The estimation of K (K ≥ 3) location parameters is considered under quadratic loss when the coordinates of the best invariant estimators are spherically symmetrically distributed. Under these stochastic mechanisms traditional Stein estimators are evaluated for finite samples and shown to have a risk performance superior to some conventional rules.

A limit theory for instrumental variables (IV) estimation that allows for possibly nonstationary processes was developed in Kitamura and Phillips (1992, Fully Modified IV, GIVE, and GMM Estimation with Possibly Non-stationary Regressors and Instruments, mimeo, Yale University). This theory covers a case that is important for practitioners, where the nonstationarity of the regressors may not be of full rank, and shows that the fully modified (FM) regression procedure of Phillips and Hansen (1990) is still applicable. FM. versions of the generalized method of moments (GMM) estimator and the generalized instrumental variables estimator (GIVE) were also developed, and these estimators (FM-GMM and FM-GIVE) were designed specifically to take advantage of potential stationarity in the regressors (or unknown linear combinations of them). These estimators were shown to deliver efficiency gains over FM-IV in the estimation of the stationary components of a model.

This paper provides an overview of the FM-IV, FM-GMM, and FM-GIVE procedures and investigates the small sample properties of these estimation procedures by simulations. We compare the following five estimation methods: ordinary least squares, crude (conventional) IV, FM-IV, FM-GMM, and FM-GIVE. Our findings are as follows, (i) In terms of overall performance in both stationary and nonstationary cases, FM-IV is more concentrated and better centered than OLS and crude IV, though it has a higher root mean square error than crude IV due to occasional outliers, (ii) Among FM-IV, FM-GMM, and FM-GIVE, (a) when applied to the stationary coefficients, FM-GIVE generally outperforms FM-IV and FM-GMM by a wide margin, whereas the difference between the latter two is quite small when the AR roots of the stationary processes are rather large; and (b) when applied to the nonstationary coefficients, the three estimators are numerically very close. The performance of the FM-GIVE estimator is generally very encouraging.

This article deals with detection of a nonconstant long memory parameter in time series. The null hypothesis presumes stationary or nonstationary time series with a constant long memory parameter, typically an I (d) series with d > −.5 . The alternative corresponds to an increase in persistence and includes in particular an abrupt or gradual change from I (d1) to I (d2), −.5 < d1 < d2. We discuss several test statistics based on the ratio of forward and backward sample variances of the partial sums. The consistency of the tests is proved under a very general setting. We also study the behavior of these test statistics for some models with a changing memory parameter. A simulation study shows that our testing procedures have good finite sample properties and turn out to be more powerful than the KPSS-based tests (see Kwiatkowski, Phillips, Schmidt and Shin, 1992) considered in some previous works.

Because of the incidental parameters problem, the fixed effects maximum likelihood estimator in a nonlinear panel data model is in general inconsistent when the time series length T is short and fixed. Even if T approaches infinity but at a rate not faster than the cross sectional sample size n, the fixed effects estimator is still asymptotically biased. This paper proposes using the standard bootstrap and k-step bootstrap to correct the bias. We establish the asymptotic validity of the bootstrap bias corrections for both model parameters and average marginal effects. Our results apply to static models as well as some dynamic Markov models. Monte Carlo simulations show that our procedures are effective in reducing the bias of the fixed effects estimator and improving the coverage accuracy of the associated confidence interval.

This paper proposes a notion of near cointegration and generalizes several existing results from the cointegration literature to the case of near cointegration. In particular, the properties of conventional cointegration methods under near cointegration are characterized, thereby investigating the robustness of cointegration methods. In addition, we obtain local asymptotic power functions of five cointegration tests that take cointegration as the null hypothesis.

It was June 1940. The Second World War was already raging in full swing in its devastation. A young man not yet 20 set out on a 500-mile train journey from the coastal city of Vishakapatnam, India, to Calcutta, the second largest city of the British Empire, after obtaining a first-class first degree in mathematics and with a glimmer of hope of finding a job in the military. The young man was not so lucky; he was deemed too young for the job. However, while in Calcutta, through a chance encounter, he visited the Indian Statistical Institute (ISI) founded in 1931 by Professor Prasanta Chandra Mahalanobis, a Cambridge-trained physicist. As a last resort he applied for a one-year training program in statistics there. Very promptly he received a positive reply from Professor Mahalanobis admitting him to the Training Section of the ISI from January 1, 1941. This young man was none other than perhaps the most well known statistician in the world today, Calyampudi Radhakrishna Rao (or simply Dr. Rao, as he was universally known to his students and colleagues at the ISI. The “professor” title was reserved for Mahalanobis). Such was the beginning of his spectacular career in statistics, a field chosen as a last resort, and he never looked back.

In this paper, some structural properties of a family of generalized autoregressive conditionally heteroskedastic (GARCH) processes are considered. First, a sufficient and necessary condition for the strict stationarity of this family of GARCH processes is given. Second, some simple conditions for the existence of the moments of the family of GARCH processes are also derived. Finally, we describe the tail of the marginal distribution of the family of GARCH processes.I am grateful to Bruce E. Hansen and an anonymous referee for helpful comments on the initial version.

We consider a panel cointegration model with latent group structures that allows for heterogeneous long-run relationships across groups. We extend Su, Shi, and Phillips (2016, Econometrica 84(6), 2215–2264) classifier-Lasso (C-Lasso) method to the nonstationary panels and allow for the presence of endogeneity in both the stationary and nonstationary regressors in the model. In addition, we allow the dimension of the stationary regressors to diverge with the sample size. We show that we can identify the individuals’ group membership and estimate the group-specific long-run cointegrated relationships simultaneously. We demonstrate the desirable property of uniform classification consistency and the oracle properties of both the C-Lasso estimators and their post-Lasso versions. The special case of dynamic penalized least squares is also studied. Simulations show superb finite sample performance in both classification and estimation. In an empirical application, we study the potential heterogeneous behavior in testing the validity of long-run purchasing power parity (PPP) hypothesis in the post–Bretton Woods period from 1975–2014 covering 99 countries. We identify two groups in the period 1975–1998 and three groups in the period 1999–2014. The results confirm that at least some countries favor the long-run PPP hypothesis in the post–Bretton Woods period.

This paper adopts a two-step technique to estimate structural vector error correction models and provides the asymptotic distribution of the impulse response functions of such a system. The method combines two popular tools in econometrics, namely, vector autoregressive cointegration analysis in the first step and structural vector autoregression analysis in the second. The proposed structural model structure is very general in the sense that all just-identifying or overidentifying schemes that can be expressed as linear restrictions on either the contemporaneous or long-run impact of the structural shocks are allowed for. The long-run restrictions complicate the derivation of the asymptotic distribution of the structural parameter estimates as these restrictions are a function of the reduced form parameters. Consequently, the asymptotic distribution involves an extra partial derivative.Useful comments by Peter Boswijk, Günter Coenen, Neil Ericsson, Sören Johansen, Klaus Neusser, Franz Palm, Paolo Paruolo, Peter van Els, Anders Warne, and ESEM 1998 participants are gratefully acknowledged. The paper also significantly benefited from suggestions by the co-editor Pentti Saikkonen and two anonymous referees.

This paper describes an estimator of the additive components of a nonparametric additive model with an unknown link function. When the additive components and link function are twice differentiable with sufficiently smooth second derivatives, the estimator is asymptotically normally distributed with a rate of convergence in probability of n−2/5. This is true regardless of the (finite) dimension of the explanatory variable. Thus, the estimator has no curse of dimensionality. Moreover, the asymptotic distribution of the estimator of each additive component is the same as it would be if the link function and the other components were known with certainty. Thus, asymptotically there is no penalty for not knowing the link function or the other components.

This paper studies the robust estimation and inference of threshold models with integrated regressors. We derive the asymptotic distribution of the profiled least squares (LS) estimator under the diminishing threshold effect assumption that the size of the threshold effect converges to zero. Depending on how rapidly this sequence converges, the model may be identified or only weakly identified and asymptotic theorems are developed for both cases. As the convergence rate is unknown in practice, a model-selection procedure is applied to determine the model identification strength and to construct robust confidence intervals, which have the correct asymptotic size irrespective of the magnitude of the threshold effect. The model is then generalized to incorporate endogeneity and serial correlation in error terms, under which, we design a Cochrane–Orcutt feasible generalized least squares (FGLS) estimator which enjoys efficiency gains and robustness against different error specifications, including both I(0) and I(1) errors. Based on this FGLS estimator, we further develop a sup-Wald statistic to test for the existence of the threshold effect. Monte Carlo simulations show that our estimators and test statistics perform well.

I attempt to present Denis Sargan's work in some kind of historical perspective, in two ways. First, I discuss some previous members of the Tooke Chair of Economic Science and Statistics, which was founded in 1859 and which Sargan held. Second, I discuss one of his articles, “Asymptotic Theory and Large Models,” in relation to modern preoccupations with semiparametric econometrics.This research was supported by a Leverhulme Trust Personal Research Professorship and ESRC grant R000239252. I thank Peter Phillips for helpful comments and Javier Hualde for obtaining material employed in Section 2. Section 2 is based partly on my Doctor Honoris Causa investiture address at Carlos III Madrid University, in October 2000.

We introduce a semiparametric estimator for the censored linear regression model. It is based on the regression version of Huber's [6] M-estimator. It includes Powell's [19] censored least absolute deviations estimator as a special case and is related to Powell's [20] symmetrically censored least-squares estimator. We prove strong consistency and derive its asymptotic distribution which is √n-consistent with an easily computable covariance matrix. A small-scale simulation study shows that it works quite well in various cases.

This paper studies the GMM estimation and inference problem that occurs when the Jacobian of the moment conditions is rank deficient of known forms at the true parameter values. Dovonon and Renault (2013) recently raised a local identification issue stemming from this type of degenerate Jacobian. The local identification issue leads to a slow rate of convergence of the GMM estimator and a nonstandard asymptotic distribution of the over-identification test statistics. We show that the known form of rank-deficient Jacobian matrix contains nontrivial information about the economic model. By exploiting such information in estimation, we provide GMM estimator and over-identification tests with standard properties. The main theory developed in this paper is applied to the estimation of and inference about the common conditionally heteroskedastic (CH) features in asset returns. The performances of the newly proposed GMM estimators and over-identification tests are investigated under the similar simulation designs used in Dovonon and Renault (2013).

We consider the problem of estimating the value ℓ(ϕ) of a linear functional, where the structural function ϕ models a nonparametric relationship in presence of instrumental variables. We propose a plug-in estimator which is based on a dimension reduction technique and additional thresholding. It is shown that this estimator is consistent and can attain the minimax optimal rate of convergence under additional regularity conditions. This, however, requires an optimal choice of the dimension parameter m depending on certain characteristics of the structural function ϕ and the joint distribution of the regressor and the instrument, which are unknown in practice. We propose a fully data driven choice of m which combines model selection and Lepski’s method. We show that the adaptive estimator attains the optimal rate of convergence up to a logarithmic factor. The theory in this paper is illustrated by considering classical smoothness assumptions and we discuss examples such as pointwise estimation or estimation of averages of the structural function ϕ.

We propose a statistical model to assess whether individuals strategically use mixed strategies in repeated games. We formulate a hidden Markov model in which the latent state space contains both pure and mixed strategies. We apply the model to data from an experiment in which human subjects repeatedly play a normal form game against a computer that always follows its part of the unique mixed strategy Nash equilibrium profile. Estimated results show significant mixed strategy play and nonstationary dynamics. We also explore the ability of the model to forecast action choice.

This article develops inferential methods for a very general class of ill-posed models in econometrics encompassing the nonparametric instrumental variable regression, various functional regressions, and the density deconvolution. We focus on uniform confidence sets for the parameter of interest estimated with Tikhonov regularization, as in Darolles et al. (2011, Econometrica 79, 1541–1565). Since it is impossible to have inferential methods based on the central limit theorem, we develop two alternative approaches relying on the concentration inequality and bootstrap approximations. We show that expected diameters and coverage properties of resulting sets have uniform validity over a large class of models, that is, constructed confidence sets are honest. Monte Carlo experiments illustrate that introduced confidence sets have reasonable width and coverage properties. Using U.S. data, we provide uniform confidence sets for Engel curves for various commodities.

Recursive relations for objects of statistical interest have long been important for computation, and they remain so even with hugely improved computing power. Such recursions are frequently derived by exploiting relations between generating functions. For example, the top-order zonal polynomials that occur in much distribution theory under normality can be recursively related to other (easily computed) symmetric functions (power-sum and elementary symmetric functions; Ruben, 1962, Annals of Mathematical Statistics 33, 542–570; Hillier, Kan, and Wang, 2009, Econometric Theory 25, 211–242). Typically, in a recursion of this type the kth object of interest, dk, say, is expressed in terms of all lower order dj’s. In Hillier et al. (2009) we pointed out that, in the case of top-order zonal polynomials and other invariant polynomials of multiple matrix argument, a fixed length recursion can be deduced. We refer to this as a short recursion. The present paper shows that the main results in Hillier et al. (2009) can be generalized and that short recursions can be obtained for a much larger class of objects/generating functions. As applications, we show that short recursions can be obtained for various problems involving quadratic forms in noncentral normal vectors, including moments, product moments, and expectations of ratios of powers of quadratic forms. For this class of problems, we also show that the length of the recursion can be further reduced by an application of a generalization of Horner’s method (cf. Brown, 1986, SIAM Journal on Scientific and Statistical Computing 7, 689–695), producing a super-short recursion that is significantly more efficient than even the short recursion.

In this paper we study the convergence to fractional Brownian motion for long memory time series having independent innovations with infinite second moment. For the sake of applications we derive the self-normalized version of this theorem. The study is motivated by models arising in economic applications where often the linear processes have long memory, and the innovations have heavy tails.

In this article, we propose a new nonparametric density estimator derived from the theory of frames and Riesz bases. In particular, we propose the so-called bi-orthogonal density estimator based on the class of B-splines and derive its theoretical properties, including the asymptotically optimal choice of bandwidth. Detailed theoretical analysis and comparisons of our estimator with existing local basis and kernel density estimators are presented. The estimator is particularly well suited for high-frequency data analysis in financial and economic markets.