Dynamic panel data u-shaped correlation

[Badiah Eljahimi]

Hi all,

I am currently analysing the influence of the financial development on economic growth using system GMM. N =43, T=8 and set of control variables
I am trying to check weather the independent variable (fdxstwo ) might have a u-shaped / inverse u-shaped correlation to the dependent variable (rgdpg) with system GMM

Here are my results

 xtabond2 rgdpg rgdpg_lag1 ihs_inigdppc fdxstwo c.fdxsquartwo ihs_inf ihs_gfcf ihs_gov ihs_trd ihs_lbor y*, gmm(rgdpg ihs_inigdppc fdxstwo c.fdxsquartwo ihs_gfcf , lag(2 5) collapse eq(diff)) iv(ihs_inf ihs_gov ihs_lbor ihs_trd , eq(diff)) gmm(rgdpg fdxstwo c.fdxsquartwo ihs_gfcf, lag(1 1) collapse eq(level)) ivstyle(y*, equation(level)) twostep robust
Favoring space over speed. To switch, type or click on mata: mata set matafavor speed, perm.
year dropped due to collinearity
yr_1 dropped due to collinearity
yr_2 dropped due to collinearity
yr_8 dropped due to collinearity
Warning: Two-step estimated covariance matrix of moments is singular.
  Using a generalized inverse to calculate optimal weighting matrix for two-step estimation.
  Difference-in-Sargan/Hansen statistics may be negative.

Dynamic panel-data estimation, two-step system GMM
------------------------------------------------------------------------------
Group variable: id                              Number of obs      =       301
Time variable : year                            Number of groups   =        43
Number of instruments = 35                      Obs per group: min =         7
Wald chi2(15) =    304.78                                      avg =      7.00
Prob > chi2   =     0.000                                      max =         7
------------------------------------------------------------------------------
             |              Corrected
       rgdpg | Coefficient  std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
  rgdpg_lag1 |   -.012188   .1191046    -0.10   0.918    -.2456288    .2212528
ihs_inigdppc |   2.709269   4.068484     0.67   0.505    -5.264814    10.68335
     fdxstwo |  -.4580117   .1786356    -2.56   0.010    -.8081311   -.1078923
 fdxsquartwo |   .0033699   .0018742     1.80   0.072    -.0003035    .0070432
     ihs_inf |  -1.241957   .3137191    -3.96   0.000    -1.856835   -.6270785
    ihs_gfcf |   15.39889   4.351249     3.54   0.000     6.870596    23.92718
     ihs_gov |  -2.381506   3.479245    -0.68   0.494    -9.200702     4.43769
     ihs_trd |   6.915461   1.889015     3.66   0.000     3.213059    10.61786
    ihs_lbor |  -22.44103   8.464955    -2.65   0.008    -39.03204   -5.850023
       year3 |  -.0806661   .0669904    -1.20   0.229     -.211965    .0506327
        yr_3 |   .2723212   .3655125     0.75   0.456    -.4440702    .9887125
        yr_4 |   .2183402   .6200907     0.35   0.725    -.9970153    1.433696
        yr_5 |  -2.630467   .6420469    -4.10   0.000    -3.888855   -1.372078
        yr_6 |   -.954281   .5092936    -1.87   0.061    -1.952478    .0439161
        yr_7 |  -.9572508   .5601684    -1.71   0.087    -2.055161    .1406591
       _cons |   174.9178   170.7205     1.02   0.306    -159.6883    509.5238
------------------------------------------------------------------------------
Instruments for first differences equation
  Standard
    D.(ihs_inf ihs_gov ihs_lbor ihs_trd)
  GMM-type (missing=0, separate instruments for each period unless collapsed)
    L(2/5).(rgdpg ihs_inigdppc fdxstwo fdxsquartwo ihs_gfcf) collapsed
Instruments for levels equation
  Standard
    year3 year yr_1 yr_2 yr_3 yr_4 yr_5 yr_6 yr_7 yr_8
    _cons
  GMM-type (missing=0, separate instruments for each period unless collapsed)
    DL.(rgdpg fdxstwo fdxsquartwo ihs_gfcf) collapsed
------------------------------------------------------------------------------
Arellano-Bond test for AR(1) in first differences: z =  -2.08  Pr > z =  0.038
Arellano-Bond test for AR(2) in first differences: z =  -0.35  Pr > z =  0.725
------------------------------------------------------------------------------
Sargan test of overid. restrictions: chi2(19)   =  92.83  Prob > chi2 =  0.000
  (Not robust, but not weakened by many instruments.)
Hansen test of overid. restrictions: chi2(19)   =  22.55  Prob > chi2 =  0.258
  (Robust, but weakened by many instruments.)

Difference-in-Hansen tests of exogeneity of instrument subsets:
  iv(ihs_inf ihs_gov ihs_lbor ihs_trd, eq(diff))
    Hansen test excluding group:     chi2(15)   =  20.73  Prob > chi2 =  0.146
    Difference (null H = exogenous): chi2(4)    =   1.82  Prob > chi2 =  0.769
  iv(year3 year yr_1 yr_2 yr_3 yr_4 yr_5 yr_6 yr_7 yr_8, eq(level))
    Hansen test excluding group:     chi2(12)   =  18.83  Prob > chi2 =  0.093
    Difference (null H = exogenous): chi2(7)    =   3.72  Prob > chi2 =  0.812


. utest fdxstwo fdxsquartwo

Specification: f(x)=x^2
Extreme point:  67.95693

Test:
     H1: U shape
 vs. H0: Monotone or Inverse U shape 

-------------------------------------------------
                 |   Lower bound      Upper bound
-----------------+-------------------------------
Interval         |     12.9673           93.312
Slope            |   -.3706156         .1708864
t-value          |    -2.55363         .7023809
P>|t|            |    .0055767         .2414922
-------------------------------------------------

Overall test of presence of a U shape:
     t-value =      0.70
     P>|t|   =      .241

.

According to P-value = 0.241 this means there is an Inverse U shape relation between financial development (fdxstwo) and economic growth (rgdpg). However, when I back to the GMM results I find the coefficient of the level financial development (fdxstwo) = -.4580117 has a negative sign and the squared one is positive (fdxsquartwo) = 0.0033699 meaning that the relationship is u-shaped AND THIS IS OPPSITE TO UTEST decision!
Do I just focus on the utest decision? If yes, why the Gmm give the opposite interpretation?

Please guide me to the correct understanding


Thank you

Badiah

[Sebastian Kripfganz]

As far as I understand the output of the utest command, under the null hypothesis it assumes either a monotone or inverse u-shape relationship. Not rejecting the null hypothesis does not imply that there is an inverse u-shape relationship. It could just be monotone. In fact, the GMM coefficient on the squared term is statistically insignificant at the 5% level, also pointing towards a monotone relationship.

[Badiah Eljahimi]

Thank you very much for the clarification