Let I(M;FYjZ;RZ) denote the identi…ed set of functions g(; ), or, with a parametric speci…ca-
tion, the identi…ed set of parameters, delivered by a modelM and the conditional distribution of
Y = (Y1; Y2) conditional on Z for each value z in the support of Z. Characterizations of the set I(M;FYjZ;RZ) are provided under various speci…cations of the modelM.
Throughout this section the restriction that there is a single scalar source of unobservable heterogeneity is maintained. This restriction can be relaxed in a GIV analysis as shown in Chesher and Rosen (2014) which studies a binary outcome random coe¢ cients model admitting multiple sources of heterogeneity.
A structural function capturing the restrictions of the threshold crossing index model is given in (11), and theU-level sets of the structural function are simply closed intervals, as follows.
U(y; z;g) = 8 > < > : [0; g(y2; z)] ; y1 = 0 [g(y2; z);1] ; y1 = 1 (60)
Since the structural function is characterized by the functiong(; ), “g”is used as the argument of theU-level set.
Theorem 3 tells us that core determining sets are connected unions of the level sets that comprise the support of the random set U(Y; Z;g). All such unions are closed intervals, subsets of the unit interval, [0;1], with either 0 as a lower endpoint or1 as an upper endpoint. The collection
Q=f[0; u];[u;1] :u2[0;1]g
is a core determining collection.
The identi…ed set of functions g(; ) is
Ig(M;FYjZ;RZ) =fg(; ) :8S 2Q; GU(S) Cg(Sjz); a.e. z2 RZg (61)
whereCg(Sjz)is the conditional containment probability for the random setU(Y; Z;g)givenZ =z.
Cg(Sjz) FYjZ(fy:U(y; z;g) Sg jz)
3 3
These three papers were completed before the development of the general approach set out in CR17. They deliver the same identi…ed sets as CR17 but provide laborious constructive proofs of sharpness, which CR17 renders unnecessary.
The probability GU(S) in (61) is simply the length of the interval S because U is normalized
uniform on[0;1].
Consider the containment functional probability Cg(Sjz) that appears in (61) and let P(Ejz)
denote the conditional probability of an event E delivered by the distributionFYjZ(jz)2 FYjZ. The randomU-level setU(Y; z;g) has realizations which are a subset of[0; u]only whenY1 = 0
and g(Y2; z) u. The random setU(Y; z;h) has realizations which are a subset of[u;1]only when
Y1 = 1 andg(Y2; z) u. There is therefore the following characterization ofIg(M;FYjZ;RZ): the
identi…ed set of functionsg(; ) under the restrictions of the model M.
Ig(M;FYjZ;RZ) =A0\ A1
where
A0 =fg(; ) :8u2(0;1) u P[Y1= 0^g(Y2; z) ujz]; a.e. zg A1 =fg(; ) :8u2(0;1) 1 u P[Y1 = 1^g(Y2; z) ujz]; a.e. zg:
This can be expressed as follows.
Ig(M;FYjZ;RZ) =fg(; ) :8u2(0;1)
1 P[Y1= 1^g(Y2; z) ujz]
u
P[Y1= 0^g(Y2; z) ujz]; a.e. zg (62)
If one uses a model employing a parametric speci…cation then these expressions de…ne identi…ed sets of parameters. In an application in Section 8.2 we consider IV probit type models in which
g(y2; z) = ( 0+ 1z1+ y2)where ( )is the standard normal distribution function.
8.1.1 The power of instruments
The upper bounding probability in (62) can be written as P[Y1 = 0_g(Y2; z)< ujz], from which
it is trivial that for any …xed (u; z), the upper bounding probability in (62) is at least equal to the lower bounding probability. It follows that these inequalities can only place restrictions on the threshold function g(; ) when, for one or more values of u, the restrictions that the model places on the threshold function and the support of Z are such that there exists variation inz for which the bounding probabilities in (62) vary while the functiong(; z) remains unchanged.
A leading case of interest is one in which there is an exclusion restriction so that, with Z
(Z1; Z2),Z2 is excluded from the threshold function which then becomesg(y2; z1).
In this case, in (62) at each value of Z1 only the in…mum of the upper bounding probabilities
for identi…cation and the identi…ed set is as follows. Ig(M;FYjZ;RZ) =fg(; ) :8u2(0;1) inf z22RZ2jz1 (1 P[Y1= 1^g(Y2; z1) ujz]) u sup z22RZ2jz1 P[Y1 = 0^g(Y2; z1) ujz]; a.e. z1g (63)
Here there is one pair of upper and lower bounding probabilities at each valuez1 in the support of
the included exogenous variables, Z1, and RZ2jz1 denotes the conditional support of excluded Z2 given Z1 =z1.
Richer support forZ2, and stronger dependence on Z2of the bounding conditional probabilities
that appear in the de…nitions lead to smaller identi…ed sets. There will typically be richer support forZ2 when there are more instruments in this list.
In the application set out in Section 8.2 there are two instrumental variables. One is quite weak and delivers quite large disconnected sets for structural features of interest; the other is strong, delivering a small, connected set. Using both instruments we get just a slight re…nement of the set obtained using the strong instrument.
8.1.2 Monotone index restriction
In many models employed in econometric practice the threshold function,g(y2; z), is restricted to be
a monotone function of a linear index, that isg(y2; z1) =s(z1 +y2 )with s( ) strictly monotone,
normalized increasing, wherez1andy2 denote row-vectors, which are possibly multivariate and are
conformable with parameter vectors and . This restriction appears for example in probit and logit models.
Under the monotone index restriction the inequalities in (63) can be written
inf z22RZ2jz1 1 P[Y1 = 1^z1 +Y2 s 1(u)jz] u sup z22RZ2jz1 P[Y1= 0^z1 +Y2 s 1(u)jz]; a.e. z1
and then after the change of variable t=s 1(u)there is the following identi…ed set for ( ; ; s( )).
I( ; ;s)(M;FYjZ;RZ) =f ; ; monotone strictly increasings( ) :8t2R inf z22RZ2jz1 (1 P[Y1 = 1^z1 +Y2 tjz]) s(t) sup z22RZ2jz1 P[Y1 = 0^z1 +Y2 tjz]; a.e. z1g (64)
In practice if s( ) is an unknown strictly monotone function, one will normalize one of the index coe¢ cients, for example setting an a priori non-zero element of equal to1.
The upper and lower bounding probabilities in (64) are weakly increasing functions of t. If for some value of ( ; ) they cross then there is no monotone function that can pass between the bounding probabilities and that value of( ; ) does not lie in the identi…ed set.
If for some value of( ; )the upper bounding probability is at least equal to the lower bounding probability for all t2R then there exists at least one monotone functions( ) that passes between the upper and lower bounding probability functions and that value of( ; )lies in the identi…ed set of parameter values. There is therefore the identi…ed set of index coe¢ cients which is a projection of the set (64), as follows.
I( ; )(M;FYjZ;RZ) =f ; :8t2R inf z22RZ2jz1 (1 P[Y1 = 1^z1 +Y2 tjz]) sup z22RZ2jz1 P[Y1 = 0^z1 +Y2 tjz]; a.e. z1g
In a parametric model, for example a probit model, the function s( ) is the standard normal distribution function. The identi…ed set of index coe¢ cients I( ; )(M;FYjZ;RZ) is then the set
of ( ; ) that satisfy the same conditions as in (64), but with s( ) …xed at ( ). When that set is empty there is no value of ( ; ) in that set such that the normal distribution function can pass between the upper and lower bounding probability functions. In that case the probit model is misspeci…ed and the identi…ed set of parameter values is empty.
8.1.3 Discrete endogenous variables
Before moving on to an application consider the case in which Y2, possibly a vector, has …nite
support, say with K points of support, (y12; : : : ; y2K). This arises in the application set out in Section 8.2.
At each value z of Z the function g(; ) is characterized by a point, (z), in the unitK-cube.
(z) ( 1(z); : : : K(z)); k(z) g(yk2; z); k2 f1; : : : ; Kg
De…ne f (z) : z 2 RZg. Consider a particular element of (z), k(z), which is not the
largest element in (z). Let k(z) denote the smallest amongst the elements of (z)that are larger than k(z), that is:
k(z) min
j6=kf j(z) : j(z)> k(z)g:
For any value u 2 [ k(z); k(z)) the event g(Y2; z) u occurs if and only if Y2 2 fyj2 : j(z) k(z)g. So only values u 2 f 1(z); : : : K(z))g are instrumental in de…ning the identi…ed set and
(62) can be written as follows.
I (M;FYjZ;RZ) =f :8k2 f1; : : : ; Kg 1 P[Y1 = 1^Y2 2 fy2j : j(z) k(z)gjz] k 2(z) P[Y1= 0^Y22 fyj2 : j(z) k(z)gjz]; a.e. z o (65) The probabilities in (65) are determined by the ordering of the elements of (z). For each of theK!possible orderings of these elements the inequalities in (62) de…ne an intersection of linear half spaces and thus a convex polytope for each element (z),z 2 RZ. The identi…ed set for is
a union of these polytopes, some of which may be empty, and this union may not be convex nor even connected.
The situation is illustrated in Chesher (2013) for an example in which Y2 has three points of
support and there is no exogenous variable a¤ecting the threshold function. The identi…ed set for the three values of the nonparametrically speci…ed threshold function comprises the union of up to 6 convex polytopes in the unit cube. Progressively increasing the predictive power of the instrumental variable causes the convex subsets of the identi…ed set associated with each ordering to become successively empty, eventually leaving just one, showing that in the case considered the ordering of the elements of , e¤ectively the shape of the threshold function, can be identi…ed.
Imposing shape restrictions will render some orderings inadmissible. When Y2 is scalar with
support y1
2 yK2 and there is a monotonicity restriction so that for all z either (z) is an
increasing sequence or a decreasing sequence then the identi…ed set of values of is as follows.
A" =f increasing:8k2 f1; : : : ; Kg 1 P[Y1 = 1^Y2 y2kjz] k(z) P[Y1 = 0^Y2 yk2jz]; a.e. z o A# =f decreasing:8k2 f1; : : : ; Kg 1 P[Y1 = 1^Y2 y2kjz] k(z) P[Y1 = 0^Y2 yk2jz]; a.e. z o
This also applies in the case withK= 2 points of support forY2 which arises in the application
studied in the next section. With strong enough instruments one of the sets A" and A# may be empty in which case the direction of the e¤ect of Y2 on the threshold function is identi…ed. Both
sets may be empty in which case the monotonicity restriction can be rejected.