Announcement

Collapse
No announcement yet.
X
  • Filter
  • Time
  • Show
Clear All
new posts

  • xtreg, xtmixed, or reg for aggregate (state-level) repeated cross-sectional data?

    Greetings,

    I'm running Stata 15.1 on MacOS. The dataset I'm working with was generated as follows: I pooled a number of large-sample cross-sectional surveys that featured the same question for the years 2008, 2010, 2011, 2012, 2014, 2015, 2016, and 2018. My goal is to examine whether states' google search interest in related topics (for those years) influences the proportion of state respondents giving a 'disagree' response. The variables I use (including the outcome variable) here was thus created as follows:

    Code:
    egen var1_state_year=mean(var1), by(state year)
    Once all the necessary variables were created, I then applied the 'collapse' command:

    Code:
    collapse (mean) var1_state_year var2_state_year, by(state year)
    My question is an attempt at clarifying an apparent inconsistency in what model the stata output suggets I run vs. what was suggested here: https://www.statalist.org/forums/for...ith-time-trend . Specifically, to capture overtime variation both within and between states, my intuition was to run a multi-level model with the xtmixed command:

    Code:
    . xtmixed whstate_favors indexZ  i.year  state:
    
    Performing EM optimization: 
    
    Performing gradient-based optimization: 
    
    Iteration 0:   log likelihood = -1305.9038  
    Iteration 1:   log likelihood = -1305.9038  
    
    Computing standard errors:
    
    Mixed-effects ML regression                     Number of obs     =        400
    Group variable: state                           Number of groups  =         50
    
    Obs per group:
    min =          8
    avg =        8.0
    max =          8
    
    Wald chi2(8)      =     275.78
    Log likelihood = -1305.9038                     Prob > chi2       =     0.0000
    
    
    whstate_favors       Coef.   Std. Err.      z    P>z     [95% Conf. Interval]
    
    indexZ    1.336521    .620664     2.15   0.031     .1200416       2.553
    
    year 
    2010    -3.011593   1.064883    -2.83   0.005    -5.098725   -.9244611
    2011    -.5052854   1.064613    -0.47   0.635    -2.591888    1.581317
    2012    -1.860114   1.064116    -1.75   0.080    -3.945744    .2255152
    2014     -1.57528   1.111708    -1.42   0.156    -3.754188    .6036278
    2015     7.134012   1.239951     5.75   0.000     4.703752    9.564272
    2016     2.535333   1.522316     1.67   0.096    -.4483511    5.519016
    2018     6.170662   1.577615     3.91   0.000     3.078594     9.26273
    
    _cons    17.14003   1.323791    12.95   0.000     14.54544    19.73461
    
    
    
    Random-effects Parameters     Estimate   Std. Err.     [95% Conf. Interval]
    
    state: Identity              
    sd(_cons)    7.348734   .7864944      5.958173    9.063834
    
    sd(Residual)     5.32021   .2012029       4.94012    5.729543
    
    LR test vs. linear model: chibar2(01) = 280.74        Prob >= chibar2 = 0.0000
    As you can see at the bottom, the lr test seems to indicate (though correct me if I'm wrong) that a multi-level model is preferable to OLS. However, in the earlier thread I linked, it was mentioned that the xt commands (whether xtreg or xtmixed) are unnecessary for pooled cross-sectional data, and that one can simply use OLS with i.year to capture time-specific effects. Can someone thus clarify how to proceed with such data?

    If it helps, here is a sample of my data (whstate_favors=outcome variable, indexZ=state's average annual google search interest score):

    Code:
    * Example generated by -dataex-. To install: ssc install dataex
    clear
    input float(whstate_favors indexZ) double(year state)
     7.680336 -1.1429285 2008  1
     24.03921  -.9920198 2008  2
     13.80334  -.7394119 2008  4
     8.201495  -.5032071 2008  5
     20.11939 -.55897766 2008  6
      18.4817 -.50976837 2008  8
    22.488667   .6318882 2008  9
     22.00062 -1.3594495 2008 10
     48.47411 -.58522266 2008 11
    14.984724  -.6737995 2008 12
     11.23602  -.3982272 2008 13
     28.74142  -.6180289 2008 15
     10.57713  -.4047884 2008 16
    17.614655 -.18826735 2008 17
      16.9839 -.28340536 2008 18
    13.688213          . 2008 19
     14.39021  -.3851047 2008 20
    11.007668 -.58850324 2008 21
     13.33329  -.1981092 2008 22
    21.126213 -1.1790153 2008 23
    16.110003  -.6049064 2008 24
     21.99363   .4941021 2008 25
    12.200358   .0971468 2008 26
    17.403385 -.26700228 2008 27
      7.49368    -.83455 2008 28
     16.29431 -.50976837 2008 29
    13.251606 -1.3594495 2008 30
    16.155375  -.3424566 2008 31
    13.729026  -.7361313 2008 32
    14.480515  -.6409932 2008 33
    13.202847  -.7590957 2008 34
     19.35502   -.486804 2008 35
     17.36421 -.35885975 2008 36
     9.305553 -.15546118 2008 37
     9.009853  -.9723361 2008 38
     10.74107  -.4441559 2008 39
     9.809873 -.56553894 2008 40
     25.01471  -.6049064 2008 41
     11.75892 -.20467043 2008 42
     20.31874   .5531533 2008 44
    12.128377  -.1915479 2008 45
    19.033066 -1.3594495 2008 46
    12.045777  -.5064877 2008 47
    11.860756  -.4539978 2008 48
    15.495923  -.7262894 2008 49
    37.404438  -.5032071 2008 50
      14.6579  -.7886212 2008 51
     23.78781 -.12265485 2008 53
     8.866741  -.9362493 2008 54
     16.52739  -.4900846 2008 55
     9.820115  -.8017437 2008 56
     6.425949 -1.0379485 2010  1
    20.993994 -1.2413472 2010  2
     15.34381 -.24075733 2010  4
      4.02392  -.6049063 2010  5
     19.68124  -.7525344 2010  6
    16.478086  -.1915479 2010  8
    17.863935  .22509107 2010  9
    17.345894 -.17842548 2010 10
    20.170927 -.42447215 2010 11
    10.718645 -.24075726 2010 12
     8.976017  -.3621403 2010 13
     30.34577 -1.0445098 2010 15
      11.6666   .1857236 2010 16
    14.709275  .30054545 2010 17
     9.224236  -.6049063 2010 18
    12.499412          . 2010 19
     8.198928   -.408069 2010 20
     13.38905  -.2243542 2010 21
     7.463886  -.6409932 2010 22
    21.378126  -.5950646 2010 23
    13.691906  -.4441559 2010 24
     20.93805    .172601 2010 25
    10.807766  .05777933 2010 26
      16.0334  .24149424 2010 27
      3.65953  -.9362493 2010 28
    13.752706  -.6705188 2010 29
     9.103647   -.850953 2010 30
     5.954556  -.1160937 2010 31
     9.344904  -.7361313 2010 32
    13.878927  -.6442739 2010 33
     16.11743   -.434314 2010 34
     20.65257  -.5491358 2010 35
     14.31302  -.2374767 2010 36
    13.121726 -.14233859 2010 37
    11.473763    -.83455 2010 38
      9.93098  -.2276348 2010 39
    11.126292  -.3030891 2010 40
      20.6783  -.6049064 2010 41
     9.692232 -.16530304 2010 42
      20.0009 -.17186426 2010 44
    10.009124 -1.0707548 2010 45
    4.7525783 -1.1822959 2010 46
     7.694825  -.6049064 2010 47
    10.644823  -.2801248 2010 48
    15.650936  -.3457372 2010 49
     31.19452  -.8607949 2010 50
    10.943864 -1.0969998 2010 51
    19.273506  -.3621403 2010 53
      5.90513  -1.198699 2010 54
    end
    label values state state
    label def state 1 "Alabama", modify
    label def state 2 "Alaska", modify
    label def state 4 "Arizona", modify
    label def state 5 "Arkansas", modify
    label def state 6 "California", modify
    label def state 8 "Colorado", modify
    label def state 9 "Connecticut", modify
    label def state 10 "Delaware", modify
    label def state 11 "District of Columbia", modify
    label def state 12 "Florida", modify
    label def state 13 "Georgia", modify
    label def state 15 "Hawaii", modify
    label def state 16 "Idaho", modify
    label def state 17 "Illinois", modify
    label def state 18 "Indiana", modify
    label def state 19 "Iowa", modify
    label def state 20 "Kansas", modify
    label def state 21 "Kentucky", modify
    label def state 22 "Louisiana", modify
    label def state 23 "Maine", modify
    label def state 24 "Maryland", modify
    label def state 25 "Massachusetts", modify
    label def state 26 "Michigan", modify
    label def state 27 "Minnesota", modify
    label def state 28 "Mississippi", modify
    label def state 29 "Missouri", modify
    label def state 30 "Montana", modify
    label def state 31 "Nebraska", modify
    label def state 32 "Nevada", modify
    label def state 33 "New Hampshire", modify
    label def state 34 "New Jersey", modify
    label def state 35 "New Mexico", modify
    label def state 36 "New York", modify
    label def state 37 "North Carolina", modify
    label def state 38 "North Dakota", modify
    label def state 39 "Ohio", modify
    label def state 40 "Oklahoma", modify
    label def state 41 "Oregon", modify
    label def state 42 "Pennsylvania", modify
    label def state 44 "Rhode Island", modify
    label def state 45 "South Carolina", modify
    label def state 46 "South Dakota", modify
    label def state 47 "Tennessee", modify
    label def state 48 "Texas", modify
    label def state 49 "Utah", modify
    label def state 50 "Vermont", modify
    label def state 51 "Virginia", modify
    label def state 53 "Washington", modify
    label def state 54 "West Virginia", modify
    label def state 55 "Wisconsin", modify
    label def state 56 "Wyoming", modify
    Thanks for your help!




  • #2
    When you have pooled cross-sectional data, then the data are, at least in that respect, single-level: that distinguishes them from panel data where the same people are observed in each wave and observations are nested within individuals.

    But that is not the end of the story. In your case, you have respondents clustered within states, and the state is another level in the data. (So, if these surveys were panel data, you would actually have a three-level model: surveys nested in respondents nested in states.)

    So OLS is not appropriate here (at least not unless you end up with essentially zero variance at the state level--which definitely did not happen in your data.)

    So you need to do this as a two level model. As between using -mixed- (if you are using version 13 or later, the name -xtmixed- has been replaced by -mixed-) and -xtreg, re-, it really makes no difference. They estimate the same model. The calculations used for the estimation are different, but the results will be the same either way, except for negligible rounding errors perhaps. -xtreg, re- might be a little faster if you had a very large data set, and there is probably a greater risk that -mixed- will not converge at all--but apparently you have already gotten it to converge. Having already run it with -mixed-, there is no reason to do it with -xtreg, re-.

    Comment


    • #3
      Hey Clyde,

      Thanks for clearing that up. One last thing: am I correct in including the i. prefix (i.year) with the time variable? I notice that the predictor variable moves from just reaching conventional levels of significance when I include it to significant at the 0.001 level when I don't:

      Code:
      . xtmixed whstate_favors indexZ year || state: 
      
      Performing EM optimization: 
      
      Performing gradient-based optimization: 
      
      Iteration 0:   log likelihood = -1342.5958  
      Iteration 1:   log likelihood = -1342.5958  
      
      Computing standard errors:
      
      Mixed-effects ML regression                     Number of obs     =        400
      Group variable: state                           Number of groups  =         50
      
                                                      Obs per group:
                                                                    min =          8
                                                                    avg =        8.0
                                                                    max =          8
      
                                                      Wald chi2(2)      =     157.31
      Log likelihood = -1342.5958                     Prob > chi2       =     0.0000
      
      --------------------------------------------------------------------------------
      whstate_favors |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
      ---------------+----------------------------------------------------------------
              indexZ |   2.581808   .5485436     4.71   0.000     1.506682    3.656934
                year |   .5347239   .1533757     3.49   0.000     .2341131    .8353346
               _cons |  -1058.298    308.695    -3.43   0.001    -1663.329   -453.2674
      --------------------------------------------------------------------------------
      
      ------------------------------------------------------------------------------
        Random-effects Parameters  |   Estimate   Std. Err.     [95% Conf. Interval]
      -----------------------------+------------------------------------------------
      state: Identity              |
                         sd(_cons) |   7.176968   .7800121      5.800028    8.880795
      -----------------------------+------------------------------------------------
                      sd(Residual) |   5.920536   .2238123      5.497729     6.37586
      ------------------------------------------------------------------------------
      LR test vs. linear model: chibar2(01) = 232.58        Prob >= chibar2 = 0.0000
      Thanks again!

      Comment


      • #4
        Zach:
        yes, you're correct in using -i.year-.
        About the variation in significance with/without -year- as a predictor, since you have a non-negligible time-series dimension in your dataset, I would add it.
        Kind regards,
        Carlo
        (Stata 19.0)

        Comment


        • #5
          My perspective is somewhat different. Once you collapsed the data from individual to the state level, you have a panel data set with N = 50, T = 8. It seems very like that a fixed effects analysis at the state level is warranted, and you should cluster your standard errors at the state level.

          Code:
          xtset state year
          xtmixed whstate favors indexZ i.year, fe vce(cluster state)
          With aggregated data like this, a random effects analysis is rarely appropriate if you're truly interested in causality. That is what the xtmixed command does (in econometrics terms) in this case. And you have not provided robust standard errors, as the FE command above. If you want to see how the results differ, use -re- in place of -fe- above. And then you can try a robust Hausman test, but be careful as you cannot test the coefficients on the year dummies.

          By the way, pooled OLS could be okay with cluster-robust standard errors. But, like xtmixed (or xtreg, re) it imposes exogeneity on the covariates with respect to the unobserved effect. That is the main reason pooled OLS is deficient -- just like the main reason xtmixed is. If you want a descriptive analysis, pooled OLS could be just as good as xtmixed.

          It seems to me that many users think xtmixed solves a fundamental endogeneity problem. It does not. It solves a second-order problem, which is panel correlation. But it does so in a very restrictive way. Your best bet with state level data is fixed effects with cluster-robust standard errors. You can compare that to RE if you want.

          Jeff

          Comment


          • #6
            Hey Jeff,

            Thank you very much for your input. I tried fixed effects with xtreg + state-clustered errors and now have an interpretation question. Two of my predictor variables--the google search interest index and state-level political ideology--are insignificant when entered individually.....

            Code:
            . xtreg whstate_favors indexZ whpolorZ  i.year , fe vce(cluster state) 
            
            Fixed-effects (within) regression               Number of obs     =        400
            Group variable: state                           Number of groups  =         50
            
            R-sq:                                           Obs per group:
                 within  = 0.4469                                         min =          8
                 between = 0.6587                                         avg =        8.0
                 overall = 0.3929                                         max =          8
            
                                                            F(9,49)           =      71.80
            corr(u_i, Xb)  = 0.2552                         Prob > F          =     0.0000
            
                                             (Std. Err. adjusted for 50 clusters in state)
            ------------------------------------------------------------------------------
                         |               Robust
            whstate_fa~s |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
            -------------+----------------------------------------------------------------
                  indexZ |   .6923516    .683187     1.01   0.316     -.680564    2.065267
                whpolorZ |   1.682276   1.218256     1.38   0.174    -.7659007    4.130453
                         |
                    year |
                   2010  |  -2.906201   .7237082    -4.02   0.000    -4.360547   -1.451855
                   2011  |  -.1382943   1.140536    -0.12   0.904    -2.430288    2.153699
                   2012  |  -1.769559   .5262186    -3.36   0.002    -2.827035   -.7120827
                   2014  |  -.7952771   .8659583    -0.92   0.363    -2.535486    .9449313
                   2015  |   8.011363   1.130108     7.09   0.000     5.740326     10.2824
                   2016  |   3.915599   1.527978     2.56   0.014     .8450117    6.986186
                   2018  |   7.687942   1.400221     5.49   0.000     4.874093    10.50179
                         |
                   _cons |   16.66123   .5473087    30.44   0.000     15.56137    17.76109
            -------------+----------------------------------------------------------------
                 sigma_u |  6.5222289
                 sigma_e |  5.3497789
                     rho |  .59780363   (fraction of variance due to u_i)
            ------------------------------------------------------------------------------
            but are 'highly' significant when entered as an interaction term...

            Code:
            . xtreg whstate_favors c.indexZ##c.whpolorZ  i.year , fe vce(cluster state) 
            
            Fixed-effects (within) regression               Number of obs     =        400
            Group variable: state                           Number of groups  =         50
            
            R-sq:                                           Obs per group:
                 within  = 0.4828                                         min =          8
                 between = 0.6498                                         avg =        8.0
                 overall = 0.3442                                         max =          8
            
                                                            F(10,49)          =      84.14
            corr(u_i, Xb)  = 0.1801                         Prob > F          =     0.0000
            
                                                    (Std. Err. adjusted for 50 clusters in state)
            -------------------------------------------------------------------------------------
                                |               Robust
                 whstate_favors |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
            --------------------+----------------------------------------------------------------
                         indexZ |  -.0663325   .4807782    -0.14   0.891    -1.032492    .8998275
                       whpolorZ |     .72824   1.746959     0.42   0.679    -2.782405    4.238885
                                |
            c.indexZ#c.whpolorZ |   1.293644   .1725034     7.50   0.000     .9469852    1.640302
                                |
                           year |
                          2010  |  -2.855888   .7786838    -3.67   0.001    -4.420712   -1.291064
                          2011  |  -.5354861   1.061284    -0.50   0.616    -2.668216    1.597244
                          2012  |  -1.863197   .5575004    -3.34   0.002    -2.983536   -.7428582
                          2014  |  -.8911267   .9001452    -0.99   0.327    -2.700036    .9177828
                          2015  |   8.382463   1.031144     8.13   0.000     6.310302    10.45462
                          2016  |   4.667336   1.468214     3.18   0.003     1.716849    7.617823
                          2018  |   8.308727   1.325704     6.27   0.000     5.644625    10.97283
                                |
                          _cons |   16.28415   .5436043    29.96   0.000     15.19173    17.37656
            --------------------+----------------------------------------------------------------
                        sigma_u |  6.9382538
                        sigma_e |  5.1806786
                            rho |  .64203972   (fraction of variance due to u_i)
            -------------------------------------------------------------------------------------
            Thus, if I'm interpreting this correctly, the effects of my search interest index on the outcome variable strengthen the more a state's population identifies as democrat/liberal.

            Code:
            . margins, dydx(indexZ)  at(whpolorZ=(-2 -1 0 1 2 3 4 5 6))
            
            Average marginal effects                        Number of obs     =        400
            Model VCE    : Robust
            
            Expression   : Linear prediction, predict()
            dy/dx w.r.t. : indexZ
            
            1._at        : whpolorZ        =          -2
            
            2._at        : whpolorZ        =          -1
            
            3._at        : whpolorZ        =           0
            
            4._at        : whpolorZ        =           1
            
            5._at        : whpolorZ        =           2
            
            6._at        : whpolorZ        =           3
            
            7._at        : whpolorZ        =           4
            
            8._at        : whpolorZ        =           5
            
            9._at        : whpolorZ        =           6
            
            ------------------------------------------------------------------------------
                         |            Delta-method
                         |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval]
            -------------+----------------------------------------------------------------
            indexZ       |
                     _at |
                      1  |   -2.65362   .6039612    -4.39   0.000    -3.837362   -1.469878
                      2  |  -1.359976   .5178812    -2.63   0.009    -2.375005   -.3449477
                      3  |  -.0663325   .4807782    -0.14   0.890     -1.00864    .8759755
                      4  |   1.227311   .5035963     2.44   0.015     .2402807    2.214342
                      5  |   2.520955   .5792976     4.35   0.000     1.385553    3.656357
                      6  |   3.814599   .6907076     5.52   0.000     2.460837    5.168361
                      7  |   5.108243    .823458     6.20   0.000     3.494295    6.722191
                      8  |   6.401887   .9688158     6.61   0.000     4.503043    8.300731
                      9  |    7.69553   1.121891     6.86   0.000     5.496664    9.894397
            ------------------------------------------------------------------------------
            However, if I add i.year to the interaction term to assess whether the effects of this relationship varies by time period....

            Code:
            . xtreg whstate_favors c.indexZ##c.whpolorZ##i.year , fe vce(cluster    state) 
            
            Fixed-effects (within) regression               Number of obs     =    400
            Group variable: state                           Number of groups  =    50
            
            R-sq:                                           Obs per group:
            within  = 0.5336                                         min =    8
            between = 0.7326                                         avg =    8.0
            overall = 0.4104                                         max =    8
            
            F(31,49)          =    170.51
            corr(u_i, Xb)  = 0.2258                         Prob > F          =    0.0000
            
            (Std. Err. adjusted for    50 clusters    in state)
                    
            Robust
            whstate_favors       Coef.   Std. Err.      t    P>t    [95% Conf.    Interval]
                    
            indexZ   -1.380751    1.86907    -0.74   0.464    -5.136787    2.375286
            whpolorZ    1.116271   2.322876     0.48   0.633    -3.551723    5.784264
            
            c.indexZ#c.whpolorZ    .1015852   1.361798     0.07   0.941    -2.63505    2.838221
            
            year 
            2010    -2.077525   1.132799    -1.83   0.073    -4.35397    .1989209
            2011    -1.034707   2.596973    -0.40   0.692    -6.25352    4.184105
            2012    -1.670061   1.431587    -1.17   0.249    -4.546944    1.206821
            2014    -.4351476   1.334684    -0.33   0.746    -3.117295    2.247
            2015     8.964804   1.682143     5.33   0.000    5.584411    12.3452
            2016     6.695549   2.496453     2.68   0.010    1.678739    11.71236
            2018     9.368482   1.900923     4.93   0.000    5.548434    13.18853
            
            year#c.indexZ 
            2010     .9485692   1.660689     0.57   0.570    -2.388709    4.285848
            2011    -1.261267   4.304157    -0.29   0.771    -9.910794    7.38826
            2012     .3596812   2.106218     0.17   0.865    -3.872922    4.592284
            2014     1.535718   1.625191     0.94   0.349    -1.730225    4.801662
            2015     .9174938   1.962762     0.47   0.642    -3.026824    4.861812
            2016     .3961546   2.189848     0.18   0.857    -4.004509    4.796818
            2018     1.303338   1.715671     0.76   0.451    -2.144431    4.751108
            
            year#c.whpolorZ 
            2010    -4.168671   3.214117    -1.30   0.201    -10.62768    2.29034
            2011       1.4173   5.442172     0.26   0.796    -9.519154    12.35375
            2012    -.5647654   1.647373    -0.34   0.733    -3.875285    2.745754
            2014    -.9360264   1.174099    -0.80   0.429    -3.295467    1.423414
            2015    -.9596184   1.589719    -0.60   0.549    -4.154279    2.235042
            2016      3.40517   2.626293     1.30   0.201    -1.872562    8.682903
            2018     .2168549   1.913815     0.11   0.910    -3.629101    4.062811
            
            year#c.indexZ#c.whpolorZ 
            2010    -3.963512   3.127274    -1.27   0.211    -10.248    2.32098
            2011     .8602187   6.568851     0.13   0.896    -12.34038    14.06082
            2012       .50964   2.696234     0.19   0.851    -4.908646    5.927926
            2014     1.696225   1.692284     1.00   0.321    -1.704546    5.096997
            2015     2.171904   1.685697     1.29   0.204    -1.215631    5.559439
            2016     1.236284    1.77716     0.70   0.490    -2.335053    4.807622
            2018     .5976175   1.352205     0.44   0.660    -2.119741    3.314976
            
            _cons    15.61515   1.154004    13.53   0.000    13.29609    17.93421
                    
            sigma_u   6.6370996
            sigma_e   5.0794014
            rho   .63063998   (fraction of variance due to    u_i)
            ...I get mostly null results....

            Code:
            . margins, dydx(indexZ)  at(whpolorZ=(-2 -1 0    1    2 3 4 5 6) year=(2008 2010 2011    2012    2014    2015    2016    2018))
            
            Average marginal effects        Number of obs     =        400
            Model VCE    : Robust
            
            Expression   : Linear prediction, predict()
            dy/dx w.r.t. : indexZ
            
            1._at        : whpolorZ        =          -2
            year            =        2008
            
            2._at        : whpolorZ        =          -2
            year            =        2010
            
            3._at        : whpolorZ        =          -2
            year            =        2011
            
            4._at        : whpolorZ        =          -2
            year            =        2012
            
            5._at        : whpolorZ        =          -2
            year            =        2014
            
            6._at        : whpolorZ        =          -2
            year            =        2015
            
            7._at        : whpolorZ        =          -2
            year            =        2016
            
            8._at        : whpolorZ        =          -2
            year            =        2018
            
            9._at        : whpolorZ        =          -1
            year            =        2008
            
            10._at       : whpolorZ        =          -1
            year            =        2010
            
            11._at       : whpolorZ        =          -1
            year            =        2011
            
            12._at       : whpolorZ        =          -1
            year            =        2012
            
            13._at       : whpolorZ        =          -1
            year            =        2014
            
            14._at       : whpolorZ        =          -1
            year            =        2015
            
            15._at       : whpolorZ        =          -1
            year            =        2016
            
            16._at       : whpolorZ        =          -1
            year            =        2018
            
            17._at       : whpolorZ        =           0
            year            =        2008
            
            18._at       : whpolorZ        =           0
            year            =        2010
            
            19._at       : whpolorZ        =           0
            year            =        2011
            
            20._at       : whpolorZ        =           0
            year            =        2012
            
            21._at       : whpolorZ        =           0
            year            =        2014
            
            22._at       : whpolorZ        =           0
            year            =        2015
            
            23._at       : whpolorZ        =           0
            year            =        2016
            
            24._at       : whpolorZ        =           0
            year            =        2018
            
            25._at       : whpolorZ        =           1
            year            =        2008
            
            26._at       : whpolorZ        =           1
            year            =        2010
            
            27._at       : whpolorZ        =           1
            year            =        2011
            
            28._at       : whpolorZ        =           1
            year            =        2012
            
            29._at       : whpolorZ        =           1
            year            =        2014
            
            30._at       : whpolorZ        =           1
            year            =        2015
            
            31._at       : whpolorZ        =           1
            year            =        2016
            
            32._at       : whpolorZ        =           1
            year            =        2018
            
            33._at       : whpolorZ        =           2
            year            =        2008
            
            34._at       : whpolorZ        =           2
            year            =        2010
            
            35._at       : whpolorZ        =           2
            year            =        2011
            
            36._at       : whpolorZ        =           2
            year            =        2012
            
            37._at       : whpolorZ        =           2
            year            =        2014
            
            38._at       : whpolorZ        =           2
            year            =        2015
            
            39._at       : whpolorZ        =           2
            year            =        2016
            
            40._at       : whpolorZ        =           2
            year            =        2018
            
            41._at       : whpolorZ        =           3
            year            =        2008
            
            42._at       : whpolorZ        =           3
            year            =        2010
            
            43._at       : whpolorZ        =           3
            year            =        2011
            
            44._at       : whpolorZ        =           3
            year            =        2012
            
            45._at       : whpolorZ        =           3
            year            =        2014
            
            46._at       : whpolorZ        =           3
            year            =        2015
            
            47._at       : whpolorZ        =           3
            year            =        2016
            
            48._at       : whpolorZ        =           3
            year            =        2018
            
            49._at       : whpolorZ        =           4
            year            =        2008
            
            50._at       : whpolorZ        =           4
            year            =        2010
            
            51._at       : whpolorZ        =           4
            year            =        2011
            
            52._at       : whpolorZ        =           4
            year            =        2012
            
            53._at       : whpolorZ        =           4
            year            =        2014
            
            54._at       : whpolorZ        =           4
            year            =        2015
            
            55._at       : whpolorZ        =           4
            year            =        2016
            
            56._at       : whpolorZ        =           4
            year            =        2018
            
            57._at       : whpolorZ        =           5
            year            =        2008
            
            58._at       : whpolorZ        =           5
            year            =        2010
            
            59._at       : whpolorZ        =           5
            year            =        2011
            
            60._at       : whpolorZ        =           5
            year            =        2012
            
            61._at       : whpolorZ        =           5
            year            =        2014
            
            62._at       : whpolorZ        =           5
            year            =        2015
            
            63._at       : whpolorZ        =           5
            year            =        2016
            
            64._at       : whpolorZ        =           5
            year            =        2018
            
            65._at       : whpolorZ        =           6
            year            =        2008
            
            66._at       : whpolorZ        =           6
            year            =        2010
            
            67._at       : whpolorZ        =           6
            year            =        2011
            
            68._at       : whpolorZ        =           6
            year            =        2012
            
            69._at       : whpolorZ        =           6
            year            =        2014
            
            70._at       : whpolorZ        =           6
            year            =        2015
            
            71._at       : whpolorZ        =           6
            year            =        2016
            
            72._at       : whpolorZ        =           6
            year            =        2018
            
                    
            Delta-method
            dy/dx   Std. Err.      z        P>z     [95% Conf. Interval]
                    
            indexZ       
            _at 
            1    -1.583921   3.664725    -0.43        0.666    -8.766651    5.598809
            2     7.291672   6.845205     1.07        0.287    -6.124683    20.70803
            3    -4.565625    12.2221    -0.37        0.709     -28.5205    19.38925
            4     -2.24352   4.578347    -0.49        0.624    -11.21692    6.729876
            5    -3.440653   2.981015    -1.15        0.248    -9.283335    2.402029
            6    -5.010235   2.184563    -2.29        0.022    -9.291901   -.7285694
            7    -3.660335   2.186809    -1.67        0.094    -7.946402    .6257326
            8    -1.475818   .9563636    -1.54        0.123    -3.350256    .3986204
            9    -1.482336   2.570472    -0.58        0.564    -6.520367    3.555696
            10     3.429745   3.665885     0.94        0.349    -3.755258    10.61475
            11    -3.603822   6.633929    -0.54        0.587    -16.60608     9.39844
            12    -1.632295   2.481111    -0.66        0.511    -6.495184    3.230594
            13    -1.642843   1.563673    -1.05        0.293    -4.707586    1.421901
            14    -2.736746   1.400446    -1.95        0.051    -5.481569    .0080773
            15    -2.322465   1.301445    -1.78        0.074    -4.873251    .2283206
            16    -.7766151   .7189051    -1.08        0.280    -2.185643    .6324129
            17    -1.380751    1.86907    -0.74        0.460     -5.04406    2.282559
            18    -.4321814   1.314413    -0.33        0.742    -3.008384    2.144022
            19    -2.642018   3.937585    -0.67        0.502    -10.35954    5.075507
            20    -1.021069   1.876374    -0.54        0.586    -4.698694    2.656555
            21     .1549678   1.050902     0.15        0.883    -1.904762    2.214698
            22    -.4632569   1.058386    -0.44        0.662    -2.537655    1.611141
            23    -.9845961   1.091886    -0.90        0.367    -3.124653    1.155461
            24    -.0774124   .6339403    -0.12        0.903    -1.319913    1.165088
            25    -1.279165   2.022006    -0.63        0.527    -5.242224    2.683893
            26    -4.294108   3.423994    -1.25        0.210    -11.00501    2.416798
            27    -1.680214   7.991648    -0.21        0.833    -17.34356    13.98313
            28    -.4098443   3.613289    -0.11        0.910    -7.491761    6.672072
            29     1.952778   2.205541     0.89        0.376    -2.370002    6.275559
            30     1.810232   1.499705     1.21        0.227    -1.129135      4.7496
            31     .3532733   1.810387     0.20        0.845    -3.195019    3.901566
            32     .6217902   .7548501     0.82        0.410    -.8576888    2.101269
            33     -1.17758   2.896995    -0.41        0.684    -6.855586    4.500426
            34    -8.156035   6.589907    -1.24        0.216    -21.07202    4.759947
            35    -.7184099   13.75117    -0.05        0.958     -27.6702    26.23338
            36     .2013809   5.895859     0.03        0.973    -11.35429    11.75705
            37     3.750589   3.704743     1.01        0.311    -3.510574    11.01175
            38     4.083721   2.312585     1.77        0.077    -.4488622    8.616305
            39     1.691143   2.819506     0.60        0.549    -3.834987    7.217273
            40     1.320993   1.010237     1.31        0.191    -.6590361    3.301022
            41    -1.075995   4.050388    -0.27        0.791    -9.014609    6.862619
            42    -12.01796   9.840585    -1.22        0.222    -31.30515     7.26923
            43     .2433939    19.7785     0.01        0.990    -38.52175    39.00854
            44     .8126061   8.284725     0.10        0.922    -15.42516    17.05037
            45     5.548399   5.261799     1.05        0.292    -4.764537    15.86134
            46      6.35721   3.227536     1.97        0.049     .0313567    12.68306
            47     3.029012   3.899815     0.78        0.437    -4.614485    10.67251
            48     2.020196   1.324471     1.53        0.127    -.5757198    4.616111
            49    -.9744098   5.303554    -0.18        0.854    -11.36919    9.420366
            50    -15.87989   13.11314    -1.21        0.226    -41.58117    9.821393
            51     1.205198   25.88725     0.05        0.963    -49.53288    51.94328
            52     1.423831   10.70899     0.13        0.894     -19.5654    22.41306
            53      7.34621   6.837297     1.07        0.283    -6.054645    20.74707
            54       8.6307   4.178029     2.07        0.039     .4419135    16.81949
            55     4.366881    5.00543     0.87        0.383    -5.443581    14.17734
            56     2.719398   1.664551     1.63        0.102    -.5430615    5.981858
            57    -.8728246   6.599903    -0.13        0.895     -13.8084    12.06275
            58    -19.74181   16.39448    -1.20        0.229     -51.8744    12.39077
            59     2.167002   32.03087     0.07        0.946    -60.61235    64.94636
            60     2.035056   13.14908     0.15        0.877    -23.73667    27.80678
            61      9.14402   8.420892     1.09        0.278    -7.360625    25.64867
            62     10.90419   5.144401     2.12        0.034     .8213474    20.98703
            63     5.704751   6.122658     0.93        0.351    -6.295437    17.70494
            64     3.418601   2.017448     1.69        0.090    -.5355255    7.372727
            65    -.7712394   7.918254    -0.10        0.922    -16.29073    14.74825
            66    -23.60374    19.6802    -1.20        0.230    -62.17623    14.96874
            67     3.128806   38.19254     0.08        0.935     -71.7272    77.98481
            68     2.646282   15.59758     0.17        0.865    -27.92441    33.21698
            69     10.94183   10.00874     1.09        0.274    -8.674942     30.5586
            70     13.17768   6.119134     2.15        0.031     1.184395    25.17096
            71      7.04262   7.246128     0.97        0.331    -7.159529    21.24477
            72     4.117804   2.377463     1.73        0.083    -.5419381    8.777545
            What can be inferred from this? That the effects are significant in the aggregate but insignificant by year? Thanks again!

            Comment

            Working...
            X