Panel fixed effects: How to avoid Dummy Variable Trap with time fixed effects

[Roman Goossens]

Hi,

I have a question about using time fixed effects in a panel data setting and avoiding dummy variable trap.

I will explain my problem to make the question clearer:

I want to determine what variables drive the speed that a fund manager invests through time. My dependent variable is a time series of "capital that a fund has invested" for every fund. This is at a quarterly frequency and from 2000 to 2017.

Note however that each fund level data time series does not span the whole period. As the funds opened at different points in time (e.g. some were launched in 2000 others in 2001, 2002, etc) the data starts at different points in time.

My concern is twofold: I want to include an explanatory variable which varies through time but is the same for all funds. For example the yearly return of the US stock market index (S&P500) as shown in the model below.

CapInvested_it = a_i + b USequityReturn_t + g timeEffects_t + e_{i t}

As the time fixed effects will be collinear with the US equity returns one of the time effects needs to be dropped when estimating the model with xtreg (see Stata output) (q. Does that mean that I can never use time fixed effects when using this type of explanatory variable (which does not vary across individual but only across time)? Is there any way around this?

. xtreg marg_called spx_yoy i.quarter , fe robust
note: 176.quarter omitted because of collinearity

Fixed-effects (within) regression               Number of obs      =     99076
Group variable: fundid_fun~e                    Number of groups   =      3663

R-sq:  within  = 0.0628                         Obs per group: min =         1
       between = 0.0654                                        avg =      27.0
       overall = 0.0149                                        max =        68

                                                F(67,3662)         =     24.30
corr(u_i, Xb)  = -0.2667                        Prob > F           =    0.0000

                     (Std. Err. adjusted for 3663 clusters in fundid_fundtype)
------------------------------------------------------------------------------
             |               Robust
 marg_called |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
     spx_yoy |   .0021898   .0047615     0.46   0.646    -.0071457    .0115252
             |
     quarter |
        165  |   .3634196   .2077487     1.75   0.080     -.043895    .7707343
        166  |   .0673388   .2092053     0.32   0.748    -.3428316    .4775092
        167  |  -.0017774   .2124482    -0.01   0.993    -.4183059    .4147511
        168  |  -.2032346   .1689662    -1.20   0.229    -.5345118    .1280426
        169  |  -.0529558   .1859052    -0.28   0.776    -.4174438    .3115321
        170  |  -.4048783   .2090664    -1.94   0.053    -.8147764    .0050198
        171  |   .0751246   .2536233     0.30   0.767    -.4221323    .5723815
        172  |  -.2271042   .2259984    -1.00   0.315    -.6701993    .2159909
        173  |  -.1562853   .1924783    -0.81   0.417    -.5336605    .2210899
        174  |  -.1072304   .1721214    -0.62   0.533    -.4446936    .2302329
        175  |  -.1459967   .1656886    -0.88   0.378    -.4708477    .1788544
        176  |          0  (omitted)
        177  |   .5829734   .2048757     2.85   0.004     .1812915    .9846552
        178  |   .2298571   .1871147     1.23   0.219    -.1370022    .5967164
        179  |   .2485688   .1906258     1.30   0.192    -.1251745    .6223121
        180  |   .3328267   .1797579     1.85   0.064    -.0196087    .6852622
        181  |   .0123119   .1684457     0.07   0.942    -.3179447    .3425685
        182  |   .0829718   .1741036     0.48   0.634    -.2583779    .4243215
        183  |   .3220636   .1859411     1.73   0.083    -.0424949     .686622
        184  |   .0987804   .1728828     0.57   0.568    -.2401757    .4377365
        185  |  -.0045399   .1704852    -0.03   0.979    -.3387952    .3297155
        186  |  -.1576511   .1608511    -0.98   0.327    -.4730176    .1577155
        187  |   .2527712   .1834247     1.38   0.168    -.1068535     .612396
        188  |  -.1552237   .1745983    -0.89   0.374    -.4975433    .1870959
        189  |  -.1698568   .1711432    -0.99   0.321    -.5054021    .1656886
        190  |  -.4140906    .173447    -2.39   0.017    -.7541529   -.0740284
        191  |   .0017222   .1907655     0.01   0.993     -.372295    .3757394
        192  |   -.801054   .1569176    -5.10   0.000    -1.108709   -.4933994
        193  |  -.8747737   .1609558    -5.43   0.000    -1.190346   -.5592018
        194  |  -.7400855   .1893973    -3.91   0.000     -1.11142   -.3687509
        195  |  -1.054512   .2637383    -4.00   0.000    -1.571601   -.5374239
        196  |  -1.379948   .2633164    -5.24   0.000    -1.896209   -.8636866
        197  |  -1.336222   .2448343    -5.46   0.000    -1.816247   -.8561972
        198  |  -1.460848    .185791    -7.86   0.000    -1.825112   -1.096583
        199  |  -1.386937   .1303834   -10.64   0.000    -1.642568   -1.131305
        200  |  -1.539267   .1863003    -8.26   0.000     -1.90453   -1.174004
        201  |  -1.267155   .1558174    -8.13   0.000    -1.572653    -.961658
        202  |  -1.355659   .1289233   -10.52   0.000    -1.608427    -1.10289
        203  |  -1.132549   .1315177    -8.61   0.000    -1.390405   -.8746941
        204  |  -1.508157   .1288626   -11.70   0.000    -1.760806   -1.255507
        205  |  -1.501627   .1281399   -11.72   0.000     -1.75286   -1.250395
        206  |  -1.429224   .1284764   -11.12   0.000    -1.681116   -1.177331
        207  |  -1.446775   .1311461   -11.03   0.000    -1.703901   -1.189648
        208  |  -1.566608   .1274366   -12.29   0.000    -1.816462   -1.316754
        209  |  -1.703131   .1240032   -13.73   0.000    -1.946253   -1.460009
        210  |  -1.760753   .1246472   -14.13   0.000    -2.005137   -1.516368
        211  |  -1.519571   .1308827   -11.61   0.000    -1.776182   -1.262961
        212  |  -1.910804   .1221832   -15.64   0.000    -2.150357    -1.67125
        213  |  -1.930304    .132662   -14.55   0.000    -2.190403   -1.670205
        214  |  -1.916775   .1334156   -14.37   0.000    -2.178351   -1.655199
        215  |   -1.92109   .1434959   -13.39   0.000     -2.20243   -1.639751
        216  |  -2.080455   .1322585   -15.73   0.000    -2.339762   -1.821147
        217  |  -2.005137   .1304181   -15.37   0.000    -2.260836   -1.749437
        218  |  -2.052398   .1303389   -15.75   0.000    -2.307942   -1.796854
        219  |  -2.019814   .1253039   -16.12   0.000    -2.265486   -1.774142
        220  |  -2.149828   .1213671   -17.71   0.000    -2.387782   -1.911875
        221  |  -2.116326   .1211521   -17.47   0.000    -2.353858   -1.878794
        222  |  -2.137921   .1286837   -16.61   0.000     -2.39022   -1.885622
        223  |  -2.177748   .1253641   -17.37   0.000    -2.423538   -1.931957
        224  |   -2.35278   .1387592   -16.96   0.000    -2.624833   -2.080727
        225  |  -2.347668   .1304164   -18.00   0.000    -2.603364   -2.091972
        226  |  -2.373192   .1225429   -19.37   0.000    -2.613451   -2.132933
        227  |  -2.389172    .120666   -19.80   0.000    -2.625751   -2.152593
        228  |  -2.611655   .1301455   -20.07   0.000     -2.86682    -2.35649
        229  |  -2.409805   .1264132   -19.06   0.000    -2.657652   -2.161958
        230  |  -2.401352     .12628   -19.02   0.000    -2.648938   -2.153766
        231  |  -2.476485   .1302434   -19.01   0.000    -2.731842   -2.221129
             |
       _cons |   2.660913   .1150825    23.12   0.000      2.43528    2.886545
-------------+----------------------------------------------------------------
     sigma_u |  2.6434856
     sigma_e |  2.7854585
         rho |  .47386677   (fraction of variance due to u_i)
------------------------------------------------------------------------------

Secondly, how can I evaluate how many regressors I can use (degrees of freedom) in panel data settings before I risk overfitting the data? I am concerned that the inclusion of time fixed effects will increase the number of regressors so much that the regression results will not unreliable. See for example comment #10 in the thread below.

https://www.statalist.org/forums/for...nexpected-sign

Thanks

[Carlo Lazzaro]

Roman:
welcome to this forum.
As far as -timevar- is concerned, it would possibly be more interesting to search for turning point, considering -quarter as continuos instead of categorical (by the way, a more parsimonius model):

xtreg marg_called spx_yoy c.quarter##.quarter , fe robust

Besides, is there any reason why you clustered SEs (autocorrelation?). Using default SEs with -fe- should allow Stata to report an F-test as a footnote of the outcome table: if it reaches statistical significance, you can go -xtreg-, otherwise you would be better off with pooled OLS.

[Roman Goossens]

Hi Carlo,

Many thanks for your prompt reply.

Just to make sure, I assume that you mean the following specification (there was a "c" missing in front of the second "quarter"). This gives me the following output (and I got rid of the clustering to show the F-test results):

 
 . xtreg marg_called spx_yoy c.quarter##c.quarter , fe  Fixed-effects (within) regression               Number of obs      =     99076 Group variable: fundid_fun~e                    Number of groups   =      3663  R-sq:  within  = 0.0567                         Obs per group: min =         1        between = 0.0847                                        avg =      27.0        overall = 0.0108                                        max =        68                                                  F(3,95410)         =   1911.73 corr(u_i, Xb)  = -0.2775                        Prob > F           =    0.0000  -------------------------------------------------------------------------------------         marg_called |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval] --------------------+----------------------------------------------------------------             spx_yoy |   .0066692   .0006482    10.29   0.000     .0053988    .0079396             quarter |  -.0648124   .0126339    -5.13   0.000    -.0895747   -.0400502                     | c.quarter#c.quarter |   .0000405   .0000311     1.30   0.194    -.0000205    .0001015                     |               _cons |   12.80593   1.275634    10.04   0.000      10.3057    15.30616 --------------------+----------------------------------------------------------------             sigma_u |  2.6565279             sigma_e |  2.7935595                 rho |  .47487287   (fraction of variance due to u_i) ------------------------------------------------------------------------------------- F test that all u_i=0:     F(3662, 95410) =     8.72         Prob > F = 0.0000

I am not sure that I follow why I should use this specification. Doesn't this just add a time trend and a squared term time?

I just want to control for time-specific events which are not captured by my other explanatory variables. That is why add separate time fixed effects.

By the way, I cluster the errors because of autocorrelation and other fund specific characteristics I want to control for.

Kind regards,
Roman

[Carlo Lazzaro]

Roman:
1) my mistake: I omitted the c. you correctly mentioned.
2) I see why you clustered your standard errors, and it seems reasonable;
3) categorical or continuous -timevar-: the reason behind my previous proposal was that, while it's perfectly legal to add -i.time- among predictors, with so many quarters I would find hard to disseminate my results.

[Roman Goossens]

Hi Carlo,

Many thanks for your answer.

Sorry, I don't quite follow what you mean for 3) when you write "I would find hard to disseminate my results"?

Do you mean that you would be worried about overfitting (i.e. too many regressors)?

Best,
Roman

[Carlo Lazzaro]

Roman:
I meant that I would found difficult to comment on so many quarters (I did not mean overfitting, which seems one of your main worries, which seems unjustified with such a low Rsq-within. I would be more concerned about a too scant handful of predictors).

[Roman Goossens]

Ok, thanks for taking the time to answer my questions Carlo,

Best,
Roman