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help on ^linest^                                      (STB-47: sg100)
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Two-stage estimation with linear constraints
--------------------------------------------

   ^linest^ [^, c^onstraints^(^clist^)^ ^e^form^(^str^)^ ^l^evel^(^#^)^ ^k^eep ^m^odify]
 
where clist is of the form   #[^-^#][^,^ #[^-^#] ...] 
       
^linest^ shares the features of all estimation commands.


Description
-----------        

^linest^ performs linear constrained estimation after unconstrained estimation
(e.g., ^regress^, ^qreq^, ^logit^, ^stcox^, ^heckman^, ^mvreg^, ...) was performed. 
^linest^ displays the table with the two-stage constrained estimates and their 
confidence intervals, and a Wald test for the hypothesis that the parameters 
satisfy the constraints.

^linest^ can be used as a variant of ^test^ and ^testparm^ that requires the 
specification of the hypotheses by means of Stata's ^constraints^.

The standard errors of the unconstrained estimators may be of the standard or 
'robust' sandwich type and the standard errors may be adjusted for clustering.

The two-stage constrained estimator is asymptotically equivalent to the 
one-stage constrained estimator (Gourieroux and Montfort 1995: Ch 10). This 
result, however, on additional linearization. Thus, if a one-stage constrained
estimator is available (^cnsreg^, ^mlogit^, ^reg3^), I expect this estimator 
to be better behaved in small samples.


Options
-------

^constraints(^clist^)^ specifies the constraint numbers of the constraints to 
    be applied. Constraints are defined using the ^constraint^ command; see 
    @help constraint@.

^level(^#^)^ specifies the confidence level in percent for confidence intervals
    of the coefficients.
    
^eform(^str^)^ specifies that the parameter estimates should be exponentiated.
    In accordance with Stata's regular behavior, the standard errors are left
    unchanged, and the CI-interval is exponentiated as well. The string 
    argument of eform specifies the name of the column with transformed 
    coefficients.

^keep^ specifies that the results for the constrained estimates are saved for
    replay, and for post-estimation commands (e.g., ^predict^, ^test^). 

^modify^ specifies that the results for the unconstrained estimates are 
    modified into the associated results for the constrained estimates. Thus,
    replaying the original estimation command with display the results for 
    the constrained estimates. ^modify^ is not permitted for 'internal'
    estimation commands (e.g., ^logit/logistic^, ^ologit^, ^cox/stcox^).
                
                
Examples
--------             

   . ^logit y x1 x2 x3 x4^
   . ^constr def 1 x1+x2+x3+x4=0^
   . ^linest, c(1)^                            (estimate model s.t. constraint 1)
   . ^const def 2  x1=x2^
   . ^linest, c(1,2)^                   (estimate model s.t. constraints 1 and 2)


Remarks
-------

Currently, Stata supports constrained estimation in a limited number of 
models, namely linear regression (^cnsreg^), multinomial logistic regression
(^mlogit^) and simultaneous equations (^reg3^). I expect that all of Stata's 
official commands support linear constraints, one day. This is the desired 
situation indeed. But for now, and for user-written command for which the 
trouble to add code to deal with constraints may be excessive, we supply a 
solution. This is based on the theory of equality-constrained estimation 
that states that that the constrained estimation program

        (1)   max l(b,x)  subject to   G b = a
        
is first-order equivalent to the 2-stage program

    (2.1) max l(b,x) without constraints
          let bu be the unconstrained l-extremum estimator of b
          with asymptotic variance matrix Vu
                                
    (2.2) minimize (bu-b)' Inv(Vu) (bu-b)  subject to G b = a
                  
For the linear regression with linear constraints case, the equivalence is 
even exact. (The estimated coefficients from ^cnsreg^ and ^regres/linest^ 
are (should be) identical, but the estimated standard errors will differ 
somewhat as a consequence of the asymptotic (large sample) nature of the 
equivalence of constrained and two-stage estimation.) For other models the
equivalence is asymptotic. 

In fact, the asymptotic equivalence applies even to smooth nonlinear 
constraints. At this moment Stata does not contain an efficient tool for 
non-linear GLS-estimation, and I don't want to make this large investment 
now. I plan to provide a number of commands for imposing for specific 
non-linear constraints: rank restrictions (e.g., factor-analytic models 
for interaction effects, Goodman's rc models, ..), Sobel's diagonal 
reference constraints.
        

Acknowledgements
----------------

This research is support by grant PGS 50-370 from the Netherlands Organization
for Scientific Research.  I thank Vince Wiggens (StataCorp) for helpful 
suggestions on linking linest-results to the rest of Stata.

 
References
----------

   Gourieroux, C. and A. Monfort (1995/1998)
      Statistics and Econometric Models. 2 Volumes.
      Cambridge University Press.
                
   Silvey, S.D. (1970)
      Statistical Inference. London: Chapman and Hall.
                               

Author
------

      Jeroen Weesie
      Utrecht University, Netherlands
      J.Weesie@@fss.uu.nl


Also See
--------

    STB:  STB-47 sg100
On-line:  ^help^ for @constraint@, @test@, @testparm@, @testnl@.