Inference for Forecasting Accuracy: Pooled versus Individual Estimators in High-dimensional Panel Data

Panels with large time $(T)$ and cross-sectional $(N)$ dimensions are a key data structure in social sciences and other fields. A central question in panel data analysis is whether to pool data across individuals or to estimate separate models. Pooled estimators typically have lower variance but may suffer from bias, creating a fundamental trade-off for optimal estimation. We develop a new inference method to compare the forecasting performance of pooled and individual estimators. Specifically, we propose a confidence interval for the difference between their forecasting errors and establish its asymptotic validity. Our theory allows for complex temporal and cross-sectional dependence in the model errors and covers scenarios where $N$ can be much larger than $T$-including the independent case under the classical condition $N/T^2 \to 0$. The finite-sample properties of the proposed method are examined in an extensive simulation study.

💡 Research Summary

This paper tackles a fundamental practical question in large‑dimensional panel data analysis: should one pool observations across cross‑sectional units or estimate separate models for each unit when the ultimate goal is forecasting? While the literature on panel homogeneity tests (e.g., Swamy, Phillips‑Sul, Pesaran‑Yamagata) provides powerful tools for detecting slope equality, such tests are often too sensitive for applied work where some degree of heterogeneity is unavoidable. The authors therefore propose a decision‑making framework that directly compares the mean‑squared‑forecast‑error (MSFE) of pooled versus individual ordinary‑least‑squares (OLS) estimators.

The underlying model is linear: (y_{i,t}=x_{i,t}^\prime\beta_i+\varepsilon_{i,t}) for (i=1,\dots,N) and (t=1,\dots,T). Crucially, the slopes (\beta_i) are treated as fixed (non‑random) parameters, avoiding the restrictive random‑coefficient assumptions of Swamy‑type models. Errors are allowed to be heteroskedastic and to exhibit both temporal and cross‑sectional dependence. Formally, the error vector (\varepsilon) has covariance (\Sigma=\Sigma_N\otimes\Sigma_T), where (\Sigma_T) captures serial dependence of a stationary process and (\Sigma_N) captures spatial (cross‑sectional) dependence. This Kronecker structure nests many familiar cases (i.i.d., heteroskedastic, pure spatial dependence) and is considerably more general than the independence assumptions that dominate most panel‑forecasting literature.

The authors define the conditional MSFE for each unit under both estimators, then aggregate across units to obtain (E_{\text{ind}}) and (E_{\text{pool}}). By algebraic manipulation (Lemma 2.2) the difference (\Delta=E_{\text{ind}}-E_{\text{pool}}) is decomposed into three components: (i) a variance term for the individual estimators ((E_1)), (ii) a bias term arising from slope heterogeneity ((E_2)), and (iii) a variance term for the pooled estimator ((E_3)). Lemma 2.4 shows that, under standard regularity conditions, (E_1=O_P(T^{-1})) and (E_3=O_P((NT)^{-1})). Hence, when (N) is large relative to (T), the pooled variance can be dramatically smaller, but the bias term (E_2) may dominate if heterogeneity is substantial.

The asymptotic framework permits both dimensions to diverge, allowing (N\gg T). The key “moderately heterogeneous” regime assumes that the dispersion of the (\beta_i) is not too large, making the bias term of order comparable to the variance terms. Under this regime, the authors establish a Gaussian approximation for the sample analogue (\widehat{\Delta}) (Theorem 2.9). The result holds under strong α‑mixing of the error field with exponential decay ((\alpha(r)\le\psi^{-r})), bounded moments up to order 16, and convergence of the scaled regressor matrices (X_i’X_i/T) to positive‑definite limits (Q_i). Importantly, the classic condition (N/T^2\to0) is relaxed; the theory remains valid even when (N/T^2) diverges, provided the moderate‑heterogeneity assumption holds.

A practical confidence interval for (\Delta) is constructed in equation (2.15) using a consistent estimator of the asymptotic variance. If the interval lies entirely above zero, the pooled estimator is statistically superior in forecasting; if below zero, the individual estimators are preferred.

Monte‑Carlo experiments explore a range of designs: varying (N/T) ratios, different covariance structures ((\Sigma_N,\Sigma_T)), and several levels of slope heterogeneity. The simulations confirm that the proposed confidence interval maintains nominal coverage across all settings, while exhibiting higher power than traditional homogeneity tests in detecting when pooling harms forecast accuracy. The method also performs well when (N) is an order of magnitude larger than (T), a scenario increasingly common in modern macro‑econometric and micro‑panel applications.

In conclusion, the paper delivers the first asymptotically valid inference tool for directly comparing pooled and individual forecasting performance in high‑dimensional panels. By allowing complex dependence, accommodating (N\gg T), and avoiding random‑coefficient assumptions, the approach aligns closely with the practical needs of researchers and policymakers who must decide whether to pool data for prediction. The authors suggest extensions to nonlinear models, endogenous regressors, and model‑selection frameworks as promising avenues for future work.