Extreme Estimates Research Articles

Functional coefficient cointegration models with Box–Cox transformation

This paper considers a functional coefficient cointegration model where the dependent variable is subject to a Box–Cox transformation. A profile method is proposed to estimate the Box–Cox transformation parameter β0. We first approximate the functional coefficients by sieve method assuming that β0 is known. Then an extremum estimator of β0 is proposed based on a loss function which measures the relative variation of the regression residual compared to the variation in the transformed dependent variable. Finally, a plug-in estimator of the functional coefficients is obtained. Asymptotic properties of the proposed estimators are developed. Numerical results demonstrate the nice performance of the estimators and corroborate the theoretical development.

Random utility models with ordered types and domains

Random utility models in which heterogeneity of preferences is modeled by means of an ordered collection of utilities, or types, provide a powerful framework for understanding a variety of economic behaviors. This paper studies the micro-foundations of ordered random utility models with the objective of meeting empirical requirements. This is done by working with arbitrary collections of ordered menus of alternatives, and by making no parametric assumptions about the type distribution. The model is characterized by a simple monotonicity axiom. Goodness-of-fit measures are proposed, with proof provided of the strong consistency of extremum estimators defined upon them. A statistical test for the model is also provided.

Non-parametric inference about mean functionals of non-ignorable non-response data without identifying the joint distribution.

We consider identification and inference about mean functionals of observed covariates and an outcome variable subject to non-ignorable missingness. By leveraging a shadow variable, we establish a necessary and sufficient condition for identification of the mean functional even if the full data distribution is not identified. We further characterize a necessary condition for -estimability of the mean functional. This condition naturally strengthens the identifying condition, and it requires the existence of a function as a solution to a representer equation that connects the shadow variable to the mean functional. Solutions to the representer equation may not be unique, which presents substantial challenges for non-parametric estimation, and standard theories for non-parametric sieve estimators are not applicable here. We construct a consistent estimator of the solution set and then adapt the theory of extremum estimators to find from the estimated set a consistent estimator of an appropriately chosen solution. The estimator is asymptotically normal, locally efficient and attains the semi-parametric efficiency bound under certain regularity conditions. We illustrate the proposed approach via simulations and a real data application on home pricing.

Wald, QLR, and score tests when parameters are subject to linear inequality constraints

This paper develops Wald-type, QLR, and score-type tests for linear equality constraints in a general class of extremum estimation problems where the parameter space is characterized by a finite number of linear equality and inequality constraints. We show that the asymptotic null distributions of the Wald and QLR statistics are discontinuous in an implicit nuisance parameter and propose an algorithm to identify it. In contrast, the asymptotic null distribution of the score statistic is not discontinuous in any model parameter but depends on a polytope projection. We present an algorithm based on the Fourier–Motzkin elimination to compute such a projection. We study the consistency and local power properties of the three tests. Finally, we present numerical results of our tests’ finite sample performance from a Monte Carlo study and conduct an empirical illustration of a Mincer earnings regression.

Proximal causal inference without uniqueness assumptions

We consider identification and inference about a counterfactual outcome mean when there is unmeasured confounding using tools from proximal causal inference (Miao et al., 2018, Tchetgen Tchetgen et al., 2020). Proximal causal inference requires existence of solutions to at least one of two integral equations. We motivate the existence of solutions to the integral equations from proximal causal inference by demonstrating that, assuming the existence of a solution to one of the integral equations, n-estimability of a linear functional (such as its mean) of that solution requires the existence of a solution to the other integral equation. Solutions to the integral equations may not be unique, which complicates estimation and inference. We construct a consistent estimator for the solution set for one of the integral equations and then adapt the theory of extremum estimators to find from the estimated set a consistent estimator for a uniquely defined solution. A debiased estimator for the counterfactual mean is shown to be root-n consistent, regular, and asymptotically semiparametrically locally efficient under additional regularity conditions.

Limit Theorems for Deviation Means of Independent and Identically Distributed Random Variables

We derive a strong law of large numbers, a central limit theorem, a law of the iterated logarithm and a large deviation theorem for so-called deviation means of independent and identically distributed random variables. (For the strong law of large numbers, we suppose only pairwise independence instead of (total) independence.) The class of deviation means is a special class of M-estimators or more generally extremum estimators, which are well studied in statistics. The assumptions of our limit theorems for deviation means seem to be new and weaker than the known ones for M-estimators in the literature. In particular, our results on the strong law of large numbers and on the central limit theorem generalize the corresponding ones for quasi-arithmetic means due to de Carvalho (Am Stat 70(3):270–274, 2016) and the ones for Bajraktarević means due to Barczy and Burai (Aequ Math 96(2):279–305, 2022).

AN AVERAGING ESTIMATOR FOR TWO-STEP M-ESTIMATION IN SEMIPARAMETRIC MODELS

In a two-step extremum estimation (M-estimation) framework with a finite-dimensional parameter of interest and a potentially infinite-dimensional first-step nuisance parameter, this paper proposes an averaging estimator that combines a semiparametric estimator based on a nonparametric first step and a parametric estimator which imposes parametric restrictions on the first step. The averaging weight is an easy-to-compute sample analog of an infeasible optimal weight that minimizes the asymptotic quadratic risk. Under Stein-type conditions, the asymptotic lower bound of the truncated quadratic risk difference between the averaging estimator and the semiparametric estimator is strictly less than zero for a class of data generating processes that includes both correct specification and varied degrees of misspecification of the parametric restrictions, and the asymptotic upper bound is weakly less than zero. The averaging estimator, along with an easy-to-implement inference method, is demonstrated in an example.

Probabilistic-based random maximum defect estimation and defect-related fatigue life prediction for laser direct deposited 316L parts

• A probabilistic-based small sample extremum estimation method was proposed for the random maximum defect of LDD parts. • The number of defects samplings was reduced with the combination of EVS and approximate defect distribution. • A defect-related fatigue life prediction model was established basing fatigue critical stress. • The high cycle fatigue life of parts was directly and reliably predicted with process-independent approach. Laser direct deposition (LDD) is a typical additive manufacture process for complex parts, while the highly complex thermal behavior during LDD results in the generation and randomness of parts’ internal defects. Among these internal defects, the maximum defect-induced fatigue cracks initiation is the most influential factor of fatigue life for the in-service performance. Apparently, how to estimate the random maximum defect size of surface polished parts is critical for predicting the fatigue life. Therefore, according to extreme value statistic (EVS) theory, an extremum probabilistic estimation method from small sample size was proposed for parts’ random maximum defect size from a sub-volume to the whole-volume. Subsequently, with the obtained maximum defect and being taken to be equivalent to cracks, a defect-related fatigue life prediction model was established based on the failure critical stress. Orthogonal experiment was carried out for obtaining the different maximum defect, and the hardness for each sample was measured as well. The results showed that: (1) The proposed method can reliably estimate the maximum defect size of LDD-316L parts under the small defect samples size and the error was within 10 %. (2) The established prediction model provided a process-independent method for directly estimating the LDD-316L parts’ fatigue life, with the accuracy being over 78 %. This research provides a novel methodology for estimating parts’ maximum defect size and fatigue life, and offers a theoretical basis for reliability and economy of parts during manufacturing and servicing process.

Constrained estimation using penalization and MCMC

We study inference for parameters defined by either classical extremum estimators or Laplace-type estimators subject to general nonlinear constraints on the parameters. We show that running MCMC on the penalized version of the problem offers a computationally attractive alternative to solving the original constrained optimization problem. Bayesian credible intervals are asymptotically valid confidence intervals in a pointwise sense, providing exact asymptotic coverage for general functions of the parameters. We allow for nonadaptive and adaptive penalizations using the ℓp for p⩾1 penalty functions. These methods are motivated by and include as special cases model selection and shrinkage methods such as the LASSO and its Bayesian and adaptive versions. A simulation study validates the theoretical results. We also provide an empirical application on estimating the joint density of U.S. real consumption and asset returns subject to Euler equation constraints in a CRRA asset pricing model.

NONLINEAR COINTEGRATING POWER FUNCTION REGRESSION WITH ENDOGENEITY

This paper develops an asymptotic theory for nonlinear cointegrating power function regression. The framework extends earlier work on the deterministic trend case and allows for both endogeneity and heteroskedasticity, which makes the models and inferential methods relevant to many empirical economic and financial applications, including predictive regression. A new test for linear cointegration against nonlinear departures is developed based on a simple linearized pseudo-model that is very convenient for practical implementation and has standard normal limit theory in the strictly exogenous regressor case. Accompanying the asymptotic theory of nonlinear regression, the paper establishes some new results on weak convergence to stochastic integrals that go beyond the usual semimartingale structure and considerably extend existing limit theory, complementing other recent findings on stochastic integral asymptotics. The paper also provides a general framework for extremum estimation limit theory that encompasses stochastically nonstationary time series and should be of wide applicability.

Econometric analysis of production networks with dominant units

This paper introduces the notions of strongly and weakly dominant units for networks, and shows that pervasiveness of shocks to a network is measured by the degree of dominance of its most pervasive unit; shown to be equivalent to the inverse of the shape parameter of the power law fitted to the network outdegrees. New cross-section and panel extremum estimators of the degree of dominance in networks are proposed, and their asymptotic properties investigated. The small sample properties of the proposed estimators are examined by Monte Carlo experiments, and their use is illustrated by an empirical application to US input–output tables.

Estimation for double-nonlinear cointegration

In recent years statistical inference for nonlinear cointegration has attracted attention from both academics and practitioners. This paper proposes a new type of cointegration in the sense that two univariate time series yt and xt are cointegrated via two (unknown) smooth nonlinear transformations, further generalizing the notion of cointegration initially revealed by Box and Tiao (1977), and more systematically studied by Engle and Granger (1987). More precisely, it holds that G(yt,β0)=g(xt)+ut, where G(⋅,β0) is strictly increasing and known up to an unknown parameter β0, g(⋅) is unknown and smooth, xt is I(1), and ut is the stationary disturbance. This setting nests the nonlinear cointegration model of Wang and Phillips (2009b) as a special case with G(y,β0)=y. It extends the model of Linton et al. (2008) to the cases with a unit-root nonstationary regressor. Sieve approximations to the smooth nonparametric function g are applied, leading to an extremum estimator for β and a plugging-in estimator for g(⋅). Asymptotic properties of the estimators are established, revealing that both the convergence rates and the limiting distributions depend intimately on the properties of the two nonlinear transformation functions. Simulation studies demonstrate that the estimators perform well even with small samples. A real data example on the environmental Kuznets curve portraying the nonlinear impact of per-capita GDP on air-pollution illustrates the practical relevance of the proposed double-nonlinear cointegration.

Bilinear form test statistics for extremum estimation

This paper develops a set of test statistics based on bilinear forms in the context of the extremum estimation framework with particular interest in nonlinear hypothesis. We show that the proposed statistic converges to a conventional chi-square limit. A Monte Carlo experiment suggests that the test statistic works well in finite samples.

Optimizing Energy Consumption for Lighting Control System via Multivariate Extremum Seeking Control With Diminishing Dither Signal

Because that the light-energy consumption (LC) is growing quite dramatically over the past decades, the light-energy efficiency is regarded as a key component of the modern energy system in major cities. In this paper, a novel and sophisticated algorism of lighting control is presented to improve light-energy efficiency and cut down energy consumption, when considering the lighting system with multilighting equipment. In this paper, a multivariate extremum seeking controller is used to manipulate, respectively, the luminance of multilighting equipment, in order to track the minimal LC efficiently. Meanwhile, a proportional–integral–differential (PID) approach is adopted to achieve the required level of illumination. The proposed extremum seeking control (ESC) is designed to ensure the amplitude of the dither signal converge exponentially to zero, by making the amplitudes change with the extremum estimation error. The presented frame helps to improve search speed for the minimal LC value and remove the steady-state oscillations. The comparative experiments reveal that, compared with the convergence velocity of the Newton- and gradient-based ESCs, the developed method for the track of the minimal LC can have higher accuracy and faster speed. Note to Practitioners —In a modern large commercial building, the exorbitant LC is an important problem for the engineering technicians and managers, and the hybrid lighting system which combines artificial light with natural light offers a solution to this problem. However, the minimum light-energy optimal control is a very key problem in manipulating multigroup lighting equipment, since the required level of illumination usually varies according to the exterior lighting environment. This paper contributes to a useful multivariate extremum seeking controller to solve these light-energy control problems for multiple-input single-output hybrid lighting system, whereas the PID controller is also designed to maintain the required level of illumination simultaneously. This paper discusses how to seek the minimal energy necessary for the required levels of illumination by separately regulating different light equipment. The research results of this paper will be beneficial for a practical application of extremum seeking control method in enhancing the lighting energy efficiency of lighting system.

Online speed optimization with feedforward of unmanned sailboat via extremum seeking without steady-state oscillation

Concentrated on the problem of steady-state oscillation of the traditional extremum seeking algorithm (ESA) that it is not beneficial to the speed optimization of unmanned sailboat, a novel speed optimization scheme with feedforward of unmanned sailboat via extremum seeking without steady-state oscillation (ESWSO) is proposed. This scheme consists of feedforward and feedback control: the feedforward part firstly presets the initial sail angle of attack according to an approximate piecewise constant function of the sailing speed polar diagram, its main function is avoid the large deviation of the longitudinal speed from the optimal value. Subsequently, the feedback part fine-tunes the speed to reach the optimal value through the extremum seeking (ES) control. In the process of continuously optimizing longitudinal speed, a functional relationship between the excitation signal amplitude and the extremum estimation error is constructed to eliminate the steady-state oscillation. Besides, the rigorous stability Proof of the scheme is given. Finally, taking a 12-m class unmanned sailboat as the simulation object, the results show that the optimization scheme can effectively eliminate steady-state oscillation and enable the sailing speed to reach the ideal optimal value, as well as has a high control accuracy and strong robustness.

Estimation and inference with a (nearly) singular Jacobian

This paper develops extremum estimation and inference results for nonlinear models with very general forms of potential identification failure when the source of this identification failure is known. We examine models that may have a general deficient rank Jacobian in certain parts of the parameter space. When identification fails in one of these models, it becomes underidentified and the identification status of individual parameters is not generally straightforward to characterize. We provide a systematic reparameterization procedure that leads to a reparametrized model with straightforward identification status. Using this reparameterization, we determine the asymptotic behavior of standard extremum estimators and Wald statistics under a comprehensive class of parameter sequences characterizing the strength of identification of the model parameters, ranging from nonidentification to strong identification. Using the asymptotic results, we propose hypothesis testing methods that make use of a standard Wald statistic and data‐dependent critical values, leading to tests with correct asymptotic size regardless of identification strength and good power properties. Importantly, this allows one to directly conduct uniform inference on low‐dimensional functions of the model parameters, including one‐dimensional subvectors. The paper illustrates these results in three examples: a sample selection model, a triangular threshold crossing model, and a collective model for household expenditures.

Nonlinear Cointegrating Power Function Regression with Endogeneity

This paper develops an asymptotic theory for nonlinear cointegrating power function regression. The framework extends earlier work on the deterministic trend case and allows for both endogeneity and heteroskedasticity, which makes the models and inferential methods relevant to many empirical economic and financial applications, including predictive regression. Accompanying the asymptotic theory of nonlinear regression, the paper establishes some new results on weak convergence to stochastic integrals that go beyond the usual semi-martingale structure and considerably extend existing limit theory, complementing other recent findings on stochastic integral asymptotics. The paper also provides a general framework for extremum estimation limit theory that encompasses stochastically nonstationary time series and should be of wide applicability.

Heteroskedasticity-consistent covariance matrix estimators for spatial autoregressive models

ABSTRACTIn the presence of heteroskedasticity, conventional test statistics based on the ordinary least squares (OLS) estimator lead to incorrect inference results for the linear regression model. Given that heteroskedasticity is common in cross-sectional data, the test statistics based on various forms of heteroskedasticity-consistent covariance matrices (HCCMs) have been developed in the literature. In contrast to the standard linear regression model, heteroskedasticity is a more serious problem for spatial econometric models, generally causing inconsistent extremum estimators of model coefficients. This paper investigates the finite sample properties of the heteroskedasticity-robust generalized method of moments estimator (RGMME) for a spatial econometric model with an unknown form of heteroskedasticity. In particular, it develops various HCCM-type corrections to improve the finite sample properties of the RGMME and the conventional Wald test. The Monte Carlo results indicate that the HCCM-type corrections can produce more accurate results for inference on model parameters and the impact effects estimates in small samples.

Subvector inference when the true parameter vector may be near or at the boundary

Extremum estimators are not asymptotically normally distributed when the estimator satisfies the restrictions on the parameter space – such as the non-negativity of a variance parameter – and the true parameter vector is near or at the boundary. This possible lack of asymptotic normality makes it difficult to construct tests for testing subvector hypotheses that control asymptotic size in a uniform sense and have good local asymptotic power irrespective of whether the true parameter vector is at, near, or far from the boundary. We propose a novel estimator that is asymptotically normally distributed even when the true parameter vector is near or at the boundary and the objective function is not defined outside the parameter space. The proposed estimator allows the implementation of a new test based on the Conditional Likelihood Ratio statistic that is easy-to-implement, controls asymptotic size, and has good local asymptotic power properties. Furthermore, we show that the test enjoys certain asymptotic optimality properties when the parameter of interest is scalar. In an application of the random coefficients logit model (Berry, Levinsohn and Pakes, 1995) to the European car market, we find that, for most parameters, the new test leads to tighter confidence intervals than the two-sided t-test commonly used in practice.

An Information - Theoretic Approach to Estimating Risk Premia

This paper shows that factor risk premia can be consistently estimated using a semi-parametric estimate of the stochastic discount factor without requiring a correctly specified linear factor model. We use a minimum discrepancy objective function to construct a stochastic discount factor from asset returns using only the economic assumption of no arbitrage. The stochastic discount factor and factor risk-premia are estimated using only data on portfolio returns and factor realizations: The same data used when evaluating linear models. The econometrics are applications of standard extremum estimator arguments and the Delta Method, making inference simple. In simulations, the estimated risk-premia have low root mean squared errors and are comparable to classic two-pass estimates even when the model is correctly specified. Empirical estimates of popular traded factors are close to their mean excess returns. For non-traded factors, we find that intermediary leverage and consumption growth carry risk-premia, while employment growth does not. A final application shows that the estimated risk-premia can be used as an extra moment condition to discipline the creation of factor mimicking portfolios.