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.

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.

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.

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.

Durbin–Watson statistic and random individual Effects.

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.

We consider conditions for strict stationarity and ergodicity of a class of multivariate BEKK processes $(X_t : t=1,2,\ldots )$ and study the tail behavior of the associated stationary distributions. Specifically, we consider a class of BEKK-ARCH processes where the innovations are assumed to be Gaussian and a finite number of lagged $X_t$ ’s may load into the conditional covariance matrix of $X_t$ . By exploiting that the processes have multivariate stochastic recurrence equation representations, we show the existence of strictly stationary solutions under mild conditions, where only a fractional moment of $X_t$ may be finite. Moreover, we show that each component of the BEKK processes is regularly varying with some tail index. In general, the tail index differs along the components, which contrasts with most of the existing literature on the tail behavior of multivariate GARCH processes. Lastly, in an empirical illustration of our theoretical results, we quantify the model-implied tail index of the daily returns on two cryptocurrencies.

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.

The limiting distribution of the least squares estimate of the derived process of a noninvertible and nearly noninvertible moving average model with infinite variance innovations is established as a functional of a Lévy process. The form of the limiting law depends on the initial value of the innovation and the stable index α. This result enables one to perform asymptotic testing for the presence of a unit root for a noninvertible moving average model through the constructed derived process under the null hypothesis. It provides not only a parallel analog of its autoregressive counterparts, but also a useful alternative to determine “over-differencing” for time series that exhibit heavy-tailed phenomena.

We estimate the density and its derivatives using a local polynomial approximation to the logarithm of an unknown density function f. The estimator is guaranteed to be non-negative and achieves the same optimal rate of convergence in the interior as on the boundary of the support of f. The estimator is therefore well-suited to applications in which non-negative density estimates are required, such as in semiparametric maximum likelihood estimation. In addition, we show that our estimator compares favorably with other kernel-based methods, both in terms of asymptotic performance and computational ease. Simulation results confirm that our method can perform similarly or better in finite samples compared to these alternative methods when they are used with optimal inputs, that is, an Epanechnikov kernel and optimally chosen bandwidth sequence. We provide code in several languages.

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.

Lyapunov exponents may be used to provide information on attractors of nonlinear models and, if strange, their aperiodic dynamics, in much the same way as eigenvalues of a linear model. An advantage of calculating these from an estimated model, rather than calculating the largest ones from a time series, is that economic theory is used to help distinguish between deterministic and stochastic behavior. Estimates of the Bergstrom/Wymer model of the United Kingdom, and of some other models of a similar form, were unstable in a classical sense, to some extent being caused by policy parameters. The Lyapunov exponents show that this model, and variants of it, have strange attractors in the neighborhood of the estimated parameter values and hence are stable but with aperiodic oscillations.

Wu introduced a new technique for proving consistency of least-squares estimators in nonlinear regression. This paper extends his results in three directions. First, we consider the minimization of arbitrary functions (M-estimators instead of least squares). Second, we use an improved type 2 inequality. Third, an extension of Kronecker's lemma yields a more powerful result.

In testing for correlation of the errors in regression models, the power of tests can be very low for strongly correlated errors. This counterintuitive phenomenon has become known as the “zero-power trap.” Despite a considerable amount of literature devoted to this problem, mainly focusing on its detection, a convincing solution has not yet been found. In this article, we first discuss theoretical results concerning the occurrence of the zero-power trap phenomenon. Then, we suggest and compare three ways to avoid it. Given an initial test that suffers from the zero-power trap, the method we recommend for practice leads to a modified test whose power converges to $1$ as the correlation gets very strong. Furthermore, the modified test has approximately the same power function as the initial test and thus approximately preserves all of its optimality properties. We also provide some numerical illustrations in the context of testing for network generated correlation.

An unobserved components and a conventional first-order autoregressive (AR) model with constant are analyzed with the adjusted profile likelihood of Cox and Reid (1987, Journal of the Royal Statistical Society, Series B 49, 1–39; 1993, Journal of the Royal Statistical Society, Series B 55, 467–71). Both the unobserved components model and the Cox–Reid adjustment can provide more accurate estimates of an AR coefficient of unity. The unobserved components model yields more powerful unit-root tests. In general the most powerful test utilizes the adjustment. Under the unobserved components model the adjusted statistics follow the asymptotic distributions better than the unadjusted.The basis of this research is my D. Phil. thesis (1997) for the University of Oxford. I am grateful to David Hendry for supervision and to Pentti Saikkonen for most helpful advice. I appreciate also the CBRT Young Economist Award with which I was honored after presenting a previous version at the 1998 erc/METU International Conference on Economics. I remain responsible for any errors. Financial support by the Yrjö Jahnsson Foundation and Academy of Finland is gratefully acknowledged.