Mundlak procedure

[Lucas LewitMendes]

Dear all,

When using the Mundlak procedure, if one wishes to include year fixed effects (i.e. a dummy variable for each year except one) to control for year-specific shocks, should one also include the mean of each year's dummy variable? If I use the "mundlak" command, stata automatically includes these means when I have year dummies as right-hand side variables. However, I'm not sure if this is correct.

Thank you!

[Joro Kolev]

Yes, in principle you must include the time averages of year dummies.

However if your panel is balanced, that is, you have the same number of periods T for each cross sectional unit i, then your time averages of the year dummies will be dropped due to multicollinearity. So in this case you do not need to bother calculating the time averages of the year dummies.

But if your panel is not balanced, you need to include the time averages of the dummies and they will not be dropped due to multicollinearity.

[Joro Kolev]

This question appears often around here, so I will provide detailed analysis.

Case 1: Balanced panel data

***Balanced sample

webuse grunfeld, clear

keep if year<1940

qui tab time, gen(year)

foreach X of varlist invest mvalue kstock year1-year5 {

egen `X'mean = mean(`X'), by(company)

}

xtreg invest mvalue kstock year2-year5, fe

reg invest mvalue kstock year2-year5 mvaluemean kstockmean year2mean-year5mean

Resulting in

. ***Balanced sample
. 
. webuse grunfeld, clear

. 
. keep if year<1940
(150 observations deleted)

. 
. qui tab time, gen(year)

. 
. foreach X of varlist invest mvalue kstock year1-year5 {
  2. 
. egen `X'mean = mean(`X'), by(company)
  3. 
. }

. 
. xtreg invest mvalue kstock year2-year5, fe

Fixed-effects (within) regression               Number of obs     =         50
Group variable: company                         Number of groups  =         10

R-sq:                                           Obs per group:
     within  = 0.6380                                         min =          5
     between = 0.5577                                         avg =        5.0
     overall = 0.5628                                         max =          5

                                                F(6,34)           =       9.99
corr(u_i, Xb)  = 0.1638                         Prob > F          =     0.0000

------------------------------------------------------------------------------
      invest |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      mvalue |   .0599551   .0117349     5.11   0.000      .036107    .0838032
      kstock |  -.3416076   .1099002    -3.11   0.004    -.5649518   -.1182634
       year2 |   10.61359   12.52288     0.85   0.403    -14.83597    36.06315
       year3 |   24.93282   14.52805     1.72   0.095    -4.591731    54.45737
       year4 |    22.7716   14.48812     1.57   0.125    -6.671813      52.215
       year5 |   17.27716   16.03763     1.08   0.289    -15.31523    49.86954
       _cons |   51.61509   13.40755     3.85   0.000     24.36768    78.86251
-------------+----------------------------------------------------------------
     sigma_u |  83.765483
     sigma_e |  26.122686
         rho |  .91136638   (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0: F(9, 34) = 31.22                      Prob > F = 0.0000

. 
. reg invest mvalue kstock year2-year5 mvaluemean kstockmean year2mean-year5mean
note: year2mean omitted because of collinearity
note: year3mean omitted because of collinearity
note: year4mean omitted because of collinearity
note: year5mean omitted because of collinearity

      Source |       SS           df       MS      Number of obs   =        50
-------------+----------------------------------   F(8, 41)        =     14.42
       Model |  557641.992         8  69705.2489   Prob > F        =    0.0000
    Residual |  198233.399        41  4834.96094   R-squared       =    0.7377
-------------+----------------------------------   Adj R-squared   =    0.6866
       Total |   755875.39        49  15426.0284   Root MSE        =    69.534

------------------------------------------------------------------------------
      invest |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      mvalue |   .0599551   .0312361     1.92   0.062    -.0031274    .1230376
      kstock |  -.3416075   .2925347    -1.17   0.250    -.9323932    .2491782
       year2 |   10.61358   33.33366     0.32   0.752    -56.70511    77.93227
       year3 |   24.93281   38.67105     0.64   0.523    -53.16497    103.0306
       year4 |   22.77159   38.56477     0.59   0.558    -55.11155    100.6547
       year5 |   17.27714   42.68928     0.40   0.688    -68.93561    103.4899
  mvaluemean |   .0201061   .0322834     0.62   0.537    -.0450916    .0853039
  kstockmean |   .5480482   .3185861     1.72   0.093    -.0953494    1.191446
   year2mean |          0  (omitted)
   year3mean |          0  (omitted)
   year4mean |          0  (omitted)
   year5mean |          0  (omitted)
       _cons |  -27.32733   30.81893    -0.89   0.380    -89.56743    34.91276
------------------------------------------------------------------------------

.

Moral of the story: the time averages of the time dummies were dropped due to multicollinearity in the balanced sample, but the computer did not break. Still the estimates of our two main variables of interest mvalue and kstock are the same across the fixed effects, and the Mundlak regression.

[Joro Kolev]

Case 2: Unbalanced panel data

***Unbalanced sample

webuse grunfeld, clear

keep if year<1940

drop if (company==1 | company==2) & year<1937

drop if (company==3 | company==4) & year>1936

drop if (company==5 | company==6) & year>1935


qui tab time, gen(year)

foreach X of varlist invest mvalue kstock year1-year5 {

egen `X'mean = mean(`X'), by(company)

}

xtreg invest mvalue kstock year2-year5, fe

reg invest mvalue kstock year2-year5 mvaluemean kstockmean year2mean-year5mean

Resulting in

. ***Unbalanced sample
.
. webuse grunfeld, clear

.
. keep if year<1940
(150 observations deleted)

.
. drop if (company==1 | company==2) & year<1937
(4 observations deleted)

.
. drop if (company==3 | company==4) & year>1936
(6 observations deleted)

.
. drop if (company==5 | company==6) & year>1935
(8 observations deleted)

.
.
. qui tab time, gen(year)

.
. foreach X of varlist invest mvalue kstock year1-year5 {
  2.
. egen `X'mean = mean(`X'), by(company)
  3.
. }

.
. xtreg invest mvalue kstock year2-year5, fe

Fixed-effects (within) regression               Number of obs     =         32
Group variable: company                         Number of groups  =         10

R-sq:                                           Obs per group:
     within  = 0.9038                                         min =          1
     between = 0.0001                                         avg =        3.2
     overall = 0.0159                                         max =          5

                                                F(6,16)           =      25.04
corr(u_i, Xb)  = -0.4799                        Prob > F          =     0.0000

------------------------------------------------------------------------------
      invest |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      mvalue |   .0415765   .0103648     4.01   0.001     .0196042    .0635488
      kstock |  -.9826048   .1415481    -6.94   0.000    -1.282673   -.6825361
       year2 |   4.082768   10.23045     0.40   0.695    -17.60482    25.77035
       year3 |   22.32429    11.8815     1.88   0.079    -2.863374    47.51195
       year4 |    24.6022   11.68686     2.11   0.051    -.1728256    49.37723
       year5 |   33.88411   13.44504     2.52   0.023     5.381908    62.38632
       _cons |   132.3256   18.28009     7.24   0.000     93.57356    171.0777
-------------+----------------------------------------------------------------
     sigma_u |  148.40276
     sigma_e |  16.729725
         rho |    .987451   (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0: F(9, 16) = 31.97                      Prob > F = 0.0000

.
. reg invest mvalue kstock year2-year5 mvaluemean kstockmean year2mean-year5mean
note: year4mean omitted because of collinearity
note: year5mean omitted because of collinearity

      Source |       SS           df       MS      Number of obs   =        32
-------------+----------------------------------   F(10, 21)       =     19.27
       Model |  447979.654        10  44797.9654   Prob > F        =    0.0000
    Residual |  48829.5933        21  2325.21873   R-squared       =    0.9017
-------------+----------------------------------   Adj R-squared   =    0.8549
       Total |  496809.248        31  16026.1048   Root MSE        =    48.221

------------------------------------------------------------------------------
      invest |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      mvalue |   .0415765   .0298746     1.39   0.179    -.0205512    .1037042
      kstock |  -.9826048   .4079879    -2.41   0.025    -1.831062   -.1341476
       year2 |   4.082767   29.48749     0.14   0.891    -57.23983    65.40536
       year3 |   22.32429   34.24637     0.65   0.522    -48.89493     93.5435
       year4 |    24.6022   33.68533     0.73   0.473    -45.45027    94.65468
       year5 |   33.88411   38.75298     0.87   0.392    -46.70712    114.4753
  mvaluemean |   .0298491   .0310428     0.96   0.347    -.0347078    .0944061
  kstockmean |    1.17485   .4251216     2.76   0.012     .2907617    2.058939
   year2mean |  -134.1457   86.95541    -1.54   0.138    -314.9794    46.68799
   year3mean |   34.52447   149.3492     0.23   0.819    -276.0643    345.1132
   year4mean |          0  (omitted)
   year5mean |          0  (omitted)
       _cons |  -.8818338   35.81467    -0.02   0.981    -75.36252    73.59885
------------------------------------------------------------------------------

Moral of the story: We again replicated the estimates from the fixed effects and the Mundlak regression on the main variables of interest, and notice that two of the time averages of the time dummies were not dropped. Two were dropped because I generated fairly elementary pattern of unbalancedness.

What happens had we omitted the time averages of the time dummies? Here:

. reg invest mvalue kstock year2-year5 mvaluemean kstockmean

      Source |       SS           df       MS      Number of obs   =        32
-------------+----------------------------------   F(8, 23)        =     21.68
       Model |  438652.336         8   54831.542   Prob > F        =    0.0000
    Residual |  58156.9119        23  2528.56139   R-squared       =    0.8829
-------------+----------------------------------   Adj R-squared   =    0.8422
       Total |  496809.248        31  16026.1048   Root MSE        =    50.285

------------------------------------------------------------------------------
      invest |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      mvalue |   .0397905   .0304812     1.31   0.205    -.0232648    .1028457
      kstock |  -1.088032   .4070048    -2.67   0.014    -1.929985    -.246078
       year2 |  -3.964683   28.29357    -0.14   0.890     -62.4944    54.56503
       year3 |   33.02513   31.11164     1.06   0.299    -31.33419    97.38445
       year4 |    38.4382   29.83349     1.29   0.210    -23.27709    100.1535
       year5 |   49.85638   33.72022     1.48   0.153    -19.89921     119.612
  mvaluemean |   .0347599   .0312413     1.11   0.277    -.0298677    .0993875
  kstockmean |   1.352167   .4171152     3.24   0.004     .4892986    2.215036
       _cons |  -36.11844   20.56722    -1.76   0.092    -78.66497    6.428094
------------------------------------------------------------------------------

The estimates on the main variables of interest are no longer equivalent in the fixed effects and the Mundlak regressions.

[Joro Kolev]

Overall moral of the story: Always include the time averages of the time dummies in your Mundlak regression, unless you are absolutely sure that your panel is balanced. Depending on your pattern of unbalancedness, some of the time averages might be dropped due to multicollinearity... but omitting them results in a procedure which is not equivalent to the fixed effects estimator.

[Lucas LewitMendes]

Thank you Joro, that is super helpful!