Solving Nonlinear Rational Expectations Models by Approximating the Stochastic Equilibrium System

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1 Solving Nonlinear Rational Expectations Models by Approximating the Stochastic Equilibrium System Michael P. Evers September 20, 2012 Revised version coming soon! Abstract Dynamic stochastic rational expectations models are commonly solved by computing the linearized solution from the deterministic equilibrium system. Therefore, the linearized solution must be independent of the distribution of the exogenous shocks. This paper proposes to compute the linearized solution from an approximated equilibrium system. This approximated equilibrium system (denoted by AES) is obtained from a kth-order Taylor series expansion of the true stochastic equilibrium system about the deterministic equilibrium system by perturbing it in the exogenous disturbances. The AES is non-stochastic and preserves the nonlinearity in the endogenous variables, but it is linear in the first k moments of the exogenous disturbances. Hence, the linearized solution to the I thank Stephanie Schmitt-Grohé, Martin Uribe, Christopher Sims, Harald Uhlig, Kenneth Judd, Jesus Fernandez-Villaverde, Matthias Paustian, Paul Pichler, Casper de Vries, Christian Matthes, and in particular Johannes Pfeifer for helpful discussions and valuable comments at different stages of this project. All errors are my responsibility. IIW, ZEI, and BGSE, Bonn University; Correspondence: Institut für Internationale Wirtschaftspolitik, Lennéstr. 37, D Bonn, Germany; mevers@uni-bonn.de. 1

2 AES fully captures the equilibrium implications of the stochastic environment up to the first k moments. The solution procedure thus allows to employ standard linear toolboxes both from mathematical economics and econometrics to study the ramifications of risk in DSGE models. Keywords: Solving dynamic general equilibrium models; Perturbation methods; Linear solution in second and higher moments of shock distribution; Accuracy JEL classification: E0, C63 1 Introduction Nonlinearities and stochastic environments are key features of modern dynamic macroeconomic modeling. The predominant solution method to solve nonlinear dynamic stochastic general equilibrium (DSGE) models is based on perturbation methods. Following Magill (1977), Kydland and Prescott (1982), and King et al. (1988), the conventional practice is to compute the linear solution to the deterministic version of the model, which is why the equilibrium interaction of the stochastic environment with nonlinearities must be neglected. This paper demonstrates how to obtain linear approximate solutions to dynamic stochastic rational expectations models which explicitly account for the stochastic environment in nonlinear models. The innovation of the approach is to compute the linear solution from an approximated equilibrium system that preserves the impact of risk on the complete solution process instead of computing the solution from the deterministic version of the equilibrium system. The approximated equilibrium system (denoted by AES henceforth) is the kth-order Taylor series approximation to the original stochastic equilibrium system which is expanded in the exogenous disturbances about the deterministic model with its deterministic solution. In these terms, the 2

3 conventional practice is to compute the linear solution from the 0th-order AES. The AES-approach has three key properties: First, the kth-order Taylor series approximation to the original stochastic equilibrium conditions is non-stochastic and preserves the nonlinearity in the endogenous variables. But it is linear in the first k moments of the distribution of exogenous shocks. The solution to the kth-order AES is parameterized by the first k moments of the exogenous disturbances. Notably, the steady state of the AES and the linear coefficients on the state vector capture the equilibrium implications of the stochastic environment in nonlinear models up to the first k moments. The practical implication for researchers is that the linear state-space representation of the solution process can be applied to study the equilibrium implications of nonlinearities in stochastic environments as for instance on the dynamic behavior and the equilibrium distribution. Most notably, the AES-approach captures the effect of risk on existence and uniqueness up to the first k moments. This is in stark contrast to the conventional perturbation approach where existence and uniqueness is generically assessed for the deterministic model (i.e. the 0th-order AES). Moreover, when taking the model to the data, the equilibrium solution process can be analyzed with standard linear econometric tools such as VARs and likelihood based approaches with linear filtering whereas nonlinear econometric tools are computationally expensive if not infeasible. Linear methods thus become applicable for a wide range of macroeconomic questions that were hitherto beyond their reach, like e.g. welfare comparisons across different policies, portfolio choice models, or the study of asset pricing in DSGE models. The same is true for studying Markov-switching models, models with recursive preferences, and the topical debate about the quantitative role of uncertainty and time-varying volatility on macroeconomic aggregates 1. 1 Compare on welfare evaluations e.g. Tesar (1995) and Kim and Kim (2003); on portfolio choice models, see e.g. Baxter and Jermann (1997) and Devereux and Sutherland (2011) in an open economy context; on asset pricing in DSGE model, see e.g. Uhlig (2007), Rudebusch and Swanson (2008), or Kliem and Uhlig (2011); on Markov-switching models in conventionally linearized setups see e.g. Farmer et al. (2009) and with a nonlinear solution method see e.g. Davig et al. (n.d.); on recursive 3

4 Second, the Euler error implied by the exact solution to the AES is simply the approximation error of the AES to the true stochastic equilibrium system. As a consequence, the accuracy of the linearized solution to the kth-order AES thus increases in the order k, too. This has important implications: i) The accuracy of the approximated key local properties of existence and uniqueness as well as the accuracy of the approximated equilibrium distribution and the dynamic behavior increases in the order k; ii) As Ackerberg et al. (2009) show, the effect of using approximated solutions on the parameter estimation based on maximum likelihood inference is limited by the approximation error to the solution. Consequently, the accuracy of parameter estimates based on the linear solution to the AES also increases in the order k; iii) Using the DSGE model for forecasting, the accuracy of forecasts based on the linear solution to the AES also increases in the order k. The third key property of the AES-approach is that the procedure requires the solution to be locally determined for the kth-order AES but not for the deterministic version of the model. This makes the perturbation method applicable to models where the solution is locally indeterminate in the deterministic version of the model. This is the case for instance in portfolio choice models. From a methodological perspective, the AES-approach regards the stochastic equilibrium system to be arising from the deterministic model by small random disturbances. This view has a long tradition in economics: Measures of risk aversion and precautionary motives following Arrow (1965) and Pratt (1964) are based on expansions in the random disturbances about some deterministic reference point. It is also the basic concept of Samuelson s fundamental approximation theorem to the mean-variance solution in portfolio choice problems (Samuelson (1970)). In this regard, the AES-approach generalizes Samuelson s approach to solving dynamic stochastic preferences in DSGE models, see e.g. Rudebusch and Swanson (2012); on the quantitative role of uncertainty and time-varying volatility in the DSGE framework, see for instance Justiniano and Primiceri (2008), Fernandez-Villaverde and Rubio-Ramirez (2007), Fernández-Villaverde et al. (2010), and Born and Pfeifer (2011). 4

5 rational expectations models. Related literature. In order to account for risk, several recent papers provide algorithms to compute higher-order Taylor series expansions to the solution of DSGE models. Schmitt-Grohé and Uribe (2004), Kim et al. (2008), and Jin and Judd (2002) provide perturbation algorithms for calculating second-order approximations to the solution of DSGE models. 2 The AES-approach is closely related to conventional higher-order approximations because they are based on the same perturbation principles. The crucial difference between the two approaches boils down to the sequencing of perturbation with respect to exogenous disturbances/risk and the vector of state variables. Most conveniently from a practical perspective, by sorting and rearranging the Taylor coefficients appropriately, the AES-approach is easily implemented into existing computer programs that compute higher-order approximations to the solution of DSGE models based on Schmitt-Grohé and Uribe (2004), Sims (2000), or the popular DYNARE/DYNARE++ (Adjemian et al. (2011)). Simply sorting and rearranging the Taylor coefficients appropriately then dissolves two major difficulties of conventional higher-order approximations: First, pruning of higher-order perturbation approximations to avoid explosive simulated solution paths is no longer required. 3 Second, nonlinear statistical methods are avoided when estimating the rational expectations model. Instead, the linear state-space representation of the solution process allows to employ the standard linear tools like for instance the Kalman filter. The relation between the AES-approach and the conventional higher-order approximation is discussed in detail in Supplement A. This paper is also closely related to recent contributions that seek a linear approxi- 2 Other contributions include Collard and Juillard (2001), Anderson and Levin (2002), Anderson et al. (2006), Lombardo and Sutherland (2007), and Gomme and Klein (2011). 3 Kim et al. (2008) and den Haan and Wind (2012) describe the problem of explosive simulated time paths for higher-order perturbation approximations and potential remedies for this problem. 5

6 mation to the solution about a risky or stochastic steady state rather than about the deterministic steady state, as e.g. in Juillard (2012), Coeurdacier et al. (2011), and Kliem and Uhlig (2011). The basic idea and common to the AES-approach is to compute a (2nd-order) Taylor series expansion to the stochastic equilibrium conditions (Euler equations). Their approach is to compute the Taylor series expansion to the equilibrium system in the endogenous and exogenous variables with the consequence that the Euler equations are stated in terms of the moments of the equilibrium distribution of the endogenous and exogenous variables instead of the moments of the exogenous shocks. Assuming the linear structure of the solution, the risky steady state and the corresponding linear coefficients are then determined numerically by iteratively computing the implied equilibrium distribution and then updating the approximated Euler equations. The risky steady state approach is rather heuristic and lacks clear theoretical foundation. Crucially, the iterative procedure might be unstable as the fixed-point algorithm need not be convergent. In contrast, the AES-approach computes the Taylor series expansion to the equilibrium system directly in terms of the moments of the distribution of the exogenous shocks taking fully into account the impact of shocks on endogenous variables. The present paper thus provides a theoretical foundation to the risky steady state approach and discusses its properties. Specifically, the risky steady state simply refers to the steady state of the AES and the linear approximation about the risky steady state refers to the linearized solution to the AES. Importantly, the AES-approach is based on standard perturbation methods: Any computational instabilities and inefficiencies involved with the iterative fixed-point algorithms are avoided and the approximated solution process is easily taken to the data. Organization of the paper. The paper is organized as follows. The next section presents the simple portfolio choice problem as a motivating example. Section 3 6

7 describes the general framework and fixes notation. The Taylor series approximation to the stochastic equilibrium model is presented in Section 4. A discussion of the approximation error of the AES and its implications for the accuracy of the approximated solution follows in Section 5. Section 6 concludes with some final remarks. Supplemental Material. The Supplemental Material contains a detailed discussion of how the AES-approach relates to conventional higher-order perturbation algorithms as in Schmitt-Grohé and Uribe (2004) (Supplement A). Supplement A also describes how the AES-approach can be implemented into existing routines. The Supplemental Material also contains two further applications of the AES-approach: Supplement B provides an algebraic application to the consumption-savings problem where it is highlighted how the AES-approach captures precautionary motives. Supplement C presents a comprehensive numerical application to an asset pricing model (Burnside (1998)) which permits an exact assessment of the accuracy of the AES-approach with respect to the Taylor coefficients of the linear solution, the implied equilibrium distribution, existence and uniqueness, and maximum likelihood estimates of the deep parameters. It also provides a comparison to the conventional 2nd-order approximation. 2 Motivating Example: The Portfolio Choice Problem Before starting with the general description of the problem and the solution procedure, it is useful to first illustrate the basic principle and the properties of the AES-approach. To this end, consider the simple portfolio choice example in Samuelson (1970). A risk averse investor with concave utility function U maximizes expected utility from investing a fraction (1 w) of initial unit wealth into cash and a fraction w into a risky asset with an excess return over cash of a. The overall return of the portfolio 7

8 investment is (1 w) + w(a + 1). The exogenous process for the excess return is simply a = σɛ where ɛ is a random shock and σ is the perturbation parameter. The solution to the portfolio investment problem is characterized by the first order condition EU (wσɛ + 1) σɛ = 0. (1) The first step to solve the portfolio choice model is to restate the stochastic equilibrium condition in terms of the moments of the exogenous excess return process. Step 1: Approximating the original stochastic equilibrium condition (computing the AES) The stochastic equilibrium condition (1) can be approximated by a Taylor series expansion in the perturbation parameter σ about σ = 0, which corresponds to the deterministic optimality condition. For instance, the second order Taylor series of (1) about σ = 0 yields EU (wσɛ + 1) σɛ = U (1)σEɛ + U (1)wσ 2 Eɛ 2 + O(σ 3 ) = 0, (2) where the remainder O(σ 3 ) captures the approximation error of the 2nd-order Taylor expansion to the original condition in (1). Step 2: Computing the solution to the AES The second step of the solution procedure is to compute the optimal portfolio fraction w of the risky asset from the approximated equilibrium condition in (2). Setting the mean and the variance as in Samuelson s example, i.e. Eɛ = σµ a and Eɛ 2 = σ 2 a, results in w = µ a U (1) σa 2 U (1) + O(σ3 ), which is the classical mean-variance approximation when omitting the remainder O(σ 3 ). The simple portfolio choice example already highlights the key aspects of the AES-approach. First, it shows how the 2nd-order Taylor series expansion about σ = 0 8

9 approximates the expectational optimality condition (1) by a non-stochastic equation that explicitly takes into account uncertainty via the first and second moments of the shocks to the excess return. The solution to the new equilibrium (2) system therefore also depends on these moments. Second, the simple example shows that the approximation error of the Taylor series expansion O(σ 3 ) is simply the Euler error of the exact solution to the new equilibrium (2), w = µa U (1). The accuracy of σa 2 U (1) the solution to the new equilibrium system AES as the approximate solution to the true problem ES thus increases in the order k of the Taylor series expansion AES. Third, the example also reveals why the procedure allows to solve problems where the solution is locally indeterminate in the deterministic version of the model. In the deterministic case when σ = 0, the optimality condition (1) is trivially satisfied so that w cannot be determined. The higher-order Taylor terms in (2), however, provide a system of equilibrium conditions that uniquely determines w. The solution procedure thereby makes local linear methods applicable to problems where the deterministic equilibrium system when σ = 0 is locally indeterminate. 4 4 Note that by construction the perturbation coefficient σ is used to scale the effective distribution of the shocks. In principle, instead of introducing it as a coefficient on the shock terms, the perturbation coefficient could have also been introduced directly into the shock distribution as coefficients on the moments. In fact, the latter approach is the underlying idea of Samuelson (1970) when he introduced a scaling parameter by which he let shrink the riskiness of the stochastic returns to zero to obtain the non-stochastic outcome. He coined the term compact probability for the distribution that is parameterized in the scaling coefficient. In fact, Samuelson did not explicitly use the scaling coefficient as the perturbation parameter, but his approach rests on Taylor series expansion in the shock terms, too. 9

10 3 The General Form of the Dynamic Rational Expectations Model The dynamic, discrete-time equilibrium model can be cast into the general form of a nonlinear stochastic vector difference equation, 0 = E t f(y t, x t, z t, y t+1, x t+1, z t+1 ), (3) for t = 0, 1,...,, where y t R ny is the vector of endogenous variables nonpredetermined at the beginning of time t, x t R nx is the vector of predetermined endogenous variables, and z t R nz is the vector of exogenous variables. 5 The stochastic equilibrium system (denoted by ES) is characterized by the function f : R 2 ny R 2 nx R 2 nz R n. It is the collection of all equilibrium conditions which determine the solution to the n = n y + n x endogenous variables. The law of motion of the exogenous variables follows a stochastic process which is implicitly described by where M : R nz R nz R nɛ 0 = M(z t+1, z t, σɛ t+1 ), (4) R nz. The process of the exogenous state variables is assumed to be dynamically stable. The disturbances to the exogenous variables are collected by the vector ɛ t R nɛ across time with finite first k moments. which is independently and identically distributed The stochastic equilibrium system (3) is conceived as the perturbation of the deterministic equilibrium system by small random shocks σɛ t+1. Taking the distribution of 5 In principle, the partition of variables into states and controls is not necessary for the proposed solution method but it is done for expositional convenience. The same procedure can also be applied to the more general formulation as in Sims (2000) and Kim et al. (2008) where the partitioning into states and controls is part of the solution procedure as it follows from the partitioning into predetermined and non-predetermined disturbances. For the reader s convenience, the exposition of the general setup follows Schmitt-Grohé and Uribe (2004) as closely as possible but allows for more general laws of motion of the exogenous variables. 10

11 shocks as given, the perturbation parameter σ on the shock vector ɛ t+1 allows to scale the risk in the equilibrium system. In (3), E t denotes the mathematical expectations operator conditional on the information available at time t, when the current state (x t, z t ) is completely observable. The Solution The solution process to (3) evolves according to the following rules for the endogenous predetermined and non-predetermined variables: x t+1 = h(x t, z t ) and y t = g(x t, z t ), with z t+1 = m(z t, σɛ t+1 ), where the equilibrium rules are vector valued functions with h : R nx R nz R nx and g : R nx R nz R ny, and where the exogenous variables are assumed to follow the law of motion with m : R nz R nɛ R nz. Substituting the solution process into the system of nonlinear stochastic equilibrium equations in (3) allows to restate the stochastic equilibrium system in terms of the current state and the corresponding solution to the endogenous variables as E t F (x t, z t, σɛ t+1 ; g( ), h( )) E t f (g(x t, z t ), x t, z t, g(h(x t, z t ), m(z t, σɛ t+1 )), h(x t, z t ), m(z t, σɛ t+1 )). The general model can then be stated for the true solution process as 0 = E t F (x t, z t, σɛ t+1 ; g( ), h( )) z t+1 = m(z t, σɛ t+1 ). (5) The deterministic model when σ = 0 can thus be stated as 0 = F (x t, z t, 0; g 0 ( ), h 0 ( )) z t+1 = m(z t, 0), 11

12 where g 0 ( ) and h 0 ( ) denote the solution to the deterministic model. Law of Motion for exogenous state variables. The law of motion for the exogenous state variables is often assumed to be linear in both z t and σɛ t+1 where usually z t is the log of a variable. This directly yields the law of motion of the form z t+1 = m(z t, σɛ t+1 ) = Az t + Bσɛ t+1 for some matrices A of size n z n z and B of size n z n ɛ. The more general form in (4) allows for nonlinear processes and interactive disturbances that enter the law of motion non-additively. This, of course, requires the approximation to the exogenous law of motion to be consistent with the approximation to the solution for the endogenous variables. The set of equations M( ) = 0 which describe the process of the exogenous variables is necessarily independent of the endogenous variables and the system of equilibrium equations in (ES). The approximate solution to m(z t, σɛ t+1 ) can therefore be easily computed in isolation using the standard perturbation approach by means of the implicit function theorem. For instance, the first order Taylor series expansion in z t and σ about some steady state z with z = m( z, 0) for σ = 0 yields m(z t, σɛ t+1 ) = z + m z ( z, 0)(z t z) + m ɛ ( z, 0)σɛ t+1, where m z ( z, 0) and m ɛ ( z, 0) denote the respective partial derivatives of m(z t, σɛ t+1 ) with respect to z t and σɛ t+1 evaluated at ( z, 0). The linear Taylor coefficients are computed by plugging the solution to the law of motion into (4) and setting the derivatives of M(m(z t, σɛ t+1 ), z t, σɛ t+1 ) with respect to z t and ɛ t+1 equal to zero. To be specific, m z and m ɛ are obtained by solving M zt+1 m z +M zt = 0 and M zt+1 m ɛ +M ɛt+1 = 0, respectively. Second and higher-order approximations are computed analogously. 6 6 Take as an example the linear form M(z t+1, z t, σɛ t+1 ) = z t+1 Az t Bσɛ t+1, where ɛ t+1 is i.i.d with mean zero. The steady state is z = 0 for ɛ = 0. The respective derivatives of M( ) are M zt+1 = I nz, M zt = A, and M ɛt+1 = Bσ, where I nz denotes the identity matrix of size n z n z. Consequently, m(z t, σɛ t+1 ) = Az t + Bσɛ t+1. 12

13 Regularity Conditions. It is assumed that f( ) is smooth in the sense that up to the order of approximation k + r all derivatives exist, where k denotes the order of the Taylor series expansion to the equilibrium system in risk and r denotes the order of the Taylor series approximation to the functional form of the solution g( ) and h( ). Moreover, since solving for the Taylor series expansion in g( ) and h( ) makes extensive use of the Implicit Function Theorem, it is assumed that the solvability conditions on the respective derivatives of f are met and that g( ) and h( ) are continuous and have the necessary k + r derivatives. The regularity conditions ensure that first, a k-th order Taylor series expansion of the equilibrium system (3) about the deterministic model is computable, and second, that the r-th order expansion of the solution to the problem h( ) and g( ) can be computed from the approximated equilibrium system using the implicit function theorem. Similar conditions apply to M( ) and m( ) for the exogenous variables. In addition to the local properties of the functional forms, the k moments of the distribution of ɛ exist and are finite. For a more detailed discussion of the regularity conditions and also about the necessity of bounds on the support of the shocks which also applies here, see Jin and Judd (2002) and Kim et al. (2008). 4 Solving the Model by Approximating the Stochastic Equilibrium System ES The aim is to find an approximation for g( ) and h( ) of the dynamic stochastic equilibrium model in (3) which is computable from the deterministic model F (x t, z t, 0; g 0 ( ), h 0 ( )) where σ = 0. The solution procedure consists of two steps: The first step is to restate the original system of stochastic equilibrium equations E t f( ) by a kth-order Taylor series expansion in the perturbation parameter about σ = 0. This corresponds to a Taylor series approximation in the stochastic disturbances about the deterministic model F (x t, z t, 0; g 0 ( ), h 0 ( )). The first step thus directly adopts the notion that the stochastic equilibrium system emanates from the perturbation 13

14 of the deterministic equilibrium system by small random shocks σɛ t+1. The result is an approximated system of equilibrium conditions AES which is non-stochastic and nonlinear in the endogenous variables, but which is linear in the first k moments of the exogenous disturbances. The second step of the solution procedure is then to compute the solution to the new equilibrium system AES as the approximation to the true solution process for g( ) and h( ). 4.1 Step 1: The Approximation of the Stochastic Equilibrium System ES (Computing the AES) The following proposition states the approximation of the stochastic equilibrium system by a Taylor series expansion about the deterministic model in terms of F ( ) as stated in (5). Proposition 1. At a given state (x t, z t ), the k-th order Taylor series expansion of the stochastic equilibrium system E t F (x t, z t, σɛ t+1 ; g( ), h( )) in the perturbation parameter about the deterministic model when σ = 0 yields E t F (x t, z t, σɛ t+1 ; g( ), h( )) = F (x t, z t, 0; g 0 ( ), h 0 ( )) + F ɛ i(x t, z t, 0; g 0 ( ), h 0 ( ))σe t ɛ i t ! F ɛ i ɛ j(x t, z t, 0; g 0 ( ), h 0 ( ))σ 2 E t ɛ i t+1ɛ j t ! F ɛ i ɛ j ɛ l(x t, z t, 0; g 0 ( ), h 0 ( ))σ 3 E t ɛ i t+1ɛ j t+1ɛ l t O(σ k+1 ), for i, j, l = 1,..., n ɛ. The Taylor series expansion is non-stochastic and nonlinear in the state (x t, z t ), and it is linear in the first k moments E t ɛ i t+1, E t ɛ i t+1ɛ j t+1,... for i, j,.. = 1,..., n ɛ of the exogenous shock distribution. F ɛ i( ), for instance, denotes the derivative of F ( ) with respect to the exogenous shock ɛ i, F ɛ i ɛ j( ) denotes the derivative of F ɛ i( ) with respect to ɛj, and so on, and it 14

15 is resorted to Tensor notation only to suppress summation signs. Proof. The proof of Proposition 1 is an application of the stochastic version of Taylor s theorem. See Appendix. The approximation of the stochastic equilibrium system by the k-th order Taylor series thus delivers a non-stochastic system of equations which preserves the nonlinearities of the equilibrium conditions in the state vector (x t, z t ), but which explicitly takes into account the stochastic environment of the rational expectations model. The respective Taylor coefficients are deterministic because the Taylor series is expanded about the deterministic model when σ = 0. By taking the expectations conditional on the observed state (x t, z t ), the k-th order Taylor series is linear in the first k moments of the shock distribution. Therefore, in the neighborhood of the deterministic model F (x t, z t, 0; g 0 ( ), h 0 ( )), the Taylor series approximation to the equilibrium system captures the effect of the stochastics on the true model up to the first k moments. Note that the kth-order Taylor series expansion in Proposition 1 is exact if either higher moments of the exogenous shock distribution are zero, or if derivatives to F ( ) of all higher-orders are zero. 7 Moreover, the Taylor series expansion converges to the exact equilibrium system in k if F (x t, z t, σɛ t+1 ; g( ), h( )) is analytic in the neighborhood of the given state (x t, z t ). Using the definition of F ( ), the approximation to the equilibrium system in 7 An example is the simple consumption-savings problem with CARA utility function (see Supplement B). 15

16 Proposition 1 can be stated explicitly as E t f(y t, x t, z t, y t+1, x t+1, z t+1 ) = f (y t, x t, z t, y t+1, x t+1, m(z t, 0)) + f ɛ i (y t, x t, z t, y t+1, x t+1, m(z t, 0)) σe t ɛ i t ! f ɛ i ɛ j (y t, x t, z t, y t+1, x t+1, m(z t, 0)) σ 2 E t ɛ i t+1ɛ j t ! f ɛ i ɛ j ɛ (y t, x l t, z t, y t+1, x t+1, m(z t, 0)) σ 3 E t ɛ i t+1ɛ j t+1ɛ l t O(σ k+1 ) f σ (y t, x t, z t, y t+1, x t+1, m(z t, 0)) + O(σ k+1 ), (6) where the law of motion for the exogenous state variables implies z t+1 = m(z t, 0) in the deterministic case when σ = 0. The associated actual law of motion for z t is appropriately approximated by the k-th order Taylor expansion in the perturbation parameter about the deterministic law, z t+1 = m(z t, 0) + m ɛ i(z t, 0)σɛ i t ! m ɛ i ɛ j(z t, 0)σ 2 ɛ i t+1ɛ j t ! m ɛ i ɛ j ɛ l(z t, 0)σ 3 ɛ i t+1ɛ j t+1ɛ l t O(σ k+1 ) m σ (z t, 0) + O(σ k+1 ). The solution to the approximated equilibrium system is denoted by x t+1 = h σ (x t, z t ) and y t = g σ (x t, z t ), with z t+1 = m σ (z t, 0). Substituting the solution process into the approximated system of nonlinear stochastic 16

17 equilibrium equations in (6) again allows to restate the system in terms of the current state and the respective solution to the endogenous variables as F σ (x t, z t ; g σ ( ), h σ ( )) f σ (g σ (x t, z t ), x t, z t, g σ (h σ (x t, z t ), m σ (z t, 0)), h σ (x t, z t ), m σ (z t, 0)). The k-th order Taylor expansion of the general model in the perturbation parameter σ about the deterministic model when σ = 0 can be stated as 0 = F σ (x t, z t ; g σ ( ), h σ ( )) z t+1 = m σ (z t, 0). (7) The approximated equilibrium system itself is now a system of n equations describing the joint equilibrium process for the endogenous variables y t, x t, y t+1, and x t+1, taking as given the process of the exogenous variables z t and z t+1 = m σ (z t, 0). The useful implication is that we can seek the solution to the new equilibrium system in (6), i.e. g σ ( ) and h σ ( ), as the proper approximation to the true solution g( ) and h( ) of the original problem in (3). Importantly, the solution to (6) is then also parameterized in the first k moments of the exogenous disturbances. Moreover, the approximation error relative to the true stochastic equilibrium system is in the order of O(σ k+1 ) (see Proposition 1). It is thus a quantitative statement about the accuracy of the approximated equilibrium system. It is also a statement about the accuracy of the solution process g σ ( ) and h σ ( ) as the approximation to the true solution because O(σ k+1 ) simply denotes the expected Euler residual of the exact solution to the new equilibrium system (6). To see this, note that E t F (x t, z t, σɛ t+1 ; g σ ( ), h σ ( )) = F σ (x t, z t ; g σ ( ), h σ ( )) + O(σ k+1 ) = O(σ k+1 ). (8) This will be exploited in Section 5 where accuracy properties are discussed. 17

18 Remark on the approximation of the stochastic equilibrium system. In the rational expectations model, approximating the stochastic equilibrium conditions requires to take into account the response of the endogenous non-predetermined variables y t+1 to exogenous disturbances ɛ t+1 and thereby changes in the exogenous state z t+1. Because the approximated equilibrium system AES is the Taylor series expansion about the deterministic model when σ = 0, the endogenous responses to changes in the exogenous shocks are computed from the solution to the deterministic model. These equilibrium responses of the endogenous non-predetermined variables to changes in the exogenous state, gz( ) 0 and gzz( ), 0 are computed from solving Fz 0 (x t, z t, 0; g 0 ( ), h 0 ( )) = 0 and Fz,z(x 0 t, z t, 0; g 0 ( ), h 0 ( )) = 0. This refers to standard approaches as explained in detail for instance in Schmitt-Grohé and Uribe (2004) and Jin and Judd (2002). A general derivation of the 2nd-order AES is provided in the Supplemental Material. For explicit examples of how to compute the 2nd-order AES, compare the applications of the AES-approach to the consumption-savings problem (Supplement B) and the asset pricing model (Supplement C). 4.2 Step 2: Computing the Linear Solution to the AES The second step of the solution procedure is to compute the functional form of the solution to the endogenous variables g σ ( ) and h σ ( ) from the non-stochastic and nonlinear vector difference equation in (6). To compute the linear solution, first it is solved for the steady state to the AES which is defined as follows: Definition 1 (Steady State). The steady state (ȳ σ, x σ, z) of the AES satisfies ȳ σ = g σ ( x σ, z) and x σ = h σ ( x σ, z), with z = m( z, 0), where the shock realization is ɛ = 0 and ȳ σ, x σ and z solve F σ ( x σ, z; g σ ( ), h σ ( )) = 0. 18

19 This definition of the steady state to the AES is the natural extension of the definition of the deterministic steady state to stochastic setups. It also corresponds to the definition of the risky or stochastic steady state as in Juillard (2012) and Coeurdacier et al. (2011). The linear solution to the AES is then computed by the first-order Taylor series approximation to g σ (x t, z t ) and h σ (x t, z t ) about the steady state (ȳ σ, x σ, z), i.e. g σ ( x σ + δ x, z + δ z ) = ȳ σ + g σ x( x σ, z)δ x + g σ z ( x σ, z)δ z, and h σ ( x σ + δ x, z + δ z ) = x σ + h σ x( x σ, z)δ x + h σ z ( x σ, z)δ z. The first-order Taylor coefficients gq σ ( ) and h σ q ( ) are computed from solving Fq σ ( x σ, z; g σ ( ), h σ ( )) = 0 for q = x, z, respectively, following the standard procedure. Similarly, higher-order approximations to g σ (x t, z t ) and h σ (x t, z t ) can be computed analogously. 5 Approximation Error of the AES, the Euler Residuals, and Accuracy Properties In this section, a discussion of the error of the Taylor series approximation of the equilibrium system AES and its implication for the accuracy of the approximated solution follows. 5.1 The Approximation Error of the AES and the Euler Residuals The accuracy properties of numerical solutions are commonly assessed by computing Euler equation residuals. Such accuracy tests are proposed in den Haan and Marcet 19

20 (1994) and Santos (2000). Applying this concept to our setup, the Euler residual associated with the solution process g σ (x t, z t ) and h σ (x t, z t ) is defined as the residual of the true stochastic equilibrium system evaluated at this solution process: Definition 2 (Euler Residuals). Let g σ (x t, z t ) and h σ (x t, z t ) denote the solution process for the endogenous variables which solve the equilibrium system F σ (x t, z t ; g σ ( ), h σ ( )) = 0. The corresponding Euler residual Ut+1 σ is defined as U σ t+1 = F (x t, z t, σɛ t+1 ; g σ ( ), h σ ( )). This definition of the Euler residual restates the usual Euler residual as in den Haan and Marcet (1994) in terms of the functional form of the solution to the approximated equilibrium. The Euler residuals are thus the prediction errors to the true expectational equilibrium conditions in F ( ) associated with the respective solution process. For the true solution process g(x t, z t ) and h(x t, z t ), in the rational expectations equilibrium the average Euler residual converges to zero. The following proposition presents the convergence statement about the average Euler residual corresponding to the exact solution to the approximated equilibrium system, g σ (x t, z t ) and h σ (x t, z t ). Proposition 2. Let g σ (x t, z t ) and h σ (x t, z t ) denote the solution to the kth-order Taylor series approximation to the stochastic equilibrium system, F σ (x t, z t ; g σ ( ), h σ ( )) = 0. Then 1 T T Ut+s σ s=1 a.s. O(σ k+1 ) as T, where O(σ k+1 ) denotes the error of the Taylor approximation to the true stochastic equilibrium system. Proof. See Appendix. 20

21 5.2 The Approximation Error of the AES and the Accuracy of the Exact Solution to the AES Recall from the discussion of Proposition 1 that the difference between the original stochastic equilibrium system E t F (x t, z t, σɛ t+1 ; g( ), h( )) and the kth-order Taylor approximation AES is simply the approximation error (see equation (8)). Proposition 2 states that the average Euler residual of the exact solution the AES, g σ (x t, z t ) and h σ (x t, z t ) with F σ (x t, z t ; g σ ( ), h σ ( )) = 0, converges to the approximation error of the AES to the original stochastic equilibrium system which is in the order of O(σ k+1 ). As a consequence, the accuracy of the approximation to the solution increases in the order k of the Taylor series approximation F σ (x t, z t ; g σ ( ), h σ ( )) to the stochastic equilibrium system, too. This is stated formally in the next Proposition. Proposition 3. Let g(x t, z t ) and h(x t, z t ) denote the true solution and let g σ (x t, z t ) and h σ (x t, z t ) denote the solution to the kth-order Taylor series approximation to the stochastic equilibrium system F σ (x t, z t ; g σ ( ), h σ ( )) = 0. Then g(x t, z t ) g σ (x t, z t ) = O(σ k+1 ) and h(x t, z t ) h σ (x t, z t ) = O(σ k+1 ), where O(σ k+1 ) denotes a term in the order of magnitude of the error of the Taylor approximation to the true stochastic equilibrium system. Proof. See Appendix. The approximation error of the solution g σ (x t, z t ) and h σ (x t, z t ) to the AES is therefore bounded by a term which is of the same order of magnitude as the Euler error and the approximation error of the AES to the true stochastic system, respectively (compare also Santos (2000)). As a result, solving nonlinear rational expectations models by first approximating the equilibrium system yields not only the moments of the exogenous shock distribution to become parameters of the model and its solution. It also increases the accuracy of the approximated solution process in the order k of 21

22 the Taylor approximation to the true stochastic equilibrium system relative to the conventional procedure when computing the solution directly from the deterministic model (k = 0). 5.3 The Approximation Error of the AES and the Accuracy of the Linearized Solution to the AES The result of Proposition 3 can be translated into accuracy statements about the steady state as well as first- and higher-order derivatives of the solution process g σ (x t, z t ) and h σ (x t, z t ) at its steady state. Specifically, define the corresponding steady state of the true stochastic system by (ȳ, x, z), where the equilibrium system is exposed to uncertainty, but where the endogenous and exogenous state variables remain unchanged and solve x = h( x, z) and M( z, z, 0) = 0 with z = m( z, 0), respectively. The steady state of the true stochastic system then solves E t F ( x, z, σɛ t+1 ; g( x, z), h( x, z)) = 0 with ȳ = g( x, z). Proposition Let (ȳ, x, z) denote the steady state of the true stochastic equilibrium system and (ȳ σ, x σ, z) the steady state of the AES. Then ȳ ȳ σ = O(σ k+1 ) and x x σ = O(σ k+1 ). 2. Let g s p,s q,...,s u( x, z) and h s p,s q,...,su( x, z) denote the r-th order derivatives of the true solution at the steady state (ȳ, x, z) and g σ s p,s q,...,s u( xσ, z) and g σ s p,s q,...,s u( xσ, z) the r-th order derivatives of the exact solution to the AES at the steady state (ȳ σ, x σ, z). Then g s p,s q,...,s u( x, z) gσ s p,s q,...,s u( xσ, z) = O(σ k+1 ) and h s p,s q,...,s u( x, z) hσ s p,s q,...,s u( xσ, z) = O(σ k+1 ), with s = [x, z ] and p, q, u = 1,..., n x + n z. 22

23 Proof. See Appendix. From Proposition 4 follows immediately that the implied error of the linearized solution to the approximated equilibrium system is also bounded by a term in the order of magnitude of O(σ k+1 ). Corollary 1. Let g(x t, z t ) and h(x t, z t ) denote the linearized solution to the true stochastic equilibrium system computed about (ȳ, x, z), and let g σ (x t, z t ) and h σ (x t, z t ) denote the linearized solution to the AES computed about (ȳ σ, x σ, z). Then g(x t, z t ) g σ (x t, z t ) = O(σ k+1 ) and h(xt, z t ) h σ (x t, z t ) = O(σ k+1 ). Proof. See Appendix. Corollary 1 states that the approximation error of the linearized solution is of the same order of magnitude as the size of the Euler error of the k-th order AES. It shows that the characteristic equilibrium interaction between the stochastic environment and nonlinearities has indeed first-order implications on the solution. Capturing this effect therefore increases the accuracy of the linearized solution. The solution to the AES linearized in the state vector thus fully captures these effects up the kth moments of the exogenous shock distribution. Note that the conventional wisdom about linear solutions is that they cannot account for uncertainty. This is trivially true if the linear solution is computed for the deterministic model where the stochastic environment is neglected. Once it is computed for an approximated equilibrium system that explicitly takes into account the stochastic environment, the linear solution to this approximated system is no longer independent of uncertainty. Discussion of the Implications. Capturing the first-order implications of risk in the linear solution might have important consequences for the analysis of the model economy, because the linear solution already determines the key properties of the model such as the dynamic behavior and the equilibrium distribution. Most notably, 23

24 because the linear solution to the kth-order AES captures equilibrium implications of nonlinearities in the stochastic environment up to the first k moments, the AESapproach also captures the effect of risk on the equilibrium properties with respect to existence and uniqueness up to the first k moments. This is in stark contrast to the conventional perturbation approach. The conventional perturbation cannot take into account the equilibrium implications of risk because existence and uniqueness of the equilibrium is generically assessed for the deterministic model (i.e. the 0thorder AES). This is true not only for the conventional linear solution but for any order of the conventional approximation approach. It is also worth pointing out the relevance of the above results for using the approximated solution to the model for simulation, estimation, and forecasting. Consider for instance likelihood based methods. Fernandez-Villaverde et al. (2006) demonstrate that for a given sample size T the likelihood associated with the approximated solution to the stochastic equilibrium model converges to the true likelihood as the approximation error to the solution becomes smaller (Proposition 1, p. 101). Moreover, in a comment on Fernandez-Villaverde et al. (2006), Ackerberg et al. (2009) show that as T increases, the difference between the maximum likelihood parameter estimates of the approximated solution and the true parameters converges to a term which is in the same order of magnitude as the approximation error, i.e. O(σ k+1 ) (Theorem 2, p. 2015). These results imply that computing the solution from the kth-order AES also increases the accuracy of the parameter inference based on the approximated solution process. Similar accuracy statements can be made about forecasts based on the linear solution to the AES. The one-step-ahead forecast based on the linear solution to the kth-order AES can be stated as y t+1 x t+2 = gσ (h σ (x t, z t ), m σ (z t )) h σ (h σ (x t, z t ), m σ (z t )) + O p ( ẑ t+1 ) + O( ˆx t+1 + σ k+1 ), 24

25 where g σ (x t, z t ) and h σ (x t, z t ) denote the linearized solution to the AES computed about (ȳ σ, x σ, z) and where ˆx t+1 = x σ x t+1 and ẑ t+1 = z z t+1 are the deviations from steady state. The term O p ( ẑ t+1 ) denotes the order of magnitude in probability. As a consequence, the accuracy of the one-step-ahead forecast based on the DSGE model is also increasing in the order k. Accuracy statements about n-step-ahead forecasts are obtained accordingly. Supplement C provides an application of the AES-approach to the asset pricing model taken from Burnside (1998). The model permits an exact assessment of the accuracy of the AES-approach. The exercise substantiates the theoretical accuracy results quantitatively: The approximation error of the linear solution to the 2ndorder AES is 1-2 orders of magnitude smaller than the approximation error of the conventional linear solution for the steady state and coefficients linear in the state variable, the equilibrium distribution, and the maximum likelihood estimates of the deep parameters. The exercise also demonstrates how the AES-approach captures the implications of risk on the existence and uniqueness of the solution which is completely absent in the conventional perturbation approach of any order. When comparing the linear solution to the 2nd-order AES to the conventional 2nd-order approximation, the numerical example indicates that it is more important to capture the equilibrium effects of nonlinearities associated with risk on the slope (linear coefficient) than to capture the nonlinearities of the solution through higher-order terms in the state-vector in the conventional 2nd-order solution. 6 Concluding Remarks This paper demonstrates how to compute a linear solution to a DSGE model that accounts for the interaction between nonlinearities and the stochastic environment. The fundamental concept of the AES-approach is to disentangle the account of the 25

26 stochastic environment of the original set of equilibrium conditions in the first step from computing the functional form of the solution processes in the second step. Once this separation is made, it is obvious that computing the functional form of the solution processes to approximated equilibrium system is not restricted to perturbation methods at all but any local or global method is applicable, too. Here, the attention is restricted to the linear solution to the AES because of the simplicity and widespread use of linear methods in solving and estimating DSGE models. 26

27 A Proofs of the Propositions Proof of Proposition 1. By the stochastic Taylor s Theorem, at a given state (x t, z t ), there exists a measurable function ξ(ɛ t+1 ) [0, σɛ t+1 ] such that for the k-th order Taylor series expansion of the stochastic equilibrium system F (x t, z t, σɛ t+1 ; g( ), h( )) in the perturbation parameter about the deterministic model when σ = 0, F (x t, z t, σɛ t+1 ; g( ), h( )) = F (x t, z t, 0; g 0 ( ), h 0 ( )) + F ɛ i(x t, z t, 0; g 0 ( ), h 0 ( ))σɛ i t ! F ɛ i,ɛ j(x t, z t, 0; g 0 ( ), h 0 ( ))σ 2 ɛ i t+1ɛ j t ! F ɛ i,ɛ j,ɛ l(x t, z t, 0; g 0 ( ), h 0 ( ))σ 3 ɛ i t+1ɛ j t+1ɛ l t R(x t, z t, ξ(ɛ t+1 ))σ k+1, for i, j, l,... = 1,..., n ɛ. Note that by assumption (Regularity condition), the k + 1-th derivatives of F ( ) exist. R(x t, z t, ξ(ɛ t+1 ))σ k+1 denotes the measurable remainder of the Taylor series which must be finite almost surely for the equilibrium system to hold. The remainder is thus in probability in the order of magnitude of O p (σ k+1 ). For instance, in case of k = 2, R(x t, z t, ξ(ɛ t+1 )) = 1 F 3! ɛ i,ɛ j,ɛ l(x t, z t, ξ(ɛ t+1 ))ɛ i t+1ɛ j t+1ɛ l t+1. Taking expectations then implies with E t R(x t, z t, ξ(ɛ t+1 )) < and thus E t R(x t, z t, ξ(ɛ t+1 ))σ k+1 = O(σ k+1 ) that E t F (x t, z t, σɛ t+1 ; g( ), h( )) = F (x t, z t, 0; g 0 ( ), h 0 ( )) + F ɛ i(x t, z t, 0; g 0 ( ), h 0 ( ))σe t ɛ i t ! F ɛ i,ɛ j(x t, z t, 0; g 0 ( ), h 0 ( ))σ 2 E t ɛ i t+1ɛ j t ! F ɛ i,ɛ j,ɛ l(x t, z t, 0; g 0 ( ), h 0 ( ))σ 3 E t ɛ i t+1ɛ j t+1ɛ l t O(σ k+1 ). 27

28 Note that by the definition of the approximated equilibrium system in (AES-Sol), the above equation can be restated as E t F (x t, z t, σɛ t+1 ; g( ), h( )) = F σ (x t, z t ; g( ), h( )) + O(σ k+1 ). Consequently, the exact solution to the AES, g σ ( ) and h σ ( ) solves F σ (x t, z t ; g σ ( ), h σ ( )) = 0 which implies E t F (x t, z t, σɛ t+1 ; g σ ( ), h σ ( )) = F σ (x t, z t ; g σ ( ), h σ ( )) + O(σ k+1 ) = O(σ k+1 ). (9) Proof of Proposition 2. Using the definition of the approximated equilibrium system in (7) and in Proposition 1, F σ (x t, z t ; g σ ( ), h σ ( )), the Euler residuals can be stated as the sum of the Euler residuals to the approximated equilibrium system plus the approximation error of the approximated equilibrium system to the true equilibrium system, i.e. U σ t+s+1 = F (x t+s, z t+s, σɛ t+s+1 ; g σ ( ), h σ ( )) = F σ (x t+s, z t+s, σɛ t+s+1 ; g σ ( ), h σ ( )) + R(x t+s, z t+s, ξ(ɛ t+s+1 ))σ k+1, where F σ (x t+s, z t+s, σɛ t+s+1 ; g σ ( ), h σ ( )) is finite for all s and independently distributed with mean zero. The approximation error R(x t+s, z t+s, ξ(ɛ t+s+1 ))σ k+1 is finite for all s, independently distributed, and its mean is in the order of magnitude of O(σ k+1 ). By the law of large numbers 1 T T F σ (x t+s, z t+s, σɛ t+s+1 ; g σ ( ), h σ ( )) s=1 a.s. 0 as T, because g σ ( ) and h σ ( ) is the exact solution to AES, and 1 T T R(x t+s, z t+s, ξ(ɛ t+s+1 ))σ k+1 s=1 a.s. O(σ k+1 ) as T, 28

29 which together imply that 1 T T Ut+s σ s=1 a.s. O(σ k+1 ) as T. Proof of Proposition 3. By the mean value theorem, there exists some finite n (n y + n x + n z ) matrix W (x t, z t, σɛ t+1 ) at (x t, z t ) for each σɛ t+1 such that f(g(x t, z t ), x t, z t, g(h(x t, z t ), m(z t, σɛ t+1 )), h(x t, z t ), m(z t, σɛ t+1 )) =f(g σ (x t, z t ), x t, z t, g σ (h σ (x t, z t ), m σ (z t, 0)), h σ (x t, z t ), m σ (z t, 0)) g(x t, z t ) g σ (x t, z t ) + W (x t, z t, σɛ t+1 ) h(x t, z t ) h σ (x t, z t ), m(z t, σɛ t+1 ) m σ (z t, 0) (10) where W (x t, z t, σɛ t+1 ) = [W g ( ) W h ( ) W m ( )] with W i g(x t, z t, σɛ t+1 ) = f i g(x t,z t)( g(x t, z t ), x t, z t, g( h(x t, z t ), m(z t, σɛ t+1 )), h(x t, z t ), m(z t, σɛ t+1 )) W i h(x t, z t, σɛ t+1 ) = f i h(x t,z t)( g(x t, z t ), x t, z t, g( h(x t, z t ), m(z t, σɛ t+1 )), h(x t, z t ), m(z t, σɛ t+1 )) W i m(x t, z t, σɛ t+1 ) = f i m(z t,σɛ t+1 )( g(x t, z t ), x t, z t, g( h(x t, z t ), m(z t, σɛ t+1 )), h(x t, z t ), m(z t, σɛ t+1 )), for g(x t, z t ) = (1 τ i )g(x t, z t )+τ i g σ (x t, z t ), h(x t, z t ) = (1 τ i )h(x t, z t )+τ i h σk (x t, z t ), and m(x t, z t ) = (1 τ i )m(z t, σɛ t+1 ) + τ i m σ (z t, 0) with τ i [0, 1] for all equations i = 1,..., n. 29