On the Solution and Application of Rational Expectations Models With

On the Solution and Application of Rational Expectations Models with Function-Valued States⇤ David Childers† November 16, 2015 Abstract Many variables of interest to economists take the form of time varying distributions or functions. This high-dimensional ‘functional’ data can be interpreted in the context of economic models with function valued endogenous variables, but deriving the implications of these models requires solving a nonlinear system for a potentially infinite-dimensional function of infinite-dimensional objects. To overcome this difficulty, I provide methods for characterizing and numerically approximating the equilibria of dynamic, stochastic, general equilibrium models with function-valued state variables by linearization in function space and representation using basis functions. These methods permit arbitrary infinite-dimensional variation in the state variables, do not impose exclusion restrictions on the relationship between variables or limit their impact to a finite-dimensional sufficient statistic, and, most importantly, come with demonstrable guarantees of consistency and polynomial time computational complexity. I demonstrate the applicability of the theory by providing an analytical characterization and computing the solution to a dynamic model of trade, migration, and economic geography. ⇤Preliminary, subject to revisions. For the latest version, please obtain from the author’s website at https://sites.google.com/site/davidbchilders/DavidChildersFunctionValuedStates.pdf †Yale University, Department of Economics. The author gratefully acknowledges financial support from the Cowles Foundation. This research has benefited from discussion and feedback from Peter Phillips, Tony Smith, Costas Arkolakis, John Geanakoplos, Xiaohong Chen, Tim Christensen, Kieran Walsh, and seminar participants at the Yale University Econometrics Seminar, Macroeconomics Lunch, and Financial Markets Reading Group. All errors are my own. 1 1Introduction In order to understand and evaluate the causes and consequences of economic heterogeneity, it is helpful to have an analytical framework in which the distribution of heterogeneity can change over time and can both affect and be affected by other variables. A perspective in which some of the state variables of an economic model are endogenous random functions allows distributions, as well as objects like demand and supply curves or policy or value functions, to be treated as data. While descriptive models and methods for function-valued time series are undergoing rapid development,1 interpreting this data requires formulating economic models capable of generating the observed functional data and deriving their implications. For models featuring forward looking decision making and endogenous aggregate variables, this derivation typically requires solving a computationally intractable infinite- dimensional system of nonlinear expectational difference equations. Although heuristic or strongly model dependent methods have been proposed, to date there appears to be no general purpose algorithm which provides a formal guarantee of even an approximate solution to rational expectations models with stochastic function-valued states. This paper provides such an algorithm. In particular, it demonstrates how the equilibrium conditions for a general class of function-valued rational expectations models, including but not limited to heterogeneous agent dynamic stochastic general equilibrium models, can be linearized directly in function space, with solutions characterized locally by a functional linear process, a tractable empirical model for function-valued time series (Bosq, 2000). Construction of a local solution requires introducing a novel infinite-dimensional extension of the generalized Schur decomposition used to solve finite-dimensional rational expectations models (Klein, 2000) and developing perturbation theory for this object, which may be contributions of independent mathematical interest. The solution can be implemented numerically by a procedure based on finite-dimensional projection approximations which converges to the local solution under mild regularity conditions. I analyze in detail a particular approximation algorithm in this class, a wavelet transform based procedure which yields an approximate solution accurate to within any desired degree in polynomial time. To demonstrate and evaluate the method, I develop a dynamic spatial model of trade, migration, and economic geography which introduces forward looking migration decisions and spatial shocks into the economic geography model of Krugman (1996). In the model, the spatial distribution of population, wages, and welfare over a continuum of locations is allowed to vary nonparametrically in response to persistent spatially correlated shocks to 1See Horváth & Kokoszka (2012); Bosq (2000); Morris (2014); Ferraty & Romain (2011) for surveys of the rapidly expanding field of functional data analysis, which focuses on modeling, estimation, and inference for series of observed or estimated functions. 2 the desirability of different locations. Due to the spatial structure of trade and production, the spatial distribution of economic activity is a determined by the distribution of population across locations, while the distribution of population is determined by forward looking migration decisions which take into account the expected distribution of economic activity. In this setting, the relationship between these two functions is not easily reduced to low- dimensional summaries or split into “local” and “global” components, but is well characterized by a functional linear model representation. By exploiting an analytical characterization of the solution to (certain parameterizations of) this model, the speed and numerical accuracy of the algorithm are evaluated in practice and shown to be in line with the strong theoretical guarantees. The core idea behind the solution method is functional linearization.Bytakingthe functional derivatives of the equations defining an equilibrium, it is possible to construct a system of equations which can be solved for the functional derivatives at a fixed point in function space of the policy operator, a map from function-valued states to function-valued endogenous variables. In this way, it is possible to recover local information about the solutions, which can then be used to construct a functional Taylor expansion of the policy operator which provides an accurate solution for all functions not too far from the function around which the model is linearized. Constructing this linear approximation of the policy operators from the functional derivatives of the model equations requires solving a system of quadratic equations in linear operators. In the case of linear or linearized finite-dimensional rational expectations models, the analogous quadratic equation can be solved using matrix decomposition. In particular, Klein (2000) demonstrated that a solution can be found using the generalized Schur (or QZ) decomposition of the matrices of derivatives. In infinite dimensions, an analogous decomposition appears to be absent from the literature, in part because the finite-dimensional version is constructed by induction using eigenvalues, which may fail to exist or have countable car- dinality in infinite-dimensional space. Nevertheless, it is possible to construct an analogous decomposition by other methods, described in detail in Appendix A. Under the conditions required for such a decomposition to exist and under further conditions analogous to the well known criterion of Blanchard & Kahn (1980) ensuring that the model has a linear solution, it is possible to solve for the first order expansion of the policy operator. Calculating this local solution numerically requires representing it in a form that can be evaluated on a computer. A standard procedure for reducing problems in function spaces to finite-dimensional objects is to approximate the functions by projecting the space onto the span of a set of basis functions, such as wavelets, splines, or trigonometric or Chebyshev polynomials, and representing operators on function space in terms of their behavior with 3 respect to the basis functions. These approaches are referred to as spectral methods and are commonly applied to solve integral and differential equations: see Boyd (2000), Chatelin (2011). If any function we are interested in can be represented reasonably accurately by a finite set of basis functions, the loss from the use of a finite set of functions may be small. The caveat here is that, unlike in classical function approximation problems where the class of functions is known, ‘the set of functions we are interested in’ is not explicitly assumed, but must be determined by the properties of the model. The issue that projection methods must overcome is that the class of functions well approximated by finite projection is in fact small in the class of all possible functions which could conceivably arise endogenously as outcomes of an implicitly defined model with function valued variables. To handle this concern, conditions must be imposed on the model which ensure both that the solutions themselves are continuous with respect to projection approximations and that the solutions are operators which have the property that they map functions which are well approximated by basis functions to functions which are well approximated by basis functions. Continuity properties of the generalized Schur decomposition are derived in Appendix

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