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

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

4. Probit model: Link function (g) induces likelihoods for each possible outcome y=g(y*)=1{y*>0}

5. Relabeling left graph for ε scale: “error link” function (h) induces likelihoods for each possible outcome y=h(ε)=1{ε>–xi'β} (h(ε)=g(x'β +ε))

6. Tobit (censored) Just change g() to get new models Ordered probit With generalization, embraces multinomial and rank-ordered probit, truncated regression…

7. Compute likelihood same way 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

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

9. 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)′.

10. 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

11. 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

12. Recursive systems y’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

13. Fact If system is Recursive Fully observed (y’s appear in RHS but never y*’s) then likelihoods developed for SUR still work Can treat y’s in RHS just like x’s suregand biprobitcan be IV estimators! Rarely understood, not proved in general in literature Greene (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 rigorously Maybe too simple for great econometricians to bother publishing

14. General recursive, fully observed system

15. 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

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

17. Emulation examples

18. Heteroskedasticity can make censored models not just inefficient but inconsistent Tobit example: error variance rises with x

19. 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.)

20. 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 d1evaluator, 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 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

21. Possible extensions Marginal effects that reflect interactions between equations (Multi-level) random effects Dropping full observability—y*’s on right Rank-ordered multinomial probit

23. 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.