A N I NTRODUCTION TO N ONLINEAR S OLUTION AND E STIMATION T ECHNIQUES Alexander W. Richter Federal Reserve Bank of Dallas Nathaniel A. Throckmorton College of William & Mary The views expressed in this presentation are our own and do not necessarily reflect the views of the Federal Reserve Bank of Dallas or the Federal Reserve System.
T OOLBOX F UNCTIONS • script.m : assigns options to O and runs the algorithm • parameters.m : assigns model parameters to P • steadystate.m : assigns steady state values to S ( P ) • variables.m : outputs a structure, V , containing indices of variables, forecast errors, and shocks and variable titles • grids.m : assigns the discretized state space to G ( O , P ) • guess.m : assigns the initial conjectures to pf ( O , P , S , G ) • linmodel.m : outputs the linear transition matrix, T , the impact matrix, M , and a 2 -element vector of flags, eu , indicating existence and uniqueness of the linear solution • eqm.m : outputs a vector, R , containing the residuals to a subsystem of expectational equations that are constrained by all of the other equations in the equilibrium system R ICHTER AND T HROCKMORTON : A N I NTRODUCTION TO N ONLINEAR S OLUTION AND E STIMATION T ECHNIQUES
E XAMPLE : R EAL B USINESS C YCLE M ODEL A social planner chooses { c t , k t +1 } ∞ t =0 to maximize: ∞ β t c 1 − σ � t E 0 1 − σ t =0 subject to c t + k t +1 = z t k α t + (1 − δ ) k t z t = (1 − ρ )¯ z + ρz t − 1 + ε t Optimality condition: 1 = βE t [( c t /c t +1 ) σ ( αz t +1 k α − 1 t +1 + 1 − δ ) ] � �� � ≡ Φ( z t +1 ) R ICHTER AND T HROCKMORTON : A N I NTRODUCTION TO N ONLINEAR S OLUTION AND E STIMATION T ECHNIQUES
D ISCRETIZED S TATE S PACE • State variables: k t , z t • Number of grid points: N k , N z • Grid boundaries: [ k min , k max ] and [ z min , z max ] • Create evenly spaced grids: x grid = linspace ( x min , x max , N x ) , x ∈ { k, z } • State space contains N = N k × N z independent nodes • Create an array for each state variable, where every position is a unique permutation of the state space: [ k gr , z gr ] = ndgrid ( k grid , z grid ) R ICHTER AND T HROCKMORTON : A N I NTRODUCTION TO N ONLINEAR S OLUTION AND E STIMATION T ECHNIQUES
F UNCTIONAL A PPROXIMATION • True RE solution only exists in special cases (e.g., δ = 1 ) • Goal: Find an approximating function that maps the state space to the optimal decision rule for consumption: c ( k, z ) ≈ P c ( k, z ) � �� � � �� � True RE Solution Approximating Function • Basic elements of the algorithm: 1. Interpolation: Linear, Least squares 2. Integration: Gauss-Hermite, Trapezoid, Rouwenhorst 3. Iteration: Time, Fixed-point R ICHTER AND T HROCKMORTON : A N I NTRODUCTION TO N ONLINEAR S OLUTION AND E STIMATION T ECHNIQUES
I NITIAL C ONJECTURE Use the linear solution as a guess for P c ( k, z ) : • Linear solution from gensys.m takes the form: Y ′ = T ˆ ˆ Y + Mε where ˆ Y = [ˆ c ] T , ˆ x , and ε ∼ N (0 , σ 2 ) . k, ˆ z, ˆ x ≡ ( x t − ¯ x ) / ¯ • Convert the state space to deviations from steady state • Compute an initial conjecture for all nodes ( i = 1 , . . . , N ): [vec(ˆ ˆ z gr )] T P c = T ( c idx , [ k idx , z idx ]) k gr ) , vec(ˆ � �� � � �� � 1 × 2 2 × N • Convert ˆ c (1 + ˆ P c ) ) and assign to pf.c P c to levels ( P c = ¯ R ICHTER AND T HROCKMORTON : A N I NTRODUCTION TO N ONLINEAR S OLUTION AND E STIMATION T ECHNIQUES
L OCAL A PPROXIMATION • Piecewise Linear Interpolation: 2 state variables ( k, z ) • Goal: Find the policy function value P c ( k ′ , z ′ ) • We have policy function values on nearest nodes [ P c ( k i , z j ) , P c ( k i , z j +1 ) , P c ( k i +1 , z j ) , P c ( k i +1 , z j +1 )] once we determine the grid indices, i, j • Locate the grid point to left of x ′ , x ∈ { k, z } dist = x ′ − x 1 step = x 2 − x 1 , loc = min( N x − 1 , max(1 , floor( dist / step ) + 1)) dist loc c loc a loc b ′ step ′ ′ x c x 1 x 2 x 3 x 4 x 5 x b x a R ICHTER AND T HROCKMORTON : A N I NTRODUCTION TO N ONLINEAR S OLUTION AND E STIMATION T ECHNIQUES
L OCAL A PPROXIMATION • Interpolate in the k direction: P c ( k ′ , z j ) = P c ( k i , z j ) + ( k ′ − k i ) P c ( k i +1 , z j ) − P c ( k i , z j ) k i +1 − k i k ′ − k i = k i +1 − k ′ P c ( k i , z j ) + P c ( k i +1 , z j ) k i +1 − k i k i +1 − k i � �� � � �� � ω ki ω ki +1 • Then interpolate in the z direction: z ′ − z j P c ( k ′ , z ′ ) = z j +1 − z ′ P c ( k ′ , z j ) + P c ( k ′ , z j +1 ) z j +1 − z j z j +1 − z j � �� � � �� � ω zj ω zj +1 • Combine these two equations: 1 1 � � P c ( k ′ , z ′ ) = ω k i + a ω z j + b P c ( k i + a , z j + b ) a =0 b =0 R ICHTER AND T HROCKMORTON : A N I NTRODUCTION TO N ONLINEAR S OLUTION AND E STIMATION T ECHNIQUES
L OCAL A PPROXIMATION • Use a nested loop or write out all of the terms in the sum to calculate the interpolated value of the policy function: nestedsum = 0; %initialize for a = 0:1 %loop for k for b = 0:1 %loop for z nestedsum = nestedsum + ... wk(1+a)*wz(1+b)*pf.c(kloc+a,zloc+b); end end • Must calculate the interpolated value for each realization of the stochastic variable(s), each of which requires calculating a different set of locations and weights • Number of loops equals the number of exogenous states R ICHTER AND T HROCKMORTON : A N I NTRODUCTION TO N ONLINEAR S OLUTION AND E STIMATION T ECHNIQUES
G LOBAL A PPROXIMATION • A general class of polynomials can be written as: n � P ( x ; η ) = η i ϕ i ( x ) . i =0 • Linear interpolation is a special case of this general class (i.e., n = 1 , ϕ i ( x ) = x i , and α is chosen appropriately) • For n > 1 , ϕ i ( x ) = x i is a collection of monomials and P ( x ; η ) = η 0 + η 1 x + η 2 x 2 + · · · + η p x p • This set of monomials may lead to multicollinearity (i.e., near linear dependence among the monomials) • Bases consisting of orthogonal polynomials fix this problem (e.g., Chebyshev and Hermite Polynomials) R ICHTER AND T HROCKMORTON : A N I NTRODUCTION TO N ONLINEAR S OLUTION AND E STIMATION T ECHNIQUES
E XAMPLE : M ONOMIALS • Consider the complete set of basis functions of order 2: P ( k, z ) = η 0 + η k k + η z z + η kk k 2 + η kz kz + η zz z 2 • Regressor matrix (subscripts denote grid indices): k 2 z 2 1 k 1 z 1 k 1 z 1 1 1 k 2 z 2 1 k 2 z 2 k 2 z 2 2 2 X = . . . . . . . . . . . . . . . . . . k 2 z 2 1 k N z N k N z N N N • Obtain coefficients using OLS: η = ( X T X ) − 1 X T vec( P c ( k, z )) ˆ P c ( k ′ , z ′ ) = X ′ ˆ η R ICHTER AND T HROCKMORTON : A N I NTRODUCTION TO N ONLINEAR S OLUTION AND E STIMATION T ECHNIQUES
I NTEGRATION : T RAPEZOID R ULE E [Φ( z )] ≈ Pr( ε 1 )Φ( z ( ε 1 )) + Pr( ε 2 )Φ( z ( ε 2 )) ∆ ε 2 + Pr( ε 2 )Φ( z ( ε 2 )) + Pr( ε 3 )Φ( z ( ε 3 )) ∆ ε + · · · 2 + Pr( ε m − 1 )Φ( z ( ε m − 1 )) + Pr( ε m )Φ( z ( ε m )) ∆ ε 2 � � m = ∆ ε � 2 Pr( ε i )Φ( z ( ε i )) − Pr( ε 1 )Φ( z ( ε 1 )) − Pr( ε m )Φ( z ( ε m )) 2 i =1 R ICHTER AND T HROCKMORTON : A N I NTRODUCTION TO N ONLINEAR S OLUTION AND E STIMATION T ECHNIQUES
I NTEGRATION : G AUSS -H ERMITE • Given a shock, ε ∼ N ( µ, σ 2 ) , � ∞ Φ( z ( ε )) e − ( ε − µ ) 2 / (2 σ 2 ) dε E [Φ( z ( ε ))] = (2 πσ 2 ) − 1 / 2 −∞ √ • Apply change of variables, υ = ( ε − µ ) / ( 2 σ ) , � ∞ √ 2 συ + µ )) e − υ 2 dυ E [Φ( z ( υ ))] = π − 1 / 2 Φ( z ( −∞ n √ � ≈ π − 1 / 2 ω i Φ( z ( 2 συ i + µ )) i =1 • ω i and υ i are Gauss-Hermite weights and nodes: ω i = 2 n +1 n ! √ π [ H n +1 ( υ i )] − 2 H n +1 is the physicists’ Hermite polynomial of order n + 1 . R ICHTER AND T HROCKMORTON : A N I NTRODUCTION TO N ONLINEAR S OLUTION AND E STIMATION T ECHNIQUES
E XAMPLE : T IME I TERATION On iteration q , solve for the P q c ( k, z ) that satisfies equilibrium 1. Use log-linear solution on each node to obtain P 0 c ◮ Local: P 1 c = P 0 c η 0 = ( X T X ) − 1 X T vec( P 0 ◮ Global: ˆ c ) so P 1 η 0 c = X ˆ 2. Solve for k ′ and z ′ , given ε ′ 3. Find P q c ( k ′ , z ′ ) given the updated state ◮ Local: use piecewise linear interpolation ◮ Global: update the basis so P q η q − 1 c ( k ′ , z ′ ) = X ′ ˆ 4. Evaluate expectations (Trapezoid rule or Gauss Hermite) c ( k ′ , z ′ ) − σ ( αz ′ k ′ α − 1 + 1 − δ )] E [Φ( z ′ )] = βE [ P q R ICHTER AND T HROCKMORTON : A N I NTRODUCTION TO N ONLINEAR S OLUTION AND E STIMATION T ECHNIQUES
E XAMPLE : T IME I TERATION 5. Use nonlinear solver to find a P q c ( k, z ) that satisfies the c ( k, z ) − σ = E [Φ( z ′ )] . consumption Euler equation, P q 6. Update policy function ◮ Local: P q +1 = P q c c η q = ( X T X ) − 1 X T vec( P q c ( k, z )) , P q +1 ◮ Global: ˆ η q ( k, z ) = X ˆ c 7. Calculate distance between updates ◮ Local: dist = P q c ( k, z ) − P q − 1 ( k, z ) c ◮ Global: dist = ˆ η q − ˆ η q − 1 8. If | dist | < tol , then stop. If not, then set q = q + 1 and repeat steps 2-7 using P q +1 as the new initial conjecture. c Advantage: Satisfies the equilibrium system on each node and nodes can be run in parallel. Disadvantage: Nonlinear solver must execute on each node. R ICHTER AND T HROCKMORTON : A N I NTRODUCTION TO N ONLINEAR S OLUTION AND E STIMATION T ECHNIQUES
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