Source code for quantecon.dle

"""
Provides a class called DLE to convert and solve dynamic linear economics 
(as set out in Hansen & Sargent (2013)) as LQ problems.
"""

import numpy as np
from .lqcontrol import LQ
from .matrix_eqn import solve_discrete_lyapunov
from .rank_nullspace import nullspace

[docs]class DLE(object): r""" This class is for analyzing dynamic linear economies, as set out in Hansen & Sargent (2013). The planner's problem is to choose \{c_t, s_t, i_t, h_t, k_t, g_t\}_{t=0}^\infty to maximize \max -(1/2) \mathbb{E} \sum_{t=0}^{\infty} \beta^t [(s_t - b_t).(s_t-b_t) + g_t.g_t] subject to the linear constraints \Phi_c c_t + \Phi_g g_t + \Phi_i i_t = \Gamma k_{t-1} + d_t k_t = \Delta_k k_{t-1} + \Theta_k i_t h_t = \Delta_h h_{t-1} + \Theta_h c_t s_t = \Lambda h_{t-1} + \Pi c_t and z_{t+1} = A_{22} z_t + C_2 w_{t+1} b_t = U_b z_t d_t = U_d z_t where h_{-1}, k_{-1}, and z_0 are given as initial conditions. Section 5.5 of HS2013 describes how to map these matrices into those of a LQ problem. HS2013 sort the matrices defining the problem into three groups: Information: A_{22}, C_2, U_b , and U_d characterize the motion of information sets and of taste and technology shocks Technology: \Phi_c, \Phi_g, \Phi_i, \Gamma, \Delta_k, and \Theta_k determine the technology for producing consumption goods Preferences: \Delta_h, \Theta_h, \Lambda, and \Pi determine the technology for producing consumption services from consumer goods. A scalar discount factor \beta determines the preference ordering over consumption services. Parameters ---------- Information : tuple Information is a tuple containing the matrices A_{22}, C_2, U_b, and U_d Technology : tuple Technology is a tuple containing the matrices \Phi_c, \Phi_g, \Phi_i, \Gamma, \Delta_k, and \Theta_k Preferences : tuple Preferences is a tuple containing the matrices \Delta_h, \Theta_h, \Lambda, \Pi, and the scalar \beta """ def __init__(self, information, technology, preferences): # === Unpack the tuples which define information, technology and preferences === # self.a22, self.c2, self.ub, self.ud = information self.phic, self.phig, self.phii, self.gamma, self.deltak, self.thetak = technology self.beta, self.llambda, self.pih, self.deltah, self.thetah = preferences # === Computation of the dimension of the structural parameter matrices === # self.nb, self.nh = self.llambda.shape self.nd, self.nc = self.phic.shape self.nz, self.nw = self.c2.shape junk, self.ng = self.phig.shape self.nk, self.ni = self.thetak.shape # === Creation of various useful matrices === # uc = np.hstack((np.eye(self.nc), np.zeros((self.nc, self.ng)))) ug = np.hstack((np.zeros((self.ng, self.nc)), np.eye(self.ng))) phiin = np.linalg.inv(np.hstack((self.phic, self.phig))) phiinc = uc.dot(phiin) phiing = ug.dot(phiin) b11 = - self.thetah.dot(phiinc).dot(self.phii) a1 = self.thetah.dot(phiinc).dot(self.gamma) a12 = np.vstack((self.thetah.dot(phiinc).dot( self.ud), np.zeros((self.nk, self.nz)))) # === Creation of the A Matrix for the state transition of the LQ problem === # a11 = np.vstack((np.hstack((self.deltah, a1)), np.hstack( (np.zeros((self.nk, self.nh)), self.deltak)))) self.A = np.vstack((np.hstack((a11, a12)), np.hstack( (np.zeros((self.nz, self.nk + self.nh)), self.a22)))) # === Creation of the B Matrix for the state transition of the LQ problem === # b1 = np.vstack((b11, self.thetak)) self.B = np.vstack((b1, np.zeros((self.nz, self.ni)))) # === Creation of the C Matrix for the state transition of the LQ problem === # self.C = np.vstack((np.zeros((self.nk + self.nh, self.nw)), self.c2)) # === Define R,W and Q for the payoff function of the LQ problem === # self.H = np.hstack((self.llambda, self.pih.dot(uc).dot(phiin).dot(self.gamma), self.pih.dot( uc).dot(phiin).dot(self.ud) - self.ub, -self.pih.dot(uc).dot(phiin).dot(self.phii))) self.G = ug.dot(phiin).dot( np.hstack((np.zeros((self.nd, self.nh)), self.gamma, self.ud, -self.phii))) self.S = (self.G.T.dot(self.G) + self.H.T.dot(self.H)) / 2 self.nx = self.nh + self.nk + self.nz self.n = self.ni + self.nh + self.nk + self.nz self.R = self.S[0:self.nx, 0:self.nx] self.W = self.S[self.nx:self.n, 0:self.nx] self.Q = self.S[self.nx:self.n, self.nx:self.n] # === Use quantecon's LQ code to solve our LQ problem === # lq = LQ(self.Q, self.R, self.A, self.B, self.C, N=self.W, beta=self.beta) self.P, self.F, self.d = lq.stationary_values() # === Construct output matrices for our economy using the solution to the LQ problem === # self.A0 = self.A - self.B.dot(self.F) self.Sh = self.A0[0:self.nh, 0:self.nx] self.Sk = self.A0[self.nh:self.nh + self.nk, 0:self.nx] self.Sk1 = np.hstack((np.zeros((self.nk, self.nh)), np.eye( self.nk), np.zeros((self.nk, self.nz)))) self.Si = -self.F self.Sd = np.hstack((np.zeros((self.nd, self.nh + self.nk)), self.ud)) self.Sb = np.hstack((np.zeros((self.nb, self.nh + self.nk)), self.ub)) self.Sc = uc.dot(phiin).dot(-self.phii.dot(self.Si) + self.gamma.dot(self.Sk1) + self.Sd) self.Sg = ug.dot(phiin).dot(-self.phii.dot(self.Si) + self.gamma.dot(self.Sk1) + self.Sd) self.Ss = self.llambda.dot(np.hstack((np.eye(self.nh), np.zeros( (self.nh, self.nk + self.nz))))) + self.pih.dot(self.Sc) # === Calculate eigenvalues of A0 === # self.A110 = self.A0[0:self.nh + self.nk, 0:self.nh + self.nk] self.endo = np.linalg.eigvals(self.A110) self.exo = np.linalg.eigvals(self.a22) # === Construct matrices for Lagrange Multipliers === # self.Mk = -2 * np.asscalar(self.beta) * (np.hstack((np.zeros((self.nk, self.nh)), np.eye( self.nk), np.zeros((self.nk, self.nz))))).dot(self.P).dot(self.A0) self.Mh = -2 * np.asscalar(self.beta) * (np.hstack((np.eye(self.nh), np.zeros( (self.nh, self.nk)), np.zeros((self.nh, self.nz))))).dot(self.P).dot(self.A0) self.Ms = -(self.Sb - self.Ss) self.Md = -(np.linalg.inv(np.vstack((self.phic.T, self.phig.T))).dot( np.vstack((self.thetah.T.dot(self.Mh) + self.pih.T.dot(self.Ms), -self.Sg)))) self.Mc = -(self.thetah.T.dot(self.Mh) + self.pih.T.dot(self.Ms)) self.Mi = -(self.thetak.T.dot(self.Mk))
[docs] def compute_steadystate(self, nnc=2): """ Computes the non-stochastic steady-state of the economy. Parameters ---------- nnc : array_like(float) nnc is the location of the constant in the state vector x_t """ zx = np.eye(self.A0.shape[0])-self.A0 self.zz = nullspace(zx) self.zz /= self.zz[nnc] self.css = self.Sc.dot(self.zz) self.sss = self.Ss.dot(self.zz) self.iss = self.Si.dot(self.zz) self.dss = self.Sd.dot(self.zz) self.bss = self.Sb.dot(self.zz) self.kss = self.Sk.dot(self.zz) self.hss = self.Sh.dot(self.zz)
[docs] def compute_sequence(self, x0, ts_length=None, Pay=None): """ Simulate quantities and prices for the economy Parameters ---------- x0 : array_like(float) The initial state ts_length : scalar(int) Length of the simulation Pay : array_like(float) Vector to price an asset whose payout is Pay*xt """ lq = LQ(self.Q, self.R, self.A, self.B, self.C, N=self.W, beta=self.beta) xp, up, wp = lq.compute_sequence(x0, ts_length) self.h = self.Sh.dot(xp) self.k = self.Sk.dot(xp) self.i = self.Si.dot(xp) self.b = self.Sb.dot(xp) self.d = self.Sd.dot(xp) self.c = self.Sc.dot(xp) self.g = self.Sg.dot(xp) self.s = self.Ss.dot(xp) # === Value of J-period risk-free bonds === # # === See p.145: Equation (7.11.2) === # e1 = np.zeros((1, self.nc)) e1[0, 0] = 1 self.R1_Price = np.empty((ts_length + 1, 1)) self.R2_Price = np.empty((ts_length + 1, 1)) self.R5_Price = np.empty((ts_length + 1, 1)) for i in range(ts_length + 1): self.R1_Price[i, 0] = self.beta * e1.dot(self.Mc).dot(np.linalg.matrix_power( self.A0, 1)).dot(xp[:, i]) / e1.dot(self.Mc).dot(xp[:, i]) self.R2_Price[i, 0] = self.beta**2 * e1.dot(self.Mc).dot( np.linalg.matrix_power(self.A0, 2)).dot(xp[:, i]) / e1.dot(self.Mc).dot(xp[:, i]) self.R5_Price[i, 0] = self.beta**5 * e1.dot(self.Mc).dot( np.linalg.matrix_power(self.A0, 5)).dot(xp[:, i]) / e1.dot(self.Mc).dot(xp[:, i]) # === Gross rates of return on 1-period risk-free bonds === # self.R1_Gross = 1 / self.R1_Price # === Net rates of return on J-period risk-free bonds === # # === See p.148: log of gross rate of return, divided by j === # self.R1_Net = np.log(1 / self.R1_Price) / 1 self.R2_Net = np.log(1 / self.R2_Price) / 2 self.R5_Net = np.log(1 / self.R5_Price) / 5 # === Value of asset whose payout vector is Pay*xt === # # See p.145: Equation (7.11.1) if isinstance(Pay, np.ndarray) == True: self.Za = Pay.T.dot(self.Mc) self.Q = solve_discrete_lyapunov( self.A0.T * self.beta**0.5, self.Za) self.q = self.beta / (1 - self.beta) * \ np.trace(self.C.T.dot(self.Q).dot(self.C)) self.Pay_Price = np.empty((ts_length + 1, 1)) self.Pay_Gross = np.empty((ts_length + 1, 1)) self.Pay_Gross[0, 0] = np.nan for i in range(ts_length + 1): self.Pay_Price[i, 0] = (xp[:, i].T.dot(self.Q).dot( xp[:, i]) + self.q) / e1.dot(self.Mc).dot(xp[:, i]) for i in range(ts_length): self.Pay_Gross[i + 1, 0] = self.Pay_Price[i + 1, 0] / (self.Pay_Price[i, 0] - Pay.dot(xp[:, i])) return
[docs] def irf(self, ts_length=100, shock=None): """ Create Impulse Response Functions Parameters ---------- ts_length : scalar(int) Number of periods to calculate IRF Shock : array_like(float) Vector of shocks to calculate IRF to. Default is first element of w """ if type(shock) != np.ndarray: # Default is to select first element of w shock = np.vstack((np.ones((1, 1)), np.zeros((self.nw - 1, 1)))) self.c_irf = np.empty((ts_length, self.nc)) self.s_irf = np.empty((ts_length, self.nb)) self.i_irf = np.empty((ts_length, self.ni)) self.k_irf = np.empty((ts_length, self.nk)) self.h_irf = np.empty((ts_length, self.nh)) self.g_irf = np.empty((ts_length, self.ng)) self.d_irf = np.empty((ts_length, self.nd)) self.b_irf = np.empty((ts_length, self.nb)) for i in range(ts_length): self.c_irf[i, :] = self.Sc.dot( np.linalg.matrix_power(self.A0, i)).dot(self.C).dot(shock).T self.s_irf[i, :] = self.Ss.dot( np.linalg.matrix_power(self.A0, i)).dot(self.C).dot(shock).T self.i_irf[i, :] = self.Si.dot( np.linalg.matrix_power(self.A0, i)).dot(self.C).dot(shock).T self.k_irf[i, :] = self.Sk.dot( np.linalg.matrix_power(self.A0, i)).dot(self.C).dot(shock).T self.h_irf[i, :] = self.Sh.dot( np.linalg.matrix_power(self.A0, i)).dot(self.C).dot(shock).T self.g_irf[i, :] = self.Sg.dot( np.linalg.matrix_power(self.A0, i)).dot(self.C).dot(shock).T self.d_irf[i, :] = self.Sd.dot( np.linalg.matrix_power(self.A0, i)).dot(self.C).dot(shock).T self.b_irf[i, :] = self.Sb.dot( np.linalg.matrix_power(self.A0, i)).dot(self.C).dot(shock).T return
[docs] def canonical(self): """ Compute canonical preference representation Uses auxiliary problem of 9.4.2, with the preference shock process reintroduced Calculates pihat, llambdahat and ubhat for the equivalent canonical household technology """ Ac1 = np.hstack((self.deltah, np.zeros((self.nh, self.nz)))) Ac2 = np.hstack((np.zeros((self.nz, self.nh)), self.a22)) Ac = np.vstack((Ac1, Ac2)) Bc = np.vstack((self.thetah, np.zeros((self.nz, self.nc)))) Cc = np.vstack((np.zeros((self.nh, self.nw)), self.c2)) Rc1 = np.hstack((self.llambda.T.dot(self.llambda), - self.llambda.T.dot(self.ub))) Rc2 = np.hstack((-self.ub.T.dot(self.llambda), self.ub.T.dot(self.ub))) Rc = np.vstack((Rc1, Rc2)) Qc = self.pih.T.dot(self.pih) Nc = np.hstack( (self.pih.T.dot(self.llambda), -self.pih.T.dot(self.ub))) lq_aux = LQ(Qc, Rc, Ac, Bc, N=Nc, beta=self.beta) P1, F1, d1 = lq_aux.stationary_values() self.F_b = F1[:, 0:self.nh] self.F_f = F1[:, self.nh:] self.pihat = np.linalg.cholesky(self.pih.T.dot( self.pih) + self.beta.dot(self.thetah.T).dot(P1[0:self.nh, 0:self.nh]).dot(self.thetah)).T self.llambdahat = self.pihat.dot(self.F_b) self.ubhat = - self.pihat.dot(self.F_f) return