Estimation of possibly misspecified semiparametric conditional moment restriction models with different conditioning variables

Newey and Powell [2003. Instrumental variable estimation of nonparametric models. Econometrica 71, 1565–1578] and Ai and Chen [2003. Efficient estimation of conditional moment restrictions models containing unknown functions. Econometrica 71, 1795–1843] propose sieve minimum distance (SMD) estimation of both finite dimensional parameter (θ)$(\theta )$ and infinite dimensional parameter (h) that are identified through a conditional moment restriction model, in which h could depend on endogenous variables. This paper modifies their SMD procedure to allow for different conditioning variables to be used in different equations, and derives the asymptotic properties when the model may be misspecified  . Under low-level sufficient conditions, we show that: (i) the modified SMD estimators of both θ$\theta$ and h   converge to some pseudo-true values in probability; (ii) the SMD estimators of smooth functionals, including the θ$\theta$ estimator and the average derivative estimator, are asymptotically normally distributed; and (iii) the estimators for the asymptotic covariances of the SMD estimators of smooth functionals are consistent and easy to compute. These results allow for asymptotically valid tests of various hypotheses on the smooth functionals regardless of whether the semiparametric model is correctly specified or not.     

Statistical estimation of multivariate Ornstein–Uhlenbeck processes and applications to co-integration

Ornstein–Uhlenbeck models are continuous-time processes which have broad applications in finance as, e.g., volatility processes in stochastic volatility models or spread models in spread options and pairs trading. The paper presents a least squares estimator for the model parameter in a multivariate Ornstein–Uhlenbeck model driven by a multivariate regularly varying Lévy process with infinite variance. We show that the estimator is consistent. Moreover, we derive its asymptotic behavior and test statistics. The results are compared to the finite variance case. For the proof we require some new results on multivariate regular variation of products of random vectors and central limit theorems. Furthermore, we embed this model in the setup of a co-integrated model in continuous time.     

Set identification via quantile restrictions in short panels

This paper studies the identifying power of conditional quantile restrictions in short panels with fixed effects. In contrast to classical fixed effects models with conditional mean restrictions, conditional quantile restrictions are not preserved by taking differences in the regression equation over time. This paper shows however that a conditional quantile restriction, in conjunction with a weak conditional independence restriction, provides bounds on quantiles of differences in time-varying unobservables across periods. These bounds carry observable implications for model parameters which generally result in set identification. The analysis of these bounds includes conditions for point identification of the parameter vector, as well as weaker conditions that result in point identification of individual parameter components.     

Bayesian Model Selection with an Uninformative Prior*

Bayesian model selection with posterior probabilities and no subjective prior information is generally not possible because of the Bayes factors being ill‐defined. Using careful consideration of the parameter of interest in cointegration analysis and a re‐specification of the triangular model of Phillips (Econometrica, Vol. 59, pp. 283–306, 1991), this paper presents an approach that allows for Bayesian comparison of models of cointegration with ‘ignorance’ priors. Using the concept of Stiefel and Grassman manifolds, diffuse priors are specified on the dimension and direction of the cointegrating space. The approach is illustrated using a simple term structure of the interest rates model.     

Verification results for age-structured models of economic–epidemics dynamics

In this paper we propose a macro-dynamic age-structured set-up for the analysis of epidemics/economic dynamics in continuous time.The resulting optimal control problem is reformulated in an infinite dimensional Hilbert space framework where we perform the basic steps of dynamic programming approach.Our main result is a verification theorem which allows to guess the feedback form of optimal strategies. This will be a departure point to discuss the behavior of the models of the family we introduce and their policy implications.     

Theorems on the core of an economy with infinitely many commodities and consumers

It is known that the classical theorems of Grodal [Grodal, B., 1972. A second remark on the core of an atomless economy. Econometrica 40, 581–583] and Schmeidler [Schmeidler, D., 1972. A remark on the core of an atomless economy. Econometrica 40, 579–580] on the veto power of small coalitions in finite dimensional, atomless economies can be extended (with some minor modifications) to include the case of countably many commodities. This paper presents a further extension of these results to include the case of uncountably many commodities. We also extend Vind’s [Vind, K., 1972. A third remark on the core of an atomless economy. Econometrica 40, 585–586] classical theorem on the veto power of big coalitions in finite dimensional, atomless economies to include the case of an arbitrary number of commodities. In another result, we show that in the coalitional economy defined by an atomless individualistic model, core–Walras equivalence holds even if the commodity space is non-separable. The above-mentioned results are also valid for a differential information economy with a finite state space. We also extend Kannai’s [Kannai, Y., 1970. Continuity properties of the core of a market. Econometrica 38, 791–815] theorem on the continuity of the core of a finite dimensional, large economy to include the case of an arbitrary number of commodities. All of our results are applications of a lemma, that we prove here, about the set of aggregate alternatives available to a coalition. Throughout the paper, the commodity space is assumed to be an ordered Banach space which has an interior point in its positive cone.     

Efficiency. of infinite dimensional M- estimators

It is well-known that maximum likelihood estimators are asymptotically normal with covariance equal to the inverse Fisher information in smooth, finite dimensional parametric models. Thus they are asymptotically efficient. A similar phenomenon has been observed for certain infinite dimensional parameter spaces. We give a simple proof of efficiency, starting from a theorem on asymptotic normality of infinite dimensional M -estimators. The proof avoids the explicit calculation of the Fisher information. We also address Hadamard differentiability of the corresponding M -functionals.     

Least squares model averaging by Mallows criterion

This paper is in response to a recent paper by Hansen (2007) who proposed an optimal model average estimator with weights selected by minimizing a Mallows criterion. The main contribution of Hansen’s paper is a demonstration that the Mallows criterion is asymptotically equivalent to the squared error, so the model average estimator that minimizes the Mallows criterion also minimizes the squared error in large samples. We are concerned with two assumptions that accompany Hansen’s approach. The first is the assumption that the approximating models are strictly nested in a way that depends on the ordering of regressors. Often there is no clear basis for the ordering and the approach does not permit non-nested models which are more realistic from a practical viewpoint. Second, for the optimality result to hold the model weights are required to lie within a special discrete set. In fact, Hansen noted both difficulties and called for extensions of the proof techniques. We provide an alternative proof which shows that the result on the optimality of the Mallows criterion in fact holds for continuous model weights and under a non-nested set-up that allows any linear combination of regressors in the approximating models that make up the model average estimator. These results provide a stronger theoretical basis for the use of the Mallows criterion in model averaging by strengthening existing findings.     

Testing for Parameter Stability in Dynamic Models across Frequencies*

This paper contributes to the econometric literature on structural breaks by proposing a test for parameter stability in vector autoregressive (VAR) models at a particular frequency ω, where ω ∈ [0, π]. When a dynamic model is affected by a structural break, the new tests allow for detecting which frequencies of the data are responsible for parameter instability. If the model is locally stable at the frequencies of interest, the whole sample size can then be exploited despite the presence of a break. The methodology is applied to analyse the productivity slowdown in the US, and the outcome is that local stability concerns only the higher frequencies of data on consumption, investment and output.     

Nonparametric maximum likelihood estimation in a non locally compact setting

Maximum likelihood estimation is considered in the context of infinite dimensional parameter spaces. It is shown that in some locally convex parameter spaces sequential compactness of the bounded sets ensures the existence of minimizers of objective functions and the consistency of maximum likelihood estimators in an appropriate topology. The theory is applied to revisit some classical problems of nonparametric maximum likelihood estimation, to study location parameters in Banach spaces, and finally to obtain Varadarajan’s theorem on the convergence of empirical measures in the form of a consistency result for a sequence of maximum likelihood estimators. Several parameter spaces sharing the crucial compactness property are identified. This research was supported by grants from the National Sciences and Engineering Research Council of Canada and the Fonds FCAR de la Province de Québec.     

Equilibria in large economies with a separable Banach commodity space and non-ordered preferences

The purpose of this paper is to provide an existence result of equilibria for economies with a measure space of agents, a non-trivial production sector and an infinite dimensional commodity space. The commodity space is modeled by an ordered separable Banach space whose positive cone has a non-empty interior. The discretization approach proposed in this paper, allows us to extend the existence results in Khan and Yannelis [Equilibrium in markets with a continuum of agents and commodities. In: Khan, M.A., Yannelis, N.C. (Eds.), Equilibrium Theory in Infinite Dimensional Spaces. Springer, Berlin, 1991] and Podczeck [Economic Theory 9 (1997) 585] to economies with a non-trivial production sector and with possibly non-ordered but convex preferences as well as partially ordered (possibly incomplete) but non-convex preferences.     

A note on the equilibrium theory of economies with asymmetric information

In this note, we give an equilibrium existence theorem for exchange economies with asymmetric information and with an infinite dimensional commodity space. In our model, we assume that preferences are represented by well behaved utility functions, the positive cone has a non empty interior and the individual rational utility set is compact. Our result complements the corresponding one in Podczeck and Yannelis (2008), in the sense that is applicable to commodity spaces in which the order intervals are (possibly) not compact with respect to any Hausdorff linear topology.     

Bayesian inference in a time varying cointegration model

There are both theoretical and empirical reasons for believing that the parameters of macroeconomic models may vary over time. However, work with time-varying parameter models has largely involved vector autoregressions (VARs), ignoring cointegration. This is despite the fact that cointegration plays an important role in informing macroeconomists on a range of issues. In this paper, we develop a new time varying parameter model which permits cointegration. We use a specification which allows for the cointegrating space to evolve over time in a manner comparable to the random walk variation used with TVP–VARs. The properties of our approach are investigated before developing a method of posterior simulation. We use our methods in an empirical investigation involving the Fisher effect.     

Finitely additive economies with free extremely desirable commodities

Continuing the investigation on finitely additive economies with infinite dimensional commodity space, we state a core-walras individualistic equivalence for an economy where preferences admit a summable map of extremely desirable commodities. The main result extends previous equivalences obtained both in the countably additive and in the finitely additive setting. Extra assumptions on the model are discussed via a pair of examples.     

Semiparametric estimation of Markov decision processes with continuous state space

We propose a general two-step estimator for a popular Markov discrete choice model that includes a class of Markovian games with continuous observable state space. Our estimation procedure generalizes the computationally attractive methodology of Pesendorfer and Schmidt-Dengler (2008) that assumed finite observable states. This extension is non-trivial as the policy value functions are solutions to some type II integral equations. We show that the inverse problem is well-posed. We provide a set of primitive conditions to ensure root-T consistent estimation for the finite dimensional structural parameters and the distribution theory for the value functions in a time series framework.     

How different is Africa? A comment on Masanjala and Papageorgiou

Masanjala and Papageorgiou (Rough and lonely road to prosperity: a reexamination of the sources of growth in Africa using Bayesian model averaging, Journal of Applied Econometrics 2008; 23 (5): 671–682) use Bayesian model averaging to evaluate the existence of parameter heterogeneity between African and non‐African countries in the framework of cross‐country growth regressions. We show that their results are not robust to the use of a prior over the model space that respects the strong heredity principle put forward by Chipman (Chipman HA. 1996. Bayesian variable selection with related predictors. Canadian Journal of Statistics 24 : 17–36) for models including interaction terms. Copyright © 2010 John Wiley & Sons, Ltd.     

Time-varying combinations of predictive densities using nonlinear filtering

We propose a Bayesian combination approach for multivariate predictive densities which relies upon a distributional state space representation of the combination weights. Several specifications of multivariate time-varying weights are introduced with a particular focus on weight dynamics driven by the past performance of the predictive densities and the use of learning mechanisms. In the proposed approach the model set can be incomplete, meaning that all models can be individually misspecified. A Sequential Monte Carlo method is proposed to approximate the filtering and predictive densities. The combination approach is assessed using statistical and utility-based performance measures for evaluating density forecasts of simulated data, US macroeconomic time series and surveys of stock market prices. Simulation results indicate that, for a set of linear autoregressive models, the combination strategy is successful in selecting, with probability close to one, the true model when the model set is complete and it is able to detect parameter instability when the model set includes the true model that has generated subsamples of data. Also, substantial uncertainty appears in the weights when predictors are similar; residual uncertainty reduces when the model set is complete; and learning reduces this uncertainty. For the macro series we find that incompleteness of the models is relatively large in the 1970’s, the beginning of the 1980’s and during the recent financial crisis, and lower during the Great Moderation; the predicted probabilities of recession accurately compare with the NBER business cycle dating; model weights have substantial uncertainty attached. With respect to returns of the S&P 500 series, we find that an investment strategy using a combination of predictions from professional forecasters and from a white noise model puts more weight on the white noise model in the beginning of the 1990’s and switches to giving more weight to the professional forecasts over time. Information on the complete predictive distribution and not just on some moments turns out to be very important, above all during turbulent times such as the recent financial crisis. More generally, the proposed distributional state space representation offers great flexibility in combining densities.     

Methods for computing marginal data densities from the Gibbs output

We introduce two estimators for estimating the Marginal Data Density (MDD) from the Gibbs output. Our methods are based on exploiting the analytical tractability condition, which requires that some parameter blocks can be analytically integrated out from the conditional posterior densities. This condition is satisfied by several widely used time series models. An empirical application to six-variate VAR models shows that the bias of a fully computational estimator is sufficiently large to distort the implied model rankings. One of the estimators is fast enough to make multiple computations of MDDs in densely parameterized models feasible.     

Bifurcations in regional migration dynamics

The tomahawk bifurcation is used by Fujita et al. [Fujita, M., Krugman P., Venables A.J., 1999, The Spatial Economy: Cities, Regions, and International Trade, MIT Press: Cambridge, MA.] in a model with two regions to explain the formation of a core–periphery urban pattern from an initial uniform distribution. Baldwin et al. [Baldwin, R., Forslid, R., Martin, P., Ottaviano, G.I.P., Robert-Nicoud, F., 2003, Economic Geography and Public Policy, Princeton University Press: Princeton, NJ.] show that the tomahawk bifurcation disappears when the two regions have an uneven population of immobile agricultural workers. Thus, the appearance of this type of bifurcation is the result of assumed exogenous model symmetry. We provide a general analysis in a regional model of the class of bifurcations that have crossing equilibrium loci, including the tomahawk bifurcation, by examining arbitrary smooth parameter paths in a higher dimensional parameter space. We find that, in a parameter space satisfying a mild rank condition, generically in all parameter paths this class of bifurcations does not appear. In other words, conclusions drawn from the use of this bifurcation to generate a core–periphery pattern are not robust.     

A Comparison of Johansen's,Bierens’ and the Subspace Algorithm Method for Cointegration Analysis*

The methods listed in the title are compared by means of a simulation study and a real world application. The aspects compared via simulations are the performance of the tests for the cointegrating rank and the quality of the estimated cointegrating space. The subspace algorithm method, formulated in the state space framework and thus applicable for vector autoregressive moving average (VARMA) processes, performs at least comparably to the Johansen method. Both the Johansen procedure and the subspace algorithm cointegration analysis perform significantly better than Bierens’ method. The real‐world application is an investigation of the long‐run properties of the one‐sector neoclassical growth model for Austria. The results do not fully support the implications of the model with respect to cointegration. Furthermore, the results differ greatly between the different methods. The amount of variability depends strongly upon the number of variables considered and huge differences occur for the full system with six variables. Therefore we conclude that the results of such applications with about five or six variables and 100 observations, which are typical in the applied literature, should possibly be interpreted with more caution than is commonly done.