Applications for solving DSGE models. October 25th, 2011

Transcription

1 MATLAB Workshop Applications for solving DSGE models Freddy Rojas Cama Marola Castillo Quinto Preliminary October 25th, 2011

2 A model to solve The model The model A model is set up in order to draw conclusions about the dynamics of the economy taking into consideration utility maximization framework. You have plenty knowledge over dynamics of states and shocks (you have made the model!) We cannot expect reality entirely ts in those models; reality is complicated but we are interested in isolating eects according to our research.

3 A model to solve The model Some facts the model can reproduce By using a neoclassical-growth model set up we can explain most of moments and autocorrelation between macroeconomics variables Dave and Dejong Table 6.4 Moments comparison RBC model σ j σ j j σy ϕ (1) ϕ j,y (0) ϕ j,y (1) y c i h

4 Percent Deviations MATLAB Workshop A model to solve The model Impulse responses The solution of the model gives us a policy function and we can study this "articial" economy by impulse responses (similar to VAR specications); Output Consumption Capital Productivity Labor Quarters

5 A model to solve The model A Policy function (an intuitive story) The Bellman equation is V (π t ) = min y t 1 2 (π t π) 2 + λy 2 t + βv (π t+1 ) (1) subject to π t+1 = π t + αy t + ε t+1 (Phillips curve) y t+1 = θy t γ (i t π t ) (IS curve) The "guess" for the value function is V (π t ) = κ κ 1(π t π) 2. For this problem the policy function is (setting λ = 0) i t = π t θ αγ (π t π) + y t (2) γ

6 A model to solve A solution A solution Suppose we have the following equations 1 = β 1 αk α 1 C t C t+1 K t+1 = Kt α C t we log-linearize them around steady state and we have c t+1 = c t (1 α) k t+1 (3) k t+1 = α Y K k t C K c t (4) Substituting expression (4) in (3 c t+1 = 1 + (1 α) C c t (1 α) α Y k t (5) K K

7 A model to solve A solution A solution Then expressions (4) in (5) are equivalent to 4c t+1 = (1 α) C K c t 4k t+1 = We can plot these expressions 1 α Y k t K (1 α) α Y k t (6) K C K c t (7)

8 A model to solve A solution A solution The standard approach is to grab expressions (6) and (7) and to set a state space in the form X t+1 = AX t " # kt+1 α Y C K K X t+1 = kt c t+1 (1 α) α Y 1 + (1 α) C c t K K (8) The eigenvalues of the matrix A are λ 1 = α and λ 2 = Y. Then a K solution is X = γ 1 b 1 (λ 1 ) t + γ 2 b 2 (λ 2 ) t.then we get rid of explosive solution (γ 2 = 0) and we have X = γ 1 b 1 α t being b 1 = α 1 eigenvalue. 1 0 the eigenvector related to the stable

9 A model to solve A solution A solution Therefore, in order to get an analtical form of the policy funtion we γ 1 = k 0 α; thus k t = k 0 α t and c t = k 0 α t+1 = αk t, nally we have that k 1 = k 0 α k 2 = k 0 α 2 = αk 1 k t+1 = αk t So, the policy function is c t = αk t..

10 A model to solve An example: Neoclassical Growth model A Neoclassical growth model We have the following functional forms U = log c t y = A t k α t (Utility function) (production function) k t+1 = (1 δ) k t + i t (Law motion of capital) ln A t = ρ ln A t 1 + ε t (shocks dynamics) y t = c t + i t (national account)

11 Setting up the model Setting up the model We have the objetive function max o fu t g β t=0e t r (x t, u t ) (9) t=0 where x t and u t are state and control variables subject to the dynamics of states x t+1 = g (x t, u t, ε t+1 ) (10) The solution of this problem gives us the policy function u t = h (x t ). So, we are going to present methods to approximate h.

12 Setting up the model A recursive formulation We can reformulate the optimization problem; basically we turn the problem of choosing sequences into a problem of nding a functional form (Bellman; 1957). We dene the Bellman equation as V (s t ) = sup c t,k t+1 fr (s t, u t ) + βv (s t+1 )g Since we do not know the "shape" of V we can explore to guess and get convergence in the functional form of V.

13 Linear Quadratic Linear Quadratic (LQ) We learned that (a sort of inspection in fact) we can easily get a closed or analitical solution of the model; two cases If the objective function is quadratic and the restrictions are linear; then the value function and policy functions are quadratic and linear respectively So, why not to approximate both target function and constrains to be quadratic and linear respectively and then solve the model with the "guess" on the value function?. Linear quadratic method provides a solution; particularly LQ makes iterations on the value function in roder to get convergence.

14 Linear Quadratic Linear Quadratic (LQ) We learned that (a sort of inspection in fact) we can easily get a closed or analitical solution of the model; two cases If the objective function is quadratic and the restrictions are linear; then the value function and policy functions are quadratic and linear respectively If the objective function is logarithmic and the restrictions are non-linear; then the value function and policy functions are logarithmic and non;inear respectively. So, why not to approximate both target function and constrains to be quadratic and linear respectively and then solve the model with the "guess" on the value function?. Linear quadratic method provides a solution; particularly LQ makes iterations on the value function in roder to get convergence.

15 Linear Quadratic Linear Quadratic (LQ) We turn the initial objective function (r) and constrain (g) into a cuadratic and linear function respectively. So, we know that if the objetive function is quadratic then we can use a ad-hoc "guess"; this is a quadratic function. The model consists of expressions (9) and (10); in order to solve the model we have the following algorithm: 1 We must calculate the steady state of the variables in the system; So, we need to get the Euler equations. 2 We must linearize the restrictions (rst order approximation) x t+1 = g (x t, u t, ε t+1 ) Ax t + Bu t + C ε t+1 3 Approximate r by using a second order approximation (Kydland and Prescot method; 1986) r (x t, u t ) x 0 trx t + u 0 tqu t + 2x 0 thu t

16 Linear Quadratic Linear Quadratic (LQ) R = 1 2 Q = 1 2 H uu (11) Hxx r x H xx x H ux u r 0 x x 0 H xx u 0 H ux q (12) q = 2r (x, u) + x 0 H xx x 2rxx 0 2ruu 0 + u 0 H uu u + 2x 0 H xu u (13) H = 1 Hxu 2 ru 0 u 0 H uu x 0 (14) H xu

17 Linear Quadratic Linear Quadratic (LQ) 4. We adjust the dynamics of states x t+1 = Ax t + Bu t + C ε t+1 5. We set up the Ricatti equation P = R + βa 0 PA βa 0 PB + H 0 Q + βb 0 PB + H 0 1 βb 0 PA + H (15) Where P solves this algebraic expression.

18 Linear Quadratic Linear Quadratic (LQ) 6. Iterate (setting and updating) P on Ricatti equation P j = R + βa 0 P j 1 A βa 0 P j 1 B + H 0 Q + βb 0 P j 1 B + H 0 1 βb 0 P j 1 A + H (16) Stop the iteration process until kp j P j 1 k < Check x (A BF ) x = 0 where F = (Q + βb 0 P B + H 0 ) 1 (βb 0 P A + H) 8. The optimal policy function is u t = Q + βb 0 P B + H 0 1 βb 0 P A + H x t (17)

19 Perturbation method Schmidt-Grohe and Uribe (2002) The linear quadratic method is ok.but; We may need other method if we are going to talk about welfare What about second moments? can volatility aect control variables?. Standard procedure results in responses which are symmetric. (related to ARCH or GARCH especications). Approximation errors may be signicant.

20 Perturbation method Schmidt-Grohe and Uribe (2002) We consider the Uribe and Schmidt-Grohe (2003)'s methodology which states that equilibrium conditions may be represented as follows E t f (y t+1, y t, x t+1, x t ) = 0 (18) where E t denotes the mathematical conditional expectation on information at time t. the vector x t of predetermined variables has size n x, this vector can be split to endogenous and exogenous determined variables. The vector y t gathers control variables and it has size n y the solution has the following shape y t = g(x t, σ) (19) x t+1 = h (x t, σ) + ησε t+1 (20)

21 Perturbation method Schmidt-Grohe and Uribe (2002) we look for an approximation of g y h around (x, σ) = (x, 0), in order to get the solution Uribe and Schmidt-Grohe (2003)perform rst and second order approximations. Uribe and Schmidt-Grohe (2003)states that it is straightforward to apply the method described thus far to higher-order approximations to the policy function. The authors mention that given rst and second-order terms of the Taylor expansions of h and g, the third-order terms can be identied by solving a linear system of equations. Authors refer Collard and Juillard, 2001; and Judd, 1998 for details. Interesting results from this methodology arises when second or third order approximation are applied to models.

22 Perturbation method Schmidt-Grohe and Uribe (2002) You can dowload the matlab codes in Uribe's web page; basically you need to modify the following.m codes neoclassical model run.m neoclassical model ss.m neoclassical model.m Also, you must download some other functions available in Uribe's web page. Something interesting is that authors provide codes for analytical and numerical derivatives.

23 Schur Decomposition method Klein's method This method has the following state space set up; MX t+1jt = NX t X t gathers the vector of predetermined variables (k t ), exogenous variables (a t ) and jumping variables (c t ); We are interested in looking for a (saddlepoint stable) solution of the form c t = Fs t and kt+1 s t+1jt = = Ps t a t To solve for F and P using Klein's programs, simply dene M, N and the number of state variables (the dimension of s t ), and then call solab.m function: [F P] = solab(m,n,ns)

24 Blanchard and Khan Blanchard and Khan First, we talk about Blanchard and Khan metodology and how Sims simplies the notation and provide a straighforward representation to implement. The Blanchard and Kahn representation states that you can easily discern the state variables from the jump (or control) variables. They suggest to write the system as; Γ BK 0 St+1 E t X t+1 = Γ BK 1 St X t + Λ + Ωz t (21) where S t is a vector of state (predetermined) variables at time t; X is a vector of jump (non-predetermined or control) variables at time t: z t is a vector of exogenous shocks (or variables) known at time t; which in many applications can be taken to be iid over time

25 Blanchard and Khan Blanchard and Khan E t X t+1 is a vector that contains the agents expectations of X t+1 at time t, Λ is a vector of constants, that in many applications is simply a vector of zeros. The state-space representation is intuitive but in order to nd the solution a recursive method is not going to work since the variables in the left hand side of (21) are diferent from those appearing at rst column vector in the right hand side. We need to note that E t X t+1 and X t+1 are two variables both known at time t that are dierent (see Michelacci, 2006).

26 Sims' method Sims's method To avoid the problem Sims (2002) suggests to write the system in the following form: St+1 St Γ 0 = Γ E t X 1 + Λ + Ψz t+1 X t + Πη t (22) t where (S t E t X t ) 0 is a (n 1) vector containing all variables determined at time t. The rst n s entries correspond to the variables that never enter the system with expectations, which generally correspond to the predetermined variables at a given point in time (endogenous state variables). The second n x entries correspond to the variables that (at least) sometimes enter with an expectational term, which generally correspond to the jump (or control) variables.

27 Sims' method Sims's method It is valuable to mention the properties of Sims's procedure: 1 In Sims's representation we do not have to specify explicitly which variables are predetermined and which are jump variables or include explicitly the expectations in the state-control vector. But, sometimes is preferable to arrange the system as in expression (22) and then having directly the policy functions. In fact under that representation we would have as solution a matrix with the last columns lled in with zeros. 2 The notation is such that the time arguments relate consistently to the information structure: variables dated t are always known at t. Expectations of variabels are not the same as thier value at time t.

28 Sims' method Sims's method 3. Sims adds an equation of the type η t = E t X t+1 X t, that denes the expectational error. 4. The expectational errors are endogenous. Dening their structure is part of the solution of the system. Michelacci, 2006 states that is an important aspect of the analysis that is hidden in the Blanchard-Kahn formulation. 5. Given the convention about the ordering, the matrix consists of zeros except for the last n x rows that consist of an identity matrix of dimension equal to the number of expectational error in the system.

29 References References Schmitt-Grohe, S. and M. Uribe Solving Dynamic General Equilibrium Models Using a Second-Order Approximation to the Policy Function. JEDC, vol. 28, January 2004, pp Dejong, D. and Ch. Dave Structural Macroeconometrics. Princeton University Press. Michelacci, C. Advanced Macroeconomics Lectures. CEMFI and UCL. Ljungqvist, L. and T. Sargent Recursive Macroeconomic Theory. Massachusetts Institute of Technology. Second Edition.

30 References References Klein, P Using the Generalized Schur Form to Solve a Multivariate Linear Rational Expectations Model. Journal of Economic Dynamics and Control 24, 2000, Sims, Ch Solving Linear Rational Expectations Models. Computational Economics 20, October 2002, 1-20.

31 MATLAB Workshop References Notes References Klein, P Using the Generalized Schur Form to Solve a Multivariate Linear Rational Expectations Model. Journal of Economic Dynamics and Control 24, 2000, Sims, Ch Solving Linear Rational Expectations Models. Computational Economics 20, October 2002, Notes