The Gibbs posterior is a useful tool for risk minimization, which adopts a Bayesian framework and can incorporate convenient computational algorithms such as Markov chain Monte Carlo. We derive risk bounds for the Gibbs posterior using some general nonasymptotic inequalities, which can be used to derive nearly optimal convergence rates and select models to optimally balance the approximation errors and the stochastic errors. These inequalities are formulated in a very general way that does not require the empirical risk to be a usual sample average over independent observations. We apply this framework to study the convergence rate of the GMM (generalized method of moments) risk and derive an oracle inequality for the ranking risk, where models are selected based on the Gibbs posterior with a nonadditive empirical risk.

This paper proposes an empirical likelihood-based estimation method for semiparametric conditional moment restriction models, which contain finite dimensional unknown parameters and unknown functions. We extend the results of Donald, Imbens, and Newey (2003, Journal of Econometrics 117, 55–93) by allowing unknown functions to be included in the conditional moment restrictions. We approximate unknown functions by a sieve method and estimate the finite dimensional parameters and unknown functions jointly. We establish consistency and derive the convergence rate of the estimator. We also show that the estimator of the finite dimensional parameters is $\sqrt n$-consistent, asymptotically normally distributed, and asymptotically efficient.

We use identification robust tests to show that difference (Dif), level (Lev), and nonlinear (NL) moment conditions, as proposed by Arellano and Bond (1991, Review of Economic Studies 58, 277–297), Ahn and Schmidt (1995, Journal of Econometrics 68, 5–27), Arellano and Bover (1995, Journal of Econometrics 68, 29–51), and Blundell and Bond (1998, Journal of Econometrics 87, 115–143) for the linear dynamic panel data model, do not separately identify the autoregressive parameter when its true value is close to one and the variance of the initial observations is large. We prove that combinations of these moment conditions, however, do so when there are more than three time series observations. This identification then solely results from a set of, so-called, robust moment conditions. These robust moments are spanned by the combined Dif, Lev, and NL moment conditions and only depend on differenced data. We show that, when only the robust moments contain identifying information on the autoregressive parameter, the discriminatory power of the Kleibergen (2005, Econometrica 73, 1103–1124) Lagrange multiplier (KLM) test using the combined moments is identical to the largest rejection frequencies that can be obtained from solely using the robust moments. This shows that the KLM test implicitly uses the robust moments when only they contain information on the autoregressive parameter.

This book, Matrix Algebra, is the first volume of the Econometric Exercises series. Other volumes of the series (Statistics [vol. 2], Econometric Theory I [vol. 3], and Empirical Applications [vol. 4]) are to appear in the near future. A variety of planned titles will also follow (e.g., Time Series, Microeconometrics, and Financial Econometrics). Although there have been a few worked-exercise books in econometrics (e.g., Phillips and Wickens, 1978), this series is the first comprehensive exercise series in modern econometrics that covers the materials for advanced undergraduate and graduate econometrics courses. According to the preface, each volume follows the same format. Chapters start with a brief overview, and then exercises and solutions follow. Solutions are provided immediately after the exercise is posed. There is a variety of exercises from proofs and applications of theorems to numerical examples. In this review I discuss only the first volume in the Econometric Exercises series and find it very useful and successful. I expect succeeding volumes in the series to be of similar quality and usefulness, given the overall objectives of the series.

In this paper, we propose two bootstrap procedures, namely parametric and block bootstrap, to approximate the finite sample distribution of change-point estimators for piecewise stationary time series. The bootstrap procedures are then used to develop a generalized likelihood ratio scan method (GLRSM) for multiple change-point inference in piecewise stationary time series, which estimates the number and locations of change-points and provides a confidence interval for each change-point. The computational complexity of using GLRSM for multiple change-point detection is as low as $O(n(\log n)^{3})$ for a series of length n. Extensive simulation studies are provided to demonstrate the effectiveness of the proposed methodology under different scenarios. Applications to financial time series are also illustrated.

Alternative definitions of the concentration ellipsoid of a random vector are surveyed, and an extension of the concentration ellipsoid of Darmois is suggested as being the most convenient and natural definition. The advantage of the proposed definition in providing substantially simplified proofs of results in (linear) estimation theory is discussed, and is illustrated by new and short proofs of two key results. A not-so-well-known, but elementary, extremal representation of a nonnegative definite quadratic form, together with the corresponding Cauchy-Schwarẓ-type inequality, is seen to play a crucial role in these proofs.

Let {X(t)} be a multivariate Gaussian stationary process with the spectral density matrix f0(ω), where θ is an unknown parameter vector. Using a quasi-maximum likelihood estimator of θ, we estimate the spectral density matrix f0(ω) by f(ω). Then we derive asymptotic expansions of the distributions of functions of f(ω). Also asymptotic expansions for the distributions of functions of the eigenvalues of f(ω) are given. These results can be applied to many fundamental statistics in multivariate time series analysis. As an example, we take the reduced form of the cobweb model which is expressed as a two-dimensional vector autoregressive process of order 1 (AR(1) process) and show the asymptotic distribution of , the estimated coherency, and contribution ratio in the principal component analysis based on in the model, up to the second-order terms. Although our general formulas seem very involved, we can show that they are tractable by using REDUCE 3.

This paper is devoted to a detailed examination of the exact sampling properties of the instrumental variables (IV) estimator of the vector of coefficients on the exogenous variables in a single structural equation. The first two moments of a linear combination of the elements of this estimator and the joint distribution of these elements are considered. Estimable bounds for the first moment that can readily be incorporated into any IV estimation package are provided. The results obtained are in terms of the same special functions as those that characterize other results for this model, allowing a unified treatment of the model.

Under normality, we obtain higher-order approximations to the distributions of the periodogram and related statistics. Our approach is based on the theorem which decomposes the periodogram into the sum of two independent random variables. It is seen that this decomposition enables us to study fairly closely the higher-order properties of not only the periodogram, but also periodogram-based statistics such as the estimators of the spectrum and prediction error variance. Some of the approximation results are graphically presented together with the exact results based on simulations.

The paper “Kernel estimation when density may not exist” (Zinde-Walsh, 2008) considered density as a generalized function given by a functional on a space of smooth functions; this made it possible to establish the limit properties of the kernel estimator without assuming the existence of the density function. This note corrects an error in that paper in the derivation of the variance of the kernel estimator. The corrected result is that in the space of generalized functions the parametric rate of convergence of the kernel density estimator to the limit Gaussian process is achievable.

This paper proposes a robust moment selection method aiming to pick the best model even if this is a moment condition model with mixed identification strength, that is, moment conditions including moment functions that are local to zero uniformly over the parameter set. We show that the relevant moment selection procedure of Hall et al. (2007, Journal of Econometrics 138, 488–512) is inconsistent in this setting as it does not explicitly account for the rate of convergence of parameter estimation of the candidate models which may vary. We introduce a new moment selection procedure based on a criterion that automatically accounts for both the convergence rate of the candidate model’s parameter estimate and the entropy of the estimator’s asymptotic distribution. The benchmark estimator that we consider is the two-step efficient generalized method of moments estimator, which is known to be efficient in this framework as well. A family of penalization functions is introduced that guarantees the consistency of the selection procedure. The finite-sample performance of the proposed method is assessed through Monte Carlo simulations.

This paper introduces a novel Itô diffusion process to model high-frequency financial data that can accommodate low-frequency volatility dynamics by embedding the discrete-time nonlinear exponential generalized autoregressive conditional heteroskedasticity (GARCH) structure with log-integrated volatility in a continuous instantaneous volatility process. The key feature of the proposed model is that, unlike existing GARCH-Itô models, the instantaneous volatility process has a nonlinear structure, which ensures that the log-integrated volatilities have the realized GARCH structure. We call this the exponential realized GARCH-Itô model. Given the autoregressive structure of the log-integrated volatility, we propose a quasi-likelihood estimation procedure for parameter estimation and establish its asymptotic properties. We conduct a simulation study to check the finite-sample performance of the proposed model and an empirical study with 50 assets among the S&P 500 compositions. Numerical studies show the advantages of the proposed model.

Teaching graduate econometrics means covering three different kinds of subject matter: a grounding in the theory of econometrics, a long laundry list of available econometric techniques, and an introduction to the fact that the practice of linking models and data is every bit as untidy as mathematical statistics is neat. I assign Econometric Theory and Methods (ETM) as a primary text in our first Ph.D. econometrics course. ETM is in charge of getting the students their theoretical grounding. I also assign Greene's excellent Econometric Analysis (2003) for its coverage of a long list of techniques. My laptop, EViews, and I, together with a whole lot of real data, are responsible for being untidy.

We present a new robust bootstrap method for a test when there is a nuisance parameter under the alternative, and some parameters are possibly weakly or nonidentified. We focus on a Bierens (1990, Econometrica 58, 1443–1458)-type conditional moment test of omitted nonlinearity for convenience. Existing methods include the supremum p-value which promotes a conservative test that is generally not consistent, and test statistic transforms like the supremum and average for which bootstrap methods are not valid under weak identification. We propose a new wild bootstrap method for p-value computation by targeting specific identification cases. We then combine bootstrapped p-values across polar identification cases to form an asymptotically valid p-value approximation that is robust to any identification case. Our wild bootstrap procedure does not require knowledge of the covariance structure of the bootstrapped processes, whereas Andrews and Cheng’s (2012a, Econometrica 80, 2153–2211; 2013, Journal of Econometrics 173, 36–56; 2014, Econometric Theory 30, 287–333) simulation approach generally does. Our method allows for robust bootstrap critical value computation as well. Our bootstrap method (like conventional ones) does not lead to a consistent p-value approximation for test statistic functions like the supremum and average. Therefore, we smooth over the robust bootstrapped p-value as the basis for several tests which achieve the correct asymptotic level, and are consistent, for any degree of identification. They also achieve uniform size control. A simulation study reveals possibly large empirical size distortions in nonrobust tests when weak or nonidentification arises. One of our smoothed p-value tests, however, dominates all other tests by delivering accurate empirical size and comparatively high power.

The work of Professor P.C.B. Phillips, even if it is focused on the area of linear nonstationary models, is enormous. So it is hard for me to explore the whole of his work in this paper. Therefore, I have decided to take up only a few results of his work. The topics chosen here are applications of the martingale approximation and the problem of choosing between stochastic and deterministic trends, which I discuss and, hopefully, extend.