A comparison of numerical methods for the. Solution of continuous-time DSGE models. Juan Carlos Parra Alvarez

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1 A comparison of numerical methods for the solution of continuous-time DSGE models Juan Carlos Parra Alvarez Department of Economics and Business, and CREATES Aarhus University, Denmark November 14, 2012 AU AARHUS UNIVERSITY BUSINESS AND SOCIAL SCIENCES DEPARTMENT OF ECONOMICS AND BUSINESS CREATES Center for Research in Econometric Analysis of Time Series

2 Outline of the presentation 1 Motivation 2 Solution methods 3 Benchmark model 4 Numerical results 5 Summary of main findings

3 Motivation f(x) "Optimal" solution of a DSGE Model: 1. Consumption 2. Labor 3. Welfare True (unknown) function X

4 Motivation Stochastic Optimal Control Problem f(x) "Optimal" solution of a DSGE Model: 1. Consumption 2. Labor 3. Welfare True (unknown) function X

5 Motivation In general, no analytical solution of these unknown functions is available. For the discrete-time case, Taylor & Uhlig (1990), Arouba et al. (2006), Binsbergen et al. (2009) and Caldara et al. (2012) have studied the relative performance of numerical approximations: Value/Policy function iteration, Projection methods: Finite elements and Spectral methods, Perturbation methods, Linear-Quadratic approximation. Little attention has been paid to stochastic optimal control models in continuous-time. Previous work: Judd (1996), Gaspar & Judd (1997), Judd (1998), Miranda & Fackler (2002), Candler (2004), Kompas & Chu (2010) and Posch & Trimborn (2011).

6 Motivation Stochastic Optimal Control Problem Consider the autonomous discounted stochastic optimal control problem: subject to ˆ V (s 0 ) = max E 0 e ρt π (s t, c t ) dt {c t } t=0 D(s) D(s)... 0 ds t = µ (s t, c t ) dt + Σ 2 1 (s t ) db t s 0 = s given where s t S R m + is a vector of state variables at time t with right-continuous sample paths, left-hand limit and initial value s 0 = s; c t D (S) R n + is a non-negative vector of control-decision variables at time t. Define Σ (s t ) = Σ 1 2 (s t ) Σ 1 2 (s t ) to be the variance-covariance matrix of the disturbances.

7 Motivation Stochastic Optimal Control Problem The associated Hamilton-Jacobi-Bellman (HJB) equation is given by: 0 = max {π (s, c) ρv (s) + µ (s, c) V (s) c D(s) trace ( Σ (s) 2 V (s) )} (1) The FOCs are: π c (s, c) + µ c (s, c) V (s) = 0 (2) for each c c, which makes the control a function of the state vector, c = P (s). The implicit maximized HJB equation reads: { } π (s, P (s)) ρv (s) + µ (s, P (s)) V (s) 0 = trace ( Σ (s) 2 V (s) ) (3) which together with the first order conditions determines the unknown functions V (s) and P (s). A solution amounts to find these unknown functions such that they solve the deterministic problem described by equation (1).

8 Motivation But why do we bother with continuous-time if the current state of art of DSGE models is in discrete-time (>30 year of research!)?: Analytical tractability. Clear distinction between flows and stocks. Decisions of agents need not to be perfectly synchronized. From a technical point of view we could (potentially) save time for policy advise!: Discrete-time Composition of unknown functions s = µ(s, c) V (s) = π(c(s)) + βv (µ(s, c(s))) π c (c(s)) = βv s (µ(s, c(s))) V s (s) = βv s (µ(s, c(s)))µ s (s, c) + multidimensional integral E{V ((µ(s, c(s)))), s ) s} Continuous-time No Composition of unknown functions ds = µ(s, c)dt ρv (s) = π(c(s)) + (µ(s, c(s)))v s (s) π c (c(s)) = V s (s) ρv s (s) = µ s (s, c(s))v s (s) + (µ(s, c(s)))v ss (s) + NO integrals V s (s)(µ(s, c(c))) + σ 2 V ss (s)

9 Motivation Performance of different numerical methods based on the resulting maximized HJB equation. Estimation of the model parameters Which approximation method is faster and more accurate to make the estimation feasible? Are we interested in the global shape of the approximated likelihood function? Better fit of the model to the data from a linear or non-linear model? Do the approximation errors affect the approximated likelihood function? Four main approaches: Linear-quadratic approximation: linear and linearized constraints, Perturbation: First and second order, Projection: Collocation, least squares residuals, Time-discretization methods: Markov-Chain approximation.

10 Solution methods Linear-quadratic approximation Use local information (known a priori) to replace the original non-linear problem with a more tractable problem for which a closed form solution can be easily derived Second order Taylor approximation of the objective function π (s t, c t ). First order Taylor approximation of the SDE describing the evolution of the state variables. The transformed problem is given by: subject to: ˆ ( V (ŝ 0 ) = max E 0 e ρt ŝ {ĉ t } t Rŝ ) t + ĉt Qĉt dt t=0 0 dŝ t = ( µ s (s ss, c ss ) ŝ t + µ c (s ss, c ss ) ĉ t ) dt + Σ 1 2 (s ss ) db t ŝ 0 = ŝ given where ŝ t = (s t s ss ) and ĉ t = (c t c ss ) and {s ss, c ss } is the set of local information used in the approximation.

11 Solution methods Linear-quadratic approximation It can be shown that the solution to this stochastic optimal control problem is given by: and ˆP (s) = c ss Q 1 [ W + µ c (s ss, c ss ) Λ 1 ] ŝ ˆV (s) = Λ 0 + ŝ Λ 1 ŝ where Λ 1 is the solution to of a continuous-time algebraic Riccati equation and Λ 0 = 1 ρ trace (Σ (sss ) Λ 1 ). Caveats: Imposition of the certainty equivalence property ˆP (s) is independent of Λ 0. Room for the modified certainty equivalence from robust control theory? (Hansen and Sargent, 2008) The slope coefficients of the policy function are very different to the true coefficients if the constraints are not linear, but instead linearized. (See Judd (1998) and Benigno and Woodford (2012)).

12 Solution methods Perturbation method 1 Express the problem in (2) and (3) as a continuum of problems parameterized by the added perturbation parameter ɛ with the ɛ = 0 case known. ds t = µ (s t, c t ) dt + ɛσ 1 2 (st ) db t where ɛ measures the amount of variance in the model. 2 Differentiate the continuum of problems with respect to each of the control variables in c, each of the state variables in s, and the perturbation parameter, ɛ. Whenever is needed use the envelope condition to simplify the system of equations. 3 Solve for the implicitly defined derivatives at s = s ss and ɛ = 0. 4 Compute the desired order of approximation by means of Taylor s theorem and set ɛ = 1.

13 Solution methods Perturbation method Approximate the true value and policy functions by a polynomial in a neighborhood of a know solution: ˆV (s, ɛ) = V ss,0 + V ss,0 i ˆP (s, ɛ) = P ss,0 + P ss,0 i (s s ss ) i + V ss,0 ɛ ɛ + V ss,0 iɛ (s s ss ) i ɛ V ss,0 ij (s s ss ) i (s s ss ) j V ɛɛ ss,0 ɛ P ss ij (s s ss ) i + P ss,0 ɛ ɛ + P ss,0 iɛ (s s ss ) i ɛ (s s ss ) i (s s ss ) j P ss ɛɛ ɛ

14 Solution methods Perturbation method: Certainty equivalence property Discrete-time: the first order terms Vɛ ss,0 and Pɛ ss,0 are zero regardless of the properties of the economic model (See Judd (1998), Binsbergen et al. (2009) and Caldara et al. (2012)). Continuous-time: the approximation only exhibits the CE property if the economic model exhibits the property itself. Example (Judd,1996) For the "stochastic Ramsey model": ˆ V (s) = max E 0 { e ρt π(c)dt}, 0 ds = (f (s) c)dt + 2ɛσ(s)dB it can be shown that at the deterministic steady state (ɛ = 0, s = s ss ) P ɛ = π cccps 2 + P ss σ(s) + σ s (s). π cc P s

15 Solution methods Projection methods Approximate the dimensional problem associated with the HJB equation with a finite dimensional problem with the following parametric approximations of the unknown functions: ˆV (a, s) = ˆP (b, s) = k a a j φ j (s) = Φ a (s) a, s S R m + j=0 k b b j φ j (s) = Φ b (s) b, s S R m + j=0 where φ j Q and Q is a family of basis functions. The transformed problem is finite dimensional: we are not looking on the function of spaces for V ( ) and P ( ). We want to compute the basis coefficients { } ka aj j=0 and { } kb b j such that R (a, b; s) 0, where j=0 [ ( π s, ˆP(s) ) ρ ˆV (a, s) + µ ( s, ˆP(s) ) ˆV (a, s) + R (a, b; s) := 2 1 tr ( Σ (s) 2 ˆV (a, s) ) ( π c s, ˆP(s) ) ( + µ c s, ˆP(s) ) ˆV (s) ]

16 Solution methods Projection methods Consider a set of test functions { g j (s) } k and a weighting function w (s) 0. j=0 Combined with the residual function they define an inner product: ˆ R (a, b; s), g (s) w (s) R (a, b; s) g j (s) ds. S The inner product induces a norm on the function space. Then, choose the set of parameters ( ) ( ) a 0,... a ka and b0,... b kb such that ˆ w (s) R (a, b; s) g j (s) ds = 0, j = 0,..., k. S where k = k a + k b. Residual functions used: Least squares: g j (s) R(a,s)/ a j and w (s) 1. ( ) Collocation: g j 1 and w (s) δ s s j, where δ is the Dirac delta function and s j for j = {1, 2}.

17 Benchmark model Stochastic neoclassical growth model with endogenous labour Model in Arouba et al. (2006): ˆ (C t (1 L t ) ψ) 1 γ max E 0 e ρt dt, ψ 0, γ > 0, γ (1 γ) ψ 0 {C t,l t } 1 γ t=0 0 subject to Y t = A t K α t L1 α t, α (0, 1) (4) da t = κ (ω A t ) dt + η A t db A,t, κ, ω > 0 and A 0 > 0 given dk t = (I t δk t ) dt + σk t db K,t, K 0 > 0 given (5) (6) Y t = C t + I t (7)

18 Benchmark model Stochastic neoclassical growth model with endogenous labour The solution is fully characterized by the maximized HJB equation: ρv = (C(1 L)ψ ) 1 γ ( ) 1 γ + AK α L 1 α δk C V K (8) +κ (ω A) V A σ2 K 2 V KK η2 AV AA and the FOCs with respect to consumption and leisure: (C (1 L) ψ) 1 γ ψ C (C (1 L) ψ) 1 γ (1 L) = V K (9) = (1 α) AK α L α V K (10)

19 Benchmark model Stochastic neoclassical growth model with endogenous labor Proposition (Constant savings function, (Posch, 2011)) Suppose that total factor productivity is constant, i.e., A t = Ā. Let ψ > 0, γ 1 and ρ = ρ with ρ = (1 αγ) (δ + 12 ) αγσ2 > 0 then, the value function is given by: and the optimal policy functions are given by: V (K t, Ā) = Γ K 1 αγ t 1 αγ C(K t, Ā) = (1 s)ākt α γ(1 α) L1 α, L(K t, Ā) = γ(1 α) ψ(1 γ), where (1 s) denotes the constant propensity to consume.

20 Numerical results Calibration Parameter ρ ψ γ α δ σ κ ω η Ā Closed Form ρ N.A. N.A. N.A No Closed Form N.A. Case η = η = σ = σ = σ = σ = γ = 0.65 M1 M3 M6 M8 γ = 2 Benchmark M4 M7 M9 γ = 10 M2 M5 Extreme I Extreme II Table: Calibrated parameters and robustness analysis. The value γ = 0.65 was chosen in such a way that the concavity condition of the utility function was fulfilled given the calibrated parameter of ψ.

21 Numerical results Closed form solution (Constant Saving Function) 100 Value function Perturbation order 1 Perturbation order 2 Colocation OLS MarkovChain (M=1000) LQ: Consumption LQ: Investment True policy function Consumption Function Labour Supply Function K/Kss Figure: Approximated vs. true value and policy functions over [0.1K ss, 1.5K ss ]

22 Numerical results Closed form solution (Constant Saving Function) E l 1 = n K n K i=1 G l (K i ) Ĝ l (K i ) G l (K i ), E l = 100 max i { G l (K i ) Ĝ l (K i ) G l (K i ) } Approximation Method Value Function Consumption Labour Supply E 1 E E 1 E E 1 E Markov Chain Markov Chain (adj.) Perturbation Perturbation Collocation Least Squares LQ (cons.) LQ (inv.) Table: Accuracy checks (%): Collard and Julliard (2001)

23 Numerical results Closed form solution (Constant Saving Function) 1 2 Log10( Relative numerical error ) Perturbation order 1 Perturbation order 2 Colocation OLS MarkovChain (M=1000) LQ: Consumption LQ: Investment K/Kss Figure: Exact numerical error for the value function (benchmark calibration)

24 Numerical results No closed form solution (Benchmark) Value function Perturbation order 1 Perturbation order 2 Colocation OLS LQ: Consumption LQ: Investment Consumption Function Labour Supply Function K/Kss Figure: Computed value and policy functions at A = A ss over [0.5K ss, 1.5K ss ]

25 Numerical results No closed form solution (Benchmark) Log10( Bellman residual + eps ) Perturbation order 1 Perturbation order 2 Colocation OLS LQ: Consumption LQ: Investment K/Kss Figure: HJB residuals at A t = A ss

26 Numerical results No Closed Form Solution (Benchmark) R l HJB ( ( ( ) N ˆV l ( )) ) K i, A j Ki, A j ρ ˆV l (K ss, A ss ) Method (l) Ẽ 1 Ẽ Markov Chain Perturbation Perturbation Collocation Least Squares LQ (cons.) LQ (inv.) Table: HJB residuals for benchmark calibration

27 Numerical results Robustness check Case Pert. 1 Pert. 2 Coll. LS LQ (cons.) LQ (inv.) M M M M M M M M M Extreme Extreme Table: Maximum HJB residuals ( ) Ẽ for alternative calibrations

28 Numerical results Computing time Method Closed No closed Arouba et al. (2006) form form (discrete-time)( ) Markov Chain 2.455E E+5 Perturbation E E-2 Perturbation E E-1 Collocation 5.928E E+0 Least Squares 1.078E E+2 LQ (cons.) 2.448E E-1 LQ (inv.) 1.613E E+0 [2.00E+0, 1.00E+1] [2.00E+1, 1.80E+2] Table: Computing time for alternative approximation methods using benchmark calibration (in seconds) ( ) These results come from the working paper version. The published paper does not report the computing time.

29 Summary of main findings All methods provide an acceptable degree of accuracy around the deterministic steady state for reasonable values of the risk aversion parameter and different levels of the volatility. For high levels of risk aversion, local methods heavily deteriorate in terms of the level of the unkown functions. Linear-quadratic policy function approximations are less accurate than perturbation, specially when the constraints are linearized. The latter does not affect the accuracy of the approximated value function. The level of the volatility has no significant effects on the accuracy of the approximations. Projection methods are the most accurate and stable approximations. However a good initial value is required compromising its reliability in terms of computational cost. Increasing the order of approximation in the perturbation method, at a low computational cost, improves substancially the goodness of fit. Linear-quadratic approximations and perturbation methods are ideal candidates for the initial guess of global methods. Continuous-time DSGE modeling could be a promising area of policy-oriented research relative to discrete-time modeling. All the approximation methods use much less computing time! The results stimulate the use of perturbation methods, preferably with high orders of approximation, and suggest the use of projection techniques whenever high accuracy and stability is needed.