Review of Economic Dynamics Computing DSGE models with

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Author s personal copy, D Caldara et al Review of Economic Dynamics 15 2012 188 206 189. Bansal and Yaron 2004 have argued that the combination of recursive preferences and SV is the key for their proposed. mechanism long run risk to be successful at explaining asset pricing. But despite the popularity and importance of these issues nearly nothing is known about the numerical properties of. the different solution methods that solve equilibrium models with recursive preferences and SV For example we do not. know how well value function iteration VFI performs or how good local approximations are compared with global ones. Similarly if we want to estimate the model we need to assess which solution method is su ciently reliable yet quick. enough to make the exercise feasible More important the most common solution algorithm in the DSGE literature log. linearization cannot be applied since it makes us miss the whole point of recursive preferences or SV the resulting log. linear decision rules are certainty equivalent and do not depend on risk aversion or volatility This paper attempts to ll. this gap in the literature and therefore it complements previous work by Aruoba et al 2006 in which a similar exercise. is performed with the neoclassical growth model with CRRA utility function and constant volatility. We solve and simulate the model using four main approaches perturbation of second and third order Chebyshev. polynomials and VFI By doing so we span most of the relevant methods in the literature Our results provide a strong. guess of how some other methods not covered here such as nite elements would work rather similar to Chebyshev. polynomials but more computationally intensive We report results for a benchmark calibration of the model and for. alternative calibrations that change the variance of the productivity shock the risk aversion and the intertemporal elasticity. of substitution In that way we study the performance of the methods both for cases close to the CRRA utility function. with constant volatility and for highly non linear cases far away from the CRRA benchmark For each method we compute. decision rules the value function the ergodic distribution of the economy business cycle statistics the welfare costs of. aggregate uctuations and asset prices Also we evaluate the accuracy of the solution by reporting Euler equation errors. We highlight four main results from our exercise First all methods provide a high degree of accuracy Thus researchers. who stay within our set of solution algorithms can be con dent that their quantitative answers are sound. Second perturbations deliver a surprisingly high level of accuracy with considerable speed Both second and third order. perturbations perform remarkably well in terms of accuracy for the benchmark calibration being competitive with VFI or. Chebyshev polynomials For this calibration a second order perturbation that runs in a fraction of a second does nearly. as well in terms of the average Euler equation error as a VFI that takes ten hours to run Even in the extreme calibration. with high risk aversion and high volatility of productivity shocks perturbation works at a more than acceptable level Since. in practice perturbation methods are the only computationally feasible method to solve the medium scale DSGE models. used for policy analysis that have dozens of state variables as in Smets and Wouters 2007 this nding has an outmost. applicability Moreover since implementing second and third order perturbations is feasible with off the shelf software. like Dynare which requires minimum programming knowledge by the user our ndings may induce many researchers to. explore recursive preferences and or SV in further detail Two nal advantages of perturbation are that often the perturbed. solution provides insights about the economics of the problem and that it might be an excellent initial guess for VFI or for. Chebyshev polynomials, Third Chebyshev polynomials provide a terri c level of accuracy with reasonable computational burden When accuracy. is most required and the dimensionality of the state space is not too high as in our model they are the obvious choice. Fourth we were disappointed by the poor performance of VFI which compared with Chebyshev could not achieve a. high accuracy even with a large grid This suggests that we should relegate VFI to solving those problems where non. differentiabilities complicate the application of the previous methods. The rest of the paper is organized as follows In Section 2 we present our test model Section 3 describes the different. solution methods used to approximate the decision rules of the model Section 4 discusses the calibration of the model. Section 5 reports numerical results and Section 6 concludes Appendix A provides some additional details. 2 The stochastic neoclassical growth model with recursive preferences and SV. We use the stochastic neoclassical growth model with recursive preferences and SV in the process for technology as. our test case We select this model for three reasons First it is the workhorse of modern macroeconomics Even more. complicated New Keynesian models with real and nominal rigidities such as those in Woodford 2003 or Christiano et al. 2005 are built around the core of the neoclassical growth model Thus any lesson learned with it is likely to have a wide. applicability Second the model is except for the form of the utility function and the process for SV the same test case as. in Aruoba et al 2006 This provides us with a set of results to compare to our ndings Three the introduction of recursive. preferences and SV make the model both more non linear and hence a challenge for different solution algorithms and. potentially more relevant for practical use For example and as mentioned in the Introduction Bansal and Yaron 2004. have emphasized the importance of the combination of recursive preferences and time varying volatility to account for. asset prices, The description of the model is straightforward and we just go through the details required to x notation There is a. representative household that has preferences over streams of consumption ct and leisure 1 lt represented by a recursive. function of the form,U t max 1 ct 1 lt 1,Et U t 1 1. Author s personal copy, 190 D Caldara et al Review of Economic Dynamics 15 2012 188 206.
The parameters in these preferences include the discount factor which controls labor supply which controls risk. aversion and, where is the EIS The parameter is an index of the deviation with respect to the benchmark CRRA utility function. when 1 we are back in that CRRA case where the inverse of the EIS and risk aversion coincide. The household s budget constraint is given by,ct i t w t lt rt kt bt. where i t is investment R t is the risk free gross interest rate bt is the holding of an uncontingent bond that pays 1 unit of. consumption good at time t 1 w t is the wage lt is labor rt is the rental rate of capital and kt is capital Asset markets. are complete and we could have also included in the budget constraint the whole set of Arrow securities Since we have. a representative household this is not necessary because the net supply of any security is zero Households accumulate. capital according to the law of motion kt 1 1 kt i t where is the depreciation rate. The nal good in the economy is produced by a competitive rm with a Cobb Douglas technology yt e zt kt lt where. zt is the productivity level that follows,zt zt 1 e t t t N 0 1. Stationarity is the natural choice for our exercise If we had a deterministic trend we would only need to adjust in. our calibration below and the results would be nearly identical If we had a stochastic trend we would need to rescale. the variables by the productivity level and solve the transformed problem However in this case it is well known that. the economy uctuates less than when 1 and therefore all solution methods would be closer limiting our ability to. appreciate differences in their performance, The innovation t is scaled by an SV level t which evolves as. t 1 t 1 t t N 0 1, where is the unconditional mean level of t is the persistence of the processes and is the standard deviation of the.
innovations to t Our speci cation is parsimonious and it introduces only two new parameters and At the same time. it captures some important features of the data see a detailed discussion in Fern ndez Villaverde and Rubio Ram rez 2010. The combination of an exponent in the productivity process e t and a level in the evolution of t generates interesting. non linear dynamics Another important point is that with SV we have two innovations an innovation to technology t. and an innovation to the standard deviation of technology t Finally the economy must satisfy the aggregate resource. constraint yt ct i t, The de nition of equilibrium is standard and we skip it in the interest of space Also both welfare theorems hold a fact. that we will exploit by jumping back and forth between the solution of the social planner s problem and the competitive. equilibrium However this is only to simplify our derivations It is straightforward to adapt the solution methods described. below to solve problems that are not Pareto optimal. Thus an alternative way to write this economy is to look at the value function representation of the social planner s. problem in terms of its three state variables capital kt productivity zt and volatility t. V kt zt t max 1 ct 1 lt 1,Et V 1 kt 1 zt 1 t 1,s t ct kt 1 e zt kt lt 1 kt. zt zt 1 e t t t N 0 1,t 1 t 1 t t N 0 1,Then we can nd the pricing kernel of the economy. Now note that,Author s personal copy, D Caldara et al Review of Economic Dynamics 15 2012 188 206 191. 1 1 ct 1 1 lt 1 1,Vt Et V t 1 Et V t 1 1 V t 1,c t 1 c t 1.
where in the last step we use the result regarding V t ct forwarded by one period Canceling redundant terms we get. 1 1 1 1 1 1 1,V t ct 1 c t 1 1 l t 1 V t 1,V t ct ct 1 lt Et Vt 1. This equation shows how the pricing kernel is affected by the presence of recursive preferences If 1 the last term. is equal to 1 and we get back the pricing kernel of the standard CRRA case If 1 the pricing kernel is twisted by 3. We identify the net return on equity with the marginal net return on investment. R kt 1 e zt 1 kt 1 lt 1,with expected return Et R kt 1. 3 Solution methods, We are interested in comparing different solution methods to approximate the dynamics of the previous model Since. the literature on computational methods is large it would be cumbersome to review every proposed method Instead we. select those methods that we nd most promising, Our rst method is perturbation introduced by Judd and Guu 1992 1997 and nicely explained in Schmitt Groh and. Uribe 2004 Perturbation algorithms build a Taylor series expansion of the agents decision rules Often perturbation meth. ods are very fast and despite their local nature highly accurate in a large range of values of the state variables Aruoba. et al 2006 This means that in practice perturbations are the only method that can handle models with dozens of state. variables within any reasonable amount of time Moreover perturbation often provides insights into the structure of the. solution and on the economics of the model Finally linearization and log linearization the most common solution methods. for DSGE models are particular cases of a perturbation of rst order. We implement a second and a third order perturbation of our model A rst order perturbation is useless for our. investigation the resulting decision rules are certainty equivalent and therefore they depend on but not on or t In. other words the rst order decision rules of the model with recursive preferences coincide with the decision rules of the. model with CRRA preferences with the same and for any value of or t We need to go at least to second order. decision rules to have terms that depend on or t and hence allow recursive preferences or SV to play a role Since. the accuracy of second order decision rules may not be high enough and in addition we want to explore time varying. risk premia we also compute a third order perturbation As we will document below a third order perturbation provides. enough accuracy without unnecessary complications Thus we do not need to go to higher orders. The second method is a projection algorithm with Chebyshev polynomials Judd 1992 Projection algorithms build. approximated decision rules that minimize a residual function that measures the distance between the left and right hand. side of the equilibrium conditions of the model Projection methods are attractive because they offer a global solution. over the whole range of the state space Their main drawback is that they suffer from an acute curse of dimensionality. that makes it challenging to extend it to models with many state variables Among the many different types of projection. methods Aruoba et al 2006 show that Chebyshev polynomials are particularly e cient Other projection methods such as. nite elements or parameterized expectations tend to perform somewhat worse than Chebyshev polynomials and therefore. in the interest of space we do not consider them, Finally we compute the model using VFI Epstein and Zin 1989 show that a version of the contraction mapping theorem.
holds in the value function of the problem with recursive preferences VFI is slow and it suffers as well from the curse of. dimensionality but it is safe reliable and well understood Thus it is a natural default al. DSGE models Recursive preferences Perturbation This paper compares different solution methods for computing the equilibrium of dynamic stochastic general equilibrium DSGE models with recursive preferences such as those in Epstein and Zin 1989 1991 and stochastic volatility Models with these two features

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