Moment Restrictions Research Articles

Inference on conditional moment restriction models with generated variables

A seminal work by Domínguez and Lobato (2004) proposed a consistent estimation method for conditional moment restrictions, which does not rely on additional identification assumptions as in the GMM estimator using unconditional moments and is free from any user-chosen number. Their methodology is further extended by Domínguez and Lobato (2015, 2020) for consistent specification testing of conditional moment restrictions, which may involve generated variables. We follow up this literature and derive the asymptotic distribution of Domínguez and Lobato’s (2004) estimator that involves generated variables. Our simulation result illustrates that ignoring proxy errors in the generated variables may cause severer distortions for the coverage or size properties of statistical inference on parameters.

A Generalized Non-Parametric Instrumental Variable-Control Function Approach to Estimation in Nonlinear Settings

Abstract When the endogenous variables enter non-parametrically into the regression equation standard linear instrumental variables approaches fail. Two existing solutions are the non-parametric instrumental variables (NPIVs) estimators, which are based on a set of conditional moment restrictions (CMRs), and the control function (CF) estimators, which use conditional mean independence (CMI) restrictions. Our first contribution is to show that – similar to CMI – the CMR place shape restrictions on the conditional expectation of the error given the instruments and endogenous variables that are sufficient for identification, and we call our new estimator based on these restrictions the CMR-CF estimator. Our second contribution is to develop an estimator for non-linear and non-parametric settings that can combine both CMR and CMI restrictions, which cannot be done in either the NPIV nor the non-parametric CF setting. This new “Generalized CMR-CF” uses both CMR and CMI restrictions together by allowing the conditional expectation of the structural error to depend on both instruments and control variables. When sieves are used to approximate both the structural function and the CF our estimator reduces to a series of least squares regressions. Our Monte Carlos illustrate that our new estimator performs well across several economic settings.

Macro‐Finance Decoupling: Robust Evaluations of Macro Asset Pricing Models

This paper shows that robust inference under weak identification is important to the evaluation of many influential macro asset pricing models, including (time‐varying) rare‐disaster risk models and long‐run risk models. Building on recent developments in the conditional inference literature, we provide a novel conditional specification test by simulating the critical value conditional on a sufficient statistic. This sufficient statistic can be intuitively interpreted as a measure capturing the macroeconomic information decoupled from the underlying content of asset pricing theories. Macro‐finance decoupling is an effective way to improve the power of the specification test when asset pricing theories are difficult to refute because of a severe imbalance in the information content about the key model parameters between macroeconomic moment restrictions and asset pricing cross‐equation restrictions. We apply the proposed conditional specification test to the evaluation of a time‐varying rare‐disaster risk model and the construction of robust model uncertainty sets.

Bayesian Estimation and Comparison of Conditional Moment Models

AbstractWe consider the Bayesian analysis of models in which the unknown distribution of the outcomes is specified up to a set of conditional moment restrictions. The non-parametric exponentially tilted empirical likelihood function is constructed to satisfy a sequence of unconditional moments based on an increasing (in sample size) vector of approximating functions (such as tensor splines based on the splines of each conditioning variable). For any given sample size, results are robust to the number of expanded moments. We derive Bernstein–von Mises theorems for the behaviour of the posterior distribution under both correct and incorrect specification of the conditional moments, subject to growth rate conditions (slower under misspecification) on the number of approximating functions. A large-sample theory for comparing different conditional moment models is also developed. The central result is that the marginal likelihood criterion selects the model that is less misspecified. We also introduce sparsity-based model search for high-dimensional conditioning variables, and provide efficient Markov chain Monte Carlo computations for high-dimensional parameters. Along with clarifying examples, the framework is illustrated with real data applications to risk-factor determination in finance, and causal inference under conditional ignorability.

Testing conditional moment restriction models using empirical likelihood

Summary An empirical likelihood test is proposed for parameters of models defined by conditional moment restrictions, such as models with nonlinear endogenous covariates, with and without heteroscedastic errors and non-separable transformation models. The number of empirical likelihood constraints is given by the size of the parameter, unlike alternative semi-parametric approaches. We show that the empirical likelihood ratio test is asymptotically pivotal, without explicit studentization. A simulation study shows that the observed size is close to the nominal level, unlike alternative empirical likelihood approaches. It also offers a major advantage over two-stage least-squares, because the relationship between endogenous and instrumental variables does not need to be known. An empirical likelihood model specification test is also proposed.

Nonparametric Tests of Conditional Independence for Time Series

We propose consistent nonparametric tests of conditional independence for time series data. Our methods are motivated from the difference between joint conditional cumulative distribution function (CDF) and the product of conditional CDFs. The difference is transformed into a proper conditional moment restriction (CMR), which forms the basis for our testing procedure. Our test statistics are then constructed using the integrated moment restrictions that are equivalent to the CMR. We establish the asymptotic behavior of the test statistics under the null, the alternative, and the sequence of local alternatives converging to conditional independence at the parametric rate. Our tests are implemented with the assistance of a multiplier bootstrap. Monte Carlo simulations are conducted to evaluate the finite sample performance of the proposed tests. We apply our tests to examine the predictability of equity risk premium using variance risk premium for different horizons and find that there exist various degrees of nonlinear predictability at mid-run and long-run horizons.

Feedback in panel data models

Much of the analysis of panel data has been based on an assumption of strict exogeneity. Distributions are specified for outcome variables conditional on a latent individual effect and conditional on observed predictor variables at all dates, with the future values of the predictor variables assumed to have no effect on the conditional distribution. The paper relaxes this assumption in order to allow for lagged dependent variables and, more generally, for feedback from lagged dependent variables to current values of the predictor variables. Such feedback would arise in an evaluation study if the treatment variable is randomly assigned only conditional on the individual effect and on previous outcomes.An information bound is derived for a semiparametric regression model with sequential moment restrictions, with the information set increasing over time. The bound is then applied to a model with a (scalar) multiplicative random effect. The mean of the random effect conditional on the predictor variables is not restricted, so that the random effect can control for various omitted variables. This conditional mean is the nonparametric component of the semiparametric regression model. There is a transformation that eliminates the random effect and leads to a set of sequential moment restrictions in which the moment function depends on only a finite-dimensional parameter. The information bound for this simpler problem coincides with that of the original problem. The form of the optimal instrumental variables is derived.The paper also considers the identification problems that arise when the random effect is a vector with two or more components.

High dimensional semiparametric moment restriction models

We consider nonlinear moment restriction semiparametric models where both the dimension of the parameter vector and the number of restrictions are divergent with sample size and an unknown smooth function is involved. We propose an estimation method based on the sieve generalized method of moments (sieve-GMM). We establish consistency and asymptotic normality for the estimated quantities when the number of parameters increases modestly with sample size. We also consider the case where the number of potential parameters/covariates is very large, i.e., increases rapidly with sample size, but the true model exhibits sparsity. We use a penalized sieve GMM approach to select the relevant variables, and establish the oracle property of our method in this case. We also provide new results for inference. We propose several new test statistics for the over-identification and establish their large sample properties. We provide a simulation study and an application to data from the NLSY79 used by Carneiro et al. (2011).

Regularised orthogonal machine learning for nonlinear semiparametric models

Summary This paper proposes a Lasso-type estimator for a high-dimensional sparse parameter identified by a single index conditional moment restriction (CMR). In addition to this parameter, the moment function can also depend on a nuisance function, such as the propensity score or the conditional choice probability, which we estimate by modern machine learning tools. We first adjust the moment function so that the gradient of the future loss function is insensitive (formally, Neyman orthogonal) with respect to the first-stage regularisation bias, preserving the single index property. We then take the loss function to be an indefinite integral of the adjusted moment function with respect to the single index. The proposed Lasso estimator converges at the oracle rate, where the oracle knows the nuisance function and solves only the parametric problem. We demonstrate our method by estimating the short-term heterogeneous impact of Connecticut’s Jobs First welfare reform experiment on women’s welfare participation decision.

Efficient Estimation of Pricing Kernels and Market-Implied Densities

This paper studies the nonparametric identification and estimation of projected pricing kernels implicit in European option prices and underlying asset returns using conditional moment restrictions. The proposed series estimator avoids computing ratios of estimated risk-neutral and physical densities. Instead, we consider efficient estimation based on the conditional Euclidean empirical likelihood or continuously-updated GMM criterion, which takes into account the informativeness of option prices of varying strike prices beyond observed conditioning variables. In a second step, we convert the implied probabilities into predictive densities by matching the informative part of cross-sections of option prices. Empirically, pricing kernels tend to be U-shaped in the S&P 500 index return given high levels of the VIX, and call and ATM options are more informative about their payoff than put and OTM options.

A kernel-based measure for conditional mean dependence

A novel metric, called kernel-based conditional mean dependence (KCMD), is proposed to measure and test the departure from conditional mean independence between a response variable Y and a predictor variable X, based on the reproducing kernel embedding and the Hilbert-Schmidt norm of a tensor operator. The KCMD has several appealing merits. It equals zero if and only if the conditional mean of Y given X is independent of X, i.e. E(Y|X)=E(Y) almost surely, provided that the employed kernel is characteristic; it can be used to detect all kinds of conditional mean dependence with an appropriate choice of kernel; it has a simple expectation form and allows an unbiased empirical estimator. A class of test statistics based on the estimated KCMD is constructed, and a wild bootstrap test procedure to the conditional mean independence is presented. The limit distributions of the test statistics and the bootstrapped statistics under null hypothesis, fixed alternative hypothesis and local alternative hypothesis are given respectively, and a data-driven procedure to choose a suitable kernel is suggested. Simulation studies indicate that the tests based on the KCMD have close powers to the tests based on martingale difference divergence in monotone dependence, but excel in the cases of nonlinear relationships or the moment restriction on X is violated. Two real data examples are presented for the illustration of the proposed method.

Empirical process results for exchangeable arrays

Exchangeable arrays are natural tools to model common forms of dependence between units of a sample. Jointly exchangeable arrays are well suited to dyadic data, where observed random variables are indexed by two units from the same population. Examples include trade flows between countries or relationships in a network. Separately exchangeable arrays are well suited to multiway clustering, where units sharing the same cluster (e.g. geographical areas or sectors of activity when considering individual wages) may be dependent in an unrestricted way. We prove uniform laws of large numbers and central limit theorems for such exchangeable arrays. We obtain these results under the same moment restrictions and conditions on the class of functions as those typically assumed with i.i.d. data. We also show the convergence of bootstrap processes adapted to such arrays.

Spatial dependence and spillover effects in customized bus demand: Empirical evidence using spatial dynamic panel models

This paper proposes customized bus (CB) demand models to investigate dynamic adjustments, spatial dependence, and spatial spillover effects using spatial dynamic panel data techniques and a balanced panel data set collected over two years. The endogeneity introduced by spatial dependence effects is resolved by extending the moment restrictions of the system GMM estimator to a spatial autoregressive dynamic panel. A wide range of variables representing factors related to 1) characteristics of service supply, 2) demographic characteristics, and 3) land use and accessibility were mined and fused to investigate their direct and spillover effects on CB demand. The results reveal the mechanism of spatial dependence such that the increments (or decrement) in demand rather than the amount of demand itself in the neighbourhood contribute to the spatial dependence effects. There is more than 43% matched demand generated in a given area for each-unit increment of demand from its neighbourhood. CB services are more attractive for passengers with long-distance trips, and therefore, increasing the focus on long-distance trips is one potential strategy for increasing ridership. Determined by the role of CB in the transportation system, areas with poor accessibility are found to have larger market niches for CB services.

Strong law of large numbers under moment restrictions in sublinear expectation spaces

In this paper, we obtain a new kind of strong law of large numbers in sublinear expectation spaces without the independent or identically distributed assumption. Instead, the strong law of large numbers is based on some restrictions on the sub-linear expectations of the partial sums. As an application, we give an interesting example in which the random variable sequence enjoys our moment restrictions, but does not enjoy the independent conditions introduced by Marinacci (1999), Peng (2009), Chen, Wu, and Li (2013), Zhang (2016b), nor Chen, Huang, and Wu (2019).

Cash Flows Discounted Using a Model-Free SDF Extracted under a Yield Curve Prior

We developed a model-free Bayesian extraction procedure for the stochastic discount factor under a yield curve prior. Previous methods in the literature directly or indirectly use some particular parametric asset-pricing models such as with long-run risks or habits as the prior. Here, in contrast, we used no such model, but rather, we adopted a prior that enforces external information about the historically very low levels of U.S. short- and long-term interest rates. For clarity and simplicity, our data were annual time series. We used the extracted stochastic discount factor to determine the stripped cash flow risk premiums on a panel of industrial profits and consumption. Interestingly, the results align very closely with recent limited information (bounded rationality) models of the term structure of equity risk premiums, although nowhere did we use any theory on the discount factor other than its implied moment restrictions.

Empirical likelihood-MM (EL-MM) estimation for the parameters of a linear regression model

ABSTRACT Parameters of a linear regression model can be estimated using empirical likelihood (EL) estimation method when some distributional assumptions on error term are not satisfied. In the EL procedure constrains are formed using the likelihood scores of maximum likelihood (ML) estimation method under normality assumption. However, since the ML estimators under normality assumption are dramatically affected from outliers due to unboundedness of the score function the resulting EL estimators with the moment restrictions borrowed from the ML score functions will also be ruined by outliers. In this paper, robust empirical likelihood estimators for the parameters of a linear regression model are proposed. This robustification will be done using robust constraints borrowed from the MM estimating equation. Simulation results and real data examples show that the performance of the proposed EL-MM estimator is remarkably superior to the performance of classical EL estimator when there are outliers in response and/or explanatory variables.

A Comment on the Practice of the Arellano-Bond/Blundell-Bond Generalized Method of Moments Estimator in IS Research

As a quick econometric solution to handle potential endogeneity issues in panel data models, the Arellano-Bond/Blundell-Bond generalized method of moments (GMM) estimator continues to gain popularity in IS research. Despite this estimator’s sensitivity to model specifications and estimation strategies, a noticeable number of IS studies that have employed this method have failed to report the detailed model specifications, robustness check results with different model specifications and estimation strategies, or test statistics, which render their empirical results less credible. Using simulated data and real data, we empirically demonstrate that passing the commonly required tests, such as the m2 test and the Sargan/Hansen test, does not guarantee the estimate’s validity because its size and statistical significance can largely depend on the estimation procedure and moment restrictions that researchers choose. We urge researchers to not only report the results of significant focal variables but also explicitly discuss model specifications and estimation strategies and provide robustness checks with different model specifications along with their complete test results.

Robust inference in deconvolution

Kotlarski's identity has been widely used in applied economic research based on repeated‐measurement or panel models with latent variables. However, how to conduct inference for these models has been an open question for two decades. This paper addresses this open problem by constructing a novel confidence band for the density function of a latent variable in repeated measurement error model. The confidence band builds on our finding that we can rewrite Kotlarski's identity as a system of linear moment restrictions. Our approach is robust in that we do not require the completeness. The confidence band controls the asymptotic size uniformly over a class of data generating processes, and it is consistent against all fixed alternatives. Simulation studies support our theoretical results.

Robust Inference for Moment Condition Models without Rational Expectations

Applied researchers using structural models under rational expectations (RE) often confront empirical evidence of misspecification. In this paper we consider a generic dynamic model that is posed as a vector of unconditional moment restrictions. We suppose that the model is globally misspecified under RE, and thus empirically in a way that is not econometrically subtle. We relax the RE restriction by allowing subjective beliefs to differ from the data-generating probability (DGP) model while still maintaining that the moment conditions are satisfied under the subjective beliefs of economic agents. We use statistical measures of divergence relative to RE to bound the set of subjective probabilities. This form of misspecification alters econometric identification and inferences in a substantial way, leading us to construct robust confidence sets for various set identified functionals.

Empirical Likelihood-Based Inference in Regressive Model with Moment Restrictions

In this paper, we consider the parameter estimation problem of linear regression model when the auxiliary information can be denoted by moment restrictions. We use the weighted least squares method to estimate the model parameters and to obtain the weights based on the auxiliary information by using the empirical likelihood method. The limiting distribution of the estimator is established, and the simulation studies are carried out to demonstrate the feasibility of our theoretical results.