Between EstimatorEdit

The between estimator is a technique in panel data analysis that isolates the variation that exists across entities (for example, individuals, firms, or countries) over time by using averages. It stands alongside other panel data estimators such as the within estimator and the first-difference estimator, offering a way to examine how differences between entities relate to differences in outcomes when time-series movement within each entity is not the focus of interest.

In practice, the between estimator regresses the cross-sectional means of the dependent variable on the cross-sectional means of the regressor. Concretely, for a model of the form y_it = α_i + β x_it + ε_it, where α_i captures time-invariant, entity-specific effects, the between estimator looks at the relation between ȳ_i = (1/T) Σ_t y_it and x̄_i = (1/T) Σ_t x_it across i. The estimated β from this between-regression reflects how differences in average regressor values across entities are associated with differences in average outcomes across those entities.

Overview and intuition - What it uses: variation between entities, aggregated over time. - What it ignores: within-entity variation over time. If a lot of the signal about y is driven by how a given entity changes over time, the between estimator may miss that story. - When it is appealing: when the research question centers on cross-sectional differences between entities and when time-invariant unobserved heterogeneity plays a role that the researcher is willing to tolerate in a cross-sectional sense. - When it is risky: when the unobserved, time-invariant effects α_i are correlated with the regressor x_it, because that correlation can bias the estimated β.

Assumptions and interpretation - Consistency hinges on the relationship between the unobserved entity effects and the regressor. If the average α_i across entities is uncorrelated with the averages x̄_i, the between estimator can be consistent. If there is a systematic correlation—such as unobserved factors that both differ across entities and influence x_it—bias can arise. - The estimator is easy to compute and interpret in a straightforward cross-sectional sense: it tells you how differences in average x across entities relate to differences in average y across entities, holding constant the notion that some entity-specific character does not shift the slope in a way captured by within variation. - It is distinct from the within estimator, which leverages deviations from an entity’s own mean to identify β, effectively removing the effect of all time-invariant entity characteristics from the estimation. The within estimator and the between estimator can give different pictures of the same underlying relationship because they rely on different sources of variation: within versus between.

Comparisons with other estimators - Within estimator: The within estimator eliminates α_i by demeaning y_it and x_it within each entity, so it identifies β based on how an entity’s deviations from its own average co-move over time. This can be more robust to time-invariant omitted variables but discards cross-entity information and can be inefficient if there is substantial between-entity variation that is informative about the relationship. - First-difference estimator: This approach uses year-to-year changes, effectively differencing out α_i. It shares the goal of removing fixed effects, but the utilization of differences can amplify measurement error and is sensitive to the exact time structure of the data. - Against pooled OLS: Pooled OLS ignores the panel structure entirely, which can bias results if α_i is correlated with x_it. The between estimator sits between pooled OLS and the within-type approaches, exploiting between-entity variation without discarding all time-series information. - Mundlak and related devices: A common move is to augment the model with the cross-sectional mean of the regressor, x̄_i, turning the model into a hybrid that separates between and within components. This Mundlak-style device helps address concerns about correlation between α_i and x_it by explicitly modeling that correlation.

Practical considerations - Data requirements: The between estimator relies on observing multiple time periods per entity to form reliable averages. If T is small, the between estimates can be unstable and more sensitive to outliers. - Heterogeneity and interpretation: If entities differ in unobserved ways that affect both y and x, the between estimate may conflate genuine cross-entity causal relationships with the effects of those unobserved factors. Researchers often complement the between estimator with within-based analyses to triangulate the signal. - Robustness to measurement error: Aggregating over time can sometimes reduce the impact of high-frequency measurement error in x_it, potentially stabilizing the estimated relationship across entities.

Controversies and debates - Causal interpretation: A central tension is whether a between-estimator-derived β should be interpreted causally. Critics point out that cross-entity differences often reflect pre-existing structural differences rather than effects of changes in x across entities. Proponents argue that, in certain contexts where policy or structural differences are of primary interest, between estimates reveal meaningful, cross-sectional relationships that are robust to short-run fluctuations. - Comparing estimators: Debates persist about when to prefer between, within, or first-difference approaches. Proponents of the between estimator emphasize leveraging stable, cross-entity variation, while critics stress the risk of omitted-variable bias from unobserved, time-invariant heterogeneity that is correlated with x_it. - Hybrid modeling: The Mundlak-style hybrid approach—explicitly modeling the relationship between α_i and x̄_i—has become a practical compromise in many applications. It allows researchers to decompose effects into between- and within-entity components and to test whether unobserved heterogeneity is correlated with the regressor.

See also - panel data - within estimator - first-difference estimator - fixed effects and random effects - Mundlak's device - causal inference