Identification and Estimation of Time-Varying Nonseparable Panel Data Models without Stayers
Takuya Ishihara
stat.ME





Identification and Estimation of Time-Varying Nonseparable Panel Data Models without Stayers
    Takuya Ishihara[University of Tokyo, Graduate School of Economics, email: ishihara-takuya143@g.ecc.u-tokyo.ac.jp.
I would like to express my appreciation to the editor, the associate editor, and an anonymous referee for their careful reading and comments.
I also would like to thank Katsumi Shimotsu, Hidehiko Ichimura, and the seminar participants at the University of Tokyo, Hakodate Conference, Otaru University of Commerce, and Nanzan University.
This research is supported by Grant-in-Aid for JSPS Research Fellow (17J03043) from the JSPS.]
    December 30, 2023
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This paper explores the identification and estimation of nonseparable panel data models.
We show that the structural function is nonparametrically identified when it is strictly increasing in a scalar unobservable variable, the conditional distributions of unobservable variables do not change over time, and the joint support of explanatory variables satisfies some weak assumptions.
To identify the target parameters, existing studies assume that the structural function does not change over time, and that there are “stayers", namely individuals with the same regressor values in two time periods.
Our approach, by contrast, allows the structural function to depend on the time period in an arbitrary manner and does not require the existence of stayers.
In estimation part of the paper, we consider parametric models and develop an estimator that implements our identification results. 
We then show the consistency and asymptotic normality of our estimator.
Monte Carlo studies indicate that our estimator performs well in finite samples.
Finally, we extend our identification results to models with discrete outcomes, and show that the structural function is partially identified.

Keywords: Nonseparable models, nonparametric identification, panel data, unobserved heterogeneity.







§ INTRODUCTION


This paper considers the identification and estimation of the following nonseparable panel data model:

    Y_it   =    g_t(X_it,U_it),      i = 1, ⋯, n ,    t=1,⋯ ,T,

where Y_it∈ℝ is a scalar response variable, X_it∈ℝ^k is a vector of explanatory variables, and U_it∈ℝ is a scalar unobservable variable.
Suppose that 𝐘_i = (Y_i1, ⋯ , Y_iT) and 𝐗_i = (X_i1', ⋯ , X_iT') are observable.
Many widely used panel data models fall into this category.
For example, this specification contains the textbook linear panel data model

    Y_it = X_it'β + α_i +ϵ_it,

because we can regard α_i + ϵ_it as U_it.
Furthermore, it contains the following nonlinear panel data models:

    Y_it   =    h^-1(X_it'β + γ_t + α_i +ϵ_it), 
    
    Y_it   =    c(U(α_i,ϵ_it)) + X_it' β(U(α_i,ϵ_it)),

where (<ref>) is the transformation model proposed by <cit.>, and (<ref>) is the random effects quantile regression model proposed by <cit.>.

The importance of unobserved heterogeneity when modeling economic behavior is widely recognized.
Nonseparable models capture the unobserved heterogeneity effect of explanatory variables on outcomes because these models allow the derivative of the structural function to depend on an unobserved variable.
Indeed, there is extensive literature on nonseparable panel data models including <cit.>,<cit.>, <cit.>, <cit.>, <cit.>, and <cit.>.

This study shows that we can nonparametrically identify g_t(x,u) when g_t(x,u) is strictly increasing in u, the conditional distributions of U_it are the same over time, and the support of 𝐗_i satisfies some weak assumptions.
To identify the target parameters, many nonseparable panel data models assume that the structural function does not change over time, and require the existence of “stayers", namely individuals with the same regressor values in two time periods.
By contrast, our approach allows g_t to depend on the time period t in an arbitrary manner and does not require the existence of stayers.

Although modeling time trends is important in research on panel data, existing nonseparable panel data models assume that the structural function does not change over time or impose some restrictions on these time trends.
For instance, <cit.> do not allow g_t to depend on time period t; <cit.> and <cit.> use additive time effects; and <cit.> and <cit.> use additive location time effects
and multiplicative-scale time effects. Moreover,
<cit.> assume that g_t(x,u) can be written as μ_t(x) + σ_t(x) ϕ(x,u).
Thus, time effects are linearly conditional on explanatory variables in this model, and as such it does not allow for nonlinear time effects.
Indeed, <cit.> allow for nonlinear time effects by assuming that g_t(x,u) can be written as m_t(h(x,u)), where m_t is a monotonic transformation.
While this transformation extends the typical additive location time effects and captures macro-shocks, it does not allow the effect of macro-shocks to depend directly on an unobserved variable, and stipulates that ∇_x g_t(x,u) / ∇_u g_t(x,u) does not depend on time. 
For example, consider the relationship between consumption and income.
We write the Engel function of the i-th household as 

    Y_it = ϕ(X_it,M_t,U_it),

where Y_it is consumption, X_it is income, U_it is the scalar unobserved heterogeneity that represents preference, and M_t is a macroeconomic variable.
However, such a model does not satisfy <cit.> since ∇_x ϕ(x,M_t,u) / ∇_u ϕ(x,M_t,u) depends on M_t.
By contrast, our assumptions can accommodate this model, because we can rewrite this as (<ref>) by treating ϕ(x,M_t,u) as g_t(x,u).

Many nonseparable panel data models require the existence of stayers:
<cit.>, <cit.>, and <cit.> require stayers in order to identify the structural functions or derivatives of the average and quantile structural functions.
In particular, to identify the structural function, <cit.> requires the support of (X_i1,X_i2) contains (x,x) for all x.
Many empirically important models do not satisfy this assumption.
For example, in standard difference-in-differences (DID) models, there are no individuals treated during both time periods.
Our approach does not require the existence of stayers and allows the support condition employed in standard DID models.

Our identification approach is based on the conditional stationary condition, that is, the conditional distribution function of U_it given 𝐗_i does not change over time.
Similar assumptions are employed in all the aforementioned papers except for <cit.>.
Indeed, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, <cit.>, and <cit.> essentially make the same assumption.
Whereas <cit.> does not impose this assumption explicitly, a similar assumption is made by considering the following model: Y_it = m(X_it,α_i)+U_it.
In this model, the unobservable variable α_i automatically satisfies the conditional stationarity because α_i does not depend on t.
By contrast, <cit.> do not assume the conditional stationarity of U_it given 𝐗_i because they consider the identification of nonseparable models using repeated cross-sections.
Rather, they assume that the conditional distribution of U_it given V_it≡ F_X_t(X_it) does not depend on time.

In the literature on nonseparable panel data models, many papers allow the unobservable variable to be a vector or do not impose monotonicity on the structural function, for example, <cit.>, <cit.>, <cit.>, <cit.>, and <cit.>.
On the other hand, our model assumes that the unobservable variable is scalar, and that the structural function is strictly increasing in the unobservable variable.
These assumptions are restrictive but crucial for our identification results.

In the estimation part of the paper, we assume that the admissible collection of structural functions is indexed by a finite-dimensional parameter.
In what follows, we develop an estimator based on the conditional stationary condition.
Our method is similar to that of <cit.>.
The estimator is obtained by minimizing the distance between the conditional distributions of U_i1 and U_i2.
We then prove the consistency and asymptotic normality of this estimator.
Because the asymptotic variance is complicated, we also show the validity of the nonparametric bootstrap.
Monte Carlo studies indicate that our estimator performs well in finite samples.

Finally, we extend our identification results to models in which outcomes are discrete.
This class of models includes many empirically important models such as binary choice panel data models.
Models in this class cannot point-identify g_t, but can partially identify it by using the suggestion developed in <cit.>.
We also allows g_t to depend on the time period t in an arbitrary manner and does not require the existence of stayers.
However, the support condition becomes stronger than it is in models with continuous outcomes.

The remainder of the paper is organized as follows.
Section 2 demonstrates the nonparametric identification of g_t when outcome variables are continuous.
In Section 3, we propose the estimator under the parametric assumption and discuss its consistency, asymptotic normality, and bootstrap.
Section 4 reports the results of several Monte Carlo simulations.
In Section 5, we consider the case where Y_it is discrete and show the partial identification of g_t.
Section 6 offers concluding remarks.
The proofs of the theorems and auxiliary lemmas are collected in the Appendix.






§ IDENTIFICATION


First, for notational convenience we drop the subscript i and let T=2.
It is straightforward to extend the results to the case with T ≥ 3.
For any random variables V and W, let F_V|W and Q_V|W denote the conditional distribution function and the conditional quantile function, respectively.
𝒳_t, 𝒳_12, and 𝒰_t denote, respectively, the supports of X_t, (X_1,X_2), and U_t.

First, we assume g_t(x,u) is strictly increasing in u, and U_t is continuously distributed.


(i) For all t, the function g_t(x,u) is continuous and strictly increasing in u for all x.
If X_t is continuously distributed, then g_t(x,u) is also continuous in x.
(ii) For all t, U_t|𝐗=𝐱 is continuously distributed for all 𝐱.



For all t and 𝐱∈𝒳_12, the conditional distribution of the Y_t conditional on 𝐗=𝐱 is continuous and strictly increasing.


Many nonseparable panel data models do not employ the strict monotonicity assumption, for example, <cit.>, <cit.>, <cit.>, and <cit.>.
These models allow for unobserved variables to be multivariate.
Hence, our model is more restrictive than theirs.
However, as noted in the previous section, our model covers many widely used panel data models, such as typical linear fixed-effects models.

Assumptions I1 and I2 rule out the case where outcomes are discrete.
In Section 5, we relax the strict monotonicity assumption by allowing g_t(x,u) to be flat inside the support of U_t, and consider the case where outcomes are discrete.

Next, we impose the normalization assumption.


For some x̅∈𝒳_1, we have g_1(x̅,u)=u for all u.


Assumption I3 is a normalization assumption common in nonseparable models (see, e.g., <cit.>).
Because we assume U_1 | 𝐗=𝐱d= U_2 | 𝐗=𝐱 below, it is sufficient to normalize g_1(x,u) exclusively.
The functions g_t(x,u) and distributions of U_t depend on the choice of x̅.
However, it is easy to show that the function

    h_t(x,τ) ≡ g_t(x,Q_U_t(τ))

does not depend on the choice of x̅.

Nevertheless, we can normalize this model in an alternative way.

\beginAssumption_I3'
For all t, the marginal distribution of U_t is uniform on [0,1].
\endAssumption_I3'

Under this normalization and additional assumptions, we can regard this structural function as the quantile function of the potential outcome considered by <cit.>.
They refer to U_t as the rank variable.
As they show, under the rank invariance or rank similarity assumption, we can think of the function g_t(x,u) as the quantile function of the potential outcome.
It is easy to show that the function g_t(x,τ) under Assumption I3' is the same as the function h_t(x,τ) under Assumption I3.

Hereafter, we use Assumption I3, but we can replace Assumption I3 with Assumption I3' and identify the structural function g_t, as we show below.

We assume the conditional stationarity of U_t by following <cit.>, <cit.>, <cit.>,  <cit.>, <cit.>, <cit.>, and <cit.>.


(i) The conditional distributions of the unobservable U_t conditional on 𝐗 is the same across t.
That is, for all 𝐱∈𝒳_12, we have

    U_1 | 𝐗=𝐱d= U_2 | 𝐗=𝐱,

which implies that 𝒰_1 = 𝒰_2 ≡𝒰.
(ii) For all t, the conditional support of U_t|𝐗=𝐱 is 𝒰.


When we can decompose U_t into time-variant and time-invariant parts, this assumption does not impose any restrictions on the dependence between the time-invariant part and 𝐗.
To see this, let U_t = U(α, ϵ_t), where α is time-invariant and ϵ_t is time-variant.
Then, Assumption I4 holds, if 

    ϵ_1|α=a, 𝐗=𝐱 d= ϵ_2|α=a, 𝐗=𝐱.

Because condition (<ref>) allows α to be correlated with 𝐗 arbitrarily, Assumption I4 imposes no restrictions on the time-invariant unobservable variables.

Indeed, <cit.> and <cit.> employ similar assumptions, although the former does not make this assumption explicitly.
By considering the model Y_it = m(X_it,α_i)+U_it, the unobservable variable α_i automatically satisfies the conditional stationarity.
Moreover, since <cit.> consider the identification using repeated cross-sections, they do not impose this assumption.
Instead, they impose the following:

    U_1|𝐕_1=𝐯d= U_2|𝐕_2=𝐯,

where 𝐕_t ≡ (F_X_t,1(X_t,1), ⋯ , F_X_t,k(X_t,k)) and 𝐯∈ (0,1)^k.

To show the identification of g_t, we introduce the following sets.
Define 𝒮_0^1 ≡{x̅}, 𝒮_0^2 ≡{x∈𝒳_2: (x̅,x) ∈𝒳_12}, namely the cross-section of 𝒳_12 at X_1=x.
For m ≥ 1, define

    𝒮_m^1    ≡   { x∈𝒳_1 : there exists x_2 ∈𝒮_m-1^2 such that (x,x_2)∈𝒳_12.}, 
    𝒮_m^2    ≡   { x∈𝒳_2 : there exists x_1 ∈𝒮_m^1 such that (x_1,x)∈𝒳_12.}.

Figure 1 illustrates these sets.
Because U_1|𝐗=𝐱d= U_2|𝐗=𝐱, for all (x_1,x_2) ∈𝒳_12, we have

    F_Y_1|𝐗( g_1(x_1,u)|x_1,x_2 )    =    P( g_1(x_1,U_1) ≤ g_1(x_1,u)|X_1=x_1,X_2=x_2 ) 
       =    P( U_1 ≤ u |X_1=x_1,X_2=x_2 ) 
       =    P( U_2 ≤ u |X_1=x_1,X_2=x_2 ) 
       =    P( g_2(x_2,U_2) ≤ g_2(x_2,u) |X_1=x_1,X_2=x_2 ) 
       =    F_Y_2|𝐗( g_2(x_2,u)|x_1,x_2 ).

Because F_Y_t|𝐗(y|x_1,x_2) is invertible in y for all (x_1,x_2) ∈𝒳_12, we obtain 

    g_1(x_1,u)    =    Q_Y_1|𝐗( F_Y_2|𝐗( g_2(x_2,u)|x_1,x_2 ) |x_1,x_2 ) 
    
    g_2(x_2,u)    =    Q_Y_2|𝐗( F_Y_1|𝐗( g_1(x_1,u)|x_1,x_2 ) |x_1,x_2 ).


Equations (<ref>) imply that if g_1(x_1,u) (or g_2(x_2,u)) is identified and (x_1,x_2) ∈𝒳_12, then g_2(x_2,u) (or g_1(x_1,u)) is also identified.
First, we can identify g_1(x̅,u) because g_1(x̅,u)=u holds by Assumption I3.
Hence, we can identify g_2(x,u) for all x ∈𝒮_0^2, because g_2(x,u) = Q_Y_2|𝐗( F_Y_1|𝐗( g_1(x̅,u)|x̅,x ) |x̅,x ) = Q_Y_2|𝐗( F_Y_1|𝐗( u|x̅,x ) |x̅,x ).
We now turn to identifying g_1(x,u) for x ∈𝒮_1^1.

First, we fix x ∈𝒮_1^1.
According to the definition of 𝒮_1^1, there exists x_2 ∈𝒮_0^2 such that (x,x_2) ∈𝒳_12.
Then, it follows from (<ref>) that 

    g_1(x,u) = Q_Y_1|𝐗( F_Y_2|𝐗( g_2(x_2,u)|x,x_2 ) |x,x_2 ),

and hence, g_1(x,u) is identified because g_2(x_2,u) is already identified.
Similarly, by using (<ref>), we can identify g_2(x,u) for all x ∈𝒮_1^2.
Repeating this argument provides the following theorem.


Suppose that Assumptions I1, I2, I3, and I4 are satisfied.
For all t, if we have 𝒳_t = ∪_m=0^∞𝒮_m^t, then the structural function g_t(x,u) is identified for all x∈𝒳_t and u∈𝒰.


We also show the identification of g_t under Assumption I3' instead of I3.


Suppose that Assumptions I1, I2, I3', and I4 are satisfied.
For all t, if 𝒳_t = ∪_m=0^∞𝒮_m^t holds for some x∈𝒳_1, then the function g_t(x,u) is identified for all x∈𝒳_t and u∈𝒰.


This identification approach is similar to that of <cit.>, <cit.>, and <cit.>, who all identify nonseparable IV models.
<cit.> and <cit.> use the same normalization as Assumption I3'.
<cit.> show that under appropriate assumptions, if for all x and x', we identify the function T_x',x(y) that is strictly increasing in y and satisfies

    g(x',u) = T_x',x(g(x,u)),

then we can identify the structural function g(x,u).
We can also construct similar functions and show that g_t is point identified.

We next introduce some examples that satisfy this support condition.

[DID model]
In standard DID models, if X_t is a treatment indicator, then we have 𝒳_12={(0,0),(0,1)}.
Because 𝒳_1={0}, we assume x̅=0.
That is, g_1(0,u)=u for all u.
Hence, we identify g_1(x,u) for all x ∈𝒳_1 and u ∈𝒰.
Then, because 𝒮^2_0={0,1}=𝒳_12, the support condition of Theorem 1 holds and we can identify g_2(x,u) for all x ∈𝒳_2 and u ∈𝒰.

Our identification approach does not require the joint distribution of (Y_1,Y_2).
Hence, if we can observe D≡1{X_2=1}, then we can identify the structural function g_t by using repeated cross-sections.
If the potential outcome Y_t(x) is equal to g_t(x,U_t), then this setting is similar to <cit.>.

Similar to <cit.>, we can also identify the counterfactual distribution even when 𝒳_12≠{(0,0),(0,1)}.
Let Y_t(x) ≡ g_t(x,U_t) denote the potential outcomes.
Then, we can identify F_Y_2(x)|X_2(y|x'), where x ≠ x'.
Suppose that there exists x_1 ∈𝒳_1 such that (x_1,x), (x_1,x') ∈𝒳_12.
In this case, it follows from (<ref>) that

    F_Y_1|𝐗=(x_1,x')( Q_Y_1|𝐗=(x_1,x)(F_Y_2|𝐗=(x_1,x)(y)) ) 
       =    F_Y_1|𝐗( g_1(x_1,g_2^-1(x,y)) |x_1,x' ) 
       =    P( g_1(x_1,U_1) ≤ g_1(x_1,g_2^-1(x,y)) | X_1=x_1, X_2=x' ) 
       =    P( g_2(x,U_1) ≤ y | X_1=x_1, X_2=x' ) 
       =    P( Y_2(x) ≤ y | X_1=x_1, X_2=x' ).

Hence, we can obtain F_Y_2(x)|X_2(y|x') by integrating out x_1.
The left-hand side is similar to the counterfactual distribution of <cit.>.
When 𝒳_12 = {(0,0), (0,1)}, this result is same as their result.


[connected support]
When the interior of 𝒳_12 is connected, the support condition of Theorem 1 holds.
Because the interior of 𝒳_12 is connected, for any x ∈𝒳_1, there exists a series (x_1^0,x_2^0),(x_1^1,x_2^0),(x_1^1,x_2^1),(x_1^2, x_2^1),(x_1^2,x_2^2) ⋯ such that x_1^0=x̅, (x_1^m,x_2^m), (x_1^m+1,x_2^m) ∈𝒳_12 for all m = 0, 1, ⋯, and lim_m →∞ x_1^m = x.
Figure 2 illustrates this result intuitively.
From the definition of 𝒮^1_m, x_1^m ∈𝒮_1^m for all m.
Hence, we have x ∈∪_m=0^∞𝒮_m^1, and the support condition of Theorem 1 holds.


The support condition of Theorem 1 rules out the case where X_1=X_2.
Hence, if the explanatory variables do not vary across time periods, such as sex or race, this support condition does not hold.

If we have panel data with more than two periods, we can relax this support condition.
Similar to the case where T=2, we define the following sets.
Define 𝒮̃^1_0 ≡{x̅}, 𝒮̃^t_0 ≡{x ∈𝒳_t : (x̅,x) ∈ supp(X_1,X_t)}, t=2, ⋯ , T, and for m ≥ 1 and t = 1, ⋯ , T,

    𝒮̃^t_m ≡⋃_s ≠ t{ x ∈𝒳_t : there exists x_s ∈𝒮̃^s_m-1 such that (x_s,x) ∈ supp(X_s,X_t).}.

Then, we show that g_t(x,u) is point-identified under a similar support condition to that of Theorem 1.


Suppose Assumptions I1, I2, I3, and I4 are satisfied for T≥ 3.
For t = 1, ⋯, T, if 𝒳_t = ∪_m=0^∞𝒮̃^t_m, then the function g_t(x,u) is identified for all x ∈𝒳_t and u ∈𝒰.








§ ESTIMATION AND INFERENCE


In the previous sections, we considered nonparametric identification.
In this section, we assume that the admissible collection of structural functions is indexed by a finite-dimensional parameter.
Consider the following parametric model:

    Y_it = g_t(X_it,U_it;θ_0)      i = 1, ⋯, n,    t=1, ⋯, T.

The outcome functions are parameterized by θ∈Θ⊂ℝ^d_θ, where θ_0 ∈Θ is the true parameter.
We assume that {(𝐘_i, 𝐗_i)}_i=1^n are independent and identically distributed.

Indeed, <cit.> consider a similar setting, and develop an estimator based on the identification result of <cit.>.
Following <cit.>, we develop a minimum distance estimator based on our identification results.

The following assumptions are the parametric versions of Assumptions I1, I2, I3, and I4.


(i) For all t, the function g_t(x,u;θ) is continuous and strictly increasing in u for all θ∈Θ.
(ii) For all t, U_it|𝐗_i=𝐱 is continuously distributed for all 𝐱.



For all t and 𝐱∈ supp(𝐗), the conditional distribution of Y_it conditional on 𝐗_i=𝐱 is continuous and strictly increasing.



(i) For some x̅∈𝒳_1, g_1(x̅,u;θ)=u holds for all u∈𝒰 and θ∈Θ.
(ii) For all θ, θ' ∈Θ with θ≠θ', we have g_t(·,·;θ) ≠ g_t(·,·;θ') for some t.



(i) For all 𝐱∈ supp(𝐗) and s,t ∈{1, ⋯, T}, we have U_is | 𝐗_i=𝐱d= U_it | 𝐗_i=𝐱.
(ii) The support of U_it|𝐗_i=𝐱 is 𝒰.


These assumptions are similar to the assumptions in Section 2.
Assumption P3(ii) allows that g_t(x,u;θ) does not depend on some part of θ.
For example, consider θ = (θ_1,θ_2, ⋯, θ_T).
Then, this condition allows that g_t depends exclusively on θ_t.

Similar to Section 2, we suppose T=2.
Under Assumptions P1–P4 and the support condition of Theorem 1, it follows from Theorem 1 that

    U_i1,θ | 𝐗_i = 𝐱d=  U_i2,θ | 𝐗_i = 𝐱  for all 𝐱  ⇔  θ = θ_0,

where U_it,θ≡ g_t^-1(X_it,Y_it;θ), t = 1,2.
Therefore, (<ref>) implies that the function

    D_θ(v)    ≡    P( U_i1,θ≤ v_u,  𝐗_i ≤ v_𝐱) - P( U_i2,θ≤ v_u,  𝐗_i ≤ v_𝐱) 
       =    E[ ( 1{Y_i1≤ g_1(X_i1,v_u;θ)}-1{Y_i2≤ g_2(X_i2,v_u;θ)}) 1{𝐗_i ≤ v_𝐱}]

is zero for all v=(v_𝐱,v_u) ∈𝒱≡𝒳_12×𝒰 if and only if θ=θ_0.
Let ·_μ denote the L_2-norm with respect to a probability measure with support 𝒱.
Then, we have D_θ_μ≥ 0 and

    D_θ_μ = 0   ⇔  θ=θ_0.

Hence, θ_0 is the value that provides a global minimum for D_θ_μ.

We construct the estimator D̂_n,θ(v) of D_θ(v) as a sample analogue of (<ref>):

    D̂_n,θ(v) ≡1/n∑_i=1^n ( 1{Y_i1≤ g_1(X_i1,v_u;θ)}-1{Y_i2≤ g_2(X_i2,v_u;θ)}) 1{𝐗_i ≤ v_𝐱}.

This is a natural estimator of D_θ(v).
We can obtain the estimator θ̂_n by minimizing D̂_n,θ_μ.
That is,

    θ̂_n = argmin_θ∈ΘD̂_n,θ_μ.

This estimator is similar to the estimators proposed by <cit.> and <cit.>.
They prove the consistency and the asymptotic normality of their estimators, and also show the consistency of the nonparametric bootstrap.
In what follow, we likewise prove the consistency and the asymptotic normality of our estimator, and show that the nonparametric bootstrap is consistent.

First, we collect the observable data together into a single vector, W_i=(𝐘_i,𝐗_i)=(Y_i1,Y_i2,X_i1,X_i2).
Next, we define 

    A_θ^v(w) ≡[ 1{y_1 ≤  g_1(x_1,v_u;θ)}-1{y_2 ≤ g_2(x_2,v_u;θ)}] 1{𝐱≤ v_𝐱},

where w = (y_1,y_2,x_1,x_2).
Then, D̂_n,θ(v) = 1/n∑_i=1^n A_θ^v(W_i).




 §.§ Consistency


Here, we demonstrate the consistency of θ̂_n.
Under condition (<ref>), the following assumptions are sufficient for θ̂_n to be consistent.


θ̂_n satisfies D̂_n,θ̂_n_μ = inf_θ∈ΘD̂_n,θ_μ.



Θ is compact.



For all θ',θ, |g_t(x,u;θ')-g_t(x,u;θ)|≤g̅(x)θ'-θ holds for some strictly positive g̅(x) with E[g̅(X_t)] ≤ K, where 0<K<∞.



For all t, Y_it is absolutely continuously distributed given 𝐗_i, with a conditional pdf f_Y_t|𝐗(y|𝐱) that is uniformly bounded above and continuous in y.



For all t, there exists an integer J_t and functions {β_j}_j^J_t such that for every θ∈Θ and u ∈𝒰 there is an α^t(θ,u) ∈ℝ^J_t with g_t(x,u)= ∑_j=1^J_tα_j^t(θ, u) β_j(x).


Assumption C1 entails that θ̂_n minimizes D̂_n,θ_μ.
Assumptions C3 and C4 imply that D_θ_μ is continuous in θ.
Assumption C5 ensures that a class of functions, {A_θ^v:θ∈Θ, v ∈𝒱}, is P-Glivenko–Cantelli.
Hence, we show that D̂_n,θ_μ uniformly convergences to D_θ_μ almost surely.
<cit.> and <cit.> also make similar assumptions.
From these results and the compactness of Θ, we show the consistency from the usual arguments of extremum estimators (e.g., <cit.>).


Under Assumptions P1–P4, C1–C5, and (<ref>), we have θ̂_n →_a.s.θ_0.





 §.§ Asymptotic Normality


Because the objective function D̂_n,θ_μ is not differentiable in θ, our approach follows from <cit.>.
Similarly, although D̂_n,θ(v) is not differentiable in θ, we also assume D_θ(v) is differentiable in θ.
We let ∇_x f denote the column vector of partial derivatives of f with respect to x.
We define Γ_θ(v) ≡∇_θ D_θ(v) and Γ_0(v) ≡Γ_θ_0(v).


θ_0 is an interior point of Θ.



For all t, g_t(x,u;θ) is continuously differentiable in θ in the neighborhood of θ_0.
In the neighborhood of θ_0, | ∇_θ g_t(x,u;θ) | is bounded by some positive function ∇g̅(x) with E|∇g̅(X_t)| < ∞.



(i) There exists c >0 such that Γ_0(v)'a _μ≥ c a for all a ∈ℝ^d_θ.
(ii) {Γ_θ(v):v∈𝒱} is equicontinuous in θ at θ_0.
(iii) ∫Γ_0(v)^2 dμ(v) < ∞.


Assumption N1 is a standard assumption.
Combined with Assumption C4, Assumption N2 implies that D_θ(v) is continuously differentiable in θ in the neighborhood of θ_0.
Assumption N3(i) is a rank condition that
corresponds to Assumption D4 in <cit.>.
Assumption N3(ii) implies that Γ_0(v)'(θ - θ_0) approximates D_θ(v) in the neighborhood of θ_0 uniformly over v.


Under assumptions P1–P4, C1–C5, N1–N3, and (<ref>),

    √(n)(θ̂_n - θ_0) ⇝ N(0,Δ_0^-1Σ_0Δ_0^-1),

where Δ_0 ≡∫Γ_0(v)Γ_0(v)'dμ(v) and

    Σ_0 ≡∫∫_𝒱×𝒱{Ψ(v,v') Γ_0(v) Γ_0(v')' } dμ(v) dμ(v')

with Ψ(v,v') ≡ E[A_θ_0^v(W)A_θ_0^v'(W)].


The proof of Theorem 3 is similar to the proof in <cit.> for their Theorem (3.3).

The asymptotic distribution of √(n)(θ̂_n-θ_0) depends on the probability measure μ.
<cit.> consider the generalized method of moments (GMM) procedure with a continuum of moment conditions, obtaining the optimal estimator.
They consider the following type of GMM estimator to minimize

    ∫∫D̂_n,θ(v) a_n(v,v') D̂_n,θ(v') dv dv',

where a_n(v,v') converges to a kernel a(v,v').
As in <cit.>, we consider only the special case where a_n(v,v')=a(v,v') and a(v,v')=0 for v≠ v'.
Although our setting appears to be similar to that of <cit.>, their approach is not directly applicable because their objective function is smooth.
Hence, we do not pursue this problem.




 §.§ Bootstrap


Let {W_in^*}_i=1^n denote a bootstrap sample drawn with replacement from {W_i}_i=1^n.
That is, {W_in^*}_i=1^n are independently and identically distributed from the empirical measure P_n, conditional on the realizations {W_i}_i=1^n.
We define

    D̂^*_n,θ(v) ≡1/n∑_i=1^n A_θ^v(W_in^*)

as the bootstrap counterpart to D̂_n,θ(v).
Next, we suppose that θ̂_n^* satisfies

    D̂^*_n,θ̂_n^*_μ = inf_θ∈ΘD̂^*_n,θ.

Then, we can obtain the following theorem.


Suppose that θ̂_n^* satisfies (<ref>).
Under the assumptions of Theorem 4, √(n)(θ̂_n^*-θ̂_n) converges weakly to the limit distribution of √(n)(θ̂_n-θ_0) in probability.


The proof of this theorem is similar to the proof of Theorem 6 in <cit.>.
From Theorem 3, we show that

    θ̂_n - θ_0    =   γ_n + o_p(n^-1/2),

where γ_n = Δ_0^-11n∑_i=1^n ∫Γ_0(v) (A_θ_0^v(W_i)-E[A_θ_0^v(W_i)]) dμ(v).
By using the bootstrap stochastic equicontinuity due to <cit.>, we show that

    √(n)θ̂_n^*-θ̂_n - γ^*_n

converges to zero in probability, conditional on almost all samples, where γ^*_n is the bootstrap counterpart of γ_n.
The term γ_n^* has the same limiting distribution as γ_n according to the bootstrap theorem for the mean in ℝ^d_θ.
Hence, we show that √(n)(θ̂_n^*-θ̂_n) converges weakly to the limit distribution of √(n)(θ̂_n-θ_0) in probability.






§ SIMULATIONS


To evaluate the finite sample performance of our estimator, we conducted two Monte Carlo experiments.


The outcome equation is given by

    g_1(x,u)    =    u + (θ_1 + θ_2 u) (x-x̅), 
    
    g_2(x,u)    =    u + (θ_1 + θ_3 u) (x-x̅),

where x̅=0.
Because g_1(x̅,u) = u for all x, the normalization Assumption I3 is satisfied.
We assume

    X_t    =    4Φ(Z_t)    t=1,2 , 
    
    U_t    =   α + ϵ_t    t=1,2,

where Φ(·) is the standard normal distribution function and

    (Z_1,Z_2,α)    ∼    N( ( [ 0; 0; 0 ]) ,( [ 1.0 0.3 0.6; 0.3 1.0 0.5; 0.6 0.5 1.0 ]) ), 
    
    (ϵ_1,ϵ_2)    ∼    N( ( [ 0; 0 ]) ,( [ (0.3)^2       0;       0 (0.3)^2 ]) ).

Because the correlations between α and (Z_1, Z_2) are not zero, X_1 and X_2 are correlated with U_t.
Because ϵ_1|𝐗=𝐱, α=a d=ϵ_2|𝐗=𝐱, α=a, the conditional stationarity assumption holds.
We used μ = Unif(0,4) ×Unif(0,4) × N(0,1) as the integrating measure.

We considered the following two settings: (i) (θ_1,θ_2,θ_3) = (0.5,1.0,0.7), (ii) θ_2 = θ_3 and (θ_1,θ_2) = (0.5,1.0).
Under Setting (i), we cannot use estimation methods of other papers, because their time effects depend on U_t.
On the other hand, under Setting (ii), there are no time effects.
Hence, we can estimate E[ ∇_x g_t(X_t,U_t)|X_1=X_2=x ] by using the method proposed in <cit.> because there are stayers.
Thus, we estimated E[ ∇_x g_t(X_t,U_t)|X_1=X_2=2 ] using our method and their method and compared the results of both.
Under Setting (ii), we have E[ ∇_x g_t(X_t,U_t)|X_1=X_2=2 ] = θ_1 +θ_2 E[U_t|X_1=X_2=2] = 0.5.
Hence, we estimated E[ ∇_x g_t(X_t,U_t)|X_1=X_2=2 ] by using θ̂_1 + θ̂_2 Ê[U_t|X_1=X_2=2].

Table 1 contains the results under Setting (i) for sample sizes of 400, 800, and 1600.
The number of replications was set to 1000 throughout.
Table 1 shows the bias, standard deviation, and the mean squared error (MSE) of the estimates of (θ_1,θ_2, θ_3), highlighting that the standard deviation and MSE decrease as the sample size increases.
In some cases, the biases of the estimates do not decrease.
However they are relatively small under all settings.

Table 2 contains the results under Setting (ii) for sample sizes of 500 and 1000.
Table 2 shows that the standard error of our estimator is smaller than that of <cit.> for all settings.
Although the bias of our estimator is larger than their estimator, the MSE of our estimator is smaller. 


[DID model]
We considered the case where 𝒳_12={(0,0),(0,1)}.
The outcome equation is given by

    g_1(x,u)    =    u  
    
    g_2(x,u)    =    (θ_1 + θ_2 u)(1-x) +(θ_3 + θ_4 u)x,

where (θ_1,θ_2,θ_3,θ_4) = (0.5,0.7,0.5,1.2).
Because g_1(x,u) does not depend on x, the normalization Assumption I3 holds for any x̅∈𝒳_1.
We assumed

    X_2    =   1{Z>0}, 
    
    U_t    =   α + ϵ_t    t=1,2,

where Φ(·) is the standard normal distribution function and

    (Z,α)    ∼    N( ( [   0; 2.0 ]) ,( [ 1.0 0.6; 0.6 1.0 ]) ), 
    
    (ϵ_1,ϵ_2)    ∼    N( ( [ 0; 0 ]) ,( [ (0.3)^2       0;       0 (0.3)^2 ]) ).

Because ϵ_1|X_2=x d=ϵ_2|X_2=x for all 𝐱, the conditional stationarity Assumption I4 holds.
When X_2=0, we have Y_2=g_2(0,U_2)=θ_1 + θ_2 U_2, and when X_2=1, we have Y_2=g_2(1,U_2)=θ_3 + θ_4 U_2.
This specification is similar to that of a typical DID model.
However, letting Y_t(x)=g_t(x,U_2) be potential outcomes, this model does not satisfy the parallel trend assumption if θ_2 ≠ 1, because E[Y_2(0)-Y_1(0)|X_2=x] = θ_1 + (θ_2-1)E[U_1|X_2=x] holds by the conditional stationarity of U_t.
Hence, we cannot estimate the average treatment effect on the treated (ATT) or the average treatment effect (ATE) by using a standard DID method.
Under this setting, we have

    ATE    ≡    E[Y_2(1)-Y_2(0)] = 1.00, 
    
    QTE_25   ≡    Q_Y_2(1)(0.25)-Q_Y_2(0)(0.25)≒ 0.65, 
    
    QTE_50   ≡    Q_Y_2(1)(0.50)-Q_Y_2(0)(0.50)=1.00, 
    
    QTE_75   ≡    Q_Y_2(1)(0.75)-Q_Y_2(0)(0.75)≒ 1.35.

We also estimated ATE and QTE as follows:

    ÂT̂Ê   =   ( θ̂_3 + θ̂_4 Ê[Û] ) - ( θ̂_1 + θ̂_2 Ê[Û] ), 
    Q̂T̂Ê_100τ   =   ( θ̂_3 + θ̂_4 Q̂_Û(τ) ) - ( θ̂_1 + θ̂_2 Q̂_Û(τ) ),

where Ê[Û] is a sample average of Û≡ (Y_i1, ⋯, Y_in, g_2^-1(X_12,Y_12;θ̂), ⋯, g_2^-1(X_n2,Y_n2;θ̂)), and Q̂_Û(τ) is a sample τ-th quantile of Û.
Because X_1=0 and X_2 is discrete, we used

    A_θ^v(w) = ( 1{y_1 ≤ v_u } - 1{ y_2 ≤ g_2(x_2,v_u; θ ) }) 1{x_2 = v_x},

where v = (v_x,v_u).
We used μ = Ber(0.5) × N(Y̅_1,s_Y_1) as the integrating measure, where Y̅_1 is the sample average of Y_1 and s_Y_1 is the standard deviation of Y_1.
Table 4 contains the results of this experiment for sample sizes of 400, 800, and 1600.
The number of replications was set to 1000 throughout.
Table 4 shows the bias, standard deviation, and MSE of the estimates of parameters, the ATE, and QTE, highlighting that the standard deviation and MSE of estimates decrease as the sample size increases.
The biases of the estimates of parameters, the ATE, and QTE_50 are relatively small, whereas the biases of the estimates of QTE_25 and QTE_75 are large.
This may be caused by the fact that the sample quantiles are biased.







§ DISCRETE OUTCOMES


In this section, we consider the case where outcomes are discrete.
In the case of discrete outcomes, we cannot point-identify g_t(x,u).
This is likewise true in <cit.>, <cit.>, and <cit.>.
They consider the case where outcomes are discrete, and instead show partial identification of the structural function.
Hence, in this section, we also consider partial identification of g_t(x,u).

First, we drop the i subscript and let T=2, as in Section 2.
Let 𝒴_t denote the support of Y_t.
The assumptions employed in Section 2 do not allow the outcomes to be discrete.
Hence, we impose the following assumptions.


(i) For all t∈{1,2}, the function g_t(x,u) is weakly increasing in u for all x.
(ii) For all t∈{1,2}, U_t|𝐗=𝐱 is continuously distributed for all 𝐱.



(i) For all t ∈{1,2}, Y_t is discretely distributed.
(ii) 𝒴_1 = 𝒴_2 ≡𝒴 with y≡inf𝒴 and y≡sup𝒴.



For all t ∈{1,2}, the marginal distribution of U_t is uniform on [0,1].



(i) For all 𝐱∈𝒳_12, U_1 | 𝐗=𝐱d= U_2 | 𝐗=𝐱 holds.
(ii) The support of U_t|𝐗=𝐱 is [0,1].


Assumption I1(i) stipulates that g_t(x,u) is strictly increasing in u.
If U_t is continuously distributed, then Y_t must be continuously distributed under Assumption I1(i).
Hence, in this section, we relax Assumption I1 by allowing g_t to be flat inside the support of U_t.
<cit.> and <cit.> also employ this weakly monotonic assumption in models with discrete outcomes.
Furthermore, when outcomes are discrete, we cannot use Assumption I3, because U_t is continuously distributed.
Hence, we use another normalization assumption.
Assumption D4 is identical to Assumption I4.

We can thus obtain the following theorem.


Suppose that Assumptions D1, D2, D3, and D4 are satisfied.
For all t ∈{1,2}, if 𝒳_12=𝒳_1 ×𝒳_2 holds, then we have

    g_t(x,u)    ≥    g^L_t(x,u) ≡inf{y ∈ [y,y] :G_t,x^L(y) ≥ u}, 
    
    g_t(x,u)    ≤    g^U_t(x,u) ≡sup{y∈ [y,y]: G^U_t,x(y) ≤ u},

where G^L_t,x and G^U_t,x are defined by (<ref>) and (<ref>), respectively.


This identification approach is similar to that in <cit.>, who considers the identification of nonseparable models with binary instruments and shows that the structural functions are partially identified when outcomes are discrete.

In Theorem 5, we assume that 𝒳_12=𝒳_1 ×𝒳_2.
Although this support condition does not require stayers, it is nevertheless stronger than that of Theorem 1.
Indeed, we can relax this condition and partially identify g_t under a weaker support condition.
However, if we do, then the bounds of g_t may be looser.

To illustrate Theorem 5, we introduce two examples.

[DID model with binary outcomes]
Suppose that the outcomes are binary and 𝒳_12 = {(0,0),(0,1)}.
Then, 𝒳_12 = 𝒳_1 ×𝒳_2, where 𝒳_1={0} and 𝒳_2 = {0,1}.
This is the usual DID setting.
Define D≡1{X_2=1}.
We consider the partial identification of g_2(0,u) and g_2(1,u).

In this case, we have 

    G^L_2,0(y) = P(D=1) F^+_Y_2|D=1(T^U_2,1,0(y)) + P(D=0) F^+_Y_2|D=0(T^U_2,0,0(y)),

where T^U_2,1,0(y) = Q^+_Y_2|D=1∘ F^+_Y_2|D=1∘ Q^+_Y_1|D=0∘ F^+_Y_1|D=0(y) and T^U_2,0,0(y)=y.
We define p_t(d) ≡ P(Y_t=1|D=d), then

    G^L_2,0(y)= 
    P(D=1,Y_2 ≤1{p_2(1) ≥ p_1(1)})    if y<0
    
    P(D=1,Y_2 ≤1{p_2(1) ≥ p_1(1)  or p_1(0) ≥ p_2(0) }) + P(D=0,Y_2=0)    if 0≤ y<1
    
    1    if y ≥ 1.

Therefore, we can obtain a lower bound

    g^L_2(0,u) = 1{u > P(Y_2=0)}   if p_1(1) > p_2(1) and p_1(0) < p_2(0)
    1{u > P(Y_2=0) + P(D=1,Y_2=1)}   if p_1(1) ≤ p_2(1) or p_1(0) ≥ p_2(0).

Similarly, we can obtain the following functions:

    g^L_2(1,u)    =   1{u > P(Y_2=0)}   if p_1(1) < p_2(1) and p_1(0) > p_2(0)
    1{u > P(D=0) + P(D=1,Y_2=0)}   if p_1(1) ≥ p_2(1) or p_1(0) ≤ p_2(0)
    
    g^U_2(0,u)    =   1{u ≥ P(D=0,Y_2=0)}   if p_1(1) ≥ p_2(1) or p_1(0) ≤ p_2(0)
    1{u ≥ P(Y_2=0)}   if p_1(1) < p_2(1) and p_1(0) > p_2(0)
    
    g^U_2(1,u)    =   1{u ≥ P(D=1,Y_2=0)}   if p_1(1) ≤ p_2(1) or p_1(0) ≥ p_2(0)
    1{u ≥ P(Y_2=0)}   if p_1(1) > p_2(1) and p_1(0) < p_2(0)


If we define the potential outcomes as Y_t(x) = g_t(x,U_t), we can partially identify the ATE.
Because g_2^L(x,u) and g_2^U(x,u) respectively denote the lower and upper bounds of g_2(x,u), we have

    E[g_2^L(x,U)] ≤ E[Y_2(x)] ≤ E[g_2^U(x,U)],   for x=1,2,

where U ∼ U(0,1).
Hence, we have

    E[g_2^L(1,U)]-E[g_2^U(0,U)] ≤μ_ATE≤ E[g_2^U(1,U)] - E[g_2^L(0,U)],

where μ_ATE≡ E[Y_2(1)-Y_2(0)].

Hence, above bounds of g_2 imply that the lower (upper) bound of ATE is not larger (smaller) than 0.
Actually, when p_1(1) < p_2(1) and p_1(0) > p_2(0), that is E[Y_1(0)|D=1] < E[Y_2(1)|D=1] and E[Y_1(0)|D=0] > E[Y_2(0)|D=0], a lower bound of ATE becomes 0.
This situation implies that the mean of the treated group increases, although the time trend effect is negative.
Hence, in this case, it is intuitive that the ATE is larger than 0.
Contrarily, when p_1(1) > p_2(1) and p_1(0) < p_2(0), that is E[Y_1(0)|D=1] > E[Y_2(1)|D=1] and E[Y_1(0)|D=0] < E[Y_2(0)|D=0], an upper bound of ATE becomes 0.
This situation implies that the mean of the treated group decreases, although the time trend effect is positive.
Hence, in this case, it is intuitive that the ATE is smaller than 0.

As an example, we consider the following case:

    E[Y_1|D=1] = 0.4,     E[Y_1|D=0] = 0.3, 
       E[Y_2|D=1] = 0.5,     E[Y_1|D=0] = 0.2, 
       P(D=1) = 0.5.

In this case, we can obtain 

    g_2^L(0,u)    =   1{u>0.9}, 
    
    g_2^L(1,u)    =   1{u>0.65}, 
    
    g_2^U(0,u)    =   1{u>0.65}, 
    
    g_2^U(1,u)    =   1{u>0.25}.

Hence, in this case, ATE is smaller than 0.65 and larger than 0.
As discussed above, because E[Y_1|D=1] < E[Y_2|D=1] and E[Y_1|D=0] > E[Y_2|D=0], a lower bound of ATE becomes 0.



We consider the following model:

    Y_t = g_t(X_t,U_t) = 1{U_t > (1+exp(α_t + β_t X_t))^-1},   t = 1,2,

where U_t = Φ(ϵ_t) and

    (X_1,X_2,ϵ_t) ∼  N( ( [ 0; 0; 0 ]) ,( [ 1.0 0.6 0.4; 0.6 1.0 0.4; 0.4 0.4 1.0 ]) ).

Hence, U_t ∼ U(0,1) for all t and U_1|𝐗=𝐱d= U_2|𝐗=𝐱 for all 𝐱.
We set (α_1,α_2,β_1,β_2)=(0,0.3,0.5,0.6).
Under this setting, we calculate g_t^L(x,u) and g_t^U(x,u) defined by Theorem 5 for x = -2, -1, 0, 1, 2.
Table 4 shows g_t^L(x,u), g_t^U(x,u), and g_t(x,u) at x=-2,-1,0,1,2.

When x is small, the lower (upper) bounds are uninformative (informative).
Contrarily, when x is large, lower (upper) bounds are informative (uninformative).
In this model, there is a positive time trend because g_1(x,u) ≤ g_2(x,u).
These bounds reflect this fact.
That is, they also satisfy g_1^L(x,u) ≤ g_2^L(x,u) and g_1^U(x,u) ≤ g_2^U(x,u).


We can extend Theorem 5 to panel data with more than two periods.


Suppose Assumptions D1, D2, D3, and D4 are satisfied for T ≥ 3.
For t=1, ⋯, T, if supp(𝐗) = 𝒳_1 ×⋯×𝒳_T, then we have

    g_t(x,u)    ≥    g_t^L(x,u) ≡inf{y ∈ [y,y] : G_t,x^L(y) ≥ u}, 
    
    g_t(x,u)    ≤    g_t^U(x,u) ≡sup{y ∈ [y,y] : G_t,x^U(y) ≤ u},

where G_t,x^L(y) and G_t,x^U(y) are defined by (<ref>).







§ CONCLUSION


This paper explored the identification and estimation of nonseparable panel data models.
We showed that the structural function is nonparametrically identified when the structural function g_t(x,u) is strictly increasing in u, the conditional distributions of U_it are the same over time, and the joint support of 𝐗_i satisfies weak assumptions.
Many nonseparable panel data models assume that the structural function does not change over time and that stayers exist.
By contrast, our approach allows the structural function to depend on the time period in an arbitrary manner, and it does not require the existence of stayers.
In estimation part of the paper, we assumed that the admissible collection of structural functions is indexed by a finite-dimensional parameter.
We developed an estimator that implements our identification results. 
We demonstrated the consistency and asymptotic normality of our estimator and showed the validity of the nonparametric bootstrap.
Monte Carlo studies indicated that our estimator performs well with finite samples.
Finally, we extended our identification results to models with discrete outcomes and showed that the structural function is partially identified.







§ APPENDIX 1: THE CASE WITH T ≥ 3


Here, we consider the estimation when T ≥ 3.
Then, similar to the case of T=2, we show that under the assumptions of Corollary 2,

    U_is,θ|𝐗_i=𝐱d= U_it,θ|𝐗_i=𝐱  for all 𝐱 and s,t  ⇔  θ = θ_0,

where U_it,θ≡ g_t^-1(X_it,Y_it;θ).
(<ref>) implies that

    E [ 1{U_it,θ≤ v_u,  𝐗_i ≤ v_𝐱}] - E [ 1{U_is,θ≤ v_u,  𝐗_i ≤ v_𝐱}]

is zero for all v ∈𝒱 and t,s ∈{1, …, T} if and only if θ = θ_0.
Therefore, we have

    D^t_θ(v) = 0,  for all v and t  ⇔  θ = θ_0,

where

    D^t_θ(v)    ≡    E [ 1{U_t,θ≤ v_u,  𝐗≤ v_𝐱}]-1/T∑_s=1^T E [ 1{U_s,θ≤ v_u,  𝐗≤ v_𝐱}].

Then, we show that 1/T∑_t=1^T  D^t_θ_μ^2 ≥ 0 and

    1/T∑_t=1^T  D^t_θ_μ^2 = 0   ⇔  θ = θ_0.

Hence, θ_0 is the value that provides a global minimum for 1/T∑_t=1^T  D^t_θ_μ^2.

We construct the estimator D̂_n,θ^t(v) of D_θ^t(v) as a sample analogue of D_θ^t(v):

    D̂_n,θ^t(v)    =   1/n∑_i=1^n 1{ Y_it≤ g_t(X_it,v_u;θ),  𝐗_i ≤ v_𝐱}
           - 1/T∑_s=1^T ( 1/n∑_i=1^n 1{ Y_is≤ g_s(X_is,v_u;θ),  𝐗_i ≤ v_𝐱}) 
       =   1/n∑_i=1^n ( 1{ Y_it≤ g_t(X_it,v_u;θ)} - 1/T∑_s=1^T 1{ Y_is≤ g_s(X_is,v_u;θ)}) 1{𝐗_i ≤ v_𝐱}.

We obtain the estimator θ̂_n of θ_0 by minimizing 1/T∑_t=1^T D̂_n,θ^t_μ^2.
When T=2, this estimator is identical to estimator (<ref>).
Define

    A_θ^t,v(W_i) ≡( 1{ Y_it≤ g_t(X_it,v_u;θ)} - 1/T∑_s=1^T 1{ Y_is≤ g_s(X_is,v_u;θ)}) 1{𝐗_i ≤ v_𝐱},

then D̂^t_n,θ(v) = 1/n∑_i=1^n A_θ^t,v(W_i).

Then, similar to Theorems 3 and 4, the following theorems hold.


Suppose that Assumptions P1–P4, C2–C5, and (<ref>) hold.
If 

    1/T∑_t=1^T D̂^t_n,θ̂_n_μ^2 = inf_θ∈Θ1/T∑_t=1^T D̂^t_n,θ_μ^2

holds, then θ̂_n →_a.s.θ_0.


We define Γ^t_θ(v) ≡∇_θ D^t_θ(v) and Γ^t_0(v) ≡Γ^t_θ_0(v).

\beginAssumption_N3'
(i) There exists c >0 such that ∑_t=1^T Γ_0^t(v)'a _μ^2 ≥ c a^2 for all a ∈ℝ^d_θ.
(ii) For all t ∈{1, ⋯, T}, {Γ_θ^t(v):v∈𝒱} is equicontinuous in θ at θ_0.
\endAssumption_N3'


Suppose that Assumptions P1–P4, C2–C5, N1, N2, N3', (<ref>), and (<ref>) hold.
Then, √(n)(θ̂_n - θ_0) ⇝ N(0,Δ_0,T^-1Σ_0,TΔ_0,T^-1) holds, where Δ_0,T≡1/T∑_t=1^T ∫Γ^t_0(v) Γ^t_0(v)' dμ(v) and

    Σ_0,T≡1/T^2∑_s=1^T ∑_t=1^T ∫∫_𝒱×𝒱{Ψ_s,t(v,v') Γ_0^s(v) Γ_0^t(v')' } dμ(v) dμ(v')

with Ψ_s,t(v,v') ≡ E[A^s,v_θ_0(W) A^t,v'_θ_0(W)].







§ APPENDIX 2: PROOFS



First, we show that g_t(x,u) is identified for all x ∈∪_m=0^∞𝒮_m^t.
By the monotonicity of g_t and (<ref>), equations (<ref>) hold for all (x_1,x_2)∈𝒳_12.
First, we can identify g_1(x,u)=u by Assumption I3.
We can also identify g_2(x_2,u) for all x_2 ∈𝒮_0^2 because (x̅,x_2) ∈𝒳_12 and we have 

    g_2(x_2,u)= Q_Y_2|𝐗( F_Y_1|𝐗( g_1(x̅,u)|x̅,x_2 ) |x̅,x_2 ) = Q_Y_2|𝐗( F_Y_1|𝐗( u|x̅,x_2 ) |x̅,x_2 ).

We now turn to identifying g_1(x_1,u) for x_1 ∈𝒮_1^1.
Fix x_1 ∈𝒮_1^1.
According to the definition of 𝒮_1^1, there exists x_2 ∈𝒮_0^2 such that (x_1,x_2) ∈𝒳_12.
Then, it follows from (<ref>) that 

    g_1(x_1,u) = Q_Y_1|𝐗( F_Y_2|𝐗( g_2(x_2,u)|x_1,x_2 ) |x_1,x_2 ),

and hence, g_1(x_1,u) is identified because g_2(x_2,u) is already identified.
Similarly, by using (<ref>), we can identify g_2(x,u) for all x ∈𝒮_1^2.
Repeating this argument gives the identification of g_t(x,u) for all x ∈∪_m=0^∞𝒮_m^t.

Next, we show that g_t(x,u) is identified for all x ∈𝒳_t.
We fix x' ∈𝒳_t \( ∪_m=0^∞𝒮_m^t ).
Since 𝒳_t = ∪_m=0^∞𝒮_m^t, there exists a sequence {x^m}_m=1^∞⊂∪_m=0^∞𝒮_m^t such that lim_m→∞ x^m = x'.
By the continuity of g_t, we have lim_m→∞ g_t(x^m,u) = g_t(x',u) for all u ∈𝒰.
Hence, we can also identify g_t(x',u) because g_t(x^m,u) is identified for all m.






First, we show that if for all x,x' ∈𝒳_t, we can identify the strictly increasing function T_t,x',x(y) that satisfies

    g_t(x',u) = T_t,x',x( g_t(x,u) ),

then, g_t(x,u) is point identified.
We define G^t_x(y) ≡∫ F_Y_t|X_t( T_t,x',x(y) | x') dF_X_t(x'), and then we have

    G^t_x( g_t(x,u) )    =   ∫ F_Y_t|X_t( g_t(x',u) | x') dF_X_t(x') 
       =   ∫ P( U_t ≤ u | X_t=x') dF_X_t(x') 
       =    P( U_t ≤ u ) = u,

where the last equality follows from Assumption I3'.
Because T_t,x',x(y) is strictly increasing in y, G^t_x(y) is invertible.
Hence, we obtain g_t(x,u) = ( G^t_x )^-1(u).
This implies that g_t(x,u) is identified if we can construct T_t,x',x(y) for all x,x' ∈𝒳_t.

To construct T_t,x',x(y), we show that for all x ∈𝒳_t, we can identify the strictly increasing function T^*_t,x(y) that satisfies 

    g_t(x,u)=T^*_t,x(g_1(x̅,u)).

For all x ∈∪_m=0^∞𝒮_m^t, the proof of Theorem 1 implies that we can construct T^*_t,x(y) that satisfies (<ref>).
Because F_Y_t|𝐗 and Q_Y_t|𝐗 are strictly increasing, T^*_t,x(y) is strictly increasing in y for all x ∈∪_m=0^∞𝒮_m^t.
We fix x' ∈𝒳_t \( ∪_m=0^∞𝒮_m^t ).
Since 𝒳_t = ∪_m=0^∞𝒮_m^t, there exists a sequence {x^m}_m=1^∞⊂∪_m=0^∞𝒮_m^t such that lim_m→∞ x^m = x'.
By the continuity of g_t and (<ref>), we have lim_m →∞T^*_t,x_m(g_1(x,u)) = g_t(x,u).
Because g_t(x,u) is strictly increasing in u, lim_m →∞T^*_t,x_m(y) is also strictly increasing in y.
Hence, for all x ∈𝒳_t, we can identify the strictly increasing function T^*_t,x(y) that satisfies (<ref>).

By using T^*_t,x(y), we identify T_t,x',x(y) that satisfies (<ref>).
Because, for x,x' ∈𝒳_t, we have

    g_t(x',u) = T^*_t,x'( (T^*_t,x)^-1 (g_t(x,u)) ),

we can construct the function T_t,x',x(y) that satisfies g_t(x',u)=T_t,x',x(g_t(x,u)).
Therefore, we can identify g_t(x,u).






The proof is the same as that for Theorem 1.






We fix δ >0.
By Lemma 1, C2, and (<ref>), there exists ϵ >0 such that θ-θ_0≥δ implies D_θ_μ≥ϵ.
Therefore, we have

    D_θ̂_n_μ < ϵ  ⇒  θ̂_n -θ_0 < δ

and it will suffice to show that D_θ̂_n_μ→_a.s. 0.
By (<ref>), we have

    sup_θD̂_n,θ-D_θ_μ≤sup_θ,v | D̂_n,θ(v)-D_θ(v) |   = o_a.s.(1).

By the triangle inequality and C1,

    D_θ̂_n_μ   ≤   D̂_n,θ̂_n - D_θ̂_n_μ + D̂_n,θ̂_n_μ
       ≤   D̂_n,θ̂_n - D_θ̂_n_μ + D̂_n,θ_0_μ.

By the uniform convergence (<ref>), D̂_n,θ̂_n - D_θ̂_n_μ = o_a.s.(1) and D̂_n,θ_0_μ = D̂_n,θ_0 - D_θ_0_μ = o_a.s.(1).
Hence, we can show that θ̂_n →_a.s.θ_0.






First, we prove the √(n)-consistency of θ̂_n.
As seen in the previous theorem, θ̂_n is a consistent estimator of θ_0.
Because θ̂_n is consistent, we can select a sequence {δ_n} that converges to zero sufficiently slowly to ensure

    P(θ̂_n-θ_0≥δ_n) → 0.

For this sequence, the supremum in (<ref>) runs over a range that includes θ̂_n.
Hence, by the triangle inequality and Lemma 4, we have

    D_θ̂_n_μ-D̂_n,θ̂_n_μ-D̂_n,θ_0_μ≤D̂_n,θ̂_n - D_θ̂_n -D̂_n,θ_0_μ = o_p(n^-1/2).

From Assumption C1,

    D_θ̂_n_μ≤ o_p(n^-1/2) + 2D̂_n,θ_0_μ.

Because E[A_θ_0^v(W)]=0 for all v and E|A_θ_0^v(W) A_θ_0^v'(W)| ≤ 1 for all v and v', we have √(n)D̂_n,θ_0(v) = 1/n∑_i=1^n A_θ^v(W_i) ⇝ N(0,Ψ(v,v)).
Since the proof for Lemma 2 shows that {A_θ_0^v: v∈𝒱} is a Donsker class, {√(n)D̂_n,θ_0(v):v∈𝒱} converges weakly in l^∞(𝒱) to a mean-zero Gaussian process with covariance function Ψ(v,v') and we have D̂_n,θ_0_μ = O_p(n^-1/2).
Hence, we have

    D_θ̂_n_μ = O_p(n^-1/2).

Because D_θ_0(v) = 0 for all v, Lemma 3 implies that for all θ in a neighborhood of θ_0,

    D_θ_μ   =   Γ_0(v)'(θ-θ_0) - ( D_θ(v) - D_θ_0(v) - Γ_0(v)'(θ-θ_0)  ) _μ
       ≥   Γ_0(v)'(θ-θ_0)_μ - D_θ(v) - D_θ_0(v) - Γ_0(v)'(θ-θ_0) _μ
       ≥    (c-o(1)) ×θ-θ_0.

Therefore, θ̂_n-θ_0≤1/c-o_p(1)D_θ̂_n_μ = O_p(n^-1/2).

Next we establish the asymptotic normality of √(n) (θ̂_n-θ_0) by approximating D̂_n,θ(v) as the linear function

    L̂_n,θ(v) ≡Γ_0(v)'(θ-θ_0) + D̂_n,θ_0(v).

We have

    D̂_n,θ̂_n - L̂_n,θ̂_n_μ   ≤   D̂_n,θ̂_n - D_θ̂_n -D̂_n,θ_0_μ + D_θ̂_n(v)-Γ_0(v)'(θ̂_n-θ_0)_μ
       ≤    o_p(n^-1/2) + o_p(θ̂_n-θ_0) = o_p(n^-1/2),

where the second inequality follows from Lemma 3 and Lemma 4, and the last equality follows from the √(n)-consistency of θ̂_n.

Let θ̃_n be the value that provides a global minimum for L̂_n,θ.
Then, Γ_0(·)'(θ̃_n-θ_0) is the L_2(μ)-projection of -D̂_n,θ_0(·) onto the subspace of L_2(μ) spanned by the components of Γ_0(·).
Because Δ_0 = ∫Γ_0(v)Γ_0(v)' dμ(v) is finite and invertible by N3, we have

    √(n) (θ̃_n-θ_0) = - Δ_0^-1∫Γ_0(v) √(n)D̂_n,θ_0(v) dμ(v).

Then, we have

    ∫Γ_0(v) √(n)D̂_n,θ_0(v) dμ(v)    =   1/√(n)∑_i=1^n∫ A_θ_0^v(W_i) Γ_0(v) dμ(v) 
       =   1/√(n)∑_i=1^nξ_i,

for ξ_i ≡∫ A_θ_0^v(W_i) Γ_0(v) dμ(v).
By Fubini's theorem, E[ ξ_i ] = ∫ E[A_θ_0^v(W_i)] Γ_0(v) dμ(v) =0, and

    E[ξ_i ξ_i']    =   ∫∫_𝒱×𝒱{ E[A_θ_0^v(W_i) A_θ_0^v'(W_i)] Γ_0(v) Γ_0(v')' } dμ(v) dμ(v') 
       =   ∫∫_𝒱×𝒱{Ψ(v,v') Γ_0(v) Γ_0(v')' } dμ(v) dμ(v'),

where all elements of E[ξξ'] are finite.
Hence, √(n)(θ̃_n-θ_0) ⇝ N(0,Σ_0) by (<ref>).
Consequently, θ̃_n=θ_0 + O_p(n^-1/2), and {δ_n} can be assumed to satisfy P(θ̃_n - θ_0≥δ_n)→ 0.
Because θ_0 is an interior point of Θ, θ̃_n lies in Θ with probability approaching one.
To simplify the argument, we assume that θ̃_n-θ_0<δ_n and θ̃_n always belongs to Θ.

Because |D_θ(v)| ≤ |Γ_0(v)'(θ-θ_0)| + o(θ-θ_0) uniformly over v by Lemma 3, we have

    D(θ̃_n)_μ≤Γ_0(v)'(θ̃_n-θ_0)_μ  + o_p(θ̃_n-θ_0) = O_p(n^-1/2).

By the triangle inequality and Lemma 4, we have D̂_n,θ̃_n_μ-D_θ̃_n_μ-D̂_n,θ_0_μ = o_p(n^-1/2), and hence D̂_n,θ̃_n_μ = O_p(n^-1/2).
Then, we can argue as for θ̂_n to deduce that

    D̂_n,θ̃_n-L̂_n,θ̃_n_μ = o_p(n^-1/2).


Above, we showed that D̂_n,θ̂_n-L̂_n,θ̂_n_μ = o_p(n^-1/2) and D̂_n,θ̃_n-L̂_n,θ̃_n_μ = o_p(n^-1/2).
Therefore, we have

    L̂_n,θ̂_n_μ - o_p(n^-1/2)    ≤   D̂_n,θ̂_n_μ
       ≤   D̂_n,θ̃_n_μ + o_p(n^-1/2) 
       ≤   L̂_n,θ̃_n_μ  + o_p(n^-1/2).

That is,

    L̂_n,θ̂_n_μ = L̂_n,θ̃_n_μ  + o_p(n^-1/2),

and by squaring both sides, we have

    L̂_n,θ̂_n_μ^2 = L̂_n,θ̃_n_μ^2  + o_p(n^-1),

where the cross product term is absorbed into o_p(n^-1) because L̂_n,θ̃_n_μ = O_p(n^-1/2).
Because L̂_n,θ̃_n(·) and Γ_0(·) are orthogonal according to the definition of θ̃_n, we can obtain

    L̂_n,θ_μ^2    =   L̂_n,θ̃_n(v) + Γ_0(v)' (θ-θ̃_n) _μ^2 
       =   L̂_n,θ̃_n_μ^2 + Γ_0(v)' (θ-θ̃_n)_μ^2.

By making θ equal to θ̂_n, we have

    o_p(n^-1) = Γ_0(v) (θ̂_n-θ̃_n)_μ^2 ≥ c^2 θ̂_n-θ̃_n^2.

Hence, √(n)(θ̂_n-θ_0)=√(n)(θ̃_n-θ_0)+o_p(1)⇝ N(0,Δ_0^-1Σ_0 Δ_0^-1).





Suppose that each W_i is a coordinate function of (∏_i=1^∞S, ∏_i=1^∞σ(S), ∏_i=1^∞ P).
Let ω denote one of the realizations of W_i, and let (W_n1^*, ⋯ , W_nn^*) denote the bootstrap sample.
Following <cit.>, we introduce the following notations.
Let {ζ^*_n} be a sequence of some bootstrap statistic: each ζ^*_n is some function f_n(W_n1^*, ⋯ , W_nn^*) of the bootstrap sample.
We write ζ^*_n = O_p^ω(a_n) if ζ^*_n, when conditioned on ω, is O_p(a_n) for almost all ω.
If ζ^*_n, when conditioned on ω, is o_p(a_n) for almost all ω, we write ζ^*_n = o_p^ω(a_n).
We write ζ^*_n = O_B(1) if, for a given subsequence {n'}, there exists a further subsequence {n”} such that ζ^*_n” = O_p^ω(1).
If for any subsequence {n'} there is a further subsequence {n”} such that ζ^*_n” = o_p^ω(1), we write ζ^*_n = o_B(1).
Note that ζ^*_n = o_B(1) if and only if ζ^*_n converges weakly to zero in probability.


The proof is similar to that of <cit.>.
First, we define

    M(θ)    ≡   ∫ D_θ(v)^2 dμ(v), 
    
    M_n(θ)    ≡   ∫D̂_n,θ(v)^2 dμ(v), 
    
    M_n^*(θ)    ≡   ∫D̂_n,θ^*(v)^2 dμ(v).

Then, for any θ→θ_0, we have

    M_n^*(θ)-M_n(θ)    =   ∫ (D̂^*_n,θ(v)-D̂_n,θ(v))^2 dμ(v) + 2 ∫D̂_n,θ(v) (D̂^*_n,θ(v)-D̂_n,θ(v)) dμ(v) 
       =   ∫ (D̂^*_n,θ_0(v)-D̂_n,θ_0(v))^2 dμ(v) + [ ∫ (D̂^*_n,θ(v)-D̂_n,θ(v))^2 dμ(v) . 
          . - ∫ (D̂^*_n,θ_0(v)-D̂_n,θ_0(v))^2 dμ(v) ] + 2 ∫ D_θ(v) (D̂^*_n,θ(v)-D̂_n,θ(v)) dμ(v) 
           + 2 ∫ (D̂_n,θ(v) - D_θ(v)) (D̂^*_n,θ(v)-D̂_n,θ(v)) dμ(v).

Suppose that θ - θ_0 ≤δ_n for δ_n ↓ 0.
By Lemma 6, we obtain √(n)(D̂^*_n,θ-D̂_n,θ)_μ - √(n)(D̂^*_n,θ_0-D̂_n,θ_0)_μ=o_B(1).
Hence, 

    ∫ (D̂^*_n,θ(v)-D̂_n,θ(v))^2 dμ(v) - ∫ (D̂^*_n,θ_0(v)-D̂_n,θ_0(v))^2 dμ(v) =o_B(n^-1).

Similarly, by the Donsker property of {A_θ^v:θ∈Θ, v ∈𝒱},

    ∫ (D̂_n,θ(v) - D_θ(v)) (D̂^*_n,θ(v)-D̂_n,θ(v)) dμ(v) 
       =   ∫D̂_n,θ_0(v) (D̂^*_n,θ(v)-D̂_n,θ(v)) dμ(v) + o_p(n^-1).

Therefore, we have

    M_n^*(θ)-M_n(θ)    =   ∫ (D̂^*_n,θ_0(v)-D̂_n,θ_0(v))^2 dμ(v) + 2 ∫D̂_n,θ_0(v) (D̂^*_n,θ(v)-D̂_n,θ(v)) dμ(v) 
           + 2 ∫ D_θ(v) (D̂^*_n,θ(v)-D̂_n,θ(v)) dμ(v) + o_B(n^-1) 
       =   ∫ (D̂^*_n,θ_0(v)-D̂_n,θ_0(v))^2 dμ(v) + 2 ∫D̂_n,θ_0(v) (D̂^*_n,θ(v)-D̂_n,θ(v)) dμ(v) 
           + 2 (θ - θ_0)' ∫Γ_0(v) (D̂^*_n,θ(v)-D̂_n,θ(v)) dμ(v) 
           + o_B(n^-1/2θ-θ_0  + n^-1).

Consequently, for θ→θ_0 and η→θ_0,

    M_n^*(θ)-M_n^*(η) 
       =    [(M_n^*-M_n)(θ)-(M_n^*-M_n)(η)] + [(M_n-M)(θ)-(M_n-M)(η)] + [M(θ)-M(η)] 
       =    2 (θ - η)' ∫Γ_0(v) [ (D̂^*_n,θ_0(v)-D̂_n,θ_0(v))-(D̂_n,θ_0(v)-D_θ_0(v)) ] dμ(v)  
           + ∫[ (θ-θ_0)' Γ_0(v)+o(θ-θ_0) ]^2 dμ(v) - ∫[ (η-θ_0)' Γ_0(v)+o(η-θ_0) ]^2 dμ(v) 
           +o_B(n^-1/2θ-θ_0+n^-1/2η-θ_0+n^-1) 
       =    2 (θ - η)' ∫Γ_0(v) [ (D̂^*_n,θ_0(v)-D̂_n,θ_0(v))-(D̂_n,θ_0(v)-D_θ_0(v)) ] dμ(v)  
           + (θ-θ_0)'Δ_0(θ-θ_0) - (η-θ_0)'Δ_0(η-θ_0) 
           +o_B(θ-θ_0^2 + η-θ_0^2 +n^-1/2θ-θ_0 + n^-1/2η-θ_0 + n^-1).

We define 

    γ_n    ≡   Δ_0^-1∫Γ_0(v)(D̂_n,θ_0(v)-D_θ_0(v)) dμ(v), 
    γ_n^*    ≡   Δ_0^-1∫Γ_0(v) (D̂^*_n,θ_0(v)-D̂_n,θ_0(v))dμ(v).

Then, we can rewrite (<ref>) by

    M_n^*(θ)-M_n^*(η)    =    2(θ-η)' Δ_0 (γ_n+γ_n^*) + (θ-θ_0)'Δ_0(θ-θ_0) - (η-θ_0)'Δ_0(η-θ_0) 
           +o_B(θ-θ_0^2 + η-θ_0^2 +n^-1/2θ-θ_0 + n^-1/2η-θ_0 + n^-1).

We take θ = θ̂_n^* and η = θ_0-(γ_n+γ_n^*).
Observe that η∈Θ for n is sufficiently large, because θ_0 is an interior point of Θ.
Hence, we have

    0    ≥    M_n^*(θ̂_n^*) - M_n^*(θ_0-(γ_n+γ_n^*)) 
       =    2[ (θ̂_n^*-θ_0) + (γ_n + γ_n^*) ]' Δ_0 (γ_n + γ_n^*) 
           + (θ̂_n^*-θ_0)' Δ_0 (θ̂_n^*-θ_0) - (γ_n + γ_n^*)' Δ_0 (γ_n + γ_n^*) 
           + o_B(θ̂_n^*-θ_0^2 + γ_n + γ_n^*^2 +n^-1/2θ̂_n^*-θ_0 + n^-1/2γ_n + γ_n^* + n^-1) 
       =   [ (θ̂_n^*-θ_0) + (γ_n + γ_n^*) ]' Δ_0 [ (θ̂_n^*-θ_0) + (γ_n + γ_n^*) ] 
           + o_B(θ̂_n^*-θ_0^2 + γ_n + γ_n^*^2 +n^-1/2θ̂_n^*-θ_0 + n^-1/2γ_n + γ_n^* + n^-1).

By the same argument in Theorem 4, we have θ̂_n^* - θ̂_n = O_B(n^-1/2).
Hence, θ̂_n^*-θ_0≤θ̂_n^* - θ̂_n + θ̂_n - θ_0 = O_B(n^-1/2) + O_P(n^-1/2).
Since γ_n = O_p(n^-1/2) and γ_n^* =O_B(n^-1/2), we have

    n θ̂_n^*-θ_0 + (γ_n + γ_n^*) ^2 = o_B(1).

Because it follows from Theorem 4 that

    θ̂_n - θ_0 = -γ_n + o_p(n^-1/2),

and we can obtain θ̂_n^*-θ̂_n= -γ_n^* + o_B(n^-1/2).
The term γ_n^* has the same limiting distribution as γ_n by the bootstrap theorem for the mean in ℝ^d_θ.
This concludes the proof.






We establish the partial identification of g_t by showing that we can identify functions T^U_t,x',x:ℝ→ℝ and T^L_t,x',x:ℝ→ℝ that satisfy

    g_t(x',u)    ≤    T^U_t,x',x( g_t(x,u) ), 
    
    g_t(x',u)    ≥    T^L_t,x',x( g_t(x,u) ).

If T^U_x',x is identified for all x,x', then we can obtain a lower bound of the function g as follows.
For any random variables V,W, we define

    F^+_V|W(v|w)    ≡    P( V ≤ v | W=w ), 
    
    F^-_V|W(v|w)    ≡    P( V < v | W=w ),

where F^+ is an usual distribution function.
In addition, we define 

    G_t,x^L(y) ≡∫ F^+_Y_t|X_t=x'( T^U_t,x',x(y) ) dF_X_t(x').

Then, we have

    G_t,x^L( g_t(x,u) )    =   ∫ F^+_Y_t|X_t=x'( T^U_t,x',x( g_t(x,u) ) ) dF_X_t(x') 
       ≥   ∫ F^+_Y_t|X_t=x'(  g_t(x',u) ) dF_X_t(x') 
       =   ∫ P( g_t(x',U_t) ≤ g_t(x',u) | X_t = x' ) dF_X_t(x') 
       ≥   ∫ P( U_t ≤ u | X_t = x' ) dF_X_t(x') = u,

where the first inequality follows from (<ref>).
Because g_t(x,u) is weakly increasing in u, we have {U_t≤ u}⊂{g_t(x,U_t) ≤ g_t(x,u)} and the second inequality of (<ref>) holds.
Hence, because G_t,x^L( g_t(x,u) ) ≥ u, we can obtain a lower bound

    g_t(x,u) ≥inf{y ∈ [y,y] :G_t,x^L(y) ≥ u}.

Similarly, we define

    G^U_t,x(y) ≡∫ F^-_Y_t|X_t=x'( T^L_t,x',x(y) ) dF_X_t(x').

Then, we have

    G^U_t,x( g_t(x,u) )    =    ∫ F^-_Y_t|X_t=x'(  T^L_t,x',x(g_t(x,u)) ) dF_X_t(x') 
       ≤   ∫ F^-_Y_t|X_t=x'(  g_t(x',u) ) dF_X_t(x') 
       =   ∫ P( g_t(x',U_t) < g_t(x',u) | X_t = x' ) dF_X_t(x') 
       ≤   ∫ P( U_t < u | X_t = x' ) dF_X_t(x') = u.

Owing to the weak monotonicity of g_t, we have {g_t(x,U_t) < g_t(x,u)}⊂{U_t < u} and the second inequality of (<ref>) holds.
Hence, similarly, we can obtain an upper bound

    g_t(x,u) ≥sup{y∈ [y,y]: G^U_t,x(y) ≤ u}.


We here describe the construction of the functions T_t,x',x^U(y) and T_t,x',x^L(y) that satisfy (<ref>).
We define 

    Q^+_Y_t|𝐗(τ | 𝐱)    ≡   sup{y ∈ [y,y] : F^-_Y_t|𝐗(y|𝐱) ≤τ}, 
    
    Q^-_Y_t|𝐗(τ | 𝐱)    ≡   inf{y ∈ [y,y] : F^+_Y_t|𝐗(y|𝐱) ≥τ}.

Because {U:U≤ u}⊂{U:g(x,U) ≤ g(x,u)} and {U:g(x,U) < g(x,u)}⊂{U:U < u}, for all (x_1,x_2) ∈𝒳_12 and t,s ∈{(1,2),(2,1)}, we have

    F^-_Y_t|𝐗( g_t(x_t,u) | x_1,x_2 )    =    P( g_t(x_t,U_t) < g_t(x_t,u) | X_1=x_1, X_2=x_2 ) 
       ≤    P( U_t < u | X_1=x_1, X_2=x_2 ) 
       =    P( U_s < u | X_1=x_1, X_2=x_2 ) 
       ≤    P( g_s(x_s,U_s) < g_s(x_s,u) | X_1=x_1, X_2=x_2 ) 
       =    F^+_Y_s|𝐗( g_s(x_s,u) | x_1,x_2 ).

For t≠ s, we define

    T̃^U,t,s_x_t,x_s(y)    ≡    Q^+_Y_t|(X_t,X_s)=(x_t,x_s)( F^+_Y_s|(X_t,X_s)=(x_t,x_s)( y ) ) 
    T̃^L,t,s_x_t,x_s(y)    ≡    Q^-_Y_t|(X_t,X_s)=(x_t,x_s)( F^-_Y_s|(X_t,X_s)=(x_t,x_s)( y ) ).

Then, we have

    g_t(x_t,u) ≤T̃^U,t,s_x_t,x_s( g_s(x_s,u) ),

because Q^+_Y_t|𝐗(F^-_Y_t|𝐗(y | 𝐱) | 𝐱 ) = sup{y' ∈ [y,y]: F^-_Y_t|𝐗(y' | 𝐱) ≤ F^+_Y_t|𝐗(y | 𝐱) }≥ y. 
Hence, if (x',x̃)∈ supp(X_t,X_s) and (x,x̃)∈ supp(X_t,X_s), then we have

    g_t(x',u) ≤T̃^U,t,s_x',x̃∘T̃^U,s,t_x̃,x( g_t(x,u) ).

Similarly, we have

    g_t(x',u) ≥T̃^L,t,s_x',x̃∘T̃^L,s,t_x̃,x( g_t(x,u) ).

We define

    T^U_t,x',x(y)    ≡   inf_x̃{T̃^U,t,s_x',x̃∘T̃^U,s,t_x̃,x(y) }   if x ≠ x'
    
    y    if x = x'
    
    T^L_t,x',x(y)    ≡   sup_x̃{T̃^L,t,s_x',x̃∘T̃^L,s,t_x̃,x(y)}   if x ≠ x'
    
    y    if x = x'.

Then, these functions satisfy (<ref>).






Similarly to (<ref>) and (<ref>), for t ≠ s, we have

    g_t(x',u)    ≤   T̃^U,t,s_x',x̃∘T̃^U,s,t_x̃,x( g_t(x,u) ), 
    
    g_t(x',u)    ≥   T̃^L,t,s_x',x̃∘T̃^L,s,t_x̃,x( g_t(x,u) ),

where 

    T̃^U,t,s_x_t,x_s(y)    ≡   inf_𝐱_-(t,s) Q^+_Y_t|(X_t,X_s,𝐗_-(t,s))=(x_t,x_s,𝐱_-(t,s))( F^+_Y_s|(X_t,X_s,𝐗_-(t,s))=(x_t,x_s,𝐱_-(t,s))( y ) ), 
    T̃^L,t,s_x_t,x_s(y)    ≡   sup_𝐱_-(t,s) Q^-_Y_t|(X_t,X_s,𝐗_-(t,s))=(x_t,x_s,𝐱_-(t,s))( F^-_Y_s|(X_t,X_s,𝐗_-(t,s))=(x_t,x_s,𝐱_-(t,s))( y ) ),

and 𝐗_-(t,s) denotes a vector of 𝐗 except X_t and X_s.
Hence,

    T̂^U_t,x',x(y)    ≡   inf_s ≠ t, x̃∈𝒳_s{T̃^U,t,s_x',x̃∘T̃^U,s,t_x̃,x(y) }   if x ≠ x'
    
    y    if x = x'

and

    T̂^L_t,x',x(y)    ≡   sup_s ≠ t, x̃∈𝒳_s{T̃^L,t,s_x',x̃∘T̃^L,s,t_x̃,x(y) }   if x ≠ x'
    
    y    if x = x'

satisfy inequality (<ref>).
Define

    Ĝ^L_t,x(y)    ≡   ∫ F^+_Y_t|X_t=x'( T̂^U_t,x',x(y) ) dF_X_t(x'), 
    Ĝ^U_t,x(y)    ≡   ∫ F^-_Y_t|X_t=x'( T̂^L_t,x',x(y) ) dF_X_t(x').

By a similar argument to the proof for Theorem 2, we have g_t(x,u) ≥inf{y ∈ [y,y] : G_t,x^L(y) ≥ u} and g_t(x,u) ≤sup{y ∈ [y,y] : G_t,x^U(y) ≤ u}.






We define μ_T as a probability measure on {1,⋯ ,T} such that μ_T({t})=1/T for all t ∈{1, ⋯ , T}. 
Let μ̃ be μ_T ×μ, then 1/T∑_t=1^T D̂^t_n,θ_μ^2 = 1/T∑_t=1^T ∫D̂^t_n,θ(v)^2 dμ(v) = D̂^t_n,θ(v)_μ̃^2, where ·_μ̃ is the L_2-norm with respect to μ̃.
Hence, the estimator θ̂_n satisfies

    D̂^t_n,θ̂_n(v) _μ̃ = inf_θ∈ΘD̂^t_n,θ(v) _μ̃.

Therefore, we can prove Theorems 6 and 7 by following arguments similar to the proofs of Theorems 3 and 4, respectively.







§ APPENDIX 3: AUXILIARY LEMMAS



Under Assumptions C3 and C4, D_θ_μ is continuous in θ.



By Assumption C4, the density f_Y_t|𝐗(y|𝐱) is bounded above by a constant C.
For any θ',θ and v, we have

    |D_θ'(v)-D_θ(v)| 
       ≤    2 max_t |E[ ( 1{Y_t≤ g_t(X_t,v_u;θ')}-1{Y_t≤ g_t(X_t,v_u;θ)}) 1{𝐗≤ v_𝐱}]| 
       ≤    2 max_t |E[ ( F_Y_t|𝐗( g_t(X_t,v_u;θ')| 𝐗) - F_Y_t|𝐗( g_t(X_t,v_u;θ)| 𝐗) ) 1{𝐗≤ v_𝐱}]| 
       ≤    2 max_t  ∫| ∫_g_t(x_t,v_u;θ)^g_t(x_t,v_u;θ') f_Y_t|𝐗(y|𝐱) dy  | dF_𝐗(𝐱)  
       ≤    2 C max_t ∫| g_t(x_t,v_u;θ')-g_t(x_t,v_u;θ) | dF_X_t(x_t) ≤ 2CK θ'-θ.

Hence, | D_θ'_μ - D_θ_μ | ≤D_θ'-D_θ_μ≤ 2CK θ'-θ, which implies the continuity of D_θ_μ.





Under Assumptions C3, C4, and C5, 

    sup_θ,v| D̂_n,θ(v) - D_θ(v) | = o_a.s.(1),

and for any δ_n ↓ 0

    sup_v ∈𝒱, θ-θ_0 < δ_n| √(n)( D̂_n,θ(v) - D_θ(v) ) - √(n)( D̂_n,θ_0(v) - D_θ_0(v) ) | = o_p(1).




The collection of indicator functions {𝐱↦1{𝐱≤ v_𝐱} : v_𝐱∈𝒳_12} is a VC-class.
By Assumption C5, the collection of indicator functions for subgraphs of {g_t(·,v_u;θ) : θ∈Θ, v_u ∈𝒰} is also a VC-class.
By reference to Examples 2.10.7 and 2.10.8 in <cit.>, {A_θ^v: θ∈Θ, v ∈𝒱} is P-Donsker, and also P-Glivenko–Cantelli.
Hence, we have (<ref>) and for any δ_n ↓ 0

    sup_(θ,v), (θ',v') : ℙ(A_θ^v - A_θ'^v')^2 < δ_n | 𝔾_n A_θ^v - 𝔾_n A_θ'^v' | = o_p(1) 
       ⇒   sup_v, θ, θ_0 : ℙ(A_θ^v - A_θ_0^v)^2 < δ_n | 𝔾_n A_θ^v - 𝔾_n A_θ_0^v | = o_p(1).

Then, we have

    ℙ(A_θ^v - A_θ_0^v)^2 
       ≤    E[ { (1{Y_1 ≤ g_1(X_1,v_u;θ)} - 1{Y_1 ≤ g_1(X_1,v_u;θ_0)}) . . 
          . . - (1{Y_2 ≤ g_2(X_2,v_u;θ)} - 1{Y_2 ≤ g_2(X_2,v_u;θ_0)}) }^2 ] 
       ≤    4 max_t E[ | 1{Y_t ≤ g_t(X_t,v_u;θ)} - 1{Y_t ≤ g_t(X_t,v_u;θ_0)}| ] 
       =    4 max_t E[ | 1{g_t(X_t,v_u;θ_0) < Y_t ≤ g_t(X_t,v_u;θ)} + 1{g_t(X_t,v_u;θ)} < Y_t ≤ g_t(X_t,v_u;θ_0)}| ] 
       ≤    8 max_t∫| F_Y_t|𝐗( g_t(x_t,v_u;θ)| 𝐱) - F_Y_t|𝐗( g_t(x_t,v_u;θ_0)| 𝐱) | dF_𝐗(𝐱) 
       ≤    8 CK θ - θ_0.

Because θ - θ_0→ 0 implies that ℙ(A_θ^v - A_θ_0^v)^2 → 0, we have (<ref>).





Under Assumptions C4, N2, and N3, D_θ(v) is continuously differentiable in θ in a neighborhood of θ_0 for all v, and |D_θ(v) - D_θ_0(v) - Γ_0(v)'(θ - θ_0)| = o(θ - θ_0) uniformly over v.



First, we show continuous differentiability of D_θ(v).
For all v and θ in the neighborhood,

    ∇_θ D_θ(v)    =   ∇_θ E[ (F_Y_1|𝐗(g_1(X_1,v_u;θ)|𝐗) - F_Y_2|𝐗(g_2(X_2,v_u;θ)|𝐗) ) 1{𝐗≤ v_𝐱}] 
       =   ∇_θ∫_{𝐱≤ v_𝐱}(F_Y_1|𝐗(g_1(x_1,v_u;θ)|𝐱) - F_Y_2|𝐗(g_2(x_2,v_u;θ)|𝐱) ) dF_𝐗(𝐱).

Let C be a constant such that f_Y_t|𝐗(y|𝐱) ≤ C.
Because |f_Y_t|𝐗(g_t(x_t,v_u;θ)|𝐱) ∇_θ g_t(x_t,v_u;θ)| is bounded by the integrable function C ∇g̅(x_t), we can interchange a differential operator with an integral.
Hence, we have

    ∇_θ D_θ(v)    =    E[f_Y_1|𝐗(g_1(X_1,v_u;θ)|𝐗) ∇_θ g_1(X_1,v_u;θ) 1{𝐗≤ v_𝐱}] 
           - E[f_Y_2|𝐗(g_2(X_2,v_u;θ)|𝐗) ∇_θ g_2(X_2,v_u;θ)1{𝐗≤ v_𝐱}].

According to the dominated convergence theorem, ∇_θ D_θ(v) is continuous in θ in a neighborhood of θ_0 for all v.

Next, we show the second statement.
Because D_θ(v) is continuously differentiable in θ, for θ in a neighborhood of θ_0, there exists θ̅_v between θ and θ_0 such that

    |D_θ(v) - D_θ_0(v)- Γ_0(v)'(θ - θ_0)|    =    | {Γ_θ̅_v(v) - Γ_0(v) } (θ - θ_0) | 
       ≤   θ - θ_0×sup_v ∈𝒱Γ_θ̅_v(v) - Γ_0(v)

It follows from Assumption N3(ii) that sup_v ∈𝒱Γ_θ̅_v(v) - Γ_0(v) → 0 as θ - θ_0→ 0.
Hence, we have |D_θ(v) - D_θ_0(v) - Γ_0(v)'(θ - θ_0)| = o(θ - θ_0) uniformly over v.





Under Assumptions C3 and C5, for every sequence {δ_n} of positive numbers that converges to zero,

    sup_θ - θ_0<δ_nD̂_n,θ - D_θ -D̂_n,θ_0_μ = o_p(n^-1/2).




Note that

    sup_θ - θ_0<δ_nD̂_n,θ - D_θ -D̂_n,θ_0_μ≤sup_v ∈𝒱, θ - θ_0<δ_n| D̂_n,θ(v) - D_θ(v) -D̂_n,θ_0(v) |.

By Lemma 2, the right-hand side is o_p(n^-1/2).
Hence, (<ref>) holds.





Under the assumptions for Theorem 3, θ̂^* →_a.sθ_0 for almost all samples W_1, ⋯, W_n.


By the triangle inequality, for almost all samples W_1, ⋯ , W_n, we have

    sup_θD̂_n,θ^* - D_θ_μ   ≤   sup_θD̂_n,θ^* - D̂_n,θ_μ + sup_θD̂_n,θ - D_θ_μ
       ≤   sup_θ, v | D̂_n,θ^*(v) - D̂_n,θ(v)| + sup_θ , v |D̂_n,θ(v) - D_θ(v)| →_a.s. 0,

since {A_θ^v: θ∈Θ, v ∈𝒱} is a Donsker class.
The reminder of the proof is same as for Theorem 3.





Suppose that the assumptions of Theorem 4 hold.
For every sequence {δ_n} of positive numbers that converges to zero, 

    sup_θ - θ_0<δ_n√(n) (D̂^*_n,θ-D̂_n,θ) - √(n) (D̂^*_n,θ_0-D̂_n,θ_0) _μ=o_B(1).



The left-hand side of (<ref>) is dominated above by

    sup_v, θ - θ_0<δ_n | √(n) (D̂^*_n,θ(v)-D̂_n,θ(v)) - √(n) (D̂^*_n,θ_0(v)-D̂_n,θ_0(v)) |.

The bootstrap equicontinuity due to <cit.> implies that this random variable is o_B(1).
Hence, we can obtain (<ref>).








§ APPENDIX 4: FIGURES AND TABLES


















ecta


