Distortions of Asymptotic Confidence Size in Locally Misspecified Moment Inequality Models

This paper studies the behavior, under local misspecification, of several confidence sets (CSs) commonly used in the literature on inference in moment (in)equality models. We propose the amount of asymptotic confidence size distortion as a criterion to choose among competing inference methods. This criterion is then applied to compare across test statistics and critical values employed in the construction of CSs. We find two important results under weak assumptions. First, we show that CSs based on subsampling and generalized moment selection (Andrews and Soares (2010)) suffer from the same degree of asymptotic confidence size distortion, despite the fact that asymptotically the latter can lead to CSs with strictly smaller expected volume under correct model specification. Second, we show that the asymptotic confidence size of CSs based on the quasi‐likelihood ratio test statistic can be an arbitrary small fraction of the asymptotic confidence size of CSs based on the modified method of moments test statistic.     

Conditional Linear Combination Tests for Weakly Identified Models

We introduce the class of conditional linear combination tests, which reject null hypotheses concerning model parameters when a data‐dependent convex combination of two identification‐robust statistics is large. These tests control size under weak identification and have a number of optimality properties in a conditional problem. We show that the conditional likelihood ratio test of Moreira, 2003 is a conditional linear combination test in models with one endogenous regressor, and that the class of conditional linear combination tests is equivalent to a class of quasi‐conditional likelihood ratio tests. We suggest using minimax regret conditional linear combination tests and propose a computationally tractable class of tests that plug in an estimator for a nuisance parameter. These plug‐in tests perform well in simulation and have optimal power in many strongly identified models, thus allowing powerful identification‐robust inference in a wide range of linear and nonlinear models without sacrificing efficiency if identification is strong.     

Hybrid and Size‐Corrected Subsampling Methods

This paper considers inference in a broad class of nonregular models. The models considered are nonregular in the sense that standard test statistics have asymptotic distributions that are discontinuous in some parameters. It is shown in Andrews and Guggenberger (2009a) that standard fixed critical value, subsampling, and m out of n bootstrap methods often have incorrect asymptotic size in such models. This paper introduces general methods of constructing tests and confidence intervals that have correct asymptotic size. In particular, we consider a hybrid subsampling/fixed‐critical‐value method and size‐correction methods. The paper discusses two examples in detail. They are (i) confidence intervals in an autoregressive model with a root that may be close to unity and conditional heteroskedasticity of unknown form and (ii) tests and confidence intervals based on a post‐conservative model selection estimator.     

Asymptotic Properties for a Class of Partially Identified Models

We propose inference procedures for partially identified population features for which the population identification region can be written as a transformation of the Aumann expectation of a properly defined set valued random variable (SVRV). An SVRV is a mapping that associates a set (rather than a real number) with each element of the sample space. Examples of population features in this class include interval‐identified scalar parameters, best linear predictors with interval outcome data, and parameters of semiparametric binary models with interval regressor data. We extend the analogy principle to SVRVs and show that the sample analog estimator of the population identification region is given by a transformation of a Minkowski average of SVRVs. Using the results of the mathematics literature on SVRVs, we show that this estimator converges in probability to the population identification region with respect to the Hausdorff distance. We then show that the Hausdorff distance and the directed Hausdorff distance between the population identification region and the estimator, when properly normalized by , converge in distribution to functions of a Gaussian process whose covariance kernel depends on parameters of the population identification region. We provide consistent bootstrap procedures to approximate these limiting distributions. Using similar arguments as those applied for vector valued random variables, we develop a methodology to test assumptions about the true identification region and its subsets. We show that these results can be used to construct a confidence collection and a directed confidence collection. Those are (respectively) collection of sets that, when specified as a null hypothesis for the true value (a subset of values) of the population identification region, cannot be rejected by our tests.     

Inference for Parameters Defined by Moment Inequalities: A Recommended Moment Selection Procedure

This paper is concerned with tests and confidence intervals for parameters that are not necessarily point identified and are defined by moment inequalities. In the literature, different test statistics, critical‐value methods, and implementation methods (i.e., the asymptotic distribution versus the bootstrap) have been proposed. In this paper, we compare these methods. We provide a recommended test statistic, moment selection critical value, and implementation method. We provide data‐dependent procedures for choosing the key moment selection tuning parameter κ and a size‐correction factor η.     

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.     

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.     

Panel Data Discrete Choice Models with Lagged Dependent Variables

In this paper, we consider identification and estimation in panel data discrete choice models when the explanatory variable set includes strictly exogenous variables, lags of the endogenous dependent variable as well as unobservable individual‐specific effects. For the binary logit model with the dependent variable lagged only once, Chamberlain (1993) gave conditions under which the model is not identified. We present a stronger set of conditions under which the parameters of the model are identified. The identification result suggests estimators of the model, and we show that these are consistent and asymptotically normal, although their rate of convergence is slower than the inverse of the square root of the sample size. We also consider identification in the semiparametric case where the logit assumption is relaxed. We propose an estimator in the spirit of the conditional maximum score estimator (Manski (1987)) and we show that it is consistent. In addition, we discuss an extension of the identification result to multinomial discrete choice models, and to the case where the dependent variable is lagged twice. Finally, we present some Monte Carlo evidence on the small sample performance of the proposed estimators for the binary response model.     

Estimating and Testing Structural Changes in Multivariate Regressions

This paper considers issues related to estimation, inference, and computation with multiple structural changes that occur at unknown dates in a system of equations. Changes can occur in the regression coefficients and/or the covariance matrix of the errors. We also allow arbitrary restrictions on these parameters, which permits the analysis of partial structural change models, common breaks that occur in all equations, breaks that occur in a subset of equations, and so forth. The method of estimation is quasi‐maximum likelihood based on Normal errors. The limiting distributions are obtained under more general assumptions than previous studies. For testing, we propose likelihood ratio type statistics to test the null hypothesis of no structural change and to select the number of changes. Structural change tests with restrictions on the parameters can be constructed to achieve higher power when prior information is present. For computation, an algorithm for an efficient procedure is proposed to construct the estimates and test statistics. We also introduce a novel locally ordered breaks model, which allows the breaks in different equations to be related yet not occurring at the same dates.     

Conditional Inference With a Functional Nuisance Parameter

This paper shows that the problem of testing hypotheses in moment condition models without any assumptions about identification may be considered as a problem of testing with an infinite‐dimensional nuisance parameter. We introduce a sufficient statistic for this nuisance parameter in a Gaussian problem and propose conditional tests. These conditional tests have uniformly correct asymptotic size for a large class of models and test statistics. We apply our approach to construct tests based on quasi‐likelihood ratio statistics, which we show are efficient in strongly identified models and perform well relative to existing alternatives in two examples.     

Dedicated Transportation Subnetworks: Design,Analysis, and Insights

A dedicated subnetwork (DSN) refers to a subset of lanes, with associated loads, in a shipper's transportation network, for which resources—trucks, drivers, and other equipment—are exclusively assigned to accomplish shipping requirements. The resources assigned to a DSN are not shared with the rest of the shipper's network. Thus, a DSN is an autonomously operated subnetwork and, hence, can be subcontracted. We address a novel problem of extracting a DSN for outsourcing to one or more subcontractors, with the objective of maximizing the shipper's savings. In their pure form, the defining conditions of a DSN are often too restrictive to enable the extraction of a sizable subnetwork. We consider two notions—deadheading and lane‐sharing—that aid in improving the size of the DSN. We show that all the optimization problems involved are both strongly NP‐hard and APX‐hard, and demonstrate several polynomially solvable special cases arising from topological properties of the network and parametric relationships. Next, we develop a network‐flow‐based heuristic that provides near‐optimal solutions to practical instances in reasonable time. Finally, using a test bed based on data obtained from a national 3PL company, we demonstrate the substantial monetary impact of subcontracting a DSN and offer useful managerial insights.     

Semi‐Nonparametric IV Estimation of Shape‐Invariant Engel Curves

This paper studies a shape‐invariant Engel curve system with endogenous total expenditure, in which the shape‐invariant specification involves a common shift parameter for each demographic group in a pooled system of nonparametric Engel curves. We focus on the identification and estimation of both the nonparametric shapes of the Engel curves and the parametric specification of the demographic scaling parameters. The identification condition relates to the bounded completeness and the estimation procedure applies the sieve minimum distance estimation of conditional moment restrictions, allowing for endogeneity. We establish a new root mean squared convergence rate for the nonparametric instrumental variable regression when the endogenous regressor could have unbounded support. Root‐n asymptotic normality and semiparametric efficiency of the parametric components are also given under a set of “low‐level” sufficient conditions. Our empirical application using the U.K. Family Expenditure Survey shows the importance of adjusting for endogeneity in terms of both the nonparametric curvatures and the demographic parameters of systems of Engel curves.     

A Conditional Likelihood Ratio Test for Structural Models

This paper develops a general method for constructing exactly similar tests based on the conditional distribution of nonpivotal statistics in a simultaneous equations model with normal errors and known reduced‐form covariance matrix. These tests are shown to be similar under weak‐instrument asymptotics when the reduced‐form covariance matrix is estimated and the errors are non‐normal. The conditional test based on the likelihood ratio statistic is particularly simple and has good power properties. Like the score test, it is optimal under the usual local‐to‐null asymptotics, but it has better power when identification is weak.     

Inference for Parameters Defined by Moment Inequalities Using Generalized Moment Selection

The topic of this paper is inference in models in which parameters are defined by moment inequalities and/or equalities. The parameters may or may not be identified. This paper introduces a new class of confidence sets and tests based on generalized moment selection (GMS). GMS procedures are shown to have correct asymptotic size in a uniform sense and are shown not to be asymptotically conservative. The power of GMS tests is compared to that of subsampling, m out of n bootstrap, and “plug‐in asymptotic” (PA) tests. The latter three procedures are the only general procedures in the literature that have been shown to have correct asymptotic size (in a uniform sense) for the moment inequality/equality model. GMS tests are shown to have asymptotic power that dominates that of subsampling, m out of n bootstrap, and PA tests. Subsampling and m out of n bootstrap tests are shown to have asymptotic power that dominates that of PA tests.     

Capacity Sharing and Cost Allocation among Independent Firms with Congestion

We analyze the benefit of production/service capacity sharing for a set of independent firms. Firms have the choice of either operating their own production/service facilities or investing in a facility that is shared. Facilities are modeled as queueing systems with finite service rates. Firms decide on capacity levels (the service rate) to minimize delay costs and capacity investment costs possibly subject to service‐level constraints on delay. If firms decide to operate a shared facility they must also decide on a scheme for sharing the capacity cost. We formulate the problem as a cooperative game and identify settings under which capacity sharing is beneficial and there is a cost allocation that is in the core under either the first‐come, first‐served policy or an optimal priority policy. We show that capacity sharing may not be beneficial in settings where firms have heterogeneous work contents and service variabilities. In such cases, we specify conditions under which capacity sharing may still be beneficial for a subset of the firms.     

Bootstrap Testing of Hypotheses on Co‐Integration Relations in Vector Autoregressive Models

It is well known that the finite‐sample properties of tests of hypotheses on the co‐integrating vectors in vector autoregressive models can be quite poor, and that current solutions based on Bartlett‐type corrections or bootstrap based on unrestricted parameter estimators are unsatisfactory, in particular in those cases where also asymptotic χ² tests fail most severely. In this paper, we solve this inference problem by showing the novel result that a bootstrap test where the null hypothesis is imposed on the bootstrap sample is asymptotically valid. That is, not only does it have asymptotically correct size, but, in contrast to what is claimed in existing literature, it is consistent under the alternative. Compared to the theory for bootstrap tests on the co‐integration rank (Cavaliere, Rahbek, and Taylor, 2012), establishing the validity of the bootstrap in the framework of hypotheses on the co‐integrating vectors requires new theoretical developments, including the introduction of multivariate Ornstein–Uhlenbeck processes with random (reduced rank) drift parameters. Finally, as documented by Monte Carlo simulations, the bootstrap test outperforms existing methods.     

Power Enhancement in High‐Dimensional Cross‐Sectional Tests

We propose a novel technique to boost the power of testing a high‐dimensional vector H : θ = 0 against sparse alternatives where the null hypothesis is violated by only a few components. Existing tests based on quadratic forms such as the Wald statistic often suffer from low powers due to the accumulation of errors in estimating high‐dimensional parameters. More powerful tests for sparse alternatives such as thresholding and extreme value tests, on the other hand, require either stringent conditions or bootstrap to derive the null distribution and often suffer from size distortions due to the slow convergence. Based on a screening technique, we introduce a “power enhancement component,” which is zero under the null hypothesis with high probability, but diverges quickly under sparse alternatives. The proposed test statistic combines the power enhancement component with an asymptotically pivotal statistic, and strengthens the power under sparse alternatives. The null distribution does not require stringent regularity conditions, and is completely determined by that of the pivotal statistic. The proposed methods are then applied to testing the factor pricing models and validating the cross‐sectional independence in panel data models.     

Nonparametric Test for Causality with Long‐range Dependence

This paper introduces a nonparametric Granger‐causality test for covariance stationary linear processes under, possibly, the presence of long‐range dependence. We show that the test is consistent and has power against contiguous alternatives converging to the parametric rate T^−1/2. Since the test is based on estimates of the parameters of the representation of a VAR model as a, possibly, two‐sided infinite distributed lag model, we first show that a modification of Hannan's (1963, 1967) estimator is root‐ T consistent and asymptotically normal for the coefficients of such a representation. When the data are long‐range dependent, this method of estimation becomes more attractive than least squares, since the latter can be neither root‐ T consistent nor asymptotically normal as is the case with short‐range dependent data.     

Robust Estimation and Inference for Jumps in Noisy High Frequency Data: A Local‐to‐Continuity Theory for the Pre‐Averaging Method

We develop an asymptotic theory for the pre‐averaging estimator when asset price jumps are weakly identified, here modeled as local to zero. The theory unifies the conventional asymptotic theory for continuous and discontinuous semimartingales as two polar cases with a continuum of local asymptotics, and explains the breakdown of the conventional procedures under weak identification. We propose simple bias‐corrected estimators for jump power variations, and construct robust confidence sets with valid asymptotic size in a uniform sense. The method is also robust to certain forms of microstructure noise.     

GMM with Weak Identification

This paper develops asymptotic distribution theory for GMM estimators and test statistics when some or all of the parameters are weakly identified. General results are obtained and are specialized to two important cases: linear instrumental variables regression and Euler equations estimation of the CCAPM. Numerical results for the CCAPM demonstrate that weak‐identification asymptotics explains the breakdown of conventional GMM procedures documented in previous Monte Carlo studies. Confidence sets immune to weak identification are proposed. We use these results to inform an empirical investigation of various CCAPM specifications; the substantive conclusions reached differ from those obtained using conventional methods.