For a general stationary ARMA(p,q) process u we derive the exact form of the orthogonalizing matrix R such that R′R = Σ−1, where Σ = E(uu′) is the covariance matrix of u, generalizing the known formulae for AR(p) processes. In a linear regression model with an ARMA(p,q) error process, transforming the data by R yields a regression model with white-noise errors. We also consider an application to semi-recursive (being recursive for the model parameters, but not for the parameters of the error process) estimation.

This paper proposes robust M-estimators of dynamic linear models with a structural break of unknown location. Rates of convergence and limiting distributions for the estimated shift point and the estimated regression parameters are derived. The analysis is carried out in the framework of possibly dependent observations and also with trending regressors. The asymptotic distribution of the break location estimator is obtained both for fixed magnitude of shift and for shift with magnitude converging to zero as the sample size increases. The latter is essential for the derivation of feasible confidence intervals for the break location. Monte Carlo simulations illustrate the performance of asymptotic inferences in practice.

Limit theory is developed for the dynamic panel IV estimator in the presence of an autoregressive root near unity. In the unit root case, Anderson–Hsiao lagged variable instruments satisfy orthogonality conditions but are well known to be irrelevant. For a fixed time series sample size (T) IV is inconsistent and approaches a shifted Cauchy-distributed random variate as the cross-section sample size n → ∞. But when T → ∞, either for fixed n or as n → ∞, IV is $\sqrt T$ consistent and its limit distribution is a ratio of random variables that converges to twice a standard Cauchy as n → ∞. In this case, the usual instruments are uncorrelated with the regressor but irrelevance does not prevent consistent estimation. The same Cauchy limit theory holds sequentially and jointly as (n, T) → ∞ with no restriction on the divergence rates of n and T. When the common autoregressive root $\rho = 1 + c/\sqrt T$ the panel comprises a collection of mildly integrated time series. In this case, the IV estimator is $\sqrt n$ consistent for fixed T and $\sqrt {nT}$ consistent with limit distribution N (0, 4) when (n, T) → ∞ sequentially or jointly. These results are robust for common roots of the form ρ = 1+c/Tγ for all γ ∈ (0, 1) and joint convergence holds. Limit normality holds but the variance changes when γ = 1. When γ > 1 joint convergence fails and sequential limits differ with different rates of convergence. These findings reveal the fragility of conventional Gaussian IV asymptotics to persistence in dynamic panel regressions.

This paper develops an asymptotic theory for estimated change-points in linear and nonlinear time series models. Based on a measurable objective function, it is shown that the estimated change-point converges weakly to the location of the maxima of a double-sided random walk and other estimated parameters are asymptotically normal. When the magnitude d of changed parameters is small, it is shown that the limiting distribution can be approximated by the known distribution as in Yao (1987, Annals of Statistics 15, 1321–1328). This provides a channel to connect our results with those in Picard (1985, Advances in Applied Probability 17, 841–867) and Bai, Lumsdaine, and Stock (1998, Review of Economic Studies 65, 395–432), where the magnitude of changed parameters depends on the sample size n and tends to zero as n → ∞. The theory is applied for the self-weighted QMLE and the local QMLE of change-points in ARMA-GARCH/IGARCH models. A simulation study is carried out to evaluate the performance of these estimators in the finite sample.

This paper considers the problem of predicting binary choices by selecting from a possibly large set of candidate explanatory variables, which can include both exogenous variables and lagged dependent variables. We consider risk minimization with the risk function being the predictive classification error. We study the convergence rates of empirical risk minimization in both the frequentist and Bayesian approaches. The Bayesian treatment uses a Gibbs posterior constructed directly from the empirical risk instead of using the usual likelihood-based posterior. Therefore these approaches do not require a correctly specified probability model. We show that the proposed methods have near optimal performance relative to a class of linear classification rules with selected variables. Such results in classification are obtained in a framework of dependent data with strong mixing.

This note considers a panel data regression model with spatial autoregressive disturbances and random effects where the weight matrix is normalized and has equal elements. This is motivated by Kelejian, Prucha, and Yuzefovich (2005, Journal of Regional Science, forthcoming), who argue that such a weighting matrix, having blocks of equal elements, might be considered when units are equally distant within certain neighborhoods but unrelated between neighborhoods. We derive a simple weighted least squares transformation that obtains generalized least squares (GLS) on this model as a simple ordinary least squares (OLS). For the special case of a spatial panel model with no random effects, we obtain two sufficient conditions where GLS on this model is equivalent to OLS. Finally, we show that these results, for the equal weight matrix, hold whether we use the spatial autoregressive specification, the spatial moving average specification, the spatial error components specification, or the Kapoor, Kelejian, and Prucha (2005, Journal of Econometrics, forthcoming) alternative to modeling panel data with spatially correlated error components.I thank Paolo Paruolo and an anonymous referee for helpful comments and suggestions.

In this paper, we prove the validity of an Edgeworth expansion to the distribution of the Whittle maximum likelihood estimator for stationary long-memory Gaussian models with unknown parameter . The error of the (s − 2)-order expansion is shown to be o(n−(s−2)/2)—the usual independent and identically distributed rate—for a wide range of models, including the popular ARFIMA(p,d,q) models. The expansion is valid under mild assumptions on the behavior of the spectral density and its derivatives in the neighborhood of the origin. As a by-product, we generalize a theorem by Fox and Taqqu (1987, Probability Theory and Related Fields 74, 213–240) concerning the asymptotic behavior of Toeplitz matrices.

Lieberman, Rousseau, and Zucker (2003, Annals of Statistics 31, 586–612) establish a valid Edgeworth expansion for the maximum likelihood estimator for stationary long-memory Gaussian models. For a significant class of models, their expansion is shown to have an error of o(n−1). The results given here improve upon those of Lieberman et al. in that the results provide an Edgeworth expansion for an asymptotically efficient estimator, as Lieberman et al. do, but the error of the expansion is shown to be o(n−(s−2)/2), not o(n−1), for a broad range of models.

We examine a kernel regression estimator for time series that takes account of the error correlation structure as proposed by Xiao et al. (2003, Journal of the American Statistical Association 98, 980–992). We show that this method continues to improve estimation in the case where the regressor is a unit root or a near unit root process.

Granger noncausality in distribution is fundamentally a probabilistic conditional independence notion that can be applied not only to time series data but also to cross-section and panel data. In this paper, we provide a natural definition of structural causality in cross-section and panel data and forge a direct link between Granger (G–) causality and structural causality under a key conditional exogeneity assumption. To put it simply, when structural effects are well defined and identifiable, G–non-causality follows from structural noncausality, and with suitable conditions (e.g., separability or monotonicity), structural causality also implies G–causality. This justifies using tests of G–non-causality to test for structural noncausality under the key conditional exogeneity assumption for both cross-section and panel data. We pay special attention to heterogeneous populations, allowing both structural heterogeneity and distributional heterogeneity. Most of our results are obtained for the general case, without assuming linearity, monotonicity in observables or unobservables, or separability between observed and unobserved variables in the structural relations.

We provide new conditions for identification of accelerated failure time competing risks models. These include Roy models and some auction models. In our setup, unknown regression functions and the joint survivor function of latent disturbance terms are all nonparametric. We show that this model is identified given covariates that are independent of latent errors, provided that a certain rank condition is satisfied. We present a simple example in which our rank condition for identification is verified. Our identification strategy does not depend on identification at infinity or near zero, and it does not require exclusion assumptions. Given our identification, we show estimation can be accomplished using sieves.

It is known that Efron’s bootstrap of the mean of a distribution in the domain of attraction of the stable laws with infinite variance is not consistent, in the sense that the limiting distribution of the bootstrap mean is not the same as the limiting distribution of the mean from the real sample. Moreover, the limiting bootstrap distribution is random and unknown. The conventional remedy for this problem, at least asymptotically, is either the m out of n bootstrap or subsampling. However, we show that both these procedures can be unreliable in other than very large samples. We introduce a parametric bootstrap that overcomes the failure of Efron’s bootstrap and performs better than the m out of n bootstrap and subsampling. The quality of inference based on the parametric bootstrap is examined in a simulation study, and is found to be satisfactory with heavy-tailed distributions unless the tail index is close to 1 and the distribution is heavily skewed.

Economic data are frequently generated by stochastic processes that can be modeled as realizations of random functions (functional data). This paper adapts the specification test for functional data developed by Bugni, Hall, Horowitz, and Neumann (2009, Econometrics Journal12, S1–S18) to the presence of missing observations. By using a worst case scenario approach, our method is able to extract the information available in the observed portion of the data while being agnostic about the nature of the missing observations. The presence of missing data implies that our test will not only result in the rejection or lack of rejection of the null hypothesis, but it may also be inconclusive.

Under the null hypothesis, our specification test will reject the null hypothesis with a probability that, in the limit, does not exceed the significance level of the test. Moreover, the power of the test converges to one whenever the distribution of the observations conveys that the null hypothesis is false.

Monte Carlo evidence shows that the test may produce informative results (either rejection or lack of rejection of the null hypothesis) even under the presence of significant amounts of missing data. The procedure is illustrated by testing whether the Burdett–Mortensen labor market model is the correct framework for wage paths constructed from the National Longitudinal Survery of Youth, 1979 survey.

The tail index as a measure of tail thickness provides information that is not captured by standard volatility measures. It may however change over time. Currently available procedures for detecting those changes for dependent data (e.g., Quintos et al., 2001) are all based on comparing Hill (1975) estimates from different subsamples. We derive tests for a wide class of other tail index estimators. The limiting distribution of the test statistics is shown not to depend on the particular choice of the estimator, while the assumptions on the dependence structure allow for sufficient generality in applications. A simulation study investigates empirical sizes and powers of the tests in finite samples.

Conditions ensuring a central limit theorem for strongly mixing triangular arrays are given. Larger samples can show longer range dependence than shorter samples. The result is obtained by constraining the rate growth of dependence as a function of the sample size, with the usual trade-off of memory and moment conditions. An application to heteroskedasticity and autocorrelation consistent estimators is proposed.

We investigate the properties of several bootstrap-based inference procedures for semiparametric density-weighted average derivatives. The key innovation in this paper is to employ an alternative asymptotic framework to assess the properties of these inference procedures. This theoretical approach is conceptually distinct from the traditional approach (based on asymptotic linearity of the estimator and Edgeworth expansions), and leads to different theoretical prescriptions for bootstrap-based semiparametric inference. First, we show that the conventional bootstrap-based approximations to the distribution of the estimator and its classical studentized version are both invalid in general. This result shows a fundamental lack of “robustness” of the associated, classical bootstrap-based inference procedures with respect to the bandwidth choice. Second, we present a new bootstrap-based inference procedure for density-weighted average derivatives that is more “robust” to perturbations of the bandwidth choice, and hence exhibits demonstrable superior theoretical statistical properties over the traditional bootstrap-based inference procedures. Finally, we also examine the validity and invalidity of related bootstrap-based inference procedures and discuss additional results that may be of independent interest. Some simulation evidence is also presented.

This paper presents a method for estimating a class of panel data duration models, under which an unknown transformation of the duration variable is linearly related to the observed explanatory variables and the unobserved heterogeneity (or frailty) with completely known error distributions. This class of duration models includes a panel data proportional hazards model with fixed effects. The proposed estimator is shown to be n1/2-consistent and asymptotically normal with dependent right censoring. The paper provides some discussions on extending the estimator to the cases of longer panels and multiple states. Some Monte Carlo studies are carried out to illustrate the finite-sample performance of the new estimator.

A model M is said to encompass another model N if the former can explain the results obtained by the latter. In this paper, we propose a general notion of encompassing that covers both classical and Bayesian viewpoints and essentially represents a concept of sufficiency among models. We introduce the parent notion of specificity that aims at measuring lack of encompassing. Tests for encompassing are discussed and the test statistics are compared to Bayesian posterior odds. Operational approximations are offered to cover situations where exact solutions cannot be obtained.

In this paper, we consider a linear regression model when relevant regressors are omitted. We derive the explicit formulae for the predictive mean squared errors (PMSEs) of the Stein-rule (SR) estimator, the positive-part Stein-rule (PSR) estimator, the minimum mean squared error (MMSE) estimator, and the adjusted minimum mean squared error (AMMSE) estimator. It is shown analytically that the PSR estimator dominates the SR estimator in terms of PMSE even when there are omitted relevant regressors. Also, our numerical results show that the PSR estimator and the AMMSE estimator have much smaller PMSEs than the ordinary least squares estimator even when the relevant regressors are omitted.

This paper considers a continuous time unobserved components model in which the cyclical component follows a differential-difference equation whereas the trend and seasonal components follow more standard differential equations. Estimation of the parameters of the model with either a stock or a flow variable is analyzed using a frequency domain Gaussian estimator whose asymptotic properties are derived paying particular attention to the role of a truncation parameter that arises in the practical computation of the spectral density function. The results of a simulation exercise, which assesses the finite sample performance of the estimator, are also provided.

The family of multiplicative error models is important for studying non-negative variables such as realized volatility, trading volume, and duration between consecutive financial transactions. Methods are developed for testing the parametric specification of a multiplicative error model, which consists of separate parametric models for the conditional mean and the error distribution. The same method can also be used for testing the specification of the error distribution provided the conditional mean is correctly specified. A bootstrap method is proposed for computing the p-values of the tests and is shown to be consistent. The proposed tests have nontrivial asymptotic power against a class of O(n−1/2)-local alternatives. The tests performed well in a simulation study, and they are illustrated using a data example on realized volatility.