what should we do if after first differences the non-stationary of panel data still remained?

[Khati Zolfaghari]

hello everyone, I use a panel data and do after running the regression I get the residual and to follow my adviser I did the test of stationarity, but even after first differences the non-stationarity still remains, I hope I recieve your advice:

 xtset ifscode year
       panel variable:  ifscode (strongly balanced)
        time variable:  year, 1996 to 2019
                delta:  1 unit

. reghdfe f(0)lnfertility l(0/2)dum_recession l(1/2)lnfertility , absorb( i.ifscode year) vce(cluster ifscode) residuals(res)
(MWFE estimator converged in 3 iterations)

HDFE Linear regression                            Number of obs   =      3,116
Absorbing 2 HDFE groups                           F(   5,    141) =    6592.37
Statistics robust to heteroskedasticity           Prob > F        =     0.0000
                                                  R-squared       =     0.9998
                                                  Adj R-squared   =     0.9998
                                                  Within R-sq.    =     0.9493
Number of clusters (ifscode) =        142         Root MSE        =     0.0190

                               (Std. Err. adjusted for 142 clusters in ifscode)
-------------------------------------------------------------------------------
              |               Robust
  lnfertility |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
--------------+----------------------------------------------------------------
dum_recession |
          --. |  -.0013254   .0015998    -0.83   0.409    -.0044881    .0018372
          L1. |  -.0015468   .0011184    -1.38   0.169    -.0037579    .0006642
          L2. |    .000273   .0010136     0.27   0.788    -.0017309    .0022769
              |
  lnfertility |
          L1. |   1.316072   .0545001    24.15   0.000     1.208329    1.423815
          L2. |    -.36521   .0512967    -7.12   0.000    -.4666201      -.2638
              |
        _cons |   .8267416   .1036365     7.98   0.000     .6218594    1.031624
-------------------------------------------------------------------------------

Absorbed degrees of freedom:
-----------------------------------------------------+
 Absorbed FE | Categories  - Redundant  = Num. Coefs |
-------------+---------------------------------------|
     ifscode |       142         142           0    *|
        year |        22           0          22     |
-----------------------------------------------------+
* = FE nested within cluster; treated as redundant for DoF computation

. predict r, resid
(292 missing values generated)

. xtunitroot fisher r, dfuller lags(0)
(292 missing values generated)

Fisher-type unit-root test for r
Based on augmented Dickey-Fuller tests
--------------------------------------
Ho: All panels contain unit roots           Number of panels       =    142
Ha: At least one panel is stationary        Avg. number of periods =  21.94

AR parameter: Panel-specific                Asymptotics: T -> Infinity
Panel means:  Included
Time trend:   Not included
Drift term:   Not included                  ADF regressions: 0 lags
------------------------------------------------------------------------------
                                  Statistic      p-value
------------------------------------------------------------------------------
 Inverse chi-squared(284)  P      2180.3843       0.0000
 Inverse normal            Z       -34.7407       0.0000
 Inverse logit t(714)      L*      -49.9196       0.0000
 Modified inv. chi-squared Pm       79.5705       0.0000
------------------------------------------------------------------------------
 P statistic requires number of panels to be finite.
 Other statistics are suitable for finite or infinite number of panels.
------------------------------------------------------------------------------

now after first differences :

. reghdfe f(0)dlnfertility l(0/2)dum_recession l(1/2)lnfertility , absorb( i.ifscode year) vce(cluster ifscode) residuals(res)
(MWFE estimator converged in 3 iterations)

HDFE Linear regression                            Number of obs   =      3,116
Absorbing 2 HDFE groups                           F(   5,    141) =      52.36
Statistics robust to heteroskedasticity           Prob > F        =     0.0000
                                                  R-squared       =     0.4073
                                                  Adj R-squared   =     0.3735
                                                  Within R-sq.    =     0.1755
Number of clusters (ifscode) =        142         Root MSE        =     0.0190

                               (Std. Err. adjusted for 142 clusters in ifscode)
-------------------------------------------------------------------------------
              |               Robust
 dlnfertility |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
--------------+----------------------------------------------------------------
dum_recession |
          --. |  -.0013254   .0015998    -0.83   0.409    -.0044881    .0018372
          L1. |  -.0015468   .0011184    -1.38   0.169    -.0037579    .0006642
          L2. |    .000273   .0010136     0.27   0.788    -.0017309    .0022769
              |
  lnfertility |
          L1. |   .3160717   .0545001     5.80   0.000     .2083288    .4238146
          L2. |    -.36521   .0512967    -7.12   0.000    -.4666201      -.2638
              |
        _cons |   .8267416   .1036365     7.98   0.000     .6218594    1.031624
-------------------------------------------------------------------------------

Absorbed degrees of freedom:
-----------------------------------------------------+
 Absorbed FE | Categories  - Redundant  = Num. Coefs |
-------------+---------------------------------------|
     ifscode |       142         142           0    *|
        year |        22           0          22     |
-----------------------------------------------------+
* = FE nested within cluster; treated as redundant for DoF computation

. predict r, resid
(292 missing values generated)

. xtunitroot fisher r, dfuller lags(0)
(292 missing values generated)

Fisher-type unit-root test for r
Based on augmented Dickey-Fuller tests
--------------------------------------
Ho: All panels contain unit roots           Number of panels       =    142
Ha: At least one panel is stationary        Avg. number of periods =  21.94

AR parameter: Panel-specific                Asymptotics: T -> Infinity
Panel means:  Included
Time trend:   Not included
Drift term:   Not included                  ADF regressions: 0 lags
------------------------------------------------------------------------------
                                  Statistic      p-value
------------------------------------------------------------------------------
 Inverse chi-squared(284)  P      2180.3843       0.0000
 Inverse normal            Z       -34.7407       0.0000
 Inverse logit t(714)      L*      -49.9196       0.0000
 Modified inv. chi-squared Pm       79.5705       0.0000
------------------------------------------------------------------------------
 P statistic requires number of panels to be finite.
 Other statistics are suitable for finite or infinite number of panels.
------------------------------------------------------------------------------

also , I use lag(0) in both, and if I use lag(5) it is station.

many thanks for your valuable time and advice,

best regards .

[Justin Niakamal]

Take a look at the helpfile for xtunitroot. You'll want to apply xtunitroot to your lnfertility variable, not the residuals.

[Khati Zolfaghari]

Justin Niakamal thank you so much for your reply.
actually I did but still non-station after first differences.

if it is possible I ask one question more, how we can do the unit root test on residuals? Because as I read we have to do the test on variable not residuals,
however my adviser told me do the test on stationarity on residuals.
Many thanks for your valuable time and advice.
Best regards.