First order approximation of stochastic models

Transcription

1 First order approximation of stochastic models Shanghai Dynare Workshop Sébastien Villemot CEPREMAP October 27, 2013 Sébastien Villemot (CEPREMAP) First order approximation of stochastic models October 27, / 53

2 Outline 1 Presentation of the problem 2 The solution function 3 The steady state 4 First order approximation 5 Example 6 Dating variables in Dynare 7 The Dynare preprocessor Sébastien Villemot (CEPREMAP) First order approximation of stochastic models October 27, / 53

3 General problem E t {f (y t+1, y t, y t 1, u t )} = 0 E(u t ) = 0 E(u t u t) = Σ u E(u t u τ ) = 0 t τ y : vector of endogenous variables u : vector of exogenous stochastic shocks Sébastien Villemot (CEPREMAP) First order approximation of stochastic models October 27, / 53

4 Computation of first order approximation Perturbation approach: recovering a Taylor expansion of the solution function from a Taylor expansion of the original model. A first order approximation is nothing else than a standard solution thru linearization. A first order approximation in terms of the logarithm of the variables provides standard log-linearization. Sébastien Villemot (CEPREMAP) First order approximation of stochastic models October 27, / 53

5 Timing assumptions E t {f (y t+1, y t, y t 1, u t )} = 0 shocks u t are observed at the beginning of period t, decisions affecting the current value of the variables y t, are function of the previous state of the system, yt 1, the shocks ut. Sébastien Villemot (CEPREMAP) First order approximation of stochastic models October 27, / 53

6 The stochastic scale variable E t {f (y t+1, y t, y t 1, u t )} = 0 At period t, the only unknown stochastic variable is y t+1, and, implicitly, u t+1. We introduce the stochastic scale variable, σ and the auxiliary random variable, ɛ t, such that u t+1 = σɛ t+1 Sébastien Villemot (CEPREMAP) First order approximation of stochastic models October 27, / 53

7 The stochastic scale variable (continued) E(ɛ t ) = 0 (1) E(ɛ t ɛ t) = Σ ɛ (2) E(ɛ t ɛ τ ) = 0 t τ (3) and Σ u = σ 2 Σ ɛ Sébastien Villemot (CEPREMAP) First order approximation of stochastic models October 27, / 53

8 Remarks E t {f (y t+1, y t, y t 1, u t )} = 0 The exogenous shocks may appear only at the current period There is no deterministic exogenous variables Not all variables are necessarily present with a lead and a lag Generalization to leads and lags on more than one period: similar to deterministic case, but more complicated for lead variables (because of expectancy operator) Sébastien Villemot (CEPREMAP) First order approximation of stochastic models October 27, / 53

9 Outline 1 Presentation of the problem 2 The solution function 3 The steady state 4 First order approximation 5 Example 6 Dating variables in Dynare 7 The Dynare preprocessor Sébastien Villemot (CEPREMAP) First order approximation of stochastic models October 27, / 53

10 Solution function y t = g(y t 1, u t, σ) where σ is the stochastic scale of the model. If σ = 0, the model is deterministic. For σ > 0, the model is stochastic. Under some conditions, the existence of g() function is proven via an implicit function theorem. See H. Jin and K. Judd Solving Dynamic Stochastic Models ( Sébastien Villemot (CEPREMAP) First order approximation of stochastic models October 27, / 53

11 Solution function (continued) Then, y t+1 = g(y t, u t+1, σ) = g(g(y t 1, u t, σ), u t+1, σ) Let s: F (y t 1, u t, ɛ t+1, σ) = f (g(g(y t 1, u t, σ), σɛ t+1, σ), g(y t 1, u t, σ), y t 1, u t ) So that the problem is redefined as: E t {F (y t 1, u t, ɛ t+1, σ)} = 0 Sébastien Villemot (CEPREMAP) First order approximation of stochastic models October 27, / 53

12 The perturbation approach Obtain a Taylor expansion of the unkown solution function in the neighborhood of a problem that we know how to solve. The problem that we know how to solve is the deterministic steady state. One obtains the Taylor expansion of the solution for the Taylor expansion of the original problem. One consider two different perturbations: 1 points in the neighborhood from the steady sate, 2 from a deterministic model towards a stochastic one (by increasing σ from a zero value). Sébastien Villemot (CEPREMAP) First order approximation of stochastic models October 27, / 53

13 The perturbation approach (continued) The Taylor approximation is taken with respect to y t 1, u t and σ, the arguments of the solution function y t = g(y t 1, u t, σ). At the deterministic steady state, all derivatives are deterministic as well. Sébastien Villemot (CEPREMAP) First order approximation of stochastic models October 27, / 53

14 Outline 1 Presentation of the problem 2 The solution function 3 The steady state 4 First order approximation 5 Example 6 Dating variables in Dynare 7 The Dynare preprocessor Sébastien Villemot (CEPREMAP) First order approximation of stochastic models October 27, / 53

15 Steady state A deterministic steady state, ȳ, for the model satisfies f (ȳ, ȳ, ȳ, 0) = 0 A model can have several steady states, but only one of them will be used for approximation. Furthermore, ȳ = g(ȳ, 0, 0) Sébastien Villemot (CEPREMAP) First order approximation of stochastic models October 27, / 53

16 Outline 1 Presentation of the problem 2 The solution function 3 The steady state 4 First order approximation 5 Example 6 Dating variables in Dynare 7 The Dynare preprocessor Sébastien Villemot (CEPREMAP) First order approximation of stochastic models October 27, / 53

17 First order approximation Around ȳ: } E t {F (1) (y t 1, u t, ɛ t+1, σ) = = 0 { ( E t f (ȳ, ȳ, ȳ, 0) + f y+ gy (g y ŷ + g u u + g σ σ) + g u σɛ + g σ σ ) } +f y0 (g y ŷ + g u u + g σ σ) + f y ŷ + f u u with ŷ = y t 1 ȳ, u = u t, ɛ = ɛ t+1, f y+ = f u = f u t, g y = g y t 1, g u = g u t, g σ = g σ. f y t+1, f y0 = f y t, f y = f y t 1, Sébastien Villemot (CEPREMAP) First order approximation of stochastic models October 27, / 53

18 Taking the expectation } E t {F (1) (y t 1, u t, ɛ t+1, σ) = f (ȳ, ȳ, ȳ, 0) + f y+ (g y (g y ŷ + g u u + g σ σ) + g σ σ) } +f y0 (g y ŷ + g u u + g σ σ) + f y ŷ + f u u = ( f y+ g y g y + f y0 g y + f y ) ŷ + (fy+ g y g u + f y0 g u + f u ) u = 0 + (f y+ g y g σ + f y0 g σ ) σ Sébastien Villemot (CEPREMAP) First order approximation of stochastic models October 27, / 53

19 Recovering g y ( fy+ g y g y + f y0 g y + f y ) ŷ = 0 Structural state space representation: [ ] [ ] 0 fy+ I g I 0 y ŷ = or [ 0 fy+ I 0 g y ] [ yt ȳ y t+1 ȳ ] = [ fy f y0 0 I [ fy f y0 0 I ] [ I g y ] ŷ ] [ yt 1 ȳ y t ȳ ] Sébastien Villemot (CEPREMAP) First order approximation of stochastic models October 27, / 53

20 Structural state space representation Dx t+1 = Ex t with x t+1 = [ yt ȳ y t+1 ȳ ] x t = [ yt 1 ȳ y t ȳ ] There are multiple solutions but we want a unique stable one. Problem when D is singular. Sébastien Villemot (CEPREMAP) First order approximation of stochastic models October 27, / 53

21 Real generalized Schur decomposition Taking the real generalized Schur decomposition of the pencil < E, D >: D = QTZ E = QSZ with T, upper triangular, S quasi-upper triangular, Q Q = I and Z Z = I. Sébastien Villemot (CEPREMAP) First order approximation of stochastic models October 27, / 53

22 Generalized eigenvalues λ i solves λ i Dx i = Ex i For diagonal blocks on S of dimension 1 x 1: T ii 0: λ i = S ii T ii T ii = 0, S ii > 0: λ i = + T ii = 0, S ii < 0: λ i = T ii = 0, S ii = 0: λ i C Sébastien Villemot (CEPREMAP) First order approximation of stochastic models October 27, / 53

23 A pair of complex eigenvalues [ When a diagonal block ] of matrix S is a 2x2 matrix of the form S ii S i,i+1 : S i+1,i S i+1,i+1 the corresponding block of matrix T is a diagonal matrix, (S i,i T i+1,i+1 + S i+1,i+1 T i,i ) 2 < 4S i+1,i S i+1,i T i,i T i+1,i+1, there is a pair of conjugate complex eigenvalues λ i, λ i+1 = S ii T i+1,i+1 + S i+1,i+1 T i,i ± (S i,i T i+1,i+1 S i+1,i+1 T i,i ) 2 + 4S i+1,i S i+1,i T i,i T i+1,i+1 2T i,i T i+1,i+1 Sébastien Villemot (CEPREMAP) First order approximation of stochastic models October 27, / 53

24 Applying the decomposition [ I D g y [ ] [ ] [ T11 T 12 Z11 Z 12 I 0 T 22 ] [ I g y ŷ = E Z 21 Z 22 = g y g y ] g y ŷ ] ŷ [ S11 S 12 0 S 22 ] [ Z11 Z 12 Z 21 Z 22 ] [ I g y ] ŷ where rows and columns are re-ordered such that: (T 11, S 11 ) contain stable generalized eigenvalues (modulus 1) (T 22, S 22 ) contain explosive generalized eigenvalues (modulus > 1) Sébastien Villemot (CEPREMAP) First order approximation of stochastic models October 27, / 53

25 Selecting the stable trajectory To exclude explosive trajectories, one imposes Z 21 + Z 22 g y = 0 g y = Z 1 22 Z 21 A unique stable trajectory exists if Z 22 is square and non-singular: there are as many roots larger than one in modulus as there are forward-looking variables in the model (Blanchard and Kahn condition) and the rank condition is satisfied. Sébastien Villemot (CEPREMAP) First order approximation of stochastic models October 27, / 53

26 An alternative algorithm: Cyclic reduction Solving Iterate A 0 + A 1 X + A 2 X 2 = 0 A (k+1) 0 = A (k) 0 (A(k) 1 ) 1 A (k) 0, A (k+1) 1 = A (k) 1 A (k) 0 (A(k) 1 ) 1 A (k) 2 A (k) 2 (A(k) 1 ) 1 A (k) 0, A (k+1) 2 = A (k) 2 (A(k) 1 ) 1 A (k) 2, Â (k+1) 1 = Â(k) 1 A (k) 2 (A(k) 1 ) 1 A (k) 0. for k = 1,... with A (1) 0 = A 0, A (1) 1 = A 1, A (1) 2 = A 2, Â(1) 1 = A 1 and until A (k) 0 < ɛ and A (k) 2 < ɛ. Then X (Â(k+1) 1 ) 1 A 0 Sébastien Villemot (CEPREMAP) First order approximation of stochastic models October 27, / 53

27 Recovering g u f y+ g y g u + f y0 g u + f u = 0 g u = (f y+ g y + f y0 ) 1 f u Hong Lan & Alexander Meyer-Gohde, Existence and Uniqueness of Perturbation Solutions to DSGE Models, SFB 649 Discussion Papers, Humboldt University, show that f y+ g y + f y0 is an invertible matrix under standard regularity and saddle stability assumptions. Sébastien Villemot (CEPREMAP) First order approximation of stochastic models October 27, / 53

28 Recovering g σ f y+ g y g σ + f y0 g σ = 0 g σ = 0 Yet another manifestation of the certainty equivalence property of first order approximation. Sébastien Villemot (CEPREMAP) First order approximation of stochastic models October 27, / 53

29 First order approximated decision function y t = ȳ + g y ŷ + g u u E {y t } = ȳ Σ y = g y Σ y g y + σ 2 g u Σ ɛ g u The variance is solved for with an algorithm for Lyapunov equations. Sébastien Villemot (CEPREMAP) First order approximation of stochastic models October 27, / 53

30 Outline 1 Presentation of the problem 2 The solution function 3 The steady state 4 First order approximation 5 Example 6 Dating variables in Dynare 7 The Dynare preprocessor Sébastien Villemot (CEPREMAP) First order approximation of stochastic models October 27, / 53

31 A simple RBC model Consider the following model of an economy. Representative agent preferences ( ) ] 1 t 1 U = E t [log (C t ) L1+γ t. 1 + ρ 1 + γ t=1 The household supplies labor and rents capital to the corporate sector. L t is labor services ρ (0, ) is the rate of time preference γ (0, ) is a labor supply parameter. Ct is consumption, wt is the real wage, r t is the real rental rate Sébastien Villemot (CEPREMAP) First order approximation of stochastic models October 27, / 53

32 RBC Model (continued) The household faces the sequence of budget constraints where K t = K t 1 (1 δ) + w t L t + r t K t 1 C t, Kt is capital at the end of period δ (0, 1) is the rate of depreciation The production function is given by the expression Y t = A t Kt 1 α ( (1 + g) t ) 1 α L t where g (0, ) is the growth rate and α and β are parameters. A t is a technology shock that follows the process A t = A λ t 1 exp (e t ), where e t is an i.i.d. zero mean normally distributed error with standard deviation σ 1 and λ (0, 1) is a parameter. Sébastien Villemot (CEPREMAP) First order approximation of stochastic models October 27, / 53

33 The household problem Lagrangian L = max C t,l t,k t ( ) 1 t 1 [ E t 1 + ρ log (C t ) L1+γ t 1 + γ ] µ t (K t K t 1 (1 δ) w t L t r t K t 1 + C t ) t=1 First order conditions ( ) L 1 t 1 ( ) 1 = µ t = 0 C t 1 + ρ C t ( ) L 1 t 1 = (L γ t µ t w t ) = 0 L t 1 + ρ ( ) L 1 t 1 ( ) 1 t = µ t + E t (µ t+1(1 δ + r t )) = 0 K t 1 + ρ 1 + ρ Sébastien Villemot (CEPREMAP) First order approximation of stochastic models October 27, / 53

34 First order conditions Eliminating the Lagrange multiplier, one obtains L γ t = w t C t 1 = 1 C t 1 + ρ E t ( ) 1 (r t δ) C t+1 Sébastien Villemot (CEPREMAP) First order approximation of stochastic models October 27, / 53

35 The firm problem max A t K α ( t 1 (1 + g) t ) 1 α L t rt K t 1 w t L t L t,k t 1 First order conditions: r t = αa t Kt 1 α 1 ( (1 + g) t ) 1 α L t w t = (1 α)a t Kt 1 α ( (1 + g) t ) 1 α L α t Sébastien Villemot (CEPREMAP) First order approximation of stochastic models October 27, / 53

36 Goods market equilibrium K t + C t = K t 1 (1 δ) + A t Kt 1 α ( (1 + g) t ) 1 α L t Sébastien Villemot (CEPREMAP) First order approximation of stochastic models October 27, / 53

37 Dynamic Equilibrium 1 = 1 ( ) 1 C t 1 + ρ E t (r t δ) C t+1 L γ t = w t C t r t = αa t Kt 1 α 1 ( (1 + g) t ) 1 α L t w t = (1 α)a t Kt 1 α ( (1 + g) t ) 1 α L α t K t + C t = K t 1 (1 δ) + A t Kt 1 α ( (1 + g) t ) 1 α L t Sébastien Villemot (CEPREMAP) First order approximation of stochastic models October 27, / 53

38 Existence of a balanced growth path There must exist a growth rates g c and g k so that (1 + g k ) t K 1 + (1 + g c ) t C 1 = (1 + g k ) t ( (1 + gk ) t ) α ( K 1 (1 δ) + A K 1 (1 + g) t ) 1 α L t 1 + g K 1 + g k So, g c = g k = g Sébastien Villemot (CEPREMAP) First order approximation of stochastic models October 27, / 53

39 Stationarized model Let s define Ĉ t = C t /(1 + g) t K t = K t /(1 + g) t ŵ t = w t /(1 + g) t Sébastien Villemot (CEPREMAP) First order approximation of stochastic models October 27, / 53

40 Stationarized model (continued) ( ) 1 Ĉ t (1 + g) = 1 t 1 + ρ E 1 t Ĉ t+1 (1 + g)(1 + g) (r t t δ) L γ t = ŵt(1 + g) t Ĉ t (1 + g) t ( (1 + g) r t = αa t K t t g ( ŵ t (1 + g) t = (1 α)a t ) α 1 ( (1 + g) t L t ) 1 α ) α ( (1 + g) t ) 1 α L α t (1 + g) K t t g ( Ĉt) Kt + (1 + g) t = K (1 + g) t t 1 (1 δ) 1 + g ( (1 + g) + A t K t ) α ( t 1 (1 + g) t ) 1 α L t 1 + g Sébastien Villemot (CEPREMAP) First order approximation of stochastic models October 27, / 53

41 Stationarized model (continued) ( ) 1 = 1 Ĉ t 1 + ρ E 1 t Ĉ t+1 (1 + g) (r t δ) L γ t = ŵt Ĉ t ) α 1 L 1 α t ( Kt 1 r t = αa t 1 + g ( ) α Kt 1 ŵ t = (1 α)a t L α t 1 + g K t + Ĉt = K t g (1 δ) + A t ( ) α Kt 1 L 1 α t 1 + g Sébastien Villemot (CEPREMAP) First order approximation of stochastic models October 27, / 53

42 Dynare implementation var C K L w r A; varexo e; parameters rho delta gamma alpha lambda g; alpha = 0.33; delta = 0.1; rho = 0.03; lambda = 0.97; gamma = 0; g = 0.015; Sébastien Villemot (CEPREMAP) First order approximation of stochastic models October 27, / 53

43 Dynare implementation (continued) model; 1/C=1/(1+rho)*(1/(C(+1)*(1+g)))*(r(+1)+1-delta); L^gamma = w/c; r = alpha*a*(k(-1)/(1+g))^(alpha-1)*l^(1-alpha); w = (1-alpha)*A*(K(-1)/(1+g))^alpha*L^(-alpha); K+C = (K(-1)/(1+g))*(1-delta) +A*(K(-1)/(1+g))^alpha*L^(1-alpha); log(a) = lambda*log(a(-1))+e; end; Sébastien Villemot (CEPREMAP) First order approximation of stochastic models October 27, / 53

44 Dynare implementation (continued) steady_state_model; A = 1; r = (1+g)*(1+rho)+delta-1; L = ((1-alpha)/(r/alpha-delta-g))*r/alpha; K = (1+g)*(r/alpha)^(1/(alpha-1))*L; C = (1-delta)*K/(1+g) +(K/(1+g))^alpha*L^(1-alpha)-K; w = C; end; steady; Sébastien Villemot (CEPREMAP) First order approximation of stochastic models October 27, / 53

45 Dynare implementation (continued) shocks; var e; stderr 0.01; end; check; stoch_simul(order=1); Sébastien Villemot (CEPREMAP) First order approximation of stochastic models October 27, / 53

46 Alternative implementation Let Dynare detrend equations for you parameters g; trend_var(growth_factor=1+g) Z; // Productivity trend var(deflator = Z) C K w; var L r A; varexo e; parameters rho delta gamma alpha lambda; model; // Declare non-detrended model equations end; Sébastien Villemot (CEPREMAP) First order approximation of stochastic models October 27, / 53

47 Decision and transition functions Dynare output: POLICY AND TRANSITION FUNCTIONS C K L w r A Constant K(-1) A(-1) e C t = ( K t 1 K ) ( A t 1 Ā) e t Sébastien Villemot (CEPREMAP) First order approximation of stochastic models October 27, / 53

48 Outline 1 Presentation of the problem 2 The solution function 3 The steady state 4 First order approximation 5 Example 6 Dating variables in Dynare 7 The Dynare preprocessor Sébastien Villemot (CEPREMAP) First order approximation of stochastic models October 27, / 53

49 Dating variables in Dynare Dynare will automatically recognize predetermined and non-predetermined variables, but you must observe a few rules: period t variables are set during period t on the basis of the state of the system at period t 1 and shocks observed at the beginning of period t. therefore, stock variables must be on an end-of-period basis: investment of period t determines the capital stock at the end of period t. Note: with the predetermined variables command, one can use a beginning-of-period convention for stocks when writing the model. However, the IRFs and other output will still be at end-of-stock convention. Sébastien Villemot (CEPREMAP) First order approximation of stochastic models October 27, / 53

50 Log-linearization Taking a log-linear approximation of a model is equivalent to take a linear approximation of a model with respect to the logarithm of the variables. In practice, it is sufficient to replace all occurences of variable X with exp(lx ) where LX = log X. It is possible to make the substitution for some variables and not anothers. You wouldn t want to take a log approximation of a variable whose steady state value is negative... There is no evidence that log-linearization is more accurate than simple linearization. In a growth model, it is often more natural to do a log-linearization. Sébastien Villemot (CEPREMAP) First order approximation of stochastic models October 27, / 53

51 Outline 1 Presentation of the problem 2 The solution function 3 The steady state 4 First order approximation 5 Example 6 Dating variables in Dynare 7 The Dynare preprocessor Sébastien Villemot (CEPREMAP) First order approximation of stochastic models October 27, / 53

52 The role of the Dynare preprocessor Dynare solves generic problems the preprocessor reads your *.mod file and translates it in specific MATLAB/Octave files filename.m: main MATLAB/Octave script for the model filename static.m: static model filename dynamic.m: dynamic model filename steadystate2.m: steady state function filename set auxiliary variables.m: auxiliary variables function Sébastien Villemot (CEPREMAP) First order approximation of stochastic models October 27, / 53

53 Overall design of Dynare Sébastien Villemot (CEPREMAP) First order approximation of stochastic models October 27, / 53