Estimating fully observed recursive
mixed-process models with cmp
David Roodman
Probit model:
Link function (g) induces likelihoods for each possible outcome
fε(y*–x'β)
Area:
1–Fε(0–x'β)
xi'β
y=g(y*)=1{y*>0}
y=1
0
y=0
Area:
Fε(0–x'β)
y* (latent)
y (observed)
Relabeling left graph for ε scale:
“error link” function (h) induces likelihoods for each possible outcome
fε(ε)
Area:
0
–xi'β
y=h(ε)=1{ε>–xi'β}
y=1
y=0
(h(ε)=g(x'β +ε))
Area:
ε (latent)
y (observed)
Just change g() to get new models
Tobit (censored)
Ordered probit
With generalization, embraces multinomial and
rank-ordered probit, truncated regression…
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
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.
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 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
If system is
Fact
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
sureg and biprobit can 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
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
200
Heteroskedasticity can make censored
models not just inefficient but inconsistent
-50
0
50
y 100
150
Tobit example:
error variance
rises with x
-20
0
20
40
60
x
y*
True model
Censored values
Fitted model
80
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.)
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 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 extensions
Marginal 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.