Non-linear constraint in a SURE estimation

[Iris Laugier]

Dear Statalist contributors,

I am trying to estimate a system of seemingly unrelated labor supplies for couples (called eq1 for women and eq2 for men) with the sureg command in Stata 13 for Windows.

When I add a linear constraint, the estimation by sureg works. However, I need to impose a non-linear constraint on four coefficients but as soon as I add this, I get the following error message:

"note: constraint number 1 caused error r(131)".

I am able to test the constraint expost with the command testnl, but I need to impose it in my estimation.

Hence, I would really appreciate if anyone could share how to introduce non-linear constraint in a sureg estimation.

Below, you can find a reproducible example:

sysuse auto.dta

constraint 1 [Eq1]mpg*[Eq2]headroom = [Eq1]headroom*[Eq2]length

sureg (Eq1: price = mpg trunk weight rep78 headroom)///
(Eq2 : price = headroom length), ///
corr constraint(1)

(note: constraint number 1 caused error r(131))
(note: constraint number 1 caused error r(131))

[Nick Cox]

Your example constraint would be linear on logarithmic scale. I don't know how much that helps.

In general, either there is a trick to make a constraint linear or you need something quite different. The help for constraint makes it quite clear: it's about linear constraints. Other kinds of constraint require quite different machinery

[Iris Laugier]

Hello,

Thanks for your reply.
I have tried to use the log to linearize my constraint but it still doesn't work.

In case it can help, to be sure that's what you were suggesting, here is what I have done:

sysuse auto.dta
constraint 2 ln([Eq1]mpg) + ln([Eq2]headroom) = ln([Eq1]headroom) + ln([Eq2]length)
sureg (Eq1: price = mpg trunk weight rep78 headroom) (Eq2 : price = headroom length), corr constraint(2)

I still get the error message 131.

Otherwise, how can I add a nonlinear constraint using another command than constraint, which is clearly not the relevant command as you highlighted?

[Andrew Musau]

constraint 1 [Eq1]mpg*[Eq2]headroom = [Eq1]headroom*[Eq2]length

How many ways can you satisfy this constraint? I can think of at least three.

[Nick Cox]

That's still a nonlinear constraint! .

What I meant -- sorry that this did not seem clear as an implication -- was that you should consider logging your variables first.

In turn, I don't know more than you do about these kinds of models, so if that isn't an answer, you will have to hope for a much better one from someone else.

[Iris Laugier]

Hello again,

Andrew Musau, what do you mean exactly ? Are you talking in terms of different ways to write the constraint ? I am not sure to understand sorry.

Thanks for your help Nick Cox, but the issue is that some of the constrained variable are already in log. Yes I really hope someone could help me finding a command that allows to add these kinds of cross-equation constraints in my estimation...

[Andrew Musau]

Here are the three ways to satisfy your constraint in your example:

sysuse auto.dta, clear
constraint 1 [Eq1]mpg = [Eq1]headroom
constraint 2 [Eq2]headroom = [Eq2]length
constraint 3 [Eq1]mpg = [Eq2]length
constraint 4 [Eq2]headroom =  [Eq1]headroom
constraint 5 [Eq1]mpg =0
constraint 6 [Eq2]headroom =0
constraint 7 [Eq1]headroom =0
constraint 8 [Eq2]length= 0



*ONE
sureg (Eq1: price = mpg trunk weight rep78 headroom) ///
(Eq2 : price = headroom length), ///
corr constraint(1/2)

di _b[Eq1:mpg]* _b[Eq2:headroom]
di _b[Eq1:headroom]*_b[Eq2:length]

*TWO
sureg (Eq1: price = mpg trunk weight rep78 headroom) ///
(Eq2 : price = headroom length), ///
corr constraint(3/4)

di _b[Eq1:mpg]* _b[Eq2:headroom]
di _b[Eq1:headroom]*_b[Eq2:length]

*THREE
sureg (Eq1: price = mpg trunk weight rep78 headroom) ///
(Eq2 : price = headroom length), ///
corr constraint(5/8)

di _b[Eq1:mpg]* _b[Eq2:headroom]
di _b[Eq1:headroom]*_b[Eq2:length]

Results:

.
. *ONE
. sureg (Eq1: price = mpg trunk weight rep78 headroom) ///
> (Eq2 : price = headroom length), ///
> corr constraint(1/2)

Seemingly unrelated regression
--------------------------------------------------------------------------
Equation             Obs   Parms        RMSE    "R-sq"       chi2        P
--------------------------------------------------------------------------
Eq1                   69       4    2420.434    0.2992      27.49   0.0000
Eq2                   69       1    2606.379    0.1874      17.21   0.0000
--------------------------------------------------------------------------

 ( 1)  [Eq1]mpg - [Eq1]headroom = 0
 ( 2)  [Eq2]headroom - [Eq2]length = 0
------------------------------------------------------------------------------
             |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
Eq1          |
         mpg |  -27.77603   30.62623    -0.91   0.364    -87.80233    32.25027
       trunk |   22.01142   34.80338     0.63   0.527    -46.20194    90.22479
      weight |   1.289644   .3228062     4.00   0.000     .6569555    1.922333
       rep78 |   178.0181   120.5124     1.48   0.140    -58.18178     414.218
    headroom |  -27.77603   30.62623    -0.91   0.364    -87.80233    32.25027
       _cons |   1997.623   1539.899     1.30   0.195    -1020.524    5015.769
-------------+----------------------------------------------------------------
Eq2          |
    headroom |   46.69754   11.25515     4.15   0.000     24.63786    68.75723
      length |   46.69754   11.25515     4.15   0.000     24.63786    68.75723
       _cons |  -2786.723   2175.619    -1.28   0.200    -7050.858    1477.412
------------------------------------------------------------------------------

Correlation matrix of residuals:

        Eq1     Eq2
Eq1  1.0000
Eq2  0.9224  1.0000

Breusch-Pagan test of independence: chi2(1) =    58.703, Pr = 0.0000

.
. di _b[Eq1:mpg]* _b[Eq2:headroom]
-1297.0725

. di _b[Eq1:headroom]*_b[Eq2:length]
-1297.0725

.
. *TWO
. sureg (Eq1: price = mpg trunk weight rep78 headroom) ///
> (Eq2 : price = headroom length), ///
> corr constraint(3/4)

Seemingly unrelated regression
--------------------------------------------------------------------------
Equation             Obs   Parms        RMSE    "R-sq"       chi2        P
--------------------------------------------------------------------------
Eq1                   69       5    2465.924    0.2726      22.20   0.0005
Eq2                   69       2    2617.456    0.1804      13.40   0.0012
--------------------------------------------------------------------------

 ( 1)  [Eq1]mpg - [Eq2]length = 0
 ( 2)  - [Eq1]headroom + [Eq2]headroom = 0
------------------------------------------------------------------------------
             |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
Eq1          |
         mpg |   39.67708   11.03947     3.59   0.000     18.04011    61.31405
       trunk |   24.26749   37.09673     0.65   0.513    -48.44077    96.97575
      weight |   1.480866   .3390225     4.37   0.000     .8163939    2.145338
       rep78 |   125.6837   121.2402     1.04   0.300    -111.9427    363.3101
    headroom |  -240.9094   364.2952    -0.66   0.508    -954.9148     473.096
       _cons |    767.985   1443.824     0.53   0.595    -2061.857    3597.827
-------------+----------------------------------------------------------------
Eq2          |
    headroom |  -240.9094   364.2952    -0.66   0.508    -954.9148     473.096
      length |   39.67708   11.03947     3.59   0.000     18.04011    61.31405
       _cons |  -602.0196   1989.094    -0.30   0.762    -4500.571    3296.532
------------------------------------------------------------------------------

Correlation matrix of residuals:

        Eq1     Eq2
Eq1  1.0000
Eq2  0.9201  1.0000

Breusch-Pagan test of independence: chi2(1) =    58.414, Pr = 0.0000

.
. di _b[Eq1:mpg]* _b[Eq2:headroom]
-9558.5818

. di _b[Eq1:headroom]*_b[Eq2:length]
-9558.5818

.
. *THREE
. sureg (Eq1: price = mpg trunk weight rep78 headroom) ///
> (Eq2 : price = headroom length), ///
> corr constraint(5/8)

Seemingly unrelated regression
--------------------------------------------------------------------------
Equation             Obs   Parms        RMSE    "R-sq"       chi2        P
--------------------------------------------------------------------------
Eq1                   69       3    2546.868    0.2240      15.08   0.0017
Eq2                   69       0    2891.259    0.0000          .        .
--------------------------------------------------------------------------

 ( 1)  [Eq1]o.mpg = 0
 ( 2)  [Eq2]o.headroom = 0
 ( 3)  [Eq1]o.headroom = 0
 ( 4)  [Eq2]o.length = 0
------------------------------------------------------------------------------
             |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
Eq1          |
         mpg |          0  (omitted)
       trunk |  -33.82572   53.17986    -0.64   0.525    -138.0563    70.40489
      weight |    1.02673   .3139409     3.27   0.001     .4114168    1.642042
       rep78 |   313.0268   189.2084     1.65   0.098    -57.81487    683.8684
    headroom |          0  (omitted)
       _cons |   2437.973   1157.528     2.11   0.035     169.2588    4706.687
-------------+----------------------------------------------------------------
Eq2          |
    headroom |          0  (omitted)
      length |          0  (omitted)
       _cons |   6146.043   348.0666    17.66   0.000     5463.845    6828.242
------------------------------------------------------------------------------

Correlation matrix of residuals:

        Eq1     Eq2
Eq1  1.0000
Eq2  0.7932  1.0000

Breusch-Pagan test of independence: chi2(1) =    43.412, Pr = 0.0000

.
. di _b[Eq1:mpg]* _b[Eq2:headroom]
0

. di _b[Eq1:headroom]*_b[Eq2:length]
0

[Iris Laugier]

Hello, thanks a lot for your precise answer! I am gonna implement it in my actual program with my data.

Best Regards,