1. INTRODUCTION
Panel data allow the possibility of controlling for unobserved
individual specific effects. In linear models, such “fixed
effects” are usually eliminated by differencing. An unintended
consequence of differencing is that it also eliminates the time-invariant
regressor, which renders its coefficient unidentified. Hausman and Taylor
(1981) used an instrumental variables (IV)
approach to overcome such a problem. They show that a variable that is
uncorrelated with individual fixed effects can be used as a valid
instrument in estimating such a coefficient.
In this paper, we generalize their intuition and develop a method of
dealing with a time-invariant regressor in the nonlinear framework. This
method requires a large number of observations per individual
(T), so its applicability is limited to the case where T
is large. Because the IV estimation requires a large number of individuals
(n), we adopt an asymptotic framework where both n and
T grow to infinity at the same rate. This result is made possible
by recent technical progress of panel analysis under such alternative
asymptotics. See, e.g., Hahn and Kuersteiner (2002, 2003), Hahn and Newey
(2004), and Woutersen (2002). We illustrate the usefulness of our result by
discussing the implication for some nonlinear models of social
interactions.
2. IV ESTIMATION
Suppose that we are given a set of moment restrictions
for some vector-valued function φ, where
yit, xit, and
wi denote the dependent variable in the
tth period, time-varying regressor in the tth period,
and time-invariant regressor. Unobserved individual specific effects are
summarized by the scalar variable γi. For example,
in the case of the linear model
with
(wi′,xit′)
strictly exogenous such that
Our primary focus is to estimate the coefficient δ0 of
the time-invariant regressor wi when
γi is possibly correlated with
wi and xit. We
should note that estimation of θ0 does not present any
substantive conceptual challenge. If both n and T grow
to infinity at the same rate, θ0 can be
-consistently estimated. Letting αi0
≡ γi0 +
wi′δ0, we can rewrite
the model as E
[φ(yit,αi0
+ xit′θ0)] = 0, to
which we can apply variants of recently developed methods discussed in
Hahn and Kuersteiner (2002, 2003), Hahn and Newey (2004), and Woutersen (2002).
To understand how δ0 could be estimated, suppose for a
moment that we observe αi0. Also suppose that we
observe an additional variable zi with
dim(zi) =
dim(wi)1
It is
easy to generalize the discussion to the overidentified case where
dim(zi) >
dim(wi). Because the primary purpose of this
paper is identification and consistent estimation of δ0, we
focus on the exactly identified case.
and such that the following
condition applies.
CONDITION 1. (i) E
[ziγi0] =
0; (ii) E [zi
wi′] is nonsingular.
Note that zi is required to be
uncorrelated with the individual fixed effects. It is clear that we can
consistently estimate δ0 by
under the mild condition that the data are independent and identically
distributed (i.i.d.) over i.
CONDITION 2.
({yi1,yi2,…},{xi1,xi2,…},zi,wi,γi0)
is i.i.d. over i.
Hausman and Taylor (1981) noted that
would remain consistent even if we replace αi0 by
an unbiased estimate. In the nonlinear context, it is difficult to come up
with such an unbiased estimator for αi0.
Therefore, Hausman and Taylor's method cannot be directly applied.
The basic intuition in this paper is that, when both n and
T grow to infinity at the same rate, we can come up with a
-consistent estimator for αi0, say,
.
CONDITION 3. n,T → ∞ such that
n/T → ρ, where 0 < ρ <
∞.
Because the estimation error becomes very small as the sample size
increases, the IV estimator
is expected to be consistent for δ0 in general. This is
quite intuitive. Note that
is an IV estimator of
on wi. Because
is a proxy for αi0, and because the
“measurement error” disappears as T → ∞,
we should expect that the distribution of
should be similar to that of
if T is large.
To come up with a
-consistent estimator
,
we will assume that
for some functions u and v, where
Xit ≡
(yi1,…,yiT,xi1,…,xiT)
and where dim(θ) = dim(u) = p and dim(α) =
dim(v) = 1. We will consider the estimator that solves
This indicates that (i) the first component u is used
throughout the sample for estimation of θ0; and (ii) the
second component v is used only for the ith individual
to estimate αi0. We do not expect this separation
to be constraining in practice. For example, in the linear model (2), we
may take
u(Xit;θ,αi)
=
xit·(yit
− αi −
xit′θ),
v(Xit;θ,αi)
= yit − αi
− xit′θ, which will result
in the usual fixed effects estimator.
Under regularity conditions discussed in Appendix A, it can be shown
that the
,
which solves (5), is uniformly consistent over i.
2 Because
is a proxy for αi0, there is a “measurement
error.” If there is a correlation between the measurement error and
the instrument zi, the resultant estimator
may be biased. Condition 4 rules this out.
CONDITION 4. E
[v(Xit;θ,αi)|zi]
= 0.3
We have
v(Xit;θ,αi) =
εit in the linear model (2), and Hausman and
Taylor's instrument zi for such a linear
model is required to satisfy E
[zi·εit]
= 0.
Theorem 1 establishes that the IV estimator
in (4) based on the proxy
of αi0 has the same asymptotic distribution as
in (3).
THEOREM 1. Assume Conditions 1–7. Further assume that E
[|zi
wi′|] < ∞ and
E
[|γi02zi
zi′|] < ∞. We
then have
Proof. See Appendix E.
3. APPLICATION: NONLINEAR MODEL OF SOCIAL
INTERACTIONS
Identification and estimation of various social effects in the
nonlinear model of social interactions based on the preceding framework
are straightforward and provide one way of dealing with the typical
identification problems that are peculiar to these models.
For a grouped cross-section of data, a model of social interactions
can have the form of a conditional likelihood f
(ygi,γg0 +
Eg
[ygi]β +
Eg
[sgi′]φ +
sgi′ζ +
xgi′θ0). Here,
ygi is the outcome/behavior of interest
for the ith individual in the gth group, and
Eg [·] denotes the mean for
the gth group. Following the classification of Manski (1993), the coefficient on
Eg
[ygi] determines the strength of
endogenous social effects in explaining individual outcomes. In
addition to ygi, we observe
sgi, a vector of individual characteristics
that also generate exogenous (contextual) social effects, and
xgi, a vector of individual characteristics
that operate at the individual level only. Finally
γg0, which is not observed by the econometrician,
captures the presence of correlated group effects.
The focus of Graham and Hahn (2003) is on
the linear-in-means model without contextual effects. They exploit the
idea of Hausman and Taylor (1981) of using IVs
to identify the parameters. Identification of the endogenous effect is
made possible by an instrument that exogenously explains the between-group
variation of the individual characteristics
xgi. To be more specific, they consider the
simplified model
and examine whether the endogenous effects can be identified in the
presence of correlated effects. For such purpose, they considered the
social equilibrium
They show that
under a standard strict exogeneity condition, where for any vector
.
They also note that θ0 /(1 − β) is the
limit of the two-stage least squares estimator applied to the social
equilibrium (6) if (i) Eg
[ygi] and
Eg
[xgi] are observed; and (ii) there
exists an instrument zg such that E
[zgγg0] = 0
and E [zg
Eg
[xgi′]] ≠ 0.
Brock and Durlauf (2001a, 2001b) are concerned with nonlinear models
without correlated group effects. To be specific, they considered
a logit model4
They actually considered the
model
where mge denotes the
(common) expectation of yg among agents in
group g. Under some auxiliary assumption including rational
expectations and common knowledge, the model is reduced to the simpler
form presented here.
with
where hg =
(Eg
[sgi′],sgi′)′.
They show that the parameter (k,β,ξ,θ0)
is identified and can be consistently estimated by maximum likelihood
estimation. Exploiting nonlinearity, they established that the endogenous
effects β can be identified from ξ. Because
hg contains Eg
[sgi], the contextual effects are
identified from the endogenous effects in nonlinear models in general.5
They note that these identification results are
still valid in the presence of multiple equilibria.
The result in the previous section can be used to identify social
effects in the presence of correlated group effects. Let
γg0 denote the group characteristic that may be
correlated with observed variables such as Eg
[ygi] or
xgi. Assume that the conditional likelihood
given as γg0,Eg
[ygi],hg,xgi
takes the form f
(ygi,γg0 +
Eg
[ygi]β +
hg′ξ +
xgi′θ0). Note that the
explanatory variables affect the outcome through the linear index
γg0 + Eg
[ygi]β +
hg′ξ +
xgi′θ0. For example, we
may have
which is a generalized version of Brock and Durlauf's logit model
that allows correlated group effects. Interpreting
Eg
[ygi] as just one of the regular
time invariant regressors and writing wg =
(Eg
[ygi],hg′)′
and wg′δ0 =
βEg
[ygi] +
hg′ξ, we get the conditional
likelihood as f
(ygi,γg0 +
wg′δ0 +
xgi′θ0). This model can
be understood to be a nonlinear panel model with group fixed effects and
some individual-invariant regressor wg, for
which identification results have been established earlier in this paper.
We note that consistent estimation of γg0 +
wg′δ0 and
θ0 can be achieved by considering the maximum likelihood
estimator, i.e., by taking
We may therefore conclude that the social effects are identified as
such.
4. SUMMARY AND EXTENSION
In this paper, we generalized the result of Hausman and Taylor (1981) to nonlinear panel models with fixed effects.
The usefulness of the result is illustrated with some nonlinear models of
social interactions. It would be interesting to generalize Hausman and
Taylor's specification test to the nonlinear setup, which is left for
future research.
APPENDIX A: REGULARITY CONDITIONS
Condition 5. (i) Given time-invariant variables
(αi0,zi,wi),
(yit,xit) is
i.i.d. over t; (ii) for every i,
G(i)(θ0,αi0)
= 0; (iii) for each η > 0, infi
inf{(θ,α):|(θ,α)−(θ0,αi0)|>η}|G(i)(θ,α)|
> 0, where
.
Remark 1. We are assuming αi are
deterministic sequence of numbers, i.e., all the results in this paper are
results conditional on αi.
Condition 6. (i) The function
g(·;θ,α) is continuous in
;
(ii) the parameter space
is compact; (iii) there exists a function
M(Xit) such that
and supi E
[M(Xit)Q]
< ∞ for some Q > 64.
Condition 7. (i) mini E
[v(Xit;θ0,αi0)2]
> 0; (ii)
,
where
.
APPENDIX B: CONSISTENCY
LEMMA 1. Assume that Wt are i.i.d.
with E [Wt] = 0 and E
[Wt2k] <
∞. Then,
for some constant C(k).
Proof. By adopting an argument in the proof of Lemma 5.1 in
Lahiri (1992), we have
where for each fixed j ∈ {1,…,2k},
[sum ]α extends over all j-tuples of positive
integers (α1,…,αj) such that
α1 + ··· + αj
= 2k and [sum ]I extends over all ordered
j-tuples
(t1,…,tj) of
integers such that 1 ≤ tj ≤ T.
Also, C(α1,…,αj)
stands for a bounded constant. Note that if j > k
then at least one of the indices αj = 1. By
independence and the fact that E
[Wt] = 0 it follows that
whenever j > k. This shows that
for some constant C(k). █
LEMMA 2. Suppose that, for each i,
{ξit,t = 1,2,…} is a sequence
of zero mean i.i.d. random variables. We assume that
{ξit,t = 1,2,3} are independent
across i. We also assume that maxi E
[|ξit|16] <
∞. Finally, we assume that n = O(T).
We then have
for every η > 0.
Proof. Using Lemma 1, we obtain
,
where C > 0 is a constant. Therefore, we have
,
or
.
█
LEMMA 3. Suppose that Conditions 3 and 6 hold. We then have for
all η > 0 that
Proof. Let η > 0 be given. We note that
Let ε > 0 be chosen such that 2ε
maxi E
[M(Xit)] <
η/3. Divide
into subsets
such that |(θ,α) −
(θ′,α′)| < ε whenever (θ,α)
and (θ′,α′) are in the same subset. Let
(θj,αj) denote
some point in ϒj for each j.
Then,
and therefore
For
,
we have
and therefore
by Lemma 2. Combining (B.2)–(B.4) and n =
O(T), we obtain the desired conclusion. █
APPENDIX C: EXPANSION
Let
be such that
.
Letting
Ui(Xit;θ,αi)
≡
u(Xit;θ,αi)
− ρi0
v(Xit;θ,αi),
ρi0 ≡ E
[∂u(Xit;θ,αi)/∂αi′](E
[∂v(Xit;θ,αi)/∂αi′])−1
(in the likelihood framework, U is the efficient score for
θ), we can recognize that
is a solution to
Let F ≡
(F1,…,Fn) denote
the collection of distribution functions Fi,
where each Fi denotes the distribution
function of Xit. Let
,
where
denotes the empirical distribution function for the stratum i.
Define F(ε) ≡ F +
εΔiT for ε ∈
[0,T−1/2], where
.
For each fixed θ and ε, let
V(Xit;θ,αi)
≡
v(Xit;θ,αi)
and
αi(θ,Fi(ε))
be the solution to the estimating equation 0 =
∫Vi
[θ,αi(θ,Fi(ε))]
dFi(ε) and let θ(ε) be the
solution to the estimating equation
.
By Taylor series expansion, we have
,
where
is somewhere in between 0 and T−1/2. We
therefore have
The last term in (C.1) can be shown to be
op(1) by the same method as in Hahn and Newey
(2004).
LEMMA 4. For every η > 0, we have
.
Proof. Only the first assertion is proved. The second
assertion can be proved similarly. Let η be given. Recall that
.
We therefore have
,
from which we find
.
By Lemma 3, we also have
.
Therefore, for every
with probability equal to 1 −
o(T−1), we have
where ε ≡ infi inf{(θ,α)
:
|(θ,α)−(θ0,αi0)|>η}|G(i)(θ,α)|
> 0. It follows that
LEMMA 5. Suppose that, for each i,
{ξit(φ),t = 1,2,…} is a
sequence of zero mean i.i.d. random variables indexed by some
parameter φ ∈ Φ. We assume that
{ξit(φ), t = 1,2,…} are
independent across i. We also assume that
supφ∈Φ|ξit(φ)|
≤ Bit for some sequence of random
variables Bit that is i.i.d. across t and
independent across i. Finally, we assume that
maxi E
[|Bit|64]
< ∞ and n = O(T). We then
have
for every υ such that
. Here, {φi} is an
arbitrary sequence in Φ.
Proof. By Markov's inequality, we obtain
By Lemma 1, we have
for some C. Therefore, we have
LEMMA 6. Suppose that
Ki(·;θ(ε),αi(θ(ε),ε))
is equal to
∂m1+m2g(Xit;θ(ε),
αi(θ(ε),ε))/∂θm1∂αim2
for some m1 + m2 ≤
1,…,5. Then, for any η > 0, we have
Also,
for some constant
.
Proof. Note that we have
Therefore, we have
the right-hand side of which can be bounded by using Lemmas 4 and 5 in
absolute value by some η > 0 with probability 1 −
o(T−1), which proves the first claim.
The second claim can be proved similarly.
As for the third claim, we can show using Lemma 5 that
maxi|∫Ki(·;θ(ε),
αi(θ(ε),ε))
dΔiT| can be bounded by in
absolute value by CT(1/10)−υ for some
constant C > 0 and υ such that
with probability 1 − o(T−1).
█
LEMMA 7.
for some constant
.
Proof. In Hahn and Kuersteiner (2003), it is shown that
Using Lemma 6, we can see that
(∫[∂Vi(·,θ,ε)/∂αi]
dFi(ε))−1 is uniformly
bounded away from zero with probability 1 −
o(T−1). We can also see that, with
probability 1 − o(T−1),
∫[∂Vi(·,θ,ε)/∂θ]
dFi(ε) is uniformly bounded by some
constant C and
∫Vi(·,θ,ε)
dΔiT is uniformly bounded by
CT(1/10)−υ. █
LEMMA 8.
for some constant
.
Proof. In Hahn and Kuersteiner (2003), it is shown that θε(ε)
is equal to
Using Lemmas 6 and 7 we can bound the denominator of
θε(ε) by some C > 0 and the numerator
by some CT(1/10)−υ with probability 1
− o(T−1). █
LEMMA 9.
for some constant
. Here,
αiθrθr′
≡ ∂2αi
/∂θr∂θr′.
We similarly define
αiθrε.
Proof. By repeatedly differentiating 0 =
∫Vi(·,θ,ε)
dFi(ε) with respect to ε, we
obtain
The result then follows by applying the same argument as in the proof
of Lemma 7. █
LEMMA 10.
for some constant
.
Sketch of Proof. By repeatedly differentiating
with respect to ε, we obtain a characterization of
θεε(ε). (For more detailed characterization,
see Hahn and Kuersteiner, 2003.) The conclusion
follows by combining it with Lemmas 6–9. █
APPENDIX D: UNIFORM CONSISTENCY OF
[circumflex]αi
THEOREM 2. Under Conditions 3, 5, 6, and 7, we
have
where
and
Pr[maxi|κi|
≥ η] = o(1) for every η > 0. Here,
vit ≡
vit(Xit,θ0,αi0).
Let
denote αi that sets
equal to zero. (In other words, let
.) We then have the expansion
for some
.
Let vi(·,ε) ≡
Vi(θ(F(ε)),αi(Fi(ε))).
The first-order condition may be written as 0 =
∫vi(·,ε)
dFi(ε). Differentiating repeatedly with
respect to ε, we obtain
Because
dvi(·,ε)/dε =
[∂vi(·,ε)/∂θ′](∂θ/∂ε)
+
[∂vi(·,ε)/∂αi](∂αi
/∂ε), (D.2) implies that
Evaluating at ε = 0 we obtain
where Viα ≡
∂v(Xit;θ,αi)/∂αi,
Viθ ≡
∂v(Xit;θ,αi)/∂θ,
and θε(0) can be deduced from (C.2). Next,
consider
such that
is characterized by
We now combine (D.1), (D.4), and
and obtain
Here, (D.6) can be obtained by evaluating (C.2) at ε = 0.
In light of (D.7), Theorem 2 can be obtained by showing that
for any
.
This follows from representation (D.5) and also from Lemmas 6, 8, and
10.
APPENDIX E: PROOF OF THEOREM 1
Theorem 2 implies that we have
Under Conditions 4 and 5, the term
is of order op(1), and substitution of
for αi0 in (3) does not affect the asymptotic
distribution of the resultant estimator (4):
