Lecture on solving1
- 1. Solution Methods for DSGE Models and Applications using Linearization Lawrence J. Christiano
- 2. Overall Outline • Perturbation and Projection Methods for DSGE Models: an Overview • Simple New Keynesian model – Formulation and log‐linear solution. – Ramsey‐optimal policy. – Using Dynare to solve the model by log‐linearization: • Taylor principle, implications of working capital, News shocks, monetary policy with the long rate. • Financial Frictions as in BGG – Risk shocks and the CKM critique of intertemporal shocks. – Dynare exercise. • Ramsey Optimal Policy, Time Consistency, Timeless Perspective.
- 3. Perturbation and Projection Methods for Solving DSGE Models Lawrence J. Christiano
- 4. Outline • A Simple Example to Illustrate the basic ideas. – Functional form characterization of model solution. – Use of Projections and Perturbations. • Neoclassical model. – Projection methods – Perturbation methods • Make sense of the proposition, ‘to a first order approximation, can replace equilibrium conditions with linear expansion about nonstochastic steady state and solve the resulting system using certainty equivalence’
- 5. Simple Example • Suppose that x is some exogenous variable and that the following equation implicitly defines y: hx, y 0, for all x ∈ X • Let the solution be defined by the ‘policy rule’, g: y gx ‘Error function’ • satisfying Rx; g ≡ hx, gx 0 • for all x ∈ X
- 6. The Need to Approximate • Finding the policy rule, g, is a big problem outside special cases – ‘Infinite number of unknowns (i.e., one value of g for each possible x) in an infinite number of equations (i.e., one equation for each possible x).’ • Two approaches: – projection and perturbation
- 7. Projection ĝx; • Find a parametric function, , where is a vector of parameters chosen so that it imitates Rx; g 0 the property of the exact solution, i.e., for all x ∈ X , as well as possible. • Choose values for so that ̂ Rx; hx, ĝx; x∈X • is close to zero for . • The method is defined by how ‘close to zero’ is ĝx; defined and by the parametric function, , that is used.
- 8. Projection, continued • Spectral and finite element approximations ĝx; – Spectral functions: functions, , in which ĝx; x∈X each parameter in influences for all example: n 1 ĝx; ∑ i Hi x, i0 n H i x x i ~ordinary polynominal (not computationaly efficient) H i x T i x, T i z : −1, 1 → −1, 1, i th order Chebyshev polynomial : X → −1, 1
- 9. Projection, continued ĝx; – Finite element approximations: functions, , ĝx; in which each parameter in influences over only a subinterval of x ∈ X ĝx; 1 2 3 4 5 6 7 4 2 X
- 10. Projection, continued • ‘Close to zero’: collocation and Galerkin x : x1 , x2 , . . . , xn ∈ X • Collocation, for n values of 1 n choose n elements of so that ̂ Rx i ; hx i , ĝx i ; 0, i 1, . . . , n – how you choose the grid of x’s matters… • Galerkin, for m>n values of x : x1 , x2 , . . . , xm ∈ X choose the n elements of 1 n m ∑ wij hxj , ĝxj ; 0, i 1, . . . , n j1
- 11. Perturbation • Projection uses the ‘global’ behavior of the functional equation to approximate solution. – Problem: requires finding zeros of non‐linear equations. Iterative methods for doing this are a pain. – Advantage: can easily adapt to situations the policy rule is not continuous or simply non‐differentiable (e.g., occasionally binding zero lower bound on interest rate). • Perturbation method uses local properties of functional equation and Implicit Function/Taylor’s theorem to approximate solution. – Advantage: can implement it using non‐iterative methods. – Possible disadvantages: • may require enormously high derivatives to achieve a decent global approximation. • Does not work when there are important non‐differentiabilities (e.g., occasionally binding zero lower bound on interest rate).
- 12. Perturbation, cnt’d x∗ ∈ X • Suppose there is a point, , where we know the value taken on by the function, g, that we wish to approximate: gx ∗ g ∗ , some x ∗ • Use the implicit function theorem to approximate g in a neighborhood of x ∗ • Note: Rx; g 0 for all x ∈ X → j R x; g ≡ d j Rx; g 0 for all j, all x ∈ X. dx j
- 13. Perturbation, cnt’d • Differentiate R with respect to and evaluate x x∗ the result at : R 1 x ∗ d hx, gx|xx ∗ h 1 x ∗ , g ∗ h 2 x ∗ , g ∗ g ′ x ∗ 0 dx ′ ∗ h 1 x ∗ , g ∗ → g x − h 2 x ∗ , g ∗ • Do it again! 2 R x ∗ d 2 hx, gx| ∗ h x ∗ , g ∗ 2h x ∗ , g ∗ g ′ x ∗ xx 11 12 dx 2 h 22 x ∗ , g ∗ g ′ x ∗ 2 h 2 x ∗ , g ∗ g ′′ x ∗ . → Solve this linearly for g ′′ x ∗ .
- 14. Perturbation, cnt’d • Preceding calculations deliver (assuming enough differentiability, appropriate invertibility, a high tolerance for painful notation!), recursively: g ′ x ∗ , g ′′ x ∗ , . . . , g n x ∗ • Then, have the following Taylor’s series approximation: gx ≈ ĝx ĝx g ∗ g ′ x ∗ x − x ∗ 1 g ′′ x ∗ x − x ∗ 2 . . . 1 g n x ∗ x − x ∗ n 2 n!
- 15. Perturbation, cnt’d • Check…. • Study the graph of Rx; ĝ x∈X – over to verify that it is everywhere close to zero (or, at least in the region of interest).
- 16. Example of Implicit Function Theorem y hx, y 1 x 2 y 2 − 8 0 2 4 gx ≃ g∗ x ∗ x − x ∗ − ∗ g ‐4 4 x ‐4 h 1 x ∗ , g ∗ x ∗ h 2 had better not be zero! g′ ∗ x − − ∗ ∗ ∗ h 2 x , g g
- 17. Neoclassical Growth Model • Objective: 1− c −1 E 0 ∑ t uc t , uc t t 1− t0 • Constraints: c t expk t1 ≤ fk t , a t , t 0, 1, 2, . . . . a t a t−1 t . fk t , a t expk t expa t 1 − expk t .
- 18. Efficiency Condition ct1 E t u ′ fk t , a t − expk t1 ct1 period t1 marginal product of capital − u ′ fk t1 , a t t1 − expk t2 f K k t1 , a t t1 0. • Here, k t , a t ~given numbers t1 ~ iid, mean zero variance V time t choice variable, k t1 • Convenient to suppose the model is the limit →1 of a sequence of models, , indexed by t1 ~ 2 V , 1.
- 19. Solution • A policy rule, k t1 gk t , a t , . • With the property: ct Rk t , a t , ; g ≡ E t u ′ fk t , a t − expgk t , a t , ct1 k t1 a t1 k t1 a t1 − u ′ f gk t , a t , , a t t1 − exp g gk t , a t , , a t t1 , k t1 a t1 fK gk t , a t , , a t t1 0, • for all a t , k t and 1.
- 20. Projection Methods • Let ĝk t , a t , ; – be a function with finite parameters (could be either spectral or finite element, as before). • Choose parameters, , to make Rk t , a t , ; ĝ – as close to zero as possible, over a range of values of the state. – use Galerkin or Collocation.
- 21. Occasionally Binding Constraints • Suppose we add the non‐negativity constraint on investment: expgk t , a t , − 1 − expk t ≥ 0 • Express problem in Lagrangian form and optimum is characterized in terms of equality conditions with a multiplier and with a complementary slackness condition associated with the constraint. • Conceptually straightforward to apply preceding method. For details, see Christiano‐Fisher, ‘Algorithms for Solving Dynamic Models with Occasionally Binding Constraints’, 2000, Journal of Economic Dynamics and Control. – This paper describes alternative strategies, based on parameterizing the expectation function, that may be easier, when constraints are occasionally binding constraints.
- 22. Perturbation Approach • Straightforward application of the perturbation approach, as in the simple example, requires knowing the value taken on by the policy rule at a point. • The overwhelming majority of models used in macro do have this property. – In these models, can compute non‐stochastic steady state without any knowledge of the policy rule, g. k∗ – Non‐stochastic steady state is such that a0 (nonstochastic steady state in no uncertainty case) 0 (no uncertainty) ∗ ∗ k g k , 0 , 0 1 – and 1− k∗ log . 1 − 1 −
- 23. Perturbation • Error function: ct Rk t , a t , ; g ≡ E t u ′ fk t , a t − expgk t , a t , ct1 − u ′ fgk t , a t , , a t t1 − expggk t , a t , , a t t1 , f K gk t , a t , , a t t1 0, k t , a t , . – for all values of • So, all order derivatives of R with respect to its arguments are zero (assuming they exist!).
- 24. Four (Easy to Show) Results About Perturbations • Taylor series expansion of policy rule: linear component of policy rule gk t , a t , ≃ k g k k t − k g a a t g second and higher order terms 1 g kk k t − k 2 g aa a 2 g 2 g ka k t − ka t g k k t − k g a a t . . . t 2 – g 0 : to a first order approximation, ‘certainty equivalence’ – All terms found by solving linear equations, except coefficient on past gk endogenous variable, ,which requires solving for eigenvalues – To second order approximation: slope terms certainty equivalent – g k g a 0 – Quadratic, higher order terms computed recursively.
- 25. First Order Perturbation • Working out the following derivatives and evaluating at k t k ∗ , a t 0 R k k t , a t , ; g R a k t , a t , ; g R k t , a t , ; g 0 ‘problematic term’ Source of certainty equivalence • Implies: In linear approximation R k u ′′ f k − e g g k − u ′ f Kk g k − u ′′ f k g k − e g g 2 f K 0 k R a u ′′ f a − e g g a − u ′ f Kk g a f Ka − u ′′ f k g a f a − e g g k g a g a f K 0 R −u ′ e g u ′′ f k − e g g k f K g 0
- 26. Technical notes for following slide u ′′ f k − e g g k − u ′ f Kk g k − u ′′ f k g k − e g g 2 f K k 0 1 f − e g g − u ′ f Kk g − f g − e g g 2 f K 0 k k u ′′ k k k k 1 f − 1 e g u ′ f Kk f f K g e g g 2 f K 0 k u ′′ k k k 1 fk − 1 u ′ f Kk f k g g 2 0 k eg fK f K u ′′ e g f K eg k 1 − 1 1 u ′ f Kk gk g2 0 k u ′′ e g f K • Simplify this further using: f K K−1 expa 1 − , K ≡ expk exp − 1k a 1 − f k expk a 1 − expk f K e g f Kk − 1 exp − 1k a f KK − 1K−2 expa − 1 exp − 2k a f Kk e −g • to obtain polynomial on next slide.
- 27. First Order, cont’d Rk 0 • Rewriting term: 1 − 1 1 u ′ f KK gk g2 0 k u ′′ f K 0 g k 1, g k 1 • There are two solutions, – Theory (see Stokey‐Lucas) tells us to pick the smaller one. – In general, could be more than one eigenvalue less than unity: multiple solutions. gk ga • Conditional on solution to , solved for Ra 0 linearly using equation. • These results all generalize to multidimensional case
- 28. Numerical Example • Parameters taken from Prescott (1986): 2 20, 0. 36, 0. 02, 0. 95, Ve 0. 01 2 • Second order approximation: 3.88 0.98 0.996 0.06 0.07 0 ĝk t , a t−1 , t , k ∗ gk k t − k ∗ ga at g 0.014 0.00017 0.067 0.079 0.000024 0.00068 1 1 g kk k t − k g aa 2 a2 t g 2 2 −0.035 −0.028 0 0 g ka k t − ka t g k k t − k g a a t
- 29. Conclusion • For modest US‐sized fluctuations and for aggregate quantities, it is reasonable to work with first order perturbations. • First order perturbation: linearize (or, log‐ linearize) equilibrium conditions around non‐ stochastic steady state and solve the resulting system. – This approach assumes ‘certainty equivalence’. Ok, as a first order approximation.
- 30. List of endogenous variables determined at t Solution by Linearization • (log) Linearized Equilibrium Conditions: E t 0 z t1 1 z t 2 z t−1 0 s t1 1 s t 0 • Posit Linear Solution: s t − Ps t−1 − t 0. z t Az t−1 Bs t Exogenous shocks • To satisfy equil conditions, A and B must: 0 A2 1 A 2 I 0, F 0 0 BP 1 0 A 1 B 0 • If there is exactly one A with eigenvalues less than unity in absolute value, that’s the solution. Otherwise, multiple solutions. • Conditional on A, solve linear system for B.