Multicollinearity and heteroskedasticity issue in small panel data

[Niara Romadha]

Hi. I am currently working on my dissertation project. I have data of 29 provinces and 11 years. I have 6 independent variables (no categorical variable at all). Initially, I found that my Fixed Effect model has heteroskedasticity and autocorrelation issues. I tried to use the robust command. The VIFs result after the xtreg robust are really high. However, after going through some discussion in the forum, I found out that with only 29 provinces, I should not use robust command and should not use Hausman test if I used robust command. I also read some advice to perform linktest by hand, but I still don't understand how it works. The reason why I used ln for several variables is because the histograms showed that the distribustions are not normal. I am also not sure if I should add i.year in my FE syntax. So I wonder if there is any advice for my problem.

. egen prov_id = group (province)

. xtset prov_id year

Panel variable: prov_id (strongly balanced)
 Time variable: year, 2010 to 2020
         Delta: 1 unit

. * data summary *
. xtsum pop65 lntfr lngdpcap lifeexp lnpopurban meanyearsedu fpaidemployed

Variable         |      Mean   Std. dev.       Min        Max |    Observations
-----------------+--------------------------------------------+----------------
pop65    overall |  5.286301   1.833573       1.54       10.5 |     N =     319
         between |              1.80598   1.931818   10.42727 |     n =      29
         within  |             .4505427   3.736301   7.326301 |     T =      11
                 |                                            |
lntfr    overall |  .9595275   .1336687   .6678294   1.302913 |     N =     319
         between |             .1239804   .7653753   1.233511 |     n =      29
         within  |             .0545853   .7779375   1.109783 |     T =      11
                 |                                            |
lngdpcap overall |  10.30609   .5308984   9.139573   12.07147 |     N =     319
         between |             .5233785   9.312996   11.86483 |     n =      29
         within  |             .1286158   9.908724   10.74314 |     T =      11
                 |                                            |
lifeexp  overall |  69.42608   2.436952      63.82      74.99 |     N =     319
         between |             2.425443   65.12727       74.6 |     n =      29
         within  |             .4908727   68.05699    70.8479 |     T =      11
                 |                                            |
lnpopu~n overall |  3.662959   .3840131   2.827314    4.60517 |     N =     319
         between |             .3872141   2.923856    4.60517 |     n =      29
         within  |              .047395   3.434371   3.832794 |     T =      11
                 |                                            |
meanye~u overall |  8.412665   .8615742       6.53       10.7 |     N =     319
         between |             .7618745   7.222727   10.56364 |     n =      29
         within  |             .4243944   7.262665   9.802665 |     T =      11
                 |                                            |
fpaide~d overall |  55.74514   6.781141       40.2       80.3 |     N =     319
         between |             6.316996   42.76364   76.41818 |     n =      29
         within  |             2.708189    47.3815    65.3815 |     T =      11

These are what I have been doing:
1. I used corr command to see if there's any correlation higher than 0.75. The result suggested no high correlation

corr pop65 lntfr lngdpcap lifeexp lnpopurban meanyearsedu fpaidemployed

. corr pop65 lntfr lngdpcap lifeexp lnpopurban meanyearsedu fpaidemployed
(obs=319)

             |    pop65    lntfr lngdpcap  lifeexp lnpopu~n meanye~u fpaide~d
-------------+---------------------------------------------------------------
       pop65 |   1.0000
       lntfr |  -0.5376   1.0000
    lngdpcap |  -0.3408  -0.2165   1.0000
     lifeexp |   0.4290  -0.6272   0.4329   1.0000
  lnpopurban |   0.2166  -0.5700   0.5091   0.6036   1.0000
meanyearsedu |   0.0143  -0.2484   0.5286   0.3607   0.4317   1.0000
fpaidemplo~d |   0.2707  -0.1676  -0.0185   0.1182   0.0852  -0.0450   1.0000

2. I used reg command then vif to check for multicollinearity, all vifs are below 10

reg pop65 lntfr lngdpcap lifeexp lnpopurban meanyearsedu fpaidemployed
vif

. reg pop65 lntfr lngdpcap lifeexp lnpopurban meanyearsedu fpaidemployed

      Source |       SS           df       MS      Number of obs   =       319
-------------+----------------------------------   F(6, 312)       =     96.22
       Model |  694.035273         6  115.672546   Prob > F        =    0.0000
    Residual |  375.077162       312  1.20217039   R-squared       =    0.6492
-------------+----------------------------------   Adj R-squared   =    0.6424
       Total |  1069.11244       318  3.36198879   Root MSE        =    1.0964

-------------------------------------------------------------------------------
        pop65 | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
--------------+----------------------------------------------------------------
        lntfr |  -4.596609   .6382869    -7.20   0.000      -5.8525   -3.340718
     lngdpcap |  -2.370571   .1511881   -15.68   0.000    -2.668048   -2.073094
      lifeexp |   .3157796   .0363818     8.68   0.000      .244195    .3873642
   lnpopurban |   .2589153   .2316014     1.12   0.264    -.1967828    .7146134
 meanyearsedu |    .268064   .0867419     3.09   0.002     .0973909    .4387371
fpaidemployed |   .0414489    .009249     4.48   0.000     .0232507    .0596471
        _cons |   6.690743   2.726136     2.45   0.015     1.326807    12.05468
-------------------------------------------------------------------------------

. vif

    Variable |       VIF       1/VIF  
-------------+----------------------
  lnpopurban |      2.09    0.477930
     lifeexp |      2.08    0.480925
       lntfr |      1.93    0.519335
    lngdpcap |      1.70    0.586788
meanyearsedu |      1.48    0.676855
fpaidemplo~d |      1.04    0.961055
-------------+----------------------
    Mean VIF |      1.72

3. I used fixed effect without robust command (Before I checked for heteroskedasticity, I am quite happy with the result since I have 2 significant variables)

xtreg pop65 lntfr lngdpcap lifeexp lnpopurban meanyearsedu fpaidemployed, fe

. xtreg pop65 lntfr lngdpcap lifeexp lnpopurban meanyearsedu fpaidemployed, fe

Fixed-effects (within) regression               Number of obs     =        319
Group variable: prov_id                         Number of groups  =         29

R-squared:                                      Obs per group:
     Within  = 0.3317                                         min =         11
     Between = 0.1525                                         avg =       11.0
     Overall = 0.1620                                         max =         11

                                                F(6, 284)         =      23.49
corr(u_i, Xb) = -0.1498                         Prob > F          =     0.0000

-------------------------------------------------------------------------------
        pop65 | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
--------------+----------------------------------------------------------------
        lntfr |  -2.311204   .6861332    -3.37   0.001    -3.661756    -.960652
     lngdpcap |   .3397021   .3815607     0.89   0.374    -.4113438    1.090748
      lifeexp |   .3442395   .1246646     2.76   0.006     .0988558    .5896233
   lnpopurban |  -.3996937   .5352642    -0.75   0.456    -1.453282    .6538948
 meanyearsedu |  -.1877593   .1341237    -1.40   0.163    -.4517621    .0762434
fpaidemployed |  -.0175182   .0089955    -1.95   0.052    -.0352244    .0001881
        _cons |  -15.87607   7.224003    -2.20   0.029    -30.09545   -1.656684
--------------+----------------------------------------------------------------
      sigma_u |  1.6829894
      sigma_e |  .38975102
          rho |  .94909934   (fraction of variance due to u_i)
-------------------------------------------------------------------------------
F test that all u_i=0: F(28, 284) = 78.04                    Prob > F = 0.0000

4. I used random effect without robust command

xtreg pop65 lntfr lngdpcap lifeexp lnpopurban meanyearsedu fpaidemployed, re

 xtreg pop65 lntfr lngdpcap lifeexp lnpopurban meanyearsedu fpaidemployed, re

Random-effects GLS regression                   Number of obs     =        319
Group variable: prov_id                         Number of groups  =         29

R-squared:                                      Obs per group:
     Within  = 0.3127                                         min =         11
     Between = 0.4259                                         avg =       11.0
     Overall = 0.4189                                         max =         11

                                                Wald chi2(6)      =     152.56
corr(u_i, X) = 0 (assumed)                      Prob > chi2       =     0.0000

-------------------------------------------------------------------------------
        pop65 | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
--------------+----------------------------------------------------------------
        lntfr |  -3.135798   .6510992    -4.82   0.000    -4.411929   -1.859667
     lngdpcap |   -.649408    .302663    -2.15   0.032    -1.242617   -.0561994
      lifeexp |   .3743336   .0849561     4.41   0.000     .2078228    .5408444
   lnpopurban |  -.0605336   .4189387    -0.14   0.885    -.8816384    .7605712
 meanyearsedu |  -.0495519   .1127146    -0.44   0.660    -.2704684    .1713646
fpaidemployed |  -.0099397   .0086585    -1.15   0.251    -.0269101    .0070307
        _cons |  -9.807787   5.293423    -1.85   0.064    -20.18271    .5671316
--------------+----------------------------------------------------------------
      sigma_u |  1.1319697
      sigma_e |  .38975102
          rho |  .89401384   (fraction of variance due to u_i)
-------------------------------------------------------------------------------

5. Hausman test (result indicated fixed effect is more appropriate)

quietly xtreg pop65 lntfr lngdpcap lifeexp lnpopurban meanyearsedu fpaidemployed, fe
estimates store fixed

quietly xtreg pop65 lntfr lngdpcap lifeexp lnpopurban meanyearsedu fpaidemployed, re
estimates store random

hausman fixed random, sigmamore

. hausman fixed random, sigmamore

                 ---- Coefficients ----
             |      (b)          (B)            (b-B)     sqrt(diag(V_b-V_B))
             |     fixed        random       Difference       Std. err.
-------------+----------------------------------------------------------------
       lntfr |   -2.311204    -3.135798        .8245942        .2795573
    lngdpcap |    .3397021     -.649408        .9891101         .252318
     lifeexp |    .3442395     .3743336        -.030094        .0967322
  lnpopurban |   -.3996937    -.0605336       -.3391601         .360623
meanyearsedu |   -.1877593    -.0495519       -.1382075        .0805032
fpaidemplo~d |   -.0175182    -.0099397       -.0075785        .0033659
------------------------------------------------------------------------------
                          b = Consistent under H0 and Ha; obtained from xtreg.
           B = Inconsistent under Ha, efficient under H0; obtained from xtreg.

Test of H0: Difference in coefficients not systematic

    chi2(6) = (b-B)'[(V_b-V_B)^(-1)](b-B)
            =  25.08
Prob > chi2 = 0.0003

6. Heteroskedasticity

xtreg pop65 lntfr lngdpcap lifeexp lnpopurban meanyearsedu fpaidemployed, fe
xttest3

. xttest3

Modified Wald test for groupwise heteroskedasticity
in fixed effect regression model

H0: sigma(i)^2 = sigma^2 for all i

chi2 (29)  =   73302.94
Prob>chi2 =      0.0000

7. Autocorrelation

xtserial pop65 lntfr lngdpcap lifeexp lnpopurban meanyearsedu fpaidemployed

. xtserial pop65 lntfr lngdpcap lifeexp lnpopurban meanyearsedu fpaidemployed

Wooldridge test for autocorrelation in panel data
H0: no first-order autocorrelation
    F(  1,      28) =    638.418
           Prob > F =      0.0000

8. I used vce corr to diagnose multicollinearity

xtreg pop65 lntfr lngdpcap lifeexp lnpopurban meanyearsedu fpaidemployed, fe 
estat vce, corr

. estat vce, corr

Correlation matrix of coefficients of xtreg model

        e(V) |    lntfr  lngdpcap   lifeexp  lnpopu~n  meanye~u  fpaide~d     _cons 
-------------+---------------------------------------------------------------------
       lntfr |   1.0000                                                             
    lngdpcap |   0.2507    1.0000                                                   
     lifeexp |   0.1683   -0.3903    1.0000                                         
  lnpopurban |  -0.2512   -0.3562   -0.0661    1.0000                               
meanyearsedu |   0.1599   -0.2257   -0.6059    0.2122    1.0000                     
fpaidemplo~d |  -0.0733   -0.2164    0.0498    0.3418    0.1803    1.0000           
       _cons |  -0.3810    0.0473   -0.8918   -0.0323    0.6079   -0.1255    1.0000

9. I also checked the VIF after xtreg

xtreg pop65 lntfr lngdpcap lifeexp lnpopurban meanyearsedu fpaidemployed, fe
vif, uncentered

. vif, uncentered

    Variable |       VIF       1/VIF  
-------------+----------------------
    lngdpcap |    633.03    0.001580
     lifeexp |    586.08    0.001706
  lnpopurban |    192.76    0.005188
meanyearsedu |    142.67    0.007009
fpaidemplo~d |     68.55    0.014588
       lntfr |     56.15    0.017811
-------------+----------------------
    Mean VIF |    279.87

10. I also used xtoverid command

xtreg pop65 lntfr lngdpcap lifeexp lnpopurban meanyearsedu fpaidemployed, fe
xtreg pop65 lntfr lngdpcap lifeexp lnpopurban meanyearsedu fpaidemployed, re
xtoverid

xtoverid

Test of overidentifying restrictions: fixed vs random effects
Cross-section time-series model: xtreg re   
Sargan-Hansen statistic  26.746  Chi-sq(6)    P-value = 0.0002

Additionally, I also performed Joint F-test and the result indicated that Fixed effect is indeed needed. LM test also indicated that Random effect is needed as well. So I figured I should go with Fixed effect.

I apologize for the long question. In summary, my problems are heteroskedasticity, autocorrelation, indication of multicollinearity (although I am not sure since the initial corr result doesn't show any high correlation between independent variables), and insignificant variables. I have tried many variables combinations and the results are the same: insignificant variables and heteroskedasticity etc.

Thank you so much for your time and assistance.

[Niara Romadha]

Another information

After I added robust command. No variable is significant in the FE robust model and only 1 variable is significant in the RE robust model. Both models have high VIFs.

[George Ford]

use reghdfe. absorb province and year. use clustered errors on province. With 29, you're probably safe. You can boottest the coefficients/stats of interest just to make sure. The typical rule of thumb is 30, at your right at it.

You don't need to worry about autocorr or hetero if you use clustered errors--it corrects for both.

I wouldn't worry about VIFS unless you're getting low t's and high R2. The reported ones above are not large, but you should include i.province i.year in that regression.

I'd include year fixed effects in all models, otherwise your estimates have time embedded in them.

[Niara Romadha]

Thank you George.

I tried this syntax. However, the F-test showed that the model is not okay.

reghdfe pop65 lngdpcap lntfr lifeexp meanyearsedu fpaidemployed lnpopurban, absorb(prov_id year) cluster(prov_id)

. reghdfe pop65 lngdpcap lntfr lifeexp meanyearsedu fpaidemployed lnpopurban, absorb(prov_id year) cluster(prov_id)
(MWFE estimator converged in 2 iterations)

HDFE Linear regression                            Number of obs   =        319
Absorbing 2 HDFE groups                           F(   6,     28) =       1.74
Statistics robust to heteroskedasticity           Prob > F        =     0.1475
                                                  R-squared       =     0.9693
                                                  Adj R-squared   =     0.9644
                                                  Within R-sq.    =     0.1925
Number of clusters (prov_id) =         29         Root MSE        =     0.3461

                                (Std. err. adjusted for 29 clusters in prov_id)
-------------------------------------------------------------------------------
              |               Robust
        pop65 | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
--------------+----------------------------------------------------------------
     lngdpcap |  -.6483645   .6460922    -1.00   0.324    -1.971824    .6750953
        lntfr |   -2.61743   1.995649    -1.31   0.200    -6.705331    1.470471
      lifeexp |  -.4342002   .3650981    -1.19   0.244     -1.18207    .3136694
 meanyearsedu |  -.8228862   .4402884    -1.87   0.072    -1.724776    .0790038
fpaidemployed |  -.0333591   .0188987    -1.77   0.088    -.0720714    .0053532
   lnpopurban |  -.1921305   1.341664    -0.14   0.887    -2.940405    2.556144
        _cons |   54.11075   26.29352     2.06   0.049     .2509161    107.9706
-------------------------------------------------------------------------------

Absorbed degrees of freedom:
-----------------------------------------------------+
 Absorbed FE | Categories  - Redundant  = Num. Coefs |
-------------+---------------------------------------|
     prov_id |        29          29           0    *|
        year |        11           1          10     |
-----------------------------------------------------+
* = FE nested within cluster; treated as redundant for DoF computation

[Niara Romadha]

I tried to absorb year only and cluster error on prov_id. The F test result is okay. However, the cons is not significant.

reghdfe pop65 lngdpcap lntfr lifeexp meanyearsedu fpaidemployed lnpopurban, absorb(year) cluster(prov_id)

. reghdfe pop65 lngdpcap lntfr lifeexp meanyearsedu fpaidemployed lnpopurban, absorb(year) cluster(prov_id)
(MWFE estimator converged in 1 iterations)

HDFE Linear regression                            Number of obs   =        319
Absorbing 1 HDFE group                            F(   6,     28) =      17.57
Statistics robust to heteroskedasticity           Prob > F        =     0.0000
                                                  R-squared       =     0.6568
                                                  Adj R-squared   =     0.6386
                                                  Within R-sq.    =     0.6490
Number of clusters (prov_id) =         29         Root MSE        =     1.1022

                                (Std. err. adjusted for 29 clusters in prov_id)
-------------------------------------------------------------------------------
              |               Robust
        pop65 | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
--------------+----------------------------------------------------------------
     lngdpcap |  -2.417531   .3793451    -6.37   0.000    -3.194585   -1.640478
        lntfr |  -3.860103    2.29263    -1.68   0.103    -8.556344    .8361372
      lifeexp |   .3183152     .11596     2.75   0.010     .0807819    .5558486
 meanyearsedu |   .1511429   .3255904     0.46   0.646    -.5157989    .8180847
fpaidemployed |   .0440367   .0288885     1.52   0.139    -.0151388    .1032122
   lnpopurban |   .5323898   .7956578     0.67   0.509    -1.097441    2.162221
        _cons |   6.129618   8.466332     0.72   0.475    -11.21288    23.47211
-------------------------------------------------------------------------------

Absorbed degrees of freedom:
-----------------------------------------------------+
 Absorbed FE | Categories  - Redundant  = Num. Coefs |
-------------+---------------------------------------|
        year |        11           0          11     |
-----------------------------------------------------+

.

[Jeff Wooldridge]

It’s not appropriate to use the method that gives you the results you want. Most of the tests you’ve done are unnecessary. This problem points to fixed effects estimation with clustered standard errors. But you should also include year fixed effects.

[Niara Romadha]

Thank you Jeff.
is this the right syntax?

xtreg pop65 lngdpcap lntfr lifeexp meanyearsedu fpaidemployed i.year, fe cluster(prov_id)

I discussed this matter with one of my lecturers. He thought I might have multicollinearity problem and suggested to do forward stepwise and transform variables.
I am a bit lost here. Does this mean I have to run the fixed effect model with year fixed effect and clustered standard errors with each variables individually? Because when I did, none of the variables is significant. So I'm not sure with which variable I have to start building the model.

[Carlo Lazzaro]

Niara:
what if you go:

 
 reghdfe pop65 lngdpcap lntfr lifeexp meanyearsedu fpaidemployed lnpopurban, absorb(prov_id year) cluster(prov_id)

As far as stepwise regression is concerned, please see Stata | FAQ: Problems with stepwise regression

[George Ford]

This is the proper form: reghdfe pop65 lngdpcap lntfr lifeexp meanyearsedu fpaidemployed lnpopurban, absorb(prov_id year) cluster(prov_id) If the F is insignificant, then the model doesn't explain pop65 very well. The constant term is irrelevant in FE regression.

[Carlo Lazzaro]

Niara:
shamelessly exploiting George's elegant assist, you may want to take a look at FAQ: Interpreting the intercept in the fixed-effects model | Stata

[Niara Romadha]

Thank you everyone for your input.

I have tried the syntax. But unfortunately, the F is insignificant. so I tried to transform everything to log. The F is significant. I suppose I don't have to run RE model and Hausman test then.
Do you think it's good if I go with this log log model instead?
There is still only 1 variable significant. However, I think I can still discuss the direction of the coefficients instead of focusing on p-value? Nonetheless, according to literatures, these variables should be significant.
And with that, is this model already immune from heteroskedasticity and autocorrelation? Should I check for multicollinearity?

reghdfe pop65 lntfr lifeexp lngdpcap meanyearsedu fpaidemployed, absorb(prov_id year) cluster(prov_id)

. reghdfe pop65 lntfr lifeexp lngdpcap meanyearsedu fpaidemployed, absorb(prov_id year) cluster(prov_id)
(MWFE estimator converged in 2 iterations)

HDFE Linear regression                            Number of obs   =        319
Absorbing 2 HDFE groups                           F(   5,     28) =       1.98
Statistics robust to heteroskedasticity           Prob > F        =     0.1120
                                                  R-squared       =     0.9693
                                                  Adj R-squared   =     0.9645
                                                  Within R-sq.    =     0.1921
Number of clusters (prov_id) =         29         Root MSE        =     0.3455

                                (Std. err. adjusted for 29 clusters in prov_id)
-------------------------------------------------------------------------------
              |               Robust
        pop65 | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
--------------+----------------------------------------------------------------
        lntfr |  -2.669191   2.027476    -1.32   0.199    -6.822288    1.483906
      lifeexp |  -.4349448   .3626482    -1.20   0.240    -1.177796    .3079063
     lngdpcap |  -.6968168   .6382243    -1.09   0.284     -2.00416    .6105264
 meanyearsedu |  -.8141222   .4619964    -1.76   0.089    -1.760479    .1322346
fpaidemployed |  -.0322921    .019881    -1.62   0.116    -.0730166    .0084323
        _cons |    53.8745   26.05559     2.07   0.048     .5020492    107.2469
-------------------------------------------------------------------------------

Absorbed degrees of freedom:
-----------------------------------------------------+
 Absorbed FE | Categories  - Redundant  = Num. Coefs |
-------------+---------------------------------------|
     prov_id |        29          29           0    *|
        year |        11           1          10     |
-----------------------------------------------------+
* = FE nested within cluster; treated as redundant for DoF computation

. 
end of do-file

.

reghdfe ln_pop65 lngdpcap ln_tfr ln_lifeexp ln_meanyearsedu ln_fpaidemployed, absorb(prov_id year) cluster(prov_id)

. reghdfe ln_pop65 lngdpcap ln_tfr ln_lifeexp ln_meanyearsedu ln_fpaidemployed, absorb(prov_id year) cluster(prov_id)
(MWFE estimator converged in 2 iterations)

HDFE Linear regression                            Number of obs   =        319
Absorbing 2 HDFE groups                           F(   5,     28) =       3.20
Statistics robust to heteroskedasticity           Prob > F        =     0.0208
                                                  R-squared       =     0.9710
                                                  Adj R-squared   =     0.9665
                                                  Within R-sq.    =     0.1587
Number of clusters (prov_id) =         29         Root MSE        =     0.0632

                                   (Std. err. adjusted for 29 clusters in prov_id)
----------------------------------------------------------------------------------
                 |               Robust
        ln_pop65 | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
-----------------+----------------------------------------------------------------
        lngdpcap |  -.3108015   .1231624    -2.52   0.018    -.5630883   -.0585147
          ln_tfr |  -.3775037   .3353599    -1.13   0.270    -1.064457    .3094499
      ln_lifeexp |  -5.121433   4.587409    -1.12   0.274    -14.51832    4.275449
 ln_meanyearsedu |  -.4852401   .8291874    -0.59   0.563    -2.183754    1.213273
ln_fpaidemployed |  -.2065409   .1978102    -1.04   0.305    -.6117367     .198655
           _cons |   28.74542    18.7995     1.53   0.137    -9.763608    67.25445
----------------------------------------------------------------------------------

Absorbed degrees of freedom:
-----------------------------------------------------+
 Absorbed FE | Categories  - Redundant  = Num. Coefs |
-------------+---------------------------------------|
     prov_id |        29          29           0    *|
        year |        11           1          10     |
-----------------------------------------------------+
* = FE nested within cluster; treated as redundant for DoF computation

[George Ford]

define these variables for us

[Carlo Lazzaro]

Niara:
you're striving to reach statistical significance, but this is not scientifis: results are what they are.
I suspect that you've an overfitting problem that you should deal with (see https://stats.stackexchange.com/ques...t-coefficients).

[Niara Romadha]

Thank you for your responses.

The variables I use:
1. Pop65: percentage of population aged 65 and over
2. lntfr: ln total fertility rate
3. lngdpcap: ln GDP per capita
4. meanyearsedu: mean of years education of adult age 20 and over
5. fpaidemployment: percentage of women with paid employment

I tried to run regdhfe for each variable.

. reghdfe pop65 lntfr, absorb(prov_id year) cluster(prov_id)
(MWFE estimator converged in 2 iterations)

HDFE Linear regression                            Number of obs   =        319
Absorbing 2 HDFE groups                           F(   1,     28) =       0.29
Statistics robust to heteroskedasticity           Prob > F        =     0.5914
                                                  R-squared       =     0.9625
                                                  Adj R-squared   =     0.9573
                                                  Within R-sq.    =     0.0140
Number of clusters (prov_id) =         29         Root MSE        =     0.3790

                               (Std. err. adjusted for 29 clusters in prov_id)
------------------------------------------------------------------------------
             |               Robust
       pop65 | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
       lntfr |  -1.209807   2.227722    -0.54   0.591    -5.773088    3.353474
       _cons |   6.447144    2.13756     3.02   0.005     2.068551    10.82574
------------------------------------------------------------------------------

Absorbed degrees of freedom:
-----------------------------------------------------+
 Absorbed FE | Categories  - Redundant  = Num. Coefs |
-------------+---------------------------------------|
     prov_id |        29          29           0    *|
        year |        11           1          10     |
-----------------------------------------------------+
* = FE nested within cluster; treated as redundant for DoF computation

. reghdfe pop65 lifeexp, absorb(prov_id year) cluster(prov_id)
(MWFE estimator converged in 2 iterations)

HDFE Linear regression                            Number of obs   =        319
Absorbing 2 HDFE groups                           F(   1,     28) =       1.48
Statistics robust to heteroskedasticity           Prob > F        =     0.2342
                                                  R-squared       =     0.9629
                                                  Adj R-squared   =     0.9577
                                                  Within R-sq.    =     0.0228
Number of clusters (prov_id) =         29         Root MSE        =     0.3773

                               (Std. err. adjusted for 29 clusters in prov_id)
------------------------------------------------------------------------------
             |               Robust
       pop65 | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
     lifeexp |  -.3858727   .3173604    -1.22   0.234    -1.035956    .2642105
       _cons |   32.07593   22.03309     1.46   0.157     -13.0568    77.20867
------------------------------------------------------------------------------

Absorbed degrees of freedom:
-----------------------------------------------------+
 Absorbed FE | Categories  - Redundant  = Num. Coefs |
-------------+---------------------------------------|
     prov_id |        29          29           0    *|
        year |        11           1          10     |
-----------------------------------------------------+
* = FE nested within cluster; treated as redundant for DoF computation

. reghdfe pop65 lngdpcap, absorb(prov_id year) cluster(prov_id)
(MWFE estimator converged in 2 iterations)

HDFE Linear regression                            Number of obs   =        319
Absorbing 2 HDFE groups                           F(   1,     28) =       1.45
Statistics robust to heteroskedasticity           Prob > F        =     0.2391
                                                  R-squared       =     0.9624
                                                  Adj R-squared   =     0.9572
                                                  Within R-sq.    =     0.0111
Number of clusters (prov_id) =         29         Root MSE        =     0.3795

                               (Std. err. adjusted for 29 clusters in prov_id)
------------------------------------------------------------------------------
             |               Robust
       pop65 | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
    lngdpcap |   -.644752   .5359797    -1.20   0.239    -1.742657    .4531527
       _cons |   11.93117   5.523854     2.16   0.039     .6160695    23.24627
------------------------------------------------------------------------------

Absorbed degrees of freedom:
-----------------------------------------------------+
 Absorbed FE | Categories  - Redundant  = Num. Coefs |
-------------+---------------------------------------|
     prov_id |        29          29           0    *|
        year |        11           1          10     |
-----------------------------------------------------+
* = FE nested within cluster; treated as redundant for DoF computation

. reghdfe pop65 meanyearsedu, absorb(prov_id year) cluster(prov_id)
(MWFE estimator converged in 2 iterations)

HDFE Linear regression                            Number of obs   =        319
Absorbing 2 HDFE groups                           F(   1,     28) =       2.13
Statistics robust to heteroskedasticity           Prob > F        =     0.1554
                                                  R-squared       =     0.9647
                                                  Adj R-squared   =     0.9598
                                                  Within R-sq.    =     0.0719
Number of clusters (prov_id) =         29         Root MSE        =     0.3677

                               (Std. err. adjusted for 29 clusters in prov_id)
------------------------------------------------------------------------------
             |               Robust
       pop65 | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
meanyearsedu |  -.6262353   .4289504    -1.46   0.155      -1.5049    .2524298
       _cons |   10.55461   3.608616     2.92   0.007     3.162694    17.94652
------------------------------------------------------------------------------

Absorbed degrees of freedom:
-----------------------------------------------------+
 Absorbed FE | Categories  - Redundant  = Num. Coefs |
-------------+---------------------------------------|
     prov_id |        29          29           0    *|
        year |        11           1          10     |
-----------------------------------------------------+
* = FE nested within cluster; treated as redundant for DoF computation

. reghdfe pop65 fpaidemployed, absorb(prov_id year) cluster(prov_id)
(MWFE estimator converged in 2 iterations)

HDFE Linear regression                            Number of obs   =        319
Absorbing 2 HDFE groups                           F(   1,     28) =       0.61
Statistics robust to heteroskedasticity           Prob > F        =     0.4426
                                                  R-squared       =     0.9626
                                                  Adj R-squared   =     0.9574
                                                  Within R-sq.    =     0.0159
Number of clusters (prov_id) =         29         Root MSE        =     0.3786

                                (Std. err. adjusted for 29 clusters in prov_id)
-------------------------------------------------------------------------------
              |               Robust
        pop65 | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
--------------+----------------------------------------------------------------
fpaidemployed |  -.0170178   .0218494    -0.78   0.443    -.0617742    .0277386
        _cons |    6.23496   1.217997     5.12   0.000     3.740007    8.729913
-------------------------------------------------------------------------------

Absorbed degrees of freedom:
-----------------------------------------------------+
 Absorbed FE | Categories  - Redundant  = Num. Coefs |
-------------+---------------------------------------|
     prov_id |        29          29           0    *|
        year |        11           1          10     |
-----------------------------------------------------+
* = FE nested within cluster; treated as redundant for DoF computation

I also run simple linear regression for each variable. Based on these results, the only variable insignificant is meanyearsedu

. regress pop65 lntfr

      Source |       SS           df       MS      Number of obs   =       319
-------------+----------------------------------   F(1, 317)       =    128.88
       Model |  309.014605         1  309.014605   Prob > F        =    0.0000
    Residual |   760.09783       317  2.39778495   R-squared       =    0.2890
-------------+----------------------------------   Adj R-squared   =    0.2868
       Total |  1069.11244       318  3.36198879   Root MSE        =    1.5485

------------------------------------------------------------------------------
       pop65 | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
       lntfr |   -7.37473   .6496235   -11.35   0.000    -8.652849   -6.096612
       _cons |   12.36256   .6293321    19.64   0.000     11.12436    13.60075
------------------------------------------------------------------------------

. regress pop65 lifeexp

      Source |       SS           df       MS      Number of obs   =       319
-------------+----------------------------------   F(1, 317)       =     71.51
       Model |  196.775225         1  196.775225   Prob > F        =    0.0000
    Residual |   872.33721       317   2.7518524   R-squared       =    0.1841
-------------+----------------------------------   Adj R-squared   =    0.1815
       Total |  1069.11244       318  3.36198879   Root MSE        =    1.6589

------------------------------------------------------------------------------
       pop65 | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
     lifeexp |   .3227934   .0381726     8.46   0.000     .2476897    .3978971
       _cons |  -17.12398   2.651803    -6.46   0.000    -22.34134   -11.90662
------------------------------------------------------------------------------

. regress pop65 lngdpcap

      Source |       SS           df       MS      Number of obs   =       319
-------------+----------------------------------   F(1, 317)       =     41.66
       Model |   124.17711         1   124.17711   Prob > F        =    0.0000
    Residual |  944.935325       317  2.98086853   R-squared       =    0.1161
-------------+----------------------------------   Adj R-squared   =    0.1134
       Total |  1069.11244       318  3.36198879   Root MSE        =    1.7265

------------------------------------------------------------------------------
       pop65 | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
    lngdpcap |  -1.177052   .1823671    -6.45   0.000    -1.535855   -.8182496
       _cons |   17.41711   1.881975     9.25   0.000     13.71437    21.11985
------------------------------------------------------------------------------

. regress pop65 meanyearsedu

      Source |       SS           df       MS      Number of obs   =       319
-------------+----------------------------------   F(1, 317)       =      0.06
       Model |  .217240191         1  .217240191   Prob > F        =    0.7998
    Residual |  1068.89519       317  3.37190913   R-squared       =    0.0002
-------------+----------------------------------   Adj R-squared   =   -0.0030
       Total |  1069.11244       318  3.36198879   Root MSE        =    1.8363

------------------------------------------------------------------------------
       pop65 | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
meanyearsedu |   .0303364   .1195175     0.25   0.800    -.2048115    .2654842
       _cons |   5.031091   1.010704     4.98   0.000     3.042556    7.019626
------------------------------------------------------------------------------

. regress pop65 fpaidemployed

      Source |       SS           df       MS      Number of obs   =       319
-------------+----------------------------------   F(1, 317)       =     25.07
       Model |  78.3430653         1  78.3430653   Prob > F        =    0.0000
    Residual |   990.76937       317  3.12545543   R-squared       =    0.0733
-------------+----------------------------------   Adj R-squared   =    0.0704
       Total |  1069.11244       318  3.36198879   Root MSE        =    1.7679

-------------------------------------------------------------------------------
        pop65 | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
--------------+----------------------------------------------------------------
fpaidemployed |   .0731954   .0146198     5.01   0.000     .0444314    .1019595
        _cons |   1.206011   .8209698     1.47   0.143    -.4092267    2.821249
-------------------------------------------------------------------------------

[Carlo Lazzaro]

Niara:
running separate regression for each predictor is not helpful, as you're not adjusting for the other independent variables that contribute to the data generating process.
And, if I may, please free yourself from significance-addiction!