estimation and inference with weak semi strong and strong identification this paper analyzes the properties of standard estimators tests and confidence sets cs s for parameters that are unidentified or weakly identified in some parts of the parameter space the paper also introduces methods to make the tests and cs s robust to such identification problems the results apply to a class of extremum estimators and corresponding tests and cs s that are based on criterion functions that satisfy certain asymptotic stochastic quadratic expansions and that depend on the parameter that determines the strength of identification this covers a class of models estimated using maximum likelihood ml least squares ls quantile generalized method of moments generalized empirical likelihood minimum distance and semi parametric estimators the consistency lack of consistency and asymptotic distributions of the estimators are established under a full range of drifting sequences of true distributions the asymptotic sizes in a uniform sense of standard and identification robust tests and cs s are established the results are applied to the arma 1 1 time series model estimated by ml and to the nonlinear regression model estimated by ls in companion papers the results are applied to a number of other models

interval forecasts and parameter uncertainty forecast intervals generalize point forecasts to represent and incorporate uncertainty forecast intervals calculated from dynamic models typically sidestep the issue of parameter estimation this paper shows how to construct asymptotic forecast intervals which incorporate the uncertainty due to parameter estimation our proposed solution is a simple proportional adjustment to the interval endpoints the adjustment factor depending on the asymptotic variance of the interval estimates our analysis is in the context of a forecasting equation with an error independent of the forecasting variables but with unknown distribution the methods are illustrated with a simulation experiment and an application to the us monthly unemployment rate

testing the order of a model using locally conic parametrization population mixtures and stationary arma processes in this paper we address the problem of testing hypotheses using the likelihood ratio test statistic in nonidentifiable models with application to model selection in situations where the parametrization for the larger model leads to nonidentifiability in the smaller model we give two major applications the case where the number of populations has to be tested in a mixture and the case of stationary arma p q processes where the order p q has to be tested we give the asymptotic distribution for the likelihood ratio test statistic when testing the order of the model in the case of order selection for armas the asymptotic distribution is invariant with respect to the parameters generating the process a locally conic parametrization is a key tool in deriving the limiting distributions it allows one to discover the deep similarity between the two problems

testing linearity against smooth transition autoregressive models summary we study a general univariate smooth transition autoregressive star model it contains as a special case the self exciting threshold autoregressive setar model we present three tests for testing linearity against star models and discuss their properties the power of the tests in small samples is investigated by simulation when the alternative is the logistic star model one of the tests is identical to tsay s 1986 test statistic and is recommended only in a special case of the two remaining tests with wider applicability one seems superior to the other in small samples it is also more powerful than the cusum test recently proposed for testing linearity against setar models

one dimensional front tracking based on high resolution wave propagation methods a simple approach to shock tracking is presented in conjunction with conservative high resolution shock capturing methods in one space dimension an underlying uniform grid is used with additional grid interfaces introduced at appropriate points for tracked shocks conservative high resolution methods based on the large time step wave propagation approach are used on the resulting nonuniform grid this method is stable even if some of the small cells created by the tracked interface are orders of magnitude smaller than the regular cells used to determine the time step a fractional step method is used to handle source terms several calculations are presented to demonstrate the effectiveness of the method including an unstable detonation wave calculation where mesh refinement in the reaction zone is required in addition to shock tracking stability and accuracy results of the method are also shown for some sample problems the basic ideas described here can be extended to two space dimensions as will be

testing for unobserved heterogeneity in exponential and weibull duration models abstract we examine the use of the likelihood ratio lr statistic to test for unobserved heterogeneity in duration models based on mixtures of exponential or weibull distributions we consider both the uncensored and censored duration cases the asymptotic null distribution of the lr test statistic is not the standard chi square as the standard regularity conditions do not hold instead there is a nuisance parameter identified only under the alternative and a null parameter value on the boundary of the parameter space as in cho and white 2007a we accommodate these and provide methods delivering consistent asymptotic critical values we conduct a number of monte carlo simulations comparing the level and power of the lr test statistic to an information matrix im test due to chesher 1984 and lagrange multiplier lm tests of kiefer 1985 and sharma 1987 our simulations show that the lr test statistic generally outperforms the im and lm tests we also revisit the work of van den berg and ridder 1998 on unemployment durations and of ghysels et al 2004 on interarrival times between stock trades and as it turns out affirm their original informal inferences

mixtures of t distributions for finance and forecasting abstract we explore convenient analytic properties of distributions constructed as mixtures of scaled and shifted t distributions a feature that makes this family particularly desirable for econometric applications is that it possesses closed formexpressions for its anti derivatives e g the cumulative density function we illustrate the usefulness of these distributions in two applications in the first application we use a scaled and shifted t distribution to produce density forecasts of u s inflation and show that these forecasts are more accurate out of sample than density forecasts obtained using normal or standard t distributions in the second application we replicate the option pricing exercise of abadir and rockinger 2003 using a mixture of scaled and shifted t distributions and obtain comparably good results while gaining analytical tractability

approximation properties of bezier summation integral type operators based on polya bernstein functions in this article we introduce the bezier variant of summation integral type operators having polya and bernstein basis functions we give a direct approximation theorem by means of the first order modulus of smoothness and the rate of convergence for absolutely continuous functions having a derivative equivalent to a function of bounded variation

inference when a nuisance parameter is not identified under the null hypothesis many econometric testing problems involve nuisance parameters which are not identified under the null hypotheses this paper studies the asymptotic distribution theory for such tests the asymptotic distributions of standard test statistics are described as functionals of chi square processes in general the distributions depend upon a large number of unknown parameters we show that a transformation based upon a conditional probability measure yields an asymptotic distribution free of nuisance parameters and we show that this transformation can be easily approximated via simulation the theory is applied to threshold models with special attention given to the so called self exciting threshold autoregressive model monte carlo methods are used to assess the finite sample distributions the tests are applied to u s gnp growth rates and we find that potter s 1995 threshold effect in this series can be possibly explained by sampling variation

statistical approximation of a kind of kantorovich type q szasz mirakjan operators abstract in the present paper we introduce a kantorovich type generalization of the q szasz mirakjan operators and investigate their a statistical approximation properties also the rate of a statistical convergence of the proposed sequence of linear positive operators is obtained by a weighted modulus of continuity

on statistical approximation of a general class of positive linear operators extended in q calculus in this paper we present a general class of positive linear operators of discrete type based on q calculus and we investigate their weighted statistical approximation properties by using a bohman korovkin type theorem we also mark out two particular cases of this general class of operators

testing when a parameter is on the boundary of the maintained hypothesis this paper considers testing problems where several of the standard regularity conditions fail to hold we consider the case where i parameter vectors in the null hypothesis may lie on the boundary of the maintained hypothesis and ii there may be a nuisance parameter that appears under the alternative hypothesis but not under the null the paper establishes the asymptotic null and local alternative distributions of quasi likelihood ratio rescaled quasi likelihood ratio wald and score tests in this case the results apply to tests based on a wide variety of extremum estimators and apply to a wide variety of models examples treated in the paper are i tests of the null hypothesis of no conditional heteroskedasticity in a garch 1 1 regression model and ii tests of the null hypothesis that some random coefficients have variances equal to zero in a random coefficients regression model with possibly correlated random coefficients

weighted statistical approximation by kantorovich type q szasz mirakjan operators abstract in the present paper we introduce a kantorovich type modification of q szasz mirakjan operators and obtain weighted statistical approximation properties of these operators also for introduced operators we give a voronovskaja type theorem related to q derivatives

on the integrated baskakov type operators the present paper deals with the study of the rate of convergence of the integrated baskakov durrmeyer type operators in simultaneous approximation we also mention some of the improvements on the recent paper of deo n deo direct results on exponential type operators applied mathematics and computation 204 1 2008 109 115

higher order approximations for testing neglected nonlinearity we illustrate the need to use higher order specifically sixth order expansions in order to properly determine the asymptotic distribution of a standard artificial neural network test for neglected nonlinearity the test statistic is a quasi likelihood ratio qlr statistic designed to test whether the mean square prediction error improves by including an additional hidden unit with an activation function violating the no zero condition in cho ishida and white 2011 this statistic is also shown to be asymptotically equivalent under the null to the lagrange multiplier lm statistic of luukkonen saikkonen and terasvirta 1988 and terasvirta 1994 in addition we compare the power properties of our qlr test to one satisfying the no zero condition and find that the latter is not consistent for detecting a dgp with neglected nonlinearity violating an analogous no zero condition whereas our qlr test is consistent

testing for regime switching we analyze use of a quasi likelihood ratio statistic for a mixture model to test the null hypothesis of one regime versus the alternative of two regimes in a markov regime switching context this test exploits mixture properties implied by the regime switching process but ignores certain implied serial correlation properties when formulated in the natural way the setting is nonstandard involving nuisance parameters on the boundary of the parameter space nuisance parameters identified only under the alternative or approximations using derivatives higher than second order we exploit recent advances by andrews 2001 and contribute to the literature by extending the scope of mixture models obtaining asymptotic null distributions different from those in the literature we further provide critical values for popular models or bounds for tail probabilities that are useful in constructing conservative critical values for regime switching tests we compare the size and power of our statistics to other useful tests for regime switching via monte carlo methods and find relatively good performance we apply our methods to reexamine the classic cartel study of porter 1983 and reaffirm porter s findings

generalized baskakov szasz type operators abstract in the present paper we introduce generalized baskakov szasz type operators and study some approximation properties of these operators e g rate of convergence in ordinary and simultaneous approximation statistical convergence and the estimate of the rate of convergence for absolutely continuous functions having a derivative coinciding a e with a function of bounded variation

durrmeyer type modification of generalized baskakov operators abstract in the present paper durrmeyer type modification of generalized baskakov operators is introduced and some direct results are established for these operators

weighted approximation by a certain family of summation integral type operators in this study we investigate the weighted approximation properties of a general sequence of summation integral type operators introduced by srivastava and gupta 1 we estimate the rate of convergence of these operators for functions of polynomial growth in terms of weighted modulus of continuity on the interval 0

improved rates and asymptotic normality for nonparametric neural network estimators we obtain an improved approximation rate in sobolev norm of r sup 1 2 spl alpha d 1 for a large class of single hidden layer feedforward artificial neural networks ann with r hidden units and possibly nonsigmoid activation functions when the target function satisfies certain smoothness conditions here d is the dimension of the domain of the target function and spl alpha spl isin 0 1 is related to the smoothness of the activation function when applying this class of anns to nonparametrically estimate train a general target function using the method of sieves we obtain new root mean square convergence rates of op n log n sup 1 2 spl alpha d 1 4 1 spl alpha d 1 op n sup 1 4 by letting the number of hidden units spl tau sub n increase appropriately with the sample size number of training examples n these rates are valid for i i d data as well as for uniform mixing and absolutely regular spl beta mixing stationary time series data in addition the rates are fast enough to deliver root n asymptotic normality for plug in estimates of smooth functionals using general ann sieve estimators as interesting applications to nonlinear time series we establish rates for ann sieve estimators of four different multivariate target functions a conditional mean a conditional quantile a joint density and a conditional density we also obtain root n asymptotic normality results for semiparametric model coefficient and average derivative estimators

specification estimation and evaluation of smooth transition autoregressive models abstract this article considers the application of two families of nonlinear autoregressive models the logistic lstar and exponential estar autoregressive models this includes the specification of the model based on simple statistical tests linearity testing against smooth transition autoregression determining the delay parameter and choosing between lstar and estar models are discussed estimation by nonlinear least squares is considered as well as evaluating the properties of the estimated model the proposed techniques are illustrated by examples using both simulated and real time series

revisiting tests for neglected nonlinearity using artificial neural networks tests for regression neglected nonlinearity based on artificial neural networks anns have so far been studied by separately analyzing the two ways in which the null of regression linearity can hold this implies that the asymptotic behavior of general ann based tests for neglected nonlinearity is still an open question here we analyze a convenient ann based quasi likelihood ratio statistic for testing neglected nonlinearity paying careful attention to both components of the null we derive the asymptotic null distribution under each component separately and analyze their interaction somewhat remarkably it turns out that the previously known asymptotic null distribution for the type 1 case still applies but under somewhat stronger conditions than previously recognized we present monte carlo experiments corroborating our theoretical results and showing that standard methods can yield misleading inference when our new stronger regularity conditions are violated

testing linearity using power transforms of regressors we develop a method of testing linearity using power transforms of regressors allowing for stationary processes and time trends the linear model is a simplifying hypothesis that derives from the power transform model in three different ways each producing its own identification problem we call this modeling difficulty the trifold identification problem and show that it may be overcome using a test based on the quasi likelihood ratio qlr statistic more specifically the qlr statistic may be approximated under each identification problem and the separate null approximations may be combined to produce a composite approximation that embodies the linear model hypothesis the limit theory for the qlr test statistic depends on a gaussian stochastic process in the important special case of a linear time trend regressor and martingale difference errors asymptotic critical values of the test are provided test power is analyzed and an empirical application to crop yield distributions is provided the paper also considers generalizations of the box cox transformation which are associated with the qlr test statistic

ridgelets estimating with ridge functions feedforward neural networks projection pursuit regression and more generally estimation via ridge functions have been proposed as an approach to bypass the curse of dimensionality and are now becoming widely applied to approximation or prediction in applied sciences to address problems inherent to these methods ranging from the construction of neural networks to their efficiency and capability candes appl comput armon anal 6 1999 197 218 developed a new system that allows the representation of arbitrary functions as superpositions of specific ridge functions the ridgelets in a nonparametric regression setting this article suggests expanding noisy data into a ridgelet series and applying a scalar nonlinearity to the coefficients damping this is unlike existing approaches based on stepwise additions of elements the procedure is simple constructive stable and spatially adaptive and fast algorithms have been developed to implement it the ridgelet estimator is nearly optimal for estimating functions with certain kinds of spatial inhomogeneities in addition ridgelets help to identify new classes of estimands corresponding to a new notion of smoothness that are well suited for ridge functions estimation while the results are stated in a decision theoretic framework numerical experiments are also presented to illustrate the practical performance of the methodology

non linear regression with discrete explanatory variables with an application to the earnings function abstract in this paper it will be shown that regression models with discrete explanatory variables take the form of a polynominal of a linear function of the regressors a two stage estimation procedure and various model specification tests will be developed in close harmony with an empirical application to the earnings function

some asymptotic results for learning in single hidden layer feedforward network models abstract we investigate the properties of a recursive estimation procedure the method of back propagation for a class of nonlinear regression models single hidden layer feedforward network models recently developed by cognitive scientists the results follow from more general results for a class of recursive m estimators obtained using theorems of ljung 1977 and walk 1977 for the method of stochastic approximation conditions are given ensuring that the back propagation estimator converges almost surely to a parameter value that locally minimizes expected squared error loss provided the estimator does not diverge and that the back propagation estimator is asymptotically normal when centered at this minimizer this estimator is shown to be statistically inefficient and a two step procedure that has efficiency equivalent to that of nonlinear least squares is proposed practical issues are illustrated by a numerical example involving approximation of the henon map