Dynamic panel data - how to test coefficient stability?

[Marta ArespaC]

I have an unbalanced panel data and I want to run an ARDL model. The longest time-span is from 1950m1 to 2019m12. I have 100 individuals.
The dependent variable is inflation volatility (the standard deviation of a 60-month overlapping-rolling-window of inflation). Among the explanatory variables I have the Lag1 of inflation volatility, inflation in levels, effective real exchange rate volatility, ...
I use the following regression:

areg sd60_inf_m L1sd60_inf_m mean_tr_hpinf_m Qadj sd60_hpEERm shelter1983 crisis, absorb(SeriesIdCode) vce(cluster SeriesIdCode)

Could I test for dynamic stability of the coefficients? I had in mind the cusum6 test, but for panel data. Does it exist?

I get the following:

. areg sd60_inf_m L1sd60_inf_m mean_tr_hpinf_m Qadj sd60_hpEERm shelter1983 crisis, abso
> rb(SeriesIdCode) vce(cluster SeriesIdCode)

Linear regression, absorbing indicators         Number of obs     =     43,497
                                                F(   5,     99)   =          .
                                                Prob > F          =          .
                                                R-squared         =     0.9980
                                                Adj R-squared     =     0.9980
                                                Root MSE          =     0.0006

                            (Std. Err. adjusted for 100 clusters in SeriesIdCode)
---------------------------------------------------------------------------------
                |               Robust
     sd60_inf_m |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
----------------+----------------------------------------------------------------
   L1sd60_inf_m |   .9961985   .0008042  1238.80   0.000     .9946029    .9977941
mean_tr_hpinf_m |   .0057533   .0030959     1.86   0.066    -.0003896    .0118962
           Qadj |    .000056   .0000197     2.84   0.005     .0000169     .000095
    sd60_hpEERm |  -.0020381   .0013724    -1.49   0.141    -.0047613     .000685
    shelter1983 |   -.000031   .0000141    -2.19   0.031     -.000059   -2.93e-06
         crisis |  -.0000117   7.99e-06    -1.47   0.145    -.0000276    4.13e-06
          _cons |   .0000634   .0000182     3.47   0.001     .0000272    .0000996
----------------+----------------------------------------------------------------
   SeriesIdCode |   absorbed                                     (100 categories) 

Thank you so much for your help.

[Jeff Wooldridge]

It looks like you have T roughly around 400, on average, for each i? Then FE should be fine, which is what you've done. Then, just interact the variables with dummies for different eras. I'd start with the two dummies you have in the equation: shelter1983 and crisis, which I assume are indicators of particular time periods.

[Marta ArespaC]

Yes. They are particular-period dummies. I will try what you suggest. Thank you.
Still, it would be nice if a test was possible...

[Jeff Wooldridge]

Use the joint Wald test on the interaction terms. That’s the test, to see if the coefficients change across regimes.

what would you do if the cumsum test rejects? You’d probably model the time-varying coefficients. So just do that and then test the null that they don’t vary.

[Marta ArespaC]

This is a good idea. Thank you so much.