Accessing individual fixed effects with xtreg?

Maurice McCourt

Join Date: Apr 2018

Posts: 15
#1

Accessing individual fixed effects with xtreg?

09 Apr 2018, 02:54

Consider the panel fixed effects model
y_it = x_itb + v_i + e_it

Using "xtreg, fe" the reported intercept is the average value of the fixed effects v_i under the constraint c1:

Σ
i=1 T_i
Σ
t=1 v_i = 0 (c1)

See: https://www.stata.com/support/faqs/s...effects-model/

However I would like to obtain each of the individual fixed effects v_i, not just the average.

Is there a way to access the v_i's using xtreg?

If not, how can I use a dummy variable fixed effects approach with the standard "regress" command so that I get the same v_i's that xtreg implicitly computes under the constraint c1?

thanks for any advice
Tags: fixed effects, panel data

Andrew Musau

Join Date: Oct 2014
Posts: 9627

10 Apr 2018, 03:02

However I would like to obtain each of the individual fixed effects v_i, not just the average.

So given the model

$$ y_{it} = a + x_{it}b + \eta_{i} + e_{it} \;\; (i= 1, \cdots, N;\; t=1, \cdots, T)$$

where $\eta_{i}$ are the individual effects, Bill Gould has explained to you in the link that you provide that the in the absence of constraints, the parameters $a$ and $\eta_{i}$ do not have a unique solution. So taking into account the constraints that Stata places on the system (also illustrated in the link), you can predict the individual effects using

Code:

predict uhat, u

Alternatively, you can predict the fixed effects residuals and average them across individuals in the panel

Code:

predict res, r
bys id: egen uhat2= mean(res)

If not, how can I use a dummy variable fixed effects approach with the standard "regress" command so that I get the same v_i's that xtreg implicitly computes under the constraint c1

From the Frisch-Waugh theorem, we know that the within estimator is equivalent to including a set of $N-1$ individual dummy variables in the regression (Least Squares Dummy Variables or LSDV). Provided that the panel is balanced (i.e., the same number of observations for each individual with no holes), the residuals from LSDV are exactly the deviations from the individual means. See the example below

Code:

webuse grunfeld, clear
quietly xtreg invest mvalue kstock, fe
predict uhat, u
predict res_fe, r
bys company: egen uhat2= mean(res_fe)
quietly reg invest mvalue kstock i.company
predict res_lsdv, r
*SHOW THAT res_lsdv =  res_fe - mean(res_fe)
gen resfe_dev =  res_fe - uhat2
list uhat uhat2 res_fe res_lsdv resfe_dev if time<4, sepby(company)

Code:

. list uhat uhat2 res_fe res_lsdv resfe_dev if time<4, sepby(company)

     +-----------------------------------------------------------+
     |      uhat       uhat2      res_fe    res_lsdv   resfe_dev |
     |-----------------------------------------------------------|
  1. | -11.55275   -11.55275    36.45965     48.0124     48.0124 |
  2. | -11.55275   -11.55275   -79.12964   -67.57689   -67.57689 |
  3. | -11.55275   -11.55275   -172.5532   -161.0005   -161.0005 |
     |-----------------------------------------------------------|
 21. |  160.6498    160.6497    101.9297   -58.72001      -58.72 |
 22. |  160.6498    160.6497    199.3809    38.73115    38.73116 |
 23. |  160.6498    160.6497    197.3009    36.65113    36.65114 |
     |-----------------------------------------------------------|
 41. | -176.8279   -176.8279   -67.39137    109.4365    109.4365 |
 42. | -176.8279   -176.8279   -150.6144    26.21345    26.21346 |
 43. | -176.8279   -176.8279   -209.3538   -32.52593   -32.52592 |
     |-----------------------------------------------------------|
 61. |  30.93464    30.93464    49.80156    18.86692    18.86692 |
 62. |  30.93464    30.93464    36.07955    5.144909    5.144909 |
 63. |  30.93464    30.93464    16.90624    -14.0284    -14.0284 |
     |-----------------------------------------------------------|
 81. | -55.87288   -55.87288    24.25344    80.12632    80.12632 |
 82. | -55.87288   -55.87288    27.73082     83.6037     83.6037 |
 83. | -55.87288   -55.87288    38.56563    94.43851    94.43851 |
     |-----------------------------------------------------------|
101. |  35.58264    35.58264    55.39412    19.81148    19.81148 |
102. |  35.58264    35.58264    56.66586    21.08323    21.08323 |
103. |  35.58264    35.58264     51.5265    15.94386    15.94386 |
     |-----------------------------------------------------------|
121. | -7.809542   -7.809542     36.9083    44.71784    44.71784 |
122. | -7.809542   -7.809542    21.15999    28.96953    28.96953 |
123. | -7.809542   -7.809542    24.23362    32.04316    32.04316 |
     |-----------------------------------------------------------|
141. |  1.198279    1.198279    50.02711    48.82883    48.82883 |
142. |  1.198279    1.198279      27.572    26.37372    26.37372 |
143. |  1.198279    1.198279     11.2192    10.02092    10.02092 |
     |-----------------------------------------------------------|
161. | -28.47834   -28.47834    3.141369    31.61971    31.61971 |
162. | -28.47834   -28.47834   -3.874476    24.60386    24.60386 |
163. | -28.47834   -28.47834   -4.239498    24.23884    24.23884 |
     |-----------------------------------------------------------|
181. |  52.17609    52.17609    52.07976   -.0963299   -.0963287 |
182. |  52.17609    52.17609    49.59924   -2.576852   -2.576851 |
183. |  52.17609    52.17609    50.46476   -1.711331    -1.71133 |
     +-----------------------------------------------------------+

Last edited by Andrew Musau; 10 Apr 2018, 03:47.

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Maurice McCourt

Join Date: Apr 2018

Posts: 15
#3

10 Apr 2018, 09:02

Cool this is great thanks!

I have another question tho - what has happened the constant a from the model that you post?

What is the expected value of a given the constraint?

Is there a way to retrieve the actual value of a using postestimation?

For example, to estimate the impact on fixed effects of a firm characteristic k, I am thinking of using two distinct regressions, a baseline (without characteristic k) and the baseline with characteristic k:
y_it = (a + v_i) + x_itb + e_{it // baseline without k}
y_itk = (a_k + v_ik) + x_itkb + k_itkb_k + e_{itk // baseline plus k} where a and a_k are constants, and v_i and v_ik are the fixed effects. I would then measure impact of characteristic k as the difference between the intercepts:
Impact_k = (a + v_i) - (a_k + v_ik) Does this make sense? Can I assume the constants a and a_k cancel out? Or can I retrieve a and a_k using postestimation and use them in the Impact_k equation?
Comment
Maurice McCourt

Join Date: Apr 2018

Posts: 15
#4

10 Apr 2018, 09:06

<repost second part of previous msg with better formatting >

For example, to estimate the impact on fixed effects of a firm characteristic k, I am thinking of using two distinct regressions, a baseline (without characteristic k) and the baseline with characteristic k:

y_it = (a + v_i) + x_itb + e_{it // baseline without k}

y_itk = (a_k + v_ik) + x_itkb + k_itkb_k + e_{itk // baseline plus k}

where a and a_k are constants, and v_i and v_ik are the fixed effects. I would then measure impact of characteristic k as the difference between the intercepts:

Impact_k = (a + v_i) - (a_k + v_ik)

Does this make sense? Can I assume the constants a and a_k cancel out? Or can I retrieve a and a_k using postestimation and use them in the Impact_k equation?
Comment
Sebastian Kripfganz

Join Date: May 2014

Posts: 2515
#5

10 Apr 2018, 09:40

It should be added that in a typical large-N, small-T setup the individual fixed effects cannot be estimated consistently and any attempt to do so is essentially meaningless. Estimates of individual fixed effects are not reliable and should not be used for statistical comparisons.

https://twitter.com/Kripfganz
Comment
Maurice McCourt

Join Date: Apr 2018

Posts: 15
#6

10 Apr 2018, 09:56

Originally posted by Sebastian Kripfganz View Post

It should be added that in a typical large-N, small-T setup the individual fixed effects cannot be estimated consistently and any attempt to do so is essentially meaningless. Estimates of individual fixed effects are not reliable and should not be used for statistical comparisons.

OK, what about comparing average fixed effects in a panel regression with, say, 1000 individuals over 250 months?

I would like to say something like, "the average fixed effects for the baseline model are x points greater than the average fixed effects for the baseline model plus characteristic k, therefore characteristic k has an impact on average fixed effects."
Comment
Andrew Musau

Join Date: Oct 2014

Posts: 9627
#7

11 Apr 2018, 02:36

Are you referring to adding a variable to your model? The constant and estimates of the individual fixed effects are meaningless. What you are interested in are the estimates of the time varying regressors and you should concentrate on how these change across different model specifications.
Comment

Maurice McCourt

Join Date: Apr 2018
Posts: 15

11 Apr 2018, 09:48

Thanks for the replies - this is all very interesting.

I understand that individual fixed effects are meaningless, but the reason I am interested in how individual fixed effects are estimated is so I can correctly estimate the average fixed effects. Average fixed effects have been used comparatively in some top journals (eg Journal of Financial Economics).

By the way, I notice that the intercept reported by xtreg does not seem to correspond with the average fixed effects estimated using the "predict" command - for example, if I execute this code:

Code:

webuse grunfeld, clear
quietly xtreg invest mvalue kstock, fe
gen intercept=_b[_cons] // average fixed effects from xtreg
predict uhat, u
predict res_fe, r
bys company: egen uhat2= mean(res_fe)
egen meanuhat = mean(uhat) // average fixed effects using predict uhat, u
egen meanuhat2 = mean(res_fe) // average fixed effects using predict res_fe, r
bys company: gen first=1 if _n==1
egen meanuhat3 = mean(uhat) if first==1

list uhat uhat2 res_fe intercept meanuhat meanuhat2 meanuhat3 if time&lt;4, sepby(company)

Then I get the output below where it is clear that the reported intercept is not the same as the mean of the predicted fixed effects.

So maybe the intercept reported by xtreg is mean(a + v_i), while the predicted fixed effects are just the v_i (under the constraint used by xtreg described in the previously linked article)?

Code:

     +---------------------------------------------------------------------------------+
     |      uhat       uhat2      res_fe   intercept   meanuhat   meanuhat2   meanuh~3 |
     |---------------------------------------------------------------------------------|
  1. | -11.55275   -11.55275    36.45965   -58.74393   6.32e-07   -7.42e-08   6.32e-07 |
  2. | -11.55275   -11.55275   -79.12964   -58.74393   6.32e-07   -7.42e-08          . |
  3. | -11.55275   -11.55275   -172.5532   -58.74393   6.32e-07   -7.42e-08          . |
     |---------------------------------------------------------------------------------|
 21. |  160.6498    160.6497    101.9297   -58.74393   6.32e-07   -7.42e-08   6.32e-07 |
 22. |  160.6498    160.6497    199.3809   -58.74393   6.32e-07   -7.42e-08          . |
 23. |  160.6498    160.6497    197.3009   -58.74393   6.32e-07   -7.42e-08          . |
     |---------------------------------------------------------------------------------|
 41. | -176.8279   -176.8279   -67.39137   -58.74393   6.32e-07   -7.42e-08   6.32e-07 |
 42. | -176.8279   -176.8279   -150.6144   -58.74393   6.32e-07   -7.42e-08          . |
 43. | -176.8279   -176.8279   -209.3538   -58.74393   6.32e-07   -7.42e-08          . |
     |---------------------------------------------------------------------------------|
 61. |  30.93464    30.93464    49.80156   -58.74393   6.32e-07   -7.42e-08   6.32e-07 |
 62. |  30.93464    30.93464    36.07955   -58.74393   6.32e-07   -7.42e-08          . |
 63. |  30.93464    30.93464    16.90624   -58.74393   6.32e-07   -7.42e-08          . |
     |---------------------------------------------------------------------------------|
 81. | -55.87288   -55.87288    24.25344   -58.74393   6.32e-07   -7.42e-08   6.32e-07 |
 82. | -55.87288   -55.87288    27.73082   -58.74393   6.32e-07   -7.42e-08          . |
 83. | -55.87288   -55.87288    38.56563   -58.74393   6.32e-07   -7.42e-08          . |
     |---------------------------------------------------------------------------------|
101. |  35.58264    35.58264    55.39412   -58.74393   6.32e-07   -7.42e-08   6.32e-07 |
102. |  35.58264    35.58264    56.66586   -58.74393   6.32e-07   -7.42e-08          . |
103. |  35.58264    35.58264     51.5265   -58.74393   6.32e-07   -7.42e-08          . |
     |---------------------------------------------------------------------------------|
121. | -7.809542   -7.809542     36.9083   -58.74393   6.32e-07   -7.42e-08   6.32e-07 |
122. | -7.809542   -7.809542    21.15999   -58.74393   6.32e-07   -7.42e-08          . |
123. | -7.809542   -7.809542    24.23362   -58.74393   6.32e-07   -7.42e-08          . |
     |---------------------------------------------------------------------------------|
141. |  1.198279    1.198279    50.02711   -58.74393   6.32e-07   -7.42e-08   6.32e-07 |
142. |  1.198279    1.198279      27.572   -58.74393   6.32e-07   -7.42e-08          . |
143. |  1.198279    1.198279     11.2192   -58.74393   6.32e-07   -7.42e-08          . |
     |---------------------------------------------------------------------------------|
161. | -28.47834   -28.47834    3.141369   -58.74393   6.32e-07   -7.42e-08   6.32e-07 |
162. | -28.47834   -28.47834   -3.874476   -58.74393   6.32e-07   -7.42e-08          . |
163. | -28.47834   -28.47834   -4.239498   -58.74393   6.32e-07   -7.42e-08          . |
     |---------------------------------------------------------------------------------|
181. |  52.17609    52.17609    52.07976   -58.74393   6.32e-07   -7.42e-08   6.32e-07 |
182. |  52.17609    52.17609    49.59924   -58.74393   6.32e-07   -7.42e-08          . |
183. |  52.17609    52.17609    50.46476   -58.74393   6.32e-07   -7.42e-08          . |
     +---------------------------------------------------------------------------------+

Comment

Andrew Musau

Join Date: Oct 2014

Posts: 9627
#9

11 Apr 2018, 12:33

I understand that individual fixed effects are meaningless, but the reason I am interested in how individual fixed effects are estimated is so I can correctly estimate the average fixed effects. Average fixed effects have been used comparatively in some top journals (eg Journal of Financial Economics)

Fine, you should consult someone who is an expert in that field.

Then I get the output below where it is clear that the reported intercept is not the same as the mean of the predicted fixed effects.

So maybe the intercept reported by xtreg is mean(a + v_i), while the predicted fixed effects are just the v_i (under the constraint used by xtreg described in the previously linked article)?

Again, from the FAQ in #1 it is explained that the constraint xtreg,fe places on the system is that the individual effects sum to 0 across all observations. The implication of this is that the average value of the fitted values equals the average value of the outcome. So from the estimates $\hat{a}$ and $\hat{b}$ in the Equation in #2, the estimates of the individual effects are obtained as

$$u_{i} = \bar{y}_{i}- \hat{a} - \bar{x}_{i}\hat{b}$$

So once you have estimates of the individual effects and knowing $\bar{u}_{i} =0$, you can back out the estimated intercept

Code:

webuse grunfeld xtreg invest mvalue kstock, fe local vars invest mvalue kstock foreach var in `vars'{ egen m`var'= mean(`var') } gen intercept= minvest- (_b[mvalue]*mmvalue + _b[kstock]*mkstock) sum intercept

Code:

_cons | -58.74393 12.45369 -4.72 0.000 -83.31086 -34.177 _________________________________________________________ . sum intercept Variable | Obs Mean Std. Dev. Min Max -------------+--------------------------------------------------------- intercept | 200 -58.74393 0 -58.74393 -58.74393

Last edited by Andrew Musau; 11 Apr 2018, 12:38.
1 like
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Maurice McCourt

Join Date: Apr 2018

Posts: 15
#10

11 Apr 2018, 14:29

Very clear thanks!

There is just one thing I am still troubled by which is the relationship between the intercept and the mean of the average individual fixed effects, where individual fixed effects are estimated using predict as described in post #2. The FAQ says the reported intercept is the average value of the fixed effects, yet the mean of the average individual fixed effects (estimated using predict) is not the same as the reported intercept (see post #8). I guess there is some relationship but I dont see it - would it be possible to clarify this?
Comment
Maurice McCourt

Join Date: Apr 2018

Posts: 15
#11

12 Apr 2018, 00:11

I will answer my own question - the average individual fixed effects (estimated using predict) is 0 by construction.

The reported intercept then includes the average "true" fixed effects,

Last edited by Maurice McCourt; 12 Apr 2018, 00:29.
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Andrew Musau

Join Date: Oct 2014
Posts: 9627

#12

12 Apr 2018, 02:30

There is just one thing I am still troubled by which is the relationship between the intercept and the mean of the average individual fixed effects, where individual fixed effects are estimated using predict as described in post #2. The FAQ says the reported intercept is the average value of the fixed effects, yet the mean of the average individual fixed effects (estimated using predict) is not the same as the reported intercept (see post #8). I guess there is some relationship but I dont see it - would it be possible to clarify this?

This implied separability of the "intercept" and the fixed effects is artificial, it results from the constraint that you place prior to estimation. It is very important that you understand this point, otherwise you might start assigning meaning to these estimates. The chosen constraint is arbitrary and If I impose a different constraint, I will get a new set of values for the intercept and the fixed effects. So without separating these, predict with option xbu gives you the prediction including the individual effect, i.e., $a +x_{it}b + u_{i}$. If you define $\gamma = a + u_{i}$, then the average value of $\gamma $ will be what you will call an intercept in xtreg, fe. See below

Code:

webuse grunfeld
xtreg invest mvalue kstock, fe
predict xbu, xbu
gen xb= _b[mvalue]*mvalue + _b[kstock]*kstock
gen gamma = xbu-xb
sum gamma

Code:

. xtreg invest mvalue kstock, fe

Fixed-effects (within) regression               Number of obs     =        200
Group variable: company                         Number of groups  =         10

R-sq:                                           Obs per group:
     within  = 0.7668                                         min =         20
     between = 0.8194                                         avg =       20.0
     overall = 0.8060                                         max =         20

                                                F(2,188)          =     309.01
corr(u_i, Xb)  = -0.1517                        Prob > F          =     0.0000

------------------------------------------------------------------------------
      invest |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      mvalue |   .1101238   .0118567     9.29   0.000     .0867345    .1335131
      kstock |   .3100653   .0173545    17.87   0.000     .2758308    .3442999
       _cons |  -58.74393   12.45369    -4.72   0.000    -83.31086     -34.177
-------------+----------------------------------------------------------------
     sigma_u |  85.732501
     sigma_e |  52.767964
         rho |  .72525012   (fraction of variance due to u_i)
------------------------------------------------------------------------------
F test that all u_i=0: F(9, 188) = 49.18                     Prob > F = 0.0000

. 
. predict xbu, xbu

. 
. gen xb= _b[mvalue]*mvalue + _b[kstock]*kstock

. 
. gen gamma = xbu-xb

. 
. sum gamma

    Variable |        Obs        Mean    Std. Dev.       Min        Max
-------------+---------------------------------------------------------
       gamma |        200   -58.74393    81.53709  -235.5718   101.9059

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Maurice McCourt

Join Date: Apr 2018

Posts: 15
#13

12 Apr 2018, 05:48

Excellent explanations and examples, many thanks.
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