Interpreting coefficients on time dummies (time fixed effects)

[lal mohan kumar]

Dear Stata Members
I have some questions regarding interpreting the coefficient on time dummies in a panel fixed effects estimation. Here are my results

 xtreg cash size debt i.year i.id, vce(r)

Random-effects GLS regression                   Number of obs     =        150
Group variable: id                              Number of groups  =          8

R-squared:                                      Obs per group:
     Within  = 0.3165                                         min =          5
     Between = 1.0000                                         avg =       18.8
     Overall = 0.5225                                         max =         21

                                                Wald chi2(7)      =          .
corr(u_i, X) = 0 (assumed)                      Prob > chi2       =          .

                                     (Std. err. adjusted for 8 clusters in id)
------------------------------------------------------------------------------
             |               Robust
        cash | Coefficient  std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
        size |  -.0386644   .0082399    -4.69   0.000    -.0548144   -.0225144
        debt |  -.1087089   .0971271    -1.12   0.263    -.2990745    .0816566
             |
        year |
       2000  |   .0108519   .0197071     0.55   0.582    -.0277733     .049477
       2001  |   .0343667   .0421997     0.81   0.415    -.0483433    .1170766
       2002  |   .0412632   .0382493     1.08   0.281    -.0337041    .1162304
       2003  |   .0136864   .0407234     0.34   0.737      -.06613    .0935028
       2004  |   .0454415    .029979     1.52   0.130    -.0133162    .1041992
       2005  |   .0235091   .0345105     0.68   0.496    -.0441301    .0911484
       2006  |   .0862778   .0374817     2.30   0.021      .012815    .1597405
       2007  |   .1512378   .0536267     2.82   0.005     .0461314    .2563443
       2008  |   .1146357   .0589621     1.94   0.052    -.0009279    .2301992
       2009  |   .1181824   .0465542     2.54   0.011     .0269379     .209427
       2010  |   .1209712   .0663762     1.82   0.068    -.0091237    .2510661
       2011  |   .1243997   .0562073     2.21   0.027     .0142355     .234564
       2012  |   .1286815   .0520854     2.47   0.013     .0265961     .230767
       2013  |   .1359156   .0556108     2.44   0.015     .0269205    .2449108
       2014  |   .0829417    .046106     1.80   0.072    -.0074243    .1733078
       2015  |   .1332168   .0681599     1.95   0.051     -.000374    .2668077
       2016  |   .1684112   .0563771     2.99   0.003     .0579142    .2789082
       2017  |   .1386612   .0472866     2.93   0.003     .0459812    .2313413
       2018  |   .0949487   .0530237     1.79   0.073    -.0089758    .1988731
       2019  |   .1144136   .0562673     2.03   0.042     .0041318    .2246954
             |
          id |
  000004.SZ  |  -.0923897   .0445574    -2.07   0.038    -.1797206   -.0050588
  000005.SZ  |   -.220671   .0224711    -9.82   0.000    -.2647135   -.1766286
  000006.SZ  |  -.0329669   .0113797    -2.90   0.004    -.0552707   -.0106631
  000007.SZ  |  -.1107981    .025728    -4.31   0.000    -.1612241   -.0603721
  000008.SZ  |  -.0935211   .0428801    -2.18   0.029    -.1775647   -.0094776
  000009.SZ  |  -.0278975   .0135077    -2.07   0.039    -.0543722   -.0014228
  000010.SZ  |  -.0649462   .0193648    -3.35   0.001    -.1029005   -.0269919
             |
       _cons |   .3961387   .0733381     5.40   0.000     .2523986    .5398787
-------------+----------------------------------------------------------------
     sigma_u |          0
     sigma_e |   .0907486
         rho |          0   (fraction of variance due to u_i)

I didnt use xtreg , fe specification as I want to get the coefficients of units (id) and year. Now question is
1) The coefficient on year 2000 is .0108519, what does this imply. Similarly for 000004.SZ which has a coefficient of -.0923897. How do we interpret this coefficients in the commonly interpreted way, 1 unit change.......
2) How to get this coefficients by hand? Is there a formula for this

Trust I made my question clear. If anyone can give me some intutution, that shall be great!

[Carlo Lazzaro]

Ial:
to get waht you've in mind you should type:

reg cash size debt i.year i.id, vce(cluster idcode)

That said, I would not sponsor what above, as it takes forever (unless you actually have 7 -id-).

Your code is -xtreg,re-: hence, there's no way that your trick can mimick the -fe- specification.

The following toy-example converts word into numbers:

. use "https://www.stata-press.com/data/r17/nlswork.dta"
(National Longitudinal Survey of Young Women, 14-24 years old in 1968)

. xtreg ln_wage c.age##c.age i.year if idcode<=3, fe vce(cluster idcode)

Fixed-effects (within) regression               Number of obs     =         39
Group variable: idcode                          Number of groups  =          3

R-squared:                                      Obs per group:
     Within  = 0.7404                                         min =         12
     Between = 0.4068                                         avg =       13.0
     Overall = 0.4014                                         max =         15

                                                F(4,2)            =          .
corr(u_i, Xb) = -0.8560                         Prob > F          =          .

                                 (Std. err. adjusted for 3 clusters in idcode)
------------------------------------------------------------------------------
             |               Robust
     ln_wage | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
         age |   .0773019   .0101936     7.58   0.017     .0334424    .1211613
             |
 c.age#c.age |  -.0045583   .0021586    -2.11   0.169    -.0138461    .0047294
             |
        year |
         69  |   .3367906   .0871839     3.86   0.061    -.0383313    .7119126
         70  |   .2089384   .2733588     0.76   0.525    -.9672295    1.385106
         71  |   .3144116   .1543689     2.04   0.179    -.3497843    .9786076
         72  |   .5888124   .4728115     1.25   0.339    -1.445531    2.623156
         73  |   .8912873   .4976548     1.79   0.215    -1.249948    3.032523
         75  |   1.246958   .5791178     2.15   0.164    -1.244785    3.738701
         77  |   1.560689   .8225333     1.90   0.198    -1.978387    5.099764
         78  |   1.941522   1.218922     1.59   0.252    -3.303077    7.186121
         80  |    2.34498   1.454951     1.61   0.248    -3.915167    8.605128
         82  |   2.698954   1.585626     1.70   0.231    -4.123442     9.52135
         83  |   2.994437   1.730077     1.73   0.226    -4.449484    10.43836
         85  |   3.538578   2.107946     1.68   0.235    -5.531183    12.60834
         87  |   3.965153      2.346     1.69   0.233     -6.12887    14.05918
         88  |    4.40786   2.563793     1.72   0.228    -6.623251    15.43897
             |
       _cons |   1.465543   .3990418     3.67   0.067    -.2513952    3.182481
-------------+----------------------------------------------------------------
     sigma_u |  .54258328
     sigma_e |  .21942548
         rho |  .85944136   (fraction of variance due to u_i)
------------------------------------------------------------------------------

. reg ln_wage c.age##c.age i.year i.idcode if idcode<=3,  vce(cluster idcode)

Linear regression                               Number of obs     =         39
                                                F(2, 2)           =          .
                                                Prob > F          =          .
                                                R-squared         =     0.8139
                                                Root MSE          =     .21943

                                 (Std. err. adjusted for 3 clusters in idcode)
------------------------------------------------------------------------------
             |               Robust
     ln_wage | Coefficient  std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
         age |   .0773019   .0106911     7.23   0.019     .0313017    .1233021
             |
 c.age#c.age |  -.0045583    .002264    -2.01   0.182    -.0142995    .0051828
             |
        year |
         69  |   .3367906   .0914392     3.68   0.066    -.0566405    .7302218
         70  |   .2089384   .2867011     0.73   0.542    -1.024637    1.442514
         71  |   .3144116   .1619035     1.94   0.192     -.382203    1.011026
         72  |   .5888124   .4958888     1.19   0.357    -1.544825     2.72245
         73  |   .8912873   .5219448     1.71   0.230     -1.35446    3.137034
         75  |   1.246958   .6073839     2.05   0.176    -1.366404     3.86032
         77  |   1.560689   .8626802     1.81   0.212    -2.151125    5.272502
         78  |   1.941522   1.278416     1.52   0.268    -3.559059    7.442103
         80  |    2.34498   1.525965     1.54   0.264    -4.220718    8.910678
         82  |   2.698954   1.663018     1.62   0.246    -4.456435    9.854344
         83  |   2.994437    1.81452     1.65   0.241    -4.812813    10.80169
         85  |   3.538578   2.210833     1.60   0.251    -5.973868    13.05102
         87  |   3.965153   2.460506     1.61   0.248    -6.621548    14.55185
         88  |    4.40786   2.688929     1.64   0.243    -7.161667    15.97739
             |
      idcode |
          2  |  -.4183815   .0165036   -25.35   0.002    -.4893909   -.3473721
          3  |   .6579353   .7215294     0.91   0.458    -2.446555    3.762426
             |
       _cons |   1.341224   .1489003     9.01   0.012     .7005575     1.98189
------------------------------------------------------------------------------

.

[lal mohan kumar]

Thanks Carlo Lazzaro for explaining the correct specifications. For interpretation purposes, I shall go with your reg model since I need to interpret the coefficients of idcodes. In your example idcode 2 has a coefficient of -.4183815 while 3 has .6579353 , what does this imply? In other words in layman's terms, what does the coefficient on units in a fe specifcation imply.

[Carlo Lazzaro]

Ial:
taking .6579353 coefficient as an example, we can say that, moving from -_cons- (that represents -id1-) to -id3- increases the regressand (other things being equal) by:

. di exp(.6579353)-1
.93080169

or 93.80%.

[lal mohan kumar]

Oh Great
So the constant/intercept represents the base id (id1), right in this setup? What is the intuition of using exponents of the coefficient -1? is that the economic significance? In general can we use it for any such coeffcients?
Sorry if I am being so obtuse person!

[Carlo Lazzaro]

Ial:
1) your intuition about -id1-=_cons is correct;
2) the -minus 1- correction is used because the -exp()- back-transformation does not work well for medium/big values of the coefficients in log-linear model. Being a good habit, it is applied even when the value of the coefficient is small
In addition, I've to correct myself as far my previous reply is concerned (I messed up brackets):
taking .6579353 coefficient as an example, we can say that, moving from -_cons- (that represents -id1-) to -id3- increases the regressand (other things being equal) by:

. di exp((.6579353)-1)
.71030225

or 71.03%.

[lal mohan kumar]

Great help Carlo Lazzaro for teaching me this important stuff

[Carlo Lazzaro]

Ial:
teaching is flattering but undeserved...let's say that I'm still learning with you!
Thanks anyway.