Cross-Section Estimation of Long-Run Relations Using Time-Compressed Data

Many empirical investigations of long-run relations are based on cross-section regressions in averaged or long differenced data that effectively have the time dimension of a T × N panel compressed. We analyze a class of time-compressed I(1) data and show that they have magnified variability stemming from the fact that the cross-section variance of a non-stationary panel ‘fans out’ with time. Cross-section regressions in time compressed data can potentially yield estimates that are super-consistent and asymptotically normal, whether the regressors are stationary, non-stationary, or highly persistent. The fastest convergence rate of √NT requires a compression scheme that not only magnifies the non-stationary signal, but also dilutes the regression noise. Omitted fixed effects preclude noise dilution but the estimates remain superconsistent. However, the fanning out effect can be weakened when the data have a strong force for mean-reversion or convergence, a problem that seems relevant for temperature data. We consider three applications and find that the long-run relation between consumption and income, and between growth/inflation and demographic variables are reasonably well determined, but the estimated relation between growth and warming temperature is fragile.