Solving LQ-Gaussian Robustness Problems in Dynare

Transcription

1 Solving LQ-Gaussian Robustness Problems in Dynare July / 20

2 Overview Why Robustness? Ellsberg s Paradox. Gambling Example. Knightian Uncertainty: there is a difference between Risk and Uncertainty. Robustness is a method that Rationalizes concerns about Knightian Uncertainty into certain preferences. What are these preferences? The General Case deals with entropy...i will show something simpler here. 2 / 20

3 Overview Why Robustness? We attempt to show that Robustness Problems in LQ-Gaussian setups are easy to solve using Dynare. Linear Quadratic Gaussian Models are very general, they can approximate dynamic preferences, habit persistence, adjustment costs, durable goods, etc. We present a matlab subroutine that does the job of writing a mod file for you once you specify the LQ-setup. The following presentation reviews the LQ setup of Hansen and Sargent, the Robustness problem, shows how the code works and uses an example. 3 / 20

4 HS-LQ Setup A first glance at LQ-Problem: Problem (LQ-Regulator) max (1/2) β t [(s t b t ) (s t b t ) + (g t ) (g t )] {c t,k t} t=0 subject to the following set of restrictions: Φ c c t + Φ g g t + Φ i i t = Γk t 1 + d t k t = k k t 1 + Θ k i t h t = h h t 1 + Θ h c t s t = Λh t 1 + Πc t and exogenous variables evolve according to a VAR structure: z t+1 = A 22 z t + C 2 w t+1, b t = U b z t and d t = U d z t w t+1 is i.i.d Standard-Normal vector. 4 / 20

5 HS-LQ Setup So we can write up a Lagrangian: To solve this program we use a Lagrangian method: (1/2) E{ β t [(s t b t ) (s t b t ) + (g t ) (g t )] +... t=0 +M d t [Φ c c t + Φ g g t + Φ i i t Γk t 1 d t ]... +M k t [k t k k t 1 Θ k i t ]... +M h t [h t h h t 1 Θ h c t ]... +M s t [s t Λh t 1 Πc t ]} 5 / 20

6 HS-LQ Setup NFOC s yield the following system The FONC for the problem, found in system of HS-LQ book are: and the restrictions: (1/2) E{ β t [(s t b t) (s t b t) + (g t) (g t)] +... t=0 +Mt d [Φ cc t + Φ g g t + Φ i i t Γk t 1 d t]... +Mt k [k t k k t 1 Θ k i t]... +Mt h [h t h h t 1 Θ h c t]... +Mt s [s t Λh t 1 Πc t]} Φ cc t + Φ g g t + Φ i i t = Γk t 1 + d t k t = k k t 1 + Θ k i t h t = h h t 1 + Θ h c t s t = Λh t 1 + Πc t b t = U b z t d t = U d z t We have a system of stochastic difference equations. Dynare is the perfect environment for solving for the optimal feedback policies. 6 / 20

7 Robustness Problem The Robustness problem is the artificial simultaneous game: min max E{ {w t+1 } {c t,k t } t=0 subject to: β t ( 1 2 ) [(s t b t) (s t b t) + (g t) (g t)] βθw t+1 w t+1 } Φ cc t + Φ g g t + Φ i i t = Γk t 1 + d t k t = k k t 1 + Θ k i t h t = h h t 1 + Θ h c t s t = Λh t 1 + Πc t b t = U b z t d t = U d z t and the exogenous process: b t = U b z t, d t = U d z t. The difference here is that minimizer chooses w t+1 : z t+1 = A 22 z t + C 2 w t+1 so w t+1 is not a Guassian vector any more. 7 / 20

8 Robustness Problem Lagrangian The LQ-Robustness problem may also be written as a Lagrangian: min max {w t+1 } {c t,k t } { t=0 (1/2) β t [(s t b t) (s t b t) + (g t) (g t)] βθ w t+1 w t+1 } 2 +M d t [Φ cc t + Φ g g t + Φ i i t Γk t 1 d t] +M k t [k t k k t 1 Θ k i t] +M h t [h t h h t 1 Θ h c t] +M s t [s t Λh t 1 Πc t] +M z t [ z t+1 + A 22 z t + C 2 w t+1 ]} 8 / 20

9 Robustness Problem Linear Equation version of the system The NFOC s are: (c t) : M d t Φ c + M h t Θ h + M s t Π = 0 (1) (g t) : g t Φ g M d t = 0 (h t) : M h t + βe (i t) : M d t Φ i + M k t Θ i = 0 (k t) : M k t k + βe (s t) : (s t b t) + M s t = 0 [ M h t+1 h + M s t+1 Λ ] = 0 [ M d t+1 Γ + Mk t+1 k ] = 0 (w t+1) : βθw t+1 + C 2 Mz t = 0 [ ] (z t+1) : βe U d Md t+1 + βe [ A [ 22 t+1] Mz βe (st+1 b t+1) U ] b Mz z t+1 M t = 0 This set of equations is then complemented by: Φ c c t + Φ g g t + Φ i i t = Γk t 1 + d t k t = k k t 1 + Θ k i t h t = h h t 1 + Θ h c t s t = Λh t 1 + Πc t and the minimizer or evil agent chooses w t+1, that will affect: where ɛ t N (0, I ). z t+1 = A 22z t + C 2 (w t+1 + ɛ t) 9 / 20

10 Robustness Problem Why is this working? The reason is that the solution to the minimizing problem is a choice of a mean + noise. This is a property of LQ-Gaussian Robust Problems. For these reasons, the solution is again a set of stochastic linear difference equations. Again, Dynare is the perfect environment for solving for the optimal feedback robust policies! 10 / 20

11 How to Run this in Dynare? How to run this? 1. Declaration of matrixes: The first thing the user has to do is declare ALL of these matrixes: Phi c, Phi g, Phi i, Gamma, Delta k, Theta k, Delta h,.theta h, Lambda, Pi, A22,C2, U b, U d; 2. Declaration of options for Dynare: Variables that go in the standard set of options for dynare should be included. For example one can include statments like: periods=5000; (for number of simulations in Dynare) irfperiods=15; (for number of periods in irf s in Dynare) 3. File Name: A statemant assigning the variable filename a script should be included:(e.g. filename = Hall Test.mod ;) 4. Optional: Assign a value for the optional dummy variable run if you decide whether or not you want to directly execute the generated dynare. Assign run=1; 5. Calling the file: Call the Hansen Sargent converter by simply writing the following statement HS Robust dyn; 11 / 20

12 How to Run this in Dynare? What does the code do? How does it does the code do it? The code does something simple. It reads the matrixes from HS-LQ setup and declares variables and equations in a mod.file using dynare syntax. The steps followed are: 1. Reading Dimensions of Matrixes For example, for the matrix Phi c, it uses the following statement to set in memory the number of rows and columns: [rr Phi c cc Phi c] =size(phi c) ; 2. Checking Consistency: The matrixes must be conformable in HS-LQ setup. The code checks for consistency along columns and rows: 3. Creating an m-file: This statement generates an object in memory that is used to record what we are writing: fid = fopen(filename, wt ); 12 / 20

13 How to Run this in Dynare? What does the code do? 1. Declaring Variables: Reads dimensions and declares a variable. 2. Declaring Parameters is done in a similar way: fprintf(fid, parameters ); fprintf(fid, beta ) ; fprintf(fid, theta ) ; for j=1:rr Phi c for i=1:cc Phi c fprintf(fid, Phi c %g%g,j,i); 13 / 20

14 How to Run this in Dynare? What does the code do? 1. Setting Parameter Values: Using a similar statement we declare parameters. The variable %g in the statement is assign the value of the variable after the second comma. n means that the lines s there: fprintf(fid, beta=%g ; n,beta) ; fprintf(fid, theta=%g ;n,theta) ; for j=1:rr Phi c for i=1:cc Phi c fprintf(fid, Phi c %g%g =%g ; n,j,i,phi c(j,i)); 14 / 20

15 How to Run this in Dynare? What does the code do? 1. Declaring Equations: Let s use an example: For the first block of equations in the system, Mt d Φ c + Mt h Θ h = 0, we use the following commands: for i=1:(rr Phi c) for j=1:(cc Phi c) fprintf(fid, Phi c %g%g*c%g +,i,j,j); for j=1:(cc Phi g) fprintf(fid, Phi g %g%g*g%g +,i,j,j); for j=1:(cc Phi i) fprintf(fid, Phi i %g%g*i%g =,i,j,j); for j=1:(cc Gamma) fprintf(fid, Gamma %g%g*k%g +,i,j,j); fprintf(fid, d%g,i); fprintf(fid, ; n ); 15 / 20

16 How to Run this in Dynare? What does the code do? 1. Running Dynare: This step simply follows the dynare syntax to call dynare according to the previously defined preferences: fprintf(fid, shocks; n ); for i=1:cc C2 fprintf(fid, var e%g; n,i); fprintf(fid, stderr 1; n ); fprintf(fid, ; n n ); fprintf(fid, n ); fprintf(fid, steady; n ); fprintf(fid, stoch simul(solve algo=3, periods=%g); n,irfperiods,periods); fclose(fid); if (run) dynare(filename) 16 / 20

17 Some Examples Hansen, Sargent and Tallarini - Review of Economic Studies 1999 The planner solves the following problem: min {w t+1 } max E{ [ ] β t (1/2) (s t b t) 2 wt+1 wt+1 βθ {ct,kt } 2 t=0 The input of the utility function are an indirect service for the household, s t, and a preference shock b t that satisfy the following: Resource constraints: s t = Λh t 1 + Πc t Productivity Shifts: c t + i t = (1 + r) k t 1 + y t y t = U y z t 17 / 20

18 Some Examples Hansen, Sargent and Tallarini - Review of Economic Studies 1999 Again, the Robust planner problem is different from the standard model in that: a minimizer or evil agent chooses w t+1, that will affect: z t+1 = A z z t + C Z w t+1 and we parameterize this to have a stable solution: γ := (1 + r) = 1 β For a pair of AR(2) process, we have the following values of A z and C Z are: ρ 22 ρ A z = and Cz = 0 0 and Uy = [5 1 1] ρ 44 ρ / 20

19 Some Examples Hansen, Sargent and Tallarini - Review of Economic Studies 1999 The following code is declares the variables and runs dynare: % Declaration of Paramteters delta=0.05; beta=1/1.05; theta=10; % Matrix Form Phi c=1 ; Phi g=0 ; Phi i=1 ; Gamma=0.1 ; Delta k=1-delta ; Theta k=1 ; Delta h=0 ; Theta h=1 ; Lambda=0 ; Pi=1 ; A22=[1 0 0; ; ] ; C2=[0 0; 1 0; 0 1] ; U b=[30 0 1] ; U d=[5 1 0] ; %% User Preferences for Dynare periods=5000; %parameters for simulation irfperiods=15; %parameters for irf s filename= Hall Test.mod ; %pick filename for mod file %Optional: decide whether or not you want to directly execute the generated dynare %code (set run=1 to execute) run=1; %% Call the Hansen Sargent converter: HS Robust dyn; 19 / 20

20 Things worth mentioning LQ is a general setup. :) My guess is that we can ext this to Robust Filtering. : ) We cannot deal yet with Optimal Robust problems with expectations. :( The code may still present some errors. :( 20 / 20