Projection Methods. Michal Kejak CERGE CERGE-EI ( ) 1 / 29

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1 Projection Methods Michal Kejak CERGE CERGE-EI ( ) 1 / 29

2 Introduction numerical methods for dynamic economies nite-di erence methods initial value problems (Euler method) two-point boundary value problems (shooting, backward shooting) numerical dynamic programming methods nite-state methods (value function iteration) discretized continuous problems linear-quadratic problems value function approximation perturbation methods projection methods supportive numerical methods approximation methods (Chebyshev polynomials) numerical integration (G-H and G-CH quadrature) nonlinear equations (Newton method) optimization NLS (Levenberg-Marquardt) CERGE-EI ( ) 2 / 29

3 Application 1: AK growth model with constant saving rate rst-order ordinary di erential equation k = sak δk initial value problem where g sa δ numerical methods k = gk, k(0) = k 0 > 0 nite-di erence methods; grid; e.g. Euler s method; sequence of values fk(i)g T i =0 projection methods; function k : [0, T ]! R ++ CERGE-EI ( ) 3 / 29

4 Application 1: AK growth model with constant saving rate II Functional equation problem functional equation problem; operator L(K ) K (t) L : C 1! C 0 K (t); L(K ) = 0, K (0) = k 0 > 0 polynomial approximation of function K residual function ˆK (t; a) = R(t; a) L( ˆK ) = n a i t i, a 0 = 1 i=0 n a i (it i 1 t i ), a 0 = 1 i=0 CERGE-EI ( ) 4 / 29

5 Application 1: AK growth model with constant saving rate III projection of residual function "nearly" zero; "goodness" of t; let n = 3 and T = 3 projection 1: least squares method ) minimization problem; Z 3 min R(t; a) 2 dt a 0 here f.o.c.; system of 3 linear equations in 3 unknown a s; analytic solution projection 2: subdomain method; zero on average in given 3 subdomains: K 1 = [0, 1], K 2 = [1, 2], and K 3 = [2, 3] Z K i R(t; a)dt = 0, i = 1, 2, 3 system of 3 equations in 3 unknown a s CERGE-EI ( ) 5 / 29

6 Application 1: AK growth model with constant saving rate IV projection of residual function II projection 3: method of moments; powers of t Z 3 R(t; a)t i dt = 0, i = 0, 1, 2 0 system of 3 equations in 3 unknown a s projection 4: Galerkin method; functions used in approximation Z 3 R(t; a)t i dt = 0, i = 1, 2, 3 0 system of 3 equations in 3 unknown a s projection 5: collocation method; set of points: t 1 = 0, t 2 = 1.5, and t = 3 R(t i ; a) = 0, i = 1, 2, 3 system of 3 equations in 3 unknown a s CERGE-EI ( ) 6 / 29

7 Example Least Squares Projection Method and " Z 3 Z 3 n 2 min R(t; a) 2 dt = a i (it i 1 t )# i dt, a 0 = 1 a 0 0 i=0 if n = 3 FOCs Z 3 0 2R(t; a) " Z 3 min a " Z R(t; a) 3 dt = 2 a 0 n i=1 1 + a i (it i 1 t i )# 2 dt 3 i=1 a i (it i 1 t i )# t 2t t 2 3t 2 t dt = CERGE-EI ( ) 7 / 29

8 Example Least Squares Projection Method II So we get system of 3 linear equations in 3 unknowns, fa 1, a 2, a 3 g Z 3 0 Z 3 0 Z 3 i.e. 0 where τ 11 = a1 (1 t) + a 2 2t t 2 + a 3 3t 2 t 3 1 (1 t) dt = a1 (1 t) + a 2 2t t 2 + a 3 3t 2 t 3 1 2t t 2 dt = a1 (1 t) + a 2 2t t 2 + a 3 3t 2 t 3 1 3t 2 t 3 dt = Z 3 τ 11 a 1 + τ 12 a 2 + τ 13 a 3 γ 1 = 0 τ 21 a 1 + τ 22 a 2 + τ 23 a 3 γ 2 = 0 τ 31 a 1 + τ 32 a 2 + τ 33 a 3 γ 3 = 0 (1 t) 2 dt = 6; τ 12 = Z 3 2t t 2 (1 t) dt = 9 2 ; etc. 0 0 CERGE-EI ( ) 8 / 29

9 Orthogonal Collocation Method with Chebyshev polynomials Chebyshev polynomials ϕ i : [ 1, 1]! [ 1, 1] orthogonal polynomials hϕ i, ϕ j i Z 1 ϕ 0 (t) = 1, ϕ 1 (t) = t, 1 ϕ i+1 (t) = 2t ϕ i (t) ϕ i 1 (t) ϕ i (t)ϕ j (t)(1 t 2 ) 1/2 dt = 0, if i 6= j functional equation in K with operator L L(K ) = 0 approximation with Chebyshev polynomials ˆK (t; a) = n a i ϕ i ( t) i=0 where t = 2 t t L t H t L 1, t L = 0, t H = 3 and n = 3 CERGE-EI ( ) 9 / 29

10 Orthogonal Collocation Method with Chebyshev polynomials II residual function R(t; a) = L( ˆK (t; a)) projection: residuals at the Chebyshev zeros of ϕ n+1 R(t i ; a) = 0 2i 1 t i = cos 2n π, i = 1,..., n system of n nonlinear equations in n unknown a i, i = 1,..., n CERGE-EI ( ) 10 / 29

11 Orthogonal Galerkin Method with Chebyshev polynomials projections: inner product with Chebyshev basis polynomials P i (a) Z th t L R(t; a)ϕ i ( t)w( t)dt, i = 1,..., n where the weight function is w( t) = 1 t 2 1/2 approximation of integral: Gauss-Chebyshev quadrature bp i (a) m R(t j ; a)ϕ i ( t j ) j=1 for m > n, where t j are the zeros of ϕ m+1 bp i (a) = 0, i = 1,..., n system of n nonlinear equations in n unknown a i, i = 1,..., n CERGE-EI ( ) 11 / 29

12 Application 2: Deterministic Optimal Growth Model with Discrete Time representative in nitely lived agent problem s.t. recursive formulation s.t. V (k 0 ) = max fc t,k t+1 β t u(c t ) gt=1 t=0 k t+1 f (k t ) c t, k 0 > 0 V (k) = max c,k 0 fu(c) + βv (k0 )g k 0 f (k) f.o.c. and envelope c.! Euler equation u 0 (c) = βu 0 (c 0 )f 0 (k 0 ) k 0 = f (k) c C (k) CERGE-EI ( ) 12 / 29

13 Application 2: Deterministic Optimal Growth Model with Discrete Time II Two-Point Boundary Value Problem k 0 > 0, lim β t u 0 (c t ) k t = 0 t! nite-di erence methods (shooting method) numerical dynamic programming methods projection methods functional forms u(c) = c 1 θ 1 θ ; f (k) = Ak α operator L; policy function C (k) L(C ) C (k) θ αβac (Ak α C (k)) θ (Ak α C (k)) α 1 functional equation in policy function C L(C ) = 0 CERGE-EI ( ) 13 / 29

14 Application 2: Deterministic Optimal Growth Model with Discrete Time III approximation with Chebyshev polynomials bc (k; a) = n a i ϕ i ( k) i=0 where k = 2 k k L k H k L 1 and n is the degree of approximation; and residual function is R(k; a) = L(bC (k; a)). CERGE-EI ( ) 14 / 29

15 Application 2: Deterministic Optimal Growth Model with Discrete Time IV Orthogonal Collocation Method (with Chebyshev polynomials) creates n + 1 equations in n + 1 unknown parameters by setting residuals equal zero at n + 1 Chebyshev zeros of ϕ n+1 R(k i ; a) = bc (k i ; a) θ αβabc (Ak α i bc (k i ; a); a) θ (Ak α i bc (k i ; a)) α 1 k i 2 zeros of ϕ n+1 (k), i.e. the system of n + 1 nolinear equations in n + 1 unknown fa i g n i= R (k 1 ; a 1, a 2,..., a n ) 0 B R (k 2 ; a 1, a 2,..., a n ) A = B 0 A R (k n+1 ; a 1, a 2,..., a n ) 0 CERGE-EI ( ) 15 / 29

16 Application 3: Deterministic Lucas Endogenous Growth Model with Continuous Time Representative In nitely-lived Agent Problem s.t. V (k 0, h 0 ) = Z max e c(t),u(t) 0 ρt c1 t θ 1 θ dt k t = Ak α t (u t h t ) 1 α c t δk t, k 0 > 0 ḣ t = φh t (1 u t ), h 0 > 0 CERGE-EI ( ) 16 / 29

17 Application 3: Deterministic Lucas Endogenous Growth Model with Continuous Time FOCs λ t = ct θ e ρt, (1) α kt µ t φh t = λ t (1 α)a h t, (2) u t h t α 1 kt λ t = λ t αa δ!, (3) u t h t and kt µ t = λ t (1 α)a u t h t k 0 > 0, h 0 > 0, α u t µ t φ(1 u t ), (4) lim k t λ t = 0, t! lim h t µ t! t = 0. CERGE-EI ( ) 17 / 29

18 Application 3: Deterministic Lucas Endogenous Growth Model with Continuous Time II Balanced Growth Path intensive variables c t, k t, h t, and y t grow along the BGP at the constant growth rate g = 1 [φ ρ] > 0, θ variable with limited domain u stay constant along BGP 0 < u = 1 g φ < 1, and the ratios of intensive variables are constant along BGP ct = δ + φ g δ, k t α 1 kt αa 1 α = u. δ + φ h t CERGE-EI ( ) 18 / 29

19 Application 3: Deterministic Lucas Endogenous Growth Model with Continuous Time III Model Transformation in order to get stationary model we transform it with x t k t /h t and q t c t /k t ẋ t = Axt α 1 ut 1 α q t δ φ(1 u t ) x t, 1 1 q t = αaxt α 1 u 1 α δ + ρ t + δ + q t q t, θ α θ δ + φ u t = δ q t φ(1 u t ) u t. α CERGE-EI ( ) 19 / 29

20 Application 3: Deterministic Lucas Endogenous Growth Model with Continuous Time IV Model Functional Equations two policy functions q = Q(x) and u = U(x) with quasi-state variable x two functional equations can be obtained from Q 0 (x) = dq/dx = q/ẋ and U 0 (x) = u/ẋ where ẋ = q = u = L 1 (Q, U) = Q 0 (x)ẋ q = 0, L 2 (Q, U) = U 0 (x)ẋ u = 0, M(x) α 1 θ δ + φ α 1 α Q(x) δ φ(1 U(x)) x, δ + ρ M(x) + δ + Q(x) Q(x), θ δ Q(x) φ(1 U(x)) U(x), CERGE-EI ( ) 20 / 29

21 Application 3: Deterministic Lucas Endogenous Growth Model with Continuous Time V Approximation and Residual Functions approximation with Chebyshev polynomials bq(x; a) = bu(x; b) = n i=0 a i ϕ i ( x), n b i ϕ i ( x), i=0 where x = 2 x x L x H x L 1 and n is the degree of approximation; and residual functions are R 1 (x; a, b) = L 1 ( bq(x; a), bu(x; b)), R 2 (x; a, b) = L 2 ( bq(x; a), bu(x; b)). Orthogonal Collocation Method with Chebyshev polynomials CERGE-EI ( ) 21 / 29

22 Application 4: Stochastic Optimal Growth Model with Discrete Time Representative In nitely-lived Agent Problem s.t. V (k 0, z 0 ) = max fc t,k t+1 g t=0 E 0 t=0 β t ct 1 θ 1 θ c t + k t+1 z t f (k t ), given k 0 > 0 and z 0 > 0 ln z t+1 = ρ ln z t + ɛ t+1, ɛ t+1 N(0, σ 2 ɛ) CERGE-EI ( ) 22 / 29

23 Application 4: Stochastic Optimal Growth Model with Discrete Time II Recursive Formulation s.t. V (k, z) = max c,k 0 f c1 θ 1 θ + βev (k0, z 0 jz)g c + k 0 zak α, z 0 = z ρ e ɛ0 equilibrium conditions (FOC and envelope c.! Euler equation) h c θ = βe c 0 θ αz 0 k 0α 1i, boundary conditions k 0 = zak α C (k, z), z 0 = z ρ e ɛ0 k 0 > 0, z 0 > 0, lim t! E 0 c θ t k t β t = 0 CERGE-EI ( ) 23 / 29

24 Application 4: Stochastic Optimal Growth Model with Discrete Time III using policy function c = C (k, z), the equilibrium conditions are getting n C (k, z) θ = βe C (k 0, z 0 ) θ αz 0 k 0α 1o, k 0 = zak α C (k, z), z 0 = z ρ e ɛ0, CERGE-EI ( ) 24 / 29

25 Application 4: Stochastic Optimal Growth Model with Discrete Time IV Model Functional Equation and functional equation in C is L(C ) = 0 using operator L where L(C ) C (k, z) θ k 0 = zak α C (k, z), Z β αaz 0 k 0 C (k 0, z 0 ) θ e x 2 /2 p dx, 2π z 0 = z ρ e σ ɛx, x N(0, 1). For the use of the Newton method in solving the system of nonlinear equations, it is useful to make the equation less nonlinear ( Z L(C ) C (k, z) β αaz 0 k 0 C (k 0, z 0 ) θ e x 2 1θ /2 p dx). 2π CERGE-EI ( ) 25 / 29

26 Application 4: Stochastic Optimal Growth Model with Discrete Time V Approximation of Policy Function two-dimensional approximation with Chebyshev polynomials bc (k, z; a) = n m i=0 j=0 a ij ϕ i ( k)ϕ j ( z) where k = 2 k k L k H k L 1, and z = 2 z z L z H z L 1; n and m are degrees od approximation along k and z dimensions, resp; approximation of integral by Gauss-Hermite quadrature Z F (y)e y q 2 dy F (y l )ω l l=1 where y l, ω l G-H quadrature nodes and weights. approximation of expected value Z I (k, z, x) e x 2 /2 p dx 2 q l=1 I (k, z, p 2x l )ω l CERGE-EI ( ) 26 / 29

27 Application 4: Stochastic Optimal Growth Model with Discrete Time VI Residual Function Approximated functional equation in C with operator bl ( β I (k, z, p ) 1 θ 2x l )ω l, where Residual function bl(c ) C (k, z) q l=1 k 0 = zak α C (k, z), z 0 = z ρ e σ ɛx, x N(0, 1). R(k, z; a) = bl(bc (k, z; a)). In order to get system of nonlinear equations in the unknown parameters fa ij g n m i=0 we again use Orthogonal Collocation j=0 Method (with Chebyshev polynomials) CERGE-EI ( ) 27 / 29

28 PROJEC Toolbox in Matlab and Gauss orthogonal collocation method with Chebyshev polynomials installation structure application source les CERGE-EI ( ) 28 / 29

29 PROJEC Toolbox in Matlab and Gauss Related Papers Kejak, M. (2000) "How to solve growth models: A User s Guide to the Collocation Method in GAUSS", DP , CERGE-EI (available at Kejak, M. (2000) "Minimum Weighted Residual Methods in Endogenous Growth Models." CERGE-EI Working Paper 155 (available at CERGE-EI ( ) 29 / 29