This page was adapted from a FAQ at the Stata Corp. FAQ page. We thank Stata
for their permission to adapt and distribute this page via our web site.
The results that xtreg, fe reports have simply been reformulated so that
the reported intercept is the average value of the fixed effects.
Intuition
One way of writing the fixed-effects model is
y = a + x b + v + e (1)
it it i it
where v_i (i=1, …, n) are simply the fixed effects to be estimated. With no
further constraints, the parameters a and v_i do not have a unique solution.
You can see that by rearranging the terms in equation (1):
y = (a + v ) + x b + e
it i it it
Consider some solution which has, say a=3. In that case, we could just as well
say that a=4 and subtract the value 1 from each of the estimated v_i.
Thus, before equation (1) can be estimated, we must place an additional constraint on
the system. Any constraint will do and the choice we make will have no effect on the
estimated b. One popular constraint is a=0 but it is important to understand
that we could just as well constrain a=3. Changing the value of a would
merely change the corresponding values of v_i. Nor do we have to constrain a;
we could place a constraint on v_i. We could, for instance, constrain v_1=0,
or v_5=3.
The constraint xtreg, fe places on the system is computationally more
difficult:
N
Sum v = 0
i=1 i
Since the constraint we choose is arbitrary, we chose a constraint that makes
interpreting results a little more convenient. The random-effects estimator proceeds under
the *ASSUMPTION* that E(v_i)=0 and is hence able to estimate an intercept. We
parameterize the fixed-effects estimator so that it proceeds under the *CONSTRAINT*
average(v_i)=0. This constraint has no implication since we had to choose some
constraint anyway.
The primary advantage of this constraint is that if you estimate some model and then
obtain the predictions
. xtreg y x1 x2 x3, fe xtpred yhat
then the average value of yhat will equal the average value of y. In
order to obtain estimates with the fixed-effects estimator, we had to impose an arbitrary
constraint and had we instead constrained a=0, xtpred yhat would have
produced yhat with average value 0. That would be the only difference; the
predictions would differ by a constant (namely, by their respective values of a).
Using the constraint Sum v_i=0 has another advantage. Let us draw a distinction
between models and estimators. The *MODEL* is
y = a + x b + v + e (1)
it it i it
Under the random-effects *MODEL*, it is assumed that E(v_i)=0 and that v_i
and x_it are uncorrelated. From that model, we can derive the random-effects
*ESTIMATOR*.
Under the fixed-effects *MODEL*, no assumptions are made about v_i except that
they are fixed parameters. From that model, we can derive the fixed-effects *ESTIMATOR*.
Now, it turns out that the fixed-effects *ESTIMATOR* is an admissible estimator for the
random-effects *MODEL*; it is merely less efficient than the random-effects *ESTIMATOR*.
That is,
| ----------------- model ---------------------
Estimator | fixed effects random effects
------------------------+---------------------------------------------------
fixed effects | appropriate appropriate
random effects | inappropriate appropriate
------------------------+---------------------------------------------------
When you use the fixed-effects *ESTIMATOR* for the random-effects *MODEL*, the
intercept a reported by xtreg, fe is the appropriate estimate for the
intercept of the random-effects model.
Derivation
The fixed-effects model is
y = a + x b + v + e (1)
it it i it
From which it follows that
_ _ _
y = a + x b + v + e (2)
i i i i
_ _ _
where y , x , and e are with averages of y , x , and e within i.
i i i it it it
Subtracting equation (2) from (1), we obtain
_ _ _
y - y = ( x - x )b + (e - e ) (3)
it i it i it i
Equation (3) is the way many people think about the fixed-effects estimator and notice
that, in this formula, a remains unestimated. From equation (1), it also follows
that
= = _ =
y = a + xb + v + e (4)
= = _ =
where y, x, v, and e are the grand averages of y , x , v , and e .
it it i it
For instance,
= n T_i
y = ( Sum Sum y_it ) / (total_number_of_observations)
i=1 t=1
Summing equations (3) and (4), we obtain
_ = _ = _ _ =
y - y + y = a + (x - x + x)b + (e - e + v) + e (5)
it i it i it i
_ xtreg, fe estimates the above equation under the constraint v=0, which is to say, it estimates
_ = _ =
y - y + y = a + (x - x + x)b + noise
it i it i
Thus, the left-hand-side variable is y_it minus the within-group means but with
the grand mean added back in and the right-hand-side variables are x_it minus the
within-group means but with the grand mean added back in. Obviously, adding in grand means
to the left- and right-hand sides has no effect on the estimated b.
Demonstration
Fixed-effects regression is supposed to produce the same coefficient estimates and
standard errors as ordinary regression when indicator (dummy) variables are included for
each of the groups. Since the fixed-effects model is
y = X b + v + e
ij ij i it
and v_i are fixed parameters to be estimated, this is the same as
y = X b + v d1 + v d2 + ... e
ij ij 1 i 2 i it
where d1 is 1 when i=1 and 0 otherwise, d2 is 1 when i=2 and 0 otherwise,
and so on. d1, d2, …, are just dummy variables indicating the groups and v_1,
v_2, …, are their regression coefficients which we must estimate.
The problem is that we typically have lots of groups — perhaps thousands — and
including logs of dummy variables is too computationally expensive. So we look for a
shortcut.
Nevertheless, we could take a little dataset with just a few groups and compare the
methods. Here is my little dataset:
. list
group x y
1. 1 0 -5
2. 1 8 23
3. 1 17 44
4. 2 10 29
5. 2 16 26
6. 3 4 17
7. 3 11 17
8. 3 5 31
9. 4 18 50
10. 4 5 26
11. 4 2 17
I’m going to show you
- what regress with group dummies reports;
- that xtreg, fe reports the same results;
- that removing the within-group means and estimating a regression on the deviations
without an intercept (as given in equation 3) produces the same coefficients but different
standard errors.
How can method 3 be wrong? Because it fails to account for the fact that the means we
removed are *ESTIMATES*. As a consequence, it understates standard errors.
1. What regress with group dummies reports
. xi: regress y x i.group
i.group Igroup_1-4 (naturally coded; Igroup_1 omitted)
Source | SS df MS Number of obs = 11
---------+------------------------------ F( 4, 6) = 4.01
Model | 1554.16667 4 388.541667 Prob > F = 0.0643
Residual | 581.833333 6 96.9722222 R-squared = 0.7276
---------+------------------------------ Adj R-squared = 0.5460
Total | 2136.00 10 213.60 Root MSE = 9.8474
------------------------------------------------------------------------------
y | Coef. Std. Err. t P>|t| [95% Conf. Interval]
---------+--------------------------------------------------------------------
x | 2 .5372223 3.723 0.010 .6854644 3.314536
Igroup_2 | -2.5 9.332493 -0.268 0.798 -25.33579 20.33579
Igroup_3 | 4.333333 8.090107 0.536 0.611 -15.46245 24.12911
Igroup_4 | 10.33333 8.040407 1.285 0.246 -9.340835 30.0075
_cons | 4 7.236455 0.553 0.600 -13.70697 21.70697
------------------------------------------------------------------------------
2. xtreg, fe reports the same results
. xtreg y x, i(group) fe
Fixed-effects (within) regression
sd(u_group) = 5.621346 Number of obs = 11
sd(e_group_t) = 9.847447 n = 4
sd(e_group_t + u_group) = 11.33895 T-bar = 2.75
corr(u_group, Xb) = -0.1939 R-sq within = 0.6979
between = 0.1716
overall = 0.6146
F( 1, 6) = 13.86
Prob > F = 0.0098
------------------------------------------------------------------------------
y | Coef. Std. Err. t P>|t| [95% Conf. Interval]
---------+--------------------------------------------------------------------
x | 2 .5372223 3.723 0.010 .6854644 3.314535
_cons | 7.545455 5.549554 1.360 0.223 -6.033815 21.12472
------------------------------------------------------------------------------
group | F(3,6) = 0.830 0.524 (4 categories)
If you compare, you will find that regress with group dummies reported the same
coefficient (2) and the same standard error (.5372223) for x as xtreg, fe just did.
In both cases, the t statistic is 3.723.
3. Estimating the deviation model reports incorrect standard errors
. egen double ybar = mean(y), by(group)
egen double xbar = mean(x), by(group)
gen yd = y-ybar
gen xd = x-xbar
reg yd xd, nocons
Source | SS df MS Number of obs = 11
---------+------------------------------ F( 1, 10) = 23.10
Model | 1343.99999 1 1343.99999 Prob > F = 0.0007
Residual | 581.833327 10 58.1833327 R-squared = 0.6979
---------+------------------------------ Adj R-squared = 0.6677
Total | 1925.83332 11 175.075756 Root MSE = 7.6278
------------------------------------------------------------------------------
yd | Coef. Std. Err. t P>|t| [95% Conf. Interval]
---------+--------------------------------------------------------------------
xd | 2 .4161306 4.806 0.001 1.072803 2.927197
------------------------------------------------------------------------------
So, to summarize:
x
| Coefficient Std. Err. t
-------------------------+------------------------------------
-regress- with dummies | 2 .5372223 3.723
-xtreg, fe- | 2 .5372223 3.723
removing the means | 2 .4161306 4.806
-------------------------+------------------------------------
regress with dummies definitionally calculates correct results.
xtreg, fe matches them.
Removing the means and estimating on the deviations with the noconstant option
produces correct coefficients but incorrect standard errors. Why? Because we did not
account for the fact that the means we removed from y and x were estimated.
This page was adapted from a FAQ at the Stata Corp. FAQ page. We thank Stata
for their permission to adapt and distribute this page via our web site.