[Limdep Nlogit List] LM, LR and Wald tests in heteroscedastic selection model

[William Greene]

Jadwiga. I do not see enough to know for sure that you have set up
the procedures correctly. But, I would say that your best bet is the
LR test. The LM test is probably not best because the starting
values actually used are not the constrained MLEs for your model.
The starting values for the probit model are not the MLEs subject
to the restriction you are testing - they are also computed assuming
RHO=0.  I can't say for sure the Wald test is right either - there is
some code missing. But, based on what I see, as noted, I would say
the LR test is the safe one.
Regards,
Bill Greene


----- Original Message -----
From: "Jadwiga Kostrzewska" <dwakotki at wp.pl>
Cc: "Limdep and Nlogit Mailing List" <limdep at limdep.itls.usyd.edu.au>
Sent: Saturday, May 23, 2009 11:11:43 AM GMT -05:00 US/Canada Eastern
Subject: [Limdep Nlogit List] LM, LR and Wald tests in heteroscedastic selection model

Dear All,
I am estimating the selection model with heteroscedasticity. I have 
chosen some variables (the set XH) to heteroscedasticity equation, and I 
am going to test statistically significance of a subset of variables 
(say AXH). I have written some procedures for three tests LM, LR and 
Wald, but I have a problem, because the results are very divergent. The 
procedure is as follows. (First my symbols are introduced.)

XH - a set of variables is heteroscedasticity equation
AXH - a subset of the XH
XHbA=XH\AXH
XHhip={XHbA,AXH} - the same variables as in the XH, but in another order:
variables from XHbA are at the beginning, then variables from AXH.

H0: all variables in AXH are statistically insignificant
H1: not all of them

PROC=hetS_zbB$
   ? model H0 (restrictive):
	   PROBIT; Lhs=SEL; Rhs=W; Hold$
	   SELECT; Lhs=ZAL; Rhs=XW; Hfn=XHbA; MLE$
	   CALC; logL0=LOGL; odch=s; roro=rho$
	? t e s t  LM:
		CALC;	j3=Col(XW)+Col(XHbA);
			j5=Col(AXH); stsw=Col(AXH)$
		MATRIX; start1=b(1:j3)$
		PROBIT; Lhs=SEL; Rhs=W; Hold$
		SELECT; Lhs=ZAL; Rhs=XW; Hfn=XHhip; MLE
			; Start=start1,j5_0,odch,roro; Maxit=0$
		CALC; statLM=LMSTAT; pvLM=1-Chi(StatLM,stsw)$
	? t e s t  LR:
	   ? model H1 (unrestrictive):
	      PROBIT; Lhs=SEL; Rhs=W; Hold$
 	      SELECT; Lhs=ZAL; Rhs=XW; Hfn=XHhip; MLE$
	      CALC; logL1=LOGL; StatLR=2*(logL1-LogL0)$
	      CALC; pvLR=1-Chi(StatLR,stsw)$
	? t e s t  Walda:
		? bhp and vbhp - are the b and the varb for tested parameters
		? jp2, jk2 - the first and the last index for AXH;
			CALC; jp2=Col(XW)+Col(XHbA)+1;	jk2=Col(XW)+Col(XHbA)+Col(AXH)$
			MATRIX; bhp=b(jp2:jk2)$
			MATRIX; vbhp=varb(jp2:jk2,jp2:jk2)$
			MATRIX; StatWald=bhp'<vbhp>bhp$
			CALC;   pvWALD=1-Chi(StatWald,stsw)$
	MATRIX; List; istLMLRW=[StatLM,pvLM/StatLR,pvLR/StatWald,pvWald]$
	CALC; Delete 
statLM,LMSTAT,statLR,pvLM,pvLR,pvWald,logL1,logL0,roro,odch,j3,j5,jp2,jk2,stsw$
	MATRIX; Delete bhp,vbhp,StatWald,start1$
ENDPROC$ ? ****** PROC=hetS_zbB$

The results are like below:
--> 	 MATRIX; List; istLMLRW$
1|   19.47718       .03460   - LM
2|   48.14571    .5837791D-06 - LR
3|    1.83534       .99745  - Wald
another example:
1|   20.36425       .06050 - LM
2|   49.46820       .1731958D-05  - LR
3|    6.79378       .87093 - Wald
another example:
1|   23.59600       .09871 - LM
2|   54.31762    .4601083D-05 - LR
3|   13.65864       .62413 - Wald

The sample is not small,  n=6987 observations:
|SEL     | Total  |     0      1   |
+--------+--------+----------------+
|       0|  1132  |  1132      0   |
|       1|  5855  |     0   5855   |
+--------+--------+----------------+
|   Total|  6987  |  1132  5       |
+--------+--------+----------------+

(I have analogous procedures for the homoscedastic selection model, and 
the results of three tests are rather similar, not like in the 
heteroscedastic selection model.)

My question:
Is the situation caused by the “two-step” MLE for the heteroscedastic 
model or by reparametrisation in likelihood function?
What is wrong? Why the results of three tests are so distant one from 
each other?
Which test should I use?
Are there any mistakes in my procedure?
;-(

Any ideas are very welcome!

Best regards,
Jadwiga Kostrzewska

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