Lieberman and Phillips (2017; LP) introduced a multivariate stochastic unit root (STUR) model, which allows for random, time varying local departures from a unit root (UR) model, where nonlinear least squares (NLLS) may be used for estimation and inference on the STUR coefficient. In a structural version of this model where the driver variables of the STUR coefficient are endogenous, the NLLS estimate of the STUR parameter is inconsistent, as are the corresponding estimates of the associated covariance parameters. This paper develops a nonlinear instrumental variable (NLIV) as well as GMM estimators of the STUR parameter which conveniently addresses endogeneity. We derive the asymptotic distributions of the NLIV and GMM estimators and establish consistency under similar orthogonality and relevance conditions to those used in the linear model. An overidentification test and its asymptotic distribution are also developed. The results enable inference about structural STUR models and a mechanism for testing the local STUR model against a simple UR null, which complements usual UR tests. Simulations reveal that the asymptotic distributions of the NLIV and GMM estimators of the STUR parameter as well as the test for overidentifying restrictions perform well in small samples and that the distribution of the NLIV estimator is heavily leptokurtic with a limit theory which has Cauchy-like tails. Comparisons of STUR coefficient and standard UR coefficient tests show that the one-sided UR test performs poorly against the one-sided STUR coefficient test both as the sample size and departures from the null rise. The results are applied to study the relationships between stock returns and bond spread changes.

In this paper we introduce a new method of projection-type inference and describe it in the context of two stage least squares–based split-sample inference on subsets of structural coefficients in a linear instrumental variables regression model. The use of the new method not only guards against the uncontrolled overrejection of the true value of the parameters of interest but also reduces the conservativeness of the usual method of projection proposed by Dufour and his coauthors (Dufour, 1997, Econometrica 65, 1365–1388; Dufour and Jasiak, 2001, International Economic Review 41, 815–843; Dufour and Taamouti, 2005, discussion paper; Dufour and Taamouti, 2005, Econometrica 73, 1351–1365; Dufour and Taamouti, 2007, Journal of Econometrics 139, 133–153).

This paper develops an estimation methodology for network data generated from a system of simultaneous equations, which allows for network interdependencies via spatial lags in the endogenous and exogenous variables, as well as in the disturbances. By allowing for higher-order spatial lags, our specification provides important flexibility in modeling network interactions. The estimation methodology builds, among others, on the two-step generalized method of moments estimation approach introduced in Kelejian and Prucha (1998, Journal of Real Estate Finance and Economics 17, 99–121; 1999, International Economic Review 40, 509–533; 2004, Journal of Econometrics 118, 27–50). The paper considers limited and full information estimators, and one- and two-step estimators, and establishes their asymptotic properties. In contrast to some of the earlier two-step estimation literature, our asymptotic results facilitate joint tests for the absence of all forms of network spillovers.

Researchers often trim observations with small values of the denominator A when they estimate moments of the form $\mathbb {E}[B/A]$ . Large trimming is common in practice to reduce variance, but it incurs a large bias. This paper provides a novel method of correcting the large trimming bias. If a researcher is willing to assume that the joint distribution between A and B is smooth, then the trimming bias may be estimated well. Along with the proposed bias correction method, we also develop an inference method. Practical advantages of the proposed method are demonstrated through simulation studies, where the data generating process entails a heavy-tailed distribution of $B/A$ . Applying the proposed method to the Compustat database, we analyze the history of external financial dependence of U.S. manufacturing firms for years 2000–2010.

We consider the problem of combining forecasts from two different levels (called “macro” and “micro”), where we have access to the forecasts and their precisions but not to the full data set. We develop a theoretical framework and provide Monte Carlo evidence in the cases of both perfect and imperfect aggregation. Our proposed procedure is simple and robust. We also extend the procedure to time series and propose a forecast model for the European zero rates, combining quarterly and monthly observations. We show that forecast accuracy is improved at both levels.We are grateful to the editors of this special issue and to two referees for their constructive comments, which greatly helped improve the paper. Earlier versions of this paper were presented at the NAKE workshop in Amsterdam, at Tilburg University, and at ESEM 2006 in Vienna. We thank the participants for their useful comments. In addition, we thank Steffan Berridge, John Einmahl, and Hans Schumacher for their constructive comments.

This paper proposes a nonparametric simultaneous test for parametric specification of the conditional mean and variance functions in a time series regression model. The test is based on an empirical likelihood (EL) statistic that measures the goodness of fit between the parametric estimates and the nonparametric kernel estimates of the mean and variance functions. A unique feature of the test is its ability to distribute natural weights automatically between the mean and the variance components of the goodness-of-fit measure. To reduce the dependence of the test on a single pair of smoothing bandwidths, we construct an adaptive test by maximizing a standardized version of the empirical likelihood test statistic over a set of smoothing bandwidths. The test procedure is based on a bootstrap calibration to the distribution of the empirical likelihood test statistic. We demonstrate that the empirical likelihood test is able to distinguish local alternatives that are different from the null hypothesis at an optimal rate.

We develop versions of the Granger–Johansen representation theorems for I(1) and I(2) vector autoregressive processes that apply to processes taking values in an arbitrary complex separable Hilbert space. This more general setting is of central relevance for statistical applications involving functional time series. An I(1) or I(2) solution to an autoregressive law of motion is obtained when the inverse of the autoregressive operator pencil has a pole of first or second order at one. We obtain a range of necessary and sufficient conditions for such a pole to be of first or second order. Cointegrating and attractor subspaces are characterized in terms of the behavior of the autoregressive operator pencil in a neighborhood of one.

Nine leading journals that publish statistical theory are used to provide a data base of institutional and individual research activity in statistics over the period 1980–1986. From this data base, we construct both institutional and individual research rankings according to standardized page counts of articles published in these journals over the stated period. The study is worldwide and we provide breakdowns of publication by country and by journal. Separate rankings are also provided for both institutions and individuals according to publication track records in the Annals of Statistics alone.

We determine the limiting behavior of near-integrated first-order random coefficient autoregressive RCA(1) time series. It is shown that the asymptotics of the finite-dimensional distributions crucially depends on how the critical value 1 is approached, which determines whether the process is near-stationary, has a unit root, or is mildly explosive. %In a second part, we derive the limit distribution of the serial correlation coefficient in the near stationary and the mildly explosive settings under very general conditions on the parameters. The results obtained are in accordance with those available for first-order autoregressive time series and can hence serve as an addition to existing literature in the area.

This paper considers linear rational expectations models from the linear systems point of view. Using a generalization of the Wiener-Hopf factorization, the linear systems approach is able to furnish very simple conditions for existence and uniqueness of both particular and generic linear rational expectations models. To illustrate the applicability of this approach, the paper characterizes the structure of stationary and cointegrated solutions, including a generalization of Granger’s representation theorem.

Semiparametric methods are widely employed in applied work where the ability to conduct inferences is important. To establish asymptotic normality for making inferences, bias control mechanisms are often used in implementing semiparametric estimators. The first contribution of this paper is to propose a mechanism that enables us to establish asymptotic normality with regular kernels. In so doing, we argue that the resulting estimator performs very well in finite samples.

Semiparametric models are commonly estimated under a single index assumption. Because the consistency of the estimator critically depends on this assumption being correct, our second objective is to develop a test for it. To ensure that the test statistic has good size and power properties in finite samples, we employ a bias control mechanism similar to that underlying the estimator. Furthermore, we structure the test so that its form adapts to the model under the alternative hypothesis. Monte Carlo results confirm that the bias control and the adaptive feature significantly improve the performance of the test statistic in finite samples.

This paper investigates the asymptotic properties of the maximum marginal likelihood estimator for a regression model with a stochastic trend component when the signal-to-noise ratio is near zero. In particular, the local level model in Harvey (1989, Forecasting, Structural Time Series Models and the Kalman Filter, Cambridge: Cambridge University Press) and its variants where a time trend or an intercept is included are considered. A local-to-zero parameterization is adopted. Two sets of asymptotic properties are presented for the local maximizer: consistency and the limiting distribution. The estimator is found to be super-consistent. The limit distribution is derived and found to possess a long tail and a mass point at zero. It yields a good approximation for samples of moderate size. Simulation also documents that the empirical distribution converges less rapidly to the limit distribution as number of regression parameters increases. The results could be viewed as a transition step toward establishing new likelihood ratio–type or Wald-type tests for the stationarity null.

The exact finite sample behavior is investigated on the bias of multiperiod leastsquares forecasts in the normal autoregressive model yt = α + βyt–1 + ut. Necessary and sufficient conditions are given for the existence of the bias and an expression is presented which we use to obtain exact numerical results for finite samples. The unit root and near unit root behavior is studied in detail and some popular preconceptions about the behavior of the bias are shown to be false.

This paper studies generalized additive partial linear models with high-dimensional covariates. We are interested in which components (including parametric and nonparametric components) are nonzero. The additive nonparametric functions are approximated by polynomial splines. We propose a doubly penalized procedure to obtain an initial estimate and then use the adaptive least absolute shrinkage and selection operator to identify nonzero components and to obtain the final selection and estimation results. We establish selection and estimation consistency of the estimator in addition to asymptotic normality for the estimator of the parametric components by employing a penalized quasi-likelihood. Thus our estimator is shown to have an asymptotic oracle property. Monte Carlo simulations show that the proposed procedure works well with moderate sample sizes.

Static models that are not identifiable in the presence of white noise measurement errors are known to be potentially identifiable when the model has dynamics. However, few results are available for the plausible case of serially correlated measurement errors. This paper provides order and rank conditions for “limited information” identification of parameters in dynamic models with measurement errors where some aspects of the probability model are not fully specified or utilized. The key is to consider a model for the contaminated data that has richer dynamics than the model for the correctly observed data. Simply counting the total number of unknown parameters in the true model relative to the estimable model will not yield an informative order condition for identification. Implications for single-equation, vector autoregressive, and panel data models are studied.

This paper estimates threshold regression models with an endogenous threshold variable using a nonparametric control function approach. Assuming diminishing threshold effects, we derive the consistency and limiting distribution of our proposed estimator constructed from the series approximation method for weakly dependent data. In addition, we propose a test for the endogeneity of the threshold variable, which is valid regardless of whether the threshold effects exist. We assess the performance of our methods using Monte Carlo simulations.

The paper develops a statistical theory for regressions with integrated regressors of unknown order and unknown cointegrating dimension. In practice, we are often unsure whether unit roots or cointegration is present in time series data, and we are also uncertain about the order of integration in some cases. This paper addresses issues of estimation and inference in cases of such uncertainty. Phillips (1995, Econometrica 63, 1023–1078) developed a theory for time series regressions with an unknown mixture of 1(0) and 1(1) variables and established that the method of fully modified ordinary least squares (FM-OLS) is applicable to models (including vector autoregressions) with some unit roots and unknown cointegrating rank. This paper extends these results to models that contain some I(0), I(1), and I(2) regressors. The theory and methods here are applicable to cointegrating regressions that include unknown numbers of I(0), I(1), and I(2) variables and an unknown degree of cointegration. Such models require a somewhat different approach than that of Phillips (1995). The paper proposes a residual-based fully modified ordinary least-squares (RBFMOLS) procedure, which employs residuals from a first-order autoregression of the first differences of the entire regressor set in the construction of the FMOLS estimator. The asymptotic theory for the RBFM-OLS estimator is developed and is shown to be normal for all the stationary coefficients and mixed normal for all the nonstationary coefficients. Under Gaussian assumptions, estimation of the cointegration space by RBFM-OLS is optimal even though the dimension of the space is unknown.

We obtain simple and generally applicable conditions for the existence of mixed moments E([X′ AX]″/[X′BX]U) of the ratio of quadratic forms T = X′ AX/X′BX where A and B are n × n symmetric matrices and X is a random n-vector. Our principal theorem is easily stated when X has an elliptically symmetric distribution, which class includes the multivariate normal and t distributions, whether degenerate or not. The result applies to the ratio of multivariate quadratic polynomials and can be expected to remain valid in most situations in which X is subject to linear constraints.

If u ≤ v, the precise distribution of X, and in particular the existence of moments of X, is virtually irrelevant to the existence of the mixed moments of T; if u > v, a prerequisite for existence of the (u, v)th mixed moment is the existence of the 2(u − v)th moment of X When Xis not degenerate, the principal further requirement for the existence of the mixed moment is that B has rank exceeding 2v.

In this paper we construct a statistic to test a specific form of the conditional density function. The main point of this work is to consider a functional explanatory variable, and the statistic is constructed following recent advances in nonparametric functional data analysis. The asymptotic behavior of the test statistic is studied under both the null hypothesis and some local alternative hypothesis. Then, the finite sample behavior of the method is studied through simulated examples. This paper is one of the first in the setting of nonparametric specification tests when functional data are involved.

This paper examines the ordinary least squares (OLS) estimator of the structural parameters in a simple macroeconomic model in which agents are boundedly rational and use an adaptive learning rule to form expectations of the endogenous variable. The popularity of learning models has recently increased amongst applied economists and policy makers who seek to estimate them empirically. Yet the econometrics of learning models is largely uncharted territory. We consider two prominent learning algorithms, namely constant gain and decreasing gain learning. For each of the two learning rules, our analysis proceeds in two stages. First, the paper derives the asymptotic properties of agents’ expectations. At the second stage, the paper derives the asymptotics of OLS in the structural model, taking the first stage learning dynamics as given. In the case of constant gain learning, the structural model effectively amounts to a stationary, cointegrating, or co-explosiveness regression. With decreasing gain learning, the regressors are asymptotically collinear such that OLS does not satisfy, in general, the Grenander conditions for consistent estimability. Nevertheless, this paper shows that the OLS estimator remains consistent in all models considered. It also shows, however, that its asymptotic distribution, and hence any inference based upon it, may be nonstandard.