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RE: st: heteroskedastic multivariate normal regression with ml, convergence problem


From   "Carlo Fezzi" <c.fezzi@uea.ac.uk>
To   <statalist@hsphsun2.harvard.edu>
Subject   RE: st: heteroskedastic multivariate normal regression with ml, convergence problem
Date   Mon, 23 Feb 2009 17:29:20 -0000

Thanks a lot, those are good suggestions.

In fact I tried increasing the number of observations to 10000 and the
algorithm converges easily after roughly 40 iterations... I guess more
experimentation is needed to find general rules..

Cheers

Carlo


-----Original Message-----
From: owner-statalist@hsphsun2.harvard.edu
[mailto:owner-statalist@hsphsun2.harvard.edu] On Behalf Of jverkuilen
Sent: 23 February 2009 17:14
To: statalist@hsphsun2.harvard.edu
Subject: RE: st: heteroskedastic multivariate normal regression with ml,
convergence problem

This isn't a Stata suggestion, but a math/numerical analysis one:

One big problem I have encountered with heteroscedastic models is the fact
that you need *much* better starting values than for homoscedastic models.
Recall that numerical optimiztion by Newton's method and its variants is
locally convergent. A model like the MVN with unconstrained covariance
matrix or other exponential family is "nice" because the log-likelihood is
globally concave, near---or exactly---quadratic, you get separability
between location and scale, etc. Add heteroscedasticity predictors and this
is no longer true generally. If you don't start close enough to the solution
you may end up in a region of the likelihood that is a local optimum, very
ill-behaved, or even singular. 

Also, you may well have a situation of empirical or even true
unidentification. 

-----Original Message-----
From: "Carlo Fezzi" <c.fezzi@uea.ac.uk>
To: statalist@hsphsun2.harvard.edu
Sent: 2/23/2009 9:32 AM
Subject: st: heteroskedastic multivariate normal regression with ml,
convergence problem

Dear all,

I am writing my code to estimate a multivariate normal regression with
heteroskedastic errors via maximum likelihood, using ?ml? with the method
?lf?. I have a question specific on my model, but I guess the answer could
be useful to a lot of beginner maximum likelihood programmers like myself. 

My multivariate normal code works really well when I estimate the model with
homoskedastic error. However, if I add a simple parametric specification for
the variance, such as log(si) = b0 + b1x1i, the maximization problem seems
much more difficult and I get unexpected results (even if I generate the
data to be homoskedastic, and so I would expect simply the parameters in the
variance equation to be non-significant). For instance, if you try to run
the example below, one of the correlation parameters will end up to have
standard error and confidence interval to be missing values. Also, the
parameters of x1 will result in being significant in the variance equations,
even if the data are generated to be homoskedastic, so I would expect them
to be non-significant.

Is there anything I can do to improve the convergence, both in the structure
of the code or in the maximization technique?

Thanks a lot for your help,

Carlo 

This is my code:

----------- begin example --------------
set seed 1

clear
set obs 1000

global n_eq = 3

matrix r = (1, 0.35, 0.5 \ 0.35, 1, 0.2 \ ///
0.5, 0.2, 1)

drawnorm u1 u2 u3, corr(r)

gen x1 = uniform() -0.5
gen x2 = 2*uniform() + 0.5

gen y1 = 0.5 + 4*x1 + u1
gen y2 = -1 -2*x1 + 3*x2 + u2
gen y3 = 3 + 3*x1 - 2*x2 + u3


capture program drop multireg_lf
program multireg_lf
	version 10.0

*** inizialize beta vectors ***	

	local mu_k "mu1"
    forvalues i = 2/$n_eq {
    local mu_k "`mu_k' mu`i'"
    }

*** inizialize sigmas ***	

	local lnsigma_k "lnsigma1"
    forvalues i = 2/$n_eq {
    local lnsigma_k "`lnsigma_k' lnsigma`i'"
    }

*** inizialize correlations ***	

	local atr_kk ""
    forvalues i = 1/$n_eq {
		forvalues j = `=`i'+1'/$n_eq {
			local atr_kk "`atr_kk' ar`i'`j'"
			
		}
	}

	args lnf `mu_k'  `lnsigma_k' `atr_kk'

**	display "`mu_k'" "`lnsigma_k'" "`atr_kk'"

*** compute residuals ***	

	forvalues i=1/$n_eq {
		tempvar e`i'
		gen double `e`i'' = (${ML_y`i'} - `mu`i'')
	}
		
	tempname S invS ip is

*** compute covariance matrix ***	
	
	matrix `S' = J($n_eq,$n_eq,0)

** variances **

	forvalues i = 1/$n_eq {
		summ `lnsigma`i'', meanonly
		matrix `S'[`i',`i']=exp(r(mean))^2
	}

** co-variances **
    
	forvalues i = 1/$n_eq {
		forvalues j = `=`i'+1'/$n_eq {
			summ `ar`i'`j'', meanonly
			matrix `S'[`i',`j'] =
sqrt(`S'[`i',`i'])*sqrt(`S'[`j',`j'])*tanh(r(mean))
			matrix `S'[`j',`i'] = `S'[`i',`j']		
		}
    }

** substitute the covariance matrix with the one at the previous step
** if not positive semidefinite
	
	capture matrix CC = cholesky(`S') 
	
	if _rc != 0	{
		matrix `S' = CC*CC'
	}

*** compute the likelihood ***

** define a names vector to do correctly the score operation **

	local nam=""
	forvalues i = 1/$n_eq {
		local nam `nam' `e`i''
		}

** compute the (x-mu)InvS(x-mu)' part of the likelihood **
	
	mat `invS' = invsym(`S')
	
	gen double `ip'=0
	
	forvalues i = 1/$n_eq {
		tempvar g`i'
		matrix `is' = `invS'[`i',1...]
		matrix colnames `is' =`nam'
		quietly matrix score `g`i''=`is'

		quietly replace `ip'= `ip' + `e`i''*`g`i''
	}

	quietly replace `lnf' =
-$n_eq/2*log(2*_pi)-1/2*log(det(`S'))-0.5*`ip'
	
end

ml model lf multireg_lf (mu1: y1 = x1 x2) (mu2: y2 = x1 x2) (mu3: y3 = x1
x2) ///
 			(lnsigma1: x1) (lnsigma2: x1) (lnsigma3:) ///
			(ar12:) (ar13:) (ar23:), technique(bhhh dfp)


quietly regress y1 x1 x2
mat b1 = e(b)
mat coleq b1 = mu1

quietly regress y2 x1 x2
mat b2 = e(b)
mat coleq b2 = mu2

quietly regress y3 x1 x2
mat b3 = e(b)
mat coleq b3 = mu3


mat b0 = b1,b2,b3
ml init b0

ml maximize, difficult

----------- end example --------------

 	





**************************************************
Carlo Fezzi
Senior Research Associate
Centre for Social Research on the Global Environment (CSERGE)
Department of Environmental Sciences
University of East Anglia
Norwich (UK) NR2 7TJ 
Telephone: +44(0)1603 591385
Fax: +44(0)1603 593739
*************************************************



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