Allow exogenous variables in pac.estimation.iterative_ols routine.
The parameters associated to these additional variables can be estimated or calibrated.time-shift
Stéphane Adjemian (Charybdis)
parent
21e6260011
commit
d00b57541e
GPG Key ID:
A6D44CB9C64CE77B
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@ -1,4 +1,4 @@
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function [pacmodl, lhs, rhs, pnames, enames, xnames, pid, eid, xid, pnames_, ipnames_, params, data, islaggedvariables, eqtag] = ...
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function [pacmodl, lhs, rhs, pnames, enames, xnames, rname, pid, eid, xid, pnames_, ipnames_, params, data, islaggedvariables, eqtag] = ...
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init(M_, oo_, eqname, params, data, range)
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% Copyright (C) 2018-2019 Dynare Team
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@ -70,7 +70,10 @@ end
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stack = dbstack;
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ipnames__ = ipnames_; % The user provided order.
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if isequal(stack(2).name, 'iterative_ols')
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ipnames_ = [M_.pac.(pacmodl).equations.(eqtag).ec.params; M_.pac.(pacmodl).equations.(eqtag).ar.params']; % The correct order.
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ipnames_ = [M_.pac.(pacmodl).equations.(eqtag).ec.params; M_.pac.(pacmodl).equations.(eqtag).ar.params'];
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if isfield(M_.pac.(pacmodl).equations.(eqtag), 'additive')
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ipnames_ = [ipnames_; M_.pac.(pacmodl).equations.(eqtag).additive.params(~isnan(M_.pac.(pacmodl).equations.(eqtag).additive.params))'];
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end
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if isfield(M_.pac.(pacmodl).equations.(eqtag), 'share_of_optimizing_agents_index')
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ipnames_ = [ipnames_; M_.pac.(pacmodl).equations.(eqtag).share_of_optimizing_agents_index];
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end
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@ -43,7 +43,7 @@ global M_ oo_
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debug = false; % If true, prints the value of SSR for the initial guess.
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[pacmodl, ~, rhs, ~, ~, ~, ~, ~, ~, ~, ipnames_, params, data, ~, eqtag] = ...
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[pacmodl, ~, rhs, ~, ~, ~, rname, ~, ~, ~, ~, ipnames_, params, data, ~, eqtag] = ...
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pac.estimate.init(M_, oo_, eqname, params, data, range);
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% Set initial condition.
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@ -52,12 +52,19 @@ params0 = cell2mat(struct2cell(params));
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% Set flag for models with non optimizing agents.
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is_non_optimizing_agents = isfield(M_.pac.(pacmodl).equations.(eqtag), 'non_optimizing_behaviour');
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% Set flag for models with exogenous variables (outside of non optimizing agents part)
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if isfield(M_.pac.(pacmodl).equations.(eqtag), 'additive')
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is_exogenous_variables = length(M_.pac.(pacmodl).equations.(eqtag).additive.vars)>1;
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else
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is_exogenous_variables = false;
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end
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if is_non_optimizing_agents
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non_optimizing_behaviour = M_.pac.(pacmodl).equations.(eqtag).non_optimizing_behaviour;
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non_optimizing_behaviour_params = NaN(length(non_optimizing_behaviour.params), 1);
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noparams = isnan(non_optimizing_behaviour.params);
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if ~all(noparams)
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% Find estimated non optimizing behaviour parameters (if any.
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% Find estimated non optimizing behaviour parameters (if any).
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non_optimizing_behaviour_estimated_params = ismember(M_.param_names(non_optimizing_behaviour.params), fieldnames(params));
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if any(non_optimizing_behaviour_estimated_params)
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error('The estimation of non optimizing behaviour parameters is not yet allowed.')
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@ -77,11 +84,32 @@ if is_non_optimizing_agents
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error('The share of optimizing agents shoud be in (0,1).')
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end
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end
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share_of_optimizing_agents_index = M_.pac.(pacmodl).equations.(eqtag).share_of_optimizing_agents_index;
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else
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share_of_optimizing_agents = 1.0;
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share_of_optimizing_agents_index = [];
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estimate_non_optimizing_agents_share = false;
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end
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if is_exogenous_variables
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additive = M_.pac.(pacmodl).equations.(eqtag).additive;
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residual_id = find(strcmp(rname, M_.exo_names));
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residual_jd = find(additive.vars==residual_id & ~additive.isendo);
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additive.params(residual_jd) = [];
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additive.vars(residual_jd) = [];
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additive.isendo(residual_jd) = [];
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additive.lags(residual_jd) = [];
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additive.scaling_factor(residual_jd) = [];
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additive.estimation = ismember(additive.params, ipnames_);
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else
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additive.params = [];
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additive.vars = [];
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additive.isendo = [];
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additive.lags = [];
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additive.scaling_factor = [];
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additive.estimation = [];
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end
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% Build PAC expectation matrix expression.
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dataForPACExpectation0 = dseries();
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listofvariables0 = {};
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@ -144,6 +172,42 @@ else
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dataForNonOptimizingBehaviour = dseries();
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end
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% Build data for exogenous variables (out of non optimizing behaviour term).
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if is_exogenous_variables
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listofvariables2 = {}; j = 0;
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dataForExogenousVariables = dseries(); % Estimated parameters
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dataForExogenousVariables_ = 0; % Calibrated parameters
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is_any_calibrated_parameter_x = false;
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is_any_estimated_parameter_x = false;
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for i=1:length(additive.vars)
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if additive.isendo(i)
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variable = M_.endo_names{additive.vars(i)};
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else
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variable = M_.exo_names{additive.vars(i)};
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end
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if additive.estimation(i)
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j = j+1;
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is_any_estimated_parameter_x = true;
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listofvariables2{j} = variable;
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dataForExogenousVariables = [dataForExogenousVariables, additive.scaling_factor(i)*data{variable}.lag(additive.lags(i))];
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else
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is_any_calibrated_parameter_x = true;
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tmp = data{variable}.lag(additive.lags(i)).data;
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if ~isnan(additive.params(i))
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tmp = M_.params(additive.params(i))*tmp;
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end
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tmp = additive.scaling_factor(i)*tmp;
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dataForExogenousVariables_ = dataForExogenousVariables_+tmp;
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end
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end
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if is_any_calibrated_parameter_x
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dataForExogenousVariables_ = dseries(dataForExogenousVariables_, data.dates(1), 'exogenous_variables_associated_with_calibrated_parameters');
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end
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else
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dataForExogenousVariables = dseries();
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dataForExogenousVariables_ = dseries();
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end
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% Reorder ec.vars locally if necessary. Second variable must be the
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% endogenous variable, while the first must be the associated trend.
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if M_.pac.(pacmodl).equations.(eqtag).ec.isendo(2)
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@ -152,7 +216,7 @@ else
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ecvars = flip(M_.pac.(pacmodl).equations.(eqtag).ec.vars);
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end
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% Build matrix for EC and AR terms.
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%% Build matrix for EC and AR terms.
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DataForOLS = dseries();
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% Error correction term is trend minus the level of the endogenous variable.
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@ -169,16 +233,39 @@ end
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XDATA = DataForOLS{listofvariables3{:}}(range).data;
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if estimate_non_optimizing_agents_share
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[~, ecm_params_id] = setdiff(ipnames_, M_.pac.(pacmodl).equations.(eqtag).share_of_optimizing_agents_index);
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[~, share_param_id] = setdiff(1:length(ipnames_), ecm_params_id);
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params0_ = params0(ecm_params_id);
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share_of_optimizing_agents = params0(share_param_id);
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if share_of_optimizing_agents>1 || share_of_optimizing_agents<0
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error('Initial value for the share of optimizing agents shoud be in (0,1).')
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end
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% Get index in params0 for share of optimizing agents parameter (if
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% not estimated, params_id_0 is empty).
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if is_non_optimizing_agents
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params_id_0 = find(ipnames_==share_of_optimizing_agents_index);
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else
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params0_ = params0;
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params_id_0 = [];
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end
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% Get indices in params0 for EC and AR parameters
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[~, params_id_1] = setdiff(ipnames_, [share_of_optimizing_agents_index, additive.params]);
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% Get indices in params0 for other parameters (optimizing agents
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% share plus parameters related to exogenous variables).
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[~, params_id_2] = setdiff(1:length(ipnames_), params_id_1);
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% Get indices in params0 for the parameters associated to the
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% exogenous variables.
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params_id_3 = setdiff(params_id_2, params_id_0);
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% Get values for EC and AR parameters
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params0_ = params0(params_id_1);
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% Get values for parameters associated to the exogenous variables.
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params0__ = params0(params_id_3);
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% Get value of the share of optimizing agents.
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if estimate_non_optimizing_agents_share
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share_of_optimizing_agents = params0(params_id_0);
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end
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% Check that the share is in (0,1)
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if share_of_optimizing_agents>1 || share_of_optimizing_agents<0
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error('Initial value for the share of optimizing agents shoud be in (0,1).')
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end
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% Update the vector of parameters.
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@ -187,19 +274,6 @@ M_.params(ipnames_) = params0;
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% Update the reduced form PAC expectation parameters and compute the expectations.
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[PacExpectations, M_] = UpdatePacExpectationsData(dataPAC0, dataPAC1, data, range, pacmodl, eqtag, M_, oo_);
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if debug
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YDATA = data{M_.endo_names{M_.pac.(pacmodl).equations.(eqtag).lhs_var}}(range).data;
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if is_non_optimizing_agents
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YDATA = YDATA-share_of_optimizing_agents*PacExpectations;
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YDATA = YDATA-(1-share_of_optimizing_agents)*(dataForNonOptimizingBehaviour(range).data*non_optimizing_behaviour_params);
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else
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YDATA = YDATA-PacExpectations;
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end
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r = YDATA-XDATA*(params0_*share_of_optimizing_agents);
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ssr = r'*r;
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fprintf('\nInitial value of the objective (SSR) is %s.\n\n', num2str(ssr));
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end
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noconvergence = true;
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counter = 0;
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@ -213,6 +287,14 @@ while noconvergence
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else
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YDATA = YDATA-PacExpectations;
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end
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if is_exogenous_variables
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if is_any_calibrated_parameter_x
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YDATA = YDATA-dataForExogenousVariables_(range).data;
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end
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if is_any_estimated_parameter_x
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YDATA = YDATA-dataForExogenousVariables{listofvariables2{:}}(range).data*params0__;
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end
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end
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% Run OLS to estimate PAC parameters (autoregressive parameters and error correction parameter).
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params1_ = (XDATA\YDATA)/share_of_optimizing_agents;
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% Compute residuals and sum of squareed residuals.
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@ -226,18 +308,53 @@ while noconvergence
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% Update the value of the share of non optimizing agents (if estimated)
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if estimate_non_optimizing_agents_share
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% First update the parameters and compute the PAC expectation reduced form parameters.
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M_.params(ipnames_(ecm_params_id)) = params0_;
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M_.params(ipnames_(params_id_1)) = params0_;
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[PacExpectations, M_] = UpdatePacExpectationsData(dataPAC0, dataPAC1, data, range, pacmodl, eqtag, M_, oo_);
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% Set vector for left handside variable.
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YDATA = data{M_.endo_names{M_.pac.(pacmodl).equations.(eqtag).lhs_var}}(range).data;
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YDATA = YDATA-dataForNonOptimizingBehaviour(range).data*non_optimizing_behaviour_params;
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if is_exogenous_variables && is_any_calibrated_parameter_x
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YDATA = YDATA-dataForExogenousVariables_(range).data;
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end
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% Set vector for regressor.
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ZDATA = XDATA*params0_+PacExpectations-dataForNonOptimizingBehaviour(range).data*non_optimizing_behaviour_params;
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if is_exogenous_variables && is_any_estimated_parameter_x
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ZDATA = [ZDATA, dataForExogenousVariables{listofvariables2{:}}(range).data];
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end
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% Update the (estimated) share of optimizing agents by running OLS
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share_of_optimizing_agents = (ZDATA\YDATA);
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beta = (ZDATA\YDATA);
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share_of_optimizing_agents = beta(1);
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if is_exogenous_variables && is_any_estimated_parameter_x
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params0__ = beta(2:end);
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end
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% Force the share of optimizing agents to be in [0,1].
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share_of_optimizing_agents = max(min(share_of_optimizing_agents, 1.0), 0.0);
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M_.params(ipnames_(share_param_id)) = share_of_optimizing_agents;
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M_.params(ipnames_(params_id_0)) = share_of_optimizing_agents;
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if is_exogenous_variables && is_any_estimated_parameter_x
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M_.params(ipnames_(params_id_3)) = params0__;
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end
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elseif is_exogenous_variables && is_any_estimated_parameter_x
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% First update the parameters and compute the PAC expectation reduced form parameters.
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M_.params(ipnames_(params_id_1)) = params0_;
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[PacExpectations, M_] = UpdatePacExpectationsData(dataPAC0, dataPAC1, data, range, pacmodl, eqtag, M_, oo_);
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% Set vector for left handside variable.
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YDATA = data{M_.endo_names{M_.pac.(pacmodl).equations.(eqtag).lhs_var}}(range).data;
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if is_any_calibrated_parameter_x
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YDATA = YDATA-dataForExogenousVariables_(range).data;
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end
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if is_non_optimizing_agents
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YDATA = YDATA-share_of_optimizing_agents*(XDATA*params0_+PacExpectations) - ...
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(1-share_of_optimizing_agents)*(dataForNonOptimizingBehaviour(range).data*non_optimizing_behaviour_params);
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else
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YDATA = YDATA-XDATA*params0_-PacExpectations;
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end
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% Set vector for regressor
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ZDATA = dataForExogenousVariables{listofvariables2{:}}(range).data;
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% Update the (estimated) parameters associated to the
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% exogenous variables.
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beta = (ZDATA\YDATA);
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params0__ = beta;
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M_.params(ipnames_(params_id_3)) = params0__;
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else
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M_.params(ipnames_) = params0_;
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[PacExpectations, M_] = UpdatePacExpectationsData(dataPAC0, dataPAC1, data, range, pacmodl, eqtag, M_, oo_);
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@ -76,7 +76,7 @@ if nargin>4 && (isequal(optimizer, 'GaussNewton') || isequal(optimizer, 'lsqnonl
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objective = 'r_';
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end
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[pacmodl, lhs, rhs, pnames, enames, xnames, pid, eid, xid, ~, ipnames_, params, data, islaggedvariables, eqtag] = ...
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[pacmodl, lhs, rhs, pnames, enames, xnames, ~, pid, eid, xid, ~, ipnames_, params, data, islaggedvariables, eqtag] = ...
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pac.estimate.init(M_, oo_, eqname, params, data, range);
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% Check that the error correction term is correct.
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@ -1 +1 @@
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Subproject commit 2312ce13dc58d3645e0f37f1a757fbe50b6932fd
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Subproject commit 5f013756f23fe89ec946fcff2cfdad7627693e12
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@ -426,6 +426,10 @@ MODFILES = \
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pac/trend-component-20-2/example.mod \
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pac/trend-component-20-3/example.mod \
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pac/trend-component-20-4/example.mod \
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pac/trend-component-21/example.mod \
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pac/trend-component-22/example.mod \
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pac/trend-component-23/example.mod \
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pac/trend-component-24/example.mod \
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ecb/backward-models/irf/solow_1.mod \
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ecb/backward-models/irf/solow_2.mod \
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dynare-command-options/ramst.mod
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@ -0,0 +1,8 @@
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#!/bin/sh
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rm -rf example
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rm -rf +example
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rm -f example.log
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rm -f *.mat
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rm -f *.m
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rm -f *.dat
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@ -0,0 +1,160 @@
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// --+ options: json=compute, stochastic +--
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var x1 x2 x1bar x2bar z y x;
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varexo ex1 ex2 ex1bar ex2bar ez ey ex;
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parameters
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rho_1 rho_2 rho_3 rho_4
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a_x1_0 a_x1_1 a_x1_2 a_x1_x2_1 a_x1_x2_2
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a_x2_0 a_x2_1 a_x2_2 a_x2_x1_1 a_x2_x1_2
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e_c_m c_z_1 c_z_2 c_z_dx2 beta
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lambda;
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rho_1 = .9;
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rho_2 = -.2;
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rho_3 = .4;
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rho_4 = -.3;
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a_x1_0 = -.9;
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a_x1_1 = .4;
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a_x1_2 = .3;
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a_x1_x2_1 = .1;
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a_x1_x2_2 = .2;
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a_x2_0 = -.9;
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a_x2_1 = .2;
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a_x2_2 = -.1;
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a_x2_x1_1 = -.1;
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a_x2_x1_2 = .2;
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beta = .2;
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e_c_m = .5;
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c_z_1 = .2;
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c_z_2 = -.1;
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c_z_dx2 = .3;
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lambda = 0.5; // Share of optimizing agents.
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trend_component_model(model_name=toto, eqtags=['eq:x1', 'eq:x2', 'eq:x1bar', 'eq:x2bar'], targets=['eq:x1bar', 'eq:x2bar']);
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pac_model(auxiliary_model_name=toto, discount=beta, model_name=pacman);
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model;
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[name='eq:y']
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y = rho_1*y(-1) + rho_2*y(-2) + ey;
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[name='eq:x']
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x = rho_3*x(-1) + rho_4*x(-2) + ex;
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[name='eq:x1']
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diff(x1) = a_x1_0*(x1(-1)-x1bar(-1)) + a_x1_1*diff(x1(-1)) + a_x1_2*diff(x1(-2)) + a_x1_x2_1*diff(x2(-1)) + a_x1_x2_2*diff(x2(-2)) + ex1;
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[name='eq:x2']
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diff(x2) = a_x2_0*(x2(-1)-x2bar(-1)) + a_x2_1*diff(x1(-1)) + a_x2_2*diff(x1(-2)) + a_x2_x1_1*diff(x2(-1)) + a_x2_x1_2*diff(x2(-2)) + ex2;
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[name='eq:x1bar']
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x1bar = x1bar(-1) + ex1bar;
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||||
|
||||
[name='eq:x2bar']
|
||||
x2bar = x2bar(-1) + ex2bar;
|
||||
|
||||
[name='zpac']
|
||||
diff(z) = lambda*(e_c_m*(x1(-1)-z(-1)) + c_z_1*diff(z(-1)) + c_z_2*diff(z(-2)) + pac_expectation(pacman)) + (1-lambda)*( y + x) + c_z_dx2*diff(x2) + ez;
|
||||
|
||||
end;
|
||||
|
||||
shocks;
|
||||
var ex1 = 1.0;
|
||||
var ex2 = 1.0;
|
||||
var ex1bar = 1.0;
|
||||
var ex2bar = 1.0;
|
||||
var ez = 1.0;
|
||||
var ey = 0.1;
|
||||
var ex = 0.1;
|
||||
end;
|
||||
|
||||
// Initialize the PAC model (build the Companion VAR representation for the auxiliary model).
|
||||
pac.initialize('pacman');
|
||||
|
||||
// Update the parameters of the PAC expectation model (h0 and h1 vectors).
|
||||
pac.update.expectation('pacman');
|
||||
|
||||
// Set initial conditions to zero. Please use more sensible values if any...
|
||||
initialconditions = dseries(zeros(10, M_.endo_nbr+M_.exo_nbr), 2000Q1, vertcat(M_.endo_names,M_.exo_names));
|
||||
|
||||
// Simulate the model for 500 periods
|
||||
TrueData = simul_backward_model(initialconditions, 5000);
|
||||
|
||||
// Define a structure describing the parameters to be estimated (with initial conditions).
|
||||
clear eparams
|
||||
eparams.e_c_m = .9;
|
||||
eparams.c_z_1 = .5;
|
||||
eparams.c_z_2 = .2;
|
||||
eparams.c_z_dx2 = .5;
|
||||
eparams.lambda = .7;
|
||||
|
||||
// Define the dataset used for estimation
|
||||
edata = TrueData;
|
||||
edata.ez = dseries(NaN(TrueData.nobs, 1), 2000Q1, 'ez');
|
||||
pac.estimate.iterative_ols('zpac', eparams, edata, 2005Q1:2005Q1+4000);
|
||||
|
||||
e_c_m_iterative_ols = M_.params(strmatch('e_c_m', M_.param_names, 'exact'));
|
||||
c_z_1_iterative_ols = M_.params(strmatch('c_z_1', M_.param_names, 'exact'));
|
||||
c_z_2_iterative_ols = M_.params(strmatch('c_z_2', M_.param_names, 'exact'));
|
||||
c_z_dx2_iterative_ols = M_.params(strmatch('c_z_dx2', M_.param_names, 'exact'));
|
||||
lambda_iterative_ols = M_.params(strmatch('lambda', M_.param_names, 'exact'));
|
||||
|
||||
disp(sprintf('Estimate of e_c_m: %f', e_c_m_iterative_ols))
|
||||
disp(sprintf('Estimate of c_z_1: %f', c_z_1_iterative_ols))
|
||||
disp(sprintf('Estimate of c_z_2: %f', c_z_2_iterative_ols))
|
||||
disp(sprintf('Estimate of c_z_dx2: %f', c_z_dx2_iterative_ols))
|
||||
disp(sprintf('Estimate of lambda: %f', lambda_iterative_ols))
|
||||
|
||||
skipline(2)
|
||||
|
||||
// Define a structure describing the parameters to be estimated (with initial conditions).
|
||||
clear eparams
|
||||
eparams.e_c_m = .9;
|
||||
eparams.c_z_1 = .5;
|
||||
eparams.c_z_2 = .2;
|
||||
eparams.c_z_dx2 = .5;
|
||||
eparams.lambda = .7;
|
||||
|
||||
edata = TrueData;
|
||||
edata.ez = dseries(NaN(TrueData.nobs, 1), 2000Q1, 'ez');
|
||||
pac.estimate.nls('zpac', eparams, edata, 2005Q1:2005Q1+4000, 'fmincon');
|
||||
|
||||
e_c_m_nls = M_.params(strmatch('e_c_m', M_.param_names, 'exact'));
|
||||
c_z_1_nls = M_.params(strmatch('c_z_1', M_.param_names, 'exact'));
|
||||
c_z_2_nls = M_.params(strmatch('c_z_2', M_.param_names, 'exact'));
|
||||
c_z_dx2_nls = M_.params(strmatch('c_z_dx2', M_.param_names, 'exact'));
|
||||
lambda_nls = M_.params(strmatch('lambda', M_.param_names, 'exact'));
|
||||
|
||||
disp(sprintf('Estimate of e_c_m: %f', e_c_m_nls))
|
||||
disp(sprintf('Estimate of c_z_1: %f', c_z_1_nls))
|
||||
disp(sprintf('Estimate of c_z_2: %f', c_z_2_nls))
|
||||
disp(sprintf('Estimate of c_z_dx2: %f', c_z_dx2_nls))
|
||||
disp(sprintf('Estimate of lambda: %f', lambda_nls))
|
||||
|
||||
if abs(e_c_m_nls-e_c_m_iterative_ols)>.01
|
||||
error('Iterative OLS and NLS do not provide consistent estimates (e_c_m)')
|
||||
end
|
||||
|
||||
if abs(c_z_1_nls-c_z_1_iterative_ols)>.01
|
||||
error('Iterative OLS and NLS do not provide consistent estimates (c_z_1)')
|
||||
end
|
||||
|
||||
if abs(c_z_2_nls-c_z_2_iterative_ols)>.01
|
||||
error('Iterative OLS and NLS do not provide consistent estimates (c_z_2)')
|
||||
end
|
||||
|
||||
if abs(c_z_dx2_nls-c_z_dx2_iterative_ols)>.01
|
||||
error('Iterative OLS and NLS do not provide consistent estimates (c_z_dx2)')
|
||||
end
|
||||
|
||||
if abs(lambda_nls-lambda_iterative_ols)>.01
|
||||
error('Iterative OLS and NLS do not provide consistent estimates (lambda)')
|
||||
end
|
||||
|
|
@ -0,0 +1,8 @@
|
|||
#!/bin/sh
|
||||
|
||||
rm -rf example
|
||||
rm -rf +example
|
||||
rm -f example.log
|
||||
rm -f *.mat
|
||||
rm -f *.m
|
||||
rm -f *.dat
|
||||
|
|
@ -0,0 +1,150 @@
|
|||
// --+ options: json=compute, stochastic +--
|
||||
|
||||
var x1 x2 x1bar x2bar z y x;
|
||||
|
||||
varexo ex1 ex2 ex1bar ex2bar ez ey ex;
|
||||
|
||||
parameters
|
||||
rho_1 rho_2 rho_3 rho_4
|
||||
a_x1_0 a_x1_1 a_x1_2 a_x1_x2_1 a_x1_x2_2
|
||||
a_x2_0 a_x2_1 a_x2_2 a_x2_x1_1 a_x2_x1_2
|
||||
e_c_m c_z_1 c_z_2 c_z_dx2 beta
|
||||
lambda;
|
||||
|
||||
rho_1 = .9;
|
||||
rho_2 = -.2;
|
||||
rho_3 = .4;
|
||||
rho_4 = -.3;
|
||||
|
||||
|
||||
a_x1_0 = -.9;
|
||||
a_x1_1 = .4;
|
||||
a_x1_2 = .3;
|
||||
a_x1_x2_1 = .1;
|
||||
a_x1_x2_2 = .2;
|
||||
|
||||
a_x2_0 = -.9;
|
||||
a_x2_1 = .2;
|
||||
a_x2_2 = -.1;
|
||||
a_x2_x1_1 = -.1;
|
||||
a_x2_x1_2 = .2;
|
||||
|
||||
beta = .2;
|
||||
e_c_m = .5;
|
||||
c_z_1 = .2;
|
||||
c_z_2 = -.1;
|
||||
c_z_dx2 = .3;
|
||||
|
||||
lambda = 0.5; // Share of optimizing agents.
|
||||
|
||||
trend_component_model(model_name=toto, eqtags=['eq:x1', 'eq:x2', 'eq:x1bar', 'eq:x2bar'], targets=['eq:x1bar', 'eq:x2bar']);
|
||||
|
||||
pac_model(auxiliary_model_name=toto, discount=beta, model_name=pacman);
|
||||
|
||||
model;
|
||||
|
||||
[name='eq:y']
|
||||
y = rho_1*y(-1) + rho_2*y(-2) + ey;
|
||||
|
||||
[name='eq:x']
|
||||
x = rho_3*x(-1) + rho_4*x(-2) + ex;
|
||||
|
||||
[name='eq:x1']
|
||||
diff(x1) = a_x1_0*(x1(-1)-x1bar(-1)) + a_x1_1*diff(x1(-1)) + a_x1_2*diff(x1(-2)) + a_x1_x2_1*diff(x2(-1)) + a_x1_x2_2*diff(x2(-2)) + ex1;
|
||||
|
||||
[name='eq:x2']
|
||||
diff(x2) = a_x2_0*(x2(-1)-x2bar(-1)) + a_x2_1*diff(x1(-1)) + a_x2_2*diff(x1(-2)) + a_x2_x1_1*diff(x2(-1)) + a_x2_x1_2*diff(x2(-2)) + ex2;
|
||||
|
||||
[name='eq:x1bar']
|
||||
x1bar = x1bar(-1) + ex1bar;
|
||||
|
||||
[name='eq:x2bar']
|
||||
x2bar = x2bar(-1) + ex2bar;
|
||||
|
||||
[name='zpac']
|
||||
diff(z) = lambda*(e_c_m*(x1(-1)-z(-1)) + c_z_1*diff(z(-1)) + c_z_2*diff(z(-2)) + pac_expectation(pacman)) + (1-lambda)*( y + x) + c_z_dx2*diff(x2) + ez;
|
||||
|
||||
end;
|
||||
|
||||
shocks;
|
||||
var ex1 = 1.0;
|
||||
var ex2 = 1.0;
|
||||
var ex1bar = 1.0;
|
||||
var ex2bar = 1.0;
|
||||
var ez = 1.0;
|
||||
var ey = 0.1;
|
||||
var ex = 0.1;
|
||||
end;
|
||||
|
||||
// Initialize the PAC model (build the Companion VAR representation for the auxiliary model).
|
||||
pac.initialize('pacman');
|
||||
|
||||
// Update the parameters of the PAC expectation model (h0 and h1 vectors).
|
||||
pac.update.expectation('pacman');
|
||||
|
||||
// Set initial conditions to zero. Please use more sensible values if any...
|
||||
initialconditions = dseries(zeros(10, M_.endo_nbr+M_.exo_nbr), 2000Q1, vertcat(M_.endo_names,M_.exo_names));
|
||||
|
||||
// Simulate the model for 500 periods
|
||||
TrueData = simul_backward_model(initialconditions, 5000);
|
||||
|
||||
// Define a structure describing the parameters to be estimated (with initial conditions).
|
||||
clear eparams
|
||||
eparams.e_c_m = .9;
|
||||
eparams.c_z_1 = .5;
|
||||
eparams.c_z_2 = .2;
|
||||
eparams.c_z_dx2 = .5;
|
||||
|
||||
// Define the dataset used for estimation
|
||||
edata = TrueData;
|
||||
edata.ez = dseries(NaN(TrueData.nobs, 1), 2000Q1, 'ez');
|
||||
pac.estimate.iterative_ols('zpac', eparams, edata, 2005Q1:2005Q1+4000);
|
||||
|
||||
e_c_m_iterative_ols = M_.params(strmatch('e_c_m', M_.param_names, 'exact'));
|
||||
c_z_1_iterative_ols = M_.params(strmatch('c_z_1', M_.param_names, 'exact'));
|
||||
c_z_2_iterative_ols = M_.params(strmatch('c_z_2', M_.param_names, 'exact'));
|
||||
c_z_dx2_iterative_ols = M_.params(strmatch('c_z_dx2', M_.param_names, 'exact'));
|
||||
|
||||
disp(sprintf('Estimate of e_c_m: %f', e_c_m_iterative_ols))
|
||||
disp(sprintf('Estimate of c_z_1: %f', c_z_1_iterative_ols))
|
||||
disp(sprintf('Estimate of c_z_2: %f', c_z_2_iterative_ols))
|
||||
disp(sprintf('Estimate of c_z_dx2: %f', c_z_dx2_iterative_ols))
|
||||
|
||||
skipline(2)
|
||||
|
||||
// Define a structure describing the parameters to be estimated (with initial conditions).
|
||||
clear eparams
|
||||
eparams.e_c_m = .9;
|
||||
eparams.c_z_1 = .5;
|
||||
eparams.c_z_2 = .2;
|
||||
eparams.c_z_dx2 = .5;
|
||||
|
||||
edata = TrueData;
|
||||
edata.ez = dseries(NaN(TrueData.nobs, 1), 2000Q1, 'ez');
|
||||
pac.estimate.nls('zpac', eparams, edata, 2005Q1:2005Q1+4000, 'fmincon');
|
||||
|
||||
e_c_m_nls = M_.params(strmatch('e_c_m', M_.param_names, 'exact'));
|
||||
c_z_1_nls = M_.params(strmatch('c_z_1', M_.param_names, 'exact'));
|
||||
c_z_2_nls = M_.params(strmatch('c_z_2', M_.param_names, 'exact'));
|
||||
c_z_dx2_nls = M_.params(strmatch('c_z_dx2', M_.param_names, 'exact'));
|
||||
|
||||
disp(sprintf('Estimate of e_c_m: %f', e_c_m_nls))
|
||||
disp(sprintf('Estimate of c_z_1: %f', c_z_1_nls))
|
||||
disp(sprintf('Estimate of c_z_2: %f', c_z_2_nls))
|
||||
disp(sprintf('Estimate of c_z_dx2: %f', c_z_dx2_nls))
|
||||
|
||||
if abs(e_c_m_nls-e_c_m_iterative_ols)>.01
|
||||
error('Iterative OLS and NLS do not provide consistent estimates (e_c_m)')
|
||||
end
|
||||
|
||||
if abs(c_z_1_nls-c_z_1_iterative_ols)>.01
|
||||
error('Iterative OLS and NLS do not provide consistent estimates (c_z_1)')
|
||||
end
|
||||
|
||||
if abs(c_z_2_nls-c_z_2_iterative_ols)>.01
|
||||
error('Iterative OLS and NLS do not provide consistent estimates (c_z_2)')
|
||||
end
|
||||
|
||||
if abs(c_z_dx2_nls-c_z_dx2_iterative_ols)>.01
|
||||
error('Iterative OLS and NLS do not provide consistent estimates (c_z_dx2)')
|
||||
end
|
||||
|
|
@ -0,0 +1,8 @@
|
|||
#!/bin/sh
|
||||
|
||||
rm -rf example
|
||||
rm -rf +example
|
||||
rm -f example.log
|
||||
rm -f *.mat
|
||||
rm -f *.m
|
||||
rm -f *.dat
|
||||
|
|
@ -0,0 +1,165 @@
|
|||
// --+ options: json=compute, stochastic +--
|
||||
|
||||
var x1 x2 x1bar x2bar z y x u;
|
||||
|
||||
varexo ex1 ex2 ex1bar ex2bar ez ey ex eu;
|
||||
|
||||
parameters
|
||||
rho_1 rho_2 rho_3 rho_4
|
||||
a_x1_0 a_x1_1 a_x1_2 a_x1_x2_1 a_x1_x2_2
|
||||
a_x2_0 a_x2_1 a_x2_2 a_x2_x1_1 a_x2_x1_2
|
||||
e_c_m c_z_1 c_z_2 c_z_dx2 c_z_u beta
|
||||
lambda;
|
||||
|
||||
rho_1 = .9;
|
||||
rho_2 = -.2;
|
||||
rho_3 = .4;
|
||||
rho_4 = -.3;
|
||||
|
||||
|
||||
a_x1_0 = -.9;
|
||||
a_x1_1 = .4;
|
||||
a_x1_2 = .3;
|
||||
a_x1_x2_1 = .1;
|
||||
a_x1_x2_2 = .2;
|
||||
|
||||
a_x2_0 = -.9;
|
||||
a_x2_1 = .2;
|
||||
a_x2_2 = -.1;
|
||||
a_x2_x1_1 = -.1;
|
||||
a_x2_x1_2 = .2;
|
||||
|
||||
beta = .2;
|
||||
e_c_m = .5;
|
||||
c_z_1 = .2;
|
||||
c_z_2 = -.1;
|
||||
c_z_dx2 = .3;
|
||||
c_z_u = .3;
|
||||
|
||||
lambda = 0.5; // Share of optimizing agents.
|
||||
|
||||
trend_component_model(model_name=toto, eqtags=['eq:x1', 'eq:x2', 'eq:x1bar', 'eq:x2bar'], targets=['eq:x1bar', 'eq:x2bar']);
|
||||
|
||||
pac_model(auxiliary_model_name=toto, discount=beta, model_name=pacman);
|
||||
|
||||
model;
|
||||
|
||||
[name='eq:u']
|
||||
u = .5*u(-1) - .2*u(-2) + eu;
|
||||
|
||||
[name='eq:y']
|
||||
y = rho_1*y(-1) + rho_2*y(-2) + ey;
|
||||
|
||||
[name='eq:x']
|
||||
x = rho_3*x(-1) + rho_4*x(-2) + ex;
|
||||
|
||||
[name='eq:x1']
|
||||
diff(x1) = a_x1_0*(x1(-1)-x1bar(-1)) + a_x1_1*diff(x1(-1)) + a_x1_2*diff(x1(-2)) + a_x1_x2_1*diff(x2(-1)) + a_x1_x2_2*diff(x2(-2)) + ex1;
|
||||
|
||||
[name='eq:x2']
|
||||
diff(x2) = a_x2_0*(x2(-1)-x2bar(-1)) + a_x2_1*diff(x1(-1)) + a_x2_2*diff(x1(-2)) + a_x2_x1_1*diff(x2(-1)) + a_x2_x1_2*diff(x2(-2)) + ex2;
|
||||
|
||||
[name='eq:x1bar']
|
||||
x1bar = x1bar(-1) + ex1bar;
|
||||
|
||||
[name='eq:x2bar']
|
||||
x2bar = x2bar(-1) + ex2bar;
|
||||
|
||||
[name='zpac']
|
||||
diff(z) = lambda*(e_c_m*(x1(-1)-z(-1)) + c_z_1*diff(z(-1)) + c_z_2*diff(z(-2)) + pac_expectation(pacman)) + (1-lambda)*( y + x) + c_z_dx2*diff(x2) + c_z_u*u + ez;
|
||||
|
||||
end;
|
||||
|
||||
shocks;
|
||||
var ex1 = 1.0;
|
||||
var ex2 = 1.0;
|
||||
var ex1bar = 1.0;
|
||||
var ex2bar = 1.0;
|
||||
var ez = 1.0;
|
||||
var ey = 0.1;
|
||||
var ex = 0.1;
|
||||
var eu = 0.05;
|
||||
end;
|
||||
|
||||
// Initialize the PAC model (build the Companion VAR representation for the auxiliary model).
|
||||
pac.initialize('pacman');
|
||||
|
||||
// Update the parameters of the PAC expectation model (h0 and h1 vectors).
|
||||
pac.update.expectation('pacman');
|
||||
|
||||
// Set initial conditions to zero. Please use more sensible values if any...
|
||||
initialconditions = dseries(zeros(10, M_.endo_nbr+M_.exo_nbr), 2000Q1, vertcat(M_.endo_names,M_.exo_names));
|
||||
|
||||
// Simulate the model for 500 periods
|
||||
TrueData = simul_backward_model(initialconditions, 5000);
|
||||
|
||||
// Define a structure describing the parameters to be estimated (with initial conditions).
|
||||
clear eparams
|
||||
eparams.e_c_m = .9;
|
||||
eparams.c_z_1 = .5;
|
||||
eparams.c_z_2 = .2;
|
||||
eparams.c_z_dx2 = .5;
|
||||
eparams.lambda = .7;
|
||||
|
||||
// Define the dataset used for estimation
|
||||
edata = TrueData;
|
||||
edata.ez = dseries(NaN(TrueData.nobs, 1), 2000Q1, 'ez');
|
||||
pac.estimate.iterative_ols('zpac', eparams, edata, 2005Q1:2005Q1+4000);
|
||||
|
||||
e_c_m_iterative_ols = M_.params(strmatch('e_c_m', M_.param_names, 'exact'));
|
||||
c_z_1_iterative_ols = M_.params(strmatch('c_z_1', M_.param_names, 'exact'));
|
||||
c_z_2_iterative_ols = M_.params(strmatch('c_z_2', M_.param_names, 'exact'));
|
||||
c_z_dx2_iterative_ols = M_.params(strmatch('c_z_dx2', M_.param_names, 'exact'));
|
||||
lambda_iterative_ols = M_.params(strmatch('lambda', M_.param_names, 'exact'));
|
||||
|
||||
disp(sprintf('Estimate of e_c_m: %f', e_c_m_iterative_ols))
|
||||
disp(sprintf('Estimate of c_z_1: %f', c_z_1_iterative_ols))
|
||||
disp(sprintf('Estimate of c_z_2: %f', c_z_2_iterative_ols))
|
||||
disp(sprintf('Estimate of c_z_dx2: %f', c_z_dx2_iterative_ols))
|
||||
disp(sprintf('Estimate of lambda: %f', lambda_iterative_ols))
|
||||
|
||||
skipline(2)
|
||||
|
||||
// Define a structure describing the parameters to be estimated (with initial conditions).
|
||||
clear eparams
|
||||
eparams.e_c_m = .9;
|
||||
eparams.c_z_1 = .5;
|
||||
eparams.c_z_2 = .2;
|
||||
eparams.c_z_dx2 = .5;
|
||||
eparams.lambda = .7;
|
||||
|
||||
edata = TrueData;
|
||||
edata.ez = dseries(NaN(TrueData.nobs, 1), 2000Q1, 'ez');
|
||||
pac.estimate.nls('zpac', eparams, edata, 2005Q1:2005Q1+4000, 'fmincon');
|
||||
|
||||
e_c_m_nls = M_.params(strmatch('e_c_m', M_.param_names, 'exact'));
|
||||
c_z_1_nls = M_.params(strmatch('c_z_1', M_.param_names, 'exact'));
|
||||
c_z_2_nls = M_.params(strmatch('c_z_2', M_.param_names, 'exact'));
|
||||
c_z_dx2_nls = M_.params(strmatch('c_z_dx2', M_.param_names, 'exact'));
|
||||
lambda_nls = M_.params(strmatch('lambda', M_.param_names, 'exact'));
|
||||
|
||||
disp(sprintf('Estimate of e_c_m: %f', e_c_m_nls))
|
||||
disp(sprintf('Estimate of c_z_1: %f', c_z_1_nls))
|
||||
disp(sprintf('Estimate of c_z_2: %f', c_z_2_nls))
|
||||
disp(sprintf('Estimate of c_z_dx2: %f', c_z_dx2_nls))
|
||||
disp(sprintf('Estimate of lambda: %f', lambda_nls))
|
||||
|
||||
if abs(e_c_m_nls-e_c_m_iterative_ols)>.01
|
||||
error('Iterative OLS and NLS do not provide consistent estimates (e_c_m)')
|
||||
end
|
||||
|
||||
if abs(c_z_1_nls-c_z_1_iterative_ols)>.01
|
||||
error('Iterative OLS and NLS do not provide consistent estimates (c_z_1)')
|
||||
end
|
||||
|
||||
if abs(c_z_2_nls-c_z_2_iterative_ols)>.01
|
||||
error('Iterative OLS and NLS do not provide consistent estimates (c_z_2)')
|
||||
end
|
||||
|
||||
if abs(c_z_dx2_nls-c_z_dx2_iterative_ols)>.01
|
||||
error('Iterative OLS and NLS do not provide consistent estimates (c_z_dx2)')
|
||||
end
|
||||
|
||||
if abs(lambda_nls-lambda_iterative_ols)>.01
|
||||
error('Iterative OLS and NLS do not provide consistent estimates (lambda)')
|
||||
end
|
||||
|
|
@ -0,0 +1,8 @@
|
|||
#!/bin/sh
|
||||
|
||||
rm -rf example
|
||||
rm -rf +example
|
||||
rm -f example.log
|
||||
rm -f *.mat
|
||||
rm -f *.m
|
||||
rm -f *.dat
|
||||
|
|
@ -0,0 +1,155 @@
|
|||
// --+ options: json=compute, stochastic +--
|
||||
|
||||
var x1 x2 x1bar x2bar z y x u;
|
||||
|
||||
varexo ex1 ex2 ex1bar ex2bar ez ey ex eu;
|
||||
|
||||
parameters
|
||||
rho_1 rho_2 rho_3 rho_4
|
||||
a_x1_0 a_x1_1 a_x1_2 a_x1_x2_1 a_x1_x2_2
|
||||
a_x2_0 a_x2_1 a_x2_2 a_x2_x1_1 a_x2_x1_2
|
||||
e_c_m c_z_1 c_z_2 c_z_dx2 c_z_u beta
|
||||
lambda;
|
||||
|
||||
rho_1 = .9;
|
||||
rho_2 = -.2;
|
||||
rho_3 = .4;
|
||||
rho_4 = -.3;
|
||||
|
||||
|
||||
a_x1_0 = -.9;
|
||||
a_x1_1 = .4;
|
||||
a_x1_2 = .3;
|
||||
a_x1_x2_1 = .1;
|
||||
a_x1_x2_2 = .2;
|
||||
|
||||
a_x2_0 = -.9;
|
||||
a_x2_1 = .2;
|
||||
a_x2_2 = -.1;
|
||||
a_x2_x1_1 = -.1;
|
||||
a_x2_x1_2 = .2;
|
||||
|
||||
beta = .2;
|
||||
e_c_m = .5;
|
||||
c_z_1 = .2;
|
||||
c_z_2 = -.1;
|
||||
c_z_dx2 = .3;
|
||||
c_z_u = .3;
|
||||
|
||||
lambda = 0.5; // Share of optimizing agents.
|
||||
|
||||
trend_component_model(model_name=toto, eqtags=['eq:x1', 'eq:x2', 'eq:x1bar', 'eq:x2bar'], targets=['eq:x1bar', 'eq:x2bar']);
|
||||
|
||||
pac_model(auxiliary_model_name=toto, discount=beta, model_name=pacman);
|
||||
|
||||
model;
|
||||
|
||||
[name='eq:u']
|
||||
u = .5*u(-1) - .2*u(-2) + eu;
|
||||
|
||||
[name='eq:y']
|
||||
y = rho_1*y(-1) + rho_2*y(-2) + ey;
|
||||
|
||||
[name='eq:x']
|
||||
x = rho_3*x(-1) + rho_4*x(-2) + ex;
|
||||
|
||||
[name='eq:x1']
|
||||
diff(x1) = a_x1_0*(x1(-1)-x1bar(-1)) + a_x1_1*diff(x1(-1)) + a_x1_2*diff(x1(-2)) + a_x1_x2_1*diff(x2(-1)) + a_x1_x2_2*diff(x2(-2)) + ex1;
|
||||
|
||||
[name='eq:x2']
|
||||
diff(x2) = a_x2_0*(x2(-1)-x2bar(-1)) + a_x2_1*diff(x1(-1)) + a_x2_2*diff(x1(-2)) + a_x2_x1_1*diff(x2(-1)) + a_x2_x1_2*diff(x2(-2)) + ex2;
|
||||
|
||||
[name='eq:x1bar']
|
||||
x1bar = x1bar(-1) + ex1bar;
|
||||
|
||||
[name='eq:x2bar']
|
||||
x2bar = x2bar(-1) + ex2bar;
|
||||
|
||||
[name='zpac']
|
||||
diff(z) = lambda*(e_c_m*(x1(-1)-z(-1)) + c_z_1*diff(z(-1)) + c_z_2*diff(z(-2)) + pac_expectation(pacman)) + (1-lambda)*( y + x) + c_z_dx2*diff(x2) + c_z_u*u + ez;
|
||||
|
||||
end;
|
||||
|
||||
shocks;
|
||||
var ex1 = 1.0;
|
||||
var ex2 = 1.0;
|
||||
var ex1bar = 1.0;
|
||||
var ex2bar = 1.0;
|
||||
var ez = 1.0;
|
||||
var ey = 0.1;
|
||||
var ex = 0.1;
|
||||
var eu = 0.05;
|
||||
end;
|
||||
|
||||
// Initialize the PAC model (build the Companion VAR representation for the auxiliary model).
|
||||
pac.initialize('pacman');
|
||||
|
||||
// Update the parameters of the PAC expectation model (h0 and h1 vectors).
|
||||
pac.update.expectation('pacman');
|
||||
|
||||
// Set initial conditions to zero. Please use more sensible values if any...
|
||||
initialconditions = dseries(zeros(10, M_.endo_nbr+M_.exo_nbr), 2000Q1, vertcat(M_.endo_names,M_.exo_names));
|
||||
|
||||
// Simulate the model for 500 periods
|
||||
TrueData = simul_backward_model(initialconditions, 5000);
|
||||
|
||||
// Define a structure describing the parameters to be estimated (with initial conditions).
|
||||
clear eparams
|
||||
eparams.e_c_m = .9;
|
||||
eparams.c_z_1 = .5;
|
||||
eparams.c_z_2 = .2;
|
||||
eparams.lambda = .7;
|
||||
|
||||
// Define the dataset used for estimation
|
||||
edata = TrueData;
|
||||
edata.ez = dseries(NaN(TrueData.nobs, 1), 2000Q1, 'ez');
|
||||
pac.estimate.iterative_ols('zpac', eparams, edata, 2005Q1:2005Q1+4000);
|
||||
|
||||
e_c_m_iterative_ols = M_.params(strmatch('e_c_m', M_.param_names, 'exact'));
|
||||
c_z_1_iterative_ols = M_.params(strmatch('c_z_1', M_.param_names, 'exact'));
|
||||
c_z_2_iterative_ols = M_.params(strmatch('c_z_2', M_.param_names, 'exact'));
|
||||
lambda_iterative_ols = M_.params(strmatch('lambda', M_.param_names, 'exact'));
|
||||
|
||||
disp(sprintf('Estimate of e_c_m: %f', e_c_m_iterative_ols))
|
||||
disp(sprintf('Estimate of c_z_1: %f', c_z_1_iterative_ols))
|
||||
disp(sprintf('Estimate of c_z_2: %f', c_z_2_iterative_ols))
|
||||
disp(sprintf('Estimate of lambda: %f', lambda_iterative_ols))
|
||||
|
||||
skipline(2)
|
||||
|
||||
// Define a structure describing the parameters to be estimated (with initial conditions).
|
||||
clear eparams
|
||||
eparams.e_c_m = .9;
|
||||
eparams.c_z_1 = .5;
|
||||
eparams.c_z_2 = .2;
|
||||
eparams.lambda = .7;
|
||||
|
||||
edata = TrueData;
|
||||
edata.ez = dseries(NaN(TrueData.nobs, 1), 2000Q1, 'ez');
|
||||
pac.estimate.nls('zpac', eparams, edata, 2005Q1:2005Q1+4000, 'fmincon');
|
||||
|
||||
e_c_m_nls = M_.params(strmatch('e_c_m', M_.param_names, 'exact'));
|
||||
c_z_1_nls = M_.params(strmatch('c_z_1', M_.param_names, 'exact'));
|
||||
c_z_2_nls = M_.params(strmatch('c_z_2', M_.param_names, 'exact'));
|
||||
lambda_nls = M_.params(strmatch('lambda', M_.param_names, 'exact'));
|
||||
|
||||
disp(sprintf('Estimate of e_c_m: %f', e_c_m_nls))
|
||||
disp(sprintf('Estimate of c_z_1: %f', c_z_1_nls))
|
||||
disp(sprintf('Estimate of c_z_2: %f', c_z_2_nls))
|
||||
disp(sprintf('Estimate of lambda: %f', lambda_nls))
|
||||
|
||||
if abs(e_c_m_nls-e_c_m_iterative_ols)>.01
|
||||
error('Iterative OLS and NLS do not provide consistent estimates (e_c_m)')
|
||||
end
|
||||
|
||||
if abs(c_z_1_nls-c_z_1_iterative_ols)>.01
|
||||
error('Iterative OLS and NLS do not provide consistent estimates (c_z_1)')
|
||||
end
|
||||
|
||||
if abs(c_z_2_nls-c_z_2_iterative_ols)>.01
|
||||
error('Iterative OLS and NLS do not provide consistent estimates (c_z_2)')
|
||||
end
|
||||
|
||||
if abs(lambda_nls-lambda_iterative_ols)>.01
|
||||
error('Iterative OLS and NLS do not provide consistent estimates (lambda)')
|
||||
end
|
||||
Loading…
Reference in New Issue