We consider the quasi-maximum likelihood estimator (qmle) obtained by replacing each transition density in the correct likelihood for a non-Markovian, stationary process by a transition density with a fixed number of lags. This estimator is of interest because it is asymptotically equivalent to the efficient method of moments estimator as typically implemented in dynamic macro and finance applications. We show that the standard regularity conditions of quasi-maximum likelihood imply that a score vector defined over the infinite past exists. We verify that the existence of a score on the infinite past implies that the asymptotic variance of the finite lag qmle tends to the asymptotic variance of the maximum likelihood estimator as the number of lags tends to infinity.

In this paper, we study functional ordinary least squares estimator and its properties in testing the hypothesis of a constant zero mean function or an unknown constant nonzero mean function. We exploit the recent work by Cho, Phillips, and Seo (Int Econ Rev 170:391–456, 2022) and show that the associated Wald test statistics have standard chi-square limiting null distributions, standard noncentral chi-square distributions for local alternatives converging to zero at a √(n) rate, and are consistent against global alternatives. These properties permit computationally convenient tests of hypotheses involving nuisance parameters. In particular, we develop new alternatives to tests for regression misspecification using the neural network model, that involves nuisance parameters identified only under the alternative. Our Monte Carlo simulations affirm the theory of the current study. Finally, we apply our methodology to the probit models for voter turnout that are estimated by Wolfinger and Rosenstone (Who votes? Yale University Press, New Haven, 1980), Nagler (Am Political Sci Rev 85:1393–1405, 1991) and test whether the models are correctly specified or not.

Researchers are often faced with the challenge of developing statistical models with incomplete data. Exacerbating this situation is the possibility that either the researcher’s complete-data model or the model of the missing-data mechanism is misspecified. In this article, we create a formal theoretical framework for developing statistical models and detecting model misspecification in the presence of incomplete data where maximum likelihood estimates are obtained by maximizing the observable-data likelihood function when the missing-data mechanism is assumed ignorable. First, we provide sufficient regularity conditions on the researcher’s complete-data model to characterize the asymptotic behavior of maximum likelihood estimates in the simultaneous presence of both missing data and model misspecification. These results are then used to derive robust hypothesis testing methods for possibly misspecified models in the presence of Missing at Random (MAR) or Missing Not at Random (MNAR) missing data. Second, we introduce a method for the detection of model misspecification in missing data problems using recently developed Generalized Information Matrix Tests (GIMT). Third, we identify regularity conditions for the Missing Information Principle (MIP) to hold in the presence of model misspecification so as to provide useful computational covariance matrix estimation formulas. Fourth, we provide regularity conditions that ensure the observable-data expected negative log-likelihood function is convex in the presence of partially observable data when the amount of missingness is sufficiently small and the complete-data likelihood is convex. Fifth, we show that when the researcher has correctly specified a complete-data model with a convex negative likelihood function and an ignorable missing-data mechanism, then its strict local minimizer is the true parameter value for the complete-data model when the amount of missingness is sufficiently small. Our results thus provide new robust estimation, inference, and specification analysis methods for developing statistical models with incomplete data.

The current article examines the limit distribution of the quasi-maximum likelihood estimator obtained from a directionally differentiable quasi-likelihood function and represents its limit distribution as a functional of a Gaussian stochastic process indexed by direction. In this way, the standard analysis that assumes a differentiable quasi-likelihood function is treated as a special case of our analysis. We also examine and redefine the standard quasi-likelihood ratio, Wald, and Lagrange multiplier test statistics so that their null limit behaviors are regular under our model framework.

Granger noncausality in distribution is fundamentally a probabilistic conditional independence notion that can be applied not only to time series data but also to cross-section and panel data. In this paper, we provide a natural definition of structural causality in cross-section and panel data and forge a direct link between Granger ( G –) causality and structural causality under a key conditional exogeneity assumption. To put it simply, when structural effects are well defined and identifiable, G –non-causality follows from structural noncausality, and with suitable conditions (e.g., separability or monotonicity), structural causality also implies G –causality. This justifies using tests of G –non-causality to test for structural noncausality under the key conditional exogeneity assumption for both cross-section and panel data. We pay special attention to heterogeneous populations, allowing both structural heterogeneity and distributional heterogeneity. Most of our results are obtained for the general case, without assuming linearity, monotonicity in observables or unobservables, or separability between observed and unobserved variables in the structural relations.

Generalized Information Matrix Tests (GIMTs) have recently been used for detecting the presence of misspecification in regression models in both randomized controlled trials and observational studies. In this paper, a unified GIMT framework is developed for the purpose of identifying, classifying, and deriving novel model misspecification tests for finite-dimensional smooth probability models. These GIMTs include previously published as well as newly developed information matrix tests. To illustrate the application of the GIMT framework, we derived and assessed the performance of new GIMTs for binary logistic regression. Although all GIMTs exhibited good level and power performance for the larger sample sizes, GIMT statistics with fewer degrees of freedom and derived using log-likelihood third derivatives exhibited improved level and power performance.

This paper proposes a nonparametric test for conditional independence that is easy to implement, yet powerful in the sense that it is consistent and achieves n −1/2 local power. The test statistic is based on an estimator of the topological “distance” between restricted and unrestricted probability measures corresponding to conditional independence or its absence. The distance is evaluated using a family of Generically Comprehensively Revealing (GCR) functions, such as the exponential or logistic functions, which are indexed by nuisance parameters. The use of GCR functions makes the test able to detect any deviation from the null. We use a kernel smoothing method when estimating the distance. An integrated conditional moment (ICM) test statistic based on these estimates is obtained by integrating out the nuisance parameters. We simulate the critical values using a conditional simulation approach. Monte Carlo experiments show that the test performs well in finite samples. As an application, we test an implication of the key assumption of unconfoundedness in the context of estimating the returns to schooling.

Recall that ̂(| ) = 01β̂ (|) where 1 is defined after (4.3). Noting that uniformly in ( ) ∈ R×U0 S () =  ()S+ ()  and B (;) =  ()BG+1 (|)+  ¡  ¢  with ̄ ≡ +  we have ̂(|)−(|) = +101S−1  BG+1 (|) +  ()−1 01S−1  V (;) + (̄) Proof of Lemma C.2. Noting that  ( ;) − ̄ ( ;) =  () and ̄ ( ;) =  ( + ) by the proof of (b) below, (a) follows from Theorem 2.1 of Su and White (2012). To prove (b), write ̄ ( ;) = 1The compact support of the kernel function in Masry (1996) can be easily relaxed, following the line of proof in Hansen (2008, Theorem 4).

This paper provides nonparametric tests for hypotheses about the effects of acontinuoustreatment variable in a nonseparable structural equation. Specifically, we consider local average responses and average partial effects and test whether these effects are identical across different levels of treatment, including testing whether they are zero for all treatment levels as a special case. Our tests are based on consistent procedures of Bierens (1982,Journal of Econometrics20, 105–134; 1990,Econometrica58, 1443–1458) and Stinchcombe and White (1998,Econometric Theory14, 295–324). The tests are easy to implement and achieven-1/2local power. Monte Carlo simulations show that the tests perform well in finite samples. We apply our tests to study the interest rate elasticities of loan demand in microfinance. We also extend our testing procedures to covariate-conditioned average effects and marginal effects.

We study the identification and estimation of covariate-conditioned average marginal effects of endogenous regressors in nonseparable structural systems when the regressors are mismeasured. We control for the endogeneity by making use of covariates as control variables; this ensures conditional independence between the endogenous causes of interest and other unobservable drivers of the dependent variable. Moreover, we recover distributions of the underlying true causes from their error-laden measurements to deliver consistent estimators. We obtain uniform convergence rates and asymptotic normality for estimators of covariate-conditioned average marginal effects, faster convergence rates for estimators of their weighted averages over instruments, and root-n consistency and asymptotic normality for estimators of their weighted averages over control variables and regressors. We investigate their finite-sample behavior using Monte Carlo simulation and apply new methods to study the impact of family income on child achievement measured by math and reading scores, using a matched mother–child subsample of the National Longitudinal Survey of Youth. Our findings suggest that these effects are considerably larger than previously recognized, and depend on parental abilities and family income. This underscores the importance of measurement errors, endogeneity of family income, nonlinearity of income effects, and interactions between causes of child achievement.

This paper proposes methods for estimation and inference in multivariate, multi-quantile models. The theory can simultaneously accommodate models with multiple random variables, multiple confidence levels, and multiple lags of the associated quantiles. The proposed framework can be conveniently thought of as a vector autoregressive (VAR) extension to quantile models. We estimate a simple version of the model using market equity returns data to analyze spillovers in the values at risk (VaR) between a market index and financial institutions. We construct impulse-response functions for the quantiles of a sample of 230 financial institutions around the world and study how financial institution-specific and system-wide shocks are absorbed by the system. We show how the long-run risk of the largest and most leveraged financial institutions is very sensitive to market wide shocks in situations of financial distress, suggesting that our methodology can prove a valuable addition to the traditional toolkit of policy makers and supervisors.

Low/ered cholesterol is linked to aggression in some study designs. Cases/series have reported reproducible aggression increases on statins, but statins also bear mechanisms that could reduce aggression. Usual statin effects on aggression have not been characterized.1016 adults (692 men, 324 postmenopausal women) underwent double-blind sex-stratified randomization to placebo, simvastatin 20mg, or pravastatin 40mg (6 months). The Overt-Aggression-Scale-Modified-Aggression-Subscale (OASMa) assessed behavioral aggression. A significant sex-statin interaction was deemed to dictate sex-stratified analysis. Exploratory analyses assessed the influence of baseline-aggression, testosterone-change (men), sleep and age.The sex-statin interaction was significant (P=0.008). In men, statins tended to decrease aggression, significantly so on pravastatin: difference=-1.0(SE=0.49)P=0.038. Three marked outliers (OASMa-change ≥40 points) offset otherwise strong significance-vs-placebo: statins:-1.3(SE=0.38)P=0.0007; simvastatin:-1.4(SE=0.43)P=0.0011; pravastatin:-1.2(SE=0.45)P=0.0083. Age≤40 predicted greater aggression-decline on statins: difference=-1.4(SE=0.64)P=0.026. Aggression-protection was emphasized in those with low baseline aggression: age<40-and-low-baseline-aggression (N=40) statin-difference-vs-placebo=-2.4(SE=0.71)P=0.0016. Statins (especially simvastatin) lowered testosterone, and increased sleep problems. Testosterone-drop on statins predicted aggression-decline: β=0.64(SE=0.30)P=0.034, particularly on simvastatin: β=1.29(SE=0.49)P=0.009. Sleep-worsening on statins significantly predicted aggression-increase: β=2.2(SE=0.55)P<0.001, particularly on simvastatin (potentially explaining two of the outliers): β=3.3(SE=0.83)P<0.001. Among (postmenopausal) women, a borderline aggression-increase on statins became significant with exclusion of one younger, surgically-menopausal woman (N=310) β=0.70(SE=0.34)P=0.039. The increase was significant, without exclusions, for women of more typical postmenopausal age (≥45): (N=304) β=0.68(SE=0.34)P=0.048 - retaining significance with modified age-cutoffs (≥50 or ≥55). Significance was observed separately for simvastatin. The aggression-increase in women on statins was stronger in those with low baseline aggression (N=175) β=0.84(SE=0.30)P=0.006. No statin effect on whole blood serotonin was observed; and serotonin-change did not predict aggression-change.Statin effects on aggression differed by sex and age: Statins generally decreased aggression in men; and generally increased aggression in women. Both findings were selectively prominent in participants with low baseline aggression - bearing lower change-variance, rendering an effect more readily evident.Clinicaltrials.gov NCT00330980.

We construct two classes of smoothed empirical likelihood ratio tests for the conditional independence hypothesis by writing the null hypothesis as an infinite collection of conditional moment restrictions indexed by a nuisance parameter. One class is based on the CDF; another is based on smoother functions. We show that the test statistics are asymptotically normal under the null hypothesis and a sequence of Pitman local alternatives. We also show that the tests possess an asymptotic optimality property in terms of average power. Simulations suggest that the tests are well behaved in finite samples. Applications to some economic and financial time series indicate that our tests reveal some interesting nonlinear causal relations which the traditional linear Granger causality test fails to detect.

Monotonicity in a scalar unobservable is a common assumption when modeling heterogeneity in structural models. Among other things, it allows one to recover the underlying structural function from certain conditional quantiles of observables. Nevertheless, monotonicity is a strong assumption and in some economic applications unlikely to hold, e.g., random coefficient models. Its failure can have substantive adverse consequences, in particular inconsistency of any estimator that is based on it. Having a test for this hypothesis is hence desirable. This paper provides such a test for cross-section data. We show how to exploit an exclusion restriction together with a conditional independence assumption, which in the binary treatment literature is commonly called unconfoundedness, to construct a test. Our statistic is asymptotically normal under local alternatives and consistent against global alternatives. Monte Carlo experiments show that a suitable bootstrap procedure yields tests with reasonable level behavior and useful power. We apply our test to study the role of unobserved ability in determining Black-White wage differences and to study whether Engel curves are monotonically driven by a scalar unobservable.

Separability is an important feature of structural equations, as it implies the absence of unobservable heterogeneity of effects and has significant implications for identification and efficiency of estimation. This paper provides a nonparametric test for separability in structural equations. The test is based on a conditional independence test recently developed by Huang et al. (2013), building on consistent procedures of Bierens (1982, 1990) and Stinchcombe and White (1998). The test is easy to implement and achieves root n local power. We apply our test to study interest rate elasticities of loan demand in microfinance and the impact of education on wages. (C) 2014 Elsevier B.V. All rights reserved.

We provide a new characterization of the equality of two positive-definite matrices A and B, and we use this to propose several new computationally convenient statistical tests for the equality of two unknown positive-definite matrices. Our primary focus is on testing the information matrix equality (e.g., White, 1982, 1994). We characterize the asymptotic behavior of our new trace-determinant information matrix test statistics under the null and the alternative and investigate their finite-sample performance for a variety of models: linear regression, exponential duration, probit, and Tobit. The parametric bootstrap suggested by Horowitz (1994) delivers critical values that provide admirable level behavior, even in samples as small as n = 50. Our new tests often have better power than the parametric-bootstrap version of the traditional IMT; when they do not, they nevertheless perform respectably.

We provide a new characterization of the equality of two positive-definite matrices A and B, and we use this to propose several new computationally convenient statistical tests for the equality of two unknown positive-definite matrices. Our primary focus is on testing the information matrix equality (e.g. White, 1982, 1994). We characterize the asymptotic behavior of our new trace-determinant information matrix test statistics under the null and the alternative and investigate their finite-sample performance for a variety of models: linear regression, exponential duration, probit, and Tobit. The parametric bootstrap suggested by Horowitz (1994) delivers critical values that provide admirable level behavior, even in samples as small as n = 50. Our new tests often have better power than the parametric-bootstrap version of the traditional IMT; when they do not, they nevertheless perform respectably.

Notions of cause and effect are fundamental to economic explanation. Although concepts such as price effects are intuitive, rigorous foundations justifying causal discourse in the wide range of economic settings remain lacking. We illustrate this deficiency using an N-bidder private-value auction, posing causal questions that cannot be addressed within existing frameworks. We extend the frameworks of Pearl (2000) and White and Chalak (2009) to introduce topological settable systems (TSS), a causal framework capable of delivering the missing answers. Particularly, TSS accommodate choices belonging to general function spaces. Our analysis suggests how TSS enable causal discourse in various areas of economics.

Policy analysis had long been a main interest of Clive Granger's. Here, we present a framework for economic policy analysis that provides a novel integration of several fundamental concepts at the heart of Granger's contributions to time-series analysis. We work with a dynamic structural system analyzed by White and Lu (2010) with well defined causal meaning; under suitable conditional exogeneity restrictions, Granger causality coincides with this structural notion. The system contains target and control subsystems, with possibly integrated or cointegrated behavior. We ensure the invariance of the target subsystem to policy interventions using an explicitly causal partial equilibrium recursivity condition. Policy effectiveness is ensured by another explicit causality condition. These properties only involve the data generating process; models play a subsidiary role. Our framework thus complements that of Ericsson et al. (1998) (EHM) by providing conditions for policy analysis alternative to weak, strong, and superexogeneity. This makes possible policy analysis for systems that may fail EHM's conditions. It also facilitates analysis of the cointegrating properties of systems subject to policymaker control. We discuss a variety of practical procedures useful for analyzing such systems and illustrate with an application to a simple model of the US macroeconomy.

This paper studies a two-stage procedure for estimating partially identified models, based on Chernozhukov, Hong, and Tamer’s (2007) theory of set estimation and inference. We consider the case where a sub-vector of parameters or their identified set can be estimated separately from the rest, possibly subject to a priori restrictions. Our procedure constructs the second-stage set estimator and confidence set by taking appropriate level sets of a criterion function, using a first-stage estimator to impose restrictions on the parameter of interest. We give conditions under which the two-stage set estimator is a set-valued random element that is measurable in an appropriate sense. We also establish the consistency of the two-stage set estimator.