def solve_portfolio_model(model, pf_names, order=2, lambda_name='lam', guess=None):
from dolo.compiler.compiler_python import GModel
if isinstance(model, GModel):
model = model.model
pf_model = model
from dolo import Variable, Parameter, Equation
import re
n_states = len(pf_model.symbols_s['states'])
states = pf_model.symbols_s['states']
steady_states = [Parameter(v.name+'_bar') for v in pf_model.symbols_s['states']]
n_pfs = len(pf_names)
pf_vars = [Variable(v) for v in pf_names]
res_vars = [Variable('res_'+str(i)) for i in range(n_pfs)]
pf_parms = [Parameter('K_'+str(i)) for i in range(n_pfs)]
pf_dparms = [[Parameter('K_'+str(i)+'_'+str(j)) for j in range(n_states)] for i in range(n_pfs)]
from sympy import Matrix
# creation of the new model
import copy
new_model = copy.copy(pf_model)
new_model.symbols_s['controls'] += res_vars
for v in res_vars + pf_vars:
new_model.calibration_s[v] = 0
new_model.symbols_s['parameters'].extend(steady_states)
for p in pf_parms + Matrix(pf_dparms)[:]:
new_model.symbols_s['parameters'].append(p)
new_model.calibration_s[p] = 0
compregex = re.compile('(.*)<=(.*)<=(.*)')
to_be_added_1 = []
to_be_added_2 = []
expressions = Matrix(pf_parms) + Matrix(pf_dparms)*( Matrix(states) - Matrix(steady_states))
for n,eq in enumerate(new_model.equations_groups['arbitrage']):
if 'complementarity' in eq.tags:
tg = eq.tags['complementarity']
[lhs,mhs,rhs] = compregex.match(tg).groups()
mhs = new_model.eval_string(mhs)
else:
mhs = None
if mhs in pf_vars:
i = pf_vars.index(mhs)
neq = Equation(mhs, expressions[i])
neq.tag(**eq.tags)
eq_res = Equation(eq.gap, res_vars[i])
eq_res.tag(eq_type='arbitrage')
to_be_added_2.append(eq_res)
new_model.equations_groups['arbitrage'][n] = neq
to_be_added_1.append(neq)
# new_model.equations_groups['arbitrage'].extend(to_be_added_1)
new_model.equations_groups['arbitrage'].extend(to_be_added_2)
new_model.update()
print("number of equations {}".format(len(new_model.equations)))
print("number of arbitrage equations {}".format( len(new_model.equations_groups['arbitrage'])) )
print('parameters_ordering')
print("number of parameters {}".format(new_model.symbols['parameters']))
print("number of parameters {}".format(new_model.parameters))
# now, we need to solve for the optimal portfolio coefficients
from trash.dolo.numeric.perturbations_to_states import approximate_controls
dr = approximate_controls(new_model)
print('ok')
import numpy
n_controls = len(model.symbols_s['controls'])
def constant_residuals(x, return_dr=False):
d = {}
for i in range(n_pfs):
p = pf_parms[i]
v = pf_vars[i]
d[p] = x[i]
d[v] = x[i]
new_model.set_calibration(d)
# new_model.parameters_values[p] = x[i]
# new_model.init_values[v] = x[i]
if return_dr:
dr = approximate_controls(new_model, order=1, return_dr=True, lambda_name='lam')
return dr
X_bar, X_s, X_ss = approximate_controls(new_model, order=2, return_dr=False, lambda_name="lam")
return X_bar[n_controls-n_pfs:n_controls]
if guess is not None:
x0 = numpy.array(guess)
else:
x0 = numpy.zeros(n_pfs)
print('Zero order portfolios')
print('Initial guess: {}'.format(x0))
print('Initial error: {}'.format( constant_residuals(x0) ))
portfolios_0 = solver(constant_residuals, x0)
print('Solution: {}'.format(portfolios_0))
print('Final error: {}'.format( constant_residuals(portfolios_0) ))
if order == 1:
dr = constant_residuals(portfolios_0, return_dr=True)
return dr
def dynamic_residuals(X, return_dr=False):
x = X[:,0]
dx = X[:,1:]
d = {}
for i in range(n_pfs):
p = pf_parms[i]
v = pf_vars[i]
d[p] = x[i]
d[v] = x[i]
for j in range(n_states):
d[pf_dparms[i][j]] = dx[i,j]
new_model.set_calibration(d)
if return_dr:
dr = approximate_controls(new_model, order=2, lambda_name='lam')
return dr
else:
[X_bar, X_s, X_ss, X_sss] = approximate_controls(new_model, order=3, return_dr=False, lambda_name='lam')
crit = numpy.column_stack([
X_bar[n_controls-n_pfs:n_controls],
X_s[n_controls-n_pfs:n_controls,:],
])
return crit
y0 = numpy.column_stack([x0, numpy.zeros((n_pfs, n_states))])
print('Initial error:')
err = (dynamic_residuals(y0))
print( abs(err).max() )
portfolios_1 = solver(dynamic_residuals, y0)
print('First order portfolios : ')
print(portfolios_1)
print('Final error:')
print(dynamic_residuals(portfolios_1))
dr = dynamic_residuals(portfolios_1, return_dr=True)
# TODO: remove coefficients of criteria
return dr
def solve_risky_ss(model, X_bar, X_s, verbose=False):
import numpy
from dolo.compiler.function_compiler import compile_function
import time
from dolo.compiler.compiler_functions import simple_global_representation
parms = model.calibration['parameters']
sigma = model.calibration['covariances']
sgm = simple_global_representation(model)
states = sgm['states']
controls = sgm['controls']
shocks = sgm['shocks']
parameters = sgm['parameters']
f_eqs = sgm['f_eqs']
g_eqs = sgm['g_eqs']
g_args = [s(-1) for s in states] + [c(-1) for c in controls] + shocks
f_args = states + controls + [v(1)
for v in states] + [v(1) for v in controls]
p_args = parameters
g_fun = compile_function(g_eqs, g_args, p_args, 2)
f_fun = compile_function(f_eqs, f_args, p_args, 3)
epsilons_0 = np.zeros((sigma.shape[0]))
from numpy import dot
from dolo.numeric.tensor import sdot, mdot
def residuals(X, sigma, parms, g_fun, f_fun):
import numpy
dummy_x = X[0:1, 0]
X_bar = X[1:, 0]
S_bar = X[0, 1:]
X_s = X[1:, 1:]
[n_x, n_s] = X_s.shape
n_e = sigma.shape[0]
xx = np.concatenate([S_bar, X_bar, epsilons_0])
[g_0, g_1, g_2] = g_fun(xx, parms)
[f_0, f_1, f_2,
f_3] = f_fun(np.concatenate([S_bar, X_bar, S_bar, X_bar]), parms)
res_g = g_0 - S_bar
# g is a first order function
g_s = g_1[:, :n_s]
g_x = g_1[:, n_s:n_s + n_x]
g_e = g_1[:, n_s + n_x:]
g_se = g_2[:, :n_s, n_s + n_x:]
g_xe = g_2[:, n_s:n_s + n_x, n_s + n_x:]
# S(s,e) = g(s,x,e)
S_s = g_s + dot(g_x, X_s)
S_e = g_e
S_se = g_se + mdot(g_xe, [X_s, numpy.eye(n_e)])
# V(s,e) = [ g(s,x,e) ; x( g(s,x,e) ) ]
V_s = np.row_stack([S_s, dot(X_s, S_s)]) # ***
V_e = np.row_stack([S_e, dot(X_s, S_e)])
V_se = np.row_stack([S_se, dot(X_s, S_se)])
# v(s) = [s, x(s)]
v_s = np.row_stack([numpy.eye(n_s), X_s])
# W(s,e) = [xx(s,e); yy(s,e)]
W_s = np.row_stack([v_s, V_s])
#return
nn = n_s + n_x
f_v = f_1[:, :nn]
f_V = f_1[:, nn:]
f_1V = f_2[:, :, nn:]
f_VV = f_2[:, nn:, nn:]
f_1VV = f_3[:, :, nn:, nn:]
# E = lambda v: np.tensordot(v, sigma, axes=((2,3),(0,1)) ) # expectation operator
F = f_0 + 0.5 * np.tensordot(
mdot(f_VV, [V_e, V_e]), sigma, axes=((1, 2), (0, 1)))
F_s = sdot(f_1, W_s)
f_see = mdot(f_1VV, [W_s, V_e, V_e]) + 2 * mdot(f_VV, [V_se, V_e])
F_s += 0.5 * np.tensordot(f_see, sigma, axes=(
(2, 3), (0, 1))) # second order correction
resp = np.row_stack(
[np.concatenate([dummy_x, res_g]),
np.column_stack([F, F_s])])
return resp
# S_bar = s_fun_init( numpy.atleast_2d(X_bar).T ,parms).flatten()
# S_bar = S_bar.flatten()
S_bar = model.calibration['states']
S_bar = np.array(S_bar)
X0 = np.row_stack(
[np.concatenate([np.zeros(1), S_bar]),
np.column_stack([X_bar, X_s])])
fobj = lambda X: residuals(X, sigma, parms, g_fun, f_fun)
if verbose:
val = fobj(X0)
print('val')
print(val)
# exit()
t = time.time()
sol = solver(fobj,
X0,
method='lmmcp',
verbose=verbose,
options={
'preprocess': False,
'eps1': 1e-15,
'eps2': 1e-15
})
if verbose:
print('initial guess')
print(X0)
print('solution')
print sol
print('initial residuals')
print(fobj(X0))
print('residuals')
print fobj(sol)
s = time.time()
if verbose:
print('Elapsed : {0}'.format(s - t))
#sol = solver(fobj,X0, method='fsolve', verbose=True, options={'preprocessor':False})
norm = lambda x: numpy.linalg.norm(x, numpy.inf)
if verbose:
print("Initial error: {0}".format(norm(fobj(X0))))
print("Final error: {0}".format(norm(fobj(sol))))
print("Solution")
print(sol)
X_bar = sol[1:, 0]
S_bar = sol[0, 1:]
X_s = sol[1:, 1:]
# compute transitions
n_s = len(states)
n_x = len(controls)
[g, dg, junk] = g_fun(np.concatenate([S_bar, X_bar, epsilons_0]), parms)
g_s = dg[:, :n_s]
g_x = dg[:, n_s:n_s + n_x]
P = g_s + dot(g_x, X_s)
if verbose:
eigenvalues = numpy.linalg.eigvals(P)
print eigenvalues
eigenvalues = [abs(e) for e in eigenvalues]
eigenvalues.sort()
print(eigenvalues)
return [S_bar, X_bar, X_s, P]
def solve_risky_ss(model, X_bar, X_s, verbose=False):
import numpy
from dolo.compiler.function_compiler import compile_function
import time
from dolo.compiler.compiler_functions import simple_global_representation
parms = model.calibration['parameters']
sigma = model.calibration['covariances']
sgm = simple_global_representation(model)
states = sgm['states']
controls = sgm['controls']
shocks = sgm['shocks']
parameters = sgm['parameters']
f_eqs = sgm['f_eqs']
g_eqs = sgm['g_eqs']
g_args = [s(-1) for s in states] + [c(-1) for c in controls] + shocks
f_args = states + controls + [v(1) for v in states] + [v(1) for v in controls]
p_args = parameters
g_fun = compile_function(g_eqs, g_args, p_args, 2)
f_fun = compile_function(f_eqs, f_args, p_args, 3)
epsilons_0 = np.zeros((sigma.shape[0]))
from numpy import dot
from dolo.numeric.tensor import sdot,mdot
def residuals(X, sigma, parms, g_fun, f_fun):
import numpy
dummy_x = X[0:1,0]
X_bar = X[1:,0]
S_bar = X[0,1:]
X_s = X[1:,1:]
[n_x,n_s] = X_s.shape
n_e = sigma.shape[0]
xx = np.concatenate([S_bar, X_bar, epsilons_0])
[g_0, g_1, g_2] = g_fun(xx, parms)
[f_0,f_1,f_2,f_3] = f_fun( np.concatenate([S_bar, X_bar, S_bar, X_bar]), parms)
res_g = g_0 - S_bar
# g is a first order function
g_s = g_1[:,:n_s]
g_x = g_1[:,n_s:n_s+n_x]
g_e = g_1[:,n_s+n_x:]
g_se = g_2[:,:n_s,n_s+n_x:]
g_xe = g_2[:, n_s:n_s+n_x, n_s+n_x:]
# S(s,e) = g(s,x,e)
S_s = g_s + dot(g_x, X_s)
S_e = g_e
S_se = g_se + mdot(g_xe,[X_s, numpy.eye(n_e)])
# V(s,e) = [ g(s,x,e) ; x( g(s,x,e) ) ]
V_s = np.row_stack([
S_s,
dot( X_s, S_s )
]) # ***
V_e = np.row_stack([
S_e,
dot( X_s, S_e )
])
V_se = np.row_stack([
S_se,
dot( X_s, S_se )
])
# v(s) = [s, x(s)]
v_s = np.row_stack([
numpy.eye(n_s),
X_s
])
# W(s,e) = [xx(s,e); yy(s,e)]
W_s = np.row_stack([
v_s,
V_s
])
#return
nn = n_s + n_x
f_v = f_1[:,:nn]
f_V = f_1[:,nn:]
f_1V = f_2[:,:,nn:]
f_VV = f_2[:,nn:,nn:]
f_1VV = f_3[:,:,nn:,nn:]
# E = lambda v: np.tensordot(v, sigma, axes=((2,3),(0,1)) ) # expectation operator
F = f_0 + 0.5*np.tensordot( mdot(f_VV,[V_e,V_e]), sigma, axes=((1,2),(0,1)) )
F_s = sdot(f_1, W_s)
f_see = mdot(f_1VV, [W_s, V_e, V_e]) + 2*mdot(f_VV, [V_se, V_e])
F_s += 0.5 * np.tensordot(f_see, sigma, axes=((2,3),(0,1)) ) # second order correction
resp = np.row_stack([
np.concatenate([dummy_x,res_g]),
np.column_stack([F,F_s])
])
return resp
# S_bar = s_fun_init( numpy.atleast_2d(X_bar).T ,parms).flatten()
# S_bar = S_bar.flatten()
S_bar = model.calibration['states']
S_bar = np.array(S_bar)
X0 = np.row_stack([
np.concatenate([np.zeros(1),S_bar]),
np.column_stack([X_bar,X_s])
])
fobj = lambda X: residuals(X, sigma, parms, g_fun, f_fun)
if verbose:
val = fobj(X0)
print('val')
print(val)
# exit()
t = time.time()
sol = solver(fobj,X0, method='lmmcp', verbose=verbose, options={'preprocess':False, 'eps1':1e-15, 'eps2': 1e-15})
if verbose:
print('initial guess')
print(X0)
print('solution')
print sol
print('initial residuals')
print(fobj(X0))
print('residuals')
print fobj(sol)
s = time.time()
if verbose:
print('Elapsed : {0}'.format(s-t))
#sol = solver(fobj,X0, method='fsolve', verbose=True, options={'preprocessor':False})
norm = lambda x: numpy.linalg.norm(x,numpy.inf)
if verbose:
print( "Initial error: {0}".format( norm( fobj(X0)) ) )
print( "Final error: {0}".format( norm( fobj(sol) ) ) )
print("Solution")
print(sol)
X_bar = sol[1:,0]
S_bar = sol[0,1:]
X_s = sol[1:,1:]
# compute transitions
n_s = len(states)
n_x = len(controls)
[g, dg, junk] = g_fun( np.concatenate( [S_bar, X_bar, epsilons_0] ), parms)
g_s = dg[:,:n_s]
g_x = dg[:,n_s:n_s+n_x]
P = g_s + dot(g_x, X_s)
if verbose:
eigenvalues = numpy.linalg.eigvals(P)
print eigenvalues
eigenvalues = [abs(e) for e in eigenvalues]
eigenvalues.sort()
print(eigenvalues)
return [S_bar, X_bar, X_s, P]