Estimating fully observed recursive mixed-process models with cmp David Roodman.

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Estimating fully observed recursive mixed-process models with cmp David Roodman

Transcript of Estimating fully observed recursive mixed-process models with cmp David Roodman.

Page 1: Estimating fully observed recursive mixed-process models with cmp David Roodman.

Estimating fully observed recursivemixed-process models with cmp

David Roodman

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Probit model:Link function (g) induces likelihoods for each possible outcome

y=g(y*)=1{y*>0}

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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))

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Tobit (censored)

Ordered probit

Just change g() to get new models

With generalization, embraces multinomial and rank-ordered probit, truncated regression…

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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

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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.

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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)′.

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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

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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

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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

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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

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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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Required. One exp for each equation. Tell cmp model type for each eq and can vary by observation

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Emulation examples

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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

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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/observationDoes 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 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

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Possible extensionsMarginal effects that reflect interactions between

equations

(Multi-level) random effects

Dropping full observability—y*’s 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.