Inverse of a matrix

Dario Maimone Ansaldo Patti

Join Date: Aug 2014
Posts: 526

Inverse of a matrix

06 Aug 2025, 04:45

Dear All,

I have the following symmetric matrix:

Code:

symmetric V_diff[11,11]
                pc_culture  urb_rate1850        school  countryname1  countryname2  countryname3  countryname4  countryname5
  pc_culture     .06447299
urb_rate1850    -.05278748     .01628029
      school    -.01874854     .01102653     .00363522
countryname1     4.5903346    -3.0965529    -1.1290617     301.73363
countryname2     2.1064027     -.7624796    -.36318082      117.9168     26.429972
countryname3    -.85005066     1.2145158     .40800901    -80.140974    -52.851368    -4.1350514
countryname4     2.3099423    -1.3619878    -.51357007     144.96865     50.376108    -45.701961     67.589369
countryname5    -.07604234     .67402019      .2123714     -28.60747    -32.117323    -17.134995    -20.746305    -21.352163
countryname6     3.1871472     -1.809098    -.69833247     198.38592     67.085994    -64.338675     91.943415    -30.135964
countryname7     1.8894841    -1.0291854    -.39089213     115.11489     36.858158    -40.093335     52.603569     -20.17471
countryname8     4.1191875    -2.9437497    -1.0622745     276.82325     113.61946    -67.176435     134.80566    -20.069622

              countryname6  countryname7  countryname8
countryname6     124.86398
countryname7     71.298498     40.350407
countryname8     184.94091     107.99118     252.38147

I want its inverse. So I use:

Code:

mat B = syminv(V_diff)

THe following is the result:

Code:

symmetric B[11,11]
                pc_culture  urb_rate1850        school  countryname1  countryname2  countryname3  countryname4  countryname5
  pc_culture             0
urb_rate1850             0             0
      school             0             0             0
countryname1             0             0             0     .00331418
countryname2             0             0             0             0             0
countryname3             0             0             0             0             0             0
countryname4             0             0             0             0             0             0             0
countryname5             0             0             0             0             0             0             0             0
countryname6             0             0             0             0             0             0             0             0
countryname7             0             0             0             0             0             0             0             0
countryname8             0             0             0             0             0             0             0             0

              countryname6  countryname7  countryname8
countryname6             0
countryname7             0             0
countryname8             0             0             0

It looks to me that something is strange. Nonetheless, if my matrix algebra is correct, if I multiply the original matrix by its inverse, I should get the identity matrix. This is what I got:

Code:

. mat C = V_diff*B

. mat list C

C[11,11]
                pc_culture  urb_rate1850        school  countryname1  countryname2  countryname3  countryname4  countryname5
  pc_culture             0             0             0      .0152132             0             0             0             0
urb_rate1850             0             0             0    -.01026254             0             0             0             0
      school             0             0             0    -.00374192             0             0             0             0
countryname1             0             0             0             1             0             0             0             0
countryname2             0             0             0     .39079768             0             0             0             0
countryname3             0             0             0    -.26560173             0             0             0             0
countryname4             0             0             0     .48045242             0             0             0             0
countryname5             0             0             0    -.09481035             0             0             0             0
countryname6             0             0             0     .65748695             0             0             0             0
countryname7             0             0             0     .38151162             0             0             0             0
countryname8             0             0             0     .91744247             0             0             0             0

              countryname6  countryname7  countryname8
  pc_culture             0             0             0
urb_rate1850             0             0             0
      school             0             0             0
countryname1             0             0             0
countryname2             0             0             0
countryname3             0             0             0
countryname4             0             0             0
countryname5             0             0             0
countryname6             0             0             0
countryname7             0             0             0
countryname8             0             0             0

I cannot understand what is going on. I assume that there is something wrong with my commands. I would be grateful for any help you may provide.

Best,

Dario

Tags: matrix, matrix operations

Joseph Coveney

Join Date: Apr 2014

Posts: 4543
#2

06 Aug 2025, 06:06

Is it positive definite?

For example, what does

Code:

mata: cholesky(V_diff) // or mata: symeigenvalues(V_diff)

give?
Comment
Joseph Coveney

Join Date: Apr 2014

Posts: 4543
#3

06 Aug 2025, 06:38

Sorry, that should be

Code:

mata: cholesky(st_matrix("V_diff")) // or mata: symeigenvalues(st_matrix("V_diff"))
Comment

Dario Maimone Ansaldo Patti

Join Date: Aug 2014
Posts: 526

06 Aug 2025, 07:04

Joseph Coveney Thanks for your reply. This is what I got:

Code:

. mata: cholesky(st_matrix("V_diff"))
         1    2    3    4    5    6    7    8    9   10   11
     +--------------------------------------------------------+
   1 |   .    0    0    0    0    0    0    0    0    0    0  |
   2 |   .    .    0    0    0    0    0    0    0    0    0  |
   3 |   .    .    .    0    0    0    0    0    0    0    0  |
   4 |   .    .    .    .    0    0    0    0    0    0    0  |
   5 |   .    .    .    .    .    0    0    0    0    0    0  |
   6 |   .    .    .    .    .    .    0    0    0    0    0  |
   7 |   .    .    .    .    .    .    .    0    0    0    0  |
   8 |   .    .    .    .    .    .    .    .    0    0    0  |
   9 |   .    .    .    .    .    .    .    .    .    0    0  |
  10 |   .    .    .    .    .    .    .    .    .    .    0  |
  11 |   .    .    .    .    .    .    .    .    .    .    .  |
     +--------------------------------------------------------+

. mata: symeigenvalues(st_matrix("V_diff"))
                  1              2              3              4              5
    +----------------------------------------------------------------------------
  1 |   868.8093179    1.08236e-13    8.19047e-14    5.15725e-14    6.39471e-15
    +----------------------------------------------------------------------------
                  6              7              8              9             10
     ----------------------------------------------------------------------------
  1    -4.96898e-17   -1.45734e-14   -6.98127e-14   -7.60789e-14   -.9766598434
     ----------------------------------------------------------------------------
                 11
     ----------------+
  1    -79.88665422  |
     ----------------+

.

So I guess that the result in the first matrix suggests that it is not positive definite. So basically, what I am doing is to replicate the Hausman test manually (I need to do it in that way in my research). Therefore, I have tried to compare the results that I get from the usage of the hausman command and the one I try to do manually.

What I did was:

Code:

ivreg Y (X1=Z1 Z2) X2 X3 countryname*,  nocon

matrix b_iv = e(b)
matrix V_iv = e(V)

reg Y X1 X2 X3 countryname*, nocon

matrix b_ols = e(b)
matrix V_ols = e(V)

matrix b_diff = b_iv - b_ols
matrix V_diff = V_iv - V_ols
matrix H = b_diff' * syminv(V_diff) * b_diff
scalar hausman_stat = H[1,1]
scalar df = colsof(b_diff)
scalar p_value = chi2tail(df, hausman_stat)
display "Hausman chi2(" df ") = " hausman_stat ", p = " p_value

If the matrix V_diff is not positive definite, the test done with hausman should ndisplay a warning, am I correct? Differently, this is what I get:

Code:

   chi2(10) = (b-B)'[(V_b-V_B)^(-1)](b-B)
            =   4.17
Prob > chi2 = 0.9392

When I try to do do it manually using the Hausman test procedure "H=(β^IV−β^OLS)′[Var(β^IV)−Var(β^OLS)]−1(β^IV−β^OLS)", I get

Code:

. matrix H = b_diff' * syminv(V_diff) * b_diff
conformability error

Hence, I was trying to understand why this happens.

Dario

Last edited by Dario Maimone Ansaldo Patti; 06 Aug 2025, 07:48.

Comment

Dario Maimone Ansaldo Patti

Join Date: Aug 2014

Posts: 526
#5

06 Aug 2025, 07:54

Please, ignore my previous post. I repeated the procedure and it works. I got exactly the same statistic and p-value
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