Re: st: Restrictions in SVAR

["David M. Drukker" <ddrukker@stata.com>]

Alvaro <aa314@cam.ac.uk> asked how

> to impose that two elements of matrix A in the SVAR estimation 
> are equal.
>
> as [a_3_1]_cons = [a_3_2]_cons
The short answer is to use the -acns()- on -svar-.

This option is complicated, so let's repeat what the help file has to say.


   -acns(matrix_acns)- specifies a matrix that defines a set of exclusion or
   cross-parameter equality constraints on A.  This matrix must be square
   with dimension equal to the number of equations in the underlying VAR.
   Each element of this matrix must be -missing-, 0, or a positive integer.
   A missing value in the (i,j) element of this matrix specifies that no
   constraint be placed on this element of A.  A zero in the (i,j) element
   of this matrix constrains the (i,j) element of A to be zero.  Any
   strictly positive integers must be in two or more elements of this
   matrix.  A strictly positive integer in the (i,j) element of this matrix
   constrains the (i,j) element of A to be equal to all the other elements
   of A that correspond to elements in this matrix that contain the same
   integer.  For example, consider the matrix

                        -          -
                    A = | .     1  |
                        | 1     0  |
                        -          -

   Specifying, -acns(A)- in a two-equation SVAR constrains
   A[2,1]=A[1,2], A[2,2]=0 while leaving A[1,1] free.

The following example uses the above A in an SVAR.


. webuse lutkepohl
(Quarterly SA West German macro data, Bil DM, from Lutkepohl 1993 Table E.1)

. matrix A = ( ., 1\ 1, 0)

. 
. svar dlincome dlconsumption, acns(A)
Estimating short-run parameters

Iteration 0:   log likelihood = -159.19754 
Iteration 1:   log likelihood =  491.87116 
Iteration 2:   log likelihood =  542.40175 
Iteration 3:   log likelihood =  572.88512 
Iteration 4:   log likelihood =  573.16306 
Iteration 5:   log likelihood =  573.17262 
Iteration 6:   log likelihood =  573.17354 
Iteration 7:   log likelihood =  573.17364 
Iteration 8:   log likelihood =  573.17365

Structural vector autoregression

Constraints:
 ( 1)  [a_2_2]_cons = 0
 ( 2) - [a_2_1]_cons + [a_1_2]_cons = 0
 ( 3)  [b_1_1]_cons = 1
 ( 4)  [b_1_2]_cons = 0
 ( 5)  [b_2_1]_cons = 0
 ( 6)  [b_2_2]_cons = 1

Sample:  1960q4   1982q4                           No. of obs      =        89
Overidentified model                               Log likelihood  =  573.1737

------------------------------------------------------------------------------
             |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      /a_1_1 |  -53.00645   11.31905    -4.68   0.000    -75.19138   -30.82152
      /a_2_1 |   103.4424   5.482438    18.87   0.000     92.69705    114.1878
      /a_1_2 |   103.4424   5.482438    18.87   0.000     92.69705    114.1878
      /a_2_2 |          0          .        .       .            .           .
-------------+----------------------------------------------------------------
      /b_1_1 |          1          .        .       .            .           .
      /b_2_1 |          0          .        .       .            .           .
      /b_1_2 |          0          .        .       .            .           .
      /b_2_2 |          1          .        .       .            .           .
------------------------------------------------------------------------------
LR test of identifying restrictions:  chi2(  1)=    10.21  Prob > chi2 = 0.001

     --David
     ddrukker@stata.com
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