def check(self):
defaults = {
'name': 'anonymous',
'init_values': {},
'parameters_values': {},
'covariances': sympy.Matrix(),
'variables_ordering': [],
'parameters_ordering': [],
'shocks_ordering': []
}
from collections import OrderedDict as odict
equations_groups = odict()
for i, eq in enumerate(self['equations']):
eq.tags['eq_number'] = i
if 'eq_type' in eq.tags:
g = eq.tags['eq_type']
if g not in equations_groups:
equations_groups[g] = []
equations_groups[g].append(eq)
self['equations_groups'] = equations_groups
for k in defaults:
if k not in self:
self[k] = defaults[k]
if not self.get('equations'):
raise Exception('No equations specified')
for n, eq in enumerate(self['equations']):
if not isinstance(eq, Equation):
self['equations'][n] = Equation(eq, 0)
def solve_recursive_block(equations):
from dolo.symbolic.symbolic import Equation
system = {eq.lhs: eq.rhs for eq in equations}
from dolo.misc.triangular_solver import solve_triangular_system
sol = solve_triangular_system(system)
solved_equations = [Equation(eq.lhs, sol[eq.lhs]) for eq in equations]
return solved_equations
def build_lagrangian(self):
vars = list(self.base_model.variables)
if self.instrument_equation != None and self.instrument_equation.tags.has_key(
'ramsey_instrument'):
ramsey_instrument = [
v for v in vars
if v.name == self.instrument_equation.tags['ramsey_instrument']
]
ramsey_instrument = ramsey_instrument[0]
self.equations.remove(self.instrument_equation)
vars.remove(ramsey_instrument)
# first we create one multiplicator by equation
n_eq = len(self.equations)
lambdas = []
for i in range(n_eq):
v = Variable('Lambda_' + str(i + 1), 0)
self.variables_ordering.append(v)
lambdas.append(v)
t_cur = self.objective + sum(
[self.equations[i].gap * lambdas[i] for i in range(n_eq)])
t_fut = time_shift(t_cur, +1)
t_past = time_shift(t_cur, -1)
beta = self.discount
lagrangian = 1 / beta * t_past + t_cur + beta * t_fut
self.lagrangian = lagrangian
print 'There are ' + str(len(self.equations)) + 'equations'
print 'There are ' + str(len(vars)) + 'variables'
if self.instrument_equation != None and self.instrument_equation.tags.has_key(
'ramsey_instrument'):
# vars = [ramsey_instrument] + vars
eq = lagrangian.diff(ramsey_instrument)
eq = Equation(eq, 0).tag(name='Derivative of lagrangian w.r.t : ' +
str(ramsey_instrument))
eq.tags.update(self.instrument_equation.tags)
self.equations.append(eq)
for v in vars:
if v in self.ignored_variables:
pass
eq = lagrangian.diff(v)
eq = Equation(eq, 0).tag(name='Derivative of lagrangian w.r.t : ' +
str(v))
self.equations.append(eq)
def portfolios_to_deterministic(model, pf_names):
#######
#######
import re
regex = re.compile('.*<=(.*)<=.*')
for i, eq in enumerate(model.equations_groups['arbitrage']):
from dolo.symbolic.symbolic import Variable, Equation
m = regex.match(eq.tags['complementarity'])
vs = m.group(1).strip()
if vs in pf_names:
v = Variable(vs)
neq = Equation(v, 0)
neq.tag(**eq.tags)
model.equations_groups['arbitrage'][i] = neq
print('Warning : initial model changed')
model.update()
return model
def portfolios_to_deterministic(model,pf_names):
#######
#######
import re
regex = re.compile('.*<=(.*)<=.*')
for i,eq in enumerate(model.equations_groups['arbitrage']):
from dolo.symbolic.symbolic import Variable, Equation
m = regex.match(eq.tags['complementarity'])
vs = m.group(1).strip()
if vs in pf_names:
v = Variable(vs)
neq = Equation(v,0)
neq.tag(**eq.tags)
model.equations_groups['arbitrage'][i] = neq
print('Warning : initial model changed')
model.update()
return model
def parse_yaml_text(txt, verbose=False):
'''
Imports the content of a modfile into the current interpreter scope
'''
txt = txt.replace('..', '-')
txt = txt.replace('--', '-')
txt = txt.replace('^', '**')
raw_dict = yaml.load(txt)
if verbose == True:
print('YAML file successfully parsed')
declarations = raw_dict['declarations']
# check
if 'controls' in declarations:
variables_groups = OrderedDict()
known_types = [
'states', 'controls', 'expectations', 'auxiliary', 'auxiliary_2'
]
for vtype in known_types:
if vtype in declarations:
variables_groups[vtype] = [
Variable(vn, 0) for vn in declarations[vtype]
]
variables_ordering = sum(variables_groups.values(), [])
else:
vnames = declarations['variables']
variables_ordering = [Variable(vn, 0) for vn in vnames]
variables_groups = None
parameters_ordering = [Parameter(vn) for vn in declarations['parameters']]
shocks_ordering = [Shock(vn, 0) for vn in declarations['shocks']]
context = [
(s.name, s)
for s in variables_ordering + parameters_ordering + shocks_ordering
]
context = dict(context)
# add some common functions
for f in [
sympy.log, sympy.exp, sympy.sin, sympy.cos, sympy.tan, sympy.asin,
sympy.acos, sympy.atan, sympy.sinh, sympy.cosh, sympy.tanh,
sympy.pi
]:
context[str(f)] = f
context['sqrt'] = sympy.sqrt
import re
# we recognize two kinds of equations:
# lhs = rhs
# lhs | comp where comp is a complementarity condition
equations = []
equations_groups = OrderedDict()
raw_equations = raw_dict['equations']
if isinstance(raw_equations,
dict): # tests whether there are groups of equations
for groupname in raw_equations.keys():
equations_groups[groupname] = []
for raw_eq in raw_equations[
groupname]: # Modfile is supposed to represent a global model. TODO: change it
teqg = raw_eq.split('|')
teq = teqg[0]
if '=' in teq:
lhs, rhs = str.split(teq, '=')
else:
lhs = teq
rhs = '0'
try:
lhs = eval(lhs, context)
rhs = eval(rhs, context)
except Exception as e:
print('Error parsing equation : ' + teq)
print str(e)
raise e
eq = Equation(lhs, rhs)
eq.tag(eq_type=groupname)
if len(teqg) > 1:
comp = teqg[1]
eq.tag(complementarity=comp)
equations.append(eq)
#equations_groups[groupname].append( eq )
else:
for teq in raw_equations:
if '=' in teq:
lhs, rhs = str.split(teq, '=')
else:
lhs = teq
rhs = '0'
try:
lhs = eval(lhs, context)
rhs = eval(rhs, context)
except Exception as e:
print('Error parsing equations : ' + teq)
print str(e)
eq = Equation(lhs, rhs)
equations.append(eq)
equations_groups = None
parameters_values = {}
init_values = {}
covariances = None
if 'calibration' in raw_dict:
calibration = raw_dict['calibration']
if 'parameters' in calibration:
parameters_values = [
(Parameter(k), eval(str(v), context))
for k, v in calibration['parameters'].iteritems()
]
parameters_values = dict(parameters_values)
#steady_state = raw_dict['steady_state']
if 'steady_state' in calibration:
init_values = [
(Variable(vn, 0), eval(str(value), context))
for vn, value in calibration['steady_state'].iteritems()
]
init_values = dict(init_values)
if 'covariances' in calibration:
context['sympy'] = sympy
covariances = eval(
'sympy.Matrix({0})'.format(calibration['covariances']),
context)
else:
covariances = None # to avoid importing numpy
model_dict = {
'variables_ordering': variables_ordering,
'parameters_ordering': parameters_ordering,
'shocks_ordering': shocks_ordering,
'variables_groups': variables_groups,
'equations_groups': equations_groups,
'equations': equations,
'parameters_values': parameters_values,
'init_values': init_values,
'covariances': covariances
}
if 'model_type' in raw_dict:
model_dict['model_type'] = raw_dict['model_type']
model_dict['original_data'] = raw_dict
model = Model(**model_dict)
model.check_consistency(auto_remove_variables=False)
return model
def parse_dynare_text(txt, add_model=True, full_output=False, debug=False):
'''
Imports the content of a modfile into the current interpreter scope
'''
# here we call "instruction group", a string finishing by a semicolon
# an "instruction group" can have several lines
# a line can be
# - a comment //...
# - an old-style tag //$...
# - a new-style tag [key1='value1',..]
# - macro-instruction @#...
# A Modfile contains several blocks (in this order) :
# - an initblock defining variables, exovariables, parameters, initialization
# inside the initblock the order of declaration doesn't matter
# - a model block with two special lines (model; end;)
# - optional blocks (like endval, shocks)
# seperated by free matlab instructions in any order;
# - all other instructions are ignored
otxt = txt
otxt = otxt.replace("\r\n", "\n")
otxt = otxt.replace("^", "**")
# first, we remove end-of-line comments : they are definitely lost
regex = re.compile("(.+)//[^#](.*)")
def remove_end_comment(line):
res = regex.search(line)
if res:
l = res.groups(1)[0]
return (l)
else:
return line
txt = str.join("\n", map(remove_end_comment, otxt.split("\n")))
name_regex = re.compile("//\s*fname\s*=\s*'(.*)'")
m = name_regex.search(txt)
if m:
fname = m.group(1)
else:
fname = None
instruction_groups = [Instruction_group(s) for s in txt.split(";")]
instructions = [ig.instruction for ig in instruction_groups]
if debug:
print('Elementary instructions')
for i in instruction_groups:
print(i)
try:
imodel = [
re.compile('model(\(.*\)|)').match(e) is not None
for e in instructions
]
imodel = imodel.index(True)
#imodel = instructions.index("model") #this doesn't work for "MODEL"
iend = instructions.index("end")
model_block = instruction_groups[imodel:(iend + 1)]
init_block = instruction_groups[0:imodel]
except:
raise Exception('Model block could not be found.')
next_instructions = instructions[(iend + 1):]
next_instruction_groups = instruction_groups[(iend + 1):]
if 'initval' in next_instructions:
iinitval = next_instructions.index('initval')
iend = next_instructions.index('end', iinitval)
matlab_block_1 = next_instruction_groups[0:iinitval]
initval_block = next_instruction_groups[iinitval:(iend + 1)]
next_instruction_groups = next_instruction_groups[(iend + 1):]
next_instructions = next_instructions[(iend + 1):]
else:
initval_block = None
matlab_block_1 = None
if 'endval' in next_instructions:
iendval = next_instructions.index('endval')
iend = next_instructions.index('end', iendval)
matlab_block_2 = next_instruction_groups[0:iendval]
endval_block = next_instruction_groups[iendval:(iend + 1)]
next_instruction_groups = next_instruction_groups[(iend + 1):]
next_instructions = next_instructions[(iend + 1):]
else:
endval_block = None
matlab_block_2 = None
# TODO : currently shocks block needs to follow initval, this restriction should be removed
if 'shocks' in next_instructions:
ishocks = next_instructions.index('shocks')
iend = next_instructions.index('end', ishocks)
matlab_block_3 = next_instruction_groups[0:ishocks]
shocks_block = next_instruction_groups[ishocks:(iend + 1)]
next_instruction_groups = next_instruction_groups[(iend + 1):]
next_instructions = next_instructions[(iend + 1):]
else:
shocks_block = None
matlab_block_3 = None
try:
init_regex = re.compile("(parameters |var |varexo |)(.*)")
var_names = []
varexo_names = []
parameters_names = []
declarations = {}
for ig in init_block:
if ig.instruction != '':
m = init_regex.match(ig.instruction)
if not m:
raise Exception("Unexpected instruction in init block : " +
str(ig.instruction))
if m.group(1) == '':
[lhs, rhs] = m.group(2).split("=")
lhs = lhs.strip()
rhs = rhs.strip()
declarations[lhs] = rhs
else:
arg = m.group(2).replace(",", " ")
names = [vn.strip() for vn in arg.split()]
if m.group(1).strip() == 'var':
dest = var_names
elif m.group(1).strip() == 'varexo':
dest = varexo_names
elif m.group(1).strip() == 'parameters':
dest = parameters_names
for n in names:
if not n in dest:
dest.append(n)
else:
raise Exception(
"symbol %s has already been defined".format(n))
except Exception as e:
raise Exception('Init block could not be read : ' + str(e))
# the following instruction set the variables "variables","shocks","parameters"
variables = []
for vn in var_names:
v = Variable(vn)
variables.append(v)
shocks = []
for vn in varexo_names:
s = Shock(vn)
shocks.append(s)
parameters = []
for vn in parameters_names:
p = Parameter(vn)
parameters.append(p)
parse_dict = dict()
for v in variables + shocks + parameters:
parse_dict[v.name] = v
special_symbols = [
sympy.exp, sympy.log, sympy.sin, sympy.cos, sympy.atan, sympy.tan
]
for s in special_symbols:
parse_dict[str(s)] = s
parse_dict['sqrt'] = sympy.sqrt
# Read parameters values
parameters_values = {}
for p in declarations:
try:
rhs = eval(declarations[p], parse_dict)
except Exception as e:
Exception("Impossible to evaluate parameter value : " + str(e))
try:
lhs = eval(p, parse_dict)
except Exception as e:
# here we could declare p
raise e
parameters_values[lhs] = rhs
# Now we read the model block
model_tags = model_block[0].tags
equations = []
for ig in model_block[1:-1]:
if ig.instruction != '':
teq = ig.instruction.replace('^', "**")
if '=' in teq:
teqlhs, teqrhs = teq.split("=")
else:
teqlhs = teq
teqrhs = '0'
eqlhs = eval(teqlhs, parse_dict)
eqrhs = eval(teqrhs, parse_dict)
eq = Equation(eqlhs, eqrhs)
eq.tags.update(ig.tags)
# if eq.tags.has_key('name'):
# eq.tags[] = ig.tags['name']
equations.append(eq)
# Now we read the initval block
init_values = {}
if initval_block != None:
for ig in initval_block[1:-1]:
if len(ig.instruction.strip()) > 0:
try:
[lhs, rhs] = ig.instruction.split("=")
except Exception as e:
print(ig.instruction)
raise e
init_values[eval(lhs, parse_dict)] = eval(rhs, parse_dict)
# Now we read the endval block
# I don't really care about the endval block !
end_values = {}
if endval_block != None:
for ig in endval_block[1:-1]:
[lhs, rhs] = ig.instruction.split("=")
end_values[eval(lhs)] = eval(rhs)
# Now we read the shocks block
covariances = None
if shocks_block != None:
covariances = sympy.zeros(len(shocks))
regex1 = re.compile("var (.*?),(.*?)=(.*)|var (.*?)=(.*)")
for ig in shocks_block[1:-1]:
m = regex1.match(ig.instruction)
if not m:
raise Exception("unrecognized instruction in block shocks : " +
str(ig.instruction))
if m.group(1) != None:
varname1 = m.group(1).strip()
varname2 = m.group(2).strip()
value = m.group(3).strip().replace("^", "**")
elif m.group(4) != None:
varname1 = m.group(4).strip()
varname2 = varname1
value = m.group(5).strip().replace("^", "**")
i = varexo_names.index(varname1)
j = varexo_names.index(varname2)
covariances[i, j] = eval(value, parse_dict)
covariances[j, i] = eval(value, parse_dict)
calibration = {}
calibration.update(parameters_values)
calibration.update(init_values)
symbols = {
'variables': variables,
'shocks': shocks,
'parameters': parameters
}
from dolo.symbolic.model import SModel
model = SModel({'dynare_block': equations}, symbols, calibration,
covariances)
return model
def parse_yaml_text(txt,verbose=False):
'''
Imports the content of a modfile into the current interpreter scope
'''
txt = txt.replace('..','-')
txt = txt.replace('--','-')
txt = txt.replace('^','**')
raw_dict = yaml.load(txt)
if verbose == True:
print('YAML file successfully parsed')
declarations = raw_dict['declarations']
# check
if 'controls' in declarations:
variables_groups = OrderedDict()
known_types = ['states','controls','expectations','auxiliary','auxiliary_2']
for vtype in known_types:
if vtype in declarations:
variables_groups[vtype] = [Variable(vn,0) for vn in declarations[vtype]]
variables_ordering = sum(variables_groups.values(),[])
else:
vnames = declarations['variables']
variables_ordering = [Variable(vn,0) for vn in vnames]
variables_groups = None
parameters_ordering = [Parameter(vn) for vn in declarations['parameters']]
shocks_ordering = [Shock(vn,0) for vn in declarations['shocks']]
context = [(s.name,s) for s in variables_ordering + parameters_ordering + shocks_ordering]
context = dict(context)
# add some common functions
for f in [sympy.log, sympy.exp,
sympy.sin, sympy.cos, sympy.tan,
sympy.asin, sympy.acos, sympy.atan,
sympy.sinh, sympy.cosh, sympy.tanh,
sympy.pi]:
context[str(f)] = f
context['sqrt'] = sympy.sqrt
import re
# we recognize two kinds of equations:
# lhs = rhs
# lhs | comp where comp is a complementarity condition
equations = []
equations_groups = OrderedDict()
raw_equations = raw_dict['equations']
if isinstance(raw_equations,dict): # tests whether there are groups of equations
for groupname in raw_equations.keys():
equations_groups[groupname] = []
for raw_eq in raw_equations[groupname]: # Modfile is supposed to represent a global model. TODO: change it
teqg = raw_eq.split('|')
teq = teqg[0]
if '=' in teq:
lhs,rhs = str.split(teq,'=')
else:
lhs = teq
rhs = '0'
try:
lhs = eval(lhs,context)
rhs = eval(rhs,context)
except Exception as e:
print('Error parsing equation : ' + teq)
print str(e)
raise e
eq = Equation(lhs,rhs)
eq.tag(eq_type=groupname)
if len(teqg)>1:
comp = teqg[1]
eq.tag(complementarity=comp)
equations.append(eq)
#equations_groups[groupname].append( eq )
else:
for teq in raw_equations:
if '=' in teq:
lhs,rhs = str.split(teq,'=')
else:
lhs = teq
rhs = '0'
try:
lhs = eval(lhs,context)
rhs = eval(rhs,context)
except Exception as e:
print('Error parsing equations : ' + teq)
print str(e)
eq = Equation(lhs,rhs)
equations.append(eq)
equations_groups = None
parameters_values = {}
init_values = {}
covariances = None
if 'calibration' in raw_dict:
calibration = raw_dict['calibration']
if 'parameters' in calibration:
parameters_values = [ (Parameter(k), eval(str(v),context)) for k,v in calibration['parameters'].iteritems() ]
parameters_values = dict(parameters_values)
#steady_state = raw_dict['steady_state']
if 'steady_state' in calibration:
init_values = [ (Variable(vn,0), eval(str(value),context)) for vn,value in calibration['steady_state'].iteritems() ]
init_values = dict(init_values)
if 'covariances' in calibration:
context['sympy'] = sympy
covariances = eval('sympy.Matrix({0})'.format( calibration['covariances'] ), context)
else:
covariances = None # to avoid importing numpy
model_dict = {
'variables_ordering': variables_ordering,
'parameters_ordering': parameters_ordering,
'shocks_ordering': shocks_ordering,
'variables_groups': variables_groups,
'equations_groups': equations_groups,
'equations': equations,
'parameters_values': parameters_values,
'init_values': init_values,
'covariances': covariances
}
if 'model_type' in raw_dict:
model_dict['model_type'] = raw_dict['model_type']
model_dict['original_data'] = raw_dict
model = Model(**model_dict)
model.check_consistency(auto_remove_variables=False)
return model
def solve_portfolio_model(model, pf_names, order=1):
pf_model = model
from dolo import Variable, Parameter, Equation
import re
n_states = len(pf_model['variables_groups']['states'])
states = pf_model['variables_groups']['states']
steady_states = [Parameter(v.name+'_bar') for v in pf_model['variables_groups']['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
print('Warning: initial model has been changed.')
new_model = copy.copy(pf_model)
new_model['variables_groups']['controls']+=res_vars
new_model['parameters_ordering'].extend(steady_states)
for p in pf_parms + Matrix(pf_dparms)[:]:
new_model['parameters_ordering'].append(p)
new_model.parameters_values[p] = 0
compregex = re.compile('(.*)<=(.*)<=(.*)')
to_be_added = []
expressions = Matrix(pf_parms) + Matrix(pf_dparms)*( Matrix(states) - Matrix(steady_states))
for eq in 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)
eq_n = eq.tags['eq_number']
neq = Equation(mhs, expressions[i])
neq.tag(**eq.tags)
new_model['equations'][eq_n] = neq
eq_res = Equation(eq.gap, res_vars[i])
eq_res.tag(eq_type='arbitrage')
to_be_added.append(eq_res)
new_model['equations'].extend(to_be_added)
new_model.check()
new_model.check_consistency(verbose=True)
print(new_model.parameters)
print(len(new_model['equations']))
print(len(new_model.equations))
print(len(new_model['equations_groups']['arbitrage']))
print('parameters_ordering')
print(new_model['parameters_ordering'])
print(new_model.parameters)
# now, we need to solve for the optimal portfolio coefficients
from dolo.numeric.perturbations_to_states import approximate_controls
dr = approximate_controls(new_model)
print('ok')
import numpy
n_controls = len(model['variables_groups']['controls'])
def constant_residuals(x):
for i in range(n_pfs):
p = pf_parms[i]
v = pf_vars[i]
new_model.parameters_values[p] = x[i]
new_model.init_values[v] = x[i]
[X_bar, X_s, X_ss] = approximate_controls(new_model, order=2, return_dr=False)
return X_bar[n_controls-n_pfs:n_controls]
x0 = numpy.zeros(n_pfs)
from dolo.numeric.solver import solver
portfolios_0 = solver(constant_residuals, x0)
print('Zero order portfolios : ')
print(portfolios_0)
print('Zero order: Final error:')
print(constant_residuals(portfolios_0))
def dynamic_residuals(X, return_dr=False):
x = X[:,0]
dx = X[:,1:]
for i in range(n_pfs):
p = pf_parms[i]
v = pf_vars[i]
model.parameters_values[p] = x[i]
model.init_values[v] = x[i]
for j in range(n_states):
model.parameters_values[pf_dparms[i][j]] = dx[i,j]
if return_dr:
dr = approximate_controls(new_model, order=2)
return dr
else:
[X_bar, X_s, X_ss, X_sss] = approximate_controls(new_model, order=3, return_dr=False)
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:')
print(dynamic_residuals(y0))
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)
return dr
def parse_yaml_text(txt,verbose=False, compiler=None):
'''
Imports the content of a modfile into the current interpreter scope
'''
txt = txt.replace('..','-')
txt = txt.replace('--','-')
txt = txt.replace('^','**')
txt = txt.replace('equilibrium:','arbitrage:')
txt = txt.replace('_|_','|')
raw_dict = yaml.load(txt)
if verbose == True:
print('YAML file successfully parsed')
declarations = raw_dict['declarations']
# check
variables_groups = OrderedDict()
for vtype in declarations.keys():
if vtype not in ('shocks','parameters'):
variables_groups[vtype] = [Variable(vn) for vn in declarations[vtype]]
variables_ordering = sum(variables_groups.values(),[])
# else:
# vnames = declarations['variables']
# variables_ordering = [Variable(vn) for vn in vnames]
# variables_groups = None
parameters_ordering = [Parameter(vn) for vn in declarations['parameters']]
shocks_ordering = [Shock(vn) for vn in declarations['shocks']]
context = [(s.name,s) for s in variables_ordering + parameters_ordering + shocks_ordering]
context = dict(context)
from dolo.symbolic.symbolic import timeshift as TS
# add some common functions
for f in [sympy.log, sympy.exp,
sympy.sin, sympy.cos, sympy.tan,
sympy.asin, sympy.acos, sympy.atan,
sympy.sinh, sympy.cosh, sympy.tanh,
sympy.pi, sympy.sign]:
context[str(f)] = f
context['sqrt'] = sympy.sqrt
context['TS'] = TS
if 'horrible_hack' in raw_dict:
tt = raw_dict['horrible_hack']
exec(tt, context)
import re
# we recognize two kinds of equations:
# lhs = rhs
# lhs | comp where comp is a complementarity condition
equations = []
equations_groups = OrderedDict()
raw_equations = raw_dict['equations']
if not isinstance(raw_equations,dict):
raw_dict['model_type'] = 'dynare'
raw_equations = {'dynare_block': raw_equations}
if True: # tests whether there are groups of equations
for groupname in raw_equations.keys():
equations_groups[groupname] = []
for raw_eq in raw_equations[groupname]: # Modfile is supposed to represent a global model. TODO: change it
teqg = raw_eq.split('|')
teq = teqg[0]
if '=' in teq:
lhs,rhs = str.split(teq,'=')
else:
lhs = teq
rhs = '0'
try:
lhs = eval(lhs,context)
rhs = eval(rhs,context)
except Exception as e:
print('Error parsing equation : ' + teq)
print( str(e) )
raise e
eq = Equation(lhs,rhs)
eq.tag(eq_type=groupname)
if len(teqg)>1:
comp = teqg[1]
eq.tag(complementarity=comp)
equations.append(eq)
equations_groups[groupname].append( eq )
else:
for teq in raw_equations:
if '=' in teq:
lhs,rhs = str.split(teq,'=')
else:
lhs = teq
rhs = '0'
try:
lhs = eval(lhs,context)
rhs = eval(rhs,context)
except Exception as e:
print('Error parsing equations : ' + teq)
print(str(e))
eq = Equation(lhs,rhs)
equations.append(eq)
equations_groups = None
parameters_values = {}
init_values = {}
covariances = None
if 'calibration' in raw_dict:
calibration = raw_dict['calibration']
if 'parameters' in calibration:
parameters_values = [ (Parameter(k), eval(str(v),context)) for k,v in iteritems(calibration['parameters']) ]
parameters_values = dict(parameters_values)
#steady_state = raw_dict['steady_state']
if 'steady_state' in calibration:
init_values = [ (Variable(vn), eval(str(value),context)) for vn,value in iteritems(calibration['steady_state']) ]
init_values = dict(init_values)
if 'covariances' in calibration:
context['sympy'] = sympy
covariances = eval('sympy.Matrix({0})'.format( calibration['covariances'] ), context)
else:
covariances = None # to avoid importing numpy
symbols = variables_groups
symbols['shocks'] = shocks_ordering
symbols['parameters'] = parameters_ordering
calibration_s = {}
calibration_s.update(parameters_values)
calibration_s.update(init_values)
from dolo.symbolic.model import SModel
model = SModel( equations_groups, symbols, calibration_s, covariances )
model.__data__ = raw_dict
return model
print('Impossible to evaluate : \n' + str(t))
raise e
residuals[gname] = [float(eq.gap.subs(dd)) for eq in geqs]
return residuals
else:
stateq = [eq.gap.subs(dd) for eq in model.equations]
residuals = [float(eq) for eq in stateq]
return residuals
def print_residuals(model):
residuals = compute_residuals(model)
print('\n{:*^90}\n'.format('Residuals'))
for category in residuals.keys():
res = residuals[category]
print category
for i, eq in enumerate(model['equations_groups'][category]):
print('\t{:03.4f}\t:\t{}'.format(res[i], eq))
if __name__ == '__main__':
from dolo.symbolic.symbolic import Variable, Equation
v = Variable('v', 0)
eq = Equation(v**2, v(1) - v(-1))
d = Model(equations=[eq])
def parse_yaml_text(txt, verbose=False, compiler=None):
'''
Imports the content of a modfile into the current interpreter scope
'''
txt = txt.replace('..', '-')
txt = txt.replace('--', '-')
txt = txt.replace('^', '**')
txt = txt.replace('equilibrium:', 'arbitrage:')
txt = txt.replace('_|_', '|')
raw_dict = yaml.load(txt)
if verbose == True:
print('YAML file successfully parsed')
declarations = raw_dict['declarations']
# check
variables_groups = OrderedDict()
for vtype in declarations.keys():
if vtype not in ('shocks', 'parameters'):
variables_groups[vtype] = [
Variable(vn) for vn in declarations[vtype]
]
variables_ordering = sum(variables_groups.values(), [])
# else:
# vnames = declarations['variables']
# variables_ordering = [Variable(vn) for vn in vnames]
# variables_groups = None
parameters_ordering = [Parameter(vn) for vn in declarations['parameters']]
shocks_ordering = [Shock(vn) for vn in declarations['shocks']]
context = [
(s.name, s)
for s in variables_ordering + parameters_ordering + shocks_ordering
]
context = dict(context)
from dolo.symbolic.symbolic import timeshift as TS
# add some common functions
for f in [
sympy.log, sympy.exp, sympy.sin, sympy.cos, sympy.tan, sympy.asin,
sympy.acos, sympy.atan, sympy.sinh, sympy.cosh, sympy.tanh,
sympy.pi, sympy.sign
]:
context[str(f)] = f
context['sqrt'] = sympy.sqrt
context['TS'] = TS
if 'horrible_hack' in raw_dict:
tt = raw_dict['horrible_hack']
exec(tt, context)
import re
# we recognize two kinds of equations:
# lhs = rhs
# lhs | comp where comp is a complementarity condition
equations = []
equations_groups = OrderedDict()
raw_equations = raw_dict['equations']
if not isinstance(raw_equations, dict):
raw_dict['model_type'] = 'dynare'
raw_equations = {'dynare_block': raw_equations}
if True: # tests whether there are groups of equations
for groupname in raw_equations.keys():
equations_groups[groupname] = []
for raw_eq in raw_equations[
groupname]: # Modfile is supposed to represent a global model. TODO: change it
teqg = raw_eq.split('|')
teq = teqg[0]
if '=' in teq:
lhs, rhs = str.split(teq, '=')
else:
lhs = teq
rhs = '0'
try:
lhs = eval(lhs, context)
rhs = eval(rhs, context)
except Exception as e:
print('Error parsing equation : ' + teq)
print(str(e))
raise e
eq = Equation(lhs, rhs)
eq.tag(eq_type=groupname)
if len(teqg) > 1:
comp = teqg[1]
eq.tag(complementarity=comp)
equations.append(eq)
equations_groups[groupname].append(eq)
else:
for teq in raw_equations:
if '=' in teq:
lhs, rhs = str.split(teq, '=')
else:
lhs = teq
rhs = '0'
try:
lhs = eval(lhs, context)
rhs = eval(rhs, context)
except Exception as e:
print('Error parsing equations : ' + teq)
print(str(e))
eq = Equation(lhs, rhs)
equations.append(eq)
equations_groups = None
parameters_values = {}
init_values = {}
covariances = None
if 'calibration' in raw_dict:
calibration = raw_dict['calibration']
if 'parameters' in calibration:
parameters_values = [
(Parameter(k), eval(str(v), context))
for k, v in iteritems(calibration['parameters'])
]
parameters_values = dict(parameters_values)
#steady_state = raw_dict['steady_state']
if 'steady_state' in calibration:
init_values = [
(Variable(vn), eval(str(value), context))
for vn, value in iteritems(calibration['steady_state'])
]
init_values = dict(init_values)
if 'covariances' in calibration:
context['sympy'] = sympy
covariances = eval(
'sympy.Matrix({0})'.format(calibration['covariances']),
context)
else:
covariances = None # to avoid importing numpy
symbols = variables_groups
symbols['shocks'] = shocks_ordering
symbols['parameters'] = parameters_ordering
calibration_s = {}
calibration_s.update(parameters_values)
calibration_s.update(init_values)
from dolo.symbolic.model import SModel
model = SModel(equations_groups, symbols, calibration_s, covariances)
model.__data__ = raw_dict
return model
def solve_portfolio_model(model, pf_names, order=2, 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 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)
return dr
X_bar, X_s, X_ss = approximate_controls(new_model,
order=2,
return_dr=False)
return X_bar[n_controls - n_pfs:n_controls]
if guess is not None:
x0 = numpy.array(guess)
else:
x0 = numpy.zeros(n_pfs)
from dolo.numeric.solver import solver
portfolios_0 = solver(constant_residuals, x0)
print('Zero order portfolios: {}'.format(portfolios_0))
print('Zero order portfolios: 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)
return dr
else:
[X_bar, X_s, X_ss, X_sss] = approximate_controls(new_model,
order=3,
return_dr=False)
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:')
print(dynamic_residuals(y0))
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)
return dr
def solve_portfolio_model(model, pf_names, order=1):
pf_model = model
from dolo import Variable, Parameter, Equation
import re
n_states = len(pf_model['variables_groups']['states'])
states = pf_model['variables_groups']['states']
steady_states = [
Parameter(v.name + '_bar')
for v in pf_model['variables_groups']['states']
]
n_pfs = len(pf_names)
pf_vars = [Variable(v, 0) for v in pf_names]
res_vars = [Variable('res_' + str(i), 0) 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
print('Warning: initial model has been changed.')
new_model = copy.copy(pf_model)
new_model['variables_groups']['controls'] += res_vars
new_model.check()
for p in pf_parms + Matrix(pf_dparms)[:]:
new_model['parameters_ordering'].append(p)
new_model.parameters_values[p] = 0
compregex = re.compile('(.*)<=(.*)<=(.*)')
to_be_added = []
expressions = Matrix(pf_parms) + Matrix(pf_dparms) * (
Matrix(states) - Matrix(steady_states))
for eq in 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)
eq_n = eq.tags['eq_number']
neq = Equation(mhs, expressions[i])
neq.tag(**eq.tags)
new_model['equations'][eq_n] = neq
eq_res = Equation(eq.gap, res_vars[i])
eq_res.tag(eq_type='arbitrage')
to_be_added.append(eq_res)
new_model['equations'].extend(to_be_added)
new_model.check()
new_model.check_consistency()
# now, we need to solve for the optimal portfolio coefficients
from dolo.numeric.perturbations_to_states import approximate_controls
import numpy
n_controls = len(model['variables_groups']['controls'])
def constant_residuals(x):
for i in range(n_pfs):
p = pf_parms[i]
v = pf_vars[i]
model.parameters_values[p] = x[i]
model.init_values[v] = x[i]
[X_bar, X_s, X_ss] = approximate_controls(new_model,
order=2,
return_dr=False)
return X_bar[n_controls - n_pfs:n_controls]
x0 = numpy.zeros(n_pfs)
from dolo.numeric.solver import solver
portfolios_0 = solver(constant_residuals, x0)
print('Zero order portfolios : ')
print(portfolios_0)
print('Zero order: Final error:')
print(constant_residuals(portfolios_0))
def dynamic_residuals(X, return_dr=False):
x = X[:, 0]
dx = X[:, 1:]
for i in range(n_pfs):
p = pf_parms[i]
v = pf_vars[i]
model.parameters_values[p] = x[i]
model.init_values[v] = x[i]
for j in range(n_states):
model.parameters_values[pf_dparms[i][j]] = dx[i, j]
if return_dr:
dr = approximate_controls(new_model, order=2, return_dr=True)
return dr
else:
[X_bar, X_s, X_ss, X_sss] = approximate_controls(new_model,
order=3,
return_dr=False)
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:')
print(dynamic_residuals(y0))
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)
return dr
def read_model(self):
if self.__transformed_model__:
return self.__transformed_model__
dmodel = SModel(**self.model) # copy the model
dmodel.check_consistency(auto_remove_variables=False)
def_eqs = [eq for eq in dmodel.equations if eq.tags['eq_type'] in ('def', 'auxiliary')]
from dolo.symbolic.symbolic import map_function_to_expression
from dolo.symbolic.symbolic import Variable
def timeshift(v,n):
if isinstance(v,Variable):
return v(n)
else:
return v
import sympy
#### build substitution dict
def_dict = {}
for eq in def_eqs:
v = eq.lhs
rhs = sympy.sympify( eq.rhs )
def_dict[v] = rhs
def_dict[v(1)] = map_function_to_expression( lambda x: timeshift(x,1), rhs)
new_equations = []
tbr = []
for i,eq in enumerate(dmodel.equations) :
if not ('def' == eq.tags['eq_type']):
lhs = sympy.sympify( eq.lhs ).subs(def_dict)
rhs = sympy.sympify( eq.rhs ).subs(def_dict)
neq = Equation(lhs,rhs).tag(**eq.tags)
new_equations.append( neq )
dmodel['equations'] = new_equations
dmodel.check_consistency()
f_eqs = [eq for eq in dmodel.equations if eq.tags['eq_type'] in ('f','arbitrage')]
g_eqs = [eq for eq in dmodel.equations if eq.tags['eq_type'] in ('g','transition')]
h_eqs = [eq for eq in dmodel.equations if eq.tags['eq_type'] in ('h','expectation')]
states_vars = [eq.lhs for eq in g_eqs]
exp_vars = [eq.lhs for eq in h_eqs]
controls = set(dmodel.variables) - set(states_vars + exp_vars)
controls = list(controls)
states_vars = [v for v in dmodel.variables if v in states_vars]
exp_vars = [v for v in dmodel.variables if v in exp_vars]
controls = [v for v in dmodel.variables if v in controls]
# now we remove the left side of equations
f_eqs = [eq.gap for eq in f_eqs]
g_eqs = [eq.rhs for eq in g_eqs]
h_eqs = [eq.rhs for eq in h_eqs]
g_eqs = [map_function_to_expression(lambda x: timeshift(x,1),eq) for eq in g_eqs]
#h_eqs = [map_function_to_expression(lambda x: timeshift(x,-1),eq) for eq in h_eqs] #no
# sub_list[v] = v.name
# read complementarity conditions
compcond = {}
of_eqs = [eq for eq in dmodel.equations if eq.tags['eq_type'] in ('f','arbitrage')]
locals = {}
import sympy
locals['inf'] = sympy.Symbol('inf')
locals['log'] = sympy.log # this should be more generic
locals['exp'] = sympy.exp
for v in dmodel.variables + dmodel.parameters:
locals[v.name] = v
import re
compregex = re.compile('(.*)<=(.*)<=(.*)')
for eq in of_eqs:
tg = eq.tags['complementarity']
[lhs,mhs,rhs] = compregex.match(tg).groups()
[lhs,mhs,rhs] = [dmodel.eval_string(x) for x in [lhs,mhs,rhs] ]
compcond[mhs] = (lhs,rhs)
complementarities = [compcond[v] for v in controls]
inf_bounds = [c[0] for c in complementarities]
sup_bounds = [c[1] for c in complementarities]
data = {
'f_eqs': f_eqs,
'g_eqs': g_eqs,
'h_eqs': h_eqs,
'controls': controls,
'states_vars': states_vars,
'exp_vars': exp_vars,
'inf_bounds': inf_bounds,
'sup_bounds': sup_bounds
}
self.__transformed_model__ = data # cache computation
return data
def solve_portfolio_model(model, pf_names, order=2, 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 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)
return dr
X_bar, X_s, X_ss = approximate_controls(new_model, order=2, return_dr=False)
return X_bar[n_controls-n_pfs:n_controls]
if guess is not None:
x0 = numpy.array(guess)
else:
x0 = numpy.zeros(n_pfs)
from dolo.numeric.solver import solver
portfolios_0 = solver(constant_residuals, x0)
print('Zero order portfolios: {}'.format(portfolios_0))
print('Zero order portfolios: 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)
return dr
else:
[X_bar, X_s, X_ss, X_sss] = approximate_controls(new_model, order=3, return_dr=False)
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:')
print(dynamic_residuals(y0))
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)
return dr