Statistical Analysis: Delta Method and Krinsky and Robb Method

LIMDEP and NLOGIT’s WALD command computes values and standard errors (using the delta method or the Krinsky and Robb method) for specified linear or nonlinear functions of your estimates. (This is one of the handiest tools in the program.) No need to program derivatives or do any matrix algebra. You provide the command with the (1) name of the vector that contains the parameters, (2) the name of the covariance matrix, (3) the names that you wish to use for the parameters in your function(s) and (4) a set of up to 50 functions of the parameters. The program returns the computed functions, estimated standard errors, asymptotic t ratios and prob values. The name of the command derives from the final computation, which is, if you wish to use it, computation of the Wald statistic for the hypothesis that all of the functions equal zero.

  • Use WALD with any model results
  • Use with any random vector provided with a covariance matrix
  • Up to 50 nonlinear functions
  • Use the delta method to estimate asymptotic covariance matrices
  • Use the Krinsky and Robb method as an alternative
  • Test hypotheses involving nonlinear functions of parameters

Example: McDonald and Moffitt Decomposition

McDonald and Moffitt’s decomposition is an analysis of the conditional mean function in the tobit model. Specifically, in this model, assuming censoring at L in the lower tail only,

E[y|x] = L*Prob(y=L) + Prob(y > L)*E[y|x,y>L)

It follows that

dE[y|x]/dx = Prob(y > L)*dE[y|x,y > L]/dx + E[y|x,Y > L]*dProb(y > L)/dx

This breaks the slope into two parts, (1) the change in y given nonlimit times the probability of being above the limit value and (2) the change in the probability of being above the limit times the conditional mean. The formal counterparts to these expressions are:

dE[y|x]/dx = Prob(y > L)*[F(1 - f/F)*(b'x/s + f/F)) + (f*(b'x/s + f/F))]*b

The following general program illustrates WALD for this computation.

?===================================================================
? Set the model up with these three commands.  The rest is standard.
? The routine computes the full set of marginal effects for the 
? estimated model, decomposed by the formula given above. Note that
? the message at the end of singular covariance matrix is to be 
? expected since the second and seventh functions are equal.
?===================================================================
NAMELIST	; x  = the list of Rhs variables $
CREATE	; y  = the dependent variable $
CALC	; li = the lower limit value (usually zero) $
?-------------------------------------------------------------------
CALC	; k 	= Col(x) $
PROC 	= McdnM(y,x,li) $
TOBIT	; Lhs = y ; Rhs=x ; Par$
WALD     ; Average ; Labels = k_b,st
       	; Start 	= b ; Var = varb 
       	; Fn1 	= alpha    = (li - b1'x)/st 	? Alpha
       	; Fn2 	= pnonzero = 1-Phi(Fn1)        	? Prob(noncensored)
       	; Fn3 	= lambda   = N01(Fn1) / Fn2    	? Lambda
       	; Fn4 	= delta    = Fn3*Fn3 - Fn1*Fn3	? Delta
       	; Fn5 	= first    = Fn2*Fn4       	? First part
       	; Fn6 	= second   = Fn2*(1-Fn4)		? Second part
	; Fn7 	= total    = Fn5 + Fn6    	$ Verify decomposition = Fn2
ENDPROC $
EXECUTE	; Proc	= McDnM(y,x,0)$

-----------------------------------------------------------------------------
WALD procedure. Estimates and standard errors
for nonlinear functions and joint test of
nonlinear restrictions.
VC matrix for the functions is singular.
Standard errors are reported, but the
Wald statistic cannot be computed.
Functions of data are averaged over the obs.
--------+--------------------------------------------------------------------
        |                  Standard            Prob.      95% Confidence
WaldFcns|  Coefficient       Error       z    |z|>Z*         Interval
--------+--------------------------------------------------------------------
   ALPHA|    -.23798***      .04729    -5.03  .0000     -.33067   -.14530
PNONZERO|     .58944***      .01675    35.19  .0000      .55661    .62227
  LAMBDA|     .67729***      .02746    24.67  .0000      .62347    .73111
   DELTA|     .57072***      .01149    49.65  .0000      .54820    .59325
   FIRST|     .31981***      .00493    64.90  .0000      .31015    .32947
  SECOND|     .26963***      .01451    18.58  .0000      .24119    .29807
   TOTAL|     .58944***      .01675    35.19  .0000      .55661    .62227
--------+--------------------------------------------------------------------
Note: ***, **, * ==>  Significance at 1%, 5%, 10% level.
-----------------------------------------------------------------------------