Role of and Advances in Stochastic Modeling

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Role of and Advances in Stochastic Modeling

• Bryan W. Brown
• Henry Hargrove Professor of Economics and
• Professor of Statistics
• Rice University
• Rice University CoEFS Conference
• November 8, 2002

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Outline

• Motivation for simulation
• Economic modeling
• Optimizing models
• Complex solutions
• Closed-form models
• Motivation for high-performance computing
• A simple rational expectations model
• Forming expectations
• Finding a fixed point solution
• Inferential complications
• A general formulation
• Parameter estimation
• Confidence intervals
• Conditional and unconditional prediction

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Economic modeling

• Goal in economics is to usefully explain
  real-world phenomena recognized in the form of
  empirical regularities of measured variables
• Modern approach is to form the explanation
  through the use of a formal mathematical model
• Form of model suggested by economic theory
• Involves both observable variables and
  unobservable stochastic components
• Inference using the model is based on statistical
  models of the stochastic components
• Any model, mathematical or otherwise, is by
  definition an abstraction or simplification of
  the real world
• The art of modeling lies in choosing first an
  appropriate level of abstraction and
• Second, given the level of abstraction making an
  informed choice among the alternative possible
  models
• Usefulness is ultimately determined by ability to
  predict most common scenarios
• Amount of useful detail in a model is restricted
  by both the length and depth of the data and the
  modeling approach
• Available data sets are increasingly longer and
  deeper
• Modeling approach used may be limiting factor

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Optimizing models

• Ideally, the form of the model is suggested by
  the optimizing behavior of individual economic
  agents with aggregation to the level of interest
• A solution form which incorporates all the
  restrictions implied in the optimizing model is a
  structural model
• A (partially) reduced form model only
  incorporates a subset of the restrictions
• Often only the list of endogenous and
  predetermined variables is suggested by the
  optimizing model but not the functional form
• A reduced form model is subject to the Lucas
  critique, whereby a model that does not fully
  incorporate the optimizing behavior will be
  misspecified when the conditions of the
  optimizing environment are changed

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Complex solutions

• As longer and deeper data sets have become
  available we have begun to recognize increasingly
  more complex economic and financial phenonmena
• Complex phenomena usually require complex models
  to be usefully approximated
• In many cases, useful modeling of these phenomena
  involves the introduction of stochastic elements
  into the optimization problem, e.g. idiosyncratic
  attributes in a random utility model or a random
  environment in finance
• When the optimization is done in a dynamic
  context and subject to constraints that introduce
  nonlinearities the solution of the problem can
  become very complex indeed

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Closed-form models

• Economics has a long tradition and fondness for
  models whose solutions can be written in closed
  form
• Familiar forms are comforting
• Facilitates comparisons and exploration of model
  properties
• Affinity for closed forms is a straight-jacket
  that severely limits the complexity of the
  phenomena studied
• Closed-form model may explain some but not all of
  the regularities hence the frequent revelation of
  puzzles
• Tweaking a closed-form model to explain a puzzle
  may reveal other puzzles
• Certain amount of cycling among highly stylized
  closed-form models each of which explains a
  different subset of phenomena
• Simulation is an attractive alternative in more
  complex optimizing models whose solutions cannot
  be represented in closed form
• Simulation is viewed with some skepticism by many
  theotists
• Increasingly accepted in macroeconomic/calibration
  and finance literatures

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A simple rational expectations model

• Aggregate consumption out of permanent income
• Ct a b Ypt et
• Yt Ct G(Yt-1)
• where a and b are parameters, Ct is aggregate
  consumption, Gt is level of government
  expenditures and depends through taxes on
  previous income levels, et is a stochastic
  error, and Ypt is permanent income as measured by
  the present discounted value of future expected
  streams of income
• As income levels fall to low levels it is highly
  likely that the marginal propensity to consume,
  as measured by b above, will rise whereupon the
  consumption relationship will be nonlinear
• Ct C(Ypt,b) et
• where C() is a nonlinear function with
  parameters b

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Forming expectations

• To operationalize this model we need permanent
  income and hence expected future levels of Yt
• For a particular t expectation depends on future
  paths of the Ys and hence Yt-1, the future draws
  of et, and the parameters of the model
• If C() and G() are linear it may be possible to
  find a closed form expression for the
  expectation, as a linear weighted average of past
  values of Y
• For nonlinear forms, however, we will have to
  repeatedly simulate the process forward by
  drawing future es and then average the
  simulations to obtain an estimate of their
  expectation

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Finding a fixed point

• Note that the all the future simulated values
  depend on the expectation used at the start of
  the simulation and the expectation depends on the
  distribution of the future values we are
  simulating
• For a particular time period t finding a future
  distribution and current expectation that are
  consistent involves finding a fixed point by
  iterating the process until convergence
• This can be a nasty problem even with simple
  models such as that given above when the process
  is nonlinear
• Need to find fixed points for all the sample in
  order to estimate the parameters
• Existence and properties such as convergence of
  the iteration process that leads to the fixed
  points are important problems in numerical
  analysis

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A general formulation

• With some simplifying assumptions we can write
  the solution for Ypt in terms of observed lagged
  values Ypt Yp(Yt, Yt-1, Yt-2,, q) where q
  (b, g) and g are additional parameters
  introduced in finding the expectations
• Substitution yields
• Ct C(Yp(Yt, Yt-1, Yt-2,, q),b) et
• C(Yt, Yt-1, Yt-2,, q) et
• A general formulation which includes models of
  this type is
• r(yt, yt-1, yt-2,, xt, q) et
• where yt is now a vector of endogenous
  variables, xt a vector of exogenous variables, et
  a vector of unobserved disturbances, q a vector
  of parameters, and indicates that the
  functional form is the result of a simulation
  process such as that outlined above

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Parameter estimation

• Note that the parameters q used in the
  simulations above are generally not available and
  must be estimated
• The same assumptions on the distribution of et
  used to obtain the expectations discussed above
  can be used to estimate the parameters
• Whatever estimation approach is needed, the
  estimation algorithm will require repeated
  evaluations of the functions at different
  parameter values as it searches for the solution
  to the estimation equations
• Each of the function evaluations requires that we
  undertake the simulation process discussed above
  for each observation
• Complexity added by this step is directly related
  to the dimensionality of q

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Resampling-based parameter inference

• Interest in these models often centers on the
  values of crucial parameters such as the MPC in
  the consumption model so we are interested in the
  distribution of the estimators in order to draw
  inferences concerning the true values
• In many cases the distributional properties, even
  asymptotic, of the estimators coming from the
  above approach are very complicated or impossible
  to obtain
• Bootstrapping, whereby we repeatedly resample
  from a postulated parametric distribution or the
  empirical distribution of the driving variables
  of the model and re-estimate the parameters in
  order to map out the distribution of the
  estimator, has been proposed as a likely approach
  for these cases
• Thus we have the basic search for expectations
  for given parameter values, embedded within an
  estimation algorithm which searches on different
  parameter values, embedded within a resampling
  procedure which repeatedly re-estimates the model
  using different draws of the driving variables
• Nasty! We need serious computing power to tackle
  this problem

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Predictional inference

• Usually the model will be used for conditional
  prediction -- either point, interval, or more
  generally in the multivariate case, region
  prediction
• For point prediction, we will be interested in
  estimating some particular property of the
  distribution of the endogenous variables such as
  the mean or median
• For interval and region prediction we need the
  entire distribution or the entire marginal
  distributions of the endogenous variables
• Given the parameters, mapping out an expectation
  or the entire distribution is the same as that
  outlined above
• If we want to account for parameter uncertainty
  we will need to embed the bootstrapping procedure
  within the prediction problem
• Nasty! Nasty!