Lecture notes on model solution techniques

Transcription

1 Lecture notes on model solution techniques HKMMW Workshop Series Jun YU, MPhil Department of Economics Hong Kong University of Science and Technology April 6, 2018

2 Outline 1 Introduction 2 Perturbation methods 3 4 Numerical dynamic programming Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

3 Overview Introduction Why we need numerical methods? Many economic models cannot be solved analytically, especially DSGE models. Quantitative answers are often needed for many economic problems, For example: How much do technology shocks contribute to business cycles? What is the impact of a major tax reform on the macro-economy? Thus, numerical methods are intensively used in macroeconomic studies. Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

4 Solution to a model Introduction What is a solution to a model? Usually, a model is characterized by a system of stochastic difference or differential equations: Γ (E t z t+1, z t, v t+1 ) = 0 (1) where z t is a vector of variables (including choice variables and state variables), v t is a vector of shocks. The solution we are seeking is characterized by following two functions: policy function and law of motion of state variables. C t = g(s t ) (2) S t = h(s t 1, v t ) (3) Where C t is a vector of choice variables, and S t is a vector of state variables. Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

5 Introduction What are the widely used methods? Generally speaking, this two function are usually nonlinear, and very difficult to solve. We may need to use some methods to approximate these two functions. Three widely used methods: Perturbation methods First order approximation:loglinearizaiton and linearization Second order approximation Global basis Spectral methods Local basis Finite elements methods Numerical dynamic programming methods Value function iteration Policy function iteration Application: solving models with occasionally binding constraints Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

6 Classification methods Introduction According to different standards, we have two ways to categorize those methods: 1 Local solution methods VS Global solution methods Local approximation: approximate a function f using the data of it around a particular point x. (e.g. Perturbation methods) Global approximation: approximate a function f over the whole interval [x, x]. (e.g. other methods) 2 Linear methods VS Nonlinear methods Linear methods: approximate a nonlinear function f by a linear function ˆf. (e.g. loglinearizaiton and linearization). Nonlinear methods: approximate a nonlinear function f by a nonlinear function ˆf. Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

7 Our focus and plan Introduction Our focus and plan Our focus: practical techniques that will be directly useful, not mathematical and theoretical proofs. Our plan Go through the basic idea of those methods Use a Simple stochastic growth model as the toy model to show how to implement these methods in Matlab or Dynare. Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

8 Toy Model Introduction Let consider a very simple stochastic growth model or RBC model: The social planner s problem E 0 β t u(c t ), β (0, 1). (4) t=0 Subject to: c t + k t+1 = e zt k α t + (1 δ)k t (5) We assume a CRRA utility function: u(c t ) = c1 γ t 1 γ (6) Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

9 Toy model Introduction The technology follows AR(1) process: Euler s equation z t+1 = ρz t + ɛ t+1 ; ɛ t N(0, σ 2 ) (7) c γ t = βe t c γ { t+1 αe z t+1 kt+1 α 1 + (1 δ)} (8) Together with the resource constraint: c t + k t+1 = e zt k α t + (1 δ)k t We have a system of non-linear difference equations. Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

10 Toy model Introduction We take parameter values as given. Setting β = 0.95, δ = 0.025, α = 0.3, ρ = 0.95, γ = 2 Steady state: c + k = e z k α + (1 δ)k (9) z = ρz (10) c γ = βc γ { αe z k α 1 + (1 δ) } (11) We can solve the steady state value of each variable as a function of those variables. Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

11 Perturbation methods 1 Introduction 2 Perturbation methods 3 4 Numerical dynamic programming Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

12 Perturbation methods Perturbation methods The basic idea The basic idea of perturbation methods is to formulate a general problem, find a special case with a known solution, and then use the special case and its solution as a reference point for computing approximate solutions to the nearby general problem. the foundation The implicit function theorem The Taylor expansion theorem Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

13 Linearization Perturbation methods Given we have solve the steady state, we want to linearize these nonlinear equations around the steady state: Let ˆv t = v t v denote the deviation of v t from the steady state value (denoted by v). Plug v t = ˆv t + v into each expression, and approximate f(v t ) around the steady state v by first order Taylors expansion: f(v t ) = f(v) + f (v) ˆv t (12) For example: liearize the resource constraint equation (5): ĉ t + ˆk t+1 = k α z t + αe z k α 1ˆkt + (1 δ)ˆk t (13) Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

14 Loglinearization Perturbation methods Now, suppose we want to loglinearize these nonlinear equations around the steady state: Let ŝ t = ln(s t /s) denote the percentage deviation of s t from the steady state value (denoted by s). Plug s t = seŝt s(1 + ŝ t ) into each expression, derive the loglinearized version of thses expressions: For example: logliearize the resource constraint equation (5): ˆk t+1 = ez k α k z t + αez k α + (1 δ)k ˆk t c k k ĉt (14) Then we will have a system of linear equations. Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

15 Mixed Case Perturbation methods For some variables that may have negative value (e.g. asset holding), we can not use loglinearization. So sometimes we will have a mixed case: some varibales are loglinearized, some are linearized. But in the end, we will still have a linearized system of equations: X t = CE t X t+1 (15) Which is called linearized State transition function. State transition function: maps the state and control variable of the model today into the state tomorrow. Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

16 Policy function Perturbation methods The vector X t contains both state variables and control variables. If we denote state variables by S t, control variables by C t, then the linearized State transition function can be writen as: [ ] [ ] C t+1 C t = C (16) S t+1 S t To do Impulse response and Simulation, we need to derive the linearized policy function, which is a mapping from state variables to control variables. C t = DS t (17) And the linearized law of motion of state variables S t+1 = GS t + ɛ t+1 (18) Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

17 Perturbation methods First order approximation We use the impulse response functions to describe the responses of endogenous variables (control variables) to exogenous shocks ɛ t+1 S t+j E t S t+j = G j 1 ɛ t+1 (19) C t+j E t C t+j = DG j 1 ɛ t+1 (20) Generally speaking, these two methods can be called First order approximation method, which delivers a linear approximation to the policy function. This method is widely used in solving DSGE models because Easy to derive and code Useful for most models Do not suffer from the curse of dimensionality. problems with a large number of state variables can be handled without much computational demands. Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

18 Perturbation methods Higher order approximation There are several reasons for us to move beyond first order approximation, and study higher order approximation methods: The requirement for higher accuracy for solving the model The limitation of first order approximation Not suitable to handle questions such as welfare evaluation and risk premia in stochastic environments. Not suitable for studying uncertainty shocks, stochastic (time varying) volatility, etc. Our focus here will be second order approximation based on Schmitt-Grohe, S. and Uribe, M. (2004, JEDC). Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

19 Perturbation methods Second order approximation In general, a large set of DSGE model can be recast in the following form: E t f(y t+1, y t, x t+1, x t ) = 0 (21) x t is the vector of predetermined variables of size n x 1. y t is the vector of non-predetermined variables of size n y 1. We define n = n x + n y, then the function f maps R ny R ny R nx R nx into R n. Let x t = [ x 1 t ; x 2 t ], the vector x 1 t consists of endogenous state variables, the vector x 2 t of exogenous state variables. Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

20 General case Perturbation methods We assume x 2 t follows the exogenous stochastic process given by: x 2 t+1 = Λx 2 t + ησɛ t+1 (22) The innovation ɛ t+1 is a vector of n ɛ 1 iid shocks with mean zero and variance/covariance matrix I. The scalar σ 0 is a perturbation parameter for the standard deviations of the exogenous shocks as an argument of the policy function. Note that when σ = 0, stochastic innovation is shut down, the model becomes deterministic. η is a n ɛ n ɛ matrix of known parameters. Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

21 General case Perturbation methods One example: Recall three expression of the toy model c γ t = βe t c γ { t+1 αe z t+1 kt+1 α 1 + (1 δ)} z t+1 = ρz t + ɛ t+1 c t + k t+1 = e zt k α t + (1 δ)k t Let y t = c t and x t = [k t ; z t ]. Then E t f(y t+1, y t, x t+1, x t ) = E t y γ t βy γ [ t+1 αe x 2t+1 x α 1 1t+1 + (1 δ)] y t + x 1t+1 e x2t x α 1t + (1 δ)x 1t x 2t+1 ρx 2t (23) Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

22 General case Perturbation methods The solution to the model is of the form: y t = g(x t, σ) (24) x t+1 = h(x t, σ) + ησɛ t+1 (25) Where g maps R ny R + into R ny, h maps R nx R + into R nx. The matrix η is of size n x n ɛ and is given by: [ ] 0 η = η (26) We define the non-stochastic steady state as vectors ( x, ȳ) such that f(ȳ, ȳ, x, x) = 0 (27) Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

23 Perturbation methods Approximating the solution It is clear that ȳ = g( x, 0), and x = h( x, 0). Note that if σ = 0, E t f = f. Typically, the function f( ) and h( ) are unknown, and in general they are non-linear and do not have exact analytical forms. Therefore, we need to obtain an approximation of them. Our approximation strategy is to find a second order Taylor expansion of them around the non-stochastic steady state. Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

24 Perturbation methods Approximating the solution Approximating around the point (x, σ) = ( x, 0) we have (for the moment to keep the notation simple, lets assume that n x = n y = 1) Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

25 Perturbation methods Approximating the solution Now, let s consider the general case, n x n y 1. Then we have: Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

26 Perturbation methods Approximating the solution Where i = 1,, n y, a, b = 1,, n x and j = 1,, n x. The unknowns of this expansion are [g xx ] i ab, [g xσ] i a, [g σx] i a, [g σσ] i, [h xx ] j ab, [h xσ] j a, [h σx] j a, [h σσ] j, where we have omitted the argument ( x, 0). Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

27 Perturbation methods How to read those expression? [g(x, σ)] i is the i th row of matrix g, [h(x, σ)] j is the j th row of matrix h: g = g 1 g 2. g ny ; h = h 1 h 2. h nx [g( x, σ)] i a is the (i, a) element of matrix g x: (28) Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

28 Perturbation methods Similarly, [h( x, σ)] j a is the (j, a) element of matrix h x: The most difficult ones are [g xx ] i ab and [h xx] j ab. Let s define g xx, h xx : h xx n x n 2 x = [ ] h x,x1, h x,xj h x,xnx (29) g xx = [ ] g x,x1, g x,xj g x,xnx n y n 2 x (30) Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

29 Perturbation methods Where we have: Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

30 Perturbation methods Then [g xx ] i ab is just the (i, ab) element of matrix g xx, similar for [h xx ] j ab. As for [g σσ ] i, [h σσ ] j, they are i th and j th row of following matrix: As long as we can solve those unknowns, we can get the desired approximation of the policy function. Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

31 Perturbation methods Second order approximation Solving a second-order approximation requires the solution to the first-order approximation. And solving a third-order approximation requires the solutions to both the first and second-order approximations,etc. Question: How to solve those unknowns? The key here is the implicit function theorem. Two steps are involved: Step 1: Solve for unknowns of the first order approximation. Step 2: Solve for unknowns of the second order approximation. Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

32 How to do it? Perturbation methods Substituting the proposed solution into the general expression, we can define: F (x, σ) E t f(g(h(x, σ) + ησɛ, σ), g(x, σ), h(x, σ) + ησɛ, x) (31) Here we use a prime to indicate variables dated in period t + 1. Because F (x; σ) must be equal to zero for any possible values of x and σ, then we must have: F x k σj(x, σ) = 0, x, σ, j, k. (32) Where where F x k σj(x, σ) denotes the derivative of F with respect to x taken k times and with respect to σ taken j times. Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

33 How to do it? Perturbation methods Step 1: Solve for unknowns of the first order approximation. F x (x, σ) = 0 implies Note that the derivatives of f evaluated at (ȳ, ȳ, x, x) are known. The above expression represents a system of n n x quadratic equations in the n n x unknowns given by the elements of g x and h x. Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

34 How to do it? Perturbation methods F σ (x, σ) = 0 implies g σ and h σ are identified as the solution to the above n equations. Note that this equation is linear and homogeneous in g σ and h σ. Thus, if a unique solution exists, we have that g σ = 0; h σ = 0 (33) Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

35 How to do it? Perturbation methods Certainty equivalent property:in a first-order approximation the expected values of x t and y t are equal to their non-stochastic steady state values x and ȳ. E(x t ) = E [x + h x (x, 0)(x t x) + h σ (x, 0)(σ 0)] = x + h x (x, 0)(E(x t ) x) + 0 (34) Which implies Ex t = x; Ey t = y (35) This property restricts the range of questions that can be addressed in a meaningful way using first-order perturbation techniques.(e.g. welfare evaluations, precautionary saving, risk premium in stochastic environments, time varying volatility.) Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

36 How to do it? Perturbation methods Step 2: Solve for unknowns of the second order approximation. We use F xx ( x, 0) = 0 to identify g xx and h xx, that is: Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

37 How to do it? Perturbation methods We use F σσ ( x, 0) = 0 to identify g σσ and h σσ. And we use F σx ( x, 0) = 0 taking into account that all terms containing either g σ and h σ are zero at ( x, 0). This is a system of n n x equations in the n n x unknowns given by the elements of g σx and h σx. the system is homogeneous in the unknowns. Thus, if a unique solution exists, it is given by g σx = 0 = h σx (36) Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

38 How to do it? Perturbation methods In general, up to second-order, the coefficients of the policy function on the terms that are linear in the state vector do not depend on the size of the variance of the underlying shocks. Higher order approximation follows the same logic and method, but it will become more complicated. For those of you who are interested in studying volatility shocks or uncertainty shocks, a third order approximation are needed, following two papers can be a good reference: J. Fernndez-Villaverde, P. Guerron-Quintana, M. Uribe, Risk matters: the real effects of volatility shocks. AER, Andrew Binning. Solving second and third-order approximations to DSGE models: A recursive Sylvester equation solution. Working paper. Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

39 Perturbation methods How to implement this method in Matlab? Matlab code are available on Uribe Martin s website: mu2166/ The part you have to modify: 1 The model part: in ( model.m) file. 2 The steady state part: in ( model ss.m) file. The result will be : Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

40 Perturbation methods How to implement this method in Dynare? The latest version of Dynare can solve for up to third order approximation, the default order is 1, to change the order: stoch simul(order = 2, periods = 2100, irf = 50) The results (the coefficient matrixs) are stored in oo.dr.. How to read the result? For each matrix result, the order of columns corresponds to the declaration order of variables. The order of the rows follows the DR order : 1 static variables (only at t) 2 purely backward variables (at t and t 1) 3 mixed variables (at t, t 1 and t + 1) 4 purely forward variables (at t and t + 1) Within each category, variables are arranged according to the declaration order. For details, see Jianjun Miao (2014), pp Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

41 1 Introduction 2 Perturbation methods 3 4 Numerical dynamic programming Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

42 : are a collection of approximation techniques for constructing function d j indexed by some coefficients that approximately solves the operator H: H(d) = 0 (37) Usually, the function d j is constructed by specifying a linear combination of degree j of basis function Ψ i (x) given coefficients θ = {θ 0, θ 1,, θ j } : d j (x θ) = j θ i Ψ i (x) (38) i=0 The we define a residual function: R(x θ) H(d j (x θ)) (39) Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

43 And we select the values of the coefficients θ that minimize the residual given some metric function (or judging criterion). The projection methods are differentiated along two dimensions: 1 the functional form of basis function Ψ i (x); 2 The metric function. Several remarks: 1 For a specific model, you first need to identify H and d. 2 This function d can have multiple dimension m 2. 3 The degree j is a number you can choose for your purpose. 4 For solving DSGE model, you can think of funtion d as the policy function, vector x as the vector of state variables. 5 Nonlinear combination approximation is possible, but difficult, our focus will be linear combination. Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

44 A Basic Projection Algorithm Step 1: Define j + 1 known linearly independent functions Ψ i : Ω R where j <. We call the Ψ 0, Ψ 1,..., Ψ j the basis functions. These basis functions depend on the vector of state variables x. { } Step 2: Define a vector of coefficients θ l = θ0 l, θl 1,..., θl j for l = 1,..., m (where recall that m is the dimension that the function d of interest maps into). Stack all coefficients on a m (j + 1) matrix θ = { θ 1 ; θ 2 ;... ; θ m}. Step 3: Define a combination of the basis functions and the θs: j d l,j ( θ l ) = θiψ l i ( ) (40) for l = 1,..., m, Then: i=0 d j ( θ) = [ d 1,j ( θ 1 ); d 2,j ( θ 2 );... ; d m,j ( θ m ) ]. (41) Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

45 A Basic Projection Algorithm Step 4: Plug d j ( θ) into the operator H( ) to find the residual equation: R( θ) H(d j ( θ)) (42) Step 5: Find the value of ˆθ that makes the residual equation as close to 0 as possible given some objective function (or metric function) Γ : J 2 J 2 R: ˆθ = arg min Γ(R( θ), 0). (43) θ R m (j+1) To ease notation, We assumed that, along each dimension of d : 1 the same basis functions Ψ i 2 the same number j + 1 of basis functions. But in practice, they are usually different. Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

46 A Basic Projection Algorithm We specify a metric function Γ to gauge how close the residual function is to zero over the domain of the state variables. We plot the residual functions for a problem with two different coefficient θ: Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

47 A Basic Projection Algorithm:example Which of the two residual functions is closer to zero over the interval? Obviously, different choices of Γ will yield different answers. Let s look at an example: How to implement this basic algorithm to our toy model? For our toy model, the system is built by the Euler equation and the resource constraint of the economy: for all k t and z t. Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

48 A Basic Projection Algorithm:example We define c t = d 1 (k t, z t ) k t+1 = d 2 (k t, z t ) (44) We have already recursively substituted k t+1 in the decision rule of consumption evaluated at t + 1. Then, we can define c t = d 1,j (k t, z t θ 1 ) = j θi 1 Ψ i (k t, z t ) (45) i=1 k t+1 = d 2,j (k t, z t θ 2 ) = j θi 2 Ψ i (k t, z t ) (46) For some Ψ 0 (k t, z t ), Ψ 1 (k t, z t ),..., Ψ j (k t, z t ). We will discuss how to choose these basis functions later. i=1 Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

49 A Basic Projection Algorithm:example The next step is to write the residual function: for all k t and z t, θ = [ θ 1 ; θ 2]. Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

50 A Basic Projection Algorithm:example Instead of approximating c t = d 1 (k t, z t ) and k t+1 = d 2 (k t, z t ), we can choose another combination: the value function and the decision rule of consumption c t = C(k t, z t ). the Bellman equation of our toy model is given by: V (k t, z t ) = max c t U(c t ) + βe t V (k t+1, z t+1 ) (47) We approximate the the decision rule of consumption by: c t = C j (k t, z t θ C ) = Approximate the value function by: j θi C Ψ i (k t, z t ) (48) i=1 V (k t, z t ) = V j (k t, z t θ V ) = j θi V Ψ i (k t, z t ) (49) i=1 Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

51 A Basic Projection Algorithm:example From the resource constraint, plug in the expression of c t, we can solve k t+1 : k t+1 = K(k t, z t ) = e zt k α t + (1 δ)k t C j (k t, z t θ C ) (50) Plug K(k t, z t ) and C j (k t, z t ) into the Euler s equation: U ( C j (k t, z t ) ) = βe t ( U ( C j (K(k t, z t ), z t+1 ) ) From the Bellman equation, we have: ( αe z t+1 K(k t, z t ) α δ ) (51) V j (k t, z t ) = U ( C j (k t, z t ) ) + βe t V j (K(k t, z t ), z t+1 ) (52) Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

52 A Basic Projection Algorithm:example From above two equations, we can define the residual function: U ( C j (k t, z t ) ) ( βe t U ( C j (K(k t, z t ), z t+1 ) ) R(k t, z t ) = ( αe z t+1 K(k t, z t ) α δ ) V j (k t, z t ) U ( C j (k t, z t ) ) βe t V j (K(k t, z t ), z t+1 ) In general, we can choose to approximate a combination of value function, policy function, and law of motion of state variables (we may have multiple choice and state variables), the number of functions to approximate depend on the model setting. There problems we need to handle in following slides: 1 How to choose the basis function? i.e. the Ψ i (k t, z t ). 2 How to get rid of the expectation operator? 3 How to choose the metric function so that we can solve the coefficients of the basis function? i.e. those θ 1, θ 2, θ V. Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

53 Relation to econometrics To gain some sense about why these methods are called projection methods, we investigate its relation to econometrics. A common way to present OLS is to think about the problem of searching for the unknown conditional expectation function: E(Y X) Given that this conditional expectation is unknown, we can approximate it by E(Y X) θ 0 + θ 1 X (53) To derive the coefficients θ, we minimize the square of the residual (the metric function) by plugging in the observed series {Y, X} t=1:t. R(Y, X θ 0, θ 1 ) 2 = (Y θ 0 θ 1 X) 2 (54) Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

54 Relation to econometrics The underlying idea is very similar: In OLS, we are projecting the conditional expectation function against the linear function of explaining variables. In projection method, we are projecting the unknown function d against some basis functions Ψ i. They all try to select the value of coefficients by minimizing some metric function. Basicly, you can think of OLS as a special case of projection methods. the differences between OLS and the projection algorithm is: In OLS, we use the observed data of {Y, X}. In projection method, we use the operator H(d) imposed by economic theory. Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

55 How to choose basis functions Generally, the basis functions (i.e.ψ 0, Ψ 1,, Ψ 2 ) can be divided into two groups: 1 Global basis: basis functions that are nonzero and smooth for most of the domain of the state variable Ω. 2 Local basis: basis functions that are zero for most of the domain of the state variable, and nonzero and smooth for only a small portion of the domain Ω. Within each group, there many kinds of basis function to choose. with a global basis are often known as spectral methods. with a local basis are also known as finite elements methods. The methods to approximate a function in projection is called Interpolation. Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

56 Spectral Bases: Uni-dimensional case We will start by introducing some of the most common spectral bases. We first deal with the uni-dimensional case where there is only one state variable. Monomials basis: 1, x, x 2,..., x j The StoneWeierstrass theorem insures that we can uniformly approximate any continuous function defined on a closed interval with linear combinations of these monomials. Drawback 1: monomials are (nearly) multicollinear. Drawback 2: monomials vary considerably in size, leading to scaling problems and the accumulation of numerical errors. Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

57 Spectral Bases: Uni-dimensional case Before we go to orthogonal polynomials, we define orthogonal. Definition Given the weighting function w(x) on [a, b] which is positive and has a finite integral over [a, b], define the inner product for integrable function over [a, b] by: f, g w = b a f(x)g(x)w(x)dx (55) The family of functions {Ψ i } on [a, b] is mutually orthogonal with respect to w(x) if Ψ i, Ψ j w = 0, i j. (56) The inner product induces a norm f w = Ψ i, Ψ j w and hence a natural metric. Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

58 Spectral Bases: Uni-dimensional case Orthogonal Polynomials: 4 widely used ones 1 Chebyshev polynomials The n th order Chebyshev polynomials on [ 1, 1] is defined recursively: T 0 (x) = 1, T 1 (x) = x, T 2 (x) = 2x 2 1, T n+1 (x) = 2xT n (x) T n 1 (x) (57) One can also equivalently define it as: T n (x) = cos(n arccos(x)) (58) They are orthogonal to each other with respect to the weighting function w(x) = (1 x 2 ) 1/2. Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

59 Chebyshev polynomials The Chebyshev polynomials of order n has n zeros, given by: ( ) 2k 1 x k = cos 2n π, k = 1,..., n. (59) This property will be useful when we discuss how to choose the metric function. Since the domain of a state variable x in a DSGE model would be, in general, different from [ 1, 1], we can use a linear translation from [a, b] into [ 1, 1]: 2 x a b a 1 (60) Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

60 Chebyshev polynomials Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

61 Chebyshev polynomials Chebyshev polynomials will generate very nice results for approxiamtion. The Chebyshev interpolant (d j = j i=0 θ iψ i (x)) coverges to d rapidly as we increase the degree of interpolation (j), this result is guaranteed by the Chebyshev interpolation theorem. Theorem (Chebyshev interpolation theorem) If d(x) [a, b], if {Ψ i (x), i = 0,...} is a system of polynomials (where Ψ i (x) is of exact degree i) orthogonal to with respect to w(x) on [a, b] and if d j = j i=0 θ iψ i (x) interpolates f(x) in the zeros of Ψ n+1 (x), then: ( lim d d j ) 2 = lim j n b a w(x) ( d(x) d j) 2 dx = 0 (61) Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

62 Chebyshev polynomials j is impossible in practice, but the the error we are accepting by truncating the approximation of d( ) after a finite (and often relatively low) j will be bounded, following the Chebyshev truncation theorem. Theorem (Chebyshev truncation theorem) The error in approximating d is bounded by the sum of the absolute values of all the neglected coefficients. In other words, if we have d j ( θ) = j θ i Ψ i ( ) (62) i=0 for any x [ 1, 1] and any j. d(x) d j ( θ) i=j+1 θ i (63) Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

63 2 Legendre polynomials The n th Legendre polynomials is given by: P n (x) = ( 1)n x n! d n dx n [ (1 x 2 ) n], P 0 (x) = 1, x [ 1, 1] (64) The corresponding weighting function is w(x) = 1 on [ 1, 1]. Other interval [a, b] can be transformed in to [ 1, 1] by z = 2 x a b a 1. 3 Laguerre polynomials The n th Laguerre polynomials is given by: L n (x) = en d n ( x n n! dx n e x), L 0 (x) = 1, x [0, ) (65) The corresponding weighting function is w(x) = ne x on [0, ). Laguerre polynomials are used to approximate functions of time in deterministic models. Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

64 Spectral Bases: Uni-dimensional case 4 Hermite polynomials The n th Hermite polynomials is given by: H n (x) = ( 1) n e x2 dn ( dx n e x2), H 0 (x) = 1, x (, + ) (66) The corresponding weighting function is w(x) = e x2 on (, + ). Hermite polynomials are often used to approximate functions of normal random variables. Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

65 Spectral Bases: Uni-dimensional case Decades of real-life applications have repeatedly shown how well Chebyshev polynomials work in a wide variety of applications. Boyd (2000), only half-jokingly, has summarized these decades of experience in what he has named his Moral Principle 1: 1 When in doubt, use Chebyshev polynomials unless the solution is spatially periodic, in which case an ordinary Fourier series is better. 2 Unless you are sure another set of basis functions is better, use Chebyshev polynomials. 3 Unless you are really, really sure another set of basis functions is better, use Chebyshev polynomials. Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

66 Spectral Bases: multidimensional case Imagine that we want to approximate a function of n state variables, d : [ 1, 1] n R with Chebyshev polynomials of degree j. d j ( θ) = j i 1 =0 j θ i1,...,i n Ψ 1 i 1 ( ) Ψ n i n ( ) (67) i n=0 Where Ψ κ i κ is the Chebyshev polynomials of degree i κ on the state variable κ and θ is the vector of coefficients θ i1,...,i n. The curse of dimensionality: as the dimension increase, the computation becomes more complicated rapidly. For example: with 5 state variables and three Chebyshev polynomials, we end up with 243 coefficients. But with 10 Chebyshev polynomials, we end up with 100,000 coefficients. Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

67 Local bases: Finite element methods Let s consider the local basis, projection methods with a local basis are known as finite elements methods. Chebyshev polynomials interpolation works well for well-behaved smooth functions. But for functions that have kinks or regions of high curvatures, it is better to use finite elements methods. Similar local behavior appears when we deal with occasionally binding constraints, kinks, or singularities. The main advantage of this class of basis functions is they can easily capture local behavior and achieve a tremendous level of accuracy even in the most challenging problems. Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

68 Local bases: Finite element methods The main disadvantage of finite elements methods is that they are hard to code and expensive to compute. For example, the decision rule for capital may look like Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

69 Local bases: Finite element methods Finite element methods usually have following four steps: 1 Step 1: Bound the domain Ω of the state variables. For Uni-dimensional case, we need to find the upper and lower bound of k t, k k t k, some care are needed in picking them. For multidimensional case, the domain may be a plane or a space. Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

70 Local bases: Finite element methods 2 Step 2: partition Ω into small, non-intersecting elements. These small sections are called elements (hence the name, finite elements). The boundaries of the elements are called nodes. For Uni-dimensional case, we can divided the bounded domain [ k, k ] into j + 1 segments with nodes given by {k0, k 1,, k j } 3 Step 3: Choose a basis for the function to be approximated (usually the policy functions) in each elements. To show the basic idea of finite element methods, We use the tent functions for i {1, j 1} to be the basis functions: k k i 1 k i k i 1, ifk [k i 1, k i ], k Ψ i (k) = i+1 k k i+1 k i, ifk [k i, k i+1 ], (68) 0, elsewhere. Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

71 Local bases: Finite element methods Together with the corresponding adjustments for the first and the last function: { k0 k Ψ 0 (k) = k 1 k 0, ifk [k 0, k 1 ], (69) 0, elsewhere. Ψ j (k) = { k kj 1 k j k j 1, ifk [k j 1, k j ], 0, elsewhere. (70) We can plot the five basis function in following figure. Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

72 Local bases: Finite element methods 4 Step 4: Approximate the function by following equation, and then plug the equation into the operator H(d), then solve for the coefficients. j d n,j ( θ n ) = θi n Ψ i ( ) (71) i=0 One important remark: the continuity of approximation function. Within the range of [k i, k i+1 ], we have ˆd ( k k i+1, k i, θi+1, n θi n ) = θ n k i+1 k i + θ n k k i i+1 (72) k i+1 k i k i+1 k i Similarly, within the range of [k i 1, k i ]we have ˆd ( k k i, k i 1, θi n, θi 1 n ) = θ n k i k i 1 + θ n k k i 1 i (73) k i k i 1 k i k i 1 Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

73 Local bases: Finite element methods When we evaluate both linear function at k i ˆd ( k k i+1, k i, θi+1, n θi n ) ( = ˆd k ki, k i 1, θi n, θi 1 n ) = θ n i (74) Therefore, by construction, the different parts of the approximating function is pasted together to ensure continuity. How good is finite element methods? Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

74 Local bases: Finite element methods As the elements become smaller and smaller, the approximation will become even more accurate. To show the basis idea of finite element methods, here we only introduce tent function which is linear in each element, it is possible to have terms with higher orders. Due to the tremendous level of accuracy in dealing with complicated, irregular problems, FEM is widely used in engineering, e.g. nuclear power, aerospace, electrical engineering, etc. For further reading about the application of FEM in economics, see McGrattan (1996), Brenner and Scott (2008), and K.Judd (1998). Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

75 How to get rid of the expectation operator? When solving stochastic dynamic programming problems, we need to compute expected values with respect to the shocks. There two methods to deal with the expectation operator: 1 Use numerical integration method (Gaussian Quadrature) to solve the continuous problem 2 Approximate the continuous shock process by a discrete Markov chain, then solve a discrete problem. If a shock follows a discrete Markov chain, then expected values are just simple summation. The well established result is that we can discretize one continuous state variable without losing much accuracy. The best example is the discretization of exogenous stochastic processes for productivity. Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

76 AR(1) Discretization We introduce a method by Tauchen(1986) to approximate an AR(1) process by a discrete markov chain. Consider following AR(1) process: z t = ρz t 1 + ɛ t (75) Where ɛ t iidn(0, σ 2 ɛ ) and z t iidn(0, σ 2 z), and σ z = Step 1: Set n, the number of potential realizations of the process z. (usually, we set n = 5, or7.) σɛ. 1 ρ 2 Step 2: Set the upper (z) and lower (z) bounds for the process. An intuitive way to set the bounds is to pick m (usually between 2 and 3) such that: z = mσ z ; z = mσ z (76) Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

77 AR(1) Discretization Step 3: Set {z i } n i=1 such that: z i = z + z z (i 1) (77) n 1 and construct the midpoints { z i } n 1 i=1, which is given by: z i = z i+1 + z i 2 (78) Step 4: The transition probability p ij P z,z (the probability of going to state z j conditional on being on state z i ), is computed according to: ( ) z1 ρz i p i1 = Φ (79) σ Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

78 AR(1) Discretization ( ) zj ρz i p ij = Φ Φ σ ( zj 1 ρz i σ ( ) zn 1 ρz i p in = 1 Φ σ ), j = 2, 3,..., n 1 (80) Where Φ( ) denotes a CDF of standard normal distribution. How to implement this method in Matlab? Use markovappr.m in matlab, first, input (lambda,sigma,m,n). lambda=0.95, sigma=1, m=3, N=5 markovappr(lambda,sigma,m,n) (81) Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

79 AR(1) Discretization Where lambda is coefficient that measures persistence of the shock. sigma is the standard deviation of the white noise ɛ t. N is the number of states, m is determines the bounds. What we have input implies following AR(1) process: Tauchen s procedure gives us: z t = 0.95z t 1 + ɛ t, ɛ t iidn(0, 1) (82) z t [ , , 0, , ] (83) P z,z = (84) Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

80 AR(1) Discretization How will this discretization help us to solve the model? Once productivity has been discretized, we can search for c(k, z m ) = d c,m,j (k θ m,c ) = j k(k, z m ) = d k,m,j (k θ m,k ) = j i=1 θm,c i i=1 θm,k i Ψ i (k) Ψ i (k) (85) Where m = 1,..., n. we search for decision rules for capital and consumption when productivity is z 1 today, decision rules for capital and consumption when productivity is z 2 today, and so on, for a total of 2 n decision rules, usually, n is 5 or 7. Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

81 AR(1) Discretization Note that since we substitute these decision rules in the Euler equation: to get: u (c t ) = βe t [ u (c t+1 ) ( αe z t+1 k α 1 t δ)] (86) Discretization allows us to substitute the integral generated by E t for the much simpler sum operator with the probabilities from the transition matrix. Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

82 Numerical integration method: Gaussian Quadrature The idea of the quadrature method is to approximate the integral by summation: b n f(x)dx w i f(x i ) (87) a where x i s are quadrature nodes, and w i s are quadrature weights. The Gaussian quadrature method is constructed with respect to a specific weight function w. For a given order of approximation, this method chooses the nodes x 1, x 2,, x n and weights w 1, w 2,, w n so as to satisfy the following 2n conditions: b n x k w(x)dx w i x k i, k = 0, 1,, 2n 1. (88) a i=1 Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99 i=1

83 Numerical integration method: Gaussian Quadrature The integral is approximately computed by forming prescribed weighted sum of function values at the prescribed nodes: b a f(x)w(x)dx n w i f(x i ). (89) i=1 For most known probability distribution functions, such as the uniform, normal, lognormal, exponential, gamma and bata distributions, there are well studied method to compute the results. Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

84 Numerical integration method: Gaussian Quadrature And there are some widely used quadrature, for example: Gauss-Hermite Quadrature which computes + f(x)e x2 dx n w i f(x i ). (90) This quadrature is very useful to compute integrals with respect to normal distributions: n Ef(Y ) = π 1/2 w i f( 2σx i + µ) (91) i=1 i=1 Where Y is normal with mean µ and variance σ 2. For further reading about Gaussian Quadrature methods, see Jianjun Miao (2014), Chapter 11, Miranda and Fackler (2002), Chapter 5. Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

85 How to choose Metric functions? Our second choice is to select a metric function Γ (R( θ), 0) to determine how we project. The most common answer to this question is given by a weighted residual: we select θ to get the residual close to 0 in the weighted integral sense. Consider a uni-dimensional case: Given some weight functions φ i : Ω R,we define the metric function: { 0, if Γ(R( θ), 0) = Ω φ i(x)r( θ)dx = 0, i = 1,..., j + 1 (92) 1, otherwise Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

86 How to choose Metric functions? Hence, the problem is to choose the θ that solves the system of integral equations: φ i (x)r( θ)dx = 0, i = 1,..., j + 1. (93) Ω Note that, for the system to have a solution, we need j + 1 weight functions. Given our choice of basis function Ψ i, the system above is in fact a standard nonlinear equations system of unknown θ. The solution of this system can be found using standard methods, such as a Newton algorithm for small problems or a Levenberg -Marquardt method for bigger ones. Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

87 How to choose Metric functions? Now the choice of metric functions becomes the choice of weight functions: There are four popular ones in economics: 1 Weight funciton I: Collocation This method is also known as pseudospectral or the method of selected points. { 1, ifx = x i, fori = 1,..., j + 1. φ i (x) = δ(x x i ) = 0, ifx x i, fori = 1,..., j + 1. (94) Where x i are the j + 1 collocation points selected by the researcher. Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

88 How to choose Metric functions? This method implies that the residual function is zero at the n collocation points. Thus, instead of having to compute complicated integrals, we only need to solve the system: R(x i θ) = 0, i = 1,..., j + 1. (95) Orthogonal collocation: A systematic way to pick collocation points is to use the zeros of the j + 1-th-order Chebyshev polynomial in each dimension of the state variable. Recall, the Chebyshev polynomials of order n has n zeros: ( ) 2k 1 x k = cos 2n π, k = 1,..., n. (96) Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

89 How to choose Metric functions? 2 Weitht function II: Least squares Least squares use as weight functions the derivatives of the residual function: φ i (x) = R(x θ) (97) θ i 1 For all i = 1,..., j This choice is motivated by the variational problem: min R 2 ( θ)dx (98) θ Ω with first order conditions: R(x θ) R( θ)dx, i = 1,..., j + 1. (99) θ i 1 Ω Least squares problems are often ill-conditioned and complicated to solve numerically. Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

90 How to choose Metric functions? 3 Weitht function III: Subdomain The subdomain approach divides the domain Ω into 1,..., j + 1 subdomain Ω i and define the j + 1 step functions: φ i (x) = { 1, ifx Ω i 0, otherwise. (100) This choice is equivalent to solving the system: R( θ)dx = 0, i = 1,..., j + 1. (101) Ω The researcher has plenty of flexibility to pick her subdomains as to satisfy her criteria of interest. Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

91 How to choose Metric functions? 4 Weitht function IV: Galerkin method This approach takes as the weight function the basis functions used in the approximation: Then we have: Ω φ i (x) = Ψ i (x) (102) Ψ i (x)r( θ)dx = 0, i = 1,..., j + 1. (103) The interpretation is that the residual has to be orthogonal to each of the basis functions. The Galerkin approach is highly accurate and robust, but difficult to code. Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

92 Concluding remarks about projection methods I can not find good Matlab code to implement projection methods online, most projection codes are written in Fortran. You will encounter many elements introduced today when you read papers, for example, Gaussian quadrature, AR(1) process discretization, Chebyshev polynomials, and so on. what will the numerical result look like? Let s look at a example borrowed from Solution and Estimation Methods for DSGE Models, Chapter 9, Handbook of Macroeconomics, 2016 Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

93 Concluding remarks about projection methods Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

94 Concluding remarks about projection methods Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

95 Numerical dynamic programming 1 Introduction 2 Perturbation methods 3 4 Numerical dynamic programming Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

96 Numerical dynamic programming Numerical dynamic programming This part is omitted! Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

97 Numerical dynamic programming Reference books Reference books: David N. DeJong and Chetan Dave, 2011, Structural Macroeconometrics, 2nd Edition, Princeton University Press, Chapter 4, 5. Jianjun Miao, 2014, Economic Dynamics in Distrete Time, MIT Press, Chapter 11. Mario J. Miranda, Paul L. Fackler, 2002, Applied Computational Economics and Finance, MIT Press. Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

98 Numerical dynamic programming Referecne papers Reference papers: J. Fernndez-Villaverde, J.F. Rubio-Ramrez,F.Schorfheide. 2016, Solution and Estimation Methods for DSGE models, Handbook of Economics, vol. 2A, Schmitt-Grohe, S., Uribe, M., Solving dynamic general equilibrium models using a second-order approximation to the policy function. J. Econ. Dyn. Control 28, McGrattan, E.R., Solving the stochastic growth model with a finite element method. J. Econ. Dyn. Control 20, Judd, K.L., for solving aggregate growth models. J. Econ. Theory 58, Lawrence J. Christiano, Jonas D.M. Fisher, 2000, Algorithms for solving dynamic models with occasionally binding constraints, Journal of Economic Dynamics Control, 24, Jun YU (HKUST) J.Yu, AER 2018 April 6, / 99

99 Thank you!