Hello,
As an example here, I am trying to estimate the following regression and compare when my dependent variable is non-routine vs routine (that is share of employees doing routine vs non-routine tasks). In reality, I also have 2 other dependent variables between which I want to compare results.
My question is the following: if there are some estimators which come up as significant in the first regression, but insignificant in the second (P>0.1), should I remove those from the second regression to make it more efficient, or just leave

Code:
. xtreg nonroutine using_computer lngva  price_computer total_internet_access sharedegre
> e sharehigher shareother, fe vce(robust)

Fixed-effects (within) regression               Number of obs     =        120
Group variable: industry1                       Number of groups  =         10

R-sq:                                           Obs per group:
     within  = 0.3276                                         min =         12
     between = 0.4580                                         avg =       12.0
     overall = 0.4408                                         max =         12

                                                F(7,9)            =      16.27
corr(u_i, Xb)  = 0.5375                         Prob > F          =     0.0002

                                   (Std. Err. adjusted for 10 clusters in industry1)
------------------------------------------------------------------------------------
                   |               Robust
           nonrout |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------------+----------------------------------------------------------------
    using_computer |   .0014271   .0004206     3.39   0.008     .0004755    .0023786
             lngva |  -.0193869   .0317928    -0.61   0.557    -.0913072    .0525334
    price_computer |   .0014037   .0009901     1.42   0.190     -.000836    .0036434
total_internet_a~s |   .0041153   .0022304     1.85   0.098    -.0009303    .0091609
       sharedegree |   .0926562   .1112741     0.83   0.427    -.1590632    .3443756
       sharehigher |  -.2771514   .1359427    -2.04   0.072    -.5846752    .0303723
        shareother |   .1583427   .0836769     1.89   0.091    -.0309475    .3476329
             _cons |    .200577   .5024723     0.40   0.699    -.9360942    1.337248
-------------------+----------------------------------------------------------------
           sigma_u |   .1681399
           sigma_e |  .01373558
               rho |  .99337076   (fraction of variance due to u_i)
---------------------------------------------------------------------------------

VS

. xtreg routsem using_computer lngva  price_computer total_internet_access sharedegre
> e sharehigher shareother, fe vce(robust)

Fixed-effects (within) regression               Number of obs     =        120
Group variable: industry1                       Number of groups  =         10

R-sq:                                           Obs per group:
     within  = 0.0458                                         min =         12
     between = 0.7000                                         avg =       12.0
     overall = 0.6918                                         max =         12

                                                F(7,9)            =       9.83
corr(u_i, Xb)  = 0.7868                         Prob > F          =     0.0014

                                   (Std. Err. adjusted for 10 clusters in industry1)
------------------------------------------------------------------------------------
                   |               Robust
           routsem |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------------+----------------------------------------------------------------
    using_computer |  -.0003031   .0002802    -1.08   0.308    -.0009371    .0003308
             lngva |  -.0181579   .0289361    -0.63   0.546     -.083616    .0473002
    price_computer |   -.000896   .0002683    -3.34   0.009     -.001503    -.000289
total_internet_a~s |  -.0018722   .0009078    -2.06   0.069    -.0039258    .0001813
       sharedegree |  -.0601111    .083996    -0.72   0.492    -.2501232     .129901
       sharehigher |  -.0041789   .1033247    -0.04   0.969    -.2379157    .2295579
        shareother |  -.0227866   .1205998    -0.19   0.854    -.2956024    .2500292
             _cons |   .7114436   .3188556     2.23   0.053     -.009858    1.432745
-------------------+----------------------------------------------------------------
           sigma_u |   .1552167
           sigma_e |  .01099854
               rho |  .99500405   (fraction of variance due to u_i)
------------------------------------------------------------------------------------

. 
end of do-file
Also - I have performed the Hausman test to ensure that I should use a fixed effects model. Are there any other tests I should consider to check for endogeneity and think about instruments? I am just learning about panel data models now.

Thank you very much.