We consider a panel cointegration model with latent group structures that allows for heterogeneous long-run relationships across groups. We extend Su, Shi, and Phillips (2016,Econometrica84(6), 2215–2264) classifier-Lasso (C-Lasso) method to the nonstationary panels and allow for the presence of endogeneity in both the stationary and nonstationary regressors in the model. In addition, we allow the dimension of the stationary regressors to diverge with the sample size. We show that we can identify the individuals’ group membership and estimate the group-specific long-run cointegrated relationships simultaneously. We demonstrate the desirable property of uniform classification consistency and the oracle properties of both the C-Lasso estimators and their post-Lasso versions. The special case of dynamic penalized least squares is also studied. Simulations show superb finite sample performance in both classification and estimation. In an empirical application, we study the potential heterogeneous behavior in testing the validity of long-run purchasing power parity (PPP) hypothesis in the post–Bretton Woods period from 1975–2014 covering 99 countries. We identify two groups in the period 1975–1998 and three groups in the period 1999–2014. The results confirm that at least some countries favor the long-run PPP hypothesis in the post–Bretton Woods period.

中文翻译：

---

识别协同面板中的潜在分组模式

我们考虑具有潜在组结构的面板协整模型，该模型允许跨组的异构长期关系。我们扩展了 Su、Shi 和 Phillips（2016，计量经济学84(6), 2215–2264) 分类器-套索 (C-Lasso) 方法适用于非平稳面板，并允许模型中的平稳和非平稳回归器都存在内生性。此外，我们允许平稳回归量的维度随样本大小而发散。我们表明，我们可以识别个人的群体成员资格并同时估计群体特定的长期协整关系。我们展示了统一分类一致性的理想属性以及 C-Lasso 估计器及其后 Lasso 版本的预言属性。还研究了动态惩罚最小二乘的特殊情况。模拟显示了在分类和估计中极好的有限样本性能。在经验应用中，我们研究了 1975 年至 2014 年布雷顿森林后时期测试长期购买力平价 (PPP) 假设有效性的潜在异质行为，涵盖 99 个国家。我们在 1975-1998 年期间确定了两组，在 1999-2014 年期间确定了三组。结果证实，至少有一些国家支持后布雷顿森林时期的长期购买力平价假说。