Estimating fully observed recursive

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Estimating fully observed recursive mixed-process
models with cmp David Roodman
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Probit model Link function (g) induces
likelihoods for each possible outcome
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Relabeling left graph for e scale error link
function (h) induces likelihoods for each
possible outcome
(h(e)g(x'ß e))
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Just change g() to get new models
With generalization, embraces multinomial and
rank-ordered probit, truncated regression
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Compute likelihood same way

• Given yi, determine feasible value(s) for e
• If just one, Li normal density at that point
• If a range, Li cumulative density over range
• For models that censor some observations (Tobit),
  L? Li combines cumulative and point densities.
• Amemiya (1973) maximizing L is consistent

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Multiple equations (SUR)
For each obs, likelihood reached as before Given
y, determine feasible set for e and integrate
normal density over it Feasible set can be point,
ray, square, half plane Cartesian product of
points, line segments, rays, lines.
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Bivariate probit

• Suppose for obs i, yi1 yi20
• Feasible range for e is
• Integral of fe(e)f(eS) over this
• Can use built-in binormal().
• Similar for y(0,1)', (1,0)', (1,1)'.

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Mixed uncensored-probit

• Suppose for obs i, we observe some y(yi1, 0)'
• Feasible range for e is a ray
• Integral of fe(e)f(eS) over this
• Integral of 2-D normal distribution over a ray.
• Hard with built-in functions
• Requires additional math

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Conditional modelingc in cmp

• Model can vary by observationdepend on data
• Worker retraining evaluation
• Model employment for all subjects
• Model program uptake only for those in cities
  where offered
• Classical Heckman selection modeling
• Model selection (probit) for every observation
• Model outcome (linear) for complete observations
• Likelihood for incomplete obs is one-equation
  probit
• Likelihood for complete obs is that on previous
  slide
• Myriad possibilities

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Recursive systems

• ys can appear on RHS in each others equations
• Matrix of y coefficients must be upper triangular
• I.e. System must have clearly defined stages.
  E.g.
• SUR (several equations, one stage)
• 2SLS
• If system is fully modeled and truly recursive,
  then estimation is FIML
• If system has simultaneity and the early equation
  stages instrument, then LIML

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Fact

• If system is
• Recursive
• Fully observed (ys appear in RHS but never ys)
• then likelihoods developed for SUR still work
• Can treat ys in RHS just like xs
• sureg and biprobit can be IV estimators!
• Rarely understood, not proved in general in
  literature
• Greene (1998) surprisinglyseem not to be
  widely known
• Wooldridge (e-mail 2009) I came to this
  realization somewhat late, although Ive known it
  for a couple of years now.
• I prove, perhaps not rigorously
• Maybe too simple for great econometricians to
  bother publishing

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General recursive, fully observed system
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• cmp can fit
• conditional recursive mixed-process systems
• Processes Linear, probit, tobit, ordered probit,
  multinomial probit, interval regression,
  truncated regression
• Can emulate
• Built-in probit, ivprobit , treatreg , biprobit,
  oprobit, mprobit, asmprobit, tobit, ivtobit,
  cnreg, intreg, truncreg, heckman, heckprob
• User-written triprobit, mvprobit, bitobit,
  mvtobit, oheckman, (partly) bioprobit

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Emulation examples
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Heteroskedasticity can make censored models not
just inefficient but inconsistent
Tobit example error variance rises with x
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Implementation innovation ghk2()

• Mata implementation of Geweke-Hajivassiliou-Keane
  algorithm for estimating cumulative normal
  densities above dimension 2.
• Differs from built-in ghkfast()
• Accepts lower as well as upper bounds
• E.g., integrate over cube a1,b1 a2,b2
  a3,b3
• (otherwise requires 23 calls instead of 1)
• Optimized for many observations few simulation
  draws/observation
• Does not pivot coordinates. Pivoting can
  improve precision, but creates discontinuities
  when draws are few. (ghkfast() now lets you turn
  off pivoting.)

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Implementation innovation lfd1

• In Stata ML, using an lf likelihood evaluator
  assumes that (A1) for each eq,
• ml computes numerically with 2 calls per eq,
• then analytically.
• And for Hessian, of calls is quadratic in of
  eq
• Using a d1 evaluator, ml does not assume A1.
• But does (A2) require evaluator to provide
  scores
• For Hessian, of calls in linear in of
  parameters
• Two unrelated changes create unnecessary
  trade-off
• ml is missing an lfd1 type that assumes A1 and
  A2would make Hessian with of calls linear in
  of eq.
• Solution pseudo-d2. d2 routine efficiently takes
  over (numerical) computation of Hessian
• Good for score-computing evaluators for which

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Possible extensions

• Marginal effects that reflect interactions
  between equations
• (Multi-level) random effects
• Dropping full observabilityys on right
• Rank-ordered multinomial probit

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References

• Roodman, David. 2009. Estimating fully observed
  recursive mixed-process models with cmp. Working
  Paper 168. Washington, DC Center for Global
  Development.
• Roodman, David, and Jonathan Morduch. 2009. The
  Impact of Microcredit on the Poor in Bangladesh
  Revisiting the Evidence. Working Paper 174.
  Washington, DC Center for Global Development.