Zhipeng Liao

Adaptive GMM Shrinkage Estimation with Consistent Moment Selection

This paper proposes a GMM shrinkage method to efficiently estimate the unknown parameters identified by some moment restrictions, when there is another set of possibly misspecified moment conditions. We show that our method enjoys oracle-like properties, i.e. it consistently selects the correct moment conditions in the second set and at the same time, its estimator achieves the semi-parametric efficiency bound implied by all correct moment conditions. For empirical implementation, we provide a simple data-driven procedure for selecting the tuning parameters of the penalty function. We also establish oracle properties of the GMM shrinkage method in the practically important scenario where the moment conditions in the first set fail to strongly identify the structural parameters. The simulation results show that the method works well in terms of correct moment selection and the finite sample properties of its estimators. As an empirical illustration, we apply our method to estimate the life-cycle labor supply equation studied in MaCurdy (1981) and Altonji (1986). Our empirical findings support the validity of the IVs used in both papers and confirm that wage is an endogenous variable in the labor supply equation.

Select the Valid and Relevant Moments: An Information-Based LASSO for GMM with Many Moments

This paper studies the selection of valid and relevant moments for the generalized method of moments (GMM) estimation. For applications with many candidate moments, our asymptotic analysis accommodates a diverging number of moments as the sample size increases. The proposed procedure achieves three objectives in one-step: (i) the valid and relevant moments are distinguished from the invalid or irrelevant ones; (ii) all desired moments are selected in one step instead of in a stepwise manner; (iii) the parameters of interest are automatically estimated with all selected moments as opposed to a post-selection estimation. The new method performs moment selection and efficient estimation simultaneously via an information-based adaptive GMM shrinkage estimation, where an appropriate penalty is attached to the standard GMM criterion to link moment selection to shrinkage estimation. The penalty is designed to signal both moment validity and relevance for consistent moment selection. We develop asymptotic results for the high-dimensional GMM shrinkage estimator, allowing for non-smooth sample moments and weakly dependent observations. For practical implementation, this one-step procedure is computationally attractive.

Automated Estimation of Vector Error Correction Models

Model selection and associated issues of post-model selection inference present well known challenges in empirical econometric research. These modeling issues are manifest in all applied work but they are particularly acute in multivariate time series settings such as cointegrated systems where multiple interconnected decisions can materially affect the form of the model and its interpretation. In cointegrated system modeling, empirical estimation typically proceeds in a stepwise manner that involves the determination of cointegrating rank and autoregressive lag order in a reduced rank vector autoregression followed by estimation and inference. This paper proposes an automated approach to cointegrated system modeling that uses adaptive shrinkage techniques to estimate vector error correction models with unknown cointegrating rank structure and unknown transient lag dynamic order. These methods enable simultaneous order estimation of the cointegrating rank and autoregressive order in conjunction with oracle-like efficient estimation of the cointegrating matrix and transient dynamics. As such they offer considerable advantages to the practitioner as an automated approach to the estimation of cointegrated systems. The paper develops the new methods, derives their limit theory, reports simulations and presents an empirical illustration with macroeconomic aggregates.

Select the Valid and Relevant Moments: A One-Step Procedure for GMM with Many Moments

This paper considers the selection of valid and relevant moments for the generalized method of moments (GMM) estimation. For applications with many candidate moments, our asymptotic analysis ccommodates a diverging number of moments as the sample size increases. The proposed procedure achieves three objectives in one-step: (i) the valid and relevant moments are selected simultaneously rather than sequentially; (ii) all desired moments are selected together instead of in a stepwise manner; (iii) the parameter of interest is automatically estimated with all selected moments as opposed to a post-selection estimation. The new moment selection method is achieved via an information-based adaptive GMM shrinkage estimation, where an appropriate penalty is attached to the standard GMM criterion to link moment selection to shrinkage estimation. The penalty is designed to signal both moment validity and relevance for consistent moment selection and efficient estimation. The asymptotic analysis allows for non -smooth sample moments and weakly dependent observations, making it generally applicable. For practical implementation, this one-step procedure is computationally attractive.

Sieve Semiparametric Two-Step GMM under Weak Dependence

This paper considers semiparametric two-step GMM estimation and inference with weakly dependent data, where unknown nuisance functions are estimated via sieve extremum estimation in the first step. We show that although the asymptotic variance of the second-step GMM estimator may not have a closed form expression, it can be well approximated by sieve variances that have simple closed form expressions. We present consistent or robust variance estimation, Wald tests and Hansen’s (1982) over-identification tests for the second step GMM that properly reflect the first-step estimated functions and the weak dependence of the data. Our sieve semiparametric two-step GMM inference procedures are shown to be numerically equivalent to the ones computed as if the first step were parametric. A new consistent random-perturbation estimator of the derivative of the expectation of the non-smooth moment function is also provided.

Series Estimation of Stochastic Processes: Recent Developments and Econometric Applications

This paper overviews recent developments in series estimation of stochastic processes and some of their applications in econometrics. Underlying this approach is the idea that a stochastic process may under certain conditions be represented in terms of a set of orthonormal basis functions, giving a series representation that involves deterministic functions. Several applications of this series approximation method are discussed. The first shows how a continuous function can be approximated by a linear combination of Brownian motions (BMs), which is useful in the study of the spurious regressions. The second application utilizes the series representation of BM to investigate the effect of the presence of deterministic trends in a regression on traditional unit-root tests. The third uses basis functions in the series approximation as instrumental variables (IVs) to perform efficient estimation of the parameters in cointegrated systems. The fourth application proposes alternative estimators of long-run variances in some econometric models with dependent data, thereby providing autocorrelation robust inference methods in these models. We review some work related to these applications and some ongoing research involving series approximation methods.

Asymptotic Properties of Penalized M Estimators with Time Series Observations

The method of penalization is a general smoothing principle to solve infinite-dimensional estimation problems. In this chapter, we study asymptotic properties of penalized M estimators with weakly dependent data. We first establish the convergence rate for any penalized M estimators of unknown functions with stationary beta mixing observations. While the existing theories on the convergence rates with i.i.d. data require that the random criteria have exponential thin tails, we allow for unbounded random criteria with finite polynomial moments. When specializing to regression and density estimation of time series models, our rates coincide with Stone (The Annals of Statistics, 10: 1040–1053, 1982) optimal rates for i.i.d. data. We then derive root-n asymptotic normality for any plug-in penalized M estimators of regular functionals, and provide consistent estimates of their long-run variances.

On Standard Inference for GMM with Local Identification Failure of Known Forms

This paper studies the GMM estimation and inference problem that occurs when the Jacobian of the moment conditions is rank deficient of known forms at the true parameter values. Dovonon and Renault (2013) recently raised a local identification issue stemming from this type of degenerate Jacobian. The local identification issue leads to a slow rate of convergence of the GMM estimator and a non-standard asymptotic distribution of the over-identification test statistics. We show that the known form of rank-deficient Jacobian matrix contains non-trivial information about the economic model. By exploiting such information in estimation, we provide GMM estimator and over-identification tests with standard properties. The main theory developed in this paper is applied to the estimation of and inference about the common conditionally heteroskedastic (CH) features in asset returns. The performances of the newly proposed GMM estimators and over-identification tests are investigated under the same simulation designs used in Dovonon and Renault (2013).

Uniform Asymptotic Risk of Averaging GMM Estimator Robust to Misspecification, Second Version

This paper studies the averaging generalized method of moments (GMM) estimator that combines a conservative GMM estimator based on valid moment conditions and an aggressive GMM estimator based on both valid and possibly misspecified moment conditions, where the weight is the sample analog of an infeasible optimal weight. It is an alternative to pre-test estimators that switch between the conservative and aggressive estimators based on model specification tests. This averaging estimator is robust in the sense that it uniformly dominates the conservative estimator by reducing the risk under any degree of misspecification, whereas the pre-test estimators reduce the risk in parts of the parameter space and increase it in other parts. To establish uniform dominance of one estimator over another, we establish asymptotic theories on uniform approximations of the finite-sample risk differences between two estimators. These asymptotic results are developed along drifting sequences of data generating processes (DGPs) that model various degrees of local misspecification as well as global misspecification. Extending seminal results on the James-Stein estimator, the uniform dominance is established in non-Gaussian semiparametric nonlinear models. The proposed averaging estimator is applied to estimate the human capital production function in a life-cycle labor supply model.