Estimating fully observed recursive mixed-process models with cmp David Roodman.
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Transcript of Estimating fully observed recursive mixed-process models with cmp David Roodman.
Estimating fully observed recursivemixed-process models with cmp
David Roodman
Probit model:Link function (g) induces likelihoods for each possible outcome
y=g(y*)=1{y*>0}
Relabeling left graph for ε scale:“error link” function (h) induces likelihoods for each possible outcome
y=h(ε)=1{ε>–xi'β}
Area:
(h(ε)=g(x'β +ε))
Tobit (censored)
Ordered probit
Just change g() to get new models
With generalization, embraces multinomial and rank-ordered probit, truncated regression…
Given yi, determine feasible value(s) for ε
– 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=Π Li combines cumulative and point densities.
Amemiya (1973): maximizing L is consistent
Compute likelihood same way
Multiple equations (SUR)
For each obs, likelihood reached as beforeGiven y, determine feasible set for ε and integrate
normal density over itFeasible set can be point, ray, square, half plane…
Cartesian product of points, line segments, rays, lines.
Bivariate probit
Suppose for obs i, yi1= yi2=0
Feasible range for ε is:
Integral of fε(ε)=φ(ε;Σ) over this:
Can use built-in binormal().
Similar for y=(0,1)′, (1,0)′, (1,1)′.
Mixed uncensored-probit
Suppose for obs i, we observe some y=(yi1, 0)′
Feasible range for ε is a ray:
Integral of fε(ε)=φ(ε;Σ) over this:
Integral of 2-D normal distribution over a ray.
Hard with built-in functions
Requires additional math
Conditional modeling—“c” in cmp
Model can vary by observation—depend 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
Recursive systemsy’s can appear on RHS in each other’s 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
If system isRecursive
Fully observed (y’s appear in RHS but never y*’s)
then likelihoods developed for SUR still workCan treat y’s in RHS just like x’s
sureg and biprobit can be IV estimators!
Rarely understood, not proved in general in literatureGreene (1998): “surprisingly”…“seem not to be widely known”
Wooldridge (e-mail 2009): “I came to this realization somewhat late, although I’ve known it for a couple of years now.”
I prove, perhaps not rigorouslyMaybe too simple for great econometricians to bother publishing
Fact
General recursive, fully observed system
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
Required. One exp for each equation. Tell cmp model type for each eq and can vary by observation
Emulation examples
Heteroskedasticity can make censored models not just inefficient but inconsistent
-50
050
100
150
200
y
-20 0 20 40 60 80x
y* Censored valuesTrue model Fitted model
Tobit example: error variance rises with x
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/observationDoes not “pivot” coordinates. Pivoting can improve precision, but creates discontinuities when draws are few. (ghkfast() now lets you turn off pivoting.)
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 scoresFor Hessian, # of calls in linear in # of parameters
Two unrelated changes create unnecessary trade-offml is missing an “lfd1” type that assumes A1 and A2—would 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
Possible extensionsMarginal effects that reflect interactions between
equations
(Multi-level) random effects
Dropping full observability—y*’s on right
Rank-ordered multinomial probit
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.