Panel data analysis with entity and time fixed effects and with small N, large T

[Jacob Neil]

Hello everybody,

I'm trying to analyze the effect of various labour market statistics (eg. unemployment rate, employment-to-population ratio, manufacturing share of employment, etc.) individually on the monthly per-capita rate of drug abuse incidents in 11 regions across 12 years (N=11, T=144). That is, I want to compare how changes in each statistic affect the rate of incidents in that region in that month by testing this relationship one labour market statistics at a time, and then perhaps with several statistics at a time that are less likely to be be collinear (eg. unemployment rate and manufacturing employment share). I want to use both region and time fixed effects to account for confounding factors that vary across regions but not over time, as well as factors that vary over time but not across regions.

So far I have been using the following regression:

(first used: xtset region yearmonth, monthly)

xtreg ratedeath labmarketvar i.yearmonth, fe robust

However, I have not had the results I expected (ie. the p-values are all extremely high). Is this the right way to go about this panel data analysis? I worry my unexpected results (ie. finding no effect) have to do with the small N, large T characteristics of my dataset given that panel data usually deals with large N, small T datasets.

Separately, I'm wondering whether it is possible to use 3, 6, or 12 month moving averages for my independent and dependent variables in this regression (hoping this would help find better results)--or would using moving averages damage the validity of this regression (eg. because of the autocorrelation inherent in moving averages)?

It is also tedious to scroll through all 143 dummy time variables while viewing the results, so if there is any way to suppress those indicator variables from being reported in the regression results that would be very much appreciated.

Thank you in advance for the help!

[Carlo Lazzaro]

Jacb:
welcome to the list.
you did not post what you got from Stata; that makes difficult (for me, at any rate) to advise positively.
Anyway, you're seemingly dealing with a small N, a bit larger T panel dataset.
I would try -xtgls-, instead.
Admittedly, I do not folow your second question concerning moving average.

[German Rodriguez]

Specifically on the question of suppressing the coefficients of 143 dummy variables, you may want to look at Sergio Correia's reghdfe available from SSC, which has the added advantage of being very fast.

[Jacob Neil]

Thank you Carlo and German for your helpful replies. My apologies for not posting my Stata results originally, I have included some below (sorry I could not figure how to copy and paste them cleaner).

German, I will check out -reghdfe-.

Carlo, I have tried -xtgls- and obtained much better results. You can ignore my question about moving averages for now. But a few questions about -xtgls-

1) Can I continue to use entity and time fixed effects with a -xtgls- panel data regression, as below?

xtgls rateed unemprate i.ymtime i.region

Cross-sectional time-series FGLS regression

Coefficients:  generalized least squares
Panels:        homoskedastic
Correlation:   no autocorrelation

Estimated covariances      =         1          Number of obs     =      1,584
Estimated autocorrelations =         0         Number of groups  =         11
Estimated coefficients     =       155          Time periods      =        144
                                                                 Wald chi2(154)    =    1877.88
Log likelihood             = -1754.705           Prob > chi2       =     0.0000

---------------------------------------------------------------------------------------------------
             rateed |      Coef.         Std. Err.      z      P>|z|     [95% Conf. Interval]
----------------------------------+----------------------------------------------------------------
      unemprate |  -4.104528   2.250921    -1.82   0.068    -8.516253    .3071963

(NB I have not included the results for the many region and time fixed effect dummy variables)

2) Is having a small N, larger T panel dataset sufficient cause for using -xtgls- (rather than -xtreg-) or must I also test for autocorrelation (if so, how should I test)?


By the way, these are the Stata results which prompted my question in my original post:

I obtained good results from this regression with -xtreg- :

xtreg rateed empratio, fe robust

Fixed-effects (within) regression               Number of obs     =      1,584
Group variable: region                              Number of groups  =         11

R-sq:                                           Obs per group:
     within  = 0.0446                                         min =        144
     between = 0.4510                                         avg =      144.0
     overall = 0.1840                                         max =        144

                                                              F(1,10)           =      18.96
corr(u_i, Xb)  = 0.2241                         Prob > F          =     0.0014

                                    (Std. Err. adjusted for 11 clusters in er)
------------------------------------------------------------------------------
                 |               Robust
      rateed |      Coef.   Std. Err.                t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
   empratio |  -8.712886      2.001      -4.35   0.001    -13.17139   -4.254379
       _cons |   7.587658   1.202233     6.31   0.000     4.908915     10.2664
-------------+----------------------------------------------------------------
     sigma_u |  .51794828
     sigma_e |   .8549781
         rho |  .26846963   (fraction of variance due to u_i)
------------------------------------------------------------------------------

But when I included time fixed effects (ymtime) the results were not good--for example, a p-value of 0.939 and much smaller coefficient than above:

 xtreg rateed empratio i.yearmonth, fe robust

Fixed-effects (within) regression               Number of obs     =      1,584
Group variable: region                              Number of groups  =         11

R-sq:                                           Obs per group:
     within  = 0.2917                                         min =        144
     between = 0.4510                                         avg =      144.0
     overall = 0.1944                                         max =        144

                                                             F(10,10)          =          .
corr(u_i, Xb)  = 0.0108                         Prob > F          =          .

                                    (Std. Err. adjusted for 11 clusters in er)
------------------------------------------------------------------------------
                 |               Robust
      rateed |          Coef.   Std. Err.           t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
  empratio |  -.2194941   2.782655    -0.08   0.939    -6.419636    5.980647

(NB Again I have not included the results for the many time fixed effect dummy variables)

I figured that perhaps the model was over-specified when including both time and region fixed effects, and/or perhaps it was an issue with using -xtreg- with a small N, larger T dataset. The results obtained using -xtgls- are much better, if that is a valid approach in my case.

Thank you again for all the help and please let me know if there is any other information I should include.

Best regards,

Jacob

[Ariel Karlinsky]

With the latest model you show (FE with year FE), what is the output produced by:
testparm i.year?

[Carlo Lazzaro]

Jacob:
a1) you shoud investigate both heteroskedasticity and autocorrelation in your data and model the latter accordingly (see -xtgls- options);
a2) when T>N, -xtgls- is the way to go; see a1) about the need of investigating autocorrelation and specifying your model accordingly.

As an aside, please take a look at -help j_robustsingular- for explanations about missing F-test in your last regression model.

[Jacob Neil]

Thank you Ariel and Carlo again for the helpful answers.

Ariel: Assuming by "i.year" you meant my time indicator variable, i.ymtime:

. testparm i.ymtime

 ( 1)  541.ymtime = 0
 ( 2)  542.ymtime = 0
 ( 3)  543.ymtime = 0
 ( 4)  544.ymtime = 0
 ( 5)  545.ymtime = 0
 ( 6)  546.ymtime = 0
 ( 7)  547.ymtime = 0
 ( 8)  548.ymtime = 0
 ( 9)  549.ymtime = 0
 (10)  550.ymtime = 0
 (11)  551.ymtime = 0
 (12)  552.ymtime = 0
 (13)  553.ymtime = 0
 (14)  554.ymtime = 0
 (15)  555.ymtime = 0
 (16)  556.ymtime = 0
 (17)  557.ymtime = 0
 (18)  558.ymtime = 0
 (19)  559.ymtime = 0
 (20)  560.ymtime = 0
 (21)  561.ymtime = 0
 (22)  562.ymtime = 0
 (23)  563.ymtime = 0
 (24)  564.ymtime = 0
 (25)  565.ymtime = 0
 (26)  566.ymtime = 0
 (27)  567.ymtime = 0
 (28)  568.ymtime = 0
 (29)  569.ymtime = 0
 (30)  570.ymtime = 0
 (31)  571.ymtime = 0
 (32)  572.ymtime = 0
 (33)  573.ymtime = 0
 (34)  574.ymtime = 0
 (35)  575.ymtime = 0
 (36)  576.ymtime = 0
 (37)  577.ymtime = 0
 (38)  578.ymtime = 0
 (39)  579.ymtime = 0
 (40)  580.ymtime = 0
 (41)  581.ymtime = 0
 (42)  582.ymtime = 0
 (43)  583.ymtime = 0
 (44)  584.ymtime = 0
 (45)  585.ymtime = 0
 (46)  586.ymtime = 0
 (47)  587.ymtime = 0
 (48)  588.ymtime = 0
 (49)  589.ymtime = 0
 (50)  590.ymtime = 0
 (51)  591.ymtime = 0
 (52)  592.ymtime = 0
 (53)  593.ymtime = 0
 (54)  594.ymtime = 0
 (55)  595.ymtime = 0
 (56)  596.ymtime = 0
 (57)  597.ymtime = 0
 (58)  598.ymtime = 0
 (59)  599.ymtime = 0
 (60)  600.ymtime = 0
 (61)  601.ymtime = 0
 (62)  602.ymtime = 0
 (63)  603.ymtime = 0
 (64)  604.ymtime = 0
 (65)  605.ymtime = 0
 (66)  606.ymtime = 0
 (67)  607.ymtime = 0
 (68)  608.ymtime = 0
 (69)  609.ymtime = 0
 (70)  610.ymtime = 0
 (71)  611.ymtime = 0
 (72)  612.ymtime = 0
 (73)  613.ymtime = 0
 (74)  614.ymtime = 0
 (75)  615.ymtime = 0
 (76)  616.ymtime = 0
 (77)  617.ymtime = 0
 (78)  618.ymtime = 0
 (79)  619.ymtime = 0
 (80)  620.ymtime = 0
 (81)  621.ymtime = 0
 (82)  622.ymtime = 0
 (83)  623.ymtime = 0
 (84)  624.ymtime = 0
 (85)  625.ymtime = 0
 (86)  626.ymtime = 0
 (87)  627.ymtime = 0
 (88)  628.ymtime = 0
 (89)  629.ymtime = 0
 (90)  630.ymtime = 0
 (91)  631.ymtime = 0
 (92)  632.ymtime = 0
 (93)  633.ymtime = 0
 (94)  634.ymtime = 0
 (95)  635.ymtime = 0
 (96)  636.ymtime = 0
 (97)  637.ymtime = 0
 (98)  638.ymtime = 0
 (99)  639.ymtime = 0
 (100)  640.ymtime = 0
 (101)  641.ymtime = 0
 (102)  642.ymtime = 0
 (103)  643.ymtime = 0
 (104)  644.ymtime = 0
 (105)  645.ymtime = 0
 (106)  646.ymtime = 0
 (107)  647.ymtime = 0
 (108)  648.ymtime = 0
 (109)  649.ymtime = 0
 (110)  650.ymtime = 0
 (111)  651.ymtime = 0
 (112)  652.ymtime = 0
 (113)  653.ymtime = 0
 (114)  654.ymtime = 0
 (115)  655.ymtime = 0
 (116)  656.ymtime = 0
 (117)  657.ymtime = 0
 (118)  658.ymtime = 0
 (119)  659.ymtime = 0
 (120)  660.ymtime = 0
 (121)  661.ymtime = 0
 (122)  662.ymtime = 0
 (123)  663.ymtime = 0
 (124)  664.ymtime = 0
 (125)  665.ymtime = 0
 (126)  666.ymtime = 0
 (127)  667.ymtime = 0
 (128)  668.ymtime = 0
 (129)  669.ymtime = 0
 (130)  670.ymtime = 0
 (131)  671.ymtime = 0
 (132)  672.ymtime = 0
 (133)  673.ymtime = 0
 (134)  674.ymtime = 0
 (135)  675.ymtime = 0
 (136)  676.ymtime = 0
 (137)  677.ymtime = 0
 (138)  678.ymtime = 0
 (139)  679.ymtime = 0
 (140)  680.ymtime = 0
 (141)  681.ymtime = 0
 (142)  682.ymtime = 0
 (143)  683.ymtime = 0
       Constraint 1 dropped
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           chi2( 10) =   95.57
         Prob > chi2 =    0.0000

I'm not sure what to conclude from this test...

Carlo: Thank you, I will test for heteroskedasticity and autocorrelation and adjust the model accordingly for the latter. The instructions here seem like a good process: https://www.stata.com/support/faqs/s...tocorrelation/

Thank you also for the advice that if T>N I should use -xtgls- rather than -xtreg-. Is it possible for you to provide me with a a reference that I can cite with this information?

[Carlo Lazzaro]

Jacob:
you may want to take a look at: https://www.stata.com/bookstore/micr...metrics-stata/ (pages 273-276).

[Jacob Neil]

Thank you Carlo for that reference, and for all the other help as well. Much appreciated!