How to interpret drastic differences in significance? (panel data)

[Guest]

I am working on my thesis about the Environmental Kuznets Curve for Latin America and Southeast Asia, with a pannel data with fixed and random effects. I am using logarithmics and robust standard errors. I would to assess the difference in the effects between the two regions.

I obtain the results attached in the pictures. I do not understand why the cubic terms lose all their significance when adding the Latin America regional dummy and interacting it with the GDPpc variables. In the first regression (the one without regional variables), the introduction of the cubic term makes all the variables significant, and in the second regression is the opposite. How does this make sense?

Depedendent variable: ln of CO2 per capita

PS: I know I should include controls, but in this first analysis I just want to analyse the effect of GDPpc and the regional different effect

These are the codes I used for each table:

est clear
eststo: xtreg ln_co2pc ln_gdppc, fe robust
est store fe_model
eststo: xtreg ln_co2pc ln_gdppc, re robust
est store re_model

eststo: xtreg ln_co2pc ln_gdppc ln_gdppc2, fe robust
eststo: xtreg ln_co2pc ln_gdppc ln_gdppc2, re robust
est store re_model1

eststo: xtreg ln_co2pc ln_gdppc ln_gdppc2 ln_gdppc3, fe robust
est store fe_model2
eststo: xtreg ln_co2pc ln_gdppc ln_gdppc2 ln_gdppc3, re robust
est store re_model2
 
esttab using "a", replace  ///
 b(3) se(3) nomtitle label star(* 0.10 ** 0.05 *** 0.01) ///
 booktabs  ///
 title("Basic regression table, 1960-2018 \label{reg1}")

est clear

eststo: xtreg ln_co2pc ln_gdppc LAgdppc, fe robust
est store fe_model
eststo: xtreg ln_co2pc ln_gdppc Latin_America LAgdppc, re robust
est store re_model

eststo: xtreg ln_co2pc ln_gdppc ln_gdppc2 LAgdppc LAgdppc2, fe robust
est store fe_model1
eststo: xtreg ln_co2pc ln_gdppc ln_gdppc2 Latin_America LAgdppc LAgdppc2, re robust
est store re_model1

eststo: xtreg ln_co2pc ln_gdppc ln_gdppc2 ln_gdppc3 LAgdppc LAgdppc2 LAgdppc3, fe robust
est store fe_model2
eststo: xtreg ln_co2pc ln_gdppc ln_gdppc2 ln_gdppc3 Latin_America LAgdppc LAgdppc2 LAgdppc3, re robust
est store re_model2

esttab using "prueba.tex", replace  ///
 b(3) se(3) nomtitle label star(* 0.10 ** 0.05 *** 0.01) ///
 booktabs  ///
 title("Basic regression table, regional effects for 1960-2018 \label{reg1}")   ///

[Carlo Lazzaro]

Guest:
the two regressions differ as far as their specification is concerned: thus, no wonder that they produce different coefficients.
It's up to you to choose which one is more consistent with the data genetating process.
In addition, how could you explain the contribution of a cubic term to variation in the dependent variable, when adjusted for the othe predictors?
Personally, I'd go with your second regression without the cubic term.
As an aside, please avoid posting screnshots to share what you typed and what Stata gave you back and use CODE delimiters instead. Thanks,

[Guest]

Thank you so much for your reply!

It's up to you to choose which one is more consistent with the data genetating process.

I am confused about which model to choose, because it is possible to find in the literature either no curve, an inverted U-shaped curve (quadratic) or a N-shaped curve (cubic).

In addition, how could you explain the contribution of a cubic term to variation in the dependent variable, when adjusted for the othe predictors?

For the first regression, I performed a Ramsey test, and I got that I should include quadratic and cubic terms of ln(GDPpc), hence why I decided to include a cubic term in the second regression as well.

I do not understand how to interpret this then: I have evidence of a N-shaped curve for both regions, but when analysing them at regional level I get evidence of an inverted U-shaped curve for Latin America and nothing for Southeast Asia?
This does not make much sense to me.....

As an aside, please avoid posting screnshots to share what you typed and what Stata gave you back and use CODE delimiters instead

Sure! I just thought that posting the tables from Latex would be more neat and clearer, hence why I included the screenshots.

[Carlo Lazzaro]

Guest:
as far as I know -estat ovtest- is not available after -xtreg-.
That said, you can check whether a quadratic term is enough with your data applying the -linktest- procedure by hand (as the Stata built-in command does not work after -xtreg-). If the squared fitted values are statistically significant, you can consider adding the cubic term and then re-run the procedure to check whether squared fitted values stop being significant.

[Carlo Lazzaro]

Guest:
to choose between -fe- and -re- specification with non-default standard errors you should consider the community-contributed module -xtoverid-.

[Guest]

Guest:
as far as I know -estat ovtest- is not available after -xtreg-.
That said, you can check whether a quadratic term is enough with your data applying the -linktest- procedure by hand (as the Stata built-in command does not work after -xtreg-). If the squared fitted values are statistically significant, you can consider adding the cubic term and then re-run the procedure to check whether squared fitted values stop being significant.

I do not really know how to apply the -linktest- procedure by hand.
However, I did the Ramsey test "by hand":

. est clear

.
. xtreg ln_co2pc ln_gdppc, fe robust

Fixed-effects (within) regression               Number of obs     =      1,714
Group variable: country                         Number of groups  =         31

R-sq:                                           Obs per group:
     within  = 0.0032                                         min =         26
     between = 0.4462                                         avg =       55.3
     overall = 0.3153                                         max =         59

                                                F(1,30)           =       0.24
corr(u_i, Xb)  = -0.5983                        Prob > F          =     0.6308

                               (Std. Err. adjusted for 31 clusters in country)
------------------------------------------------------------------------------
             |               Robust
    ln_co2pc |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
    ln_gdppc |  -.0734233   .1511927    -0.49   0.631       -.3822    .2353534
       _cons |   8.282819   1.295602     6.39   0.000     5.636848    10.92879
-------------+----------------------------------------------------------------
     sigma_u |  2.7353398
     sigma_e |   .5689028
         rho |  .95853684   (fraction of variance due to u_i)
------------------------------------------------------------------------------

.
. predict fitted, xb
(96 missing values generated)

. g sq_fitted=fitted^2
(96 missing values generated)

. g cu_fitted=fitted^3
(96 missing values generated)

.
. xtreg ln_co2pc ln_gdppc sq_fitted, fe robust

Fixed-effects (within) regression               Number of obs     =      1,714
Group variable: country                         Number of groups  =         31

R-sq:                                           Obs per group:
     within  = 0.0039                                         min =         26
     between = 0.1783                                         avg =       55.3
     overall = 0.0931                                         max =         59

                                                F(2,30)           =       0.12
corr(u_i, Xb)  = -0.3397                        Prob > F          =     0.8876

                               (Std. Err. adjusted for 31 clusters in country)
------------------------------------------------------------------------------
             |               Robust
    ln_co2pc |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
    ln_gdppc |  -3.311171   10.65314    -0.31   0.758    -25.06779    18.44545
   sq_fitted |  -2.878917   9.410758    -0.31   0.762    -22.09825    16.34042
       _cons |   204.6855   642.5999     0.32   0.752    -1107.679    1517.049
-------------+----------------------------------------------------------------
     sigma_u |  2.7128624
     sigma_e |  .56886901
         rho |  .95788077   (fraction of variance due to u_i)
------------------------------------------------------------------------------

. test sq_fitted

 ( 1)  sq_fitted = 0

       F(  1,    30) =    0.09
            Prob > F =    0.7618

.
end of do-file

I need to add a cubic term in order to get a significant quadratic term (i still do not understand the logic behind this), as table 1 from my regression displays:

.
. xtreg ln_co2pc ln_gdppc ln_gdppc2, fe robust

Fixed-effects (within) regression               Number of obs     =      1,714
Group variable: country                         Number of groups  =         31

R-sq:                                           Obs per group:
     within  = 0.0039                                         min =         26
     between = 0.1785                                         avg =       55.3
     overall = 0.0932                                         max =         59

                                                F(2,30)           =       0.12
corr(u_i, Xb)  = -0.3399                        Prob > F          =     0.8877

                               (Std. Err. adjusted for 31 clusters in country)
------------------------------------------------------------------------------
             |               Robust
    ln_co2pc |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
    ln_gdppc |   .1903111   .8059023     0.24   0.815    -1.455561    1.836183
   ln_gdppc2 |   -.015511   .0507332    -0.31   0.762    -.1191221    .0881001
       _cons |    7.17775   3.248234     2.21   0.035       .54397    13.81153
-------------+----------------------------------------------------------------
     sigma_u |  2.7128766
     sigma_e |  .56886925
         rho |  .95788115   (fraction of variance due to u_i)
------------------------------------------------------------------------------

.
. predict fitted1, xb
(96 missing values generated)

.
.
. g sq_fitted1=fitted1^2
(96 missing values generated)


. g cu_fitted1=fitted1^3
(96 missing values generated)

.
. xtreg ln_co2pc ln_gdppc ln_gdppc2 cu_fitted1, fe robust

Fixed-effects (within) regression               Number of obs     =      1,714
Group variable: country                         Number of groups  =         31

R-sq:                                           Obs per group:
     within  = 0.0417                                         min =         26
     between = 0.3009                                         avg =       55.3
     overall = 0.0821                                         max =         59

                                                F(3,30)           =       3.23
corr(u_i, Xb)  = -0.3600                        Prob > F          =     0.0362

                               (Std. Err. adjusted for 31 clusters in country)
------------------------------------------------------------------------------
             |               Robust
    ln_co2pc |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
    ln_gdppc |  -38.76917   16.09545    -2.41   0.022    -71.64047   -5.897868
   ln_gdppc2 |   3.182528   1.315425     2.42   0.022     .4960707    5.868984
  cu_fitted1 |   1.205776   .4977642     2.42   0.022     .1892055    2.222346
       _cons |  -437.8118     183.31    -2.39   0.023    -812.1807   -63.44279
-------------+----------------------------------------------------------------
     sigma_u |  2.7667764
     sigma_e |  .55812847
         rho |  .96089813   (fraction of variance due to u_i)
------------------------------------------------------------------------------

. test ln_gdppc2

 ( 1)  ln_gdppc2 = 0

       F(  1,    30) =    5.85
            Prob > F =    0.0218

.
end of do-file

PD: I am aware of my missing values, but they belong to Southeast Asian countries which were colonies and went through civil wars and territorial changes, hence why data for this country is more limited in the earlier years. (I have already explained all this in my thesis)

[Guest]

Guest:
to choose between -fe- and -re- specification with non-default standard errors you should consider the community-contributed module -xtoverid-.

Sorry, I had not seen this message.
I performed the xtoverid test, and if I understood it correct, I have evidence that I should use Fixed Effects (p-value<5%)

. est clear

.
. xtreg ln_co2pc ln_gdppc, fe robust

Fixed-effects (within) regression               Number of obs     =      1,714
Group variable: country                         Number of groups  =         31

R-sq:                                           Obs per group:
     within  = 0.0032                                         min =         26
     between = 0.4462                                         avg =       55.3
     overall = 0.3153                                         max =         59

                                                F(1,30)           =       0.24
corr(u_i, Xb)  = -0.5983                        Prob > F          =     0.6308

                               (Std. Err. adjusted for 31 clusters in country)
------------------------------------------------------------------------------
             |               Robust
    ln_co2pc |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
    ln_gdppc |  -.0734233   .1511927    -0.49   0.631       -.3822    .2353534
       _cons |   8.282819   1.295602     6.39   0.000     5.636848    10.92879
-------------+----------------------------------------------------------------
     sigma_u |  2.7353398
     sigma_e |   .5689028
         rho |  .95853684   (fraction of variance due to u_i)
------------------------------------------------------------------------------

. xtreg ln_co2pc ln_gdppc, re robust

Random-effects GLS regression                   Number of obs     =      1,714
Group variable: country                         Number of groups  =         31

R-sq:                                           Obs per group:
     within  = 0.0032                                         min =         26
     between = 0.4462                                         avg =       55.3
     overall = 0.3153                                         max =         59

                                                Wald chi2(1)      =       0.16
corr(u_i, X)   = 0 (assumed)                    Prob > chi2       =     0.6904

                               (Std. Err. adjusted for 31 clusters in country)
------------------------------------------------------------------------------
             |               Robust
    ln_co2pc |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
    ln_gdppc |  -.0600058   .1506409    -0.40   0.690    -.3552566     .235245
       _cons |   7.841673   1.489644     5.26   0.000     4.922024    10.76132
-------------+----------------------------------------------------------------
     sigma_u |  2.0314109
     sigma_e |   .5689028
         rho |  .92727416   (fraction of variance due to u_i)
------------------------------------------------------------------------------

. xtoverid

Test of overidentifying restrictions: fixed vs random effects
Cross-section time-series model: xtreg re  robust cluster(country)
Sargan-Hansen statistic  10.016  Chi-sq(1)    P-value = 0.0016

[Carlo Lazzaro]

Guest:
1) your Rsq within is low;
2) you're dealing with a T>N panel datasets. Therefore, you should switch from -xtreg- to -xtregar- or -xtgls-;
3) provided 2) your -xtoverid- results point you out to -fe- specification (thah might be -xtregar,fe-).
-

[Guest]

1) your Rsq within is low;

1. How can I fix this then? The aim of this model was to check for the existance of the Environmental Kuznets Curve, so maybe when I add controls in my other model I will get better Rsq?

2) you're dealing with a T>N panel datasets. Therefore, you should switch from -xtreg- to -xtregar- or -xtgls-;
3) provided 2) your -xtoverid- results point you out to -fe- specification (thah might be -xtregar,fe-).

2. I do not understand "xtregar fits cross-sectional time-series regression models when the disturbance term is first-order autoregressive"
-xtset- told me that my panel data is strongly balanced, so why I should I use something with an AR(1)?

3. Plus, I tried to run xtregar but I cannot use robust standard errors (and I should use them)

. est clear

xtregar ln_co2pc ln_gdppc ln_gdppc2 ln_gdppc3, fe robust
option robust not allowed

Nevertheless, ln_gdppc is significant if I remove "robust" (but neither the cuadratic nor the cubic terms are)

. est clear

.
. xtregar ln_co2pc ln_gdppc ln_gdppc2 ln_gdppc3, fe

FE (within) regression with AR(1) disturbances  Number of obs     =      1,683
Group variable: country                         Number of groups  =         31

R-sq:                                           Obs per group:
     within  = 0.0132                                         min =         25
     between = 0.5020                                         avg =       54.3
     overall = 0.3946                                         max =         58

                                                F(3,1649)         =       7.37
corr(u_i, Xb)  = 0.6191                         Prob > F          =     0.0001

------------------------------------------------------------------------------
    ln_co2pc |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
    ln_gdppc |   1.011816   .5973827     1.69   0.091    -.1598928    2.183524
   ln_gdppc2 |  -.0952462   .1151743    -0.83   0.408    -.3211496    .1306571
   ln_gdppc3 |   .0033707   .0061209     0.55   0.582    -.0086348    .0153762
       _cons |   3.733334   .0904777    41.26   0.000      3.55587    3.910797
-------------+----------------------------------------------------------------
      rho_ar |  .89587178
     sigma_u |  2.5780939
     sigma_e |  .24936573
     rho_fov |  .99073102   (fraction of variance because of u_i)
------------------------------------------------------------------------------
F test that all u_i=0: F(30,1649) = 28.02                    Prob > F = 0.0000

And if I use random effects, all of them are significant:

. xtregar ln_co2pc ln_gdppc ln_gdppc2 ln_gdppc3, re

RE GLS regression with AR(1) disturbances       Number of obs     =      1,714
Group variable: country                         Number of groups  =         31

R-sq:                                           Obs per group:
     within  = 0.0028                                         min =         26
     between = 0.5168                                         avg =       55.3
     overall = 0.4060                                         max =         59

                                                Wald chi2(4)      =      13.29
corr(u_i, Xb)      = 0 (assumed)                Prob > chi2       =     0.0099

------------------- theta --------------------
  min      5%       median        95%      max
0.7136   0.7375     0.7804     0.7804   0.7804

------------------------------------------------------------------------------
    ln_co2pc |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
    ln_gdppc |   6.219129    3.06162     2.03   0.042     .2184634    12.21979
   ln_gdppc2 |  -.7004757   .3868706    -1.81   0.070    -1.458728    .0577767
   ln_gdppc3 |   .0266936   .0160624     1.66   0.097    -.0047881    .0581753
       _cons |  -11.24441   7.984561    -1.41   0.159    -26.89386    4.405039
-------------+----------------------------------------------------------------
      rho_ar |  .89587178   (estimated autocorrelation coefficient)
     sigma_u |  1.3766474
     sigma_e |  .28170866
     rho_fov |  .95980803   (fraction of variance due to u_i)
------------------------------------------------------------------------------

4. However, in this ocasion, I cannot use xtoverid to choose between FE and RE.... how can I perform the Hausman test then?

5. Moreover, I cannot use time-fixed effects (i.year) with xtregar, and I really need to use them for my analysis.

Sorry if these are dumb questions, but my knowledge about panel data is very limited, and I do not really understand the -xtregar- manual.

[Carlo Lazzaro]

Guest:
1) let's hope so;
2) and 3) you can use -xtgls- with -i.panelid- in the right-hand side of your regression equation. -xtgls- allows cluster-robust standard errors;
4) you may want to consided a Mundlak approach, that you have to replicate by hand, though.

[Guest]

Guest:
1) let's hope so;
2) and 3) you can use -xtgls- with -i.panelid- in the right-hand side of your regression equation. -xtgls- allows cluster-robust standard errors;
4) you may want to consided a Mundlak approach, that you have to replicate by hand, though.

First of all, thanks again for all your help, I really appreciate it.

So, just to organize my thoughts:

- I cannot use xtreg because I have a long (T>N) panel dataset => I should use xtrgar or xtgls
- I have heteroskedasticity => I should use robust standard errors => however, i cannot include neither "robust" nor "vce()" with xtgls
- I have entity-fixed effects and time-fixed effects => I should include i.country and i.year if I am using xtgls
- If I want a random-effect model, I do not include neither i.country nor i.year

Is this correct?

I have more questions though:

1. As I said above, how can I control heteroskedasticity in xtgls?
2. Do I perform the Mundlak approach for xtreg, xtregar or xtgls?
3. Do I perform the Ramsey test for xtreg, xtregar or xtgls?
4. Does the interpretation of these results vary with respect to regular panel data? (for instance, in the case of regional dummy variables and their interactions, etc.)

[Carlo Lazzaro]

Guest:
1) T>N rules out -xtreg- and calls for -xtregar- or -xtgls- instead,
2) -xtgls- has its own options for cluster-robust standa4d errors. Take a look at -xtgls- entry, Stata .pdf manual;
3) yes. You should include both -i.panelid- and -i.timevar-;
4) I meant Mundlak for -xtgls-: if -test- shows the panel-specific means of time-invariant variables to reach statistical significance, -fe- is the way to go (see Mundlak worked out example in Stata blog);
5) you should apply the -linktest- procedure by hand, as -xt- commands do not support -estat ovtest-;
6) the interpretation of the coefficients does not vary.

[Guest]

Guest:
1) T>N rules out -xtreg- and calls for -xtregar- or -xtgls- instead,
2) -xtgls- has its own options for cluster-robust standa4d errors. Take a look at -xtgls- entry, Stata .pdf manual;
3) yes. You should include both -i.panelid- and -i.timevar-;
4) I meant Mundlak for -xtgls-: if -test- shows the panel-specific means of time-invariant variables to reach statistical significance, -fe- is the way to go (see Mundlak worked out example in Stata blog);
5) you should apply the -linktest- procedure by hand, as -xt- commands do not support -estat ovtest-;
6) the interpretation of the coefficients does not vary.

Thanks so much for your time and dedication.

2) I have checked out the PDF. Will "panels(heteroskedastic)" be enough? Because I cannot apply "panels(correlated)" due to missing values probably.
5) I will! But I have to perform this test after xtgls, not xtreg, right? Just making sure

[Carlo Lazzaro]

Guest:
2) -panels(correlated)- is the way to go; -panels(heteroskedastic)- accounts for heterosckedasticity only;
5) yes, after -xtgls-; with a T>N panel dataset -xtreg- is not a viable option.

[Guest]

Guest:
2) -panels(correlated)- is the way to go; -panels(heteroskedastic)- accounts for heterosckedasticity only;
5) yes, after -xtgls-; with a T>N panel dataset -xtreg- is not a viable option.

Thanks again for your help.

However, after trying to use "panels(correlated)", I get the following error message: panels must be balanced
I have missing values, but when I apply xtset, I obtain a "strongly balanced" message.

If I cannot use "panels(correlated)", are there other options?