enough information about every relevant shared factor. The observed portions of each target series must also contain enough variation to estimate how it relates to those factors.

The fitted model supplies estimates of missing conditional means. But filling gaps with means alone would miss variation in the remaining unit-specific components. Tao therefore also estimates how those components move together from observed pairs, restoring their contribution to the quantities used by the projection. The calculation preserves observed values rather than replacing them all with fitted ones.

The paper calls the bias-corrected version MRLP-BC. Its corrections address biases associated with removing unit averages and estimating the factor model. To quantify uncertainty, Tao develops a localized multiplier bootstrap, which uses correlated random weights on an estimated statistical score. Under additional dependence restrictions, it supports intervals at individual horizons and simultaneous bands across a fixed set of horizons.

These results have important conditions. The required pairs must be observed often enough for each unit, and their remaining within-unit covariances must be stable over time after conditioning on the common component and observation pattern. Missingness driven by an unobserved unit-specific disturbance can violate the restrictions. Quarterly reports, for example, may help identify a monthly conditional mean without supplying the adjacent-month pairs needed for an unrestricted lagged covariance.

Simulations and an investment application

In simulations, Tao compares the method with alternatives under several missing-data patterns. In the reported periodic-gap designs, the proposed estimator can be evaluated even when no complete regression case remains. In a separate common-coefficient design, its accuracy is similar to a split-panel correction when histories are longer. Results are not uniformly better: some shorter-panel simulations show confidence-interval undercoverage.

The empirical application revisits earlier work on geopolitical risk and firm investment. Its monthly analysis estimates a trailing-three-month investment-intensity measure between quarterly reporting dates, using monthly financial-market series as auxiliary information. It is not an estimate of investment spending in each individual month.

For a 10% increase in the geopolitical risk measure, the estimated six-month difference between high- and low-exposure firms becomes more negative as the nested samples expand: from minus 0.075 percentage points in the 232-firm complete-quarterly-history sample to minus 0.246 points among 1,174 eligible firms. The latter estimate has a nominal 95% pointwise interval of minus 0.371 to minus 0.121 points.

That pattern does not establish that the groups' underlying responses differ across samples: the firms and their projection dates change together. A sensitivity restriction requiring more observed factor variation produces a smaller six-month estimate, minus 0.106 points, with an interval including zero. The paper also notes finite-sample coverage concerns and that the sensitivity intervals do not account for selection using estimated factors.

The method recovers a specified statistical projection under its assumptions; it does not by itself establish that a change in geopolitical risk caused the estimated investment response.