Testing Identifying Assumptions in Nonseparable Panel Data … · 2017. 10. 23. · Motivation I By allowing us to observe the same individuals over time, panel data allow us to identify

Testing Identifying Assumptions inNonseparable Panel Data Models

Dalia Ghanem, UC Davis

May 22, 2015

Motivation

I By allowing us to observe the same individuals over time, panel data allow usto identify causal effects under weaker assumptions.

I Nonparametric identification in nonseparable panel data models has receivedconsiderable attention in econometrics.

“Focused”: Altonji and Matzkin (2005), Athey and Imbens (2006), Bester andHansen (2009), Hoderlein and White (2012), Graham and Powell (2012),Chernozhukov et al (2013)“Structural”: Evdokimov (2010), Evdokimov (2011)

I This paper proposes specification tests of identifying assumptions (“generalizedfirst-differencing”).

Motivation

Example: Effect of a Master’s Degree

1 2

Time Period

Ave

rage

Ear

ning

s

Group AGroup B

“Generalized First-Differencing”Average Partial Effect will be identified from

E [Yi2 − Yi1|Group = A]− E [Yi2 − Yi1|Group = B]

through restrictions imposed on the model, Yit = ξt(Xit ,Ai ,Uit)

Motivation

Example: Effect of a Master’s Degree

1 2

Time Period

Ave

rage

Ear

ning

s

Group AGroup B

“Generalized First-Differencing”Average Partial Effect will be identified from

E [Yi2 − Yi1|Group = A]− E [Yi2 − Yi1|Group = B]

through restrictions imposed on the model, Yit = ξt(Xit ,Ai ,Uit)

Motivation

I Identification through “Generalized First-Differencing”- Restrictions on Distribution of Ai ,Uit |Xi and ξt behavior across t

e.g. Time homogeneity (Chernozhukov et al, 2013), Correlated random effects(Altonji and Matzkin, 2005; Bester and Hansen, 2009)

- TESTABLE restrictions on the distribution of the outcome variable

Related Literature: Athey and Imbens (2006), Evdokimov (2010, 2011)

I Specification Tests- Extension of the classical two-sample problem to panel data with demeaning- Bootstrap-adjusted Kolmogorov-Smirnov (KS) and Cramer-von-Mises (CM)

statisticsRelated LiteratureNonparametric Specification Testing: monotonicity (Hoderlein Su and White, 2013), separability(Hoderlein and Mammen, 2009; Lu and White, 2014; Su, Tu and Ullah, 2015)Bootstrap Adjusted KS and CM: Andrews (1997), Abadie (2002)

I Empirical Relevance- Nonparametric Identification: no functional form assumptions on the

relationship between outcome variable, regressors and unobservables(“model-free”)

- Observational Panel Data: Test of Time Homogeneity- “Experimental” Panel Data: Test of Conditional Random Effects

Road Map

“Generalized First-Differencing”Model and Object of InterestIdentification and Testable Restrictions

Testing Identifying AssumptionsAsymptotic ResultsMonte Carlo Study

Empirical Illustration: Returns to Schooling

Concluding Remarks

Application

“Generalized First-Differencing”: Model and Object of Interest

ModelFor i = 1, 2, . . . , n, t = 1, 2

Yit = ξt(Xit ,Ai ,Uit)

ξt is the structural function, Xit has finite support, X.{Yit ,Xit}i,t are observable.

Object of InterestAPE of a Subpopulation Xi = (Xi1,Xi2) = (x , x ′) “Switchers”

βt(x → x ′|Xi = (x , x ′)) = E [Yitx′ − Yit

x |Xi = (x , x ′)]

=

∫(ξt(x

′, a, u)− ξt(x , a, u))︸︷︷︸∆ Str. Function due to ∆x

d FAi ,Uit |Xi(a, u|(x , x ′))︸︷︷︸

Unobs Distr of Switchers

Identification of APE = appropriately averaging over ∆Y due to ∆X

holding Str. Function and Unobs. Distr. Fixed

(ceteris paribus)

“Generalized First-Differencing”: Identification and Testable Restrictions

1 2Time

Ave

rage

Ear

ning

s

Group A

Group B

Lemma 2.1 “Generalized First-Differencing”Let Assumptions 2.1 and 2.2 in the paper hold.

βt(x → x′|Xi = (x, x′)) = E [Yi2 − Yi1|Xi = (x, x′)]︸︷︷︸Average ∆Y for Switchers

− E [Yi2 − Yi1|Xi = (x, x)]︸︷︷︸Average ∆Y for Stayers

if and only if

∫ ∆Y due to ∆t holding x fixed︷︸︸︷(ξ2(x, a, u2)− ξ1(x, a, u1))

× (fAi ,Ui1,Ui2|Xi (a, u1, u2|(x, x′))− fAi ,Ui1,Ui2|Xi (a, u1, u2|(x, x))︸︷︷︸∆ Unobserv. Distr. due to ∆X (Switchers vs. Stayers)

)d(a, u1, u2) = 0.


1 2Time

Ave

rage

Ear

ning

s

Group A

Group B

Lemma 2.1 “Generalized First-Differencing”Let Assumptions 2.1 and 2.2 in the paper hold.

βt(x → x′|Xi = (x, x′)) = E [Yi2 − Yi1|Xi = (x, x′)]︸︷︷︸Average ∆Y for Switchers

− E [Yi2 − Yi1|Xi = (x, x)]︸︷︷︸Average ∆Y for Stayers

if and only if

∫ ∆Y due to ∆t holding x fixed︷︸︸︷(ξ2(x, a, u2)− ξ1(x, a, u1))

× (fAi ,Ui1,Ui2|Xi (a, u1, u2|(x, x′))− fAi ,Ui1,Ui2|Xi (a, u1, u2|(x, x))︸︷︷︸∆ Unobserv. Distr. due to ∆X (Switchers vs. Stayers)

)d(a, u1, u2) = 0.


Theorem 2.1Fixed Effects: Identification and Testable Restrictions (T = 2)Let Assumptions 2.1 and 2.2 in the paper hold.If, in addition,

Yi t = ξ(Xit ,Ai ,Uit) + λt(Xit) (Stationarity in Unobservables)

Ui1|Xi ,Aid= Ui2|Xi ,Ai , (Time Homogeneity)

then(i) (Identification of APE)∫

(ξ2(x , a, u2)− ξ1(x , a, u1))

× (fAi ,Ui1,Ui2|Xi(a, u1, u2|(x , x ′))− fAi ,Ui1,Ui2|Xi

(a, u1, u2|(x , x)))d(a, u1, u2) = 0.

(ii) (Testable Restriction)

FYi1−λ1(x)|Xi(.|(x , x)) = FYi2−λ2(x)|Xi

(.|(x , x)), ∀x ∈ X


Theorem 2.1Fixed Effects: Identification and Testable Restrictions (T = 2)Let Assumptions 2.1 and 2.2 in the paper hold.If, in addition,

Yi t = ξ(Xit ,Ai ,Uit) + λt(Xit) (Stationarity in Unobservables)

Ui1|Xi ,Aid= Ui2|Xi ,Ai , (Time Homogeneity)


(ξ2(x , a, u2)− ξ1(x , a, u1))


(a, u1, u2|(x , x)))d(a, u1, u2) = 0.

“Intution”: ξ2(x , a, u)− ξ1(x , a, u) = λ2(x)− λ1(x), identified from Stayers

(ii) (Testable Restrictions)

FYi1−λ1(x)|Xi(.|(x , x)) = FYi2−λ2(x)|Xi

(.|(x , x)), ∀x ∈ X

“Intution”: λt(x) is the only source of ∆ across t forstayers’ distribution of outcome variable


Time Homogeneity in our Example

Density of Appropriately Demeaned Earnings ofGroup A (Switchers) and Group B (Stayers)

t = 1 t = 2

I Group A’s demeaned earnings distribution changed across time due to Master’sDegree (Identification of APE)

I Group B’s demeaned earnings distribution does not change across time(Testable Restriction)

- Suitable for observational panel data- Extends the notion of “fixed effects” to nonseparable models (Chernozhukov et al,2013; Hoderlein and White, 2012)


Time Homogeneity in our Example

Density of Appropriately Demeaned Earnings ofGroup A (Switchers) and Group B (Stayers)

t = 1 t = 2

I Group A’s demeaned earnings distribution changed across time due to Master’sDegree (Identification of APE)

I Group B’s demeaned earnings distribution does not change across time(Testable Restriction)

- Suitable for observational panel data- Extends the notion of “fixed effects” to nonseparable models (Chernozhukov et al,2013; Hoderlein and White, 2012)


Theorem 2.2Correlated Random Effects: Identification & Testable Restrictions(T = 2)Let Assumptions 2.1 and 2.2 in the paper hold (Yi t = ξt(Xit ,Ai ,Uit)).

If Ai ,Ui1,Ui2|Xid= Ai ,Ui1,Ui2|Xi1,


(ξ2(x , a, u2)− ξ1(x , a, u1))


(a, u1, u2|(x , x)))d(a, u1, u2) = 0


FYi1|Xi(.|(x , x ′)) = FYi1|Xi

(.|(x , x)), ∀x , x ′ ∈ X, x 6= x ′


Theorem 2.2Correlated Random Effects: Identification & Testable Restrictions(T = 2)Let Assumptions 2.1 and 2.2 in the paper hold (Yi t = ξt(Xit ,Ai ,Uit)).

If Ai ,Ui1,Ui2|Xid= Ai ,Ui1,Ui2|Xi1, (Conditional-Random-Effects)


(ξ2(x , a, u2)− ξ1(x , a, u1))


(a, u1, u2|(x , x))︸︷︷︸=0

)d(a, u1, u2) = 0

“Intuition”: The two subpopulations are the same in terms of unobservables. Xi2

is randomly assigned conditional on Xi1


FYi1|Xi(.|(x , x ′)) = FYi1|Xi

(.|(x , x)), ∀x , x ′ ∈ X, x 6= x ′

“Intuition”: Distribution of the outcome variable is the same for switchers andstayers before the ‘switch.’


Conditional Random Effects in our Example

Density of Earnings ofGroup A (Switchers) and Group B (Stayers)

t = 1 t = 2

I Group A and Group B are the same subpopulation in terms of unobservabledistribution. (Identification of APE)

I The distribution of the outcome variable across time is changing for both groups.

I Group A and Group B have the same distribution whenever they have the sametreatment status. (Testable Restriction)

- Suitable for experimental and quasi-experimental settings, e.g. test for attrition

Testing Identifying Assumptions

Identifying Assumptions⇒ Equality restrictions on the conditional distribution of the outcome variableFYit |Xi

(.|x) t = 1, 2, ...,T , x ∈ XT

Basic Testing ProblemExtension of the classical two-sample problem

I Time Homogeneity: dependence across time and demeaningI Conditional Random Effects: K-sample problem

Bootstrap-Adjusted P-values for StatisticsKolmogorov-Smirnov (sup-norm ‖f (y)‖∞,Y = supy∈Y |f (y)|)Cramer-von-Mises (L2-norm ‖f (y)|2,φ =

∫Y f (y)2φ(y)dy)

Classical Two-Sample

Testing Identifying Assumptions: Basic Bootstrap Procedure

Let Yi = (Yi1, . . . ,YiT ) and Xi = (Xi1, . . . ,XiT ).

1. Compute the statistic Jn for {{Y1,X1}, ..., {Yn,Xn}}, hereinafter the originalsample.

2. Resample n observations with replacement from the original sample. Computethe centered statistic, Jbn .

3. Repeat 1-2 B times.

4. Calculate

pn =B∑

b=1

1{Jbn > Jn}.

Reject if p-value is smaller than some significance level α.

Intuition:

(1) cross-sectional i.i.d. assumption

(2) resampling the entire sample, reflecting randomness in stayers and movers

(3) paired-sample problem is an extension of one-sample problem

Testing Identifying Assumptions: Asymptotic Results(Time Homogeneity)

Time Homogeneity with a Parallel Trend

Hpt0 : FYi1|∆Xi

(.|0) = FYi2−∆λ|∆Xi(.|0) (aggregates over all stayers)

Statistics of the Original Sample

KSptn,Y =

∥∥Fn,Yi1|∆Xi(y |0)− Fn,Yi2−∆λn|∆Xi

(y |0)∥∥∞,Y

CMptn,φ =

∥∥Fn,Yi1|∆Xi(y |0)− Fn,Yi2−∆λn|∆Xi

(y |0)∥∥

2,φ

Centered Stats

TheoremGiven that {(Yi ,X

′i )}ni=1 is an iid sequence, |X | = K , P(∆Xi = 0) > 0,

FYit |∆Xi(.|0) is non-degenerate for t = 1, 2, and supy∈Y |fYit

(y)| <∞ for t = 1, 2,

the procedure described in 1-4 for KSptn,Y and CMpt

n,φ to test H10

(i) provides correct asymptotic size α

(ii) is consistent against any fixed alternative.

Hadamard

∆λ = E [Yi2 − Yi1|∆Xi = 0], ∆λn is its sample analogue, and ∆λn,b is its bth bootstrap sample analogue.

fYitis the marginal density of Yit .

Testing Identifying Assumptions: Asymptotic Results(Correlated-Random-Effects)

Example: Conditional-Random-Effects with Binary Regressor

Hcre0 : FYi1|Xi

(.|(x , x ′)) = FYi1|Xi(.|(x , x ′′)),

∀x , x ′, x ′′ ∈ {0, 1}, x ′ 6= x ′′

Statistics of the Original Sample

KS0n,Y = sup

y∈Y

∣∣Fn,Yi1|Xi(y |(0, 0))− Fn,Yi1|Xi

(y |(0, 1))∣∣

KS1n,Y = sup

y∈Y

∣∣Fn,Yi1|Xi(y |(1, 0))− Fn,Yi1|Xi

(y |(1, 1))∣∣

KScren,Y = Pn(Xi1 = 0)KS0

n,Y + Pn(Xi1 = 1)KS1n,Y

Centered Stats

TheoremGiven that {(Yi ,X

′i )}ni=1 is an iid sequence, |X | = K , P(Xi = x) > 0 ∀x ∈ XT ,

FYit |Xiis non-degenerate for t = 1, 2, the procedure described in 1-4 for KScre

n,Y andCMcre

n,φ to test Hcre0

(i) provides correct asymptotic size α

(ii) is consistent against any fixed alternative.

Testing Identifying Assumptions: Monte Carlo Study

Table: Models Considered in the Monte Carlo Design

Model Ai Uit Yit

A Ai = 0.5√TXT + 0.5ψi Uit = εit XT Yit = ξ(Xit ,Ai ,Uit)

B ” ” Yit = ξ(Xit ,Ai ,Uit) + λtC ” ” Yit = ξ(Xit ,Ai ,Uit) + λtsign(Xit)

D Ai = 0.5√TXi1 + 0.5ψi Uit = Xi1εit Yit = ξ(Xit ,Ai ,Uit)σt + λt

Notes: ξ(x, a, u) = µ0 + a + (2 + a)x + u; Xit = {Zit − p(K − 1)}/√

(K − 1)p(1− p),

Ziti.i.d.∼ Binomial(K-1, p); εit

i.i.d.∼ N(0, 1); ψii.i.d.∼ N(0,1); XT ≡

∑Tt=1 Xit/T ; sign(g(x))

= 1{g(x) > 0} − 1{g(x) < 0}; λ1 = 0, σ1 = 1.

(A) Time Homogeneity with no Trend

(B) Time Homogeneity with Nonstochastic Time Effect

(C) Time Homogeneity with Generalized Time Effect

(D) Conditional Random Effects

For each model, we compute the KS and CM statistics assuming

(nt) time homogeneity with no trend (λt(x) = 0 ∀x)

(pt) time homogeneity with nonstochastic time effect (λt(x) = λt ∀x)

(gt) time homogeneity with generalized time effect λt(x)

(cre) conditional random effects


Table: Baseline Simulation Results: T = K = 2 (Models A-B)

n=500 n=2000

KS CM KS CM

α 0.025 0.05 0.10 0.025 0.05 0.10 0.025 0.05 0.10 0.025 0.05 0.10

Model A

nt 0.035 0.056 0.110 0.032 0.049 0.114 0.033 0.057 0.101 0.029 0.054 0.108pt 0.013 0.024 0.052 0.022 0.054 0.093 0.021 0.035 0.070 0.032 0.060 0.112gpt 0.003 0.011 0.030 0.011 0.031 0.062 0.007 0.025 0.050 0.033 0.057 0.095cre 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

Model B (λ2 = 0.25)

nt 0.366 0.492 0.630 0.360 0.467 0.590 0.933 0.971 0.989 0.945 0.971 0.993pt 0.014 0.020 0.053 0.019 0.042 0.101 0.024 0.032 0.069 0.034 0.059 0.111gpt 0.003 0.007 0.029 0.018 0.030 0.068 0.007 0.023 0.050 0.029 0.057 0.098cre 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

Model C (λ2 = 0.25)

nt 0.308 0.440 0.598 0.264 0.377 0.511 0.929 0.968 0.988 0.926 0.959 0.982pt 0.342 0.450 0.584 0.344 0.452 0.567 0.970 0.986 0.993 0.967 0.986 0.993gpt 0.003 0.008 0.029 0.017 0.033 0.067 0.009 0.026 0.049 0.032 0.054 0.099cre 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

Model D (λ2 = 0.5, σ2 = 1.1)

nt 0.985 0.995 1.000 0.925 0.961 0.983 1.000 1.000 1.000 1.000 1.000 1.000pt 0.409 0.531 0.666 0.107 0.165 0.273 0.989 0.993 0.996 0.555 0.692 0.800gpt 0.014 0.032 0.059 0.075 0.130 0.195 0.108 0.169 0.279 0.320 0.428 0.549cre 0.035 0.063 0.114 0.025 0.044 0.106 0.029 0.052 0.109 0.027 0.043 0.092

Model D (λ2 = 0.5, σ2 = 1.2)

nt 1.000 1.000 1.000 0.944 0.971 0.995 1.000 1.000 1.000 1.000 1.000 1.000pt 0.975 0.988 0.996 0.397 0.521 0.665 1.000 1.000 1.000 0.995 1.000 1.000gpt 0.050 0.095 0.173 0.253 0.349 0.456 0.570 0.714 0.833 0.907 0.948 0.982cre 0.035 0.061 0.114 0.025 0.053 0.103 0.035 0.060 0.108 0.029 0.042 0.085

Notes: The table reports the rejection probabilities across 1, 000 simulations for the bootstrap adjustments for the statistics. Bold fontindicates that the model considered satisfies the null hypothesis for the statistic in question. The CM statistic is implemented usingφ as the standard normal density.

Statistics


Practical Considerations Considered in MC Study

I Choice of Density in the CM Statisticsimulations suggest improved power by overweighing center of the distribution

I Aggregated vs. Disaggregated Statistics for Time Homogeneity

I Simulation Study Resembling NLSY 1983-1987

“Generalized First-Differencing”Model and Object of InterestIdentification and Testable Restrictions

Testing Identifying AssumptionsAsymptotic ResultsMonte Carlo Study


Concluding Remarks

Empirical Illustration

Over-identification Test for the Linear Fixed Effects Model(Chamberlain, 1984, MCS)

Yit = X ′itβ +Ai + UitE [Uit |Xi ,Ai ] = 0

Ai = X ′i1η1 + · · ·+ X ′iT ηT + Ei

Note that the last equation is just a predictive relationship.

3SLS equivalent for the MCS procedure (Angrist and Newey, 1991)

- NLSY subsample of young men, 1983-1987

- test linear fixed effects model for union-wage and returns to schooling equations(reject the latter)

Empirical Illustration

This paper

- revisit the returns to schooling example using the same NLSY subsample

- test of time homogeneity in the presence of a parallel trend

Table: Descriptive Statistics: Subsample of the NLSY, 1983-1987

1983 1984 1985 1986 1987

Race 0.12Age 21.84

(2.22)HGC 12.34 12.45 12.57 12.57 12.61

(1.77) (1.83) (1.94) (1.94) (1.98)South 0.29 0.30 0.30 0.30 0.30Urban 0.76 0.77 0.76 0.77 0.76Log Hourly Wage 6.31 6.39 6.50 6.61 6.72

(0.48) (0.49) (0.49) (0.49) (0.50)


Time Homogeneity with a Paralel Trend

Yit = ξ(Sit ,Ai ,Uit) + λt

Uit |Si ,Aid= Ui1|Si ,Ai

Yit is log income, Sit is schooling.

For T = 2,

⇒ H0 : FYi1|∆Si(.|0) = FYi2−∆λ|∆Si

(.|0)

Statistics: KS and CM (φ as standard normal)


Table: Testing Time Homogeneity with a Parallel Trend of Log Earnings (Sit)

Statistic KS CM CM Fφ N(6.5, 0.25) N(6.5, 0.5)

Full Sample 0.30 0.34 0.12

1983-84 0.49 0.16 0.08 0.111984-85 0.12 0.23 0.13 0.091985-86 0.46 0.61 0.56 0.251986-87 0.43 0.40 0.15 0.46

Notes: The above reports p-values of the tests for timehomogeneity using the NLSY subsample, n = 1, 087,where schooling Sit is the sole regressor.


Table: APE of Schooling on Log Hourly Wage

Subsample APE S.E. t-Stat

1983 -84 122 -0.012 0.043 -0.2671984-85 73 0.095 0.055 1.7231985-86 58 0.226 0.060 3.7371986-87 41 -0.012 0.068 -0.179

I violations of linearity

I heterogeneity of response to schooling

⇒ Rejection of the linear FE assumption in Angrist and Newey (1991) may bedriven by misspecification of the linear model.

Concluding Remarks

I Specification Testing for Identifying Assumptions (“GeneralizedFirst-Differencing”)

- Asymptotically valid bootstrap procedure- Bootstrap-adjusted statistics perform well in finite samples

I Empirical Relevance:- “Generalized first-differencing” strategies are natural extensions of the

classical difference-in-difference approach.- Intuition of nonparametric identification is whether the data can identify the

effect of interest, as opposed to a model imposed on the data.- Two potentially useful tests here are time homogeneity and conditional

random effects.

THANK YOU!

Time Homogeneity

Centered Statistics of the bth Bootstrap Sample

KSbn,Y = sup

y∈Y

∣∣∣F n,bYi1|∆Xi

(y |0)− F n,bYi2|∆Xi

(y |0)−{F nYi1|∆Xi

(y |0)− F nYi2|∆Xi

(y |0)}∣∣∣

CMbn,φ =

∫ (F n,bYi1|∆Xi

(y |0)− F n,bYi2|∆Xi

(y |0)−{F nYi1|∆Xi

(y |0)− F nYi2|∆Xi

(y |0)})2

φ(y)dy

Time Homogen

Time Homogeneity with Time Effect

Centered Statistics of the bth Bootstrap Sample

KSbn,Y =

∥∥∥∥F n,bYi1|∆Xi

(y |0)− F n,b

Yi2−∆λn,b|∆Xi(y |0)−

{F nYi1|∆Xi

(y |0)− F nYi2−∆λn|∆Xi

(y |0)}∥∥∥∥∞,Y

CMbn,φ =

∥∥∥∥F n,bYi1|∆Xi

(y |0)− F n,b

Yi2−∆λn,b|∆Xi(y |0)−

{F nYi1|∆Xi

(y |0)− F nYi2−∆λn|∆Xi

(y |0)}∥∥∥∥

2,φ

Time Homogen with TE

Exclusion Restriction

KS0,bn,Y = sup

y∈Y

∣∣∣F n,bYi1|Xi

(y |(0, 0))− F n,bYi1|Xi

(y |(0, 1))−{F nYi1|Xi

(y |(0, 0))− F nYi1|Xi

(y |(0, 1))}∣∣∣

KS1,bn,Y = sup

y∈Y

∣∣∣F n,bYi1|Xi

(y |(1, 0))− F n,bYi1|Xi

(y |(1, 1))−{F nYi1|Xi

(y |(1, 0))− F nYi1|Xi

(y |(1, 1))}∣∣∣

KSbn,Y = Pn(Xi1 = 0)KS

0,bn,Y + Pn(Xi1 = 1)KS

1,bn,Y

Exclusion Restr

Hadamard Derivative

φ(Ft(.|x), λ) = Ft(.− λ|x) (1)

The Hadamard derivative is given byφ′Ft (.|x),λ(x)

(g , ε) = g(.− λ(x))− εft(.− λ(x)|Xi1 = Xi2 = x), where the subscript

Ft(.|x) denotes FYit |Xi(.|(x , x)).

Time Homo TE

Monte Carlo Study: Statistics

KSntn,Y = ‖F1,n(.|∆Xi = 0) − F2,n(.|∆Xi = 0)‖∞,Y (2)

KSptn,Y = ‖F1,n(.|∆Xi = 0) − F2,n(. − λn|∆Xi = 0)‖∞,Y (3)

KSgtn,Y = ‖F1,n(.|∆Xi = 0) − F2,n(., Λn|∆Xi = 0)‖∞,Y (4)

KSexcln,Y = Pn(Xi1 = 0)‖F1,n(.|Xi = (0, 0)) − F1,n(.|Xi = (0, 1))‖∞,Y

+ Pn(Xi1 = 1)‖F1,n(.|Xi = (1, 0)) − F1,n(.|Xi = (1, 1))‖∞,Y (5)

CMntn,φ = ‖F1,n(.|∆Xi = 0) − F2,n(.|∆Xi = 0)‖2,φ (6)

CMptn,φ

= ‖F1,n(.|∆Xi = 0) − F2,n(. − λn|∆Xi = 0)‖2,φ (7)

CMgtn,φ

= ‖F1,n(.|∆Xi = 0) − F2,n(., λ|∆Xi = 0)‖2,φ (8)

CMexcln,φ = Pn(Xi1 = 0)‖F1,n(.|Xi = (0, 0)) − F1,n(.|Xi = (0, 1))‖2,φ

+ Pn(Xi1 = 1)‖F1,n(.|Xi = (1, 0)) − F1,n(.|Xi = (1, 1))‖2,φ. (9)

‖.‖∞,Y is the L∞-norm, ‖.‖2,φ is the L2 norm taken wrt φ.

Models A-D

Testing Identifying Assumptions

Kolmogorov-Smirnov (KS) Statistic

KSn,m = maxj≤n+m

∣∣∣∣∣n∑

i=1

1{Yi1 ≤W j}n

−m∑i=1

1{Yi1 ≤W j}m

∣∣∣∣∣≡ ‖F n

1 (.)− Fm2 (.)‖∞,n (10)

Cramer-von Mises (CM) Statistic

CMn,m =1

n + m

n+m∑j=1

(n∑

i=1

1{Yi1 ≤W j}n

−m∑i=1

1{Yi1 ≤W j}m

)2

≡ ‖F n1 (.)− Fm

2 (.)‖2,n (11)

{Wj}n+mj=1

are the order statistics of the pooled sample.

Fnt (.) is the empirical cdf of Yit with sample size n. Classical Two-Sample

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Testing Identifying Assumptions in Nonseparable Panel Data … · 2017. 10. 23. · Motivation I By allowing us to observe the same individuals over time, panel data allow us to identify