This paper provides a novel mechanism for identifying and estimating latent group structures in panel data using penalized techniques. We consider both linear and nonlinear models where the regression coefficients are heterogeneous across groups but homogeneous within a group and the group membership is unknown. Two approaches are considered -penalized profile likelihood (PPL) estimation for the general nonlinear models without endogenous regressors, and penalized GMM (PGMM) estimation for linear models with endogeneity. In both cases we develop a new variant of Lasso called classifier-Lasso (C-Lasso) that serves to shrink individual coefficients to the unknown group-specific coefficients. CLasso achieves simultaneous classification and consistent estimation in a single step and the classification exhibits the desirable property of uniform consistency. For PPL estimation C-Lasso also achieves the oracle property so that group-specific parameter estimators are asymptotically equivalent to infeasible estimators that use individual group identity information. For PGMM estimation the oracle property of C-Lasso is preserved in some special cases. Simulations demonstrate good finite-sample performance of the approach both in classification and estimation. Empirical applications to both linear and nonlinear models are presented.JEL Classification: C33, C36, C38, C51

In this appendix, we state and prove some technical lemmas that are used in the proofs of the main results in Section 2. We first state an exponential inequality for strong mixing processes.LEMMA S1.1: Let {ζ t t = 1 2 } be a zero-mean strong mixing process, not necessarily stationary, with the mixing coefficients satisfying α(τ) ≤ c α ρ τ for some c α > 0 and ρ ∈ (0 1). If sup 1≤t≤T |ζ t | ≤ M T , then there exists a constant C 0 depending on c α and ρ such that for any T ≥ 2 and > 0,PROOF: Merlevède, Peilgrad, and Rio (2009, Theorem 2) proved (i) under the condition α(τ) ≤ exp(−2cτ) for some c > 0. If c α = 1, we can take ρ = exp(−2c) and apply the theorem to obtain the claim. Q.E.D.The above lemma is used in the proof of the following lemma.LEMMA S1.2: Let ξ(w it ; φ) be a R d ξ -valued function indexed by the parameter φ ∈ Φ, where Φ is a convex compact set in R p+1 and E[ξ(w it ; φ)] = 0 for all i, 1 Acknowledgments made in the leading footnote of the main paper apply also to this supplement.

This paper provides a novel mechanism for identifying and estimating latent group structures in panel data using penalized techniques. We consider both linear and nonlinear models where the regression coefficients are heterogeneous across groups but homogeneous within a group and the group membership is unknown. Two approaches are considered -penalized profile likelihood (PPL) estimation for the general nonlinear models without endogenous regressors, and penalized GMM (PGMM) estimation for linear models with endogeneity. In both cases we develop a new variant of Lasso called classifier-Lasso (C-Lasso) that serves to shrink individual coefficients to the unknown group-specific coefficients. CLasso achieves simultaneous classification and consistent estimation in a single step and the classification exhibits the desirable property of uniform consistency. For PPL estimation C-Lasso also achieves the oracle property so that group-specific parameter estimators are asymptotically equivalent to infeasible estimators that use individual group identity information. For PGMM estimation the oracle property of C-Lasso is preserved in some special cases. Simulations demonstrate good finite-sample performance of the approach both in classification and estimation. Empirical applications to both linear and nonlinear models are presented.JEL Classification: C33, C36, C38, C51

The Hodrick-Prescott (HP) filter is one of the most widely used econometric methods in applied macroeconomic research. The technique is nonparametric and seeks to decompose a time series into a trend and a cyclical component unaided by economic theory or prior trend specification. Like all nonparametric methods, the HP filter depends critically on a tuning parameter that controls the degree of smoothing. Yet in contrast to modern nonparametric methods and applied work with these procedures, empirical practice with the HP filter almost universally relies on standard settings for the tuning parameter that have been suggested largely by experimentation with macroeconomic data and heuristic reasoning about the form of economic cycles and trends. As recent research (Phillips and Jin, 2015) has shown, standard settings may not be adequate in removing trends, particularly stochastic trends, in economic data. This paper proposes an easy-to-implement practical procedure of iterating the HP smoother that is intended to make the filter a smarter smoothing device for trend estimation and trend elimination. We call this iterated HP technique the boosted HP filter in view of its connection to L 2 -boosting in machine learning. The paper develops limit theory to show that the boosted HP (bHP) filter asymptotically recovers trend mechanisms that involve unit root processes, deterministic polynomial drifts, and polynomial drifts with structural breaks, thereby covering the most common trends that appear in macroeconomic data and current modeling methodology. In doing so, the boosted filter provides a new mechanism for consistently estimating multiple structural breaks even without knowledge of the number of such breaks. A stopping criterion is used to automate the iterative HP algorithm, making it a data-determined method that is ready for modern data-rich environments in economic research. The methodology is illustrated using three real data examples that highlight the differences between simple HP filtering, the data-determined boosted filter, and an alternative autoregressive approach. These examples show that the bHP filter is helpful in analyzing a large collection of heterogeneous macroeconomic time series that manifest various degrees of persistence, trend behavior, and volatility.The principle adopted here in the construction of a trend for a time series consists in minimizing a linear combination of two sums of squares, of which one refers to the second differences of the trend values, the other to the deviations of the observations from the trend values ... this procedure seems a particularly natural one when dealing with economic time series. The resulting family of trends may be described as quasi-linear trends. Leser (1961) Our statistical approach does not utilize standard time series analysis. The maintained hypothesis based on growth theory considerations is that the growth component of aggregate economic times series varies smoothly over time. Hodrick and Prescott (1997)

We propose a procedure of iterating the HP filter to produce a smarter smoothing device, called the boosted HP (bHP) filter, based on L 2 -boosting in machine learning. Limit theory shows that the bHP filter asymptotically recovers trend mechanisms that involve integrated processes, deterministic drifts, and structural breaks, covering the most common trends that appear in current modeling methodology. A stopping criterion automates the algorithm, giving a data-determined method for data-rich environments. The methodology is illustrated in simulations and with three real data examples that highlight the differences between simple HP filtering, the bHP filter, and an alternative autoregressive approach.