Is a spatial errorlag co-efficient greater than 1 problematic?

[Harry O'Rahilly]

Hello all,

I am using Stata 15.1

I have created a Spectrally Normalized Spatially Weighted Inverse-distance Cross-Sectional Matrix - using proprietary distance data - by means of spmatrix fromdata, i.e. not using methods employing a shapefile or co-ordinate variables.

The Matrix is 66x66, non-symmetric, and hollow (diagonal elements = 0).

spregress run on this Matrix with a gs2sls (generalized spatial two-stage least-squares) estimate produces an errorlag co-efficient greater than 1. [1.586804]

As discussed on page 147 of the Stata Spatial Autoregressive Models Ref Manual the errorlag co-efficient (rho [hat] )

should be between −1 and 1 unless the solution is explosive.

To note, the Stata example for Spatial autoregressive models provided also appears to have an errorlag co-efficient greater than 1 [3.247298]

I have two questions please.

1. Given the steps taken below, is an errorlag co-efficient [rho hat] greater than 1 problematic?*
2. If so, is there a remedy for this?

*By which I mean the results cannot be used to reject H₀

Happy to PM matrix data/provide clarity. Regards,

Harry

PS Following spregress I have run estat impact for completeness.
PPS dataex linesize limit exceeded by matrix

 
. use "C:\Users\Atlan\OneDrive\PC\UCD\Matrix\STATA\2020 08 19 distance matrix.dta"

. spmatrix fromdata WmeM = NP_22050-NP_18454, normalize(spectral) idistance replace

. spmatrix export WmeM using WmeM.txt
  (matrix WmeM saved in file WmeM.txt)

. save    "C:\Users\Atlan\OneDrive\PC\UCD\Matrix\STATA\2020 08 19 distance matrix.dta", replace
file C:\Users\Atlan\OneDrive\PC\UCD\Matrix\STATA\2020 08 19 distance matrix.dta saved

. clear

. use     "C:\Users\Atlan\OneDrive\PC\UCD\Matrix\STATA\2019 12 27 Cluster Analysis_5.dta"

. regress LE_PET_DIE_1 SUMBC660sSQ

      Source |       SS           df       MS      Number of obs   =        66
-------------+----------------------------------   F(1, 64)        =     28.13
       Model |  .005390905         1  .005390905   Prob > F        =    0.0000
    Residual |  .012265551        64  .000191649   R-squared       =    0.3053
-------------+----------------------------------   Adj R-squared   =    0.2945
       Total |  .017656456        65  .000271638   Root MSE        =    .01384

------------------------------------------------------------------------------
LE_PET_DIE_1 |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
 SUMBC660sSQ |  -.0001403   .0000265    -5.30   0.000    -.0001932   -.0000875
       _cons |   .0746009   .0035903    20.78   0.000     .0674285    .0817732
------------------------------------------------------------------------------

. spset catnumber
  Sp dataset 2019 12 27 Cluster Analysis_5.dta
                data:  cross sectional
     spatial-unit id:  _ID (equal to catnumber)
         coordinates:  none
    linked shapefile:  none

. estat moran, errorlag(WmeM)

Moran test for spatial dependence
         Ho: error is i.i.d. 
         Errorlags:  WmeM

         chi2(1)      =    10.22
         Prob > chi2  =   0.0014

. spregress LE_PET_DIE_1 SUMBC660sSQ, gs2sls errorlag(WmeM) 
  (66 observations)
  (66 observations (places) used)
  (weighting matrix defines 66 places)

Estimating rho using 2SLS residuals: 

initial:       GMM criterion =  6.965e-10
alternative:   GMM criterion =  1.179e-10
rescale:       GMM criterion =  3.067e-12
Iteration 0:   GMM criterion =  3.067e-12  
Iteration 1:   GMM criterion =  2.534e-13  

Estimating rho using GS2SLS residuals: 

Iteration 0:   GMM criterion =  .01728762  
Iteration 1:   GMM criterion =  .01186374  
Iteration 2:   GMM criterion =  .01175648  
Iteration 3:   GMM criterion =  .01175648  

Spatial autoregressive model                    Number of obs     =         66
GS2SLS estimates                                Wald chi2(1)      =      10.00
                                                Prob > chi2       =     0.0016
                                                Pseudo R2         =     0.3053

--------------------------------------------------------------------------------
  LE_PET_DIE_1 |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
---------------+----------------------------------------------------------------
LE_PET_DIE_1   |
   SUMBC660sSQ |   -.000081   .0000256    -3.16   0.002    -.0001312   -.0000308
         _cons |   .0679234   .0042821    15.86   0.000     .0595307    .0763162
---------------+----------------------------------------------------------------
WmeM           |
e.LE_PET_DIE_1 |   1.586804    .554908     2.86   0.004      .499204    2.674403
--------------------------------------------------------------------------------
Wald test of spatial terms:          chi2(1) = 8.18       Prob > chi2 = 0.0042

. estat impact 

progress   :100% 

Average impacts                                 Number of obs     =         66

------------------------------------------------------------------------------
             |            Delta-Method
             |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
direct       |
 SUMBC660sSQ |   -.000081   .0000256    -3.16   0.002    -.0001312   -.0000308
-------------+----------------------------------------------------------------
indirect     |
 SUMBC660sSQ |          0  (omitted)
-------------+----------------------------------------------------------------
total        |
 SUMBC660sSQ |   -.000081   .0000256    -3.16   0.002    -.0001312   -.0000308
------------------------------------------------------------------------------

. 
end of do-file