Solving models with (lots of) idiosyncratic risk

Transcription

1 Solving models with (lots of) idiosyncratic risk Wouter J. Den Haan November 8, 2009

2 Models with heterogeneous agents 1 Lots of idiosyncratic risk 2 If #2 leads to strong non-linearities in the policy function, then also high dimensional state space

3 Outline Projection methods and non-linearities Perturbation methods and non-linearities keeping problem well behaved exploiting idiosyncratic risk to nd perturbation points

4 Projection methods - functional forms Splines versus polynomials Splines be smart about grid points, e.g., log scale Polynomials be smart about the transformation of the variables

5 Projection methods - nding solution Possible choices Equation solver or minimization routine di cult to use for splines (too many coe cients) Iteration procedure xed-point iteration; easier but worse convergence properties time iteration; possibly a bit harder but better convergence properties

6 Projection methods - time versus xed-point iteration P

7 Projection methods - Choosing state variable Possible choices standard choice: k endogenous grid points: k 0 makes time iteration cheap

8 Projection methods - Endogenous grid points P

9 Perturbation and idiosyncratic risk Given the speed of perturbation it is naturally suited for models with heterogenous agents

10 Perturbation and idiosyncratic risk Given the speed of perturbation it is naturally suited for models with heterogenous agents Except...

11 Perturbation and idiosyncratic risk Given the speed of perturbation it is naturally suited for models with heterogenous agents Except... the non-linearities can easily create problems

12 Perturbation and idiosyncratic risk Given the speed of perturbation it is naturally suited for models with heterogenous agents Except... the non-linearities can easily create problems not just some inaccuracies but odd and explosive behavior

13 Perturbation and nonlinearities Limited radius of convergence (approximation to truth) Oscillating patterns; not shape preserving Regular polynomials: explosive behavior of dynamic systems x +1 = h(x) p N (x) p N (x) lim x! x = if if p N (x) lim x!+ x lim x!+ p N (x) x = + =) no global convergence = =) function must turn negative

14 ln(x) & Taylor series expansions at x = th th ln(x) st nd x 2

15 0.2 0 ln(x) &Taylor series expansions at x = x st nd x ln(x) 1 st 2 nd 5 th 25 th

16 limited radius of convergence and projection methods Less restricted to focus on particular perturbation point Chebyshev nodes & compact interval =) uniform convergence

17 ln(x) & uniform convergence in [0,2] U = 2 25 th th ln(x) st nd x 2

18 0.2 0 ln(x) & uniform convergence in [0,3] x U = nd st x ln(x) 1 st 2 nd 5 th 25 th

19 Problems within radius of convergence di culties in preserving shape h(x) = 0.5x α + 0.5x α is an integer, so h(x) is a polynomial

20 Perturbation solution & preserving shape nd th x true value 2 nd 10 th

21 Projection solution & preserving shape nodes nodes x true value 3 nodes 11 nodes

22 Problems within radius of convergence stability problems a = h(x) = α 0 + x + α 1 e α2x. x +1 = a + shock +1

23 Perturbation solution & stability 4 x** 2 nd x* x true value 2 nd 45-degree

24 Model max fc t,a t gt=1 t=1 β t 1 c1 ν t 1 1 ν P(a t ) s.t. c t + a t 1 + r = a t 1 + θ t, θ t = θ + ε t and ε t N(0, σ 2 ), a 0 given.

25 Penalty function Penalty function corresponding to commonly used inequality constraint: We use: P(a) = if a < 0 0 if a 0 P(a) = η 1 η 0 exp( η 0 a) η 2 a.

26 Penalty function functional form can be approximated globally with Taylor series expansion consider di erent values for curvature parameter, η 0 we do not think of penalty function as a way to implement inequality constraint η 1 and η 2 chosen to match mean and standard deviation of a t

27 FOC c ν t 1 + r + P(a t) a t = βe t c ν t+1

28 Penalty function we do not think of penalty function as a way to implement inequality constraint our calibration procedure and accurate solution =) many properties of "a 0" model similar to properties of "penalty-fcn" model

29 Perturbation solutions when η 0 = rd th a st x "Truth" 1 st 2 nd 3 rd 4 th 5 th

30 Perturbation solutions when η 0 = 20 3 rd 2 nd th a st th x "Truth" 1 st 2 nd 3 rd 4 th 5 th

31 Perturbation and higher uncertainty oscillations more problematic when σ " but higher-order perturbation solution adjust when σ "

32 Perturbation and more uncertainty th = a = x

33 Simulating 2nd & 3rd explode 4th & 5th are inaccurate

34 Pruning - summary simple generates stable solutions for sure just a trick generates policy correspondence not function

35 Pruning - procedure 1 Split up perturbation solution into two parts p N,pert (a t 1, θ t ) ā N = linear part γ N,k (a t 1 ā N ) + γ N,θ θ t θ nonlinear part + p N,pert (a t 1 ā N, θ t θ)

36 Pruning - procedure 2. Simulate a t using Simulate a prune,t using a t ā N = γ N,k a t 1 ā N + γn,θ θ t θ. a prune,t = γ N,k (a prune,t 1 ā N ) + γ N,θ θ t θ + p N,pert (a t 1 ā N, θ t θ). ā N

37 Pruning - state variables a prune,t is determined by: a prune,t 1 and a t 1 Thus, a prune,t is no longer a function of the regular set of state variables

38 Pruning - graphs Our model only has one state variable, x t = a t 1 + θ t Generate fa t g T t=1 and plot simulated a t as function of x t

39 Pruning - second-order

40 Pruning - third-order

41 Pruning - fourth-order

42 Pruning - fth-order

43 General idea Perturbation does not require you to use polynomials Suppose you are given h n (k) k n for n = 0, 1,, N x= x You would like to use g(k) = a 0 g 0 (k) + a 1 g 1 (k) + + a N g N (k) Solve for the values of a from the following N + 1 equations 2 3 h n (k) k n = [a 0, a 1,, a N ] 6 k= k 4 g n 0 (k) k n. g n N (k) k n k= k k= k 7 5

44 Trivial example 1/x Fourth-order Taylor series expansion 1/x 1 (x 1) + 2 (x 1) 2 6 (x 1) (x 1) 4 Alternative 1/x a 0 e 2x + a 1 e 2x x + a 2 e 2x x 2 + a 3 e 2x x 3 + a 4 e 2x x 4 note that this is not a transformation

45 Standard Taylor expansion

46 Alternative Taylor expansion

47 Properties of DSGE models The true solution of DSGE models typically satisfy monotoniciy stability Can perturbation be modi ed to impose this?

48 Solutions All solutions proposed satisfy the following 1 Use smooth di erentiable functions 2 Satisfy the perturbation principle Solutions considered 1 Change of variables 2 Other basis functions 3 True shape preserving

49 Change of variables - idea p N (x): regular N th -order perturbation solution x +1 = p N (x) x +1 = x +1 p 1 (x) x +1 = 2 γ 1 + exp( ˆx +1 ) γ

50 Change of variables - idea From p N (x) p 1 (x) = 2 γ 1 + exp( ˆx +1 ) obtain regular perturbation solution for ˆx Use as the alternative perturbation solution γ x +1 = p 1 (x) + x +1 p 1 (x) + 2 γ 1 + exp( ˆp N (x)) γ.

51 Change of variables -idea Properties For γ small enough close to p 1 (x) and, thus, monotone and stable

52 Change of variables - implementation Standard perturbation system (with p 1 (k) given) 1 (k α + (1 δ)k k +1 ) ν = k +1 p 1 (k) = β αk+1 α δ k+1 α + (1 δ)k ν, +1 k +2 2 γ 1 + exp( ˆk+1 ) γ, Use k +1 = p 1 (k) + 2 γ 1 + exp( ˆp N (k)) γ.

53 Pictures d

54 True underlying problem solved? Problem with standard polynomial basis functions higher-order terms always dominate away from steady state Still true for x +1 p 1 (x) + 2 γ 1 + exp( ˆp N (x)) Extension: use separate squashing function for each basis order γ

55 Alternative basis functions First-order: Higher-order: b 1 (x; γ 1,N ) = γ 1,N (x x). b n (x; γ n,n, γ n,n ) = 2γ n,n n γn,n 1 + exp n!γn,n (x x) no γ n,n Simple version: b n (x; γ n, γ ) = 2γ 1 + exp n γn n!γ (x x) no γ

56 Properties basis functions 1 Zero property at x = x. 8N and 8n N we have b n (0; γ n, γ ) = 0. 2 Levels are bounded from above and below. 3 Derivatives at x = x do not depend on γ =) solutions for γ n coe cients do not depend on γ 4 Even-order basis functions are not monotonic 5 γ n could have the wrong sign!

57 Properties approximation 1 x t cannot go to in nity for any choice of γ 2 γ can always be chosen small enough to ensure monotonicity and a unique xed point 3 Letting γ depend on n and N obviously has advantages reduce γ n,n when n is odd or if b n,n has the wrong sign

58 Solving for the coe cients More properties of the basis functions i b n (x; γ n,n, γn,n ) x i = 0 for i < n x= x i b n (x; γ n,n, γn,n ) 6= 0 for i > n x= x This is enough to solve coe cient recursively x i

59 Solving for the coe cients h 1 = γ 1,N h 2 = γ 2 b 2 (x;,) 2,N x 2 h 3 = γ 2,N 3 b 2 (x;,) x 3. h N = γ N b 2 (x;,) 2,N x N x= x + γ 3,N x= x 3 b 3 (x;,) x x= x 3 x= x + γ 3,N N b 3 (x;,) x N + + γ N,N x= x N b N (x;,) x N

60 Multivariate version x is a J 1 vector First-order Higher-order b j,1 (x; Γ j,1,n ) = Γ 0 j,1,n (x x) b j,n (x; Γ n,n, γ n,n ) = 1 + exp 2γ j,n,n γ j,n,n 1 n!γj,n,n Q n x; Γ j,n,n Q n (x, Γ j,n,n ) is a n th -order polynomial basis function of the vector x with coe cients Γ j,n,n.

61 Multivariate version - 2nd order b j,1 (x; Γ j,1,2 ) = Γ 0 j,1,2 (x x), b j,2 (x; Γ j,2,2, γ j,2,2 ) = 1 + exp 2γ j,2,2 γ 1 2!γj,2,2 (x x) 0 j,2,2, Γ j,2,2 (x x)

62 True shape preserving d