# [R] sem: negative parameter variances

From: Denis Fomchenko <fomchenko_at_iet.ru>
Date: Tue 18 Jul 2006 - 04:39:37 EST

Dear Spencer and Prof. Fox,

Thank you for your replies. I'll very appreciate, if you have any ideas concerning the problem described below.

First, I'd like to describe the model in brief.

In general I consider a model with three equations.

First one is for annual GRP growth - in general it looks like:

1. GRP growth per capita = G(investment, migration, initial GRP per capita, spatial lag on GRP growth + some additional explanatory variables to control for regional disparities),

where spatial lag on GRP growth in year t is simply spatial weights matrix W(n*n) (weights - are inverse square distances between regional centers; with zeros on the main diagonal) times GRP growth in year t, (n*1). I compute it manually for every year.

Two others are for migration (labor supply) and investment (capital supply):

2) Migration = M(factors, explaining migration)

3) Investment = I(factors, explaining investment)

I consider GRP growth, migration and investment to be endogenous and all others vars to be exogenous to the model. The data are over 77 regions (n=77) and 8 years (t = 1997..2004), so that I have total 616 observations.

Actually, the final goal of the study is to estimate a model 1) simultaneously by FIML 2) with Mundlak (1978, 1981) specification of panels to capture for fixed and between effects 3) with spatial lag on GRP growth. As for spatial lag - the model needs to be estimated by maximum likelihood (as it is shown in Anselin 1988, for example)

The main problems with Mundlak are increasing number of variables: as well as dimension of a model (additional between equations). So to begin with, I try to estimate pool, which is obviously easier, and proceed as follows:

Consider the following specification:

model.3eq.33 <- specify.model( )

```ln.inv.p0.pc                              ->        ln.grp.phvi.pc   ,           beta13             ,           NA

ln.migr.new                              ->        ln.grp.phvi.pc   ,           beta12             ,           NA

ln.grp.corr.pc97                       ->        ln.grp.phvi.pc   ,           gamma11         ,           NA

ln.sh.ind.rawwide                     ->        ln.grp.phvi.pc   ,           gamma12         ,           NA

port                                         ->        ln.grp.phvi.pc   ,           gamma13         ,           NA

t12                                          ->        ln.grp.phvi.pc   ,           gamma14         ,           NA

wd.ln.grp.phvi.pc                     ->        ln.grp.phvi.pc   ,           rho                   ,           NA

(Intercept)                               ->        ln.grp.phvi.pc   ,           alpha1              ,           NA

ln.income.pc.fcb97                   ->        ln.migr.new      ,           gamma21         ,           NA

ln.unempl.level97                     ->        ln.migr.new      ,           gamma26         ,           NA

avertemp.jan.cs                        ->        ln.migr.new      ,           gamma22         ,           NA

ln.pop.gr.89.26mbe                 ->        ln.migr.new      ,           gamma23         ,           NA

ln.pass.railway.percap              ->        ln.migr.new      ,           gamma24         ,           NA

ln.city                                      ->        ln.migr.new      ,           gamma25         ,           NA

(Intercept)                              ->        ln.migr.new      ,           alpha2              ,           NA

ln.grp.corr.pc97                       ->        ln.inv.p0.pc      ,           gamma31         ,           NA

avertemp.jan.cs                        ->        ln.inv.p0.pc      ,           gamma32         ,           NA

permafrost                               ->        ln.inv.p0.pc      ,           gamma33         ,           NA

ln.indoutput.fuel.p0.pc              ->        ln.inv.p0.pc      ,           gamma34         ,           NA

ln.phone1995                           ->        ln.inv.p0.pc      ,           gamma35         ,           NA

(Intercept)                               ->        ln.inv.p0.pc      ,           alpha3              ,           NA

ln.grp.phvi.pc                           <->      ln.grp.phvi.pc   ,           sigma11           ,           NA

ln.grp.phvi.pc                           <->      ln.migr.new      ,           sigma12           ,           NA

ln.grp.phvi.pc                           <->      ln.inv.p0.pc      ,           sigma13           ,           NA

ln.migr.new                              <->      ln.migr.new      ,           sigma22           ,           NA

ln.migr.new                              <->      ln.inv.p0.pc      ,           sigma23           ,           NA

ln.inv.p0.pc                              <->      ln.inv.p0.pc      ,           sigma33           ,           NA

```

So I consider a model with 3 equations, 3 endogenous and 13 exogenous variables and 6 double-arrow relations for error covariance.

Then, I use raw moments to capture (Intercept) in each equation:

raw.3eq.33 <- raw.moments(~ ln.grp.phvi.pc + ln.inv.p0.pc + ln.migr.new + ln.grp.corr.pc97 + ln.sh.ind.rawwide + port + t12 + wd.ln.grp.phvi.pc + ln.income.pc.fcb97 + ln.unempl.level97 + avertemp.jan.cs + ln.pop.gr.89.26mbe + ln.pass.railway.percap + ln.city + permafrost + ln.indoutput.fuel.p0.pc + ln.phone1995, data=pool.spat)

Next, I estimate model by sem-function (here, I declare fixed exogenous variables, including spatial lag 'wd.ln.grp.phvi.pc' ... though I am not sure about that)

model.3eq.33.estim <- sem(model.3eq.33, raw.3eq.33, 616, fixed.x=c('ln.grp.corr.pc97', 'ln.sh.ind.rawwide', 'port', 't12', 'wd.ln.grp.phvi.pc', 'ln.income.pc.fcb97', 'ln.unempl.level97', 'avertemp.jan.cs', 'ln.pop.gr.89.26mbe', 'ln.pass.railway.percap', 'ln.city', 'permafrost', 'ln.indoutput.fuel.p0.pc', 'ln.phone1995', '(Intercept)'), raw=TRUE)

Then, R returns:

Warning message:
Negative parameter variances.
Model is probably underidentified.
in: sem.default(ram = ram, S = S, N = N, param.names = pars, var.names = vars, :

Obtained summary:

summary(model.3eq.33.estim)

Model fit to raw moment matrix.

Model Chisquare = 398.81 Df = 24 Pr(>Chisq) = 0

Goodness-of-fit index = 0.944

RMSEA index = 0.15923 90 % CI: (0.14569, 0.17316)

BIC = 175.29 Normalized Residuals

Min. 1st Qu. Median Mean 3rd Qu. Max.

-0.9120 0.0000 0.0000 0.0268 0.0000 1.5600

```      Parameter
Estimate
Std Error
z value
Pr(>|z|)

beta13
0.0151
0.0031
4.9157
0.0000
ln.grp.phvi.pc

<---
ln.inv.p0.pc

beta12
0.9983
0.4831
2.0663
0.0388
ln.grp.phvi.pc

<---
ln.migr.new

gamma11
-0.0268
0.0074
-3.6420
0.0003
ln.grp.phvi.pc

<---
ln.grp.corr.pc97

gamma12
0.0120
0.0174
0.6867
0.4923
ln.grp.phvi.pc

<---
ln.sh.ind.rawwide

gamma13
0.0116
0.0062
1.8803
0.0601
ln.grp.phvi.pc

<---
port

gamma14
-0.0063
0.0058
-1.0996
0.2715
ln.grp.phvi.pc

<---
t12

rho
0.7542
0.0479
15.7449
0.0000
ln.grp.phvi.pc

<---
wd.ln.grp.phvi.pc

alpha1
0.1488
0.0667
2.2323
0.0256
ln.grp.phvi.pc

<---
(Intercept)

gamma25
0.0027
0.0013
2.1277
0.0334
ln.migr.new

<---
ln.income.pc.fcb97

gamma26
0.0038
0.0012
3.1503
0.0016
ln.migr.new

<---
ln.unempl.level97

gamma27
0.0003
0.0000
9.7510
0.0000
ln.migr.new

<---
avertemp.jan.cs

gamma28
-0.0028
0.0003
-9.6679
0.0000
ln.migr.new

<---
ln.pop.gr.89.26mbe

gamma29
0.0020
0.0009
2.2828
0.0224
ln.migr.new

<---
ln.pass.railway.percap

gamma210
0.0030
0.0004
8.2058
0.0000
ln.migr.new

<---
ln.city

alpha2
-0.0244
0.0041
-5.9237
0.0000
ln.migr.new

<---
(Intercept)

gamma31
0.7727
0.1070
7.2237
0.0000
ln.inv.p0.pc

<---
ln.grp.corr.pc97

gamma37
0.0123
0.0063
1.9596
0.0500
ln.inv.p0.pc

<---
avertemp.jan.cs

gamma311
0.1208
0.1118
1.0799
0.2802
ln.inv.p0.pc

<---
permafrost

gamma312
0.1805
0.0246
7.3497
0.0000
ln.inv.p0.pc

<---
ln.indoutput.fuel.p0.pc

gamma313
0.3888
0.1360
2.8591
0.0042
ln.inv.p0.pc

<---
ln.phone1995

alpha3
-1.1867
0.8601
-1.3798
0.1677
ln.inv.p0.pc

<---
(Intercept)

sigma11
0.0025
0.0001
17.3196
0.0000
ln.grp.phvi.pc
<-->
ln.grp.phvi.pc

sigma12
0.0000
0.0000
-1.3206
0.1867
ln.migr.new
<-->
ln.grp.phvi.pc

sigma13
0.0001
NaN
NaN
NaN
ln.inv.p0.pc
<-->
ln.grp.phvi.pc

sigma22
0.0000
0.0000
17.5513
0.0000
ln.migr.new
<-->
ln.migr.new

sigma23
-0.0001
0.0001
-0.5696
0.5690
ln.inv.p0.pc
<-->
ln.migr.new

sigma33
0.5859
0.0334
17.5470
0.0000
ln.inv.p0.pc
<-->
ln.inv.p0.pc

```

Iterations = 205

Aliased parameters: sigma13

Warning message:

NaN in: sqrt(diag(object\$cov))

So R estimates the model with NaN for sigma13: I'm not sure what it means... As you can see, maximum is occurred at a negative value for sigma13. At the same time one can check for rank condition - if I'm not mistaken, all three equations are exact.

After that, I estimate another specification, for example, including 'ln.postgrad.students.pc.be' (post-graduate students) instead of 't12' (dummy for poor and depressed regions), so that these two specifications differ only in one exogenous variable. I proceed in the same manner as before (including 'ln.postgrad.students.pc.be' instead of 't12' in specifying the model, computing raw moments and declaring new var to be fixed exogenous). So, I do not provide sem's code for the sake of space saving.

Now R estimates the system "correctly", without any warning messages:

Model fit to raw moment matrix.

Model Chisquare = 405.73 Df = 24 Pr(>Chisq) = 0

Goodness-of-fit index = 0.94308

RMSEA index = 0.16069 90 % CI: (0.14715, 0.17462)

BIC = 182.21 Normalized Residuals

Min. 1st Qu. Median Mean 3rd Qu. Max.

-0.9210 0.0000 0.0000 0.0286 0.0000 1.6000

```      Parameter
Estimate
Std Error
z value
Pr(>|z|)

beta13
0.0157
0.0036
4.3162
0.0000
ln.grp.phvi.pc

<---
ln.inv.p0.pc

beta12
0.9143
0.5337
1.7132
0.0867
ln.grp.phvi.pc

<---
ln.migr.new

gamma11
-0.0268
0.0068
-3.9642
0.0001
ln.grp.phvi.pc

<---
ln.grp.corr.pc97

gamma12
0.0221
0.0171
1.2919
0.1964
ln.grp.phvi.pc

<---
ln.sh.ind.rawwide

gamma13
0.0141
0.0069
2.0499
0.0404
ln.grp.phvi.pc

<---
port

gamma14
0.0061
0.0032
1.8769
0.0605
ln.grp.phvi.pc

<---

rho
0.7483
0.0480
15.5823
0.0000
ln.grp.phvi.pc

<---
wd.ln.grp.phvi.pc

alpha1
0.1433
0.0591
2.4256
0.0153
ln.grp.phvi.pc

<---
(Intercept)

gamma25
0.0026
0.0014
1.8682
0.0617
ln.migr.new

<---
ln.income.pc.fcb97

gamma26
0.0038
0.0013
2.8600
0.0042
ln.migr.new

<---
ln.unempl.level97

gamma27
0.0003
0.0000
9.3817
0.0000
ln.migr.new

<---
avertemp.jan.cs

gamma28
-0.0028
0.0003
-9.5709
0.0000
ln.migr.new

<---
ln.pop.gr.89.26mbe

gamma29
0.0021
0.0010
2.1621
0.0306
ln.migr.new

<---
ln.pass.railway.percap

gamma210
0.0030
0.0004
7.9666
0.0000
ln.migr.new

<---
ln.city

alpha2
-0.0242
0.0042
-5.7363
0.0000
ln.migr.new

<---
(Intercept)

gamma31
0.7733
0.1096
7.0574
0.0000
ln.inv.p0.pc

<---
ln.grp.corr.pc97

gamma37
0.0123
0.0067
1.8285
0.0675
ln.inv.p0.pc

<---
avertemp.jan.cs

gamma311
0.1209
0.1201
1.0065
0.3142
ln.inv.p0.pc

<---
permafrost

gamma312
0.1806
0.0250
7.2323
0.0000
ln.inv.p0.pc

<---
ln.indoutput.fuel.p0.pc

gamma313
0.3879
0.1379
2.8129
0.0049
ln.inv.p0.pc

<---
ln.phone1995

alpha3
-1.1881
0.8754
-1.3572
0.1747
ln.inv.p0.pc

<---
(Intercept)

sigma11
0.0025
0.0001
17.2039
0.0000
ln.grp.phvi.pc
<-->
ln.grp.phvi.pc

sigma12
0.0000
0.0000
-1.1470
0.2514
ln.migr.new
<-->
ln.grp.phvi.pc

sigma13
-0.0001
0.0010
-0.0820
0.9346
ln.inv.p0.pc
<-->
ln.grp.phvi.pc

sigma22
0.0000
0.0000
17.5416
0.0000
ln.migr.new
<-->
ln.migr.new

sigma23
-0.0001
0.0002
-0.4366
0.6624
ln.inv.p0.pc
<-->
ln.migr.new

sigma33
0.5859
0.0334
17.5370
0.0000
ln.inv.p0.pc
<-->
ln.inv.p0.pc

```

Iterations = 137

In fact, I've estimated about 30 specifications trying to experiment with different additional explanatory variables in the first equation, and about two thirds of 30 are turned out to be estimated with that warning message (Negative parameter variances with NaN for sigma's variances). I've not been clarified yet with the final specification... though I am not sure that's the case. Perhaps, I do something wrong concerning specification of error covariance structure.

Do you have any idea? I'll be pleased by any suggestion.

Sorry to trouble you.

Denis Fomchenko

Research fellow

Department for Economic Development Problems

Institute for the Economy in Transition

5, Gazetny lane, Moscow 125993, Russia

e-mail: fomchenko@iet.ru

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