In recent decades, in the research community of macroeconometric time series analysis, we have observed growing interest in the smoothing method known as the Hodrick–Prescott (HP) filter. This article examines the properties of an alternative smoothing method that looks like the HP filter, but is much less well known. We show that this is actually more like the exponential smoothing filter than the HP filter although it is obtainable through a slight modification of the HP filter. In addition, we also show that it is also like the low-frequency projection of Müller and Watson (2018, Econometrica 86, 775–804). We point out that these results derive from the fact that all three similar smoothing methods can be regarded as a type of graph spectral filter whose graph Fourier transform is discrete cosine transform. We then theoretically reveal the relationship between the similar smoothing methods and provide a way of specifying the smoothing parameter that is necessary for its application. An empirical examination illustrates the results.

We propose a model-free test for structural changes in factor models. The basic idea is to regress the data on commonly estimated factors by local smoothing and compare the fitted values of time-varying factor loadings with those of time-invariant factor loadings estimated via principal component analysis. By construction, the test is designed to be powerful against both smooth structural changes and sudden structural breaks with a possibly unknown number of breaks and unknown break dates in the factor loadings. No restrictions on the form of alternatives or trimming of boundary regions near the beginning or end of the sample period is required for the test. The test has power to detect the usual nonparametric rate of local alternatives. Monte Carlo studies demonstrate excellent power of the test in detecting both smooth and sudden structural changes in the factor loadings. In an application using U.S. asset returns, we find significant evidence against time-invariant factor loadings.

In this paper it is shown that a subprocess of a Markov process is markovian if a suitable condition of noncausality is satisfied. Furthermore, a markovian condition is shown to be a natural condition when analyzing the role of the horizon (finite or infinite) in the property of noncausality. We also give further conditions implying that a process is both jointly and marginally markovian only if there is both finite and infinite noncausality and that a process verifies both finite and infinite noncausality only if it is markovian. Counterexamples are also given to illustrate the cases where these further conditions are not satisfied.

Censored and truncated regression models with unknown distribution are important in econometrics. This paper characterizes the class of all conditional moment restrictions that lead to √n-consistent estimators for these models. The semiparametric efficiency bound for each conditional moment restriction is derived. In the case of a nonzero bound it is shown how an estimator can be constructed and that an appropriately weighted version can attain the efficiency bound. These estimators also work when the disturbance is independent of the regressors. The paper discusses combining conditional moment restrictions for more efficient estimation in this case.

This paper proposes a new two-stage estimation and inference procedure for a class of partially identified models. The procedure can be considered an extension of classical minimum distance estimation procedures to accommodate inequality constraints and partial identification. It involves no tuning parameter, is nonconservative, and is conceptually and computationally simple. The class of models includes models of interest to applied researchers, including the static entry game, a voting game with communication, and a discrete mixture model. Besides, a technical contribution is an implicit correspondence lemma which generalizes the implicit function theorem to multivalued implicit maps.

Locally asymptotically optimal tests are derived for the null hypothesis of traditional AR dependence, with unspecified AR coefficients and unspecified innovation densities, against an alternative of periodically correlated AR dependence. Parametric and nonparametric rank-based versions are proposed. Local powers and asymptotic relative efficiencies (with respect, e.g., to the corresponding Gaussian Lagrange multiplier tests proposed in Ghysels and Hall [1992, “Lagrange Multiplier Tests for Periodic Structures,” unpublished manuscript, CRDE, Montreal] and Liitkepohl [1991, Introduction to Multiple Time Series Analysis, Berlin: Springer-Verlag; 1991, pp. 243–264, in W.E. Griffiths, H. Liitkepohl, & M.E. Block (eds.), Readings in Econometric Theory and Practice, Amsterdam: North-Holland] are computed explicitly; a rank-based test of the van der Waerden type is proposed, for which this ARE is uniformly larger than 1. The main technical tool is Le Cam's local asymptotic normality property.

This paper develops a general procedure to check the bootstrap validity in M-estimation. We apply the procedure in discontinuous threshold regression to show the inconsistency of the nonparametric bootstrap for inference on the threshold point. Especially, the conditional weak limit of the nonparametric bootstrap is shown not to exist. By comparing with two other boundaries in the literature, we show the fact that the threshold point is a boundary of the covariate that makes its bootstrap inference so different. The remedies to the bootstrap failure in the literature are summarized, and the nonparametric posterior interval is suggested by some simulation studies.

While differencing transformations can eliminate nonstationarity, they typically reduce signal strength and correspondingly reduce rates of convergence in unit root autoregressions. The present paper shows that aggregating moment conditions that are formulated in differences provides an orderly mechanism for preserving information and signal strength in autoregressions with some very desirable properties. In first order autoregression, a partially aggregated estimator based on moment conditions in differences is shown to have a limiting normal distribution that holds uniformly in the autoregressive coefficient ρ, including stationary and unit root cases. The rate of convergence is when and the limit distribution is the same as the Gaussian maximum likelihood estimator (MLE), but when ρ = 1 the rate of convergence to the normal distribution is within a slowly varying factor of n. A fully aggregated estimator (FAE) is shown to have the same limit behavior in the stationary case and to have nonstandard limit distributions in unit root and near integrated cases, which reduce both the bias and the variance of the MLE. This result shows that it is possible to improve on the asymptotic behavior of the MLE without using an artificial shrinkage technique or otherwise accelerating convergence at unity at the cost of performance in the neighborhood of unity. Confidence intervals constructed from the FAE using local asymptotic theory around unity also lead to improvements over the MLE.

We derive the limit distribution of the number of crossings of a level by a random walk with continuously distributed increments, using a Brownian motion local time approximation. This complements the well-known result for the random walk on the integers. Use of the frequency of level crossings to test for a unit root is examined.

An influential paper by Kleibergen (2005, Econometrica 73, 1103–1123) introduces Lagrange multiplier (LM) and conditional likelihood ratio-like (CLR) tests for nonlinear moment condition models. These procedures aim to have good size performance even when the parameters are unidentified or poorly identified. However, the asymptotic size and similarity (in a uniform sense) of these procedures have not been determined in the literature. This paper does so.

This paper shows that the LM test has correct asymptotic size and is asymptotically similar for a suitably chosen parameter space of null distributions. It shows that the CLR tests also have these properties when the dimension p of the unknown parameter θ equals 1. When p ≥ 2, however, the asymptotic size properties are found to depend on how the conditioning statistic, upon which the CLR tests depend, is weighted. Two weighting methods have been suggested in the literature. The paper shows that the CLR tests are guaranteed to have correct asymptotic size when p ≥ 2 when the weighting is based on an estimator of the variance of the sample moments, i.e., moment-variance weighting, combined with the Robin and Smith (2000, Econometric Theory 16, 151–175) rank statistic. The paper also determines a formula for the asymptotic size of the CLR test when the weighting is based on an estimator of the variance of the sample Jacobian. However, the results of the paper do not guarantee correct asymptotic size when p ≥ 2 with the Jacobian-variance weighting, combined with the Robin and Smith (2000, Econometric Theory 16, 151–175) rank statistic, because two key sample quantities are not necessarily asymptotically independent under some identification scenarios.

Analogous results for confidence sets are provided. Even for the special case of a linear instrumental variable regression model with two or more right-hand side endogenous variables, the results of the paper are new to the literature.

In this article we aim to establish intuitively appealing and verifiable conditions for the first-order efficiency and asymptotic normality of ML estimators in a multi-parameter framework, assuming joint normality but neither the independence nor the identical distribution of the observations. We present five theorems (and a large number of lemmas and propositions), each being a special case of its predecessor.

Econometric analysis of continuous time models has drawn the attention of Peter Phillips for 40 years, resulting in many important publications by him. In these publications he has dealt with a wide range of continuous time models and the associated econometric problems. He has investigated problems from univariate equations to systems of equations, from asymptotic theory to finite sample issues, from parametric models to nonparametric models, from identification problems to estimation and inference problems, and from stationary models to nonstationary and nearly nonstationary models. This paper provides an overview of Peter Phillips’ contributions in the continuous time econometrics literature. We review the problems that have been tackled by him, outline the main techniques suggested by him, and discuss the main results obtained by him. Based on his early work, we compare the performance of three asymptotic distributions in a simple setup. Results indicate that the in-fill asymptotics significantly outperforms the long-span asymptotics and the double asymptotics.

Many economic models yield conditional moment inequalities that can be used for inference on parameters of these models. In this paper, I construct new tests of parameter hypotheses in conditional moment inequality models based on studentized kernel estimates of moment functions. The tests automatically adapt to the unknown smoothness of the moment functions, have uniformly correct asymptotic size, and are rate-optimal against certain classes of alternatives. Some existing tests have nontrivial power against n−1/2-local alternatives of a certain type whereas my methods only allow for nontrivial testing against (n / log n)−1/2-local alternatives of this type. There exist, however, large classes of sequences of well-behaved alternatives against which the tests developed in this paper are consistent and those tests are not.

We propose tests of the null of spurious relationship against the alternative of fractional cointegration among the components of a vector of fractionally integrated time series. Our test statistics have an asymptotic chi-square distribution under the null and rely on generalized least squares–type of corrections that control for the short-run correlation of the weak dependent components of the fractionally integrated processes. We emphasize corrections based on nonparametric modelization of the innovations' autocorrelation, relaxing important conditions that are standard in the literature and, in particular, being able to consider simultaneously (asymptotically) stationary or nonstationary processes. Relatively weak conditions on the corresponding short-run and memory parameter estimates are assumed. The new tests are consistent with a divergence rate that, in most of the cases, as we show in a simple situation, depends on the cointegration degree. Finite-sample properties of the tests are analyzed by means of a Monte Carlo experiment.We thank Helmut Lütkepohl and two referees for helpful comments and suggestions. We also thank participants at the NSF/NBER Time Series Conference at the University of Heidelberg, Germany, at the Unit Root and Cointegration Testing Conference at the University of Algarve, Faro, Portugal, and seminar participants at the Universidad de Navarra and Ente Luigi Einaudi for helpful comments. Javier Hualde's research is supported by the Spanish Ministerio de Educación y Ciencia through Juan de la Cierva and Ramón y Cajal contracts and ref. SEJ2005-07657/ECON. Carlos Velasco's research is supported by the Spanish Ministerio de Educación y Ciencia, ref. SEJ2004-04583/ECON.

We study the asymptotic covariance function of the sample mean and quantile, and derive a new and surprising characterization of the normal distribution: the asymptotic covariance between the sample mean and quantile is constant across all quantiles, if and only if the underlying distribution is normal. This is a powerful result and facilitates statistical inference. Utilizing this result, we develop a new omnibus test for normality based on the quantile-mean covariance process. Compared to existing normality tests, the proposed testing procedure has several important attractive features. Monte Carlo evidence shows that the proposed test possesses good finite sample properties. In addition to the formal test, we suggest a graphical procedure that is easy to implement and visualize in practice. Finally, we illustrate the use of the suggested techniques with an application to stock return datasets.

This note presents a procedure which enables the user to check whether a monotonic transformation can change the sign of a regression coefficient. One possible use of this procedure is to examine the robustness of key regression coefficients.

Methods of maximum-likelihood search (eg., Hildreth-Lu) for single equations models with autocorrelated disturbances are not easily extended to the multi-equation systems encountered in econometric studies of consumer demand. The obstacle is the large dimension of the space of independent serial correlation coefficients to be partitioned and searched. This paper treats stochastic disturbance vectors in demand/expenditure systems as functions of random disturbances in consumer utility functions. It specifies marginal utilities to be products of a systematic function and a nonnegative random disturbance. Under this specification, three hypotheses that readily come to mind in the context of consumer theory drastically reduce the number of independent elements of serial covariance matrices. This in turn makes practicable the extension of maximum-likelihood search methods in the estimation of parameters of large demand systems.

A general approach for the estimation of cointegration vectors with linear restrictions is described. In the special case of zero restrictions, the cointegration relations of the paper are formally similar to the structural form of a traditional simultaneous equation model. The proposed estimation procedures require a conventional rank condition of identification but no exogeneity assumption. In place of exogenous variables there are series that are not cointegrated and can therefore describe the common trends in the system. The estimators of the paper are flexible and simple to use. They can be combined with several recent estimators developed for cointegration regressions which in the present context are formally similar to the reduced form of a simultaneous equation model. After the coefficient matrix of a cointegration regression has been estimated, the estimators of the paper can be obtained by simple generalized least squares. Both single equation estimators and more efficient system estimators are developed. The asymptotic distributions of the estimators are shown to be mixed normal so that Wald tests with asymptotic chi-square distributions under the null hypothesis can be obtained in the usual way. Convenient test procedures for checking the validity of overidentification restrictions are also provided.

We consider the following linear regression model:

where are independent and identically distributed random variables, Yi, is real, Zi has values in Rm, Ui, is independent of Zi, and θ0 is an m-dimensional parameter to be estimated. The Lp estimator of θ0 is the value 6n such that

Here, we will give the exact Bahadur-Kiefer representation of θn, for each p ≥ 1. Explicitly, we will see that, under regularity conditions,

where and c is a positive constant, which depends on p and on the random variable X.

This paper investigates the global self-weighted least absolute deviation (SLAD) estimator for finite and infinite variance ARMA(p, q) models. The strong consistency and asymptotic normality of the global SLAD estimator are obtained. A simulation study is carried out to assess the performance of the global SLAD estimators. In this paper the asymptotic theory of the global LAD estimator for finite and infinite variance ARMA(p, q) models is established in the literature for the first time. The technique developed in this paper is not standard and can be used for other time series models.