Dealing with singular covariance matrix in Seemingly Unrelated Regression (sureg)

[Sangyeon Hwang]

Dear all :

Thank you for your help in advance.

I’m the user of Stata MP/16.0.

I have a panel data set consisting of 16 regions from year 2001 to 2017. There are four equations to estimate using Seemingly Unrelated Regression model.
In each equation, there is a single identical regressor which is GRP (Gross Regional Product).
In the system of four equations, there are four different dependent variables.
Y1, Y2, and Y3 are different measures of regional income, Y4 is regional consumption.

Therefore, the system of four equations are as follows.

1st equation: Y1=c1+b1*GRP+e1
2nd equation: Y2=c2+b2*GRP+e2
3rd equation: Y3=c3+b3*GRP+e3
4th equation: Y4=c4+b4*GRP+e4

Theoretically, b1+b2+b3+b4=1 but I do not impose the theoretical restriction in estimation.
Without any restriction on b’s, I estimated the above system using 'sureg' command and got an error message as follows.

. sureg (y1 grp) (y2 grp) (y3 grp) (y4 grp)
Covariance matrix of errors is singular

symmetric __00000C[4,4]
           __000006   __000007   __000008   __000009
__000006  1.1193292
__000007  .96680075  1.2204782
__000008   .9282527  .96859259  .99608893
__000009          0          0          0          0
r(506);

end of do-file

r(506);

However, when I dropped the 4th equation, I got the below result without any error.

. sureg (y1 grp) (y2 grp) (y3 grp) 

Seemingly unrelated regression
--------------------------------------------------------------------------
Equation             Obs   Parms        RMSE    "R-sq"       chi2        P
--------------------------------------------------------------------------
y1                   272       1    2.043316    0.0953      28.65   0.0000
y2                   272       1    1.952064    0.1004      30.36   0.0000
y3                   272       1    2.547562    0.0532      15.29   0.0001
--------------------------------------------------------------------------

------------------------------------------------------------------------------
             |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
y1           |
         grp |   .3481933   .0650525     5.35   0.000     .2206929    .4756938
       _cons |   .0815574   .1239898     0.66   0.511    -.1614581    .3245729
-------------+----------------------------------------------------------------
y2           |
         grp |   .3424472   .0621473     5.51   0.000     .2206407    .4642537
       _cons |  -.1918022   .1184526    -1.62   0.105     -.423965    .0403606
-------------+----------------------------------------------------------------
y3           |
         grp |   .3171816    .081106     3.91   0.000     .1582168    .4761464
       _cons |   .0436708   .1545878     0.28   0.778    -.2593157    .3466573
------------------------------------------------------------------------------

. 
end of do-file

When I dropped the 3rd equation, I got the below result without any error.

. sureg (y1 grp) (y2 grp) (y4 grp)

Seemingly unrelated regression
--------------------------------------------------------------------------
Equation             Obs   Parms        RMSE    "R-sq"       chi2        P
--------------------------------------------------------------------------
y1                   272       1    2.043316    0.0953      28.65   0.0000
y2                   272       1    1.952064    0.1004      30.36   0.0000
y4                   272       1    1.011226    0.0002       0.06   0.8080
--------------------------------------------------------------------------

------------------------------------------------------------------------------
             |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
y1           |
         grp |   .3481933   .0650525     5.35   0.000     .2206929    .4756938
       _cons |   .0815574   .1239898     0.66   0.511    -.1614581    .3245729
-------------+----------------------------------------------------------------
y2           |
         grp |   .3424472   .0621473     5.51   0.000     .2206407    .4642537
       _cons |  -.1918022   .1184526    -1.62   0.105     -.423965    .0403606
-------------+----------------------------------------------------------------
y4           |
         grp |  -.0078221   .0321941    -0.24   0.808    -.0709215    .0552772
       _cons |    .066574   .0613619     1.08   0.278    -.0536931    .1868412
------------------------------------------------------------------------------

. 
end of do-file

When I dropped the 1st equation, I got the below result without any error.

. sureg (y2 grp) (y3 grp) (y4 grp)

Seemingly unrelated regression
--------------------------------------------------------------------------
Equation             Obs   Parms        RMSE    "R-sq"       chi2        P
--------------------------------------------------------------------------
y2                   272       1    1.952064    0.1004      30.36   0.0000
y3                   272       1    2.547562    0.0532      15.29   0.0001
y4                   272       1    1.011226    0.0002       0.06   0.8080
--------------------------------------------------------------------------

------------------------------------------------------------------------------
             |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
y2           |
         grp |   .3424472   .0621473     5.51   0.000     .2206407    .4642537
       _cons |  -.1918022   .1184526    -1.62   0.105     -.423965    .0403606
-------------+----------------------------------------------------------------
y3           |
         grp |   .3171816    .081106     3.91   0.000     .1582168    .4761464
       _cons |   .0436708   .1545878     0.28   0.778    -.2593157    .3466573
-------------+----------------------------------------------------------------
y4           |
         grp |  -.0078221   .0321941    -0.24   0.808    -.0709215    .0552772
       _cons |    .066574   .0613619     1.08   0.278    -.0536931    .1868412
------------------------------------------------------------------------------

. 
end of do-file

My findings are as follows.
(1) It seems that the above system of four equations has the problem of singularity of the residual covariance.
(2) However, the coefficient estimates and their standard errors do not change across different systems of 3 equations.


Based on these findings, I'm going to proceed as follows.

(1) I need to report all four coefficient estimates. To do that, I get estimates of b1, b2, and b3 from the system of the 1st, 2nd, and 3rd equation. And I get estimates of b4 from the system of the 1st, 2nd, and 4th equation.
(2) Since I need the equality test among b1, b2, and b3 (but not b4), I just estimate the system of the 1st, 2nd, and 3rd equation and use the estimations result for this purpose.

However, I'm not confident in my approach. So, I'm looking for any comment that I have to consider.
Is it ok to use the above approach? If not, what kind of concerns do you have about this approach?


Thank you for your help.

Sangyeon

[Joro Kolev]

In principle, when the same set of regressors are used in each equation, -sureg- and -mvreg- give the same results. So you can just use mvreg.

When one of your equations is implied by the others, e.g., when you estimate shares which sum up to 1, it is a standard procedure to drop one of the equations, and to infer its coefficients from the other estimated equations.

[Sangyeon Hwang]

Btw, the estimate of b4 is invariant to different systems of equations, which is reported in the second and the third result in my original post.
Then, is there any reason why I cannot simply use the estimate of b4 in the second and third result instead of inferring it from the first result?
In addition, I'm wondering why the estimates of b's are invariant to different systems in Stata as shown in my origianl post.
Thank you for your help, Joro.