Projection Methods A Brief Introduction Craig Burnside Duke University November 2006 Craig Burnside (Duke University) Projection Methods November2006 1/14 Projection Methods Judd (1992, JET ) describes a family of methods in widespread use in the sciences, and described in more detail in his textbook. Basic idea is to approximate the solution function (or policy function) by a member of a class of parameterized functions This makes projection methods equivalent to parameterized expectations for the asset pricing model where the integral and the solution function are the same object. Variants of the method use di¤erent criteria for choosing the approximation. Galerkin method Collocation method Craig Burnside (Duke University) Projection Methods November2006 2/14 Chebyshev polynomials Judd emphasizes the use of Chebyshev polynomials The Chebyshev polynomials are orthogonal polynomials that 2 1/2 correspond to the weighting function f (z) = (1 z ) and are de…ned on the interval [ 1, 1]. They obey the recursion T0(z) = 1 T1(z) = z Tn(z) = 2zTn 1(z) Tn 2(z) for n 2. If you work with a variable x de…ned in an interval [x, x] de…ne a change of variables z = 2(x x)/(x x) 1. Presumably select x = µ + cσx and x = µ cσx for some c > 0, where µ = E (x) and σx = var(x): z = (x µ)/ (cσx ) Craig Burnside (Duke University) Projection Methods November2006 3/14 Modi…ed-Hermite polynomials Why not use the modi…ed-Hermite polynomials as we did in discussing discrete state space methods They obey the recursion φ0(z) = 1 φ1(z) = z 1/2 1/2 φn(z) = (1/n) zφn 1(z) [(n 1)/n] φn 2(z) for n 2. 2 If you work with an variable x N(µ, σx ), de…ne a change of variables z = (x µ)/σx . Craig Burnside (Duke University) Projection Methods November2006 4/14 Approximating the Price-Dividend Ratio Chebyshev method The Euler equation for the price-dividend ratio is given by v(x) = β exp(αy)[v(y) + 1]f (y x)dy. Z j Posit an approximate solution v˜ (x) de…ned for x [x, x]: 2 N v˜ (x) = ∑ ai Ti 1 [(x µ)/ (cσx )] , i=1 Craig Burnside (Duke University) Projection Methods November2006 5/14 Approximating the Price-Dividend Ratio Modi…ed-Hermite method Posit an approximate solution v˜ (x) de…ned for x ( ∞, +∞): 2 N v˜ (x) = ∑ ai φi 1 [(x µ)/σx ] , i=1 The rest of both solution methods deals with how to pick the ai coe¢ cients in so that the approximate solution is as accurate as possible. Craig Burnside (Duke University) Projection Methods November2006 6/14 The Euler Equation Error Judd (1992) suggests forming the Euler equation error corresponding to v˜: v˜ (x) β exp(αy)[v˜ (y) + 1]f (y x)dy. Z j which can be written out as 1 1 v˜ (x) β exp(αy)[v˜ (y) + 1] exp [y µ(1 ρ) ρx]2 dy. p2 2σ2 Z πσ Since y = µ(1 ρ) + ρx + e, Judd uses the change of variables transformation z = [y µ(1 ρ) ρx]/σ with dz = dy/σ. Rewrite the Euler equation error as v˜ (x) β [exp α[µ(1 ρ) + ρx + σz] Z f g 1/2 2 v˜ [µ(1 ρ) + ρx + σz] + 1 (2π) exp( z /2) dz f g i Craig Burnside (Duke University) Projection Methods November2006 7/14 The Euler Equation Error Approximation Using Quadrature Approximate the residual by performing Gaussian quadrature: M R(x) v˜ (x) β ∑ [exp α[µ(1 ρ) + ρx + σzj ] j=1 f g v˜ [µ(1 ρ) + ρx + σzj ] + 1 wj ] . f g Chebyshev method: M R(x) v˜ (x) β ∑ [exp α[µ(1 ρ) + ρx + σzj ] j=1 f g N ∑ ai Ti 1 [ρ(x µ) + σzj ] / (cσx ) + 1 wj . (i=1 f g ) # Craig Burnside (Duke University) Projection Methods November2006 8/14 The Euler Equation Error Approximation Using Quadrature, ... Approximate the residual by performing Gaussian quadrature: M R(x) v˜ (x) β ∑ [exp α[µ(1 ρ) + ρx + σzj ] j=1 f g v˜ [µ(1 ρ) + ρx + σzj ] + 1 wj ] . f g Modi…ed-Hermite method: M R(x) v˜ (x) β ∑ [exp α[µ(1 ρ) + ρx + σzj ] j=1 f g N ∑ ai φi 1 [ρ(x µ) + σzj ] /σx + 1 wj . (i=1 f g ) # Craig Burnside (Duke University) Projection Methods November2006 9/14 Picking the as The Collocation and Galerkin Methods The true function r(x) = v(x) β exp(αy)[v(y) + 1]f (y x)dy Z j obviously has the property r(x) = 0 for all x. Collocation picks a by setting R(x) = 0 at N points Also r(x)f (x) = 0 for any other function f (x), which implies r(x)f (x)dx = 0 R Galerkin method: choose a so that R is orthogonal to N mutually orthogonal functions. Craig Burnside (Duke University) Projection Methods November2006 10/14 The Galerkin Method Using Chebyshev Polynomials Judd suggests choosing the vector of coe¢ cients a so that R(x, a)Ti 1 [(x µ)/ (cσx )] dx = 0 for i = 1 to N. Z I have made the dependence of the Euler error on a explicit. Since this integral is hard to evaluate explicitly, instead, form m ∑ R(x`, a)Ti 1 [(x` µ)/ (cσx )] = 0 for i = 1 to N `=1 The m points x` are chosen so that they correspond to the zeros of the mth ordered-Chebyshev polynomial; i.e. x` = µ + cσx z` Note that the weights for doing quadrature with the Chebyshevs are always 1 (check). Craig Burnside (Duke University) Projection Methods November2006 11/14 The Galerkin Method Using Modi…ed-Hermite Polynomials Choose the vector of coe¢ cients a so that R(x, a)φi 1 [(x µ)/σx ] dx = 0 for i = 1 to N. Z To approximate this form m ∑ R(x`, a)φi 1 [(x` µ)/σx ] wi = 0 for i = 1 to N `=1 The m points x` are chosen so that they correspond to the zeros of the mth ordered-Modi…ed-Hermite polynomial; i.e. x` = µ + σx z`. So we have m ∑ R(µ + σx z`, a)φi 1(z`)wi = 0 for i = 1 to N. `=1 Craig Burnside (Duke University) Projection Methods November2006 12/14 The Collocation Method Set R(x) to zero at N points, thus determining the vector a. Usually the points chosen would be the N zeroes of the relevant orthogonal polynomials. Chebyshev: set R(x) = 0 at x` = µ + cσx z`, ` = 1, ... , N where z` represents a root of the Nth ordered polynomial. Modi…ed-Hermite: set R(x) = 0 at x` = µ + σx z`, ` = 1, ... , N where z` represents a root of the Nth ordered polynomial. Craig Burnside (Duke University) Projection Methods November2006 13/14 Everything is Linear At …rst glance this stu¤ seems hard, but it’sactually not too hard in the asset pricing case. Notice that the equations are linear in the as. In principle, this is just a linear algebra problem. Craig Burnside (Duke University) Projection Methods November2006 14/14.