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Using the Method of Simulated Moments (MSM)
for Identification of Dynamic Models
Presenter: Mohammad S. Jalali
Advisor: Dr. Hazhir Rahmandad
25th MIT-UAlbany PhD Colloquium, System Dynamics Society
Nov 16, 2012
Introduction
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 2
Increasingly, dynamic modelers face problems where
estimating model parameters from numerical empirical
data is a requirement.
The field of system dynamics (SD) was ahead of many
approaches available in social sciences in using advanced
analytical methods of the time (Forrester, 1961), and kept
this edge for many years.
For example advanced filtering and estimation methods
were introduced into the SD literature in the 70s (Peterson,
1975).
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Introduction
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 3
Increasingly, dynamic modelers face problems where
estimating model parameters from numerical empirical
data is a requirement.
The field of system dynamics (SD) was ahead of many
approaches available in social sciences in using advanced
analytical methods of the time (Forrester, 1961), and kept
this edge for many years.
For example advanced filtering and estimation methods
were introduced into the SD literature in the 70s (Peterson,
1975).
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Introduction
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 4
Increasingly, dynamic modelers face problems where
estimating model parameters from numerical empirical
data is a requirement.
The field of system dynamics (SD) was ahead of many
approaches available in social sciences in using advanced
analytical methods of the time (Forrester, 1961), and kept
this edge for many years.
For example advanced filtering and estimation methods
were introduced into the SD literature in the 70s (Peterson,
1975).
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Introduction
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 5
However, over the last three decades the research in SD
has largely focused on diverse applications of the original
toolbox, with limited methodological expansions in
parameter estimation domain.
In contrast, research in econometrics and other related
fields have provided many relevant tools over this period
(Greene, 2012).
As a result, currently many studies in SD do not report
formal parameter estimates common in social science
research, or when calibration is pursued, typical measures
of confidence in estimated parameter values are not
reported.
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Introduction
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 6
However, over the last three decades the research in SD
has largely focused on diverse applications of the original
toolbox, with limited methodological expansions in
parameter estimation domain.
In contrast, research in econometrics and other related
fields have provided many relevant tools over this period
(Greene, 2012).
As a result, currently many studies in SD do not report
formal parameter estimates common in social science
research, or when calibration is pursued, typical measures
of confidence in estimated parameter values are not
reported.
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Introduction
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 7
However, over the last three decades the research in SD
has largely focused on diverse applications of the original
toolbox, with limited methodological expansions in
parameter estimation domain.
In contrast, research in econometrics and other related
fields have provided many relevant tools over this period
(Greene, 2012).
As a result, currently many studies in SD do not report
formal parameter estimates common in social science
research, or when calibration is pursued, typical measures
of confidence in estimated parameter values are not
reported.
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Common Techniques
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 8
Modelers typically compare time series data against the
same variables in a model,
and
minimize the weighted sum of a function of the error term
by
changing the uncertain parameters until best fitting
estimates are found through a nonlinear optimization
algorithm (Oliva, 2003).
The error function is frequently defined as the squared
error but absolute error and absolute percent error terms
are also common (Sterman, 2000).
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Common Techniques
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 9
Modelers typically compare time series data against the
same variables in a model,
and
minimize the weighted sum of a function of the error term
by
changing the uncertain parameters until best fitting
estimates are found through a nonlinear optimization
algorithm (Oliva, 2003).
The error function is frequently defined as the squared
error but absolute error and absolute percent error terms
are also common (Sterman, 2000).
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Common Techniques
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 10
Weights for different data points are given based on the
confidence the researcher has in the accuracy of the data
and its relevance to the problem at hand.
When reported, confidence intervals are calculated using
normality and independence assumption for error terms
which,
with weights proportional to the reciprocal of error
variance, would turn least squared error estimates into
maximum-likelihood estimates (MLE).
Bootstrapping methods are also sometimes used for
estimating confidence intervals (Dogan, 2007).
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Common Techniques
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 11
Weights for different data points are given based on the
confidence the researcher has in the accuracy of the data
and its relevance to the problem at hand.
When reported, confidence intervals are calculated using
normality and independence assumption for error terms
which,
with weights proportional to the reciprocal of error
variance, would turn least squared error estimates into
maximum-likelihood estimates (MLE).
Bootstrapping methods are also sometimes used for
estimating confidence intervals (Dogan, 2007).
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Common Techniques
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 12
Weights for different data points are given based on the
confidence the researcher has in the accuracy of the data
and its relevance to the problem at hand.
When reported, confidence intervals are calculated using
normality and independence assumption for error terms
which,
with weights proportional to the reciprocal of error
variance, would turn least squared error estimates into
maximum-likelihood estimates (MLE).
Bootstrapping methods are also sometimes used for
estimating confidence intervals (Dogan, 2007).
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Some Challenges
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 13
Challenges with the current parameter estimation
techniques?
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Some Challenges
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 14
While the common approaches cover many important
estimation challenges, they each include some
shortcomings.
Ad hoc selection of the error term and the weights for
different data points reduces the consistency of the
methods and their ability to provide confidence intervals.
Normality and independence may regularly be violated
which negate the benefits of MLE when using squared
errors.
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Some Challenges
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 15
While the common approaches cover many important
estimation challenges, they each include some
shortcomings.
Ad hoc selection of the error term and the weights for
different data points reduces the consistency of the
methods and their ability to provide confidence intervals.
Normality and independence may regularly be violated
which negate the benefits of MLE when using squared
errors.
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Some Challenges
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 16
While the common approaches cover many important
estimation challenges, they each include some
shortcomings.
Ad hoc selection of the error term and the weights for
different data points reduces the consistency of the
methods and their ability to provide confidence intervals.
Normality and independence may regularly be violated
which negate the benefits of MLE when using squared
errors.
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Some Challenges
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 17
Bootstrapping, while flexible, increases the computational
costs significantly and as a result may prove infeasible for
many realistically-sized problems.
All these methods rely on having time series data, and
cannot extract from distributions in cross-sectional data
the dynamics that have led to those distributions.
Note: Cross-sectional data is data collected at the same point of time.
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Some Challenges
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 18
Bootstrapping, while flexible, increases the computational
costs significantly and as a result may prove infeasible for
many realistically-sized problems.
All these methods rely on having time series data, and
cannot extract from distributions in cross-sectional data
the dynamics that have led to those distributions.
Note: Cross-sectional data is data collected at the same point of time.
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Estimation Procedures’ Requirements
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 19
The following characteristics signify the estimation
procedures ideal for dynamic modelers:
Model Independence: given that most dynamic models
do not follow a fixed structural form (e.g. linearity),
estimation procedures that are independent of model
structure are most beneficial.
Analytical confidence intervals: Ability to find
confidence intervals analytically is important because of
the computational costs of optimizing none-linear dynamic
models and replications needed for bootstrapping
methods.
[4] Soll & Mannes, 2011
[5] Herzog & Hertwig, 2009; Hogarth, 1978; Larrick & Soll, 2006; Lee, Zhang, & Shi, 2011; Rauhut & Lorenz, 2011; Soll &
Larrick, 2009; Wright & Rowe, 2011; Yaniv, 2004a, 2004b
[6] Herzog & Hertwig, 2009; Larrick & Soll, 2006; Soll & Larrick, 2009; Soll & Mannes, 2011
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Estimation Procedures’ Requirements
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 20
The following characteristics signify the estimation
procedures ideal for dynamic modelers:
Model Independence: given that most dynamic models
do not follow a fixed structural form (e.g. linearity),
estimation procedures that are independent of model
structure are most beneficial.
Analytical confidence intervals: Ability to find
confidence intervals analytically is important because of
the computational costs of optimizing none-linear dynamic
models and replications needed for bootstrapping
methods.
[4] Soll & Mannes, 2011
[5] Herzog & Hertwig, 2009; Hogarth, 1978; Larrick & Soll, 2006; Lee, Zhang, & Shi, 2011; Rauhut & Lorenz, 2011; Soll &
Larrick, 2009; Wright & Rowe, 2011; Yaniv, 2004a, 2004b
[6] Herzog & Hertwig, 2009; Larrick & Soll, 2006; Soll & Larrick, 2009; Soll & Mannes, 2011
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Estimation Procedures’ Requirements
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 21
The following characteristics signify the estimation
procedures ideal for dynamic modelers:
Model Independence: given that most dynamic models
do not follow a fixed structural form (e.g. linearity),
estimation procedures that are independent of model
structure are most beneficial.
Analytical confidence intervals: Ability to find
confidence intervals analytically is important because of
the computational costs of optimizing none-linear dynamic
models and replications needed for bootstrapping
methods.
[4] Soll & Mannes, 2011
[5] Herzog & Hertwig, 2009; Hogarth, 1978; Larrick & Soll, 2006; Lee, Zhang, & Shi, 2011; Rauhut & Lorenz, 2011; Soll &
Larrick, 2009; Wright & Rowe, 2011; Yaniv, 2004a, 2004b
[6] Herzog & Hertwig, 2009; Larrick & Soll, 2006; Soll & Larrick, 2009; Soll & Mannes, 2011
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Estimation Procedures’ Requirements
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 22
The Assumption-free error terms: Independence and
distributional assumptions on error terms for dynamic
models are not always easy to justify, so methods with
fewer such assumptions are preferred.
Applicability to diverse data types: including both time
series and cross-sectional data.
No single method fully satisfies all these requirements,
therefore modelers need to choose from a menu of
available estimation methods to match their problem
requirements.
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Estimation Procedures’ Requirements
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 23
The Assumption-free error terms: Independence and
distributional assumptions on error terms for dynamic
models are not always easy to justify, so methods with
fewer such assumptions are preferred.
Applicability to diverse data types: including both time
series and cross-sectional data.
No single method fully satisfies all these requirements,
therefore modelers need to choose from a menu of
available estimation methods to match their problem
requirements.
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Estimation Procedures’ Requirements
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 24
The Assumption-free error terms: Independence and
distributional assumptions on error terms for dynamic
models are not always easy to justify, so methods with
fewer such assumptions are preferred.
Applicability to diverse data types: including both time
series and cross-sectional data.
No single method fully satisfies all these requirements,
therefore modelers need to choose from a menu of
available estimation methods to match their problem
requirements.
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Method of Simulated Moments
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 25
The basic idea of this method is to define appropriate
moments of data and, by changing uncertain parameters,
minimize the difference between those moments and their
simulated counterpart coming from the model.
Moments could be any function of data points available.
However for analytical confidence intervals to be available,
one needs these moments to be normally distributed,
often meaning that:
each moment is an average across a function of multiple
independent observations coming from the same
underlying distribution (then normality follows from the central limit theorem).
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Method of Simulated Moments
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 26
The basic idea of this method is to define appropriate
moments of data and, by changing uncertain parameters,
minimize the difference between those moments and their
simulated counterpart coming from the model.
Moments could be any function of data points available.
However for analytical confidence intervals to be available,
one needs these moments to be normally distributed,
often meaning that:
each moment is an average across a function of multiple
independent observations coming from the same
underlying distribution (then normality follows from the central limit theorem).
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Method of Simulated Moments
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 27
The basic idea of this method is to define appropriate
moments of data and, by changing uncertain parameters,
minimize the difference between those moments and their
simulated counterpart coming from the model.
Moments could be any function of data points available.
However for analytical confidence intervals to be available,
one needs these moments to be normally distributed,
often meaning that:
each moment is an average across a function of multiple
independent observations coming from the same
underlying distribution (then normality follows from the central limit theorem).
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
History of MSM
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 28
McFadden (1989) was the first who proposed using
simulation instead of trying to solve the moment conditions
analytically.
He believes that an unbiased simulator is used in the MSM
where the simulation errors are independent across
observations, and the variance introduced by simulation
will be controlled by the law of large numbers operating
across observations.
Lee (1991) and Duffie (1993) extended the framework and
provided a rigorous treatment of MSM estimators for time-
series and panel-data cases and provided relevant
statistics for making tests.
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
History of MSM
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 29
McFadden (1989) was the first who proposed using
simulation instead of trying to solve the moment conditions
analytically.
He believes that an unbiased simulator is used in the MSM
where the simulation errors are independent across
observations, and the variance introduced by simulation
will be controlled by the law of large numbers operating
across observations.
Lee (1991) and Duffie (1993) extended the framework and
provided a rigorous treatment of MSM estimators for time-
series and panel-data cases and provided relevant
statistics for making tests.
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
History of MSM
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 30
McFadden (1989) was the first who proposed using
simulation instead of trying to solve the moment conditions
analytically.
He believes that an unbiased simulator is used in the MSM
where the simulation errors are independent across
observations, and the variance introduced by simulation
will be controlled by the law of large numbers operating
across observations.
Lee (1991) and Duffie (1993) extended the framework and
provided a rigorous treatment of MSM estimators for time-
series and panel-data cases and provided relevant
statistics for making tests.
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
When to Use MSM
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 31
MSM is best fitted for dynamic modeling problems when
some of the following problem characteristics are present:
Population data:
MSM is suitable for estimation of generic models to
population data. Different units of data such as individuals,
firms, and countries could be available. For each unit one
or more data items (e.g. weight, height, and age for
individual data) could be available.
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
When to Use MSM
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 32
MSM is best fitted for dynamic modeling problems when
some of the following problem characteristics are present:
Population data:
MSM is suitable for estimation of generic models to
population data. Different units of data such as individuals,
firms, and countries could be available. For each unit one
or more data items (e.g. weight, height, and age for
individual data) could be available.
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
When to Use MSM
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 33
Role of random processes:
MSM could be a good choice when models include
stochastic processes that drive the model, and their impact
on the model behavior is reflected in the data against
which the model is to be calibrated, e.g. when we are
trying to match the variance observed across multiple
units.
Confidence Intervals:
MSM would be a suitable choice when analytical
confidence intervals are sought.
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
When to Use MSM
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 34
Role of random processes:
MSM could be a good choice when models include
stochastic processes that drive the model, and their impact
on the model behavior is reflected in the data against
which the model is to be calibrated, e.g. when we are
trying to match the variance observed across multiple
units.
Confidence Intervals:
MSM would be a suitable choice when analytical
confidence intervals are sought.
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
When to Use MSM
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 35
Cross-sectional data:
MSM applies to both cross-sectional and time series data.
Where-as time series data include multiple data points for
the same unit over time, cross-sectional data includes data
points for multiple units at the same time.
MSM may be the only viable choice for estimating dynamic
models when data is cross-sectional as it allows us to
extract the information about the historical trajectories of
units hidden in their cross-sectional distributions.
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
When to Use MSM
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 36
Cross-sectional data:
MSM applies to both cross-sectional and time series data.
Where-as time series data include multiple data points for
the same unit over time, cross-sectional data includes data
points for multiple units at the same time.
MSM may be the only viable choice for estimating dynamic
models when data is cross-sectional as it allows us to
extract the information about the historical trajectories of
units hidden in their cross-sectional distributions.
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Recipe for MSM
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 37
Step 1: Choose the Moment Conditions
Usually the first and second moments of model’s
outcomes (mean and variance) are good candidates.
Remember that the number of moment conditions should
be (equal to or) larger than the number of unknown
parameters.
The most informative moments are the ones that
1) Are sensitive to at least one of the unknown
parameters
2) Have rather small variances.
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Recipe for MSM
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 38
Step 1: Choose the Moment Conditions
Usually the first and second moments of model’s
outcomes (mean and variance) are good candidates.
Remember that the number of moment conditions should
be (equal to or) larger than the number of unknown
parameters.
The most informative moments are the ones that
1) Are sensitive to at least one of the unknown
parameters
2) Have rather small variances.
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Recipe for MSM
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 39
Step 1: Choose the Moment Conditions
Usually the first and second moments of model’s
outcomes (mean and variance) are good candidates.
Remember that the number of moment conditions should
be (equal to or) larger than the number of unknown
parameters.
The most informative moments are the ones that
1) Are sensitive to at least one of the unknown
parameters
2) Have rather small variances.
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Recipe for MSM
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 40
Step 1: Choose the Moment Conditions
The right choice of moment conditions is the most crucial
step in identifying the model and recovering model
parameters.
Identification of a model using MSM requires that the
model-generated moment conditions should fit their
empirical counterparts if and only if the structural
parameters equal their true values.
Furthermore, the sufficient condition for identification is a
one-to-one mapping between the structural parameters
and a subset of the moment restrictions of the same
dimension.
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Recipe for MSM
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 41
Step 1: Choose the Moment Conditions
The right choice of moment conditions is the most crucial
step in identifying the model and recovering model
parameters.
Identification of a model using MSM requires that the
model-generated moment conditions should fit their
empirical counterparts if and only if the structural
parameters equal their true values.
Furthermore, the sufficient condition for identification is a
one-to-one mapping between the structural parameters
and a subset of the moment restrictions of the same
dimension.
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Recipe for MSM
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 42
Step 1: Choose the Moment Conditions
The right choice of moment conditions is the most crucial
step in identifying the model and recovering model
parameters.
Identification of a model using MSM requires that the
model-generated moment conditions should fit their
empirical counterparts if and only if the structural
parameters equal their true values.
Furthermore, the sufficient condition for identification is a
one-to-one mapping between the structural parameters
and a subset of the moment restrictions of the same
dimension.
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Recipe for MSM
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 43
Step 1: Choose the Moment Conditions
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
- The moment represented by the dashed–line is not informative.
- It is not very sensitive to changes in parameter value.
- We can not even be sure that these small changes are due to true
response of the model to various parameter value or are the artifact of
computational or sampling errors.
Recipe for MSM
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 44
Step 1: Choose the Moment Conditions
Example:
A simple model of individual’s body mass, consisting of fat
mass and fat free mass, is developed by Rahmandad
(2011).
The model included a few uncertain parameters. In
absence of time series data, those parameters were
estimated from cross sectional data on individual weights
coming from the National Health and Nutrition Examination
Survey (NHANES).
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Recipe for MSM
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 45
Step 1: Choose the Moment Conditions
Example:
A simple model of individual’s body mass, consisting of fat
mass and fat free mass, is developed by Rahmandad
(2011).
The model included a few uncertain parameters. In
absence of time series data, those parameters were
estimated from cross sectional data on individual weights
coming from the National Health and Nutrition Examination
Survey (NHANES).
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Recipe for MSM
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 46
Step 1: Choose the Moment Conditions
Example:
NHANES 2005-2006 population of 5,971 subjects was
categorized into 110 subpopulations based on:
different ethnicities (5 ethnicities),
genders (2 genders),
age (11 age groups).
For each population group two moments, average body
weight and variance of body weight, were calculated as
the moments to be matched, leading to a total of 220
moments to match.
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Recipe for MSM
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 47
Step 1: Choose the Moment Conditions
Example:
NHANES 2005-2006 population of 5,971 subjects was
categorized into 110 subpopulations based on:
different ethnicities (5 ethnicities),
genders (2 genders),
age (11 age groups).
For each population group two moments, average body
weight and variance of body weight, were calculated as
the moments to be matched, leading to a total of 220
moments to match.
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Recipe for MSM
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 48
Step 2: Weighting Matrix
A two-step procedure should be developed to calculate the
optimal weighting matrix (W) (More technical information will be presented
in the forthcoming book chapter.)
The right choice of weighting matrix ensures that we get
the smallest asymptotic variance for our estimated values.
In the simplest case, the identity matrix is used as the
initial value of the weighting matrix in the first step.
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Recipe for MSM
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 49
Step 2: Weighting Matrix
A two-step procedure should be developed to calculate the
optimal weighting matrix (W) (More technical information will be presented
in the forthcoming book chapter.)
The right choice of weighting matrix ensures that we get
the smallest asymptotic variance for our estimated values.
In the simplest case, the identity matrix is used as the
initial value of the weighting matrix in the first step.
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Recipe for MSM
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 50
Step 2: Weighting Matrix
A two-step procedure should be developed to calculate the
optimal weighting matrix (W) (More technical information will be presented
in the forthcoming book chapter.)
The right choice of weighting matrix ensures that we get
the smallest asymptotic variance for our estimated values.
In the simplest case, the identity matrix is used as the
initial value of the weighting matrix in the first step.
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Recipe for MSM
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 51
Step 2: Weighting Matrix
A two-step procedure should be developed to calculate the
optimal weighting matrix (W) (More technical information will be presented
in the forthcoming book chapter.)
The right choice of weighting matrix ensures that we get
the smallest asymptotic variance for our estimated values.
In the simplest case, the identity matrix is used as the
initial value of the weighting matrix in the first step.
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Recipe for MSM
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 Slide: 52
Step 3: Initial Value and Simulation
We need to generate a large sample of model outcome as
the base for estimating model’s moments.
In equilibrium models, it is recommended that we discard
earlier observations (in the time-series sense) since these
observations are sensitive to the initial value of the model
and are not in the steady-state.
If we need observations over time, it is recommended to
generate a vector of simulations and discard the first
observation.
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Recipe for MSM
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 Slide: 53
Step 3: Initial Value and Simulation
We need to generate a large sample of model outcome as
the base for estimating model’s moments.
In equilibrium models, it is recommended that we discard
earlier observations (in the time-series sense) since these
observations are sensitive to the initial value of the model
and are not in the steady-state.
If we need observations over time, it is recommended to
generate a vector of simulations and discard the first
observation.
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Recipe for MSM
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 Slide: 54
Step 3: Initial Value and Simulation
We need to generate a large sample of model outcome as
the base for estimating model’s moments.
In equilibrium models, it is recommended that we discard
earlier observations (in the time-series sense) since these
observations are sensitive to the initial value of the model
and are not in the steady-state.
If we need observations over time, it is recommended to
generate a vector of simulations and discard the first
observation.
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Recipe for MSM
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 Slide: 55
Step 3: Initial Value and Simulation
We need to generate a large sample of model outcome as
the base for estimating model’s moments.
In equilibrium models, it is recommended that we discard
earlier observations (in the time-series sense) since these
observations are sensitive to the initial value of the model
and are not in the steady-state.
If we need observations over time, it is recommended to
generate a vector of simulations and discard the first
observation.
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Recipe for MSM
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 Slide: 56
Step 4: Optimization Routine and Iteration
This is the most computationally-challenging step of MSM
procedure. We need to minimize the weighted distance of
model-generated moments from empirical moments.
N is the number of simulations,
d is the vector of data,
h is the moment function of interest,
yj(B) is the vector of simulated values corresponding to vector d.
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
𝛽 = 𝑎𝑟𝑔𝑚𝑖𝑛𝛽∈𝑅𝑘 ℎ 𝑑 −1
𝑁 ℎ 𝑦𝑗 𝛽
𝑁
𝑗=1
′
.𝑊−1 . ℎ 𝑑 −1
𝑁 ℎ 𝑦𝑗 𝛽
𝑁
𝑗=1
Recipe for MSM
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 Slide: 57
Step 4: Optimization Routine and Iteration
This is the most computationally-challenging step of MSM
procedure. We need to minimize the weighted distance of
model-generated moments from empirical moments.
N is the number of simulations,
d is the vector of data,
h is the moment function of interest,
yj(B) is the vector of simulated values corresponding to vector d.
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
𝛽 = 𝑎𝑟𝑔𝑚𝑖𝑛𝛽∈𝑅𝑘 ℎ 𝑑 −1
𝑁 ℎ 𝑦𝑗 𝛽
𝑁
𝑗=1
′
.𝑊−1 . ℎ 𝑑 −1
𝑁 ℎ 𝑦𝑗 𝛽
𝑁
𝑗=1
Recipe for MSM
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 Slide: 58
Step 4: Optimization Routine and Iteration
We need to use numerical optimization routines to find the
minimum of the total error function. A smart choice of initial
values for parameters may facilitate the quicker
convergence of the optimization routine significantly.
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Recipe for MSM
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 Slide: 59
Step 5: Interpretation and Making Conclusions
The results we will get from previous steps are just a point
estimation of true parameter values. We need to do further
statistical and computational analysis to make sure that
our results are robust and reliable.
The first step is to look at the confidence interval of
estimated values. The confidence interval estimation will
tell us about the range of probable parameter values.
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Recipe for MSM
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 Slide: 60
Step 5: Interpretation and Making Conclusions
The results we will get from previous steps are just a point
estimation of true parameter values. We need to do further
statistical and computational analysis to make sure that
our results are robust and reliable.
The first step is to look at the confidence interval of
estimated values. The confidence interval estimation will
tell us about the range of probable parameter values.
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Recipe for MSM
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 Slide: 61
Step 5: Interpretation and Making Conclusions
The results we will get from previous steps are just a point
estimation of true parameter values. We need to do further
statistical and computational analysis to make sure that
our results are robust and reliable.
The first step is to look at the confidence interval of
estimated values. The confidence interval estimation will
tell us about the range of probable parameter values.
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Discussions
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 Slide: 62
Adda (2003) believed that for implementing the MSM, a
large number of simulations is needed to compute the
standard errors of the estimator.
Using Monte-Carlo analysis, Ruge-Murcia (2012)
presented that even when simulated series are short, the
MSM works and it can deliver accurate estimates;
however, the distribution of the estimates is not efficiently
approximated by the asymptotic normal distribution.
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Discussions
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 Slide: 63
Adda (2003) believed that for implementing the MSM, a
large number of simulations is needed to compute the
standard errors of the estimator.
Using Monte-Carlo analysis, Ruge-Murcia (2012)
presented that even when simulated series are short, the
MSM works and it can deliver accurate estimates;
however, the distribution of the estimates is not efficiently
approximated by the asymptotic normal distribution.
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Discussions
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 Slide: 64
There are ways to address some of these concerns.
- Providing graphs showing the sensitivity of the
likelihood function to parameter values is one way to
convince the others that the moments were indeed
informative.
- Providing the data (if they are not propriety) and the
programming codes enables others to check the
results independently.
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Discussions
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 Slide: 65
There are ways to address some of these concerns.
- Providing graphs showing the sensitivity of the
likelihood function to parameter values is one way to
convince the others that the moments were indeed
informative.
- Providing the data (if they are not propriety) and the
programming codes enables others to check the
results independently.
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Discussions
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 Slide: 66
There are ways to address some of these concerns.
- Providing graphs showing the sensitivity of the
likelihood function to parameter values is one way to
convince the others that the moments were indeed
informative.
- Providing the data (if they are not propriety) and the
programming codes enables others to check the
results independently.
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Discussions
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 67
MLE requires distributional assumptions on error terms.
When such assumptions are justified (either empirically,
based on the observed error, or based on theoretical
considerations), MLE provides a more efficient estimation
method than MSM.
When error terms do not follow any well-established
distribution, MSM may be a better choice.
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Discussions
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 68
MLE requires distributional assumptions on error terms.
When such assumptions are justified (either empirically,
based on the observed error, or based on theoretical
considerations), MLE provides a more efficient estimation
method than MSM.
When error terms do not follow any well-established
distribution, MSM may be a better choice.
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Discussions
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 Slide: 69
MSM provides a large degrees freedom for choosing the
moment conditions which serve the interests of the
modeler.
By imposing different moment conditions one may recover
different structural parameters and as long as there are
enough meaningful moment conditions the researcher has
the luxury of using an arbitrary subset of moments.
This critic compares MSM method to standard
econometrics techniques which is more easily available for
outsiders and provide a better chance of independent
verification of the estimation results.
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Discussions
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 Slide: 70
MSM provides a large degrees freedom for choosing the
moment conditions which serve the interests of the
modeler.
By imposing different moment conditions one may recover
different structural parameters and as long as there are
enough meaningful moment conditions the researcher has
the luxury of using an arbitrary subset of moments.
This critic compares MSM method to standard
econometrics techniques which is more easily available for
outsiders and provide a better chance of independent
verification of the estimation results.
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
Discussions
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 Slide: 71
MSM provides a large degrees freedom for choosing the
moment conditions which serve the interests of the
modeler.
By imposing different moment conditions one may recover
different structural parameters and as long as there are
enough meaningful moment conditions the researcher has
the luxury of using an arbitrary subset of moments.
This critic compares MSM method to standard
econometrics techniques which is more easily available for
outsiders and provide a better chance of independent
verification of the estimation results.
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
References
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 Slide: 72
• Adda, J., & Cooper, Russell W. (2003). Dynamic economics : quantitative methods and applications.
Cambridge, Mass.: MIT Press.
• Dogan, G. (2007). Bootstrapping for confidence interval estimation and hypothesis testing for parameters
of system dynamics models. System Dynamics Review, 23(4), 415-436.
• Duffie, D., & Singleton, K. J. (1993). Simulated Moments Estimation of Markov-Models of Asset Prices.
Econometrica, 61(4), 929-952.
• Forrester, Jay W. (1961). Industrial Dynamics (1 ed.). Cambridge: The M.I.T. Press.
• Greene, William H. (2012). Econometric analysis (7th ed.). Boston: Prentice Hall.
• Lee, Bong-Soo, & Ingram, Beth Fisher. (1991). Simulation estimation of time-series models. Journal of
Econometrics, 47(2–3), 197-205. doi: 10.1016/0304-4076(91)90098-X
• Mcfadden, D. (1989). A Method of Simulated Moments for Estimation of Discrete Response Models
without Numerical-Integration. Econometrica, 57(5), 995-1026.
• Oliva, R. (2003). Model calibration as a testing strategy for system dynamics models. European Journal
of Operational Research, 151(3), 552-568. doi: Doi 10.1016/S0377-2217(02)00622-7
• Peterson, D.W. (1975). Hypothesis, estimation, and validation of dynamic social models- Energy demand
modeling. (Ph.D.), Massachusetts Institute of Technology, Cambridge, MA.
• Rahmandad, H., & Sabounchi, N. (2011). Building and estimating a dynamic model of weight gain and
loss for individuals and populations. Paper presented at the the 29th International Conference of the
System Dynamics Society, Washington, DC.
• Ruge-Murcia, Francisco. (2012). Estimating nonlinear DSGE models by the simulated method of
moments: With an application to business cycles. Journal of Economic Dynamics and Control, 36(6), 914-
938. doi: 10.1016/j.jedc.2012.01.008
• Sterman, John. (2000). Business dynamics: systems thinking and modeling for a complex world (1 ed.).
Boston: McGraw-Hill/Irwin.
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions
25th MIT-UAlbany System Dynamics Ph.D. Colloquium, Nov 2012 Slide: 73
Thank you!
Any suggestion?
mj @ vt . edu
Contents:
- Introduction
- Common
Techniques
- Some
Challenges
- Estimation
Procedures’
Requirements
- MSM
- History of
MSM
- When to Use
MSM
- Recipe for
MSM
- Discussions