Testing for common trends in semi‐parametric panel data models with fixed effects

Abstract

Summary This paper proposes a non-parametric test for common trends in semi-parametric panel data models with fixed effects based on a measure of non-parametric goodness-of-fit (R2). We first estimate the model under the null hypothesis of common trends by the method of profile least squares, and obtain the augmented residual which consistently estimates the sum of the fixed effect and the disturbance under the null. Then we run a local linear regression of the augmented residuals on a time trend and calculate the non-parametric R2 for each cross-section unit. The proposed test statistic is obtained by averaging all cross-sectional non-parametric R2s, which is close to 0 under the null and deviates from 0 under the alternative. We show that after appropriate standardization the test statistic is asymptotically normally distributed under both the null hypothesis and a sequence of Pitman local alternatives. We prove test consistency and propose a bootstrap procedure to obtain P-values. Monte Carlo simulations indicate that the test performs well in finite samples. Empirical applications are conducted exploring the commonality of spatial trends in UK climate change data and idiosyncratic trends in OECD real GDP growth data. Both applications reveal the fragility of the widely adopted common trends assumption.