Cosmetic change (remvove trailing spaces).
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@ -7641,82 +7641,82 @@ Estimation based on moments
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===========================
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Provided that you have observations on some endogenous variables, it
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is possible to use Dynare to estimate some or all parameters using a
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method of moments approach. Both the Simulated Method of Moments (SMM)
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and the Generalized Method of Moments (GMM) are available. The general
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idea is to minimize the distance between unconditional model moments
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and corresponding data moments (so called orthogonality or moment
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conditions). For SMM, Dynare computes model moments via stochastic
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simulations based on the perturbation approximation up to any order,
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whereas for GMM model moments are computed in closed-form based on the
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pruned state-space representation of the perturbation solution up to third
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order. The implementation of SMM is inspired by *Born and Pfeifer (2014)*
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and *Ruge-Murcia (2012)*, whereas the one for GMM is adapted from
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*Andreasen, Fernández-Villaverde and Rubio-Ramírez (2018)* and *Mutschler
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(2018)*. Successful estimation heavily relies on the accuracy and efficiency of
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the perturbation approximation, so it is advised to tune this as much as
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is possible to use Dynare to estimate some or all parameters using a
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method of moments approach. Both the Simulated Method of Moments (SMM)
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and the Generalized Method of Moments (GMM) are available. The general
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idea is to minimize the distance between unconditional model moments
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and corresponding data moments (so called orthogonality or moment
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conditions). For SMM, Dynare computes model moments via stochastic
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simulations based on the perturbation approximation up to any order,
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whereas for GMM model moments are computed in closed-form based on the
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pruned state-space representation of the perturbation solution up to third
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order. The implementation of SMM is inspired by *Born and Pfeifer (2014)*
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and *Ruge-Murcia (2012)*, whereas the one for GMM is adapted from
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*Andreasen, Fernández-Villaverde and Rubio-Ramírez (2018)* and *Mutschler
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(2018)*. Successful estimation heavily relies on the accuracy and efficiency of
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the perturbation approximation, so it is advised to tune this as much as
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possible (see :ref:`stoch-sol-simul`). The method of moments estimator is consistent
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and asymptotically normally distributed given certain regularity conditions
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(see *Duffie and Singleton (1993)* for SMM and *Hansen (1982)* for GMM).
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For instance, it is required to have at least as many moment conditions as
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estimated parameters (over-identified or just identified). Moreover, the
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Jacobian of the moments with respect to the estimated parameters needs to
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and asymptotically normally distributed given certain regularity conditions
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(see *Duffie and Singleton (1993)* for SMM and *Hansen (1982)* for GMM).
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For instance, it is required to have at least as many moment conditions as
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estimated parameters (over-identified or just identified). Moreover, the
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Jacobian of the moments with respect to the estimated parameters needs to
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have full rank. :ref:`identification-analysis` helps to check this regularity condition.
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In the over-identified case of declaring more moment conditions than estimated parameters, the
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choice of :opt:`weighting_matrix <weighting_matrix = ['WM1','WM2',...,'WMn']>`
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matters for the efficiency of the estimation, because the estimated
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orthogonality conditions are random variables with unequal variances and
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usually non-zero cross-moment covariances. A weighting matrix allows to
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re-weight moments to put more emphasis on moment conditions that are
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more informative or better measured (in the sense of having a smaller
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variance). To achieve asymptotic efficiency, the weighting matrix needs to
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be chosen such that, after appropriate scaling, it has a probability limit
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proportional to the inverse of the covariance matrix of the limiting
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distribution of the vector of orthogonality conditions. Dynare uses a
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Newey-West-type estimator with a Bartlett kernel to compute an estimate of this
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so-called optimal weighting matrix. Note that in this over-identified case,
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it is advised to perform the estimation in at least two stages by setting
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In the over-identified case of declaring more moment conditions than estimated parameters, the
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choice of :opt:`weighting_matrix <weighting_matrix = ['WM1','WM2',...,'WMn']>`
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matters for the efficiency of the estimation, because the estimated
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orthogonality conditions are random variables with unequal variances and
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usually non-zero cross-moment covariances. A weighting matrix allows to
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re-weight moments to put more emphasis on moment conditions that are
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more informative or better measured (in the sense of having a smaller
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variance). To achieve asymptotic efficiency, the weighting matrix needs to
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be chosen such that, after appropriate scaling, it has a probability limit
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proportional to the inverse of the covariance matrix of the limiting
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distribution of the vector of orthogonality conditions. Dynare uses a
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Newey-West-type estimator with a Bartlett kernel to compute an estimate of this
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so-called optimal weighting matrix. Note that in this over-identified case,
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it is advised to perform the estimation in at least two stages by setting
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e.g. :opt:`weighting_matrix=['DIAGONAL','DIAGONAL'] <weighting_matrix = ['WM1','WM2',...,'WMn']>`
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so that the computation of the optimal weighting matrix benefits from the
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consistent estimation of the previous stages. The optimal weighting matrix
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is used to compute standard errors and the J-test of overidentifying
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restrictions, which tests whether the model and selection of moment
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conditions fits the data sufficiently well. If the null hypothesis of a
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"valid" model is rejected, then something is (most likely) wrong with either your model
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so that the computation of the optimal weighting matrix benefits from the
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consistent estimation of the previous stages. The optimal weighting matrix
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is used to compute standard errors and the J-test of overidentifying
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restrictions, which tests whether the model and selection of moment
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conditions fits the data sufficiently well. If the null hypothesis of a
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"valid" model is rejected, then something is (most likely) wrong with either your model
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or selection of orthogonality conditions.
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In case the (presumed) global minimum of the moment distance function is
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located in a region of the parameter space that
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is typically considered unlikely (`dilemma of absurd parameters`), you may
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opt to choose the :opt:`penalized_estimator <penalized_estimator>` option.
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Similar to adding priors to the likelihood, this option incorporates prior
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knowledge (i.e. the prior mean) as additional moment restrictions and
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weights them by their prior precision to guide the minimization algorithm
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to more plausible regions of the parameter space. Ideally, these regions are
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characterized by only slightly worse values of the objective function. Note that
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In case the (presumed) global minimum of the moment distance function is
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located in a region of the parameter space that
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is typically considered unlikely (`dilemma of absurd parameters`), you may
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opt to choose the :opt:`penalized_estimator <penalized_estimator>` option.
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Similar to adding priors to the likelihood, this option incorporates prior
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knowledge (i.e. the prior mean) as additional moment restrictions and
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weights them by their prior precision to guide the minimization algorithm
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to more plausible regions of the parameter space. Ideally, these regions are
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characterized by only slightly worse values of the objective function. Note that
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adding prior information comes at the cost of a loss in efficiency of the estimator.
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.. command:: varobs VARIABLE_NAME...;
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|br| Required. All variables used in the :bck:`matched_moments` block
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|br| Required. All variables used in the :bck:`matched_moments` block
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need to be observable. See :ref:`varobs <varobs>` for more details.
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.. block:: matched_moments ;
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|br| This block specifies the product moments which are used in estimation.
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Currently, only linear product moments (e.g.
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:math:`E[y_t], E[y_t^2], E[x_t y_t], E[y_t y_{t-1}], E[y_t^3 x^2_{t-4}]`)
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are supported. For other functions like :math:`E[\log(y_t)e^{x_t}]` you
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need to declare auxiliary endogenous variables.
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Currently, only linear product moments (e.g.
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:math:`E[y_t], E[y_t^2], E[x_t y_t], E[y_t y_{t-1}], E[y_t^3 x^2_{t-4}]`)
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are supported. For other functions like :math:`E[\log(y_t)e^{x_t}]` you
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need to declare auxiliary endogenous variables.
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Each line inside of the block should be of the form::
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VARIABLE_NAME(LEAD/LAG)^POWER*VARIABLE_NAME(LEAD/LAG)^POWER*...*VARIABLE_NAME(LEAD/LAG)^POWER;
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where `VARIABLE_NAME` is the name of a declared observable variable,
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`LEAD/LAG` is either a negative integer for lags or a positive one
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for leads, and `POWER` is a positive integer indicating the exponent on
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where `VARIABLE_NAME` is the name of a declared observable variable,
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`LEAD/LAG` is either a negative integer for lags or a positive one
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for leads, and `POWER` is a positive integer indicating the exponent on
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the variable. You can omit `LEAD/LAG` equal to `0` or `POWER` equal to `1`.
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*Example*
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@ -7729,36 +7729,36 @@ adding prior information comes at the cost of a loss in efficiency of the estima
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matched_moments;
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c;
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y;
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c*c;
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c*c;
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c*y;
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y^2;
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c*c(3);
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y(1)^2*c(-4)^3;
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c(-5)^3*y(0)^2;
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end;
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*Limitations*
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1. For GMM, Dynare can only compute the theoretical mean, covariance, and
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1. For GMM, Dynare can only compute the theoretical mean, covariance, and
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autocovariances (i.e. first and second moments). Higher-order moments are only supported for SMM.
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2. By default, the product moments are not demeaned, unless the
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:opt:`prefilter <prefilter = INTEGER>` option is set to 1. That is, by default,
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2. By default, the product moments are not demeaned, unless the
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:opt:`prefilter <prefilter = INTEGER>` option is set to 1. That is, by default,
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`c*c` corresponds to :math:`E[c_t^2]` and not to :math:`Var[c_t]=E[c_t^2]-E[c_t]^2`.
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*Output*
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Dynare translates the :bck:`matched_moments` block into a cell array
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Dynare translates the :bck:`matched_moments` block into a cell array
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``M_.matched_moments`` where:
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* the first column contains a vector of indices for the chosen variables in declaration order
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* the second column contains the corresponding vector of leads and lags
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* the third column contains the corresponding vector of powers
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During the estimation phase, Dynare will eliminate all redundant or duplicate
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orthogonality conditions in ``M_.matched_moments`` and display which
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conditions were removed. In the example above, this would be the case for the
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last row, which is the same as the second-to-last one. The original block is
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During the estimation phase, Dynare will eliminate all redundant or duplicate
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orthogonality conditions in ``M_.matched_moments`` and display which
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conditions were removed. In the example above, this would be the case for the
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last row, which is the same as the second-to-last one. The original block is
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saved in ``M_.matched_moments_orig``.
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.. block:: estimated_params ;
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@ -7775,12 +7775,12 @@ adding prior information comes at the cost of a loss in efficiency of the estima
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.. command:: method_of_moments (OPTIONS...);
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|br| This command runs the method of moments estimation. The following
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|br| This command runs the method of moments estimation. The following
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information will be displayed in the command window:
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* Overview of options chosen by the user
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* Estimation results for each stage and iteration
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* Value of minimized moment distance objective function
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* Value of minimized moment distance objective function
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* Result of the J-test
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* Table of data moments and estimated model moments
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@ -7793,28 +7793,28 @@ adding prior information comes at the cost of a loss in efficiency of the estima
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.. option:: datafile = FILENAME
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The name of the file containing the data. See
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The name of the file containing the data. See
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:opt:`datafile <datafile = FILENAME>` for the meaning and syntax.
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*Options common for SMM and GMM*
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.. option:: order = INTEGER
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Order of perturbation approximation. For GMM only orders 1|2|3 are
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supported. For SMM, you can choose an arbitrary order. Note that the
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order set in other functions will not overwrite the default.
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Order of perturbation approximation. For GMM only orders 1|2|3 are
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supported. For SMM, you can choose an arbitrary order. Note that the
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order set in other functions will not overwrite the default.
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Default: ``1``.
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.. option:: pruning
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Discard higher order terms when iteratively computing simulations
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of the solution. See :opt:`pruning <pruning>` for more details.
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Discard higher order terms when iteratively computing simulations
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of the solution. See :opt:`pruning <pruning>` for more details.
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Default: not set for SMM, always set for GMM.
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.. option:: penalized_estimator
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This option includes deviations of the estimated parameters from the
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prior mean as additional moment restrictions and weights them by
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This option includes deviations of the estimated parameters from the
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prior mean as additional moment restrictions and weights them by
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their prior precision.
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Default: not set.
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@ -7831,9 +7831,9 @@ adding prior information comes at the cost of a loss in efficiency of the estima
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``OPTIMAL``
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Uses the optimal weighting matrix computed by a Newey-West-type
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estimate with a Bartlett kernel. At the first
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stage, the data-moments are used as initial estimate of the
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model moments, whereas at subsequent stages the previous estimate
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estimate with a Bartlett kernel. At the first
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stage, the data-moments are used as initial estimate of the
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model moments, whereas at subsequent stages the previous estimate
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of model moments will be used when computing
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the optimal weighting matrix.
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@ -7844,8 +7844,8 @@ adding prior information comes at the cost of a loss in efficiency of the estima
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``FILENAME``
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The name of the mat-file (extension ``.mat``) containing a
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user-specified weighting matrix. The file must include a positive definite
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The name of the mat-file (extension ``.mat``) containing a
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user-specified weighting matrix. The file must include a positive definite
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square matrix called `weighting_matrix` with both dimensions
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equal to the number of orthogonality conditions.
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@ -7860,17 +7860,17 @@ adding prior information comes at the cost of a loss in efficiency of the estima
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.. option:: bartlett_kernel_lag = INTEGER
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Bandwidth of kernel for computing the optimal weighting matrix.
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Bandwidth of kernel for computing the optimal weighting matrix.
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Default: ``20``.
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.. option:: se_tolx = DOUBLE
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Step size for numerical differentiation when computing standard
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Step size for numerical differentiation when computing standard
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errors with a two-sided finite difference method.
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Default: ``1e-5``.
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.. option:: verbose
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Display and store intermediate estimation results in ``oo_.mom``.
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Default: not set.
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@ -7878,7 +7878,7 @@ adding prior information comes at the cost of a loss in efficiency of the estima
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.. option:: burnin = INTEGER
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Number of periods dropped at the beginning of simulation.
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Number of periods dropped at the beginning of simulation.
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Default: ``500``.
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.. option:: bounded_shock_support
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@ -7893,32 +7893,32 @@ adding prior information comes at the cost of a loss in efficiency of the estima
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.. option:: simulation_multiple = INTEGER
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Multiple of data length used for simulation.
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Multiple of data length used for simulation.
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Default: ``7``.
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*GMM-specific options*
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.. option:: analytic_standard_errors
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Compute standard errors using analytical derivatives of moments
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Compute standard errors using analytical derivatives of moments
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with respect to estimated parameters.
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Default: not set, i.e. standard errors are computed using a two-sided
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Default: not set, i.e. standard errors are computed using a two-sided
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finite difference method, see :opt:`se_tolx <se_tolx = DOUBLE>`.
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*General options*
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.. option:: dirname = FILENAME
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.. option:: dirname = FILENAME
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Directory in which to store ``estimation`` output.
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Directory in which to store ``estimation`` output.
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See :opt:`dirname <dirname = FILENAME>` for more details.
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Default: ``<mod_file>``.
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.. option:: graph_format = FORMAT
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Specify the file format(s) for graphs saved to disk.
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Specify the file format(s) for graphs saved to disk.
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See :opt:`graph_format <graph_format = FORMAT>` for more details.
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Default: ``eps``.
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.. option:: nodisplay
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See :opt:`nodisplay`. Default: not set.
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.. option:: plot_priors = INTEGER
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Control the plotting of priors.
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Control the plotting of priors.
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See :opt:`plot_priors <plot_priors = INTEGER>` for more details.
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Default: ``1``, i.e. plot priors.
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@ -7955,13 +7955,13 @@ adding prior information comes at the cost of a loss in efficiency of the estima
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.. option:: nobs = INTEGER
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See :opt:`nobs <nobs = INTEGER>`.
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See :opt:`nobs <nobs = INTEGER>`.
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Default: all observations are considered.
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.. option:: prefilter = INTEGER
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A value of 1 means that the estimation procedure will demean each data
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series by its empirical mean and each model moment by its theoretical
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A value of 1 means that the estimation procedure will demean each data
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series by its empirical mean and each model moment by its theoretical
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mean. See :opt:`prefilter <prefilter = INTEGER>` for more details.
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Default: `0`, i.e. no prefiltering.
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@ -7971,7 +7971,7 @@ adding prior information comes at the cost of a loss in efficiency of the estima
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.. option:: xls_sheet = QUOTED_STRING
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See :opt:`xls_sheet <xls_sheet = QUOTED_STRING>`.
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See :opt:`xls_sheet <xls_sheet = QUOTED_STRING>`.
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.. option:: xls_range = RANGE
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@ -7991,13 +7991,13 @@ adding prior information comes at the cost of a loss in efficiency of the estima
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.. option:: additional_optimizer_steps = [INTEGER|FUNCTION_NAME,INTEGER|FUNCTION_NAME,...]
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Vector of additional minimization algorithms run after
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``mode_compute``. If :opt:`verbose` option is set, then the additional estimation
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Vector of additional minimization algorithms run after
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``mode_compute``. If :opt:`verbose` option is set, then the additional estimation
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results are saved into the ``oo_.mom`` structure prefixed with `verbose_`.
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Default: no additional optimization iterations.
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.. option:: optim = (NAME, VALUE, ...)
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See :opt:`optim <optim = (NAME, VALUE, ...)>`.
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.. option:: silent_optimizer
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@ -8008,17 +8008,17 @@ adding prior information comes at the cost of a loss in efficiency of the estima
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*Numerical algorithms options*
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.. option:: aim_solver
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See :opt:`aim_solver <aim_solver>`. Default: not set.
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.. option:: k_order_solver
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See :opt:`k_order_solver <k_order_solver>`.
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See :opt:`k_order_solver <k_order_solver>`.
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Default: disabled for order 1 and 2, enabled for order 3 and above.
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.. option:: dr = OPTION
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See :opt:`dr <dr = OPTION>`. Default: ``default``, i.e. generalized
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See :opt:`dr <dr = OPTION>`. Default: ``default``, i.e. generalized
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Schur decomposition.
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.. option:: dr_cycle_reduction_tol = DOUBLE
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@ -8038,7 +8038,7 @@ adding prior information comes at the cost of a loss in efficiency of the estima
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.. option:: lyapunov = OPTION
|
||||
|
||||
See :opt:`lyapunov <lyapunov = OPTION>`. Default: ``default``, i.e.
|
||||
See :opt:`lyapunov <lyapunov = OPTION>`. Default: ``default``, i.e.
|
||||
based on Bartlets-Stewart algorithm.
|
||||
|
||||
.. option:: lyapunov_complex_threshold = DOUBLE
|
||||
|
|
@ -8079,12 +8079,12 @@ adding prior information comes at the cost of a loss in efficiency of the estima
|
|||
|
||||
.. option:: schur_vec_tol = DOUBLE
|
||||
|
||||
Tolerance level used to find nonstationary variables in Schur decomposition
|
||||
Tolerance level used to find nonstationary variables in Schur decomposition
|
||||
of the transition matrix. Default: ``1e-11``.
|
||||
|
||||
.. option:: mode_check
|
||||
|
||||
Plots univariate slices through the moments distance objective function around the
|
||||
Plots univariate slices through the moments distance objective function around the
|
||||
computed minimum for each estimated parameter. This is
|
||||
helpful to diagnose problems with the optimizer.
|
||||
Default: not set.
|
||||
|
|
@ -8107,14 +8107,14 @@ adding prior information comes at the cost of a loss in efficiency of the estima
|
|||
|
||||
*Output*
|
||||
|
||||
``method_of_moments`` stores user options in a structure called
|
||||
`options_mom_` in the global workspace. After running the estimation,
|
||||
``method_of_moments`` stores user options in a structure called
|
||||
`options_mom_` in the global workspace. After running the estimation,
|
||||
the parameters ``M_.params`` and the covariance matrices of the shocks
|
||||
``M_.Sigma_e`` and of the measurement errors ``M_.H`` are set to the
|
||||
parameters that minimize the quadratic moments distance objective
|
||||
``M_.Sigma_e`` and of the measurement errors ``M_.H`` are set to the
|
||||
parameters that minimize the quadratic moments distance objective
|
||||
function. The estimation results are stored in the ``oo_.mom`` structure
|
||||
with the following fields:
|
||||
|
||||
|
||||
.. matvar:: oo_.mom.data_moments
|
||||
|
||||
Variable set by the ``method_of_moments`` command. Stores the mean
|
||||
|
|
@ -8127,30 +8127,30 @@ adding prior information comes at the cost of a loss in efficiency of the estima
|
|||
|
||||
Variable set by the ``method_of_moments`` command. Stores the selected
|
||||
empirical moments at each point in time. NaN values due to leads/lags
|
||||
or missing data are replaced by the corresponding mean of the moment.
|
||||
or missing data are replaced by the corresponding mean of the moment.
|
||||
Matrix of dimension time periods times number of orthogonality conditions.
|
||||
|
||||
|
||||
.. matvar:: oo_.mom.Sw
|
||||
.. matvar:: oo_.mom.Sw
|
||||
|
||||
Variable set by the ``method_of_moments`` command. Stores the
|
||||
Variable set by the ``method_of_moments`` command. Stores the
|
||||
Cholesky decomposition of the currently used weighting matrix.
|
||||
Square matrix of dimensions equal to the number of orthogonality
|
||||
conditions.
|
||||
|
||||
|
||||
.. matvar:: oo_.mom.model_moments
|
||||
.. matvar:: oo_.mom.model_moments
|
||||
|
||||
Variable set by the ``method_of_moments`` command. Stores the implied
|
||||
Variable set by the ``method_of_moments`` command. Stores the implied
|
||||
selected model moments given the current parameter guess. Model moments
|
||||
are computed in closed-form from the pruned state-space system for GMM,
|
||||
whereas for SMM these are based on averages of simulated data. Vector of dimension equal
|
||||
to the number of orthogonality conditions.
|
||||
|
||||
|
||||
.. matvar:: oo_.mom.Q
|
||||
.. matvar:: oo_.mom.Q
|
||||
|
||||
Variable set by the ``method_of_moments`` command. Stores the scalar
|
||||
Variable set by the ``method_of_moments`` command. Stores the scalar
|
||||
value of the quadratic moment's distance objective function.
|
||||
|
||||
|
||||
|
|
@ -8159,7 +8159,7 @@ adding prior information comes at the cost of a loss in efficiency of the estima
|
|||
Variable set by the ``method_of_moments`` command. Stores the analytically
|
||||
computed Jacobian matrix of the derivatives of the model moments with
|
||||
respect to the estimated parameters. Only for GMM with :opt:`analytic_standard_errors`.
|
||||
Matrix with dimension equal to the number of orthogonality conditions
|
||||
Matrix with dimension equal to the number of orthogonality conditions
|
||||
times number of estimated parameters.
|
||||
|
||||
|
||||
|
|
@ -8167,18 +8167,18 @@ adding prior information comes at the cost of a loss in efficiency of the estima
|
|||
.. matvar:: oo_.mom.smm_stage_*_mode
|
||||
.. matvar:: oo_.mom.verbose_gmm_stage_*_mode
|
||||
.. matvar:: oo_.mom.verbose_smm_stage_*_mode
|
||||
|
||||
|
||||
Variables set by the ``method_of_moments`` command when estimating
|
||||
with GMM or SMM. Stores the estimated values at stages 1, 2,....
|
||||
with GMM or SMM. Stores the estimated values at stages 1, 2,....
|
||||
The structures contain the following fields:
|
||||
|
||||
|
||||
- ``measurement_errors_corr``: estimated correlation between two measurement errors
|
||||
- ``measurement_errors_std``: estimated standard deviation of measurement errors
|
||||
- ``parameters``: estimated model parameters
|
||||
- ``shocks_corr``: estimated correlation between two structural shocks.
|
||||
- ``shocks_std``: estimated standard deviation of structural shocks.
|
||||
|
||||
If the :opt:`verbose` option is set, additional fields prefixed with
|
||||
If the :opt:`verbose` option is set, additional fields prefixed with
|
||||
``verbose_`` are saved for all :opt:`additional_optimizer_steps<additional_optimizer_steps = [INTEGER|FUNCTION_NAME,INTEGER|FUNCTION_NAME,...]>`.
|
||||
|
||||
.. matvar:: oo_.mom.gmm_stage_*_std_at_mode
|
||||
|
|
@ -8187,28 +8187,28 @@ adding prior information comes at the cost of a loss in efficiency of the estima
|
|||
.. matvar:: oo_.mom.verbose_smm_stage_*_std_at_mode
|
||||
|
||||
Variables set by the ``method_of_moments`` command when estimating
|
||||
with GMM or SMM. Stores the estimated standard errors at stages 1, 2,....
|
||||
with GMM or SMM. Stores the estimated standard errors at stages 1, 2,....
|
||||
The structures contain the following fields:
|
||||
|
||||
|
||||
- ``measurement_errors_corr``: standard error of estimated correlation between two measurement errors
|
||||
- ``measurement_errors_std``: standard error of estimated standard deviation of measurement errors
|
||||
- ``parameters``: standard error of estimated model parameters
|
||||
- ``shocks_corr``: standard error of estimated correlation between two structural shocks.
|
||||
- ``shocks_std``: standard error of estimated standard deviation of structural shocks.
|
||||
|
||||
If the :opt:`verbose` option is set, additional fields prefixed with
|
||||
If the :opt:`verbose` option is set, additional fields prefixed with
|
||||
``verbose_`` are saved for all :opt:`additional_optimizer_steps<additional_optimizer_steps = [INTEGER|FUNCTION_NAME,INTEGER|FUNCTION_NAME,...]>`.
|
||||
|
||||
|
||||
.. matvar:: oo_.mom.J_test
|
||||
|
||||
Variable set by the ``method_of_moments`` command. Structure where the
|
||||
value of the test statistic is saved into a field called ``j_stat``, the
|
||||
degress of freedom into a field called ``degrees_freedom`` and the p-value
|
||||
value of the test statistic is saved into a field called ``j_stat``, the
|
||||
degress of freedom into a field called ``degrees_freedom`` and the p-value
|
||||
of the test statistic into a field called ``p_val``.
|
||||
|
||||
|
||||
|
||||
|
||||
Model Comparison
|
||||
================
|
||||
|
||||
|
|
|
|||
Loading…
Reference in New Issue