Unrestricted Estimator Research Articles

Trends in Factor Price Elasticities Adding Energy to Capital and Labor in the US Economy

This paper estimates trends in factor price elasticities adding energy Btu input to fixed capital assets and the labor force in annual US data from 1949 to 2013. Second order effects improve estimates of the production function in log differences. The unrestricted estimates test for concavity and constant returns to scale. Adding energy input reduces the apparent productivity of labor and reveals strong capital-labor factor price elasticities with pronounced trends. Energy and labor trend to become weak complements. These trends offer insight into recent economic history and keys to predicting the future as well.

Time-invariant restrictions of volatility functionals: Efficient estimation and specification tests

This paper studies the estimation and inference problems of time-invariant restrictions on certain known functions of the stochastic volatility process. We first develop a more efficient GMM estimator and derive the efficiency bound under such restrictions. Then we construct an integrated Hausman-type test by summing up the squared differences between this more efficient estimator and the unrestricted estimator computed at different time points. Although less efficient under the null, the latter estimator is consistent under both the null and the alternative. The efficient GMM estimator can also be used to update an existing Bierens-type test and simplify the calculation of the asymptotic variance. Since the quadratic function puts more weight on large values, the Hausman-type test can have superior power than the Bierens-type test, which is based on a linear function of the differences. The simulation study shows that except for very small local window sizes, the Hausman-type test has good size and superior power. We finally apply these tests to studying the constant beta hypothesis using empirical data and find substantial evidence against this hypothesis.

Time-Invariant Specification Test for Volatility Functionals

This paper studies the estimation and inference problems for time-invariant restrictions on certain functions of the stochastic volatility process. We first develop a more efficient GMM estimator and derive the efficiency bound under such restrictions. Then we construct an integrated Hausman-type test by summing up the standardized squared differences between this more efficient estimator and the unrestricted estimator, which is less efficient under the null but consistent under both the null and the alternative, at different time points. This efficient GMM estimator can also be used to update an existing Bierens-type test and simplifies the calculation of the asymptotic variance. Since quadratic function puts more weights on large values, the Hausman-type test can have superior power than the Bierens-type test, which is based on a linear function of the differences. Simulation study shows that except for very small local window size, the Hausman-type test has good size and superior power. We finally apply these tests to studying the constant beta hypothesis using empirical data and find substantial evidence against this hypothesis.

How to Use Economic Theory to Improve Estimators: Shrinking Toward Theoretical Restrictions

Abstract We propose to use economic theories to construct shrinkage estimators that perform well when the theories' empirical implications are approximately correct but perform no worse than unrestricted estimators when the theories' implications do not hold. We implement this construction in various settings, including labor demand and wage inequality, and estimation of consumer demand. We provide asymptotic and finite sample characterizations of the behavior of the proposed estimators. Our approach is an alternative to the use of theory as something to be tested or to be imposed on estimates. Our approach complements uses of theory for identification and extrapolation.

Inference in a multivariate generalized mean-reverting process with a change-point

In this paper, we study inference problem about the drift parameter matrix in multivariate generalized Ornstein–Uhlenbeck processes with an unknown change-point. In particular, we consider the case where the parameter matrix may satisfy some restrictions. Thus, we generalize in five ways some recent findings about univariate generalized Ornstein–Uhlenbeck processes. First, the target parameter is a matrix and we derive a sufficient condition for the existence of the unrestricted estimator (UE) and the restricted estimator (RE). Second, we establish the joint asymptotic normality of the UE and the RE under a collection of local alternatives. Third, we construct a test for testing the uncertain restriction. The proposed test is also useful for testing the absence of the change-point. Fourth, we derive the asymptotic power of the proposed test and we prove that it is consistent. Fifth, we propose the shrinkage estimators (SEs) and we prove that SEs dominate the UE. Finally, we conduct some simulation studies which corroborate our theoretical findings.

Estimation of parameters in linear mixed measurement error models with stochastic linear restrictions

In this paper, we concentrate on the estimation of parameters in the linear mixed models (LMMs) with stochastic linear restrictions on the fixed and random effects, when the fixed effects are measured with error. In addition, the asymptotic properties of the estimators are derived and their performances are compared with the estimators for no restrictions cases, in the sense of mean square error matrix (MSEM). Finally, a simulation study and numerical example have been conducted to show the superiority of the restricted estimator over the unrestricted estimator.

Estimation of parameters in the tree order restriction by a randomized decision

ABSTRACTFor estimating the smallest location parameter in the location family of distributions which are constrained by the tree ordering for , the restricted maximum likelihood estimator diverges to as and therefore fails to dominate the corresponding unrestricted estimator in terms of the bias and hence the mean squared error (MSE). In this article, we propose a new procedure for the estimation of the location parameters based on a randomized decision. The proposed randomized estimator of is improved via the smooth approach to construct the better estimator which remains bounded and decreases the growth rate of its bias and MSE. We show in the case of normal distributions that the MSE of the proposed estimator of is less than that of the corresponding unrestricted estimator. By using a simulation study, the performance of the improved estimators is compared with that of the other restricted estimators in terms of three criteria (bias, MSE and coverage probability). The results show that the proposed estimator of is substantially better than that of the alternative estimators. Unlike the other procedures, the proposed method for estimating performs well.

Asymptotic theory and wild bootstrap inference with clustered errors

We study inference based on cluster-robust variance estimators for regression models with clustered errors, focusing on the wild cluster bootstrap. We state conditions under which asymptotic and bootstrap tests and confidence intervals are asymptotically valid. These conditions put limits on the rates at which the cluster sizes can increase as the number of clusters tends to infinity. We also derive Edgeworth expansions for the asymptotic and bootstrap test statistics. Simulation experiments illustrate the theoretical results and suggest that alternative variants of the wild cluster bootstrap may perform quite differently. The Edgeworth expansions explain the overrejection of asymptotic tests and shed light on the choice of auxiliary distribution and whether to use restricted or unrestricted estimates in the bootstrap data-generating process.

Estimation strategy of multilevel model for ordinal longitudinal data

This paper considers the shrinkage estimation of multilevel models that are appropriate for ordinal longitudinal data. These models can accommodate multiple random effects and, additionally, allow for a general form of model covariates that are related to the overall level of the responses and changes to the response over time. The likelihood inference for multilevel models is computationally burdensome due to intractable integrals. A maximum marginal likelihood (MML) method with Fisher’s scoring procedure is therefore followed to estimate the random and fixed effects parameters. In real life data, researchers may have collected many covariates for the response. Some of these covariates may satisfy certain constraints which can be used to produce a restricted estimate from the unrestricted likelihood function. The unrestricted and restricted MMLs can then be combined optimally to form the pretest and shrinkage estimators. Asymptotic properties of these estimators including biases and risks will be discussed. A simulation study is conducted to assess the performance of the estimators with respect to the unrestricted MML estimator. Finally, the relevance of the proposed estimators will be illustrated with a real data set.

Estimation of normal means in the tree order model by the weighting methods

Consider k + 1 independent normal populations with the tree order restriction on the mean parameters. For the tree order model, the restricted estimator of control group parameter is dominated by the unrestricted estimator when the number of treatment groups is large. We discuss two techniques for reducing of mean squared error via to the two weighting methods which are dissimilarity and conditional Bayesian criteria. Based on the bias and mean squared error criteria, the performance of the proposed estimators is compared with the alternative estimators in order to search for a better estimator. Although the superior estimator that uniformly dominates the others does not exist in general, but the proposed estimators dominate the corresponding unrestricted estimator and compete very well with the other alternative estimators introduced by the authors.

Improved inference in generalized mean-reverting processes with multiple change-points

In this paper, we consider inference problem about the drift parameter vector in generalized mean reverting processes with multiple and unknown change-points. In particular, we study the case where the parameter may satisfy uncertain restriction. As compared to the results in literature, we generalize some findings in five ways. First, we consider the model which incorporates the uncertain prior knowledge. Second, we derive the unrestricted estimator (UE) and the restricted estimator (RE) and we study their asymptotic properties. Third, we derive a test for testing the hypothesized restriction and we derive its asymptotic local power. We also prove that the proposed test is consistent. Fourth, we construct a class of shrinkage type estimators (SEs) which encloses the UE, the RE and classical SEs. Fifth, we derive the relative risk dominance of the proposed estimators. More precisely, we prove that the SEs dominate the UE. Finally, we present some simulation results which corroborate the established theoretical findings.

Isotonic Regression Discontinuity Designs

This paper studies the estimation and inference for the isotonic regression at the boundary point, an object that is particularly interesting and required in the analysis of monotone regression discontinuity designs. We show that the isotonic regression is inconsistent in this setting and derive the asymptotic distributions of boundary corrected estimators. Interestingly, the boundary corrected estimators can be bootstrapped without subsampling or additional nonparametric smoothing which is not the case for the interior point. The Monte Carlo experiments indicate that shape restrictions can improve dramatically the finite-sample performance of unrestricted estimators. Lastly, we apply the isotonic regression discontinuity designs to estimate the causal effect of incumbency in the U.S. House elections.

Non-penalty shrinkage estimation of random effect models for longitudinal data with AR(1) errors

In this paper, we consider the non-penalty shrinkage estimation method of random effect models with autoregressive errors for longitudinal data when there are many covariates and some of them may not be active for the response variable. In observational studies, subjects are followed over equally or unequally spaced visits to determine the continuous response and whether the response is associated with the risk factors/covariates. Measurements from the same subject are usually more similar to each other and thus are correlated with each other but not with observations of other subjects. To analyse this data, we consider a linear model that contains both random effects across subjects and within-subject errors that follows autoregressive structure of order 1 (AR(1)). Considering the subject-specific random effect as a nuisance parameter, we use two competing models, one includes all the covariates and the other restricts the coefficients based on the auxiliary information. We consider the non-penalty shrinkage estimation strategy that shrinks the unrestricted estimator in the direction of the restricted estimator. We discuss the asymptotic properties of the shrinkage estimators using the notion of asymptotic biases and risks. A Monte Carlo simulation study is conducted to examine the relative performance of the shrinkage estimators with the unrestricted estimator when the shrinkage dimension exceeds two. We also numerically compare the performance of the shrinkage estimators to that of the LASSO estimator. A longitudinal CD4 cell count data set will be used to illustrate the usefulness of shrinkage and LASSO estimators.

Nonparametric tests for conditional symmetry

We propose omnibus tests for symmetry of the conditional distribution of a time series process about a nonparametric regression function. The test statistic is a weighted version of the integrated squared difference between the restricted and unrestricted estimators of the joint characteristic function of nonparametric residuals and explanatory variables, whose critical values are estimated with the assistance of a bootstrap technique. The test is sensitive to local alternatives converging to the null at the parametric rate T−1∕2, with T the sample size. We investigate the finite sample performance of the test by means of Monte Carlo experiments and two empirical applications to test whether losses are more likely than gains in financial markets, and whether expansions and contractions are equally likely in business cycles, given the relevant information.

Nonparametric estimation of median survival times with applications to multi-site or multi-center studies.

We propose a nonparametric shrinkage estimator for the median survival times from several independent samples of right-censored data, which combines the samples and hypothesis information to improve the efficiency. We compare efficiency of the proposed shrinkage estimation procedure to unrestricted estimator and combined estimator through extensive simulation studies. Our results indicate that performance of these estimators depends on the strength of homogeneity of the medians. When homogeneity holds, the combined estimator is the most efficient estimator. However, it becomes inconsistent when homogeneity fails. On the other hand, the proposed shrinkage estimator remains efficient. Its efficiency decreases as the equality of the survival medians is deviated, but is expected to be as good as or equal to the unrestricted estimator. Our simulation studies also indicate that the proposed shrinkage estimator is robust to moderate levels of censoring. We demonstrate application of these methods to estimating median time for trauma patients to receive red blood cells in the Prospective Observational Multi-center Major Trauma Transfusion (PROMMTT) study.

Estimation of monotone bivariate quantile residual life

ABSTRACTThe bivariate quantile residual life function can play an important role in statistical reliability and survival analysis. In many situations assuming a decreasing form for it is recommended. Here, we propose a new non-parametric estimator of this measure under such restriction. It has been shown that the new estimator is consistent and, with proper normalization, weakly converges to a bivariate Gaussian process. A simulation study shows that the proposed estimator is an alternative to the unrestricted estimator when the bivariate quantile residual life is decreasing. Finally, the new estimators are applied to two real data sets.

Cognitive Diagnostic Models With Attribute Hierarchies: Model Estimation With a Restricted Q-Matrix Design.

Attribute hierarchy is a common assumption in the educational context, where the mastery of one attribute is assumed to be a prerequisite to the mastery of another one. The attribute hierarchy can be incorporated through a restricted Q matrix that implies the specified structure. The latent class-based cognitive diagnostic models (CDMs) usually do not assume a hierarchical structure among attributes, which means all profiles of attributes are possible in a population of interest. This study investigates different estimation methods to the classification accuracy for a family of CDMs when they are combined with a restricted Q-matrix design. A simulation study is used to explain the misclassification caused by an unrestricted estimation procedure. The advantages of the restricted estimation procedure utilizing attribute hierarchies for increased classification accuracy are also further illustrated through a real data analysis on a syllogistic reasoning diagnostic assessment. This research can provide guidelines for educational and psychological researchers and practitioners when they use CDMs to analyze the data with a restricted Q-matrix design and make them be aware of the potentially contaminated classification results if ignoring attribute hierarchies.

Simultaneous estimation of several CDF’s: homogeneity constraint

ABSTRACTLet {xij(1 ⩽ j ⩽ ni)|i = 1, 2, …, k} be k independent samples of size nj from respective distributions of functions Fj(x)(1 ⩽ j ⩽ k). A classical statistical problem is to test whether these k samples came from a common distribution function, F(x) whose form may or may not be known. In this paper, we consider the complementary problem of estimating the distribution functions suspected to be homogeneous in order to improve the basic estimator known as “empirical distribution function” (edf), in an asymptotic setup. Accordingly, we consider four additional estimators, namely, the restricted estimator (RE), the preliminary test estimator (PTE), the shrinkage estimator (SE), and the positive rule shrinkage estimator (PRSE) and study their characteristic properties based on the mean squared error (MSE) and relative risk efficiency (RRE) with tables and graphs. We observed that for k ⩾ 4, the positive rule SE performs uniformly better than both shrinkage and the unrestricted estimator, while PTEs works reasonably well for k < 4.

Shrinkage and penalized estimators in weighted least absolute deviations regression models

ABSTRACTIn this paper, we consider the estimation problem of the weighted least absolute deviation (WLAD) regression parameter vector when there are some outliers or heavy-tailed errors in the response and the leverage points in the predictors. We propose the pretest and James–Stein shrinkage WLAD estimators when some of the parameters may be subject to certain restrictions. We derive the asymptotic risk of the pretest and shrinkage WLAD estimators and show that if the shrinkage dimension exceeds two, the asymptotic risk of the shrinkage WLAD estimator is strictly less than the unrestricted WLAD estimator. On the other hand, the risk of the pretest WLAD estimator depends on the validity of the restrictions on the parameters. Furthermore, we study the WLAD absolute shrinkage and selection operator (WLAD-LASSO) and compare its relative performance with the pretest and shrinkage WLAD estimators. A simulation study is conducted to evaluate the performance of the proposed estimators relative to that of the unrestricted WLAD estimator. A real-life data example using body fat study is used to illustrate the performance of the suggested estimators.

On the ridge regression estimator with sub-space restriction

ABSTRACTIn the linear regression model with elliptical errors, a shrinkage ridge estimator is proposed. In this regard, the restricted ridge regression estimator under sub-space restriction is improved by incorporating a general function which satisfies Taylor’s series expansion. Approximate quadratic risk function of the proposed shrinkage ridge estimator is evaluated in the elliptical regression model. A Monte Carlo simulation study and analysis based on a real data example are considered for performance analysis. It is evident from the numerical results that the shrinkage ridge estimator performs better than both unrestricted and restricted estimators in the multivariate t-regression model, for some specific cases.