انتخاب شبیه سازی مبتنی بر رقابت الگوهای اقتصادسنجی ساختاری

This paper proposes a formal model selection test for choosing between two competing structural econometric models. The procedure is based on a novel lack-of-fit criterion, namely, the simulated mean squared error of predictions (SMSEP), taking into account the complexity of structural econometric models. It is asymptotically valid for any fixed number of simulations, and allows for any estimator which has a View the MathML sourcen asymptotic normality or is nαnα-consistent for α>1/2α>1/2. The test is bi-directional and applicable to non-nested models which are both possibly misspecified. The asymptotic distribution of the test statistic is derived. The proposed test is general, regardless of whether the optimization criteria for estimation of competing models are the same as the SMSEP criterion used for model selection. A Monte Carlo study demonstrates good power and size properties of the test. An empirical application using timber auction data from Oregon illustrates the usefulness and generality of the proposed testing procedure.

Model selection is an important component of statistical inference. It involves comparing competing models based on some appropriately defined goodness-of-fit or selection criterion. For the competing models that can be estimated by (conditional) maximum likelihood estimation (MLE), there has been a vast literature on model selection procedures, such as the Akaike, 1973 and Akaike, 1974 information criterion (AIC), the Cox test (1961) and the Vuong (1989) likelihood ratio test, to name only a few. Another important development is the use of the encompassing principle in testing non-nested models assuming that one of them is correctly specified. See, e.g., Mizon and Richard (1986), and Wooldridge (1990), among others. For a comprehensive review of the literature, see Gourieroux and Monfort (1994) and Pesaran and Weeks (2001). In light of the development of new estimation methods in econometrics such as the generalized method of moments (GMM) and empirical likelihood estimation methods, which offer robust alternatives to the conventional MLE, recent work in model selection has attempted to develop procedures that can be used for models estimated by other methods than the MLE. For example, see Smith (1992) for extensions of the Cox test and the encompassing test to non-nested regression models that are both estimated by instrumental variables, Rivers and Vuong (2002) for the extension of Vuong’s (1989) test to dynamic models, Kitamura (2002) for using empirical likelihood ratio-type statistics for testing non-nested conditional models, and Chen et al. (2003) for likelihood ratio tests between parametric and (unconditional) moment condition models. These model selection tests have been found useful in some of structural microeconometric models, which have been developed in the last two decades and applied in such fields of modern economics as labor and industrial organization.1 For example Vuong’s (1989) likelihood ratio test has been used to select structural models both of which are estimated by MLE. See, e.g., Gasmi et al. (1992) for testing collusive behavior, Wolak (1994) for testing asymmetric information, and Li (2005) for testing binding reservation prices in first-price auctions, to name only a few. Also, Chen et al. (2003) develop a test to distinguish between a parametric model which can be estimated by the MLE and an unconditional moment model which can be estimated by the empirical likelihood method, and then apply their procedure to choose between a sequential search model and a non-sequential model. Despite these interesting applications of the aforementioned model selection tests, there are many other situations in which these model selection tests may not be applicable.2 Such a gap can be mainly attributed to the complexity associated with the nature of structural econometric models. Model selection criteria are formulated in such ways that they are calculated using sample information and compared between competing models. Most of the structural econometric models, however, are constructed based on economic theory which defines maps between the latent variable of interest or/and its distribution and the observables. For instance, in structural auction models, it is assumed that the observed bids are Nash–Bayesian equilibrium strategies which are strictly increasing functions of bidders’ private valuations whereas identifying and estimating the private values distribution is one of the main objectives of the structural approach. The presence of latent variables and the complex relationship between the latent and observed variables defined by structural models make the formulation of a well-defined model selection criterion more involved. Moreover, in many cases, structural econometric models are constructed through moment conditions, meaning that they are estimated not by MLE but by GMM or method of simulated moments (MSM). Therefore, to accommodate these specific features arising from the nature of structural models and the estimation methods, new model selection tests need to be developed. Developing model selection procedures suitable in distinguishing between competing structural models is especially relevant in using the structural approach to analyze economic data and make policy evaluations. In the structural approach, policy analysis and the resulting recommendations are based on a structural model that is closely derived from economic theory assuming that the involved economic agents are in the environment described by the theory and behave according to the theory. As a result, it is pivotal to validate the structural model under consideration. For example, when analyzing auction data using the structural approach, an econometrician faces choices among different paradigms such as a private value model or a common value model. Even within a chosen paradigm, the econometrician may also need to determine an appropriate parametric functional form for the latent distribution. Furthermore, the researcher sometimes needs to choose between different equilibria if multi-equilibria exist, as is the case for models of two-stage dynamic games which yield a large number of Bayesian perfect equilibria (Laffont and Maskin, 1990). The goal of this paper is thus to propose a new model selection test in discriminating between competing structural econometric models. Our test is based on a comparison of the predictability of competing structural models. In time series literature, there has been a rich set of papers since Diebold and Mariano (1995) and West (1996) in using predictability for model evaluation. More recently, a general model selection framework based on predictability is developed in Rivers and Vuong (2002). Our test falls within this framework, and uses a similar MSEP criterion. On the other hand, given that structural econometric models usually contain some latent variables that are unobserved, we propose to simulate these latent variables in order to make the predictions on the equilibrium outcomes. Also, since the simulation is used, when formulating the sample analog to those population quantities, we need to correct for the asymptotic bias term caused by the simulation, and hence propose a simulated MSEP (SMSEP) as a consistent sample analog to the population predictability criterion. 3 As a result, while those using simulation based prediction for model evaluations in time series framework usually require that the number of simulations tend to infinity, ours works for any fixed number of simulations. Moreover, our model selection test allows for any estimators that are View the MathML sourcen asymptotically normally distributed, or are nn-consistent that can arise from some structural microeconometric models such as auction models and job search models (Donald and Paarsch, 1993, Donald and Paarsch, 1996, Donald and Paarsch, 2002, Hong, 1998, Chernozhukov and Hong, 2004 and Hirano and Porter, 2003). Lastly, in a similar spirit to that of Vuong (1989) and Rivers and Vuong (2002), the test is bi-directional and applicable to non-nested structural models which are both possibly misspecified. This adds a considerable advantage to the proposed test because in real applications, structural econometric models can be best considered an approximation, but not exact modeling of the true data generating process. Nevertheless, with two possibly misspecified models, our model selection procedure enables one to tell which one is closer to the truth. While some empirical work has used predictions from structural models to validate a particular choice of the model, because of the lack of a formal test, it has been based on an ad-hoc comparison of the closeness between the predictions and the observed outcomes. The statistical significance of such a closeness is not assessed. In contrast, our testing procedure provides a formal framework in which the statistical significance of the difference in predictability of competing structural models can be assessed. The asymptotic distribution of the test statistic is derived. The proposed test is general regardless of whether the optimization criteria for estimation of competing models are the same as the SMSEP criterion used for model selection. We conduct Monte Carlo experiments to study size and power properties of the test. An empirical application using timber auction data from Oregon is used to illustrate the usefulness and generality of the proposed testing procedure. It is worth noting that most of the recent work in model selection tests has been based on comparing the Kullback–Leibler Information Criterion (KLIC) between two competing models. See, e.g., Kitamura, 2000 and Kitamura, 2002, and Chen et al. (2003). Our approach is different, as it is based on the simulated mean squared errors of predictions, a lack-of-fit criterion. This is motivated by the fact that many structural econometric models are estimated by GMM or MSM other than the MLE, thus the KLIC cannot be used as a model selection criterion.4 Our model selection criterion, on the other hand, can be used for any estimation methods that yield estimators with root-nn asymptotic normality, or with nαnα-consistency for α>1/2α>1/2, and hence has an appealing generality. This paper is organized as follows. Section 2 describes the general model selection framework for structural econometric models using the SMSEP criterion. The hypotheses for model selection are formulated. The asymptotic properties of the proposed test statistic are established. Section 3 is devoted to Monte Carlo experiments in investigating finite sample properties of the tests, and Section 4 considers an empirical application of the proposed test to structural auction models. Section 5 concludes.

This paper develops a general framework for testing between competing non-nested structural econometric models. Our method allows for any estimators that are either root-nn asymptotically normally distributed or consistent at a rate faster than root-nn, and can be used for distinguishing between two models that are both possibly misspecified. The statistical significance of the difference between two models under consideration is assessed through a simulation based lack-of-fit criterion, taking into account the complex nature of structural econometric models. As such, our approach provides a new model selection method for choosing between competing structural models. Our Monte Carlo studies demonstrate the good size and power properties of the test. We apply our testing procedure to determine the number of potential bidders in the timber auctions in Oregon. Such an application illustrates the usefulness and generality of our test, and also demonstrates the importance of developing model selection tests in structural econometric models. This paper has been focusing on selection of fully parametric structural models. While the results in the paper do not apply to nonparametric structural models, it is possible to extend them to the semiparametric framework provided that the conditional expectation of the dependent variable can be simulated with the structural parameter estimates. Also, as previously mentioned, this paper is motivated by the need to develop a general model selection test for structural models when the estimation methods used do not allow one to use the existing procedures. On the other hand, because of its generality, our proposed method can also be applied to the cases in which the existing model selection tests work as well. For instance, when two competing structural models are estimated by the MLE, we can use the Vuong (1989) likelihood ratio test as well as our test for model selection. Thus, it would be interesting in this case to compare the asymptotic properties of both tests as well as their finite sample performances. These are left for future research.