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This paper derives a second-order approximation to the solution of a general class of discrete-time rational expectations models. The main theoretical contribution is to show that for any model belonging to that class, the coefficients on the terms linear and quadratic in the state vector in a second-order expansion of the decision rule are independent of the volatility of the exogenous shocks. In addition, the paper presents a set of MATLAB programs that implement the proposed second-order approximation method and applies it to a number of model economies.

Since the seminal papers of Kydland and Prescott (1982) and King et al. (1988), it has become commonplace in macroeconomics to approximate the solution to non-linear, dynamic, stochastic, general equilibrium models using linear methods. Linear approximation methods are useful to characterize certain aspects of the dynamic properties of complicated models. In particular, if the support of the shocks driving aggregate fluctuations is small and an interior stationary solution exists, first-order approximations provide adequate answers to questions such as local existence and determinacy of equilibrium and the size of the second moments of endogenous variables. However, first-order approximation techniques are not well suited to handle questions such as welfare comparisons across alternative stochastic or policy environments. For example, Kim and Kim (in press) show that in a simple two-agent economy, a welfare comparison based on an evaluation of the utility function using a linear approximation to the policy function may yield the spurious result that welfare is higher under autarky than under full risk sharing. The problem here is that some second- and higher-order terms of the equilibrium welfare function are omitted while others are included. Consequently, the resulting criterion is inaccurate to order two or higher. The same problem arises under the common practice in macroeconomics of evaluating a second-order approximation to the objective function using a first-order approximation to the decision rules. For in this case, too, some second-order terms of the equilibrium welfare function are ignored while others are not.1 In general, a correct second-order approximation of the equilibrium welfare function requires a second-order approximation to the policy function. In this paper, we derive a second-order approximation to the policy function of a general class of dynamic, discrete-time, rational expectations models. A strength of our approach is not to follow a value function formulation. This allows us to tackle easily a wide variety of model economies that do not lend themselves naturally to the value function specification. To obtain an accurate second-order approximation, we use a perturbation method that incorporates a scale parameter for the standard deviations of the exogenous shocks as an argument of the policy function. In approximating the policy function, we take a second-order Taylor expansion with respect to the state variables as well as this scale parameter. This technique was formally introduced by Fleming (1971) and has been applied extensively to economic models by Judd and co-authors (see Judd, 1998, and the references cited therein). The main theoretical contributions of the paper are: First, it shows analytically that in general the first derivative of the policy function with respect to the parameter scaling the variance/covariance matrix of the shocks is zero at the steady state regardless of whether the model displays the certainty-equivalence property or not.2 Second, it proves that in general the cross derivative of the policy function with respect to the state vector and with respect to the parameter scaling the variance/covariance matrix of the shocks evaluated at the steady state is zero. This result implies that for any model belonging to the general class considered in this paper, the coefficients on the terms linear and quadratic in the state vector in a second-order expansion of the decision rule are independent of the volatility of the exogenous shocks. In other words, these coefficients must be the same in the stochastic and the deterministic versions of the model. Thus, up to second order, the presence of uncertainty affects only the constant term of the decision rules. The usefulness of our theoretical results can be illustrated by relating them to recent work on second-order approximation techniques by Collard and Juillard 2001a and Collard and Juillard 2001b and Sims (2000). We follow Collard and Juillard closely in notation and methodology. However, an important difference separates our paper from their work. Namely, Collard and Juillard apply a fixed-point algorithm, which they call ‘bias reduction procedure,’ to capture the fact that the policy function depends on the variance of the underlying shocks. Their procedure makes the coefficients of the approximated policy rule that are linear and quadratic in the state vector functions of the size of the volatility of the exogenous shocks. By the main theoretical result of this paper, those coefficients are, up to second order, independent of the standard deviation of the shocks. It follows that the bias reduction procedure of Collard and Juillard is not equivalent to a second-order Taylor approximation to the decision rules.3 Sims (2000) also derives a second-order approximation to the policy function for a wide class of discrete-time models. In his derivation, Sims (2000) correctly assumes that the coefficients on the terms linear and quadratic in the state vector do not depend on the volatility of the shock and obtains a second-order approximation to the policy function that is valid only under this assumption. However, he does not provide the proof that this must be the case. Our paper provides this proof in a general setting. At a practical level, our paper contributes to the existing literature by providing MATLAB code to compute second-order approximations for any rational expectations model whose equilibrium conditions can be written in the general form considered in this paper. We demonstrate the ability of this code to deliver accurate second-order approximations by applying it to a number of example economies. The first example considered is the standard, one-sector, stochastic growth model. Sims (2000) computes a second-order approximation to this economy, which we are able to replicate. The second example applies our code to the two-country growth model with complete asset markets studied by Kim and Kim (in press). This economy features multiple state variables. Kim and Kim have derived analytically the second-order approximation to the policy function of this economy. We use this example to verify that our code delivers correct answers in a multi-state environment. Finally, we apply our code to the asset-pricing model of Burnside (1998). This example is also analyzed in Collard and Juillard (2001b). Burnside solves this model analytically. Thus, we can derive analytically the second-order approximation to the policy function. This example serves two purposes. First, it gives support to the validity of our code. Second, it allows us to quantify the differences between the Taylor second-order approximation and the bias reduction procedure of Collard and Juillard 2001a and Collard and Juillard 2001b. The remainder of the paper is organized as follows. In the next section we present the model. In Section 3 we derive first- and second-order approximations to the policy function. In Section 4 we describe the Matlab computer code designed to implement the second-order approximation to the policy rules. Section 5 closes the paper with applications of the algorithm developed in this paper to three example economies.

Most models used in modern macroeconomics are too complex to allow for exact solutions. For this reason, researchers have appealed to numerical approximation techniques. One popular and widely used approximation technique is a first-order perturbation method delivering a linear approximation to the policy function. One reason for the popularity of first-order perturbation techniques is that they do not suffer from the ‘curse of dimensionality.’ That is, problems with a large number of state variables can be handled without much computational demands. However, an important limitation of this approximation technique is that the solution displays the certainty equivalence property. In particular, the first-order approximation to the unconditional means of endogenous variables coincides with their non-stochastic steady state values. This limitation restricts the range of questions that can be addressed in a meaningful way using first-order perturbation techniques. Two such questions are welfare evaluations and risk premia in stochastic environments. Within the family of perturbation methods an obvious way to overcome these limitations is to perform a higher-order approximation to the policy function. This is precisely what this paper accomplishes. We build on previous work by Collard and Juillard, Sims, and Judd among others. In particular, this paper derives a second-order approximation to the solution of a general class of discrete-time rational expectations models. The main theoretical contribution of the paper is to show that for any model belonging to the general class considered, the coefficients on the terms linear and quadratic in the state vector in a second-order expansion of the decision rule are independent of the volatility of the exogenous shocks. In other words, these coefficients must be the same in the stochastic and the deterministic versions of the model. Thus, up to second order, the presence of uncertainty affects only the constant term of the decision rules. But the fact that only the constant term is affected by the presence of uncertainty is by no means inconsequential. For it implies that up to second order the unconditional mean of endogenous variables can in general be significantly different from their non-stochastic steady state values. Thus, second-order approximation methods can in principle capture important effects of uncertainty on average rate of return differentials across assets with different risk characteristics and on the average level of consumer welfare. An additional advantage of higher-order perturbation methods is that like their first-order counterparts, they do not suffer from the curse of dimensionality. This is because given the first-order approximation to the policy function, finding the coefficients of a second-order approximation simply entails solving a system of linear equations. In addition to fully characterizing the second-order approximation to the policy function analytically, the paper presents a set of MATLAB programs designed to compute the coefficients of the second-order approximation. This code is publicly available. The validity and applicability of the proposed method is illustrated by solving the dynamics of a number of model economies. Our computer code coexists with others that have been developed recently by Chris Sims and Fabrice Collard and Michel Juillard to accomplish the same task. We believe that the availability of this set of independently developed codes, which have been shown to deliver identical results for a number of example economies, helps build confidence across potential users. A number of important aspects of higher-order approximations to the policy function remain to be explored. At the forefront stands the problem of characterizing second-order accurate approximations to artificial time series as well as to conditional and unconditional moments of endogenous variables. We leave this task for future research.