1. INTRODUCTION
Models with endogenous regressors arise frequently in
microeconometrics. For example, suppose we want to estimate the cost
function of a competitive firm; i.e., we want to estimate the model
y = μ*(p,q) + ε, where y is
the observed cost of production, μ* the firm's cost function,
(p,q) the vector of factor prices and output, and
ε an unobserved error term. Because the firm is assumed to be a
price taker in its input markets, it is reasonable to assume that the
factor prices are exogenously set and are uncorrelated with ε. On
the other hand, because an inefficient or high-cost firm will, ceteris
paribus, tend to produce less output than an efficient firm, q
may be correlated with ε. Hence, q is endogenous.
Similarly, endogenous regressors may also arise in production function
estimation. For instance, suppose we want to estimate the model y
= μ*(l,k) + ε, where y is the
firm's output, μ* the production function, and
(l,k) the vector of labor and capital factor inputs. In
some cases it may be reasonable to believe that the firm's usage of
certain inputs (say, labor) may depend upon the unobserved quality of
management. In that case, such factors will be endogenous. Endogeneity can
also be encountered in estimating wage equations of the form y =
μ*(s,c) + ε, where y is log of wage
rate, s is the years of schooling, and c denotes agent
characteristics such as experience and ethnicity. Because years of
schooling are correlated with unobservable factors such as ability and
family background, s is endogenous. Another classic example of
endogeneity is due to simultaneity. For instance, suppose we want to
estimate the market demand for a certain good given by y =
μ*(p,d) + ε, where y is the quantity
demanded in equilibrium, p the equilibrium price, d a
vector of demand shifters, and μ* the market demand function. Because
prices and quantities are determined simultaneously in equilibrium,
p is endogenous. Several additional examples of regression models
with endogenous regressors can be found in econometrics texts; see, e.g.,
Wooldridge (2002).
These models can be written generically as follows. Let y
denote a response variable and x a vector of explanatory
variables. Suppose that, corresponding to y, there exist an
unknown function μ*(x) (we temporarily suppress the
dependence of μ* on y for pedagogical convenience) and an
unobservable random variable ε such that y =
μ*(x) + ε. The parameter of interest in this model is
μ*, and its interpretation in terms of the distribution of
(y,x) depends upon the assumptions regarding the joint
distribution of x and ε; e.g., if
then
.
In this paper we investigate models defined by more general conditions on
the distribution of (x,ε). In particular, we allow some or
all of the explanatory variables to be endogenous, i.e., correlated with
ε, so that the mean independence of ε and x does not
hold.
In the parametric case, i.e., when μ* is known up to a
finite-dimensional parameter, it is well known how to handle endogeneity.
Basically, if we have instrumental variables w that suffice to
identify μ*, then we can use two-stage least squares (2SLS), if μ*
is linear, or the more efficient generalized method of moments (GMM) to
estimate μ*. For instance, in the cost function example described
previously, the size of the market served by the firm can serve as an
instrument for q; in the production function example we could
take the wage paid by the firm as an instrument for l if the
former is exogenously set; when estimating the wage equation,
mother's education can be used to instrument for years of schooling;
and, in the market demand example, variables that shift the market supply
function but are uncorrelated with ε, such as weather or other
exogenous supply shocks, can serve as instruments for p in the
demand equation.
Recently there has been a surge of interest in studying nonparametric
(i.e., where the functional form of μ* is completely unknown) models
with endogenous regressors; see, e.g., Darolles, Florens, and Renault
(2002), Ai and Chen (2003), Blundell and Powell (2003), Newey and Powell (2003), and the references therein. In endogenous
nonparametric regression models it is typically assumed that μ* lies
in L2(x), the set of functions of x
that are square integrable with respect to the distribution of x,
and the instruments w satisfy the conditional moment restriction
.
However, in this paper we allow the parameter space for μ* to be
different from L2(x) (see Section 2 for the
motivation). Hence, our results are applicable to any endogenous
nonparametric linear model and not just to the regression models described
earlier. Apart from this, the main contributions of our paper are as
follows. (i) We develop the properties of the function that maps the
reduced form into the structural form in a very general setting under
minimal assumptions. For instance, we show that it is a closed map (i.e.,
its graph is closed) although it may not be continuous. Although lack of
continuity of this mapping has been noted in earlier papers, the result
that it is closed and further characterization of its continuity
properties as done in Lemma 2.4 seem to be new to the literature. (ii)
Newey and Powell (2003) characterize
identification of μ* in terms of the completeness of the conditional
distribution of x given w. But, in the absence of any
parametric assumptions on the conditional distribution of x given
w, it is not clear how completeness can be verified in practice.
In fact, as Blundell and Powell (2003) point
out, the existing literature in this area basically assumes that μ* is
identified and focuses on estimating it. Because failure of identification
is not easily detected in nonparametric models (in Section 3 we provide
some interesting examples showing that μ* can be unidentified in
relatively simple designs), we investigate what happens if the
identification condition for μ* fails to hold or cannot be easily
checked by showing how to determine the “identifiable part” of
μ* by projecting onto an appropriately defined subspace of the
parameter space, something that does not seem to have been done earlier in
the literature. (iii) In Section 4 we examine the identification of linear
functionals of μ* when μ* itself may not be identified. We relate
identification of μ* to the identification of its linear functionals
by showing that μ* is identified if and only if all bounded linear
functionals of μ* are identified. To the best of our knowledge, the
results in this section are also new to the literature.
We do not focus on estimation in this paper. In addition to the papers
mentioned earlier, readers interested in estimating endogenous
nonparametric models should see, e.g., Pinkse (2000), Das (2001), Linton,
Mammen, Nielsen, and Tanggaard (2001), Carrasco,
Florens, and Renault (2002), Florens (2003), Hall and Horowitz (2003), Newey, Powell, and Vella (2003), and the references therein. Additional works
related to this literature include Li (1984) and
Roehrig (1988). Note that our identification
analysis is global in nature because the nonparametric models we consider
are linear in μ*. We hope our results will motivate other researchers
to study local properties of nonlinear models of the kind considered by
Blundell and Powell (2003) and Newey and Powell
(2003).
2. IDENTIFICATION IN A GENERAL SETTING
The introduction was motivated by looking at endogenous nonparametric
regression models of the form y = μ*(x) + ε,
where
.
But in many cases the parameter space for μ* can be a linear function
space different than L2(x). For instance,
suppose that x = (x1,x2)
and μ* is additive in the components
1 A
good discussion of these models, though without any endogeneity, can be
found in Hastie and Tibshirani (1990).
of
x; i.e., μ*(
x) =
μ
1*(
x1) +
μ
2*(
x2), where
μ
j* lies in
L2(
xj) for
j =
1,2. Notice that once μ* is identified, we can recover the components
up to an additive constant by marginal integration; i.e.,
and a similar operation can be carried out to recover μ2*.
An alternative model may be based on the assumption that
μ*(x) = x1′θ* +
μ2*(x2), where θ* is a
finite-dimensional parameter and μ2* ∈
L2(x2). This leads to an
endogenous version of the partially linear model proposed by Engle,
Granger, Rice, and Weiss (1986) and Robinson
(1988). Sometimes we may have information
regarding the differentiability of μ* that we want to incorporate into
the model; in this case, we might assume that μ* is an element of a
Sobolev space. We could also allow for μ* to have certain shape
restrictions. In particular, because we assume that μ* belongs to a
linear space, shape restrictions such as homogeneity and symmetry are
permissible for μ*. These variations clearly illustrate the advantage
of framing our problem in a general setting. So we now frame our problem
in a general Hilbert space setting. The geometric nature of Hilbert spaces
allows us to derive a lot of mileage from a few relatively simple
concepts.
Let y denote the response variable that is assumed to be an
element of
,
a separable Hilbert space with inner product
〈·,·〉 and induced norm ∥·∥. Also,
let
denote a known linear subspace of
(note that
is not assumed to be closed). Assume that, corresponding to y,
there exists an element
.
The vector μy* is a summarization of the
distribution of y and may be viewed as the parameter of interest.
If y − μy* is orthogonal to
,
then μy* is simply the orthogonal projection of
y onto
.
Here we assume instead that there exists a known linear subspace of
,
denoted by
,
such that 〈y −
μy*,w〉 = 0 for all
;
i.e., y − μy* is orthogonal to
,
which we write as
We call
the “model space” and
the “instrument space.” The symbol
denotes the set of all
for which the model holds; i.e., for each
there exists a
such that (2.1) holds. Because there is a one-to-one correspondence
between random variables and distribution functions,
can also be interpreted as the set of all distributions for which (2.1)
holds. Note that because
always includes
,
it is nonempty. Also, continuity of the inner product implies that
whenever (2.1) holds, y − μy* is
orthogonal to the closure of
.
Therefore,
can be assumed to be closed without loss of generality.
Clearly, the endogenous nonparametric regression models described in
the introduction are a special case of (2.1) by letting
,
and
be the set of random variables of the form y =
μy*(x) + ε, where
.
It is easy to see that μy* is identified,
i.e., uniquely defined, if and only if the following condition holds.
Condition (I). If
satisfies
,
then m = 0.
Henceforth, we refer to Condition (I) as the “identification
condition.” Let
denote orthogonal projection from
onto
using the inner product 〈·,·〉. Then the
identification condition can be alternatively stated as follows: if
satisfies
,
then m = 0.
Example 2.1 (Linear regression)
Let
be the familiar Hilbert space of random variables with finite second
moments equipped with the usual inner product
.
Also, x (the s × 1 vector of explanatory
variables) and w (the d × 1 vector of instrumental
variables) are random vectors whose coordinates are elements of
.
Moreover,
(resp.
) is the linear space spanned by the coordinates of x
(resp. w). Note that in this example
are both finite-dimensional subspaces of
.
By (2.1), for a given
there exists a linear function μy*(x) =
x′θy* such that 〈y
−
x′θy*,w′α〉
= 0 for all
;
i.e.,
.
Condition (I) states that if
〈x′θy*,w′α〉
= 0 for all
,
then θy* = 0. Hence, μy*
or, equivalently, θy* are uniquely defined if and
only if
has full column rank. Obviously, the order condition d ≥
s is necessary for
to have full column rank.
Example 2.2 (Nonparametric regression)
Again,
is the Hilbert space of random variables with finite second moments
equipped with the usual inner product and (x,w) are
random vectors whose components are elements of
.
But, unlike the previous example,
are now infinite-dimensional linear subspaces of
consisting of square integrable functions. By (2.1), for a given
y in
there exists a function μy* in
L2(x) such that
holds for all g ∈ L2(w).
Condition (I) states that if a function f ∈
L2(x) satisfies
for all g ∈ L2(w), then
f = 0 w.p.1;
2 Because
L2(x) and L2(w)
are equivalence classes of functions, equality statements in
L2(x) and L2(w)
hold w.p.1.
i.e., if
for any f in L2(x), then f
= 0 w.p.1. But this corresponds to the completeness of pdf
(x|w). Therefore, μy* is
uniquely defined if and only if the conditional distribution of
x|w is complete, a result obtained earlier by
Florens, Mouchart, and Rolin (1990, Ch. 5) and
Newey and Powell (2003). To get some intuition
behind the notion of completeness, observe that if x and
w are independent, then completeness fails (of course, if
w is independent of the regressors then it is not a good
instrument and cannot be expected to help identify
μy*). On the other extreme, if x is fully
predictable by w then completeness is satisfied trivially, and
the endogeneity and identification problems disappear altogether. In fact,
we can show the following result.
LEMMA 2.1. The conditional distribution of x|w is
complete if and only if for each function f (x) such
that
, there exists a function g(w) such that
f (x) and g(w) are
correlated.
Hence, in the context of nonparametric regression we can think of
completeness as a measure of the correlation between the model space
L2(x) and the instrument space
L2(w).
Let us assume that Condition (I) holds for the remainder of
Section 2. Hence, for each
there exists a unique
such that
.
It follows that
is a linear subspace of
and the map y [map ] μy* is a linear
transformation on
.
Therefore, from now on we write Vy for
μy* so that
denotes a linear map such that μy* = Vy.
Employing well-known terminology, V is just the function that
maps the reduced form into the structural form. Hence, a clear description
of the properties of V is central to understanding the
identification and estimation problems in nonparametric linear models with
endogenous regressors.
We now study the properties of V. Define
.
Because it is straightforward to show that
is the smallest linear subspace of
satisfying Condition (I), we may view
as the “minimal” instrument space. Let
.
Because
,
by definition of
we know that
.
But, letting I denote the identity operator, we can write
y = Vy + (I − V)y,
where
.
Hence,
.
Furthermore, because
,
we have
.
This shows that when applied to elements of
,
the projection
has the same properties as orthogonal projection on
.
Next, let
denote the restriction of
.
Then
is a continuous linear mapping from
with inverse
3 Because
is bounded, its restriction to
is also bounded and, hence, continuous. Now let m1
and m2 denote elements of
.
Suppose w1 = w2. Then
.
Hence,
.
It follows from Condition (I) that m1 =
m2 so that
is one-to-one and, by definition, the range of
.
Therefore, because
is one-to-one and onto, it has an inverse
.
.
Clearly,
is also a linear map. Therefore, we can characterize V as
The next example describes how V looks in some familiar
settings.
Example 2.3
In Example 2.1,
is the linear space spanned by the coordinates of w. Hence,
corresponds to the best linear predictor given w; i.e.,
.
It is easy to show that
.
Therefore, the map
is given by
.
But because (Vy)(x) can be written as
x′θy*, it follows that
θy* here is just the population version of the
usual 2SLS estimator. By contrast,
in Example 2.2 is the infinite-dimensional space
L2(w). Hence,
is the best prediction operator
.
Therefore, in Example 2.2 we have
.
Before describing additional properties of V, in Lemma 2.2 we
propose a series-based approach for determining V. As illustrated
by the examples given subsequently, this approach may also be useful as
the basis of a practical computational method for estimating V.
However, as noted earlier, a full consideration of estimation issues is
beyond the scope of the current paper. Instead, the reader is referred to
Pinkse (2000), Darolles et al. (2002), Ai and Chen (2003),
Hall and Horowitz (2003), and Newey and Powell
(2003) for series estimation of endogenous
nonparametric models.
LEMMA 2.2. Let
m0,m1,m2,…
be a basis for
such that
for i ≠ j. Then,
This result is similar in spirit to the eigenvector-based
decomposition of Darolles et al. (2002) although
we use a different basis in our representation. It demonstrates that if
μy* is identified then it can be explicitly
characterized in the population by a series representation using a special
set of basis vectors (if
so that endogeneity disappears, then Vy is just the projection
onto
as expected). The basis functions needed in Lemma 2.2 can be constructed
from an arbitrary basis by using the well-known Gram–Schmidt
procedure as follows. Let
d0,d1,d2,…
be a basis for
.
Define m0 = d0 and let
Then
m0,m1,m2,…
is a basis for
satisfying
for i ≠ j.
The following example illustrates the usefulness of Lemma 2.2.
Example 2.4
Let x, w, and ε be real-valued random
variables such that x and ε are correlated,
,
and (x,w) has a bivariate normal distribution with mean
zero and variance
where ρ ∈ (−1,1)[setmn ]{0}. Suppose y =
μy*(x) + ε, where
μy* is unknown and
.
Because
,
the conditional distribution of x given w is complete.
Hence, μy* is identified. Now let φ be the
standard normal probability density function (p.d.f.) and
denote Hermite polynomials that are orthogonal with respect to the
usual inner product on L2(x). From Granger
and Newbold (1976, p. 202) we know that if
,
then
.
This result ensures that the Hermite basis satisfies the requirement in
Lemma 2.2.
4 Basis vectors that satisfy Lemma
2.2 for more general bivariate distributions can be constructed by using
some of the results discussed in Buja (1990).
It, plus the facts that
,
shows that we can write μy* explicitly as
There are some interesting consequences of (2.4). For instance, if
happens to be a polynomial of degree p, then
μy* will also be a polynomial of degree p
because
for all j > p. As a particular example, suppose
that
.
Then it is easily seen that μy*(x) =
a − c/ρ2 +
bx/ρ + cx2/ρ2. It
is also clear from (2.4) that an estimator for μy*
can be based on the truncated series for Vy. This is discussed in
the next example.
Example 2.5 (Example 2.4 cont.)
As mentioned earlier, an estimator for μy* can
be obtained by truncating the series in (2.4). Suppose we have a random
sample
(y1,x1,w1),…,(yn,xn,wn)
from the distribution of (y,x,w). Let
denote the sample analog of
based on these observations; i.e.,
.
By (2.4), an estimator of μy* is given by
where kn is a function of the sample size
such that kn ↑ ∞ as n
↑ ∞. In this example we show that
is mean-square consistent and derive its rate of convergence. Suppose for
convenience that
are bounded. Then, as shown in the Appendix, for some α > 0 the
mean integrated squared error (MISE) of
is given by
Although (2.5) holds for a stylized setup, it is very informative;
e.g., it is clear that the MISE is asymptotically negligible if
kn ↑ ∞ sufficiently slowly.
Hence,
is mean-square consistent for μy*, though its
rate of convergence is slow. It converges even more slowly if the
instrument is “weak,” i.e., if |ρ| is small.
In fact, because the MISE converges to zero if and only if
kn log ρ−2 + log
kn − log n ↓
−∞, it follows that kn must be
O(log n) or smaller. Therefore, even in this simple
setting where the joint normality of regressors and instruments is known
and imposed in constructing an estimator, the best attainable rate of
decrease for the MISE is only O({log
n}−α). This suggests that rates of
convergence that are powers of 1/log n, rather than
1/n, are relevant for endogenous nonparametric regression
models when the distribution of (x,w) is unknown. Rates
better than O({log n}−α) can be
obtained by imposing additional restrictions on
μy*; e.g., Darolles et al. (2002, Thm. 4.2) and Hall and Horowitz (2003, Thm. 4.1) achieve faster rates by making the
eigenvalues of certain integral operators decay to zero at a fast enough
rate, thereby further restricting μy*
implicitly.
Example 2.6 (Endogenous nonparametric additive
regression)
Let y = μ1*(x) +
μ2*(z) + ε, where μ1* and
μ2* are unknown functions such that
;
i.e., x is the only endogenous regressor. In this example, the
model space is L2(x) +
L2(z), and the instrument space is
L2(w,z). Assume that
(x,z,w) is trivariate normal with mean zero and
positive definite variance-covariance matrix
.
Because the conditional distribution of x given
(w,z) is normal with mean depending on
(w,z) and the family of one-dimensional Gaussian
distributions with varying mean is complete,
μ1*(x) + μ2*(z) is
identified. We now use the approach of Lemma 2.2 to recover
μ1* and μ2*. Let
μ*(x,z) = μ1*(x) +
μ2*(z). Note that
for constants
{αj}j=0∞ and
{βj}j=0∞. But
because
,
we have
.
Solving these simultaneous equations for each j, it follows
that
Therefore, using the fact that
for j ≥ 1,
Hence, μ1* and μ2* are identified.
Next, we consider an iterative scheme for determining V.
5 See Petryshyn (1963)
for a detailed treatment of recursive methods of this type.
The
advantage of this approach is that we do not have to explicitly calculate
the inverse operator
.
We only need
,
where the latter denotes orthogonal projection onto the closure of
.
In contrast, the series approach of Lemma 2.2 did not require any
knowledge of
.
LEMMA 2.3. Fix
and consider the equation
for
. Let m0 denote its solution and
define
. If there exist a constant a ≠ 0 and an
such that
converges to m* as n ↑ ∞, then m* =
m0.
This result, which is related to the Landweber–Fridman procedure
described in Kress (1999, Ch. 15) and Carrasco
et al. (2002), shows that if the sequence
mn converges, then it converges to
m0. Therefore, given y, we can obtain
Vy by applying this procedure to
.
Because
,
convergence in Lemma 2.3 is ensured if there exists a nonzero constant
a such that the partial sum
converges pointwise for each
.
A well-known sufficient condition for this to happen is that
.
Of course, if
so that there is no endogeneity, then
,
and there are no further adjustments to m1.
Example 2.7
The iterative procedure of Lemma 2.3 also works for Example 2.4. To
see this, let a = 1 and note that
,
where
.
Hence,
,
and by (2.4) it follows that
as n ↑ ∞; i.e., mn
converges in mean-square to μy*.
Before ending this section, we comment briefly on the pervasiveness of
“ill-posed” endogenous nonparametric models. Recall that
Condition (I) guarantees that for each
the vector μy* is uniquely defined, i.e.,
V : y [map ] μy* is a function
from
into
.
But Condition (I) is not strong enough to ensure that this function is
continuous;
6 Discontinuity of V
means that slight perturbations in the response variable can lead to
unbounded changes in μy*, the parameter of
interest associated with it. This lack of stability makes precise the
sense in which some endogenous nonparametric models can be called
“ill-posed.” Note that sometimes a statistical problem is said
to be ill-posed because of data issues; e.g., classic nonparametric
regression itself can be called ill-posed because we cannot estimate the
graph of an unknown function using only a finite amount of data. However,
the notion of ill-posedness described here has nothing to do with sample
information but is inherent to the model.
i.e., the
identification condition by itself is not strong enough to ensure that the
problem is well-posed. However, it can be shown that
V is a
closed linear operator. To see this, let
y1,
y2,… denote a sequence
in
such that
as n ↑ ∞ and suppose that
as n ↑ ∞. To show that V is closed, it
suffices to show that
.
Note that, for each
and, because yn −
Vyn → y − m as
.
Hence, by definition of
.
The next result characterizes the continuity of V.
LEMMA 2.4. The following statements are equivalent: (i) V is
continuous on
; (ii)
is closed; (iii) if
m1,m2,… is a sequence
in
such that
as n ↑ ∞, then mn →
0 as n ↑ ∞; (iv)
is closed; (v) there exists a closed linear
subspace
of
such that
.
The restrictive nature of this lemma reveals that well-posed
endogenous nonparametric models are an exception rather than the rule;
e.g., even the simple Gaussian setting of Example 2.4 is not sufficient to
make the problem there well-posed. To see this, let
denote the normalized nth Hermite polynomial. It is then easy to
verify that
converges to zero in mean-square whereas fn
does not. Therefore, (iii) does not hold, and, hence, V is not
continuous. Of course, if
is finite-dimensional (as in parametric models, or, in nonparametric
models where the regressors are discrete random variables with finite
support),
7 See, e.g., Blundell and Powell
(2003) and Florens and Malavolti (2003).
then (iii) holds and
V is
continuous. Similarly, if
is finite-dimensional then
will be finite-dimensional and, hence, closed, implying that V
is continuous. But these are clearly very special cases. A practical
consequence of ill-posedness is that some type of
“regularization” is needed in estimation procedures to produce
estimators with good asymptotic properties. For instance, a
truncation-based regularization ensures convergence of the estimator
described in Example 2.5. For more about the different regularization
schemes used in the literature, see, e.g., Wahba (1990, Ch. 8), Kress (1999,
Ch. 15), Carrasco et al. (2002), Loubes and
Vanhems (2003), and the references therein.
3. UNDERIDENTIFICATION
In this section, we investigate the case where
μy* in (2.1) fails to be uniquely defined. As
mentioned earlier in Example 2.2, Newey and Powell (2003) and others have characterized identification of
the endogenous nonparametric regression model in terms of completeness of
the conditional distribution of x given w. They also
point out that it is sufficient to restrict pdf
(x|w) to the class of full rank exponential
densities for it to be complete. However, Examples 3.2 and 3.3 illustrate
that this sufficient condition can fail to hold in relatively simple
cases. Furthermore, if the distribution of x|w is
not assumed to be parametric, completeness can be very hard to verify.
Hence, it is important to know what happens when completeness fails or
cannot be checked. We now focus on this issue.
Let
be the set of all “identification-destroying” perturbations
of μy*. From Condition (I) it follows that
μy* is identified if and only if
.
Note that
is a closed linear subspace of
.
The properties of
play an important role in the identification of
μy*.
Example 3.1 (Underidentification in linear
regression)
We maintain the setup of Example 2.1. For linear instrumental
variables regression it is easily seen that
.
Hence, the identification condition fails to hold, i.e.,
,
if
is not of full column rank.
Example 3.2 (Underidentification in nonparametric
regression)
Let y = μy*(x) + ε,
where μy* ∈
L2(x) is unknown. The regressor is
endogenous, but we have an instrument w satisfying
.
Suppose that x = w + v, where
.
Hence,
for a.a. w ∈ [−½,½]}.
Because
,
it is straightforward to show that
holds for a.a. w ∈ [−½,½] if
and only if f is periodic in the sense that f
(x) = f (1 + x) for a.a.
.
Thus,
can be explicitly characterized as
for a.a.
.
Because
is clearly not equal to {0}, Condition (I) does not hold. Therefore,
μy* is not uniquely defined and, hence, cannot be
estimated even for the simple design given in this example.
Example 3.3 (Underidentification in nonparametric
additive regression)
Let y = μ1*(x) +
μ2*(z) + ε, where μ1* and
μ2* are unknown functions
L2(x) and L2(z),
respectively, and
;
i.e., both x and z are endogenous, but we only have one
instrument w. Obviously, here the model space is
L2(x) + L2(z),
but the instrument space is L2(w). As in
Example 2.6, assume that (x,z,w) are jointly
normal with mean zero and variance Ω. Because the conditional
distribution of x,z|w is not complete,
it follows that μ1*(x) +
μ2*(z) is not identified. In fact, it can be shown
that
,
where
.
8 Because
for each j; i.e.,
.
Next, let
.
Then m0(x,z) = f
(x) + g(z) for some f ∈
L2(x) and g ∈
L2(z) such that
;
i.e.,
.
Hence, writing
,
it follows that
if and only if
.
By the completeness of Hermite polynomials in
L2(w), this implies that
αj
ρxwj +
βj
ρzwj = 0 for each j.
Therefore,
;
i.e.,
.
Suppose that a μy* satisfying (2.1) is not
uniquely defined. Loosely speaking, this means that the model space is
“too large”; i.e., it contains more than one element
satisfying (2.1). Hence, to obtain identifiability, we may choose a
smaller model space. This approach is analogous to eliminating redundant
regressors in an underidentified linear regression model. We now formalize
this intuition. For a given
,
define
.
Identification holds when
consists of a single element. Otherwise,
is a collection of elements that cannot be distinguished based on (2.1).
A nice property of
is that each of its elements has the same projection onto
,
the orthogonal complement of
.
Hence,
is a natural choice for the reduced model space;
9 There is an analogy to
in the specification testing literature. Suppose we want to test the null
hypothesis
against the alternative that it is false. Consider the alternative
,
where δ denotes a deviation from the null. It is obvious that no test
will be able to reject the null if δ is a linear function of
x. The only detectable perturbations are those that are
orthogonal to linear functions, i.e., those satisfying
.
i.e., if μ
y** is the orthogonal
projection of an arbitrarily chosen element of
onto
,
then μy** can be regarded as the
“identifiable part” of μy*. In
technical terms, when Condition (I) does not hold, the “true
parameter” of the model is, in effect, an equivalence class of
elements of
,
which we have denoted by the symbol
.
This class of true parameters may be described in terms of their common
features as follows. Because
,
each
may be decomposed into two components
.
But because
is the same for all
,
each equivalence class
may be described by a single element μy**, which
we refer to as the identifiable part of μy*. We
may take this canonical element to be
.
It is easy to show that μy** is an element of
.
10 Let
be arbitrary. Then
.
But
by definition of
,
and
because
.
Therefore,
.
Because
,
it follows that
.
The remaining elements of
are those
such that
;
i.e., all
of the form
.
Example 3.4 (Example 3.1 cont.)
Suppose that
is not of full column rank so the identification condition fails to hold.
Here,
.
Because
,
where
,
we have
.
Hence, we can only identify linear functions of the form
x′Aθ. Of course, if the identification
condition holds, i.e.,
is of full column rank, then A reduces to the identity matrix
and
with θ* as defined in Example 2.3.
Example 3.5 (Example 3.2 cont.)
The identifiable part of μy* is given by
projecting
onto
,
where
for a.a. w ∈ [−½,½]}. Recall
that x has the triangular distribution on [−1,1];
i.e., the p.d.f. of x is given by h(x) = 1 +
x for −1 ≤ x ≤ 0 and h(x)
= 1 − x for 0 ≤ x ≤ 1. It can be shown that
,
where
11 Showing
is easy. Next, by the projection theorem,
Now let
.
Because g is an element of
,
its projection onto
is the zero function. Hence, using the expression for
,
it follows that
;
i.e.,
.
Therefore,
.
Hence, the identifiable part of μy* is a
function f satisfying f
(x)h(x) = −f (x +
1)h(x + 1) + c for a.a. x ∈
[−1,0] and some constant c.
Example 3.6 (Example 3.3 cont.)
Now let us determine the identifiable part of the underidentified
model in Example 3.3. It can be shown that
,
where
.
12 Recall that
.
Now let
.
Because
for all
.
Therefore,
.
Next, let
.
Then m1(x,z) = f
(x) + g(z) for some f ∈
L2(x) and g ∈
L2(z). But because m1 is
orthogonal to
by definition,
for all k. Hence, writing
,
we have
.
Thus (ρxzj
ρxwj −
ρzwj)αj
+
βj(ρxwj
− ρxzj
ρzwj) = 0 for j ≥
1, and
.
It follows that
;
i.e.,
.
Hence, we can identify only those additive functions whose
Hermite representation is of the form
,
where c denotes a constant. For example, suppose that
ρxw = ρzw; i.e., the
instrument has the same correlation with each regressor. Then
consists of functions of the form
;
i.e., only elements of L2(x) +
L2(z) of the form c + f
(x) + f (z) are identified.
As mentioned earlier, underidentification may be viewed as a
consequence of the fact that the model space
is too big. Hence, to obtain identifiability, we may choose a smaller
model space. A natural choice for this reduced model space is
.
But to use
in place of
,
we must verify that it satisfies two conditions. The first is that for
each
there exists a
such that
.
The second is that
satisfies Condition (I). It is easy to see that both
these conditions are satisfied: Fix
.
Then by (2.1), there exists
such that
.
Let
.
Then
.
Because
is an element of
and, hence, orthogonal to
,
it follows that
.
This shows that
satisfies the first requirement. Next, let m denote an element
of
that is orthogonal to
.
By definition, there exists
such that
.
Because
,
for any
we have
.
It follows that
and, hence, that
.
Therefore, for
implies that m = 0, proving that
also satisfies the second requirement.
Note that to describe μy**, we can use
in place of
in the theory developed in Section 2. Because Condition (I) is satisfied
by
,
all of the previous results hold with respect to this choice and
μy** = Vy, where V is now based
on
.
5. LINEAR MOMENT CONDITIONS AND INSTRUMENTAL
VARIABLES
We now formulate (2.1) in terms of moment conditions generated by
linear operators and also provide an example to illustrate the usefulness
of this characterization. Although this formulation may seem different
from the manner in which (2.1) is stated, we show that the two
representations are in fact logically equivalent. Let y denote an
element of
and let
be a known linear subspace of
.
Suppose that corresponding to y is an element of
,
denoted by μy*, defined as follows: “There
exists a linear subspace of
,
denoted by
,
and a continuous linear operator
such that T(y − μy*) =
0.” Let
denote the set of
for which this model holds; i.e., for each
there exists
such that T(y − μy*) = 0.
Note that because
,
the domain of T may be taken to be
.
Condition (I-M). If
satisfies Tm = 0, then m = 0.
Condition (I-M) is necessary and sufficient for
μy* to be uniquely defined (the proof is
straightforward and hence is omitted). We say that
is a “moment-condition” model if there is a linear subspace
and a continuous linear function
such that for each
there exists
satisfying T(y − m) = 0 and Condition
(I-M) holds. We call T the “identification function.”
Similarly, we say that
is an “instrumental variables” model if there is a closed
linear subspace
such that for each
there exists
satisfying
and Condition (I) holds. In fact, it can be easily shown that
is a moment-condition model if and only if it is an instrumental
variables model.
The following example illustrates a situation where the nature of the
available information makes it easier to write an endogenous nonparametric
regression model as a moment-condition model.
Example 5.1
Let y = μy*(x) + ε,
where
for s > 1 and μy* ∈
L2(x) is unknown. The regressors are
correlated with the error term such that the conditional distribution of
ε given x satisfies the index restriction
for some known function h with dim
h(x) < s. This, e.g., is related to the
exclusion restriction assumption maintained in Florens, Heckman, Meghir,
and Vytlacil (2002). Let
.
Our model has content if the linear moment condition
T(y − μy*) = 0 holds for
some μy* ∈
L2(x). For μy* to be
uniquely defined, by Condition (I-M) we need that
only for μy*(x) = 0 w.p.1. This reveals
that μy*'s of the form
μy*(x) = f
(h(x)) are not identifiable. Therefore, letting
denote the set of all functions in L2(x)
that are not functions of h(x), it follows that
is a moment-condition model with identification function T.
Next, we show how to write
as an instrumental variables model. Let
be the null space of T; i.e.,
is the set of all random variables ε such that
.
Because
is a moment-condition model with identification function T, by
definition there exists a unique
such that
.
It follows that
is also an instrumental variables model with instrument space
,
where
.
13 Define
and let v and ε denote arbitrary elements of
,
respectively. Then by the properties of ε and v,
;
i.e.,
,
implying that
.
Next, let
.
Because the random variable
satisfies
,
we obtain that
is an element of
.
Hence,
,
which implies that
.
Therefore, u ⊥
L2(h(x)). Now write u =
u1 + u2, where
u1 ∈ L2(x) and
u2 ∈
L2⊥(x). Because
,
it follows that
;
i.e.,
0. But because
implies that u2 = 0 w.p.1. Hence, u =
u1 ∈ L2(x). Thus
u ∈ L2(x) and u ⊥
L2(h(x)); i.e.,
.
Because u was chosen arbitrarily in
,
we conclude that
.
APPENDIX: Proofs
Proof of (2.5). By the orthogonality of Hermite polynomials,
Hence,
But because
and the maps
are bounded by assumption,
where c is a generic constant. Thus,
.
It follows that the MISE is majorized by
.
But
for some α > 0 under some smoothness conditions on
μy*; see, e.g., Milne (1929, Cor. I). The desired result follows. █
Proof of Lemma 2.1. Let S denote the statement “For
each function f (x) such that
,
there exists a function g(w) such that f
(x) and g(w) are correlated.” First,
suppose that S is not true. Hence, there exists a nonzero
function f (x) satisfying
such that
for all g(w). But this implies that
;
i.e., pdf (x|w) is not complete. Next,
suppose that the conditional distribution of x|w
is not complete. Hence, there exists a function f (x)
such that
but f (x) ≠ 0; i.e., var f (x)
> 0. Clearly, this implies that f (x) is uncorrelated
with all functions of w. The desired result follows. █
Proof of Lemma 2.2. We first consider the series expansion for
Vy. Let
.
Hence, there exists a unique (because Condition (I) is assumed to hold)
such that
.
This implies that
.
Because
,
we can write
for some constants α0,α1,…. Hence,
,
and using the fact that
for i ≠ j, it follows that
.
Hence,
because
.
Therefore,
.
Next, we consider the series for
.
Let
be any element of
and let
.
Note that
by definition of
.
Because
is contained in
,
by the previous result
But by (2.2),
.
The desired result follows. █
Proof of Lemma 2.3. Note that
.
If limn↑∞ mn
exists, then mn+1 −
mn → 0 as n ↑ ∞ so
that
as n ↑ ∞. Because
as n ↑ ∞, it follows that
.
Let m = m0 − m*; then
.
Hence,
so that
.
It follows from Condition (I) that m = 0, i.e., that m*
= m0, proving the result. █
Proof of Lemma 2.4. We show that (i) ⇒ (iii) ⇒ (ii) ⇒
(i) and (i) ⇒ (iv) ⇒ (v) ⇒ (i).
First, suppose V is continuous. Let
m1,m2,… be a sequence in
such that
and let
.
Note that for each
so that
.
Hence, mn =
Vyn. Because yn
→ 0 as n ↑ ∞, it follows that
mn → 0. Therefore, (iii) holds.
Next, assume that (iii) holds. Let
y1,y2,… denote a sequence
in
such that limn↑∞
yn = y for some
.
We need to show that
.
Observe that because
by (2.2) and
is continuous, it follows that
as n ↑ ∞. But as
is in
,
by (iii) we have
as n ↑ ∞. This shows that
yn − Vyn
− (y − Vy) → 0 as n ↑
∞. Because
,
we know that
for each n. Thus, by the continuity of the inner product,
〈y − Vy,w〉 = 0 for all
.
Hence,
is closed.
Next, suppose that
is closed. Then because V is a closed linear operator with
domain
,
continuity of V follows by the closed graph theorem; see, e.g.,
Kreyszig (1978).
Now, let (i) hold. This implies that
is closed. Let w1,w2,…
denote a sequence in
with limit
and, for each n = 1,2,…, let
satisfy
.
Because
,
it follows that
.
Thus
because
is closed. Next,
by continuity of V. Hence,
,
so that
as n ↑ ∞. Because
,
we know that
.
Hence,
is closed.
Next, assume that
is closed. Because
,
the result follows by taking
.
Finally, suppose that (v) holds. Then the restriction of
is closed, and, because
is closed, that restriction is continuous by the closed graph theorem.
Let m1,m2,… denote a
sequence in
such that
as n ↑ ∞ and let
so that w1,w2,… is a
sequence in
such that wn → 0 as n ↑
∞. Note that
for each n, implying that
;
i.e.,
because
is the restriction of
.
Hence, mn = Vwn
by (2.2). Because V is continuous on
,
it follows that limn↑∞
mn = V
limn↑∞ wn = 0.
Thus (iii) holds. But we have already shown that (iii) implies (i). The
desired result follows. █
Proof of Theorem 4.1. ρ(μy*) is identified
if and only if all
for which
yield the same value of ρ(μy*). Suppose that
Condition (I-F) holds and that
m1,m2 are elements of
satisfying
for j = 1,2. Then
,
so, by Condition (I-F), ρ(m1 −
m2) = ρ(m1) −
ρ(m2) = 0. Hence,
ρ(μy*) is identified. Next, suppose that all
m for which
yield the same value of ρ(m). Suppose
satisfies
.
Then
.
Because ρ(μy*) is identified,
ρ(m1 + m) = ρ(m1)
+ ρ(m) = ρ(m1) so that
ρ(m) = 0. Hence, Condition (I-F) holds. █
Proof of Theorem 4.2. Suppose Condition (I) holds and
satisfies
.
Then, by Condition (I), m = 0 so that ρ(m) = 0
for any linear functional ρ. Next, suppose that Condition (I-F)
holds for any continuous linear functional ρ and let
satisfy
.
Consider the bounded linear functional ρ(m1) =
〈m,m1〉, where
.
Because Condition (I-F) holds, it follows that
〈m,m〉 = 0. But this implies that m =
0. Hence, Condition (I) holds. █
