U-shaped - Sasabuchi test?

[Petko Bachvarov]

Dear Statalisters,


I am trying to test for an inversed U-shaped between credit risk index and adjusted Lerner index(A measure of market power in the banking market).
To test for a U-shaped pattern between edit risk index and adjusted Lerner index I used a random-effects negative binomial model (supported by a Hausman test - xtnberg) including the direct (significant and positive) and the squared term of my Lerner Index. To corroborate this pattern of findings I would like to conduct a Sasabuchi (1980) test for an inverted U-shape, however I am not sure whether the right command was used to perform Sasabuchi test.

Fixed-effect (Hausman test - xtnberg)
xtnbreg llrgl car  adjlerner adjlerner2 insitution  ownership_concentration cir deposit_asset loan_asset otherearningassets incomediversity size  tier1 fundingragility luqidasset logz gdp_growth inflation crisis_d listed_d, fe

Followed by

Lind and Mehlum's procedure for testing U-shaped relationships or Sasabuchi test
utest  adjlerner adjlerner2, prefix(llrgl)

My results are the following:

Conditional FE negative binomial regression     Number of obs      =      3124
Group variable: y                               Number of groups   =        14

                                                Obs per group: min =       223
                                                               avg =     223.1
                                                               max =       225

                                                Wald chi2(19)      =     95.47
Log likelihood  = -481.71589                    Prob > chi2        =    0.0000

------------------------------------------------------------------------------
       llrgl |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         car |   .0480899   .3682911     0.13   0.896    -.6737474    .7699271
   adjlerner |  -5.870174   1.569279    -3.74   0.000    -8.945904   -2.794444
  adjlerner2 |   7.036934   1.943191     3.62   0.000      3.22835    10.84552
  insitution |  -.1541609   .1247337    -1.24   0.216    -.3986344    .0903125
ownership~on |   .0115567   .2493661     0.05   0.963    -.4771918    .5003052
         cir |   .0020299    .002605     0.78   0.436    -.0030758    .0071356
deposit_as~t |    .686342   .4698969     1.46   0.144    -.2346389    1.607323
  loan_asset |  -2.129252   .5312894    -4.01   0.000     -3.17056   -1.087944
otherearni~s |    .216634    .346283     0.63   0.532    -.4620682    .8953361
incomedive~y |    .142058   .1615877     0.88   0.379     -.174648     .458764
        size |   .0747654    .039163     1.91   0.056    -.0019926    .1515234
       tier1 |   .0816059   .2788698     0.29   0.770     -.464969    .6281807
fundingrag~y |   .0959053   .4705931     0.20   0.839    -.8264402    1.018251
  luqidasset |    .860141   .4811945     1.79   0.074     -.082983    1.803265
        logz |   .0032852   .0652955     0.05   0.960    -.1246917    .1312621
  gdp_growth |  -5.295671   1.911873    -2.77   0.006    -9.042874   -1.548468
   inflation |   -.130405   1.155299    -0.11   0.910     -2.39475     2.13394
    crisis_d |   .0278075   1.120135     0.02   0.980    -2.167616    2.223231
    listed_d |    .356337   .1745903     2.04   0.041     .0141463    .6985277
       _cons |   13.40707   203.6382     0.07   0.948    -385.7164    412.5306
------------------------------------------------------------------------------

. utest  adjlerner adjlerner2, prefix(llrgl)
(325 missing values generated)

Specification: f(x)=x^2
Extreme point:  .4170974

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

-------------------------------------------------
                 |   Lower bound      Upper bound
-----------------+-------------------------------
Interval         |   -.1606019         .9939588
Slope            |   -8.130464          8.11867
t-value          |   -3.729346          3.42437
P>|t|            |    .0000977         .0003121
-------------------------------------------------

Overall test of presence of a U shape:
     t-value =      3.42
     P>|t|   =   .000312

.

Any idea on how to interpret these results? Does this mean that since we reject the null hypothesis based on p-value, that we should accept the H1 (direct U-shape) which is confirmed by the regressions (positive coefficient of Lerner^2):
Can you please confirm that this is the way to perform Sasabuchi test or I have to run another/different test in order to get the required result?


Many thanks in advance for your always precious help,

Petko Bachvarov

[Carlo Lazzaro]

Petko:
as confirmed by the coefficient of your regression, there's evidence of a non-linear relationship between the regressand and -adjlerner-.
The usual formula to calculate the turning point comes from the first derivative:

. di -(-5.870174  )/(2*7.036934)
.41709742

Hence, you seem to have a minimum to deal with (U shape, or the smiling parabola).
As an aside, you could also have used the -fvvarlist- notation to create the squared term (and more comfortably so):

c.adjlerner##c.adjlerner

[Petko Bachvarov]

Dear Carlo Lazzaro,

Thank you for your reply.
If you don't mind I would like to ask you is this the way to run Sasabuchi test or there is a different command to be used.
I am trying to replicate a paper in which the results seems to be different to what I have. Therefore, I was wondering if there is any other way to test for u-shape.

Thank you in advance for the help.
Kind Regards,
Petko Bachvarov

[Carlo Lazzaro]

Petko:
unfortunately, -search Sasabuchi- does not give back any entry.
I'm also unfamiliar with both the tests you mentioned, probably because, being a bit of an old-fogey,usuallly I calculate the turning point by hand.
That said, you may want to take a look at the following paper: https://www.econstor.eu/bitstream/10.../547244843.pdf.
As an aside, please call me Carlo, like all on (and many more off) this list do. Thanks.

[Justine Borja]

Hello Stata listers,

I have executed the Utest in Stata to strengthen my claim in EKC modeling, but I don't know how to interpret it. Please someone provide insights. Thank you, any help is deeply appreciated.

  utest lnto lnto2

 The following are the obtained results:
Specification: f(x)=x^2
Extreme point: -.1606954

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

-------------------------------------------------
                 |   Lower bound      Upper bound
-----------------+-------------------------------
Interval         |   -1.695541         .0979904
Slope            |   -1.369733         .2308575
t-value          |   -3.682233         1.902704
P>|t|            |    .0009239         .0370783
-------------------------------------------------

Overall test of presence of a U shape:
     t-value =      1.90
     P>|t|   =     .0371

[Carlo Lazzaro]

Justine:
the test outcome of the community-contributed module -utest- (as you're kindly asked to mention it for good reasons reported in the FAQ. Thanks) supports the evidence of a U-shaped relationship between regressand and predictor -x-.

[Justine Borja]

Thanks sir Carlo! However, I am curious where do we based the decision in rejecting the null hypothesis? What does the extreme point mean?

Regards,
Justine

[Carlo Lazzaro]

Justine:
1) for details on the thory underpinning the community-contributed module -utest-, please see https://onlinelibrary.wiley.com/doi/....2009.00569.x;
2) if the -H0: Monotone or Inverse U shape- and it is rejected, it supports the evidence of an U-shaped relationship between regressand and predictor -x-.
As an aside, thanks, but sir is really redundant. Carlo is enough.

[Justine Borja]

why it is rejected? because the p-value is less than 0.05? thank you again, carlo.

[Carlo Lazzaro]

Justine:
correct, as 0.0371<0.05 (setting aside any usual consideration about the arbitrariness of 0.05 value).

[Justine Borja]

Carlo, a question again. The extreme point on my utest is a negative. So does that mean that it is an inverted U-shaped? I am confused. Thank you.

[Carlo Lazzaro]

Justine:
could you please share your regression code along with the outcome table that Stata gave back? Thanks.

[Justine Borja]

These are the results:

  ardl lnco2cb_pc lnto lnto2 lnrgdp_pc lnfec, lags(1 2 2 1 0) ec

ARDL(1,2,2,1,0) regression

Sample:     1992 -     2019                     Number of obs     =         28
                                                R-squared         =     0.8648
                                                Adj R-squared     =     0.7853
Log likelihood =  63.988203                     Root MSE          =     0.0316

------------------------------------------------------------------------------
D.lnco2cb_pc |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
ADJ          |
  lnco2cb_pc |
         L1. |  -1.032176   .1714712    -6.02   0.000    -1.393949   -.6704037
-------------+----------------------------------------------------------------
LR           |
        lnto |   -.125041   .0682137    -1.83   0.084    -.2689594    .0188773
       lnto2 |   .0725624   .0374376     1.94   0.069     -.006424    .1515488
   lnrgdp_pc |   .3498542   .0482717     7.25   0.000     .2480097    .4516987
       lnfec |   .5828316    .144235     4.04   0.001     .2785224    .8871409
-------------+----------------------------------------------------------------
SR           |
        lnto |
         D1. |   .2718069   .1180811     2.30   0.034     .0226775    .5209362
         LD. |   .2591184   .1310488     1.98   0.064    -.0173703    .5356072
             |
       lnto2 |
         D1. |   .3607252   .1147127     3.14   0.006     .1187026    .6027478
         LD. |   .3251187   .1246687     2.61   0.018     .0620907    .5881467
             |
   lnrgdp_pc |
         D1. |   .5827885   .3643544     1.60   0.128    -.1859321    1.351509
             |
       _cons |  -3.082382   .9177255    -3.36   0.004    -5.018614    -1.14615
------------------------------------------------------------------------------

  ardl lnco2cb_pc lnto lnto2 lnrgdp_pc lnfec, lags(1 2 2 1 0) regstore(ecreg)

ARDL(1,2,2,1,0) regression

Sample:     1992 -     2019                     Number of obs     =         28
                                                F(  10,     17)   =      34.04
                                                Prob > F          =     0.0000
                                                R-squared         =     0.9524
                                                Adj R-squared     =     0.9245
Log likelihood =  63.988203                     Root MSE          =     0.0316

------------------------------------------------------------------------------
  lnco2cb_pc |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
  lnco2cb_pc |
         L1. |  -.0321764   .1714712    -0.19   0.853    -.3939491    .3295963
             |
        lnto |
         --. |   .1427425   .1111385     1.28   0.216    -.0917392    .3772242
         L1. |  -.0126885   .1459105    -0.09   0.932    -.3205327    .2951558
         L2. |  -.2591184   .1310488    -1.98   0.064    -.5356071    .0173703
             |
       lnto2 |
         --. |   .4356224   .1189462     3.66   0.002     .1846678     .686577
         L1. |  -.0356065   .1579928    -0.23   0.824    -.3689421    .2977291
         L2. |  -.3251187   .1246687    -2.61   0.018    -.5881467   -.0620907
             |
   lnrgdp_pc |
         --. |   .9438997   .3353069     2.82   0.012     .2364641    1.651335
         L1. |  -.5827884   .3643544    -1.60   0.128    -1.351509    .1859322
             |
       lnfec |   .6015851   .1689412     3.56   0.002     .2451502    .9580199
       _cons |  -3.082382   .9177256    -3.36   0.004    -5.018614    -1.14615
------------------------------------------------------------------------------

 utest lnto lnto2

Specification: f(x)=x^2
Extreme point: -.1638374

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

-------------------------------------------------
                 |   Lower bound      Upper bound
-----------------+-------------------------------
Interval         |   -1.695541         .0979904
Slope            |   -1.334488         .2281161
t-value          |    -3.57816         1.864236
P>|t|            |    .0011581         .0398256
-------------------------------------------------

Overall test of presence of a U shape:
     t-value =      1.86
     P>|t|   =     .0398

The linear and squared term in my long-run run coefficients are negative and positive respectively, thus suggesting a U-shaped relationship in the long-run. I converted the ARDL results into regression as given by -regstore ecreg-. However, doing that my coefficients have changed, affecting the results of my utest which shows an inverted U-shaped as denoted by the negative extreme point.
.

[Carlo Lazzaro]

Justine:
the community-contributed module -utest- applies the well-known formula:

turning point=-b/2a

that, populated with your 2nd model coefficients, becomes:

. di -.1427425/(2*.4356224)
-.16383742
.

confirming the -utest- outcome:

Extreme point: -.1638374

That said:
1) check whether this value falls within the range of -lnto- variable. If this is not the case, you do not have a turning point;
2) there something more concerning about your 2nd model coefficients: you've a sky-rocketing R-sq but many coefficients do not reach ststistical significance. This might be a sign of quasi-extreme multicollinearity. What does -estat vce, corr- tell you?

[Nicola Rossi]

Dear Carlo Lazzaro,

I read from the paper by Lind and Mehlum (2010) that to test for the presence of a U-shaped relationship:

1. the estimated extremum point should be within the data range;
2. In addition, we need to test whether the relationship decreases at low values within this interval and increases at high values within the interval.

Regarding point 1, I noted that you used a similar phrase, "value falls within the range of -lnto- variable." I was wondering what exactly it means that the extremum point falls within the range of the variable. How can I check this from the Stata command utest?

I have similar doubts regarding point 2. How can I understand that "the relationship is decreasing at low values within this interval and increasing at high values within the interval"?

Any suggestions would be immensely appreciated.

Best Regards,
NR