Firm and Period Fixed Effects

[Bob Hernandez]

I have a longitudinal dataset of individuals nested in 10 firms. I am regressing (xtreg) an individual-level characteristic on various individual level predictors, using individual as the grouping variable. I'd also like to include dummies to account for firm and year effects. It strikes me that I could include one set of dummies for firm and another for year, or I could create a single set of dummies for firm*year (e.g., FirmA-2001, FirmA-2002, FirmA-2003, FirmB-2001, FirmB-2002, etc.). What is not clear to me is how the interpretation of the estimates would differ under these two approaches.

Hoping someone can provide some guidance.
Thanks.

[William Lisowski]

I first want to point out that, based on the notation you used in stating your problem, you are perhaps unaware of Stata's "factor variable" notation, that eliminates the need to create dummy variables. If so, see the output of help fvvarlist for more details, as well as the more detailed material in Chapter 11 on the Stata User's Guide.

With that said, in general if you want to include an interaction dummy like "FirmA in 2001" (which would be identical to the product of individual dummies for FirmA and 2001), you should also be including the individual dummies for FirmA and 2001. Otherwise you're ruling out a priori the possibilities that the year effects are independent of the firm effects, and vice versa. Wouldn't you prefer, if possible, so be able to make statements like "the effect of the year on [individual characteristic] was independent of (or was dependent on) the individual's firm"? Really, the issue of interactions for dummy variables is no different than interactions for any other variable.

So using Stata's factor variable notation the two alternatives you expressed would be models including among the independent variables

i.firm i.year

and

i.firm#i.year

while my suggestion is

i.firm##i.year

[Carlo Lazzaro]

Bob:
I second William's helpful insight and would add that things may differ if you use -fe- or -re- specification (a detail that is not reported in your post), because -fe- cancels out time-invariant variables:

. use "http://www.stata-press.com/data/r13/nlswork.dta", clear
(National Longitudinal Survey.  Young Women 14-26 years of age in 1968)

. xtreg ln_wage i.yea##i.race, fe allbasel
note: 2.race omitted because of collinearity
note: 3.race omitted because of collinearity

Fixed-effects (within) regression               Number of obs      =     28534
Group variable: idcode                          Number of groups   =      4711

R-sq:  within  = 0.1094                         Obs per group: min =         1
       between = 0.0420                                        avg =       6.1
       overall = 0.0439                                        max =        15

                                                F(42,23781)        =     69.56
corr(u_i, Xb)  = -0.0699                        Prob > F           =    0.0000

------------------------------------------------------------------------------
     ln_wage |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
        year |
         68  |          0  (base)
         69  |   .0770373    .014778     5.21   0.000     .0480714    .1060031
         70  |   .0666541   .0137877     4.83   0.000     .0396292    .0936789
         71  |   .1121696   .0137247     8.17   0.000     .0852683    .1390709
         72  |    .118821   .0140417     8.46   0.000     .0912984    .1463435
         73  |   .1331448   .0136827     9.73   0.000     .1063258    .1599638
         75  |   .1254407   .0136183     9.21   0.000     .0987479    .1521335
         77  |   .1825516   .0136557    13.37   0.000     .1557856    .2093176
         78  |   .2165712   .0139386    15.54   0.000     .1892508    .2438917
         80  |   .2304686   .0141335    16.31   0.000     .2027661    .2581711
         82  |   .2516737   .0137848    18.26   0.000     .2246546    .2786927
         83  |   .2790521   .0139634    19.98   0.000     .2516831    .3064212
         85  |   .3351751   .0137973    24.29   0.000     .3081316    .3622186
         87  |   .3543588   .0136673    25.93   0.000     .3275701    .3811474
         88  |   .4230864   .0136295    31.04   0.000     .3963716    .4498012
             |
        race |
      white  |          0  (base)
      black  |          0  (omitted)
      other  |          0  (omitted)
             |
   year#race |
   68#white  |          0  (base)
   68#black  |          0  (base)
   68#other  |          0  (base)
   69#white  |          0  (base)
   69#black  |   .0369524    .028591     1.29   0.196    -.0190878    .0929925
   69#other  |   .1723561   .1335697     1.29   0.197     -.089449    .4341613
   70#white  |          0  (base)
   70#black  |   .0294008   .0269925     1.09   0.276    -.0235062    .0823078
   70#other  |      .0145   .1325885     0.11   0.913    -.2453819    .2743819
   71#white  |          0  (base)
   71#black  |    .050927   .0265687     1.92   0.055    -.0011493    .1030032
   71#other  |   .1310062   .1357516     0.97   0.335    -.1350756     .397088
   72#white  |          0  (base)
   72#black  |   .0646022   .0275709     2.34   0.019     .0105614    .1186429
   72#other  |   .1239196   .1387271     0.89   0.372    -.1479943    .3958336
   73#white  |          0  (base)
   73#black  |   .0699154   .0266143     2.63   0.009     .0177497    .1220812
   73#other  |   .0488875    .137499     0.36   0.722    -.2206193    .3183942
   75#white  |          0  (base)
   75#black  |   .1442596   .0261804     5.51   0.000     .0929443    .1955749
   75#other  |    .100443   .1254399     0.80   0.423    -.1454272    .3463133
   77#white  |          0  (base)
   77#black  |   .1372045   .0262186     5.23   0.000     .0858143    .1885947
   77#other  |   .1450303    .123663     1.17   0.241     -.097357    .3874176
   78#white  |          0  (base)
   78#black  |   .1468236   .0267648     5.49   0.000     .0943628    .1992843
   78#other  |   .0618562   .1273841     0.49   0.627    -.1878248    .3115372
   80#white  |          0  (base)
   80#black  |   .1265801   .0270099     4.69   0.000      .073639    .1795211
   80#other  |   .0555611   .1321605     0.42   0.674    -.2034819    .3146041
   82#white  |          0  (base)
   82#black  |   .1263715   .0266211     4.75   0.000     .0741926    .1785505
   82#other  |   .0207876   .1241005     0.17   0.867    -.2224573    .2640325
   83#white  |          0  (base)
   83#black  |   .1160774   .0269742     4.30   0.000     .0632062    .1689486
   83#other  |   .0272039    .125974     0.22   0.829    -.2197132    .2741211
   85#white  |          0  (base)
   85#black  |   .0970325   .0268944     3.61   0.000     .0443179    .1497472
   85#other  |   .0519716   .1266758     0.41   0.682    -.1963211    .3002643
   87#white  |          0  (base)
   87#black  |   .0891829   .0269447     3.31   0.001     .0363696    .1419963
   87#other  |   .0523172   .1254108     0.42   0.677     -.193496    .2981304
   88#white  |          0  (base)
   88#black  |   .0514973   .0269219     1.91   0.056    -.0012712    .1042659
   88#other  |  -.0454668   .1269136    -0.36   0.720    -.2942256     .203292
             |
       _cons |   1.441443   .0092383   156.03   0.000     1.423335    1.459551
-------------+----------------------------------------------------------------
     sigma_u |  .41598047
     sigma_e |  .30252368
         rho |  .65406539   (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0:     F(4710, 23781) =     8.70         Prob > F = 0.0000

. xtreg ln_wage i.yea##i.race, re allbasel

Random-effects GLS regression                   Number of obs      =     28534
Group variable: idcode                          Number of groups   =      4711

R-sq:  within  = 0.1092                         Obs per group: min =         1
       between = 0.0947                                        avg =       6.1
       overall = 0.0904                                        max =        15

                                                Wald chi2(44)      =   3394.03
corr(u_i, X)   = 0 (assumed)                    Prob > chi2        =    0.0000

------------------------------------------------------------------------------
     ln_wage |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
        year |
         68  |          0  (base)
         69  |    .079035   .0145298     5.44   0.000     .0505572    .1075128
         70  |   .0670047   .0135118     4.96   0.000      .040522    .0934873
         71  |   .1109216   .0134088     8.27   0.000     .0846408    .1372025
         72  |   .1193255   .0137055     8.71   0.000     .0924631    .1461878
         73  |    .135243    .013334    10.14   0.000     .1091088    .1613772
         75  |   .1260171   .0132597     9.50   0.000     .1000285    .1520057
         77  |   .1899368   .0132857    14.30   0.000     .1638972    .2159764
         78  |   .2278853   .0135654    16.80   0.000     .2012976     .254473
         80  |   .2406971   .0137636    17.49   0.000      .213721    .2676733
         82  |   .2581406   .0134041    19.26   0.000      .231869    .2844122
         83  |     .28792   .0135665    21.22   0.000     .2613302    .3145099
         85  |   .3457167   .0133856    25.83   0.000     .3194814    .3719521
         87  |   .3621531   .0132497    27.33   0.000     .3361842    .3881221
         88  |   .4290616   .0131671    32.59   0.000     .4032545    .4548687
             |
        race |
      white  |          0  (base)
      black  |  -.1956089   .0235379    -8.31   0.000    -.2417424   -.1494755
      other  |  -.0260184   .1122429    -0.23   0.817    -.2460104    .1939736
             |
   year#race |
   68#white  |          0  (base)
   68#black  |          0  (base)
   68#other  |          0  (base)
   69#white  |          0  (base)
   69#black  |   .0237189   .0280892     0.84   0.398     -.031335    .0787727
   69#other  |    .191005   .1315988     1.45   0.147    -.0669239    .4489338
   70#white  |          0  (base)
   70#black  |   .0152205   .0263952     0.58   0.564    -.0365131    .0669541
   70#other  |   .0399471   .1290989     0.31   0.757    -.2130821    .2929764
   71#white  |          0  (base)
   71#black  |   .0348447    .025893     1.35   0.178    -.0159046     .085594
   71#other  |   .2016216   .1319195     1.53   0.126    -.0569359    .4601792
   72#white  |          0  (base)
   72#black  |   .0490708   .0268628     1.83   0.068    -.0035793    .1017209
   72#other  |   .1402147   .1329035     1.06   0.291    -.1202713    .4007007
   73#white  |          0  (base)
   73#black  |   .0486448    .025891     1.88   0.060    -.0021007    .0993902
   73#other  |   .0810307   .1328649     0.61   0.542    -.1793797    .3414411
   75#white  |          0  (base)
   75#black  |   .1240287    .025429     4.88   0.000     .0741889    .1738686
   75#other  |   .1483527   .1210206     1.23   0.220    -.0888433    .3855486
   77#white  |          0  (base)
   77#black  |   .1088644   .0254328     4.28   0.000      .059017    .1587118
   77#other  |   .1945567   .1191359     1.63   0.102    -.0389455    .4280588
   78#white  |          0  (base)
   78#black  |   .1157913   .0259916     4.45   0.000     .0648487    .1667339
   78#other  |   .1125353   .1225346     0.92   0.358    -.1276282    .3526988
   80#white  |          0  (base)
   80#black  |   .0995713   .0262352     3.80   0.000     .0481513    .1509913
   80#other  |   .1083772   .1273831     0.85   0.395    -.1412891    .3580436
   82#white  |          0  (base)
   82#black  |   .1020749   .0258178     3.95   0.000     .0514729     .152677
   82#other  |   .0849712   .1198371     0.71   0.478    -.1499052    .3198476
   83#white  |          0  (base)
   83#black  |   .0919558   .0261464     3.52   0.000     .0407098    .1432018
   83#other  |   .1370795   .1212876     1.13   0.258    -.1006398    .3747988
   85#white  |          0  (base)
   85#black  |   .0740004   .0260576     2.84   0.005     .0229285    .1250724
   85#other  |   .1420938   .1221972     1.16   0.245    -.0974083     .381596
   87#white  |          0  (base)
   87#black  |   .0718918   .0260857     2.76   0.006     .0207647    .1230189
   87#other  |   .1435271   .1209883     1.19   0.236    -.0936057    .3806599
   88#white  |          0  (base)
   88#black  |   .0311204   .0259647     1.20   0.231    -.0197694    .0820103
   88#other  |   .0452119   .1211986     0.37   0.709     -.192333    .2827568
             |
       _cons |   1.479397   .0120531   122.74   0.000     1.455774    1.503021
-------------+----------------------------------------------------------------
     sigma_u |  .36208147
     sigma_e |  .30252368
         rho |  .58889988   (fraction of variance due to u_i)
------------------------------------------------------------------------------

As an aside, your chances of getting (more) helpful replies are conditional on posting exactly what you typed and what Stata gave you back (as per FAQ). Thanks..

[Bob Hernandez]

William: I appreciate the explanation. However, suppose I'm less interested in whether firm and year depend on each other and instead want to remove the influence of unobserved firm attributes (constant within year) from estimates associated with my other predictors. Wouldn't this be achieved by including a dummy for firm*year (omitting the "main effects")? I notice that when I include a firm-level attribute (e.g., profit or firm size) that is constant within year, the predictor drops out of the estimation when I group by a firm*year variable (i.e., id):

xtreg ws profit size, fe i(id)
note: profit omitted because of collinearity
note: size omitted because of collinearity

Fixed-effects (within) regression               Number of obs     =      1,166
Group variable: id                                   Number of groups  =        206

R-sq:                                           Obs per group:
within  =      .                                            min =          1
between =      .                                         avg =        5.7
overall =      .                                            max =         12

F(0,960)          =       0.00
corr(u_i, Xb)  =      .                                 Prob > F          =          .


ws              Coef.            Std. Err.        t        P>t       [95% Conf. Interval]

profit           0  (omitted)
size            0  (omitted)
_cons          2.440566     .0737189    33.11   0.000     2.295897    2.585235

sigma_u   1.3310006
sigma_e   2.5172614
rho   .21849111   (fraction of variance due to u_i)

F test that all u_i=0: F(205, 960) = 1.19                    Prob > F = 0.0519

This suggests to me that a dummy that identifies firm and year is accounting for variance due to firm characteristics for a given year. However, when I include separate dummies for firm and year, the predictors do not drop out, indicating that these two separate dummies are not equivalent to a single firm*year dummy.

. xi:reg ws profit size i.firm i.year
i.firm             _Ifirm_1-30              (_Ifirm_1 for firm==WAL omitted)
i.year            _Iyear_2006-2012    (naturally coded; _Iyear_2006 omitted)

Source        SS           df                                 MS      Number of obs   =     1,166
                                                                     F(37, 1128)     =      4.41
Model   964.298957        37   26.062134           Prob > F        =    0.0000
Residual   6660.65227     1,128  5.90483357     R-squared       =    0.1265
                                                                     Adj R-squared   =    0.0978
Total   7624.95123     1,165  6.54502251           Root MSE        =      2.43


ws                      Coef.   Std. Err.      t    P>t     [95% Conf. Interval]

profit         .0434737   .0079726     5.45   0.000     .0278309    .0591166
size         -.097757   .1664725    -0.59   0.557    -.4243875    .2288736
_Ifirm_2   -.4143314   .5902837    -0.70   0.483    -1.572509    .7438461
_Ifirm_3   -.5363976   .5825401    -0.92   0.357    -1.679382    .6065864
...

Assume I had an additional X in the models above that varied within firmyear. I am confused about how the interpretation of the estimate would differ across models. My best guess: In Model 1, the X effect would be net of the influence of stable factors within firmyear, but not of those factors stable within firm and within year. On the other hand, effects estimated in Model 2 would be net of stable firm factors and stable year factors but not of stable firmyear factors.