Test for significance of Time Dummies

[Alex Miller]

Hi,

I am doing a FE and RE Regression and I am not sure if I should use Time Dummies (i.date) or a linear time trend (c.date).

xtreg BEVsalesshare2 $varying $invariant i.date, fe vce(cluster county)

Fixed-effects (within) regression               Number of obs     =      1,296
Group variable: county                          Number of groups  =         18

R-sq:                                           Obs per group:
     within  = 0.8461                                         min =         72
     between = 0.5139                                         avg =       72.0
     overall = 0.4984                                         max =         72

                                                F(20,17)          =          .
corr(u_i, Xb)  = -0.8236                        Prob > F          =          .

                                (Std. Err. adjusted for 18 clusters in county)
------------------------------------------------------------------------------
             |               Robust
BEVsalessh~2 |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       HOVkm |   .0035468   .0009032     3.93   0.001     .0016412    .0054524
       NTPKM |   1.457356   .5944069     2.45   0.025     .2032672    2.711445
       NFPKM |  -.0001876   .0002327    -0.81   0.431    -.0006786    .0003034
   lCHRoadKm |   .0003344   .0001517     2.20   0.042     .0000143    .0006546
     DGPrice |   1.457764   1.039166     1.40   0.179    -.7346851    3.650213
 EnergyPrice |   .0000216   .0000378     0.57   0.575    -.0000582    .0001015
  Arbeitslos |   .0156132   .0080518     1.94   0.069    -.0013746    .0326009
        AVKT |    .000037   .0000282     1.31   0.207    -.0000225    .0000966
   Einkommen |   7.03e-07   9.98e-07     0.70   0.491    -1.40e-06    2.81e-06
   KuestenKm |          0  (omitted)
    MuMShare |          0  (omitted)
        Temp |          0  (omitted)
             |
        date |
        613  |  -.2939739   .2121634    -1.39   0.184    -.7415996    .1536518
        614  |  -.9261776   .6646269    -1.39   0.181    -2.328418    .4760626
        615  |  -1.207906    .861059    -1.40   0.179    -3.024582    .6087697
        616  |  -.9949058    .710164    -1.40   0.179    -2.493221    .5034093
        617  |  -1.020068   .7257672    -1.41   0.178    -2.551303    .5111668
        618  |  -1.027555   .7295912    -1.41   0.177    -2.566858    .5117477
        619  |  -.8171673   .5847958    -1.40   0.180    -2.050979    .4166441
        620  |  -.6837897   .4917226    -1.39   0.182    -1.721234    .3536543
        621  |  -.6799051   .4899869    -1.39   0.183    -1.713687    .3538769
        622  |  -.7046242   .5125549    -1.37   0.187     -1.78602    .3767721
        623  |  -.6535136    .470383    -1.39   0.183    -1.645935    .3389078
        624  |  -1.313945   .9263289    -1.42   0.174    -3.268328     .640438
        625  |  -1.623553   1.146794    -1.42   0.175    -4.043077    .7959706
        626  |  -1.858993   1.315581    -1.41   0.176    -4.634626    .9166396
        627  |   -2.22194   1.575141    -1.41   0.176    -5.545197    1.101316
        628  |  -1.514201   1.070677    -1.41   0.175    -3.773132    .7447305
        629  |  -1.019642   .7174562    -1.42   0.173    -2.533342    .4940579
        630  |  -.9345289   .6548755    -1.43   0.172    -2.316195    .4471376
        631  |  -1.403574   .9984869    -1.41   0.178    -3.510197    .7030493
        632  |  -1.823847      1.306    -1.40   0.181    -4.579265    .9315715
        633  |  -1.473502   1.045445    -1.41   0.177    -3.679199    .7321946
        634  |  -.9520076   .6716685    -1.42   0.174    -2.369104     .465089
        635  |  -.7893453   .5522139    -1.43   0.171    -1.954415    .3757241
        636  |  -1.031465   .7057578    -1.46   0.162    -2.520484    .4575538
        637  |  -1.326484   .9191686    -1.44   0.167     -3.26576    .6127924
        638  |  -1.241148   .8566794    -1.45   0.166    -3.048584    .5662872
        639  |  -.8768857   .5970589    -1.47   0.160     -2.13657    .3827985
        640  |  -1.009364    .690045    -1.46   0.162    -2.465231     .446504
        641  |  -1.062054   .7318846    -1.45   0.165    -2.606196    .4820875
        642  |  -1.513326   1.047113    -1.45   0.167     -3.72254    .6958884
        643  |  -1.677617   1.184271    -1.42   0.175    -4.176211    .8209776
        644  |  -1.772249   1.263255    -1.40   0.179    -4.437484    .8929865
        645  |  -1.589766    1.12864    -1.41   0.177    -3.970989    .7914568
        646  |  -1.313958   .9574884    -1.37   0.188    -3.334082     .706166
        647  |   -1.38575   1.006986    -1.38   0.187    -3.510305    .7388046
        648  |  -1.681797   1.192454    -1.41   0.176    -4.197656    .8340617
        649  |  -1.570769   1.127175    -1.39   0.181      -3.9489    .8073629
        650  |  -1.249355   .9409421    -1.33   0.202     -3.23457    .7358591
        651  |  -1.152704   .8185926    -1.41   0.177    -2.879784    .5743752
        652  |  -1.534667   1.093846    -1.40   0.179     -3.84248    .7731458
        653  |  -1.360255    .979134    -1.39   0.183    -3.426047    .7055375
        654  |  -1.759904   1.262651    -1.39   0.181    -4.423864    .9040554
        655  |  -1.617958   1.178995    -1.37   0.188    -4.105421    .8695047
        656  |  -1.393409   .9957378    -1.40   0.180    -3.494232    .7074141
        657  |  -.9588679   .6830983    -1.40   0.178    -2.400079    .4823435
        658  |  -.9905187   .7153963    -1.38   0.184    -2.499873    .5188354
        659  |  -.1240522   .0973825    -1.27   0.220    -.3295112    .0814069
        660  |  -.0512563   .0597748    -0.86   0.403      -.17737    .0748574
        661  |   .3494894   .2271063     1.54   0.142    -.1296629    .8286418
        662  |  -.1340338   .1525497    -0.88   0.392    -.4558856     .187818
        663  |  -.5874017   .4276434    -1.37   0.187     -1.48965     .314847
        664  |   .0913795   .0551258     1.66   0.116    -.0249259    .2076848
        665  |  -.8334221    .621054    -1.34   0.197    -2.143732    .4768874
        666  |  -1.008883   .7212533    -1.40   0.180    -2.530595     .512828
        667  |   .1646014   .0987002     1.67   0.114    -.0436379    .3728406
        668  |   .3036344   .1969627     1.54   0.142    -.1119206    .7191894
        669  |   .4383204   .3030653     1.45   0.166    -.2010915    1.077732
        670  |   .4428938   .3001054     1.48   0.158    -.1902732    1.076061
        671  |    .806903   .5671724     1.42   0.173    -.3897263    2.003532
        672  |   .9135457   .6428917     1.42   0.173    -.4428371    2.269929
        673  |   1.648068   1.173547     1.40   0.178    -.8279002    4.124037
        674  |   .6626571   .4521357     1.47   0.161    -.2912657     1.61658
        675  |   .8644334   .6236019     1.39   0.184    -.4512516    2.180118
        676  |     .95344    .708503     1.35   0.196    -.5413708    2.448251
        677  |   .1065229   .0872987     1.22   0.239    -.0776612    .2907069
        678  |   .0950838   .1061755     0.90   0.383    -.1289269    .3190944
        679  |    .630049   .4534669     1.39   0.183    -.3266824     1.58678
        680  |   .8275755    .573103     1.44   0.167    -.3815662    2.036717
        681  |   .9732132   .7018173     1.39   0.183    -.5074919    2.453918
        682  |   .3737313   .2477754     1.51   0.150    -.1490292    .8964918
        683  |          0  (omitted)
             |
       _cons |  -19.84632   13.11662    -1.51   0.149    -47.51997    7.827334
-------------+----------------------------------------------------------------
     sigma_u |  .08053934
     sigma_e |  .02521736
         rho |  .91071741   (fraction of variance due to u_i)
------------------------------------------------------------------------------

You can see that my individual Time dummies are insignificant, so I wanted to check for the jointly significance of the Time Dummies with a normal F.Test, but I can see that a lot of constraints are dropped.

What does this mean?

Is the joint test result still significant and I should therefore use Time dummies instead of a Time Trend?

. testparm i.date

 ( 1)  613.date = 0
 ( 2)  614.date = 0
 ( 3)  615.date = 0
 ( 4)  616.date = 0
 ( 5)  617.date = 0
 ( 6)  618.date = 0
 ( 7)  619.date = 0
 ( 8)  620.date = 0
 ( 9)  621.date = 0
 (10)  622.date = 0
 (11)  623.date = 0
 (12)  624.date = 0
 (13)  625.date = 0
 (14)  626.date = 0
 (15)  627.date = 0
 (16)  628.date = 0
 (17)  629.date = 0
 (18)  630.date = 0
 (19)  631.date = 0
 (20)  632.date = 0
 (21)  633.date = 0
 (22)  634.date = 0
 (23)  635.date = 0
 (24)  636.date = 0
 (25)  637.date = 0
 (26)  638.date = 0
 (27)  639.date = 0
 (28)  640.date = 0
 (29)  641.date = 0
 (30)  642.date = 0
 (31)  643.date = 0
 (32)  644.date = 0
 (33)  645.date = 0
 (34)  646.date = 0
 (35)  647.date = 0
 (36)  648.date = 0
 (37)  649.date = 0
 (38)  650.date = 0
 (39)  651.date = 0
 (40)  652.date = 0
 (41)  653.date = 0
 (42)  654.date = 0
 (43)  655.date = 0
 (44)  656.date = 0
 (45)  657.date = 0
 (46)  658.date = 0
 (47)  659.date = 0
 (48)  660.date = 0
 (49)  661.date = 0
 (50)  662.date = 0
 (51)  663.date = 0
 (52)  664.date = 0
 (53)  665.date = 0
 (54)  666.date = 0
 (55)  667.date = 0
 (56)  668.date = 0
 (57)  669.date = 0
 (58)  670.date = 0
 (59)  671.date = 0
 (60)  672.date = 0
 (61)  673.date = 0
 (62)  674.date = 0
 (63)  675.date = 0
 (64)  676.date = 0
 (65)  677.date = 0
 (66)  678.date = 0
 (67)  679.date = 0
 (68)  680.date = 0
 (69)  681.date = 0
 (70)  682.date = 0
       Constraint 1 dropped
       Constraint 2 dropped
       Constraint 3 dropped
       Constraint 4 dropped
       Constraint 5 dropped
       Constraint 6 dropped
       Constraint 7 dropped
       Constraint 8 dropped
       Constraint 10 dropped
       Constraint 11 dropped
       Constraint 12 dropped
       Constraint 13 dropped
       Constraint 14 dropped
       Constraint 16 dropped
       Constraint 17 dropped
       Constraint 18 dropped
       Constraint 19 dropped
       Constraint 20 dropped
       Constraint 21 dropped
       Constraint 22 dropped
       Constraint 23 dropped
       Constraint 24 dropped
       Constraint 25 dropped
       Constraint 26 dropped
       Constraint 27 dropped
       Constraint 28 dropped
       Constraint 29 dropped
       Constraint 30 dropped
       Constraint 31 dropped
       Constraint 32 dropped
       Constraint 34 dropped
       Constraint 35 dropped
       Constraint 36 dropped
       Constraint 37 dropped
       Constraint 39 dropped
       Constraint 40 dropped
       Constraint 41 dropped
       Constraint 45 dropped
       Constraint 47 dropped
       Constraint 54 dropped
       Constraint 55 dropped
       Constraint 56 dropped
       Constraint 57 dropped
       Constraint 58 dropped
       Constraint 59 dropped
       Constraint 62 dropped
       Constraint 63 dropped
       Constraint 64 dropped
       Constraint 66 dropped
       Constraint 69 dropped

       F( 20,    17) = 3.5e+06
            Prob > F =    0.0000

Thanks a lot for your help

Cheers
Alex

[Clyde Schechter]

You only have 18 clusters in your data, and since you are using -vce(cluster)- that gives you only 17 degrees of freedom for tests. But you have far more than 17 time variables, so it is not possible to test them jointly.

You might want to reconsider whether to use the cluster vce in this model. The cluster robust vce is only valid with a sufficiently large number of clusters. There is no consensus about how many is sufficient, and 18 falls in a grey area that some investigators would say is adequate and others would say is not. If you were to skip using the cluster robust vce, you would be able to do this test.

That said, hypothesis testing on variables is really not a good way to decide whether they should be included in the model. Particularly when the question is whether to use a linear time trend vs time indicators, this question should be decided either by a good theory of the data generating process that tells you whether the outcome variable drifts in a linear way over time or whether it just bounces around with idiosyncratic shocks from one time period to the next. If there is no theory to go on, do some graphical exploration of outcome vs time and see whether it looks like there is a linear trend or just jumping around. (Based on the results you got I would guess that you won't see much, if anything, in the way of a linear trend and that it's just jumping around.)

[Alex Miller]

Hi Clyde,

thanks a lot for the explanation. OK ,then I guess I will include both analysis (with and without clustered std. errors) in my thesis. The reason why I chose clustered std. errors is that I've tested the model for heteroskedasticity using the Breusch-Pagan (1979) test, see below. As well I have plotted the residuals and I can see a clear sign for heteroskedasticity (in line with the B-P test).
Do you agree?

 reg BEVsalesshare2 HOVkm NTPKM NFPKM lCHRoadKm DGPrice EnergyPrice Arbeitslos AVKT Einkommen KuestenKm MuM
> Share Temp 

      Source |       SS           df       MS      Number of obs   =     1,296
-------------+----------------------------------   F(12, 1283)     =    202.81
       Model |  4.32919186        12  .360765988   Prob > F        =    0.0000
    Residual |  2.28227608     1,283  .001778859   R-squared       =    0.6548
-------------+----------------------------------   Adj R-squared   =    0.6516
       Total |  6.61146794     1,295  .005105381   Root MSE        =    .04218

------------------------------------------------------------------------------
BEVsalessh~2 |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       HOVkm |  -.0000696   .0002899    -0.24   0.810    -.0006383     .000499
       NTPKM |   .5656652   .0770676     7.34   0.000     .4144729    .7168576
       NFPKM |   .0000148   .0000193     0.76   0.445    -.0000232    .0000527
   lCHRoadKm |    .000972    .000113     8.60   0.000     .0007504    .0011936
     DGPrice |  -.0196511   .0022284    -8.82   0.000    -.0240227   -.0152795
 EnergyPrice |  -.0000684   .0000118    -5.81   0.000    -.0000915   -.0000453
  Arbeitslos |    .034868   .0023238    15.00   0.000     .0303091    .0394269
        AVKT |   .0000148   3.93e-06     3.77   0.000     7.11e-06    .0000225
   Einkommen |   1.10e-06   5.91e-08    18.65   0.000     9.86e-07    1.22e-06
   KuestenKm |   2.03e-06   2.60e-07     7.82   0.000     1.52e-06    2.54e-06
    MuMShare |   .0007303   .0000669    10.92   0.000     .0005992    .0008615
        Temp |   .0012747   .0008112     1.57   0.116    -.0003167    .0028661
       _cons |  -.4990756   .0820384    -6.08   0.000    -.6600198   -.3381314
------------------------------------------------------------------------------

. rvfplot, yline(0)

. estat hettest // Breusch-Pagan (1979) version

Breusch-Pagan / Cook-Weisberg test for heteroskedasticity 
         Ho: Constant variance
         Variables: fitted values of BEVsalesshare2

         chi2(1)      =   252.73
         Prob > chi2  =   0.0000

.

Regarding the use of time dummies or a linear trend. I have plotted the outcome variable on a monthly basis and to me it is more jumping around than a linear trend... or even an exponential trend?




Thanks a lot for your help.

Regards,
Alex

[Carlo Lazzaro]

Alex:
as an aside to Clyde's excellen advice, please note that, if -fe- specification is the way to go (by the way, how did you test that hypothesis with clustered standard errors?), using -regress- as a work around is out of debate, as pooled OLS is consistent with the -re- specification only. Moreover, your -regress- code considers the observations as independent, which is not the case given the panel structure of your dataset.
If you detected heteroskedsticity and/or autocorrelation you're right in invoking cluster-robust standard errors under -xtreg-.
I share Clyde's point about investigating the existence of a linear time trend; I would also add a quadratic term, just to check whether a turning point exists.
Eventually, given that you have a T>N panel dataset, -xtreg- should be replaced with -xtregar-.

[Alex Miller]

Carlo,

the -regress- code above was used to test for heteroskedasticity and not as my main model! Since there is no other way (I am aware of) to test for heteroskedasticity other than using the -regress- (without time dummies) and after that -estat hettest- which gave me the result to reject the H0 of constant variance. -> I think I have to use cluster robust std errors under -xtreg-

And what do you mean by?:

(by the way, how did you test that hypothesis with clustered standard errors?)

Are you wondering how I tested for RE or FE including clustered standard errors?


I tried investigating about the existence of a linear time trend, see post #3. But I don't know what to do further....
Can you explain how to add a quadratic time trend? I know that the linear time trend can be added by c.date (in my case).

Thanks

[Carlo Lazzaro]

Alex:
- with such an high number of observations, visual inspection after -xtreg- is enough to suspect/detect heteroskedasticity (that in your case seems to be an issue);
- my previous reply implied that you cannot test -fe- vs -re- via -hausman- if you impose non-default standard errors (see -search xtoverid- instead);
- to plug in both linear and squared terms for -timevar-, just type:

c.date##c.date

;
- tthe issue of T>N panel dataset still remains; I would switch from -xtreg- to -xtgregar-.

[Alex Miller]

Hi again,

thanks for your quick replies!

- How can I visually inspect the residuals after -xtreg-? I only know the -rvfplot- command which only works after -regress-
- I am aware of the downwards of Hausman test. Problem with -xtoverrid- is that I was able to include cluster robust std. errors, but not Time dummies (i.date in my case) --> This gave me a significant test result indicating to use of the FE model instead of RE.

Therefore I also run a manually Auxiliary regression by generating all time averages of all time-varying regressors:

foreach variable of varlist $varying {
   egen `variable'_mean = mean(`variable'), by (county)
}

followed by the Auxiliary regression (including i.date and cluster std. errors)

reg BEVsalesshare2 $varying $invariant *_mean i.date, vce(cluster county)

Then I ran a joint Wald test:

test HOVkm_mean NTPKM_mean NFPKM_mean lCHRoadKm_mean DGPrice_mean EnergyPrice_mean Arbeitslos_mean AVKT_mean Einkommen_mean

with the following result:

( 1)  HOVkm_mean = 0
 ( 2)  NTPKM_mean = 0
 ( 3)  NFPKM_mean = 0
 ( 4)  lCHRoadKm_mean = 0
 ( 5)  o.DGPrice_mean = 0
 ( 6)  EnergyPrice_mean = 0
 ( 7)  Arbeitslos_mean = 0
 ( 8)  AVKT_mean = 0
 ( 9)  Einkommen_mean = 0
       Constraint 5 dropped

       F(  8,    17) =    2.16
            Prob > F =    0.0862

--> This test result is in contrast with the one from -xtoverrid- and states that I cannot reject H0 and therefore RE is more appropriate.
What do you think about this?

- I have never heard about -xtregar- command this far. But I can see that I cannot include any time variables there (i.date in my case), so why do you recommend switching to this command?

Alex

[Carlo Lazzaro]

Alex:
1) For the sake of simplicity, let's comapre -xtoverid- and -hausman- outcomes with default standard errrors and -timevar-. As you correctly report, being a buit old-fashioned, the wonderful -xtioverid- does not support the more recent -fvvarlist- notation; the trick is to use the old-fogey -xi:- prefix:

. use "http://www.stata-press.com/data/r15/nlswork.dta"
(National Longitudinal Survey.  Young Women 14-26 years of age in 1968)

. quietly xi: xtreg ln_wage race year, fe

. estimates store fe

. quietly xi: xtreg ln_wage race year, re

. estimates store re

. xtoverid

Test of overidentifying restrictions: fixed vs random effects
Cross-section time-series model: xtreg re  
Sargan-Hansen statistic  18.796  Chi-sq(1)    P-value = 0.0000

. hausman fe re

                 ---- Coefficients ----
             |      (b)          (B)            (b-B)     sqrt(diag(V_b-V_B))
             |       fe           re         Difference          S.E.
-------------+----------------------------------------------------------------
        year |    .0182116     .0186377       -.0004262        .0001006
------------------------------------------------------------------------------
                           b = consistent under Ho and Ha; obtained from xtreg
            B = inconsistent under Ha, efficient under Ho; obtained from xtreg

    Test:  Ho:  difference in coefficients not systematic

                  chi2(1) = (b-B)'[(V_b-V_B)^(-1)](b-B)
                          =       17.96
                Prob>chi2 =      0.0000

Now, let's impose cluster/robust standard errors:

. quietly xi: xtreg ln_wage race year, re robust

. xtoverid

Test of overidentifying restrictions: fixed vs random effects
Cross-section time-series model: xtreg re  robust cluster(idcode)
Sargan-Hansen statistic  15.474  Chi-sq(1)    P-value = 0.0001

In both cases, we should go -fe-.

2) The main issue with your data is that the time-series dimension prevails on tyhe cross-sectional one. The -xtreg- machinery is conceived for short panels (thiose with N>T).
It's true that -xtregar- does not accept -timevar-among predictors. The idea can be to play the machine around and use, as time predictor a clone of the -timevar- itself:

use http://www.stata-press.com/data/r15/grunfeld

. g alfa=year

. xtset company alfa
       panel variable:  company (strongly balanced)
        time variable:  alfa, 1 to 20
                delta:  1 unit

. xtregar invest mvalue kstock i.year, fe

FE (within) regression with AR(1) disturbances  Number of obs     =        190
Group variable: company                         Number of groups  =         10

R-sq:                                           Obs per group:
     within  = 0.6513                                         min =         19
     between = 0.7838                                         avg =       19.0
     overall = 0.7798                                         max =         19

                                                F(20,160)         =      14.94
corr(u_i, Xb)  = -0.1416                        Prob > F          =     0.0000

------------------------------------------------------------------------------
      invest |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      mvalue |   .0938865   .0104378     8.99   0.000     .0732729    .1145001
      kstock |   .4076907   .0373936    10.90   0.000      .333842    .4815393
             |
        year |
       1936  |   22.60707   15.23516     1.48   0.140    -7.480873    52.69502
       1937  |   29.02835   20.63253     1.41   0.161    -11.71886    69.77557
       1938  |   32.19837   23.69301     1.36   0.176    -14.59298    78.98973
       1939  |   17.30833   25.51947     0.68   0.499    -33.09011    67.70676
       1940  |   50.88403   27.01934     1.88   0.061    -2.476513    104.2446
       1941  |   79.24101   27.76837     2.85   0.005     24.40121    134.0808
       1942  |   73.97942   28.25514     2.62   0.010      18.1783    129.7805
       1943  |   56.66199   28.31439     2.00   0.047     .7438482    112.5801
       1944  |   58.95475   28.56968     2.06   0.041     2.532449     115.377
       1945  |   49.74642   28.40551     1.75   0.082    -6.351667    105.8445
       1946  |   75.78518   28.05301     2.70   0.008     20.38324    131.1871
       1947  |   57.76681   26.94116     2.14   0.034     4.560676    110.9729
       1948  |   51.01013   26.04522     1.96   0.052    -.4266119    102.4469
       1949  |   20.19058   24.97134     0.81   0.420    -29.12536    69.50651
       1950  |   18.35979   23.75434     0.77   0.441     -28.5527    65.27228
       1951  |   36.28607   21.77871     1.67   0.098    -6.724731    79.29688
       1952  |   32.36863   18.82811     1.72   0.088    -4.815034    69.55229
       1953  |   30.71474   14.03026     2.19   0.030     3.006356    58.42313
       1954  |          0  (omitted)
             |
       _cons |  -124.2761   10.59512   -11.73   0.000    -145.2004   -103.3518
-------------+----------------------------------------------------------------
      rho_ar |  .68127018
     sigma_u |  96.029876
     sigma_e |  39.732352
     rho_fov |  .85383317   (fraction of variance because of u_i)
------------------------------------------------------------------------------
F test that all u_i=0: F(9,160) = 12.27                      Prob > F = 0.0000

[Carlo Lazzaro]

Alex:
as far as visual inspection of residuals after -xtregar- is concerned:

. predict res, ue

. kdensity res

. qnorm res

[Alex Miller]

Hi Carlo,

1) I don't really understand how the -xi:- prefix solves the problem of time dummies....
I've tried your example and still I don't see any time dummies here... only a trend variable year? (this variable could also be created with c.year and that's not what I want right?)

.  xi: xtreg ln_wage race year, fe
note: race omitted because of collinearity

Fixed-effects (within) regression               Number of obs     =     28,534
Group variable: idcode                          Number of groups  =      4,711

R-sq:                                           Obs per group:
     within  = 0.1022                                         min =          1
     between = 0.0804                                         avg =        6.1
     overall = 0.0709                                         max =         15

                                                F(1,23822)        =    2712.80
corr(u_i, Xb)  = 0.0300                         Prob > F          =     0.0000

------------------------------------------------------------------------------
     ln_wage |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
        race |          0  (omitted)
        year |   .0182116   .0003497    52.08   0.000     .0175262    .0188969
       _cons |   .2551579   .0273177     9.34   0.000     .2016134    .3087023
-------------+----------------------------------------------------------------
     sigma_u |  .40800642
     sigma_e |  .30347936
         rho |  .64380983   (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0: F(4710, 23822) = 8.91                 Prob > F = 0.0000

2) Thanks for your example on how to trick -xtregar-. Still, I don't really understand what it does ? -- and how do you add cluster std. errors with -xtregar-?

3) Thanks

[Carlo Lazzaro]

Alex:
1) I forgot the -i- before year in my previous example:

. use "http://www.stata-press.com/data/r15/nlswork.dta"
(National Longitudinal Survey.  Young Women 14-26 years of age in 1968)

quietly xi: xtreg ln_wage race i.year, fe robust

. . estimates store fe

. . quietly xi: xtreg ln_wage race i.year, re robust

. . estimates store re

. xtoverid

2) The trick should allow you to estimate the time effect within the same panel. Admittedly, -xtregar- does not allow SE because it focuses on AR1 process, due to the T>N structure of its panel data. -xtgls- and -xtpcse- explicitly consider heteroskeadsticity (and autocorrelation)

[Alex Miller]

Carlo,

1) I don't know what's happening here but I cannot get it to work with my data (for your example it works).
Here's what I did:

quietly xi: xtreg BEVsalesshare2 HOVkm NTPKM NFPKM lCHRoadKm DGPrice EnergyPrice Arbeitslos AVKT Einkommen
>  KuestenKm MuMShare Temp i.date, fe robust

. estimates store fe

. quietly xi: xtreg BEVsalesshare2 HOVkm NTPKM NFPKM lCHRoadKm DGPrice EnergyPrice Arbeitslos AVKT Einkommen
>  KuestenKm MuMShare Temp i.date, re robust

. estimates store re

. xtoverid
o. operator not allowed
r(101);

- first of all: I seem to loose date dummies due to collinearity (see below) and I don't understand why. (with the normal -xtreg- without the -xi- it is not the case)

.  xi: xtreg BEVsalesshare2 HOVkm NTPKM NFPKM lCHRoadKm DGPrice EnergyPrice Arbeitslos AVKT Einkommen Kueste
> nKm MuMShare Temp i.date, fe robust
i.date            _Idate_600-695      (naturally coded; _Idate_600 omitted)
note: KuestenKm omitted because of collinearity
note: MuMShare omitted because of collinearity
note: Temp omitted because of collinearity
note: _Idate_601 omitted because of collinearity
note: _Idate_602 omitted because of collinearity
note: _Idate_603 omitted because of collinearity
note: _Idate_604 omitted because of collinearity
note: _Idate_605 omitted because of collinearity
note: _Idate_606 omitted because of collinearity
note: _Idate_607 omitted because of collinearity
note: _Idate_608 omitted because of collinearity
note: _Idate_609 omitted because of collinearity
note: _Idate_610 omitted because of collinearity
note: _Idate_611 omitted because of collinearity
note: _Idate_682 omitted because of collinearity
note: _Idate_683 omitted because of collinearity
note: _Idate_684 omitted because of collinearity
note: _Idate_685 omitted because of collinearity
note: _Idate_686 omitted because of collinearity
note: _Idate_687 omitted because of collinearity
note: _Idate_688 omitted because of collinearity
note: _Idate_689 omitted because of collinearity
note: _Idate_690 omitted because of collinearity
note: _Idate_691 omitted because of collinearity
note: _Idate_692 omitted because of collinearity
note: _Idate_693 omitted because of collinearity
note: _Idate_694 omitted because of collinearity
note: _Idate_695 omitted because of collinearity

Fixed-effects (within) regression               Number of obs     =      1,296
Group variable: county                          Number of groups  =         18

R-sq:                                           Obs per group:
     within  = 0.8461                                         min =         72
     between = 0.5139                                         avg =       72.0
     overall = 0.4984                                         max =         72

                                                F(17,17)          =          .
corr(u_i, Xb)  = -0.8236                        Prob > F          =          .

                                (Std. Err. adjusted for 18 clusters in county)
------------------------------------------------------------------------------
             |               Robust
BEVsalessh~2 |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       HOVkm |   .0035468   .0009032     3.93   0.001     .0016412    .0054524
       NTPKM |   1.457356   .5944068     2.45   0.025     .2032673    2.711445
       NFPKM |  -.0001876   .0002327    -0.81   0.431    -.0006786    .0003034
   lCHRoadKm |   .0003344   .0001517     2.20   0.042     .0000143    .0006546
     DGPrice |  -.0994497   .0331268    -3.00   0.008    -.1693411   -.0295582
 EnergyPrice |   .0000216   .0000378     0.57   0.575    -.0000582    .0001015
  Arbeitslos |   .0156132   .0080518     1.94   0.069    -.0013746    .0326009
        AVKT |    .000037   .0000282     1.31   0.207    -.0000225    .0000966
   Einkommen |   7.03e-07   9.98e-07     0.70   0.491    -1.40e-06    2.81e-06
   KuestenKm |          0  (omitted)
    MuMShare |          0  (omitted)
        Temp |          0  (omitted)
  _Idate_601 |          0  (omitted)
  _Idate_602 |          0  (omitted)
  _Idate_603 |          0  (omitted)
  _Idate_604 |          0  (omitted)
  _Idate_605 |          0  (omitted)
  _Idate_606 |          0  (omitted)
  _Idate_607 |          0  (omitted)
  _Idate_608 |          0  (omitted)
  _Idate_609 |          0  (omitted)
  _Idate_610 |          0  (omitted)
  _Idate_611 |          0  (omitted)
  _Idate_612 |  -.0934328   .0619434    -1.51   0.150    -.2241219    .0372563
  _Idate_613 |  -.0681779    .062751    -1.09   0.292    -.2005709    .0642151
  _Idate_614 |  -.0229936   .0685358    -0.34   0.741    -.1675914    .1216042
  _Idate_615 |  -.0088514   .0700164    -0.13   0.901    -.1565732    .1388704
  _Idate_616 |  -.0216471   .0675063    -0.32   0.752    -.1640729    .1207786
  _Idate_617 |  -.0234514   .0676963    -0.35   0.733     -.166278    .1193753
  _Idate_618 |  -.0231524   .0662402    -0.35   0.731     -.162907    .1166022
  _Idate_619 |  -.0307743   .0652927    -0.47   0.643    -.1685299    .1069813
  _Idate_620 |   -.037546   .0646474    -0.58   0.569      -.17394    .0988481
  _Idate_621 |  -.0336613   .0641663    -0.52   0.607    -.1690404    .1017178
  _Idate_622 |  -.0272362   .0649467    -0.42   0.680    -.1642617    .1097893
  _Idate_623 |  -.0384142   .0639221    -0.60   0.556    -.1732779    .0964496
  _Idate_624 |   .0019005   .0590657     0.03   0.975    -.1227173    .1265182
  _Idate_625 |   .0193072   .0665318     0.29   0.775    -.1210627     .159677
  _Idate_626 |   .0408079   .0654495     0.62   0.541    -.0972785    .1788943
  _Idate_627 |   .0671638   .0710951     0.94   0.358    -.0828337    .2171614
  _Idate_628 |   .0196548   .0601353     0.33   0.748    -.1072196    .1465292
  _Idate_629 |  -.0152394   .0549266    -0.28   0.785    -.1311243    .1006456
  _Idate_630 |  -.0235588   .0537298    -0.44   0.667    -.1369187    .0898011
  _Idate_631 |   .0212767   .0596946     0.36   0.726    -.1046679    .1472212
  _Idate_632 |   .0603821   .0663183     0.91   0.375    -.0795374    .2003015
  _Idate_633 |    .021423   .0605377     0.35   0.728    -.1063003    .1491463
  _Idate_634 |  -.0176793   .0547392    -0.32   0.751     -.133169    .0978104
  _Idate_635 |  -.0340966   .0534923    -0.64   0.532    -.1469555    .0787622
  _Idate_636 |  -.0192761   .0390972    -0.49   0.628     -.101764    .0632118
  _Idate_637 |    .004934   .0441517     0.11   0.912    -.0882181     .098086
  _Idate_638 |  -.0031633   .0431512    -0.07   0.942    -.0942044    .0878779
  _Idate_639 |  -.0282042   .0382798    -0.74   0.471    -.1089675    .0525592
  _Idate_640 |  -.0205329   .0395224    -0.52   0.610    -.1039178    .0628521
  _Idate_641 |  -.0109347   .0407038    -0.27   0.791    -.0968122    .0749428
  _Idate_642 |   .0127437   .0459285     0.28   0.785     -.084157    .1096443
  _Idate_643 |   .0508906   .0513784     0.99   0.336    -.0575082    .1592895
  _Idate_644 |   .0730495   .0522959     1.40   0.180    -.0372853    .1833843
  _Idate_645 |   .0530946    .050701     1.05   0.310     -.053875    .1600643
  _Idate_646 |   .0719625   .0468665     1.54   0.143    -.0269172    .1708421
  _Idate_647 |   .0780306   .0448922     1.74   0.100    -.0166837    .1727448
  _Idate_648 |   .0778547   .0421444     1.85   0.082    -.0110623    .1667717
  _Idate_649 |    .087664   .0419919     2.09   0.052    -.0009313    .1762592
  _Idate_650 |    .128779   .0376518     3.42   0.003     .0493406    .2082174
  _Idate_651 |   .0463503   .0328113     1.41   0.176    -.0228754     .115576
  _Idate_652 |   .0770493   .0391701     1.97   0.066    -.0055925    .1596911
  _Idate_653 |   .0801682   .0347023     2.31   0.034     .0069526    .1533837
  _Idate_654 |   .1087521   .0433061     2.51   0.022     .0173843    .2001199
  _Idate_655 |   .1183353   .0447323     2.65   0.017     .0239584    .2127123
  _Idate_656 |    .070372   .0397738     1.77   0.095    -.0135434    .1542875
  _Idate_657 |   .0377489   .0310179     1.22   0.240    -.0276932    .1031911
  _Idate_658 |   .0528145   .0319044     1.66   0.116    -.0144979     .120127
  _Idate_659 |  -.0072611   .0221002    -0.33   0.747    -.0538885    .0393662
  _Idate_660 |   .0266044   .0121931     2.18   0.043     .0008793    .0523295
  _Idate_661 |  -.0008837   .0155677    -0.06   0.955    -.0337286    .0319613
  _Idate_662 |   .0839761   .0127352     6.59   0.000     .0571073     .110845
  _Idate_663 |   .0432699   .0164152     2.64   0.017     .0086369     .077903
  _Idate_664 |  -.0020534   .0121236    -0.17   0.868    -.0276319    .0235252
  _Idate_665 |   .0853341   .0206006     4.14   0.001     .0418706    .1287975
  _Idate_666 |    .065594   .0252981     2.59   0.019     .0122197    .1189684
  _Idate_667 |     .00888   .0141383     0.63   0.538    -.0209492    .0387092
  _Idate_668 |  -.0000223   .0174539    -0.00   0.999    -.0368469    .0368023
  _Idate_669 |  -.0210577   .0146531    -1.44   0.169     -.051973    .0098576
  _Idate_670 |  -.0164843   .0146352    -1.13   0.276     -.047362    .0143933
  _Idate_671 |  -.0495646    .020695    -2.40   0.028    -.0932273    -.005902
  _Idate_672 |  -.0519268   .0227217    -2.29   0.035    -.0998655   -.0039881
  _Idate_673 |  -.1115834   .0375474    -2.97   0.009    -.1908016   -.0323652
  _Idate_674 |  -.0147309   .0172368    -0.85   0.405    -.0510972    .0216355
  _Idate_675 |  -.0698949   .0206623    -3.38   0.004    -.1134886   -.0263012
  _Idate_676 |  -.1054654   .0234522    -4.50   0.000    -.1549453   -.0559856
  _Idate_677 |  -.0180542   .0082928    -2.18   0.044    -.0355505   -.0005579
  _Idate_678 |  -.0606376   .0084616    -7.17   0.000      -.07849   -.0427853
  _Idate_679 |   -.047339   .0152066    -3.11   0.006    -.0794221   -.0152559
  _Idate_680 |  -.0288921   .0195494    -1.48   0.158    -.0701377    .0123534
  _Idate_681 |  -.0779062   .0229521    -3.39   0.003    -.1263308   -.0294815
  _Idate_682 |          0  (omitted)
  _Idate_683 |          0  (omitted)
  _Idate_684 |          0  (omitted)
  _Idate_685 |          0  (omitted)
  _Idate_686 |          0  (omitted)
  _Idate_687 |          0  (omitted)
  _Idate_688 |          0  (omitted)
  _Idate_689 |          0  (omitted)
  _Idate_690 |          0  (omitted)
  _Idate_691 |          0  (omitted)
  _Idate_692 |          0  (omitted)
  _Idate_693 |          0  (omitted)
  _Idate_694 |          0  (omitted)
  _Idate_695 |          0  (omitted)
       _cons |   .5064686   .8400738     0.60   0.555    -1.265932    2.278869
-------------+----------------------------------------------------------------
     sigma_u |  .08053934
     sigma_e |  .02521736
         rho |  .91071741   (fraction of variance due to u_i)
------------------------------------------------------------------------------

- and second: the -xtoverid- doesn't work anymore (maybe due to the error before?)

2) Alright. My problem is that my dataset suffers from autocorrelation, as well as heteroskedasticity. Do I understand it correctly that the -xtregar- only corrects for autocorrelation and is therefore not good for my date (T>N) and that you suggest -xtgls- and -xtpcse-?

[Carlo Lazzaro]

Alex:
1) very strange indeed. Usually, I get the same omitted variable due to collinearity.
Your diagnosis about -xtoverid- error message is correct. There's a quite old Stata thread on this topic, but I fear iis too cumbersome to aplly to your code:https://www.stata.com/statalist/arch...msg00833.html;
2) you may want to consider -xtgls- with a categorical variable as predictot for -i.county- (instead than a -i.time-). -xtgls- can accomodate both autocorrelation and heteroskedasticity with dedicated options.

[Alex Miller]

Carlo:

1) hmm...do you know another way to test for FE or RE including robust std. errors and time dummies? My approach with a Wald test on all time averages of all time-varying regressors results in RE (incl. cluster robust std. errors and time dummies) and in FE (only time dummies but no cluster robust std. errors).

2) I thought of using something like this

. xtgls BEVsalesshare2 $varying $invariant i.date, panels(hetero) corr(ar1)
note: 683.date omitted because of collinearity

Cross-sectional time-series FGLS regression

Coefficients:  generalized least squares
Panels:        heteroskedastic
Correlation:   common AR(1) coefficient for all panels  (0.6681)

Estimated covariances      =        18          Number of obs     =      1,296
Estimated autocorrelations =         1          Number of groups  =         18
Estimated coefficients     =        83          Time periods      =         72
                                                Wald chi2(82)     =    2595.61
                                                Prob > chi2       =     0.0000

------------------------------------------------------------------------------
BEVsalessh~2 |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       HOVkm |   .0015247   .0004277     3.56   0.000     .0006863    .0023631
       NTPKM |   .2252295   .1147146     1.96   0.050      .000393     .450066
       NFPKM |   .0000462   .0000205     2.26   0.024     6.04e-06    .0000863
   lCHRoadKm |   .0005126   .0002087     2.46   0.014     .0001035    .0009218
     DGPrice |    1.53958   .2188946     7.03   0.000     1.110554    1.968605
 EnergyPrice |  -6.07e-06   .0000239    -0.25   0.799    -.0000529    .0000408
  Arbeitslos |   .0153191   .0028753     5.33   0.000     .0096836    .0209547
        AVKT |   .0000184   5.38e-06     3.42   0.001     7.84e-06    .0000289
   Einkommen |   3.42e-07   9.81e-08     3.49   0.000     1.50e-07    5.35e-07
   KuestenKm |   1.68e-06   3.93e-07     4.28   0.000     9.10e-07    2.45e-06
    MuMShare |   .0004653   .0000908     5.13   0.000     .0002874    .0006432
        Temp |   .0063341   .0012288     5.15   0.000     .0039257    .0087426
             |
        date |
        613  |  -.3176895   .0412696    -7.70   0.000    -.3985765   -.2368026
        614  |  -.9848912   .1360999    -7.24   0.000    -1.251642   -.7181404
        615  |  -1.281226   .1768548    -7.24   0.000    -1.627855   -.9345966
        616  |  -1.058508   .1440269    -7.35   0.000    -1.340796   -.7762206
        617  |  -1.084492   .1469281    -7.38   0.000    -1.372465    -.796518
        618  |  -1.095114    .145945    -7.50   0.000    -1.381161   -.8090672
        619  |  -.8745847   .1150781    -7.60   0.000    -1.100134   -.6490358
        620  |   -.733017   .0963248    -7.61   0.000    -.9218102   -.5442238
        621  |  -.7348754    .092371    -7.96   0.000    -.9159194   -.5538315
        622  |  -.7554864   .1000637    -7.55   0.000    -.9516077   -.5593652
        623  |  -.7023329   .0909203    -7.72   0.000    -.8805335   -.5241324
        624  |  -1.395064   .1884671    -7.40   0.000    -1.764453   -1.025675
        625  |  -1.715536   .2391412    -7.17   0.000    -2.184244   -1.246828
        626  |  -1.972167   .2703103    -7.30   0.000    -2.501965   -1.442368
        627  |  -2.357983   .3243859    -7.27   0.000    -2.993768   -1.722198
        628  |  -1.609486   .2178175    -7.39   0.000    -2.036401   -1.182572
        629  |  -1.088155   .1437807    -7.57   0.000     -1.36996   -.8063504
        630  |  -.9962486   .1303327    -7.64   0.000    -1.251696   -.7408011
        631  |  -1.498822   .2009487    -7.46   0.000    -1.892674    -1.10497
        632  |  -1.942876   .2669882    -7.28   0.000    -2.466163   -1.419589
        633  |  -1.568519   .2129806    -7.36   0.000    -1.985954   -1.151085
        634  |  -1.016154   .1349004    -7.53   0.000    -1.280554   -.7517541
        635  |  -.8410136   .1109178    -7.58   0.000    -1.058408   -.6236187
        636  |   -1.09783   .1449468    -7.57   0.000     -1.38192   -.8137393
        637  |  -1.410609   .1904946    -7.40   0.000    -1.783972   -1.037246
        638  |  -1.318225   .1776237    -7.42   0.000    -1.666361   -.9700885
        639  |  -.9310263   .1238929    -7.51   0.000    -1.173852   -.6882006
        640  |  -1.072604   .1429091    -7.51   0.000      -1.3527   -.7925072
        641  |  -1.131314   .1508733    -7.50   0.000     -1.42702    -.835608
        642  |  -1.603682    .217259    -7.38   0.000    -2.029502   -1.177863
        643  |  -1.782499   .2461116    -7.24   0.000    -2.264869   -1.300129
        644  |   -1.88511    .262415    -7.18   0.000    -2.399434   -1.370786
        645  |  -1.689431   .2347787    -7.20   0.000    -2.149589   -1.229274
        646  |   -1.40595   .1980052    -7.10   0.000    -1.794033   -1.017867
        647  |  -1.479543   .2094076    -7.07   0.000    -1.889974   -1.069111
        648  |  -1.795469   .2494823    -7.20   0.000    -2.284446   -1.306493
        649  |  -1.678916   .2358069    -7.12   0.000    -2.141089   -1.216743
        650  |  -1.347091   .1961319    -6.87   0.000    -1.731502   -.9626793
        651  |   -1.23301   .1706759    -7.22   0.000    -1.567529   -.8984912
        652  |  -1.637812   .2278341    -7.19   0.000    -2.084359   -1.191265
        653  |  -1.454103   .2034762    -7.15   0.000    -1.852909   -1.055297
        654  |  -1.874713   .2645155    -7.09   0.000    -2.393154   -1.356272
        655  |  -1.726562   .2463829    -7.01   0.000    -2.209464    -1.24366
        656  |  -1.486773   .2083411    -7.14   0.000    -1.895114   -1.078432
        657  |   -1.02834   .1428743    -7.20   0.000    -1.308368   -.7483112
        658  |  -1.063779   .1479262    -7.19   0.000    -1.353709   -.7738493
        659  |  -.1442573   .0206613    -6.98   0.000    -.1847527    -.103762
        660  |   -.071215   .0150801    -4.72   0.000    -.1007715   -.0416585
        661  |   .3528022    .047719     7.39   0.000     .2592746    .4463298
        662  |  -.1609771   .0329406    -4.89   0.000    -.2255394   -.0964148
        663  |   -.628675   .0902109    -6.97   0.000    -.8054851   -.4518649
        664  |   .0811148   .0148261     5.47   0.000     .0520561    .1101734
        665  |   -.895318   .1297407    -6.90   0.000    -1.149605   -.6410309
        666  |  -1.079883   .1510515    -7.15   0.000    -1.375938   -.7838273
        667  |   .1550646   .0249173     6.22   0.000     .1062276    .2039016
        668  |   .3011359    .044328     6.79   0.000     .2142546    .3880172
        669  |    .450916   .0655806     6.88   0.000     .3223803    .5794516
        670  |   .4559774   .0636533     7.16   0.000     .3312193    .5807355
        671  |    .841219   .1195364     7.04   0.000      .606932    1.075506
        672  |   .9494684   .1359367     6.98   0.000     .6830374    1.215899
        673  |   1.726945   .2471776     6.99   0.000     1.242486    2.211405
        674  |   .6870768   .0950597     7.23   0.000     .5007633    .8733904
        675  |   .9034216   .1307941     6.91   0.000     .6470698    1.159773
        676  |   1.006063    .147997     6.80   0.000     .7159938    1.296131
        677  |   .1072453   .0185974     5.77   0.000      .070795    .1436957
        678  |   .0961742   .0226108     4.25   0.000     .0518579    .1404904
        679  |   .6577877   .0948192     6.94   0.000     .4719455    .8436299
        680  |   .8627259   .1204354     7.16   0.000     .6266768    1.098775
        681  |    1.02473   .1474458     6.95   0.000      .735741    1.313718
        682  |   .3921202   .0516955     7.59   0.000     .2907989    .4934414
        683  |          0  (omitted)
             |
       _cons |    -20.504   2.849867    -7.19   0.000    -26.08963   -14.91836
------------------------------------------------------------------------------

why do you recommend the i.county instead of the i.date? I mean under -xtreg- the i.county is already specified by setting the panel with -xtset-, is this different under -xtgls-?

[Carlo Lazzaro]

Alex:
1) No, unfortunately. The literature and related methodological aopproaches to deal with T>N panel datasets are probably less developed than the ones conceived for N>T panel datasets (as microeconometrics interests many more research fields, even far from economics).
2) your -xtgls- regression makes sense;
3) since in T>N panel datasets the time-series dimension is pretty long, you would be better off with replacing a categorical -i.date- with a continuos -c.date-, checking via a quadratic term whether a turning point is included in your data. By adding -i.county- you can estimate an individual effect related to you cross-sectional dimension.
For more details, I refer you to https://www.stata.com/bookstore/micr...metrics-stata/, pages 271-279.