Jackknife and Analytical Bias Reduction for Nonlinear ...Mcfadden/E242_f02/Hahn.pdfJackknife and...

Jackknife and Analytical Bias Reduction

for Nonlinear Panel Models

Jinyong Hahn Whitney NeweyBrown University MIT

June, 2002

Acknowledgment: The second authors research was partially supported by the NSF. Helpful com-

ments by G. Chamberlain, J. Hausman, Guido Kuersteiner, and seminar participants of Bristol, Exeter,

and Harvard/MIT are appreciated.

Abstract

Fixed effects estimator of panel models can be severely biased because of the well-known incidental

parameter problems. It is shown that such bias can be reduced as T grows with n by using an analyt-

ical bias correction or by using a panel jacknife. We describe both of these approaches. We consider

asymptotics where n and T grow at the same rate as an approximation that allows us to compare bias

properties. Under these asymptotics the bias corrected estimators are centered at the truth, whereas the

Þxed effects estimator is not. This asymptotic theory shows the bias reduction given by the analytical or

jacknife correction.

1 Introduction

Panel data, consisting of observations across time for different individual economic agents, allows the

possibility of controlling for unobserved individual heterogeneity. Such heterogeneity can be an important

phenomenon, and failure to control for it can result in misleading inferences. This problem is particularly

severe when the unobserved heterogeneity is correlated with explanatory variables. Such situation arises

naturally when some of the explanatory variables are decision variables.

Models and methods of controlling for unobserved heterogeneity in linear models are well established.

A partial list of references is Amemiya and MaCurdy (1986), Anderson and Hsiao (1982), Bhargava and

Sargan (1983), Chamberlain (1982), Hausman and Taylor (1981), and Mundlak (1978). Controlling for

unobserved heterogeneity is much more difficult in nonlinear models. Conditional maximum likelihood

can be used in the rare instance that there is a sufficient statistic for the individual effect. In a few cases

the individual effects can be removed by transformations. However, such cases are the exception rather

than the rule. Generally it is not possible to Þnd a transformation or a sufficient statistic to remove the

individual effect.

One way of attempting to control for individual effects in nonlinear models is to treat each effect as a

separate parameter to be estimated. Unfortunately, such estimators are typically subject to the incidental

parameters problem noted by Neyman and Scott (1948). The estimators of the parameters of interest

will be inconsistent if the number of individuals n goes to inÞnity while the number of time periods T

is held Þxed. This inconsistency occurs because only a Þnite number of observations are available to

estimate each individual effect, so that the estimate of the individual effects are random, even in the

limit, and this randomness contaminates the estimates of the parameters of interest.

Even in static models the incidental parameters bias can be severe when there are few observations,

e.g. see Chamberlain (1982) and Abrevaya (1997). With many time series observations the bias in

static models may be small, as in the Monte Carlo studies of Heckman (1981), although the bias is more

pronounced in dynamic models, e.g. Heckman (1981) and Nickell (1981). This bias is so severe that

estimators are even asymptotically biased if T grows at the same rate as n. This problem is exempliÞed

by the results of Hahn and Kuersteiner (2001), who Þnd asymptotic bias in Þxed effects estimators of

linear dyanamic models under such asymptotics. We Þnd here an analogous Þxed effects asymptotic bias

for nonlinear models.

The purpose of this note is to consider two approaches to reducing the bias from Þxed effects estimation

in nonlinear models. One approach is an analytical bias correction using the the bias formula obtained

from an asymptotic expansion as T grows, similarly to Hahn and Kuersteiner (2001). This correction

is obtained by generalizing a bias formula from Waterman et. al. (2000), estimating the bias term,

and using this to correct the estimator. The second approach is based on a jacknife in the time series

dimension of the panel. The idea here is to use the variation in the Þxed effects estimators as a time

period is dropped to form a bias corrected estimator. We do this by applying the Quenouilles (1956) and

Tukey (1958) jacknife formula to the Þxed effects estimators that drop each time period. This produces

an estimator with properties similar to the analytical approach without explicit computation of the bias

term.

We analyze the properties of both of these estimators under asymptotics as n and T grow at the same

1

rate. We Þnd that both approaches yield an estimator that is asymptotically normal and centered at the

truth. This large n, large T asymptotics provides a way to formalize the idea that the bias corrected

estimators should be closer to the truth. As usual, the asymptotic theory is primarily intended as an

approximation to the Þnite sample distribution of the estimator, here especially to its bias. Under our

asymptotic approximation, bias correction does not increase the asymptotic variance. This suggests that

the beneÞt of bias reduction substantially dominates any potential increase of variance.

We conjecture that the bias corrected estimators have an even stronger property, being asymptotically

normal and correctly centered as long as T grows faster than n1/3. This property would provide further

justiÞcation for the bias corrected estimators. It would mean they have little bias even when T is allowed

to be much smaller than n, a situation that is often encountered in practice.

In Section 2 we use some comparatively simple calculations to describe how the bias corrections work.

Section 3 gives the form of the analytical bias correction. Section 4 describes the jacknife bias correction.

Asymptotic theory is given in Section 5.

2 The Effect of Bias Correction

Some expansions can be used to motivate the bias results. Let θ denote an R-dimensional parameter

vector of interest with true value θ0, and let θ[T ] denote the limiting value, as n → ∞ with T Þxed, of

a Þxed effects estimator bθ of θ0. A consequence of the incidental parameters problem is that typically

θ[T ] 6= θ0. Note that the bias should be small for large enough T , i.e., limT→∞ θ[T ] = θ0. Indeed, it can

be shown that in some generality,

θ[T ] = θ0 +β

T+O

µ1

T 2

¶.

To see the resultant bias of the Þxed effects estimator, suppose that as n/T → ρ it is the case that√nT³bθ − θ[T ]´ d→ N (0,Ω). Then

√nT³bθ − θ0´ = √nT ³bθ − θ[T ]´+√nT β

T+O

µrn

T 3

¶d→ N(β

√ρ,Ω).

Thus we see that even when T grows as fast as n the Þxed effects estimator has asymptotic bias, with its

limit distribution not being centered at zero.

The Þrst approach to bias correction consists of estimating β by some bβ and then forming a biascorrected estimatorbbθ ≡ bθ − bβ

T, (1)

This estimator should be less biased than the Þxed effects estimator bθ. SpeciÞcally, when n and T growat the same rate as before and bβ p→ β,

√nT

µbbθ − θ0¶ = √nT ³bθ − θ[T ]´−rn

T

³bβ − β´+Oµr n

T 3

¶d→ N (0,Ω) , (2)

so that the use of bbθ eliminates that bias. Thus, all we need to do is Þnd a consistent estimator of β. Wedo this by plugging in consistent esitimators of components of the formula for β. We show that equation

(2) holds for the bβ we use.2

The panel jacknife provides an alternative approach that avoids estimation of β. To describe it, letbθ(t) be the Þxed effects estimator based on the subsample excluding the observations of the tth period.The jackknife estimator is

eθ ≡ Tbθ − (T − 1) TXt=1

bθ(t)/T (3)

To explain the bias correction from this estimator it is helpful to consider a further expansion

θ[T ] = θ +β

T+γ

T 2+O

µ1

T 3

¶. (4)

To see how the jacknife affects the bias we can consider the limit of eθ for Þxed T and see how it changeswith T . The estimator eθ will converge in probability to

Tθ[T ] − (T − 1) θ[T−1] = θ +

µ1

T− 1

T − 1¶γ +O

µ1

T 2

¶= θ − 1

T (T − 1)γ +Oµ1

T 2

¶= θ +O

µ1

T 2

¶.

Thus we see that the bias of the jacknife corrected value is of order 1/T 2. Furthermore, as shown below,

the jackknife estimator eθ has the same property as bbθ under the large n and T approximation, namely√nT³eθ − θ0´ d→ N (0,Ω).

Our conjecture, that the analytical bias correction works if T grows faster than n1/3, can also be

explained by an analogous expansion. Suppose now that β is√nT consistent, so that bβ = β+O ³1/√nT´ .

Then expanding as in equation (2) gives

√nT

µbbθ − θ0¶ = √nT ³bθ − θ[T ]´−√nT ³bβ − β´ /T +Oµr n

T 3

¶d→ N(0,Ω).

Although we do not verify this conjecture here, this expansion strongly suggests that the bias correction

removes asymptotic bias under the condition that T grows at least as fast as n1/3. This conjecture

suggests that the bias corrected Þxed effects estimator should be approximately unbiased when T is much

smaller than n.

3 Analytical Bias Correction

We Þrst describe the model we consider. Let the data observations be denoted by xit, (t = 1, ..., T ; i =

1, ..., n). We assume that these observations are mutually independent and that there is a density function

f (x; θ,α) such that

xit ∼ f (·; θ0,αi0) , (t = 1, ..., T ; i = 1, ..., n)

We assume that dim (θ) = R ≥ 1 and that dim (α) = 1. Thus, here XiT ≡ (xi1, ..., xiT ) is i.i.d. over t,while the distribution of observations across i differs only due to αi0. The Þxed effects estimator is given

3

by

bθ = argmaxθ

nXi=1

q (XiT , θ) ,

q (XiT , θ) ≡TXt=1

log f (xit; θ, bαi (θ)) ,bαi (θ) ≡ argmax

αi

TXt=1

log f (xit; θ,α) .

By the usual results for extremum estimators, for Þxed T the estimator bθ will be consistent forθ[T ] = argmax

θE [q (XiT , θ)] = argmax

θE [log f (xit; θ, bαi (θ))] .

The incidental parameters problem, with θ[T ] 6= θ0, arises because of randomness of bαi (θ). If bαi (θ) werereplaced by αi(θ) ≡ argmaxαi E [log f (xit; θ,α)], i.e. if the sum over T were replaced by the expectation,then it is straightforward to show that θ[T ] = θ0.

To describe the analytical bias correction, we need to deÞne some additional derivatives. Let

sθ (xit; θ,αi) ≡ ∂

∂θlog f (xit; θ,αi) , V (xit; θ,αi) ≡ ∂

∂αilog f (xit; θ,αi) ,

V2 (xit; θ,αi) ≡ f (xit; θ,αi)−1 ∂2

∂α2if (xit; θ,αi) .

Also, for the Þxed effect estimator bθ and bα1, ..., bαn, letbsit ≡ sθ

³xit;bθ, bαi´ , bVit ≡ V ³xit;bθ, bαi´ , bV2it ≡ V2 ³xit;bθ, bαi´ ,

bUit ≡ bsit − bVit Ã TXu=1

bVitbsit!,ÃTXu=1

bV 2it!.

Then the bias term estimator is

bβ = −12

Ã1

nT

nXi=1

TXt=1

bUit bU 0it!−1Ã

1

n

nXi=1

ÃTXt=1

bUitbV2it, TXt=1

bV 2it!!

.

The bias corrected estimator can then be formed as in equation (1).

This object can be interpreted as local bias correction for random parameters. Note that 1/PTt=1

bV 2itis an information approxmation to the variance of bαi. From Chesher (1984), we know that the score for

a random αi with variance 1/PTt=1

bV 2it would bebDit = 1

2

Ã1PT

t=1bV 2it! bV2it.

Applying the Kiefer and Skoog (1984) general formula for local bias of the MLE under misspeciÞcation

to the special case of random parameters then gives bβ. Thus we see that bβ is an estimator of the localbias that results from random parameters, and so bias correcting with it makes the bias go away faster.

4

4 Jacknife Bias Correction

The jacknife bias correction is constructed by re-estimating θ while dropping each time series observation

in turn. To describe it, for a given θ let the Þxed effect estimator of αi from all observations but the tth

be given by

bαi(t) (θ) ≡ argmaxα

TXs6=t,s=1

log f (xis; θ,α) .

The Þxed effects estimator of θ excluding time period t is then given by

bθ(t) = argmaxθ

nXi=1

Xs6=t

log f¡xis; θ, bαi(t) (θ)¢

The jacknife bias corrected estimator is then as given in equation (3).

The jacknife bias correction requires recomputing the estimator T times, but avoids the computation

of higher-order derivatives needed for the analytic bias correction. The Þxed effects estimator bθ, bα1, ..., bαnprovide a good starting value for these computations. Starting at the Þxed effects and iterating to-

wards bθ(t) should be straightforward computationally, requiring no additional software. Thus, the jackifeprovides a straightforward approach to bias correction.

A simple example may serve to illustrate this jacknife bias correction. Suppose

xiti.i.d.∼ N (αi0, θ0) t = 1, . . . , T : i = 1, . . . , n

so that

log f (xit; θ,αi) = C − 12log θ − (xit − αi)

2

2θ

The Þxed effects MLE is

bαi =1

T

TXt=1

xit ≡ xi,

bθ =1

nT

nXi=1

TXt=1

(xit − bαi)2 = 1

nT

nXi=1

TXt=1

(xit − xi)2

As is well known, Ehbθi = T−1

T θ0. It is straightforward to show that

eθ ≡ Tbθ − T − 1T

TXt=1

bθ(t) = 1

n (T − 1)nXi=1

TXt=1

(xit − xi)2 = T

(T − 1)bθ.

In this example, the jacknife bias corrected estimator is unbiased.

5 Asymptotic Theory

The Þrst result we consider is the asymptotic distribution of the Þxed effects MLE when n, T → ∞ at

the same rate. We impose the following conditions:

5

Condition 1 n,T →∞ such that nT → ρ, where 0 < ρ <∞.

Condition 2 (i) The function lnf (·; θ,α) is continuous in (θ,α) ∈ Υ; (ii) The parameter space Υ is com-pact; (iii) There exists a function functionM (xit) such that |log f (xit; θ,αi)| ≤M (xit),

¯∂ log f(xit;θ,αi)

∂(θ,αi)

¯≤

M (xit), and supiEhM (xit)

33i< 0.

Condition 3 For each η > 0, infihG(i) (θ0,αi0)− sup(θ,α):|(θ,α)−(θ0,αi0)|>ηG(i) (θ,α)

i> 0, where

bG(i) (θ,αi) ≡ T−1 TXt=1

log f (xit; θ,αi) ≡ T−1TXt=1

g (xit; θ,αi) , G(i) (θ,αi) ≡ E [log f (xit; θ,αi)]

Let

Uit ≡ sθ (xit; θ0,αi0)− ρi0 · Vit, Vit ≡ V (xit; θ0,αi0),ρi0 ≡ E[sθ(xit; θ0,αi0)Vit]/E[V

2it], V2it ≡ V2(xit; θ0,αi0),

Ii ≡ E [UitU0it] .

Condition 4 (i) There exists some M (xit) such that¯∂m1+m2 log f (xit; θ,αi)

∂θm1∂αm2i

¯≤M (xit) 0 ≤ m1 +m2 ≤ 1, . . . , 6

and supiEhM (xit)

Qi<∞ for some Q > 64; (ii) limn→∞ 1

n

Pni=1E [UitU

0it] > 0; (iii) miniE

£V 2i¤> 0.

Under these conditions we can obtain the asymptotic distribution of the Þxed effects MLE as n and

T grow at the same rate.

Theorem 1 Under Conditions 1, 2, 3, and 4, we have

√nT³bθ − θ0´→ N

β√ρ,Ã limn→∞

1

n

nXi=1

Ii!−1

where

β ≡ −12

Ãlimn→∞

1

n

nXi=1

Ii!−1Ã

limn→∞

1

n

nXi=1

E [V2itUit]

E [V 2it]

!

Proof. See Appendix B.

Waterman et al (2000) establish the asymptotic distribution of√nT³bθ − θ0´ assuming that bθ andbαi ³bθ´ are consistent for θ0 and αi0, and dim (θ) = 1. We show that bθ and bαi are consistent for θ0 and

αi0: Theorems 4 and 5 in Appendix A establish such consistency under Conditions 1, 2, and 3. We also

allow for dim (θ) to be bigger than 1.

Next, we show that the analytic bias correction eliminates the asymptotic bias term.

Theorem 2 Under Conditions 1, 2, 3, and 4, we have bβ = β + op (1)Proof. See Appendix E.

6

Corollary 1 Suppose that Conditions 1, 2, 3, and 4 hold. Letbbθ ≡ bθ − 1

Tbβ

We then have

√nT

µbbθ − θ0¶→ N

0,Ã limn→∞

1

n

nXi=1

Ii!−1 .

Proof. The conclusion easily follows from

√nT · 1

Tbβ =rn

Tbβ = β√ρ+ op (1)

Remark 1 By equation (22) in Appendix E, we can see that³1nT

Pni=1

PTt=1

bUit bU 0it´−1 is a consistentestimator of the asymptotic variance

¡limn→∞ 1

n

Pni=1 Ii

¢−1of√nT

µbbθ − θ0¶.To compare our result with Waterman et. al. (2000), let eVit ≡ (Vit, V2it)0 and

eρi ≡³EheViteV 0iti´−1E heVitUiti ,

U∗ (xit; θ,α) = sθ (xit; θ,αi)− (V (xit; θ,αi) , V2 (xit; θ,αi))eρi.They showed the same conclusion as above with scalar θ for a nonfeasible estimator obtained by solving

nXi=1

TXt=1

U∗i (xit; θ,α) = 0,TXt=1

Vi (xit; θ,αi) = 0.

This estimator is not feasible because eρi is unknown. In contrast, our result shows that a feasible biascorrected estimator satisÞes the conclusion.

The jacknife bias corrected estimator has the same limiting distribution as the analytic bias corrected

estimator. In order to simplify the proof we only show this for scalar θ.

Theorem 3 Suppose that dim (θ) = 1. Also suppose that Conditions 1, 2, 3, and 4, hold. We then have

√nT³eθ − θ0´→ N

µ0,

1

limn→∞ 1n

Pni=1E [U

2i ]

¶.

Proof. See Appendix I.

6 Summary

We developed two methods of reducing bias of the Þxed effects maximum likelihood estimator for nonlinear

panel models with Þxed effects, an analytic method and an automatic method based on a time series

jacknife. It might be of interest consider the extra term γ in the expansion of equation (4), and seek to

remove this terrm by analytical and/or jacknife methods. We expect that such could be done under yet

another alternative approximation where n and T grow to inÞnity at a different rate. Such analysis is

expected to be substantially complicated, and we leave it to future research.

7

References

[1] Abrevaya, J. (1997), The Equivalence of Two Estimators of the Fixed-Effects Logit Model,

Economics Letters, 55, 41-43.

[2] Amemiya, T., and T.E. MaCurdy (1986), Instrumental Variables Estimation of an Error-

Component Model, Econometrica, 54, 869-880.

[3] Anderson, E. (1970), Asymptotic Properties of Conditional Maximum Likelihood Estimators,

Journal of the Royal Statistical Society, Series B, 32, 283-301.

[4] Anderson, T., and C. Hsiao (1982), Formulation and Estimation of Dynamic Models Using

Panel Data, Journal of Econometrics, 18, 47-82.

[5] Bhargava, A., and J.D. Sargan (1983), Estimating Dynamic Random Effects Models from

Panel Data, Econometrica, 51, 1635-1659.

[6] Chamberlain, G. (1980), Analysis of Covariance with Qualitative Data, Review of Economic

Studies, 47, 225-238.

[7] Chamberlain, G. (1982), Multivariate Regression Models for Panel Data, Journal of Economet-

rics, 18, 5-46.

[8] Chamberlain, G. (1984), Panel Data, in Z. Griliches and M. Intriligator eds Handbook

of Econometrics. Amsterdam: North-Holland.

[9] Chesher, A. (1984), Testing for Neglected Heterogeneity, Econometrica, 52, 865-872.

[10] Hahn, J., and G. Kuersteiner (2001), Asymptotically Unbiased Inference for a Dynamic Panel

Model with Fixed Effects When Both n and T are Large, Econometrica, forthcoming.

[11] Hall, P., and J. Horowitz (1996), Bootstrap Critical Values for Tests Based on Generalized-

Method-of-Moments Estimators, Econometrica, 64, 891-916.

[12] Hausman, J.A. and W.E. Taylor (1981), Panel Data and Unobservable Individual Effects,

Econometrica, 49, 1377-1398.

[13] Heckman, J. (1981), The Incidental Parameters Problem and the Problem of Initial Conditions in

Esitmating a Discrete Time-Discrete Data Stochastic Process and Some Monte Carlo Evidence, in

McFadden, D. and C. Manski eds Structural Analysis of Discrete Data. Cambridge: MIT Press.

[14] Kiefer, N.M. and G. Skoog (1984), Local Asymptotic SpeciÞcation Error Analysis, Econo-

metrica, 52, 873-885.

[15] Lahiri, S. N. (1992), Edgeworth Correctin by Moving Block Bootstrap for Stationary and Non-

stationary Data in R. LePage and L. Billard eds Exploring the Limits of Bootstrap. Wiley:

New York.

8

[16] Neyman, J., and E.L. Scott, (1948), Consistent Estimates Based on Partially Consistent Ob-

servations, Econometrica, 16, 1-32.

[17] Nickell, S. (1981), Biases in Dynamic Models with Fixed Effects, Econometrica 49, pp. 1417 -

1426.

[18] Quenouille, M.H. (1956), Notes on Bias in Estimation, Biometrika, 43, 353-360.

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Mathematical Statistics, 29, 614.

[20] Waterman, R., B. Lindsay, H. Li, and B. Li (2000), Projected Score Methods for Nuisance

Parameters: Asymptotics and Neyman-Scott Problems, unpublished manuscript.

9

Appendix

A Consistency

Throughout this section, we assume that Conditions 1, 2, and 3 hold.

Lemma 1 Assume that Wt are iid with E [Wt] = 0 and E£W 2kt

¤<∞. Then,

E

·³PTt=1Wt

´2k¸= C(k)T k + o(T k)

for some constant C(k).

Proof. By adopting an argument in the proof of Lemma 5.1 in Lahiri (1992), we have

E

·³PTt=1Wt

´2k¸=

2kPj=1

PαC (α1, ...,αj)

PIE

·jQs=1

Wαsts

¸, (5)

where for each Þxed j ∈ 1, ..., 2k ,Pα extends over all j-tuples of positive integers (α1, ...,αj) such that

α1 + ...+ αj = 2k andPI extends over all ordered j-tuples (t1, ..., tj) of integers such that 1 ≤ tj ≤ T.

Also, C(α1, ...,αj) stands for a bounded constant. Note, that if j > k then at least one of the indices

αj = 1. By independence and the fact that E [Wt] = 0 it follows that EhQj

s=1Wαsts

i= 0 whenever j > k.

This shows that¯E

·³PTt=1Wt

´2k¸¯≤ C (k)T k ·E £W 2k

t

¤for some constant C(k).

Lemma 2 Suppose that, for each i, ξit, t = 1, 2, . . . is a sequence of zero mean i.i.d. random variables.We assume that ξit, t = 1, 2, 3 are independent across i. We also assume that maxiE

h|ξit|16

i< ∞.

Finally, we assume that n = O (T ). We then have

maxiPr

"¯¯ 1T

TXt=1

ξit

¯¯ > η

#= o

¡T−2

¢for every η > 0.

Proof. Using Lemma 1, we obtain

E

¯¯TXt=1

ξit

¯¯16 ≤ CT 8 · E £ξ16it ¤ ,

where C > 0 is a constant. Therefore, we have

T 2 Pr

"¯¯ 1T

TXt=1

ξit

¯¯ > η

#≤ T 2 CT

8

T 16η16E£ξ16it¤

or

maxiT 2 Pr

"¯¯ 1T

TXt=1

ξit

¯¯ > η

#≤ C

T 6η16maxiE£ξ16it¤= o (1) .

10

Lemma 3 Suppose that Conditions 1 and 2 hold. We then have for all η > 0 that

Pr

"max1≤i≤n

sup(θ,α)

¯ bG(i) (θ,α)−G(i) (θ,α)¯ ≥ η#= o

¡T−1

¢Proof. Let η > 0 be given. We note that

Pr

"max1≤i≤n

sup(θ,α)

¯ bG(i) (θ,α)−G(i) (θ,α)¯ ≥ η#≤

nXi=1

Pr

"sup(θ,α)

¯ bG(i) (θ,α)−G(i) (θ,α)¯ ≥ η#. (6)

Let ε > 0 be chosen such that 2εmaxiE [M (xit)] <η3 . Divide Υ into subsets Υ1,Υ2, . . . ,ΥM(ε) such

that¯(θ,α)− ¡θ0,α0¢¯ < ε whenever (θ,α) and ¡θ0,α0¢ are in the same subset. Let (θj ,αj) denote some

point in Υj for each j. Then,

sup(θ,α)

¯ bG(i) (θ,α)−G(i) (θ,α)¯ = maxjsupΥj

¯ bG(i) (θ,α)−G(i) (θ,α)¯ ,and therefore

Pr

"sup(θ,α)

¯ bG(i) (θ,α)−G(i) (θ,α)¯ > η#≤M(ε)Xj=1

Pr

"supΥj

¯ bG(i) (θ,α)−G(i) (θ,α)¯ > η#

(7)

For (θ,α) ∈ Υj , we have¯ bG(i) (θ,α)−G(i) (θ,α)¯ ≤ ¯ bG(i) (θj ,αj)−G(i) (θj ,αj)¯+ εT

¯¯TXt=1

(M (xit)−E [M (xit)])

¯¯+2εE [M (xit)]

Then,

Pr

"supΥj

¯ bG(i) (θ,α)−G(i) (θ,α)¯ > η#

≤ Prh¯ bG(i) (θj ,αj)−G(i) (θj ,αj)¯ > η

3

i+Pr

"1

T

¯¯TXt=1

(M (xit)−E [M (xit)])

¯¯ > η

3ε

#= o

¡T−2

¢(8)

by Lemma 2. Combining (6), (7), (8), and n = O (T ), we obtain the desired conclusion.

Theorem 4 Under Conditions 1, 2, and 3, Prh¯bθ − θ0 ¯ ≥ ηi = o ¡T−1¢ for every η > 0.

Proof. Let η be given, and let ε ≡ infihG(i) (θ0,αi0)− sup(θ,α):|(θ,α)−(θ0,αi0)|>ηG(i) (θ,α)

i> 0.

With probability equal to 1− o ¡ 1T ¢, we havemax

|θ−θ0|>η,α1,...,αnn−1

nXi=1

bG(i) (θ,αi) ≤ max|(θ,α)−(θ0,αi0)|>η

n−1nXi=1

bG(i) (θ,αi)< max

|(θ,α)−(θ0,αi0)|>ηn−1

nXi=1

G(i) (θ,αi) +1

3ε

< n−1nXi=1

G(i) (θ0,αi0)− 23ε

< n−1nXi=1

bG(i) (θ0,αi0)− 13ε,11

where the second and fourth inequalities are based on Lemma 3. Because

maxθ,α1,...,αn

n−1nXi=1

bG(i) (θ,αi) ≥ n−1 nXi=1

bG(i) (θ0,αi0)by deÞnition, we can conclude that Pr

h¯bθ − θ0 ¯ ≥ ηi = o ¡ 1T ¢.Theorem 5 Under Conditions 1, 2, and 3,

T Pr

·max1≤i≤n

|bαi − αi0| ≥ η¸ = o (1)Proof. We Þrst prove that

T Pr

·max1≤i≤n

supα

¯ bG(i) ³bθ,α´−G(i) (θ0,α)¯ ≥ η¸ = o (1) (9)

for every η > 0. Note that

max1≤i≤n

supα

¯ bG(i) ³bθ,α´−G(i) ³bθ,α´¯≤ max

1≤i≤nsupα

¯ bG(i) ³bθ,α´−G(i) ³bθ,α´¯+ max1≤i≤n

supα

¯G(i)

³bθ,α´−G(i) (θ0,α)¯≤ max

1≤i≤nsup(θ,α)

¯ bG(i) (θ,α)−G(i) (θ,α)¯+ max1≤i≤n

E [M (xit)] ·¯bθ − θ0 ¯ .

Therefore,

T Pr

·max1≤i≤n

supα

¯ bG(i) ³bθ,α´−G(i) (θ0,α)¯ ≥ η¸ ≤ T Pr

"max1≤i≤n

sup(θ,α)

¯ bG(i) (θ,α)−G(i) (θ,α)¯ ≥ η

2

#

+T Pr

·¯bθ − θ0 ¯ ≥ η

2 (1 +max1≤i≤nE [M (xit)])

¸= o (1)

by Lemma 3 and Theorem 4.

We now get back to the proof of Theorem 5. It suffices to prove that

T Pr

·max1≤i≤n

|bαi − αi0| ≥ η¸ = o (1)for every η > 0. Let η be given, and let ε ≡ infi

hG(i) (θ0,αi0)− supαi:|αi−αi0|>ηG(i) (θ0,αi)

i> 0.

Condition on the eventnmax1≤i≤n supα

¯ bG(i) ³bθ,α´−G(i) (θ0,α)¯ ≤ 13εo, which has a probability equal

to 1− o ¡ 1T ¢ by (9). We then havemax

|αi−αi0|>ηbG(i) ³bθ,αi´ < max

|αi−αi0|>ηG(i) (θ0,αi) +

1

3ε < G(i) (θ0,αi0)− 23ε <

bG(i) ³bθ,αi0´− 13εThis is inconsistent with bG(i) ³bθ, bαi´ ≥ bG(i) ³bθ,αi0´, and therefore, |bαi − αi0| ≤ η for every i.

12

B Proof of Theorem 1

We can see that the MLE bθ also solves0 =

nXi=1

TXt=1

U³xit;bθ, bαi ³bθ´´ .

Let F ≡ (F1, . . . , Fn) denote the collection of distribution functions. Let bF ≡³ bF1, . . . , bFn´, where bFi

denotes the empirical distribution function for the stratum i. DeÞne F (²) ≡ F + ²√T³ bF − F´ for

² ∈ £0, T−1/2¤. For each Þxed θ and ², let αi (θ, Fi (²)) be the solution to the estimating equation0 =

ZVi [θ,αi (θ, Fi (²))]dFi (²) ,

and let θ (F (²)) be the solution to the estimating equation

0 =nXi=1

ZUi (xit; θ (F (²)) ,αi (θ (Fi (²)) , Fi (²))) dFi (²) .

By Taylor series expansion, we have

θ³ bF´− θ (F ) = 1√

Tθ² (0) +

1

2

µ1√T

¶2θ²² (0) +

1

6

µ1√T

¶3θ²²² (e²) ,

where θ² (²) ≡ dθ (F (²))/ d², θ²² (²) ≡ d2θ (F (²))±d²2,. . ., and e² is somewhere in between 0 and T−1/2.

We therefore have

√nT³θ³ bF´− θ (F )´ = √nT 1√

Tθ² (0) +

√nT1

2

µ1√T

¶2θ²² (0) +

1

6

rn

T

1√Tθ²²² (e²) . (10)

Theorem 6 in Appendix D establishes that last term in (10) is op (1) under Condition 4. It is shown later

in Appendix C that

√nT

1√Tθ² (0) → N

0,Ã limn→∞

1

n

nXi=1

Ii!−1 ,

√nT1

2

µ1√T

¶2θ²² (0) = −1

2

rn

T

Ã1

n

nXi=1

Ii!−1Ã

1

n

nXi=1

E [V2itUi]

E [V 2i ]

!+ op (1) ,

from which Theorem 1 follows.

C Derivatives of θ

Let

hi (·, ²) ≡ Ui (·; θ (F (²)) ,αi (θ (F (²)) , Fi (²))) (11)

The Þrst order condition may be written as

0 =1

n

nXi=1

Zhi (·, ²) dFi (²) (12)

13

Differentiating repeatedly with respect to ², we obtain

0 =1

n

nXi=1

Zdhi (·, ²)d²

dFi (²) +1

n

nXi=1

Zhi (·, ²) d∆iT (13)

0 =1

n

nXi=1

Zd2hi (·, ²)d²2

dFi (²) + 21

n

nXi=1

Zdhi (·, ²)d²

d∆iT (14)

0 =1

n

nXi=1

Zd3hi (·, ²)d²3

dFi (²) + 31

n

nXi=1

Zd2hi (·, ²)d²2

d∆iT (15)

where ∆iT ≡√T³ bFi − Fi´.

C.1 θ² (0)

Because

dhi (·, ²)d²

=∂hi (·, ²)∂θ0

∂θ

∂²+∂hi (·, ²)∂αi

∂αi∂θ0

∂θ

∂²+∂hi (·, ²)∂αi

∂αi∂²

we may rewrite (13) as

0 =1

n

nXi=1

Z µ∂hi (·, ²)∂θ0

∂θ

∂²+∂hi (·, ²)∂αi

∂αi∂θ0

∂θ

∂²+∂hi (·, ²)∂αi

∂αi∂²

¶dFi (²) +

1

n

nXi=1

Zhi (·, ²) d∆iT (16)

Evaluating at ² = 0, and noting that E [Uαii ] = 0, we obtain

θ² (0) =

Ã1

n

nXi=1

Ii!−1Ã

1

n

nXi=1

ZUid∆iT

!(17)

We therefore have

√nT

1√Tθ² (0) =

Ã1

n

nXi=1

Ii!−1Ã

1√n√T

nXi=1

TXt=1

Ui

!→ N

0,Ã limn→∞

1

n

nXi=1

Ii!−1

C.2 αθi and α²i

In the ith stratum, αi (θ, Fi (²)) solves the estimating equationZVi (·; θ,αi (θ, Fi (²))) dFi (²) = 0 (18)

Differentiating the LHS with respect to θ and ², we obtain

0 =

Z∂Vi (·, θ, ²)

∂θdFi (²) +

µZ∂Vi (·, θ, ²)∂αi

dFi (²)

¶∂αi (θ, Fi (²))

∂θ,

0 =

µZ∂Vi (·, θ, ²)∂αi

dFi (²)

¶∂αi (θ, Fi (²))

∂²+

ZVi (·, θ, ²) d∆iT .

Observe that

∂αi (θ, Fi (²))

∂θ= −

µZ∂Vi (·, θ, ²)∂αi

dFi (²)

¶−1µZ∂Vi (·, θ, ²)

∂θdFi (²)

¶,

∂αi (θ, Fi (²))

∂²= −

µZ∂Vi (·, θ, ²)∂αi

dFi (²)

¶−1µZVi (·, θ, ²) d∆iT

¶.

14

Equating these equations to zero and solving for derivatives of αi evaluated at ² = 0 gives

αθi = −E£∂Vi∂θ

¤Eh∂Vi∂αi

i = E£∂Vi∂θ

¤E [V 2i ]

= O (1) , (19)

α²i = −PTt=1 Vit√

TEh∂Vi∂αi

i = T−1/2PTt=1 Vit

E [V 2i ]= Op (1) , (20)

where

αθi ≡∂αi (θ, Fi (0))

∂θ, α²i ≡

∂αi (θ, Fi (0))

∂².

C.3 θ²² (0)

Note that

d2hi (·, ²)d²2

= Gi (·, ²) + ∂2hi (·, ²)∂θ0∂αi

µ∂αi∂θ0

∂θ

∂²

¶∂θ

∂²+∂2hi (·, ²)∂θ0∂αi

∂αi∂²

∂θ

∂²

+∂hi (·, ²)∂θ0

∂2θ

∂²2

+∂2hi (·, ²)∂θ0∂αi

∂θ

∂²

∂αi∂θ0

∂θ

∂²+∂2hi (·, ²)∂α2i

µ∂αi∂θ0

∂θ

∂²

¶2+∂2hi (·, ²)∂α2i

µ∂αi∂θ0

∂θ

∂²

¶∂αi∂²

+∂hi (·, ²)∂αi

µ∂θ0

∂²

∂2αi∂θ∂θ0

¶∂θ

∂²+∂hi (·, ²)∂αi

∂2αi∂θ0∂²

∂θ

∂²+∂hi (·, ²)∂αi

∂αi∂θ0

∂2θ

∂²2

+∂2hi (·, ²)∂θ0∂αi

∂θ

∂²

∂αi∂²

+∂2hi (·, ²)∂α2i

µ∂αi∂θ0

∂θ

∂²

¶∂αi∂²

+∂2hi (·, ²)∂α2i

µ∂αi∂²

¶2+∂hi (·, ²)∂αi

∂2αi∂²∂θ0

∂θ

∂²+∂hi (·, ²)∂αi

∂2αi∂²2

,

where Gi (·, ²) is a R-dimensional column vector such that its r-th element G(r)i (·, ²) is equal to

G(r)i (·, ²) = ∂θ (·, ²)0∂²

∂2h(r)i (·, ²)∂θ∂θ0

∂θ (·, ²)∂²

,

and h(r)i (·, ²) denotes the r-th element of hi.Evaluating each term of (14) at ² = 0, and noting that E [Uαii ] = 0, we obtain

0 =1

n

nXi=1

E

·∂Ui∂θ0

¸θ²² (0)

+2

n

nXi=1

α²i · E·∂2Ui∂θ0∂αi

¸θ² (0) +

2

n

nXi=1

α²i¡θ² (0)0 αθi

¢ ·E ·∂2Ui∂α2i

¸

+G + 2

n

nXi=1

θ² (0)0 αθi ·E·∂2Ui∂θ0∂αi

¸θ² (0) +

1

n

nXi=1

¡θ² (0)0 αθi

¢2 ·E ·∂2Ui∂α2i

¸

+1

n

nXi=1

(α²i)2 · E

·∂2Ui∂α2i

¸

+2

n

nXi=1

µZ∂Ui∂θ0

d∆iT

¶θ² (0) +

2

n

nXi=1

¡θ² (0)0 αθi

¢ · Z ∂Ui∂αi

d∆iT +2

n

nXi=1

α²i ·Z∂Ui∂αi

d∆iT

15

where

G =

θ² (0)0

µ1n

Pni=1E

·∂2U

(1)i

∂θ∂θ0

¸¶θ² (0)

...

θ² (0)0µ1n

Pni=1E

·∂2U

(R)i

∂θ∂θ0

¸¶θ² (0)

from which we obtainÃ

1

n

nXi=1

Ii!θ²² (0) =

1

n

nXi=1

(α²i)2 · E

·∂2Ui∂α2i

¸+2

n

nXi=1


d∆iT

+2

Ã1

n

nXi=1

α²i · E·∂2Ui∂θ0∂αi

¸!θ² (0) +

Ã1

n

nXi=1

α²iE

·∂2Ui∂α2i

¸ ¡αθi¢0!

θ² (0)

+2

Ã1

n

nXi=1

Z∂Ui∂θ0

d∆iT

!θ² (0) +

Ã1

n

nXi=1

Z∂Ui∂αi

d∆iT ·¡αθi¢0!

θ² (0)

+G + 2

n

nXi=1

θ² (0)0 αθi ·E·∂2Ui∂θ0∂αi

¸θ² (0) +

1

n

nXi=1

¡θ² (0)0 αθi

¢2 · E ·∂2Ui∂α2i

¸(21)

Because

1

n

nXi=1

(α²i)2 · E

·∂2Ui∂α2i

¸=

1

n

nXi=1

E [Uαiαii ]

Ã PTt=1 Vit√TE [V 2i ]

!2nXi=1


d∆iT =1

n

nXi=1

PTt=1 Vit√TE [V 2i ]

Ã1√T

TXt=1

(Uαii −E [Uαii ])!

Ã1

n

nXi=1

α²i ·E·∂2Ui∂θ0∂αi

¸!θ² (0) = Op

µ1√n

¶Op

µ1√n

¶= Op

µ1

n

¶Ã1

n

nXi=1

α²iE

·∂2Ui∂α2i

¸ ¡αθi¢0!

θ² (0) = Op

µ1√n

¶Op

µ1√n

¶= Op

µ1

n

¶Ã1

n

nXi=1

Z∂Ui∂θ0

d∆iT

!θ² (0) = Op

µ1√n

¶Op

µ1√n

¶= Op

µ1

n

¶Ã1

n

nXi=1

Z∂Ui∂αi

d∆iT ·¡αθi¢0!

θ² (0) = Op

µ1√n

¶Op

µ1√n

¶= Op

µ1

n

¶and

θ² (0)0Ã1

n

nXi=1

E

"∂2U

(r)i

∂θ∂θ0

#!θ² (0) = Op

µ1√n

¶O (1)Op

µ1√n

¶= Op

µ1

n

¶1

n

nXi=1

θ² (0)0 αθi · E·∂2Ui∂θ0∂αi

¸θ² (0) = Op

µ1√n

¶O (1)Op

µ1√n

¶= Op

µ1

n

¶1

n

nXi=1

¡θ² (0)0 αθi

¢2 · E ·∂2Ui∂α2i

¸= Op

µ1√n

¶2O (1) = Op

µ1

n

¶

16

we may write

θ²² (0) =

Ã1

n

nXi=1

Ii!−1

1

n

nXi=1

E [Uαiαii ]

Ã PTt=1 Vit√TE [V 2i ]

!2

+2

Ã1

n

nXi=1

Ii!−1

1

n

nXi=1

PTt=1 Vit√TE [V 2i ]

Ã1√T

TXt=1

(Uαii −E [Uαii ])!

+Op

µ1

n

¶= 2

Ã1

n

nXi=1

Ii!−1

1

n

nXi=1

" PTt=1 Vit√TE [V 2i ]

#"1√T

TXt=1

µUαii +

E [Uαiαii ]

2E [V 2i ]Vit

¶#+ op (1)

Therefore, we have

√nT1

2

µ1√T

¶2θ²² (0) =

rn

T

Ã1

n

nXi=1

Ii!−1

1

n

nXi=1

"1√T

TXt=1

VitE [V 2i ]

#"1√T

TXt=1

µUαii +

E [Uαiαii ]

2E [V 2i ]Vit

¶#+op (1)

The terms in square parentheses have joint limiting normal distributions by CLT. Their product is a

quadratic form with mean

E [VitUαii ]

E [V 2i ]+E [Uαiαii ]

2E [V 2i ]=E [VitU

αii ]

E [V 2i ]− E [VitU

αii ]

2E [V 2i ]=E [VitU

αii ]

2E [V 2i ]= −E [V2itUi]

2E [V 2i ].

Therefore, we have

√nT1

2

µ1√T

¶2θ²² (0) =

1

2

rn

T

Ã1

n

nXi=1

Ii!−1Ã

− 1n

nXi=1

E [V2itUi]

E [V 2i ]

!+ op (1)

D Remainder in (10)

D.1 αθθi , αθ²i , and α²²i

Second order differentiation³∂2

∂θ2, ∂2

∂θ∂² ,∂2

∂²2

´of (18) yields

0 =

Z∂2Vi (·, θ, ²)∂θ∂θ0

dFi (²) +∂αi (θ, Fi (²))

∂θ

µZ∂2Vi (·, θ, ²)∂αi∂θ

0 dFi (²)

¶+

µZ∂2Vi (·, θ, ²)∂αi∂θ

dFi (²)

¶∂αi (θ, Fi (²))

∂θ0

+

µZ∂Vi (·, θ, ²)∂αi

dFi (²)

¶∂2αi (θ, Fi (²))

∂θ∂θ0+

µZ∂2Vi (·, θ, ²)

∂α2idFi (²)

¶∂αi (θ, Fi (²))

∂θ

∂αi (θ, Fi (²))

∂θ0,

0 =

µZ∂2Vi (·, θ, ²)∂θ∂αi

dFi (²)

¶∂αi (θ, Fi (²))

∂²

+

µZ∂Vi (·, θ, ²)∂αi

dFi (²)

¶∂2αi (θ, Fi (²))

∂θ∂²+

µZ∂2Vi (·, θ, ²)

∂α2idFi (²)

¶∂αi (θ, Fi (²))

∂θ

∂αi (θ, Fi (²))

∂²

+

Z∂Vi (·, θ, ²)

∂θd∆iT +

µZ∂Vi (·, θ, ²)∂αi

d∆iT

¶∂αi (θ, Fi (²))

∂θ,

and

0 =

µZ∂Vi (·, θ, ²)∂αi

dFi (²)

¶∂2αi (θ, Fi (²))

∂²2+

µZ∂2Vi (·, θ, ²)

∂α2idFi (²)

¶µ∂αi (θ, Fi (²))

∂²

¶2+2

µZ∂Vi (·, θ, ²)∂αi

d∆iT

¶∂αi (θ, Fi (²))

∂².

These three equalities characterizes ∂2αi(θ,Fi(²))∂θ∂θ0 , ∂

2αi(θ,Fi(²))∂θ∂² , and ∂2αi(θ,Fi(²))

∂²2 .

17

D.2 Some Lemmas

Lemma 4 Suppose that, for each i, ξit, t = 1, 2, . . . is a sequence of zero mean i.i.d. random variables.We assume that ξit, t = 1, 2, 3 are independent across i. We also assume that maxiE

h|ξit|16

i< ∞.

Finally, assume that n = O (T ). We then have

Pr

"maxi

¯¯ 1T

TXt=1

ξit

¯¯ > η

#= o

µ1

T

¶for every η > 0.

Proof. The conclusion follows from combining Lemma 2 with the observation that

T Pr

"maxi

¯¯ 1T

TXt=1

ξit

¯¯ > η

#≤ T

nXi=1

Pr

"¯¯ 1T

TXt=1

ξit

¯¯ > η

#≤ max

iT 2 Pr

"¯¯ 1T

TXt=1

ξit

¯¯ > η

#.

Lemma 5 Suppose that, for each i, ξit (φ) , t = 1, 2, . . . is a sequence of zero mean i.i.d. random

variables indexed by some parameter φ ∈ Φ. We assume that ξit (φ) , t = 1, 2, 3 are independent acrossi. We also assume that supφ∈Φ |ξit (φ)| ≤ Bit for some sequence of random variables Bit that is i.i.d.

across t and independent across i. Finally, we assume that maxiEh|Bit|64

i< ∞, and n = O (T ). We

then have

Pr

"maxi

¯¯ 1√T

TXt=1

ξit (φi)

¯¯ > T 1

10−υ#= o

µ1

T

¶for every υ such that 0 ≤ υ < 1

160 . Here, φi is an arbitrary sequence in Φ.

Proof. This proof is a modiÞcation of Hall and Horowitz (1996, Lemma 1). Note that

T 2 Pr

·max1≤t≤T

|Bit| > T 1/16¸≤ T 2

TXt=1

Prh|Bit| > T 1/16

i= T 3 Pr

h|Bit| > T 1/16

i

≤ T 3Eh|Bit|64

i¡T 1/16

¢64 ≤maxiE

h|Bit|64

iT

= o (1) .

Condition on max1≤t≤T |Bit| ≤ T 1/16. By Markovs inequality,

Pr

"supφ∈Φ

¯¯ 1√T

TXt=1

ξit (φi)

¯¯ > T 1

10−υ#= Pr

"supφ∈Φ

¯¯TXt=1

ξit (φi)

¯¯ > T 3

5−υ#

≤E

·supφ∈Φ

¯PTt=1 ξit (φi)

¯64¸T

35×64−64υη64

=

supφ∈ΦE·¯PT

t=1 ξit (φi)¯64¸

T1925 −64υη64

≤E

·¯PTt=1Bit

¯64¸T

1925 −64υη64

,

where the last equality is based on dominated convergence. By Lahiri (1992, Lemma 5.1), we have

E

¯¯TXt=1

Bit

¯¯64 ≤ CT 36,

18

where C > 0 is a constant. Therefore, we have

T 2 Pr

"¯¯ 1√T

TXt=1

ξit (φi)

¯¯ > T 1

10−υ#≤ T 2 CT 36

T1925 −64υη64

= O³T−

25+64υ

´,

and

T Pr

"maxi

¯¯ 1√T

TXt=1

ξit (φi)

¯¯ > T 1

10−υ#≤ T

nXi=1

Pr

"¯¯ 1√T

TXt=1

ξit (φi)

¯¯ > T 1

10−υ#= nT · CT 36

T1925 −64υη64

= o (1) .

Lemma 6 Under Conditions 1, 2, and 3,

T Pr

"max

0≤²≤ 1√T

¯bθ (²)− θ0 ¯ ≥ η# = o (1)and

T Pr

"max1≤i≤n

max0≤²≤ 1√

T

|bαi (²)− αi0| ≥ η# = o (1)for every η > 0.

Proof. Only the Þrst assertion is proved. The second assertion can be proved similarly. Let η be

given, and let ε ≡ infihG(i) (θ0,αi0)− sup(θ,α):|(θ,α)−(θ0,αi0)|>ηG(i) (θ,α)

i> 0. Recall that

F (²) ≡ F + ²√T³ bF − F´ , ² ∈

·0,

1√T

¸We haveZ

g (·; θ,αi (θ)) dFi (²) =³1− ²

√T´G(i) (θ,αi) + ²

√T bG(i) (θ,αi)

and¯Zg (·; θ,αi (θ)) dFi (²)−G(i) (θ,αi)

¯≤³1− ²

√T´ ¯ bG(i) (θ,α)−G(i) (θ,α)¯ ≤ ¯ bG(i) (θ,α)−G(i) (θ,α)¯ .

By Lemma 3, we have

Pr

"max

0≤²≤ 1√T

max1≤i≤n

sup(θ,α)

¯Zg (·; θ,αi (θ)) dFi (²)−G(i) (θ,α)

¯≥ η

#= o

¡T−1

¢Therefore, for every 0 ≤ ² ≤ 1√

Twith probability equal to 1− o ¡ 1T ¢, we have

max|θ−θ0|>η,α1,...,αn

n−1nXi=1

Zg (·; θ,αi (θ)) dFi (²) ≤ max

|(θ,α)−(θ0,αi0)|>ηn−1

nXi=1

Zg (·; θ,αi (θ)) dFi (²)

< max|(θ,α)−(θ0,αi0)|>η

n−1nXi=1

G(i) (θ,αi) +1

3ε

< n−1nXi=1

G(i) (θ0,αi0)− 23ε

< n−1nXi=1

Zg (·; θ0,αi0) dFi (²)− 1

3ε.

19

We also have

maxθ,α1,...,αn

n−1nXi=1

Zg (·; θ,αi) dFi (²) ≥ n−1

nXi=1

Zg (·; θ0,αi0) dFi (²)

by deÞnition. It follows that

max|θ−θ0|>η,α1,...,αn

n−1nXi=1

Zg (·; θ,αi (θ)) dFi (²) < max

θ,α1,...,αnn−1

nXi=1

Zg (·; θ,αi) dFi (²)− 1

3ε

for every 0 ≤ ² ≤ 1√T. We therefore obtain that Pr

hmax0≤²≤ 1√

T

¯bθ (²)− θ0 ¯ ≥ ηi = o ¡ 1T ¢.Lemma 7 Assume that Condition 4 holds. Suppose that Ki (·; θ (²) ,αi (θ (²) , ²)) is equal to

∂m1+m2 log f (xit; θ (²) ,αi (θ (²) , ²))

∂θm1∂αm2i

for some m1 +m2 ≤ 1, . . . , 5. Then, for any η > 0, we have

Pr

"max

0≤²≤ 1√T

¯¯ 1n

nXi=1

ZKi (·; θ (²) ,αi (θ (²) , ²)) dFi (²)− 1

n

nXi=1

E [Ki (xit; θ0,αi0)]

¯¯ > η

#= o

µ1

T

¶and

Pr

"maxi

max0≤²≤ 1√

T

¯ZKi (·; θ (²) ,αi (θ (²) , ²)) dFi (²)−E [Ki (xit; θ0,αi0)]

¯> η

#= o

µ1

T

¶.

Also,

Pr

"maxi

max0≤²≤ 1√

T

¯ZKi (·; θ (²) ,αi (θ (²) , ²)) d∆iT

¯> CT

110−υ

#= o

µ1

T

¶for some constant C > 0 and υ such that 0 ≤ υ < 1

160 .

Proof. Note that we may writeZKi (·; θ (²) ,αi (θ (²) , ²)) dFi (²)−

ZKi (·; θ (²) ,αi (θ (²) , ²)) dFi

=

ZKi (·; θ (²) ,αi (θ (²) , ²)) dFi (²)−

ZKi (·; θ (0) ,αi (θ (0) , 0)) dFi (²)

+

ZKi (·; θ (0) ,αi (θ (0) , 0)) dFi (²)−

ZKi (·; θ (0) ,αi (θ (0) , 0)) dFi

=

Z∂Ki (·; θ∗,α∗i )

∂θ0· (θ (²)− θ) dFi (²)

+

Z∂Ki (·; θ∗,α∗i )

∂αi· (αi (θ (²) , ²)− αi) dFi (²)

+²√T

ZKi (·; θ (0) ,αi (θ (0) , 0)) d

³ bFi − Fi´

20

where (θ∗,α∗i ) are between (θ,αi) and (θ (²) ,αi (θ (²) , ²)). Therefore, we have¯¯ 1n

nXi=1

ZKi (·; θ (²) ,αi (θ (²) , ²)) dFi (²)− 1

n

nXi=1


¯¯

≤ |θ (²)− θ| · 1n

nXi=1

ÃE [M (xit)] +

1

T

TXt=1

M (xit)

!

+

Ã1

n

nXi=1

(αi (θ (²) , ²)− αi)2!1/2 1

n

nXi=1

ÃE [M (xit)] +

1

T

TXt=1

M (xit)

!21/2

+

¯¯ 1n

nXi=1

Ã1

T

TXt=1

M (xit)−E [M (xit)]

!¯¯

where M (·) is deÞned in Condition 4. Using Lemmas 4 and 6, we can bound

max0≤²≤ 1√

T

¯¯ 1n

nXi=1

ZKi (·; θ (²) ,αi (θ (²) , ²)) dFi (²)− 1

n

nXi=1


¯¯

in absolute value by some η > 0 with probability 1− o ¡ 1T ¢. Because¯ZKi (·; θ (²) ,αi (θ (²) , ²)) dFi (²)−E [Ki (xit; θ0,αi0)]

¯≤ |θ (²)− θ| ·

ÃE [M (xit)] +

1

T

TXt=1

M (xit)

!

+ |αi (θ (²) , ²)− αi| ·ÃE [M (xit)] +

1

T

TXt=1

M (xit)

!

+

¯¯ 1T

TXt=1

M (xit)−E [M (xit)]

¯¯ ,

we can bound

maxi

max0≤²≤ 1√

T

¯ZKi (·; θ (²) ,αi (θ (²) , ²)) dFi (²)−E [Ki (xit; θ0,αi0)]

¯in absolute value by some η > 0 with probability 1− o ¡ 1T ¢.Using Condition 4 and Lemmas 5, we can also show that

maxi

¯ZKi (·; θ (²) ,αi (θ (²) , ²)) d∆iT

¯can be bounded by in absolute value by CT

110−υ for some constant C > 0 and υ such that 0 ≤ υ < 1

160

with probability 1− o ¡ 1T ¢.D.3 Bounds

Lemma 8 Suppose that Condition 4 holds. Then, we have

Pr

"maxi

max0≤²≤ 1√

T

¯αθi (²)

¯> C

#= o

¡T−1

¢Pr

"maxi

max0≤²≤ 1√

T

|α²i (²)| > CT110−υ

#= o

¡T−1

¢21

for some constant C > 0 and υ such that 0 ≤ υ < 1160 .

Proof. From Appendix C.2, we obtain

∂αi (θ, Fi (²))

∂θ= −

µZ∂Vi (·, θ, ²)∂αi

dFi (²)

¶−1µZ∂Vi (·, θ, ²)

∂θdFi (²)

¶,

∂αi (θ, Fi (²))

∂²= −

µZ∂Vi (·, θ, ²)∂αi

dFi (²)

¶−1µZVi (·, θ, ²) d∆iT

¶.

Using Lemma 7 and Condition 4, we can see thatµZ∂Vi (·, θ, ²)∂αi

dFi (²)

¶−1is uniformly bounded away from zero. We can also see thatZ

∂Vi (·, θ, ²)∂θ

dFi (²)

is uniformly bounded by some constant C, andZVi (·, θ, ²) d∆iT

is uniformly bounded by CT110−υ for some constant C and υ such that 0 ≤ υ < 1

160 .


Pr

"max

0≤²≤ 1√T

|θ² (²)| > CT 110−υ

#= o

¡T−1

¢for some constant C > 0 and υ such that 0 ≤ υ < 1

160 .

Proof. From (16), we have

θ² (²) = −"1

n

nXi=1

Z µ∂hi (·, ²)∂θ0

+∂hi (·, ²)∂αi

∂αi∂θ0

¶dFi (²)

#−1"1

n

nXi=1

∂αi∂²

µZ∂hi (·, ²)∂αi

dFi (²)

¶+1

n

nXi=1

Zhi (·, ²) d∆iT

#Using Lemmas 7, 8, and Condition 4, we can bound the denominator of θ² (²) by some C > 0, and the

numerator by some CT110−υ with probability 1− o ¡T−1¢.


Pr

"maxi

max0≤²≤ 1√

T

¯αθrθr0i (²)

¯> C

#= o

¡T−1

¢Pr

"maxi

max0≤²≤ 1√

T

¯αθr²i (²)

¯> CT

110−υ

#= o

¡T−1

¢Pr

"maxi

max0≤²≤ 1√

T

|α²²i (²)| > C³T

110−υ

´2#= o

¡T−1


160 . Here, αθrθr0i ≡ ∂2αi

∂θr∂θr0. We similarly deÞne αθr²i .

22

Proof. From Appendix D.1, we have

0 =

Z∂2Vi (·, θ, ²)∂θ∂θ0

dFi (²) +∂αi (θ, Fi (²))

∂θ

µZ∂2Vi (·, θ, ²)∂αi∂θ

0 dFi (²)

¶+

µZ∂2Vi (·, θ, ²)∂αi∂θ

dFi (²)

¶∂αi (θ, Fi (²))

∂θ0

+

µZ∂Vi (·, θ, ²)∂αi

dFi (²)

¶∂2αi (θ, Fi (²))

∂θ∂θ0+

µZ∂2Vi (·, θ, ²)

∂α2idFi (²)

¶∂αi (θ, Fi (²))

∂θ

∂αi (θ, Fi (²))

∂θ0,

0 =

µZ∂2Vi (·, θ, ²)∂θ∂αi

dFi (²)

¶∂αi (θ, Fi (²))

∂²

+

µZ∂Vi (·, θ, ²)∂αi

dFi (²)

¶∂2αi (θ, Fi (²))

∂θ∂²+

µZ∂2Vi (·, θ, ²)

∂α2idFi (²)

¶∂αi (θ, Fi (²))

∂θ

∂αi (θ, Fi (²))

∂²

+

Z∂Vi (·, θ, ²)

∂θd∆iT +

µZ∂Vi (·, θ, ²)∂αi

d∆iT

¶∂αi (θ, Fi (²))

∂θ,

and

0 =

µZ∂Vi (·, θ, ²)∂αi

dFi (²)

¶∂2αi (θ, Fi (²))

∂²2+

µZ∂2Vi (·, θ, ²)

∂α2idFi (²)

¶µ∂αi (θ, Fi (²))

∂²

¶2+2

µZ∂Vi (·, θ, ²)∂αi

d∆iT

¶∂αi (θ, Fi (²))

∂².

The result then follows by applying the same argument as in the proof of Lemma 8.


Pr

"max

0≤²≤ 1√T

|θ²² (²)| > C³T

110−υ

´2#= o

¡T−1


160 .

Proof. The conclusion follows by using the characterization of θ²² (²) in Appendix C.3, and Lemmas

7, 8, 9, and 10.


Pr

"maxi

max0≤²≤ 1√

T

¯αθrθr0θr00i (²)

¯> C

#= o

¡T−1

¢Pr

"maxi

max0≤²≤ 1√

T

¯αθrθr0 ²i (²)

¯> CT

110−υ

#= o

¡T−1

¢Pr

"maxi

max0≤²≤ 1√

T

¯αθr²²i (²)

¯> C

³T

110−υ

´2#= o

¡T−1

¢Pr

"maxi

max0≤²≤ 1√

T

|α²²²i (²)| > C³T

110−υ

´3#= o

¡T−1


160 .

Proof. It was seen in Appendix D.1 that

0 =

µZ∂2Vi (·, θ, ²)∂θr∂αi

dFi (²)

¶∂αi (θ, Fi (²))

∂²

+

µZ∂Vi (·, θ, ²)∂αi

dFi (²)

¶∂2αi (θ, Fi (²))

∂θr∂²+

µZ∂2Vi (·, θ, ²)

∂α2idFi (²)

¶∂αi (θ, Fi (²))

∂θr

∂αi (θ, Fi (²))

∂²

+

Z∂Vi (·, θ, ²)∂θr

d∆iT +

µZ∂Vi (·, θ, ²)∂αi

d∆iT

¶∂αi (θ, Fi (²))

∂θr,

23

and

0 =

µZ∂Vi (·, θ, ²)∂αi

dFi (²)

¶∂2αi (θ, Fi (²))

∂²2+

µZ∂2Vi (·, θ, ²)

∂α2idFi (²)

¶µ∂αi (θ, Fi (²))

∂²

¶2+2

µZ∂Vi (·, θ, ²)∂αi

d∆iT

¶∂αi (θ, Fi (²))

∂².

We therefore obtain

0 =

Z∂3Vi (·, θ, ²)∂θr∂θr0∂θr00

dFi (²) +

µZ∂3Vi (·, θ, ²)∂αi∂θr∂θr0

dFi (²)

¶∂αi (θ, Fi (²))

∂θr00

+∂2αi (θ, Fi (²))

∂θr∂θr00

µZ∂2Vi (·, θ, ²)∂αi∂θr0

dFi (²)

¶+∂αi (θ, Fi (²))

∂θr

µZ∂3Vi (·, θ, ²)∂αi∂θr0∂θr00

dFi (²)

¶+∂αi (θ, Fi (²))

∂θr

∂αi (θ, Fi (²))

∂θr00

µZ∂3Vi (·, θ, ²)∂α2i∂θr0

dFi (²)

¶+

µZ∂3Vi (·, θ, ²)∂αi∂θr∂θr00

dFi (²)

¶∂αi (θ, Fi (²))

∂θr0+

µZ∂3Vi (·, θ, ²)∂α2i∂θr

dFi (²)

¶∂αi (θ, Fi (²))

∂θr0

∂αi (θ, Fi (²))

∂θr00

+

µZ∂2Vi (·, θ, ²)∂αi∂θr

dFi (²)

¶∂2αi (θ, Fi (²))

∂θr0∂θr00

+

µZ∂2Vi (·, θ, ²)∂αi∂θr00

dFi (²)

¶∂2αi (θ, Fi (²))

∂θr∂θr0+

µZ∂2Vi (·, θ, ²)

∂α2idFi (²)

¶∂αi (θ, Fi (²))

∂θr00

∂2αi (θ, Fi (²))

∂θr∂θr0

+

µZ∂Vi (·, θ, ²)∂αi

dFi (²)

¶∂3αi (θ, Fi (²))

∂θr∂θr0∂θr00

+

µZ∂3Vi (·, θ, ²)∂α2i ∂θr00

dFi (²)

¶∂αi (θ, Fi (²))

∂θr

∂αi (θ, Fi (²))

∂θr0

+

µZ∂3Vi (·, θ, ²)

∂α3idFi (²)

¶∂αi (θ, Fi (²))

∂θr

∂αi (θ, Fi (²))

∂θr0

∂αi (θ, Fi (²))

∂θr00

+

µZ∂2Vi (·, θ, ²)

∂α2idFi (²)

¶∂2αi (θ, Fi (²))

∂θr∂θr00

∂αi (θ, Fi (²))

∂θr0

+

µZ∂2Vi (·, θ, ²)

∂α2idFi (²)

¶∂αi (θ, Fi (²))

∂θr

∂2αi (θ, Fi (²))

∂θr0∂θr00,

24

which characterizes ∂3αi(θ,Fi(²))∂θr∂θr0∂θr00

,

0 =

µZ∂3Vi (·, θ, ²)∂θr∂θr0∂αi

dFi (²)

¶∂αi (θ, Fi (²))

∂²+

µZ∂3Vi (·, θ, ²)∂θr∂α2i

dFi (²)

¶∂αi (θ, Fi (²))

∂θr0

∂αi (θ, Fi (²))

∂²

+


dFi (²)

¶∂2αi (θ, Fi (²))

∂θr0∂²

+

µZ∂2Vi (·, θ, ²)∂θr0∂αi

dFi (²)

¶∂2αi (θ, Fi (²))

∂θr∂²+

µZ∂2Vi (·, θ, ²)

∂α2idFi (²)

¶∂αi (θ, Fi (²))

∂θr0

∂2αi (θ, Fi (²))

∂θr∂²

+

µZ∂Vi (·, θ, ²)∂αi

dFi (²)

¶∂3αi (θ, Fi (²))

∂θr∂θr0∂²

+

µZ∂3Vi (·, θ, ²)∂θr0∂α2i

dFi (²)

¶∂αi (θ, Fi (²))

∂θr

∂αi (θ, Fi (²))

∂²

+

µZ∂3Vi (·, θ, ²)

∂α3idFi (²)

¶∂αi (θ, Fi (²))

∂θr0

∂αi (θ, Fi (²))

∂θr

∂αi (θ, Fi (²))

∂²

+

µZ∂2Vi (·, θ, ²)

∂α2idFi (²)

¶∂2αi (θ, Fi (²))

∂θr∂θr0

∂αi (θ, Fi (²))

∂²

+

µZ∂2Vi (·, θ, ²)

∂α2idFi (²)

¶∂αi (θ, Fi (²))

∂θr

∂2αi (θ, Fi (²))

∂θr0∂²

+

Z∂2Vi (·, θ, ²)∂θr∂θr0

d∆iT +


d∆iT

¶∂αi (θ, Fi (²))

∂θr0

+

µZ∂2Vi (·, θ, ²)∂θr0∂αi

d∆iT

¶∂αi (θ, Fi (²))

∂θr+

µZ∂2Vi (·, θ, ²)

∂α2id∆iT

¶∂αi (θ, Fi (²))

∂θr

∂αi (θ, Fi (²))

∂θr0

+

µZ∂Vi (·, θ, ²)∂αi

d∆iT

¶∂2αi (θ, Fi (²))

∂θr∂θr0,

which characterizes ∂3αi(θ,Fi(²))∂θr∂θr0∂²

,

0 =


dFi (²)

¶∂2αi (θ, Fi (²))

∂²2+

µZ∂2Vi (·, θ, ²)

∂α2idFi (²)

¶∂αi (θ, Fi (²))

∂θr

∂2αi (θ, Fi (²))

∂²2

+

µZ∂Vi (·, θ, ²)∂αi

dFi (²)

¶∂3αi (θ, Fi (²))

∂θr∂²2

+

µZ∂3Vi (·, θ, ²)∂θr∂α2i

dFi (²)

¶µ∂αi (θ, Fi (²))

∂²

¶2+

µZ∂3Vi (·, θ, ²)

∂α3idFi (²)

¶∂αi (θ, Fi (²))

∂θr

µ∂αi (θ, Fi (²))

∂²

¶2+2

µZ∂2Vi (·, θ, ²)

∂α2idFi (²)

¶∂αi (θ, Fi (²))

∂²

∂2αi (θ, Fi (²))

∂θr∂²

+2


d∆iT

¶∂αi (θ, Fi (²))

∂²+ 2

µZ∂2Vi (·, θ, ²)

∂α2id∆iT

¶∂αi (θ, Fi (²))

∂θr

∂αi (θ, Fi (²))

∂²

+2

µZ∂Vi (·, θ, ²)∂αi

d∆iT

¶∂2αi (θ, Fi (²))

∂θr∂²,

25

which characterizes ∂3αi(θ,Fi(²))∂θr∂²2

, and

0 =

µZ∂2Vi (·, θ, ²)

∂α2idFi (²)

¶∂αi (θ, Fi (²))

∂²

∂2αi (θ, Fi (²))

∂²2+

µZ∂Vi (·, θ, ²)∂αi

d∆iT

¶∂2αi (θ, Fi (²))

∂²2

+

µZ∂Vi (·, θ, ²)∂αi

dFi (²)

¶∂3αi (θ, Fi (²))

∂²3

+

µZ∂3Vi (·, θ, ²)

∂α3idFi (²)

¶µ∂αi (θ, Fi (²))

∂²

¶3+

µZ∂2Vi (·, θ, ²)

∂α2id∆iT

¶µ∂αi (θ, Fi (²))

∂²

¶2+2

µZ∂2Vi (·, θ, ²)

∂α2idFi (²)

¶∂αi (θ, Fi (²))

∂²

∂2αi (θ, Fi (²))

∂²2

+2

µZ∂2Vi (·, θ, ²)

∂α2id∆iT

¶µ∂αi (θ, Fi (²))

∂²

¶2+ 2

µZ∂Vi (·, θ, ²)∂αi

d∆iT

¶∂2αi (θ, Fi (²))

∂²2,

which characterizes ∂3αi(θ,Fi(²))∂²3 . Inspecting these derivatives and applying Lemmas 7, 8, and 10, we

obtain the desired result.

Theorem 6 Suppose that Condition 4 holds. Then, we have

Pr

"max

0≤²≤ 1√T

|θ²²² (²)| > C³T

110−υ

´3#= o

¡T−1


160 .

Proof. From (15), we have

0 =1

n

nXi=1

Zd3hi (·, ²)d²3

dFi (²) + 31

n

nXi=1

Zd2hi (·, ²)d²2

d∆iT

Combining Lemmas 7, 8, 9, 10, and 11, we can bound 1n

Pni=1

R d2hi(·,²)d²2 d∆iT by C

³T

110−υ

´3with

probability 1 − o ¡T−1¢. It was seen in Appendix C.3 that the r-th component d2h(r)i (·,²)d²2 of d

2hi(·,²)d²2 is

equal to

d2h(r)i (·, ²)d²2

=∂θ (·, ²)0∂²

∂2h(r)i (·, ²)∂θ∂θ0

∂θ (·, ²)∂²

+∂2h

(r)i (·, ²)∂θ0∂αi

µ∂αi∂θ0

∂θ

∂²

¶∂θ

∂²+∂2h

(r)i (·, ²)∂θ0∂αi

∂αi∂²

∂θ

∂²

+∂h

(r)i (·, ²)∂θ0

∂2θ

∂²2

+∂2h

(r)i (·, ²)∂θ0∂αi

∂θ

∂²

∂αi∂θ0

∂θ

∂²+∂2h

(r)i (·, ²)∂α2i

µ∂αi∂θ0

∂θ

∂²

¶2+∂2h

(r)i (·, ²)∂α2i

µ∂αi∂θ0

∂θ

∂²

¶∂αi∂²

+∂h

(r)i (·, ²)∂αi

µ∂θ0

∂²

∂2αi∂θ∂θ0

¶∂θ

∂²+∂h

(r)i (·, ²)∂αi

∂2αi∂θ0∂²

∂θ

∂²+∂h

(r)i (·, ²)∂αi

∂αi∂θ0

∂2θ

∂²2

+∂2h

(r)i (·, ²)∂θ0∂αi

∂θ

∂²

∂αi∂²

+∂2h

(r)i (·, ²)∂α2i

µ∂αi∂θ0

∂θ

∂²

¶∂αi∂²

+∂2h

(r)i (·, ²)∂α2i

µ∂αi∂²

¶2+∂h

(r)i (·, ²)∂αi

∂2αi∂²∂θ0

∂θ

∂²+∂h

(r)i (·, ²)∂αi

∂2αi∂²2

.

Using Lemmas 7, 8, 9, 10, and 11 again, we can conclude that 1n

Pni=1

R d3hi(·,²)d²3 dFi (²) is equal to³

1n

Pni=1

R ∂hi(·,²)∂θ0 dFi (²)

´∂3θ∂²3 plus terms that can all be bounded by

1n

Pni=1

R d2hi(·,²)d²2 d∆iT byC

³T

110−υ

´326

with probability 1− o ¡T−1¢. Because ³ 1nPni=1

R ∂hi(·,²)∂θ0 dFi (²)

´−1is bounded away from 0 by Lemma 7

and Condition 4, we obtain the desired conclusion.

E Proof of Theorem 2

Note that ¯¯ 1T

TXt=1

V2it

³xit;bθ, bαi´ sθ ³xit;bθ, bαi´−E [V2it (xit; θ0,αi0) sθ (xit; θ0,αi0)]

¯¯

≤¯¯ 1T

TXt=1

V2it³xit;bθ, bαi´ sθ ³xit;bθ, bαi´− 1

T

TXt=1

V2it (xit; θ0,αi0) sθ (xit; θ0,αi0)

¯¯

+

¯¯ 1T

TXt=1

V2it (xit; θ0,αi0) sθ (xit; θ0,αi0)−E [V2it (xit; θ0,αi0) sθ (xit; θ0,αi0)]¯¯

≤ÃmaxiEhMit (xit)

2i+

¯¯ 1T

TXt=1

³Mit (xit)

2 − EhMit (xit)

2i´¯¯!³¯bθ − θ0 ¯+max

i|bαi − αi0|´

+

¯¯ 1T

TXt=1

V2it (xit; θ0,αi0) sθ (xit; θ0,αi0)−E [V2it (xit; θ0,αi0) sθ (xit; θ0,αi0)]¯¯

By Lemma 2, we obtain

maxi

¯¯ 1T

TXt=1

³Mit (xit)

2 −EhMit (xit)

2i´¯¯ = op (1)

and

maxi

¯¯ 1T

TXt=1

V2it (xit; θ0,αi0) sθ (xit; θ0,αi0)−E [V2it (xit; θ0,αi0) sθ (xit; θ0,αi0)]¯¯ = op (1)

Combined with Lemmas 4 and 5, we obtain

maxi

¯¯ 1T

TXt=1

V2it

³xit;bθ, bαi´ sθ ³xit;bθ, bαi´−E [V2it (xit; θ0,αi0) sθ (xit; θ0,αi0)]

¯¯ = op (1)

We can similarly show that

maxi

¯¯ 1T

TXt=1

Vit³xit;bθ, bαi´2 −E hVit (xit; θ0,αi0)2i

¯¯ = op (1)

maxi

¯¯ 1T

TXt=1

V2it³xit;bθ, bαi´Vit ³xit;bθ, bαi´−E [V2it (xit; θ0,αi0)Vit (xit; θ0,αi0)]

¯¯ = op (1)

maxi

¯¯ 1T

TXt=1

sθ

³xit;bθ, bαi´2 −E hsθ (xit; θ0,αi0)2i

¯¯ = op (1)

maxi

¯¯ 1T

TXt=1

sθ

³xit;bθ, bαi´Vit ³xit;bθ, bαi´− sθ (xit; θ0,αi0)Vit (xit; θ0,αi0)

¯¯ = op (1)

27

It follows that ¯¯ 1nT

nXi=1

TXt=1

bUit bU 0it − 1

n

nXi=1

Ii¯¯ = op (1)¯

¯ 1nnXi=1

1T

PTt=1

bUit bV2it1T

PTt=1

bV 2it − 1

n

nXi=1

E [V2itUit]

E [V 2it]

¯¯ = op (1)

from which we obtain

1

nT

nXi=1

TXt=1

bUit bU 0it = limn→∞

1

n

nXi=1

Ii + op (1) (22)

1

n

nXi=1

1T

PTt=1

bUitbV2it1T

PTt=1

bV 2it = limn→∞

1

n

nXi=1

E [V2itUit]

E [V 2it]+ op (1) (23)

F V-Statistics

F.1 Properties of Normalized V-Statistics

Consider a statistic of the form

Wi,T ≡ 1

Tm/2

TXt1=1

TXt2=1

· · ·TX

tm=1

k1 (xi,t1) k2 (xi,t2) · · · km (xi,tm)

≡ Tm/2ki,1ki,2 · · ·ki,m≡ Tm/2Ki,T

where E [kj (xi,t)] = 0. We will call the average

1

n

nXi=1

Wi,T

of such Wi,T the normalized V-statistic of order m. We will focus on normalized V-statistic up to order

4.

Condition 5 (i) n = O (T ); (ii) E [kj (xi,t)] = 0; (iii) |kj (xi,t)| ≤ CM (xi,t) such that supiEhM (xi,t)

8i<

∞, where C denotes a generic constant.

Lemma 13 Suppose that Condition 5 holds. When m = 1, 2, 3, 4, then 1n

Pni=1Wi,T = Op (1).

Proof. Suppose that m = 1. By Chebyshevs inequality, we have

Pr

"¯¯ 1√n

nXi=1

Wi,T

¯¯ ≥M

#= Pr

"¯¯ 1√n

nXi=1

Ã1√T

TXt=1

k1 (xi,t)

!¯¯ ≥M

#

≤E

·¯1√n

Pni=1

³1√T

PTt=1 k1 (xi,t)

´¯2¸(M)2

=

1nT

Pni=1

PTt=1E

hk1 (xi,t)

2i

(M)2

≤C2 supiE

hM (xi,t)

2i

(M)2

28

It therefore follows that 1√n

Pni=1Wi,T = Op (1), from which we obtain the desired conclusion.

Now suppose that m = 2. By Markovs inequality, we have

Pr

1n

nXi=1

Ã1√T

TXt=1

k1 (xi,t)

!2≥M

≤E

·1n

Pni=1

³1√T

PTt=1 k1 (xi,t)

´2¸M

=

1nT

Pni=1

PTt=1E

hk1 (xi,t)

2i

M

=C2 supiE

hM (xi,t)

2i

MIt therefore follows that 1n

Pni=1

³1√T

PTt=1 k1 (xi,t)

´2= Op (1). Likewise, we have 1n

Pni=1

³1√T

PTt=1 k2 (xi,t)

´2=

Op (1). Because¯¯ 1n

nXi=1

Ã1√T

TXt=1

k1 (xi,t)

!Ã1√T

TXt=1

k2 (xi,t)

!¯¯ ≤

1n

nXi=1

Ã1√T

TXt=1

k1 (xi,t)

!21/2 1n

nXi=1

Ã1√T

TXt=1

k2 (xi,t)

!21/2

,

we obtain the desired conclusion.

Now, suppose that m = 3. By Markovs inequality, we have

Pr

1n

nXi=1

Ã1√T

TXt=1

k1 (xi,t)

!4≥M

≤ E

·1n

Pni=1

³1√T

PTt=1 k1 (xi,t)

´4¸M

Because

E

Ã 1√T

TXt=1

k1 (xi,t)

!4 =TE

hk1 (xi,t)

4i+ 6T (T−1)2 E

hk1 (xi,t)

2i2

T 2

=Ehk1 (xi,t)

4i

T+ 3

(T − 1)Ehk1 (xi,t)

2i2

T

≤EhM (xi,t)

4i

T+ 3

(T − 1)EhM (xi,t)

4i

T< ∞

we have 1n

Pni=1

³1√T

PTt=1 k1 (xi,t)

´4= Op (1). Likewise, we have 1

n

Pni=1

³1√T

PTt=1 k2 (xi,t)

´4=

Op (1), and 1n

Pni=1

³1√T

PTt=1 k2 (xi,t)

´4= Op (1). This implies that

1

n

nXi=1

¯¯ 1√T

TXt=1

k1 (xi,t)

¯¯3

= Op (1) ,1

n

nXi=1

¯¯ 1√T

TXt=1

k2 (xi,t)

¯¯3

= Op (1) ,1

n

nXi=1

¯¯ 1√T

TXt=1

k2 (xi,t)

¯¯3

= Op (1)

by Jensens inequality. Because¯¯ 1n

nXi=1

Ã1√T

TXt=1

k1 (xi,t)

!Ã1√T

TXt=1

k2 (xi,t)

!Ã1√T

TXt=1

k3 (xi,t)

!¯¯

≤ 1n

nXi=1

¯¯ 1√T

TXt=1

k1 (xi,t)

¯¯31/3 1

n

nXi=1

¯¯ 1√T

TXt=1

k2 (xi,t)

¯¯31/3 1

n

nXi=1

¯¯ 1√T

TXt=1

k3 (xi,t)

¯¯31/3

29

by repeated application of Holders inequality, we obtain the desired conclusion for m = 3. We obtain

the same conclusion for m = 4 by the same method.

F.2 JackkniÞng Normalized V-Statistics

We consider the average of delete-t estimators

1

T

TXt=1

Wi,T,(−t) =1

T

TXt=1

(T − 1)m/2Ki,T,(−t)

= (T − 1)m/2 1T

TXt=1

1

T − 1TXt1 6=t

k1 (xi,t1)

1

T − 1TXt2 6=t

k2 (xi,t2)

· · · 1

T − 1TX

tm 6=tkm (xi,tm)

= (T − 1)m/2 1

T

TXt=1

Tki,1 − k1 (xi,t)T − 1

Tki,2 − k2 (xi,t)T − 1 · · · Tki,m − km (xi,t)

T − 1

Lemma 14 Suppose that Condition 5 holds. Whenm = 1, then 1T

PTt=1

¡1n

Pni=1Wi,T,(−t)

¢= (T−1)1/2

T 1/21n

Pni=1Wi,T .

Whenm = 2, then 1T

PTt=1

¡1n

Pni=1Wi,T,(−t)

¢= T−2

T−11n

Pni=1Wi,T+

1T (T−1)

1n

Pni=1

PTt=1 k1 (xi,t) k2 (xi,t).

When m = 3 or 4, then 1T

PTt=1

¡1n

Pni=1Wi,T,(−t)

¢= 1

n

Pni=1Wi,T + op

³1√T

´.

Proof. Lemma 14 would follow if (i) when m = 1,

1

T

TXt=1

Wi,T,(−t) =1

T

TXt=1

(T − 1)1/2Ki,T,(−t) = (T − 1)1/2Ki,T = (T − 1)1/2T 1/2

Wi,T

(ii) when m = 2,

1

T

TXt=1

Wi,T,(−t) =T − 2T − 1Wi,T +

1

T (T − 1)TXt=1

k1 (xi,t) k2 (xi,t)

(iii) when m = 3, we have

1

T

TXt=1

Wi,T,(−t) =T 3 − 3T 2

T 3/2 (T − 1)3/2Wi,T

+T 1/2

(T − 1)3/2³T 1/2ki,1

´Ã 1T

TXt=1

k2 (xi,t) k3 (xi,t)

!

+T 1/2

(T − 1)3/2³T 1/2ki,2

´Ã 1T

TXt=1

k3 (xi,t) k1 (xi,t)

!

+T 1/2

(T − 1)3/2³T 1/2ki,3

´Ã 1T

TXt=1

k1 (xi,t) k2 (xi,t)

!

− 1

(T − 1)3/2Ã1

T

TXt=1

k (xi,t) k2 (xi,t) k3 (xi,t)

!

30

(iv) when m = 4, we have

1

T

TXt=1

Wi,T,(−t) =T 2 − 4T(T − 1)2Wi,T

+T

(T − 1)2³T 1/2ki,1

´³T 1/2ki,2

´Ã 1T

TXt=1

k3 (xi,t) k4 (xi,t)

!

+T

(T − 1)2³T 1/2ki,1

´³T 1/2ki,3

´Ã 1T

TXt=1

k2 (xi,t) k4 (xi,t)

!

+T

(T − 1)2³T 1/2ki,1

´³T 1/2ki,4

´Ã 1T

TXt=1

k2 (xi,t) k3 (xi,t)

!

+T

(T − 1)2³T 1/2ki,2

´³T 1/2ki,3

´Ã 1T

TXt=1

k1 (xi,t) k4 (xi,t)

!

+T

(T − 1)2³T 1/2ki,2

´³T 1/2ki,4

´Ã 1T

TXt=1

k1 (xi,t) k3 (xi,t)

!

+T

(T − 1)2³T 1/2ki,3

´³T 1/2ki,4

´Ã 1T

TXt=1

k1 (xi,t) k2 (xi,t)

!

− T 1/2

(T − 1)2³T 1/2ki,1

´Ã 1T

TXt=1

k2 (xi,t) k3 (xi,t) k4 (xi,t)

!

− T 1/2

(T − 1)2³T 1/2ki,2

´Ã 1T

TXt=1


!

− T 1/2

(T − 1)2³T 1/2ki,3

´Ã 1T

TXt=1


!

− T 1/2

(T − 1)2³T 1/2ki,4

´Ã 1T

TXt=1


!

+1

(T − 1)2Ã1

T

TXt=1

k1 (xi,t) k2 (xi,t) k3 (xi,t) k4 (xi,t)

!In order to prove the preceding assertions, we note the following.For m = 1, it suffices to note that

1

T

TXt=1

Ki,T,(−t) = ki,1 = Ki,T

For m = 2, it suffices to note that

1

T

TXt=1

Ki,T,(−t) =T 2 − 2T(T − 1)2 ki,1ki,2 +

1

T (T − 1)2TXt=1

k1 (xi,t) k2 (xi,t)

31


1

T

TXt=1

Ki,T,(−t) =T 3 − 3T 2(T − 1)3 ki,1ki,2ki,3

+1

(T − 1)3 ki,1TXt=1

k2 (xi,t) k3 (xi,t) +1

(T − 1)3 ki,2TXt=1

k3 (xi,t) k1 (xi,t)

+1

(T − 1)3 ki,3TXt=1

k1 (xi,t) k2 (xi,t)− 1

T (T − 1)3TXt=1



1

T

TXt=1

Ki,T,(−t) =T 4 − 4T 3(T − 1)4 ki,1ki,2ki,3ki,4

+ki,1ki,2T

(T − 1)4TXt=1

k3 (xi,t) k4 (xi,t) + ki,1ki,3T

(T − 1)4TXt=1

k2 (xi,t) k4 (xi,t)

+ki,1ki,4T

(T − 1)4TXt=1


(T − 1)4TXt=1

k1 (xi,t) k4 (xi,t)

+ki,2ki,4T

(T − 1)4TXt=1


(T − 1)4TXt=1

k1 (xi,t) k2 (xi,t)

− 1

(T − 1)4 ki,1TXt=1

k2 (xi,t) k3 (xi,t) k4 (xi,t)− 1

(T − 1)4 ki,2TXt=1


− 1

(T − 1)4 ki,3TXt=1

k1 (xi,t) k2 (xi,t) k4 (xi,t)− 1

(T − 1)4 ki,4TXt=1


+1

T (T − 1)4TXt=1

k1 (xi,t) k2 (xi,t) k3 (xi,t) k4 (xi,t)

F.3 JackkniÞng Functions of Normalized V-Statistic

Now, consider a statistic of the form

WT =W[1],TW[2],T · · ·W[L],T

where

W[`],T ≡ 1

n

nXi=1

1

Tm`/2

TXt1=1

TXt2=1

· · ·TX

tm`=1

k[`,1] (xi,t1) k[`,2] (xi,t2) · · · k[`,m`] (xi,tm)

≡ Tm`/21

n

nXi=1

ki,[`,1]ki,[`,2] · · · ki,[`,m`]

≡ Tm`/21

n

nXi=1

Ki,[`],T

≡ Tm`/2K[`],T .

32

Therefore, WT is the product of m1, · · · ,mL normalized V-statistics. We will also call them normalized

V-statistic of orderPL`=1m`.

We consider the average of delete-t estimators

1

T

TXt=1

WT,(−t) =1

T

TXt=1

W[1],T,(−t)W[2],T,(−t) · · ·W[L],T,(−t)

where

W[`],T,(−t) ≡ 1

n

nXi=1

1

(T − 1)m`/2

TXt1 6=t

TXt2 6=t

· · ·TX

tm`6=tk[`,1] (xi,t1) k[`,2] (xi,t2) · · · k[`,m`] (xi,tm)

= (T − 1)m`/2 1

n

nXi=1

Tki,[`,1] − k[`,1] (xi,t)T − 1

Tki,[`,2] − k[`,2] (xi,t)T − 1 · · · Tki,[`,m`] − k[`,m`] (xi,t)

T − 1

=1

(T − 1)m`/2

1

n

nXi=1

¡Tki,[`,1] − k[`,1] (xi,t)

¢ ¡Tki,[`,2] − k[`,2] (xi,t)

¢ · · · ¡Tki,[`,m`] − k[`,m`] (xi,t)¢

Below, we characterize properties of jackknifed normalized V-statistic of order up to 4.

Lemma 15 Suppose that Condition 5 holds. WhenP`m` = 2,

P`m` = 2, L = 2,m1 = m2 = 1, we

have

1

T

TXt=1

WT,(−t) =T − 2T − 1WT +Op

µ1

nT

¶.

Proof. We have

W[1],T = T1/2 1

n

nXi=1

ki,[1,1], W[2],T = T1/2 1

n

nXi=1

ki,[2,1]

and therefore,

WT = T1

n2

nXi1=1

nXi2=1

ki1,[1,1]ki2,[2,1]

and

1

T

TXt=1

WT,(−t) =1

T (T − 1)1

n2

nXi1=1

nXi2=1

TXt=1

¡Tki1,[1,1] − k[1,1] (xi1,t)

¢ ¡Tki2,[2,1] − k[2,1] (xi2,t)

¢=

T 2 − 2TT − 1

1

n2

nXi1=1

nXi2=1

ki1,[1,1]ki2,[2,1]

+1

T (T − 1)1

n2

nXi1=1

nXi2=1

TXt=1

k[1,1] (xi1,t) k[2,1] (xi2,t)

=T − 2T − 1WT +

1

T (T − 1)1

n2

nXi1=1

nXi2=1

TXt=1

k[1,1] (xi1,t) k[2,1] (xi2,t)

Note that

WT =1

n

Ã1√n

nXi1=1

³√Tki1,[1,1]

´!Ã 1√n

nXi2=1

³√Tki2,[2,1]

´!= Op

µ1

n

¶

33

and

1

T (T − 1)1

n2

nXi1=1

nXi2=1

TXt=1

k[1,1] (xi1,t) k[2,1] (xi2,t)

=1

n (T − 1)1

T

TXt=1

Ã1√n

nXi1=1

k[1,1] (xi1,t)

!Ã1√n

nXi2=1

k[2,1] (xi2,t)

!Because

E

¯¯ 1TTXt=1

Ã1√n

nXi1=1

k[1,1] (xi1,t)

!Ã1√n

nXi2=1

k[2,1] (xi2,t)

!¯¯2

≤ 1

T

TXt=1

E

Ã 1√n

nXi1=1

k[1,1] (xi1,t)

!2Ã1√n

nXi2=1

k[2,1] (xi2,t)

!2≤ E

Ã 1√n

nXi1=1

k[1,1] (xi1,t)

!41/2EÃ 1√

n

nXi2=1

k[2,1] (xi2,t)

!41/2

≤nmaxiE

hk1 (xi,t)

4i+ 6n(n−1)2 maxiE

hk1 (xi,t)

2i2

n2

< ∞

we have

1

T (T − 1)1

n2

nXi1=1

nXi2=1

TXt=1

k[1,1] (xi1,t) k[2,1] (xi2,t) = Op

µ1

nT

¶Therefore, we have

1

T

TXt=1

WT,(−t) =T − 2T − 1WT +Op

µ1

nT

¶

Lemma 16 Suppose that Condition 5 holds. WhenP`m` = 3, L = 2,m1 = 1,m2 = 2, we have

1

T

TXt=1

WT,(−t) =WT + op

µ1√T

¶.

Proof. We have

W[1],T = T1/2 1

n

nXi=1

ki,[1,1], W[2],T = T1

n

nXi=1

ki,[2,1]ki,[2,2]

and therefore,

WT = T3/2 1

n2

nXi1=1

nXi2=1

ki1,[1,1]ki2,[2,1]ki2,[2,2]

34

and

1

T

TXt=1

WT,(−t)

=1

T (T − 1)3/21

n2

nXi1=1

nXi2=1

TXt=1

¡Tki1,[1,1] − k[1,1] (xi1,t)

¢ ¡Tki2,[2,1] − k[2,1] (xi2,t)

¢ ¡Tki2,[2,2] − k[2,2] (xi2,t)

¢=

T 3 − 3T 2T 3/2 (T − 1)3/2

WT

+T 1/2

(T − 1)3/21

n2

nXi1=1

nXi2=1

³T 1/2ki1,[1,1]

´Ã 1T

TXt=1

k[2,1] (xi2,t) k[2,2] (xi2,t)

!

+T 1/2

(T − 1)3/21

n2

nXi1=1

nXi2=1

³T 1/2ki2,[2,1]

´Ã 1T

TXt=1

k[1,1] (xi1,t) k[2,2] (xi2,t)

!

+T 1/2

(T − 1)3/21

n2

nXi1=1

nXi2=1

³T 1/2ki2,[2,2]

´Ã 1T

TXt=1

k[1,1] (xi1,t) k[2,1] (xi2,t)

!

− 1

(T − 1)3/21

T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t)

!Ã1

n

nXi2=1

k[2,1] (xi2,t) k[2,2] (xi2,t)

!(24)

We now show that

T 1/2

(T − 1)3/21

n2

nXi1=1

nXi2=1

³T 1/2ki1,[1,1]

´Ã 1T

TXt=1

k[2,1] (xi2,t) k[2,2] (xi2,t)

!= op

µ1√T

¶(25)

For this purpose, it suffices to note that¯¯ 1n2

nXi1=1

nXi2=1

³T 1/2ki1,[1,1]

´Ã 1T

TXt=1

k[2,1] (xi2,t) k[2,2] (xi2,t)

!¯¯

≤¯¯ 1n

nXi1=1

³T 1/2ki1,[1,1]

´¯¯¯¯ 1n

nXi2=1

Ã1

T

TXt=1

k[2,1] (xi2,t) k[2,2] (xi2,t)

!¯¯

= Op (1)

by Lemma 13.

We now show that

T 1/2

(T − 1)3/21

n2

nXi1=1

nXi2=1

³T 1/2ki2,[2,1]

´Ã 1T

TXt=1

k[1,1] (xi1,t) k[2,2] (xi2,t)

!= op

µ1√T

¶,

T 1/2

(T − 1)3/21

n2

nXi1=1

nXi2=1

³T 1/2ki2,[2,2]

´Ã 1T

TXt=1

k[1,1] (xi1,t) k[2,1] (xi2,t)

!= op

µ1√T

¶. (26)

35

For this purpose, we observe that¯¯ 1n2

nXi1=1

nXi2=1

³T 1/2ki2,[2,1]

´Ã 1T

TXt=1

k[1,1] (xi1,t) k[2,2] (xi2,t)

!¯¯2

=

¯¯ 1T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t)

!Ã1

n

nXi2=1

³T 1/2ki2,[2,1]

´k[2,2] (xi2,t)

!¯¯2

≤ 1

T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t)

!2· 1T

TXt=1

Ã1

n

nXi2=1

³T 1/2ki2,[2,1]

´k[2,2] (xi2,t)

!2

≤ 1

T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t)

!2· 1T

TXt=1

Ã1

n

nXi2=1

³T 1/2ki2,[2,1]

´2k[2,2] (xi2,t)

2

!

≤ 1

T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t)2

!· 1n

nXi2=1

³T 1/2ki2,[2,1]

´2Ã 1T

TXt=1

k[2,2] (xi2,t)2

!

≤ 1

T

TXt=1

Ã1

n

nXi1=1

M (xi1,t)2

!· 1n

nXi2=1

³T 1/2ki2,[2,1]

´2Ã 1T

TXt=1

M (xi2,t)2

!,

where the far RHS has an expectation equal to

1

nT

TXt=1

nXi1=1

EhM (xi1,t)

2i· 1n

nXi2=1

Ehk[2,1] (xit)

2iÃ 1

T

TXt=1

EhM (xi2,t)

2i!

= O (1)

Therefore, we have

1

n2

nXi1=1

nXi2=1

³T 1/2ki2,[2,1]

´Ã 1T

TXt=1

k[1,1] (xi1,t) k[2,2] (xi2,t)

!= Op (1)

Likewise, we have

1

n2

nXi1=1

nXi2=1

³T 1/2ki2,[2,2]

´Ã 1T

TXt=1

k[1,1] (xi1,t) k[2,1] (xi2,t)

!= Op (1)

We now show that

− 1

(T − 1)3/21

T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t)

!Ã1

n

nXi2=1

k[2,1] (xi2,t) k[2,2] (xi2,t)

!= op

µ1√T

¶(27)

For this purpose, it suffices to note that

1

T 3/2

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t)

!Ã1

n

nXi2=1

k[2,1] (xi2,t) k[2,2] (xi2,t)

!= Op (1)

Using equations (24) and (25), (26), (27), we conclude that

1

T

TXt=1

WT,(−t) =WT + op

µ1√T

¶

36

Lemma 17 Suppose that Condition 5 holds. WhenP`m` = 3, L = 3,m1 = 1,m2 = 1,m3 = 1, we have

1

T

TXt=1

WT,(−t) =WT + op

µ1√T

¶.

Proof. We have

W[1],T = T1/2 1

n

nXi=1

ki,[1,1], W[2],T = T1/2 1

n

nXi=1

ki,[2,1], W[3],T = T1/2 1

n

nXi=1

ki,[3,1]

and therefore,

WT = T3/2 1

n2

nXi1=1

nXi2=1

ki1,[1,1]ki2,[2,1]

and

1

T

TXt=1

WT,(−t)

=1

T (T − 1)3/21

n3

nXi1=1

nXi2=1

nXi3=1

TXt=1

¡Tki1,[1,1] − k[1,1] (xi1,t)

¢ ¡Tki2,[2,1] − k[2,1] (xi2,t)

¢ ¡Tki3,[3,1] − k[3,1] (xi3,t)

¢=

T 3 − 3T 2T 3/2 (T − 1)3/2

WT

+T 1/2

(T − 1)3/2Ã1

n

nXi1=1

T 1/2ki1,[1,1]

!1

T

TXt=1

ÃÃ1

n

nXi2=1

k[2,1] (xi2,t)

!Ã1

n

nXi3=1

k[3,1] (xi3,t)

!!

+T 1/2

(T − 1)3/2Ã1

n

nXi2=1

T 1/2ki2,[2,1]

!1

T

TXt=1

ÃÃ1

n

nXi1=1

k[1,1] (xi1,t)

!Ã1

n

nXi3=1

k[3,1] (xi3,t)

!!

+T 1/2

(T − 1)3/2Ã1

n

nXi3=1

T 1/2ki3,[3,1]

!1

T

TXt=1

ÃÃ1

n

nXi1=1

k[1,1] (xi1,t)

!Ã1

n

nXi2=1

k[2,1] (xi2,t)

!!

− 1

(T − 1)3/21

T

TXt=1

ÃÃ1

n

nXi1=1

k[1,1] (xi1,t)

!Ã1

n

nXi2=1

k[2,1] (xi2,t)

!Ã1

n

nXi3=1

k[3,1] (xi3,t)

!!(28)

We now show that

T 1/2

(T − 1)3/2Ã1

n

nXi1=1

T 1/2ki1,[1,1]

!1

T

TXt=1

ÃÃ1

n

nXi2=1

k[2,1] (xi2,t)

!Ã1

n

nXi3=1

k[3,1] (xi3,t)

!!= op

µ1√T

¶T 1/2

(T − 1)3/2Ã1

n

nXi2=1

T 1/2ki2,[2,1]

!1

T

TXt=1

ÃÃ1

n

nXi1=1

k[1,1] (xi1,t)

!Ã1

n

nXi3=1

k[3,1] (xi3,t)

!!= op

µ1√T

¶T 1/2

(T − 1)3/2Ã1

n

nXi3=1

T 1/2ki3,[3,1]

!1

T

TXt=1

ÃÃ1

n

nXi1=1

k[1,1] (xi1,t)

!Ã1

n

nXi2=1

k[2,1] (xi2,t)

!!= op

µ1√T

¶1

(T − 1)3/21

T

TXt=1

ÃÃ1

n

nXi1=1

k[1,1] (xi1,t)

!Ã1

n

nXi2=1

k[2,1] (xi2,t)

!Ã1

n

nXi3=1

k[3,1] (xi3,t)

!!= op

µ1√T

¶(29)

Only the Þrst assertion is proved. Others are proved similarly. It suffices to prove that 1nPni1=1

T 1/2ki1,[1,1] =

37

Op (1) and 1T

PTt=1

¡¡1n

Pni2=1

k[2,1] (xi2,t)¢ ¡

1n

Pni3=1

k[3,1] (xi3,t)¢¢= Op (1). Because

E

Ã 1n

nXi1=1

T 1/2ki1,[1,1]

!2 =1

n2

nXi1=1

E

·³T 1/2ki1,[1,1]

´2¸=

1

n2T

nXi1=1

Eh¡k[1,1] (xi1,t)

¢2i≤ 1

nTmaxiEhM (xi,t)

2i

we have

1

n

nXi1=1

T 1/2ki1,[1,1] = Op

µ1√nT

¶and the Þrst part is proved. We also have

E

"¯¯ 1T

TXt=1

ÃÃ1

n

nXi2=1

k[2,1] (xi2,t)

!Ã1

n

nXi3=1

k[3,1] (xi3,t)

!!¯¯#

≤ 1

n2T

TXt=1

nXi2=1

nXi3=1

E£¯k[2,1] (xi2,t)

¯ ¯k[3,1] (xi3,t)

¯¤≤ 1

n2T

TXt=1

nXi2=1

nXi3=1

E [M (xi2,t)M (xi3,t)]

= O (1)

from which the second part follows.

We now get back to the proof of Lemma 17. Using equations (28) and (29), we conclude that

1

T

TXt=1

WT,(−t) =WT + op

µ1√T

¶


1

T

TXt=1

WT,(−t) =WT + op

µ1√T

¶.

Proof. We have

W[1],T = T1/2 1

n

nXi=1

ki,[1,1], W[2],T = T3/2 1

n

nXi=1

ki,[2,1]ki,[2,2]ki,[2,3]

and therefore,

WT = T2 1

n2

nXi1=1

nXi2=1

ki1,[1,1]ki2,[2,1]ki2,[2,3]

38

and

1

T

TXt=1

WT,(−t)

=1

T (T − 1)21

n2

nXi1=1

nXi2=1

TXt=1

¡Tki1,[1,1] − k[1,1] (xi1,t)

¢ ¡Tki2,[2,1] − k[2,1] (xi2,t)

¢× ¡Tki2,[2,2] − k[2,2] (xi2,t)¢ ¡Tki2,[2,3] − k[2,3] (xi2,t)¢

=T 2 − 4T(T − 1)2WT

+T

(T − 1)2Ã1

n

nXi1=1

T 1/2ki1,[1,1]

!1

n

nXi2=1

Ã³T 1/2ki2,[2,1]

´Ã 1T

TXt=1

k[2,2] (xi2,t) k[2,3] (xi2,t)

!!

+T

(T − 1)2Ã1

n

nXi1=1

T 1/2ki1,[1,1]

!1

n

nXi2=1

Ã³T 1/2ki2,[2,2]

´Ã 1T

TXt=1

k[2,1] (xi2,t) k[2,3] (xi2,t)

!!

+T

(T − 1)2Ã1

n

nXi1=1

T 1/2ki1,[1,1]

!1

n

nXi2=1

Ã³T 1/2ki2,[2,3]

´Ã 1T

TXt=1

k[2,1] (xi2,t) k[2,2] (xi2,t)

!!

+T

(T − 1)21

n

nXi2=1

³T 1/2ki2,[2,1]

´³T 1/2ki2,[2,2]

´Ã 1T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t)

!k[2,3] (xi2,t)

!

+T

(T − 1)21

n

nXi2=1

³T 1/2ki2,[2,1]

´³T 1/2ki2,[2,3]

´Ã 1T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t)

!k[2,2] (xi2,t)

!

+T

(T − 1)21

n

nXi2=1

³T 1/2ki2,[2,2]

´³T 1/2ki2,[2,3]

´Ã 1T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t)

!k[2,1] (xi2,t)

!

− T 1/2

(T − 1)21

n

nXi1=1

³T 1/2ki1,[1,1]

´ 1n

nXi2=1

Ã1

T

TXt=1

k[2,1] (xi2,t) k[2,2] (xi2,t) k[2,3] (xi2,t)

!

− T 1/2

(T − 1)21

n

nXi2=1

³T 1/2ki2,[2,1]

´Ã 1T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t)

!k[2,2] (xi2,t) k[2,3] (xi2,t)

!

− T 1/2

(T − 1)21

n

nXi2=1

³T 1/2ki2,[2,2]

´Ã 1T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t)

!k[2,1] (xi2,t) k[2,3] (xi2,t)

!

− T 1/2

(T − 1)21

n

nXi2=1

³T 1/2ki2,[2,3]

´Ã 1T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t)

!k[2,1] (xi2,t) k[2,2] (xi2,t)

!

+1

(T − 1)2Ã1

T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t)

!Ã1

n

nXi2=1

k[2,1] (xi2,t) k[2,2] (xi2,t) k[2,3] (xi2,t)

!!(30)

39

We now show that

T

(T − 1)2Ã1

n

nXi1=1

T 1/2ki1,[1,1]

!1

n

nXi2=1

Ã³T 1/2ki2,[2,1]

´Ã 1T

TXt=1

k[2,2] (xi2,t) k[2,3] (xi2,t)

!!= op

µ1√T

¶T

(T − 1)2Ã1

n

nXi1=1

T 1/2ki1,[1,1]

!1

n

nXi2=1

Ã³T 1/2ki2,[2,2]

´Ã 1T

TXt=1

k[2,1] (xi2,t) k[2,3] (xi2,t)

!!= op

µ1√T

¶T

(T − 1)2Ã1

n

nXi1=1

T 1/2ki1,[1,1]

!1

n

nXi2=1

Ã³T 1/2ki2,[2,3]

´Ã 1T

TXt=1

k[2,1] (xi2,t) k[2,2] (xi2,t)

!!= op

µ1√T

¶(31)

Only the Þrst claim is proved. Others can be proved similarly. For this purpose, it suffices to prove thatÃ1

n

nXi1=1

T 1/2ki1,[1,1]

!1

n

nXi2=1

Ã³T 1/2ki2,[2,1]

´Ã 1T

TXt=1

k[2,2] (xi2,t) k[2,3] (xi2,t)

!!= Op (1)

Because 1n

Pni1=1

T 1/2ki1,[1,1] = Op (1), we need only to prove that

1

n

nXi2=1

Ã³T 1/2ki2,[2,1]

´Ã 1T

TXt=1

k[2,2] (xi2,t) k[2,3] (xi2,t)

!!= Op (1)

Because¯¯ 1n

nXi2=1

Ã³T 1/2ki2,[2,1]

´Ã 1T

TXt=1

k[2,2] (xi2,t) k[2,3] (xi2,t)

!!¯¯ ≤ 1

n

nXi2=1

¯T 1/2ki2,[2,1]

¯ ¯¯ 1TTXt=1

k[2,2] (xi2,t) k[2,3] (xi2,t)

¯¯ ,

we have

E

"¯¯ 1n

nXi2=1

Ã³T 1/2ki2,[2,1]

´Ã 1T

TXt=1

k[2,2] (xi2,t) k[2,3] (xi2,t)

!!¯¯#

≤ 1

n

nXi2=1

µE

·³T 1/2ki2,[2,1]

´2¸¶1/2EÃ 1

T

TXt=1

k[2,2] (xi2,t) k[2,3] (xi2,t)

!21/2

≤ 1

n

nXi2=1

³Ehk[2,1] (xi2,t)

2i´1/2

E

Ã 1T

TXt=1

M (xi2,t)2

!2= O (1) .

The conclusion follows by Markov inequality.

We now show that

T

(T − 1)21

n

nXi2=1

³T 1/2ki2,[2,1]

´³T 1/2ki2,[2,2]

´Ã 1T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t)

!k[2,3] (xi2,t)

!= op

µ1√T

¶T

(T − 1)21

n

nXi2=1

³T 1/2ki2,[2,1]

´³T 1/2ki2,[2,3]

´Ã 1T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t)

!k[2,2] (xi2,t)

!= op

µ1√T

¶T

(T − 1)21

n

nXi2=1

³T 1/2ki2,[2,2]

´³T 1/2ki2,[2,3]

´Ã 1T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t)

!k[2,1] (xi2,t)

!= op

µ1√T

¶(32)

Only the Þrst claim is proved. Others can be proved similarly. For this purpose, it suffices to prove that

1

n

nXi2=1

³T 1/2ki2,[2,1]

´³T 1/2ki2,[2,2]

´Ã 1T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t)

!k[2,3] (xi2,t)

!= Op (1)

40

Note that ¯¯ 1n

nXi2=1

³T 1/2ki2,[2,1]

´³T 1/2ki2,[2,2]

´Ã 1T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t)

!k[2,3] (xi2,t)

!¯¯

≤ 1

n

nXi2=1

¯T 1/2ki2,[2,1]

¯ ¯T 1/2ki2,[2,2]

¯ Ã 1T

TXt=1

Ã1

n

nXi1=1

¯k[1,1] (xi1,t)

¯! ¯k[2,3] (xi2,t)

¯!

≤ 1

n

nXi2=1

¯T 1/2ki2,[2,1]

¯ ¯T 1/2ki2,[2,2]

¯ Ã 1

nT

TXt=1

nXi1=1

M (xi1,t)M (xi2,t)

!Therefore,

E

"¯¯ 1n

nXi2=1

³T 1/2ki2,[2,1]

´³T 1/2ki2,[2,2]

´Ã 1T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t)

!k[2,3] (xi2,t)

!¯¯#

≤ 1

n

nXi2=1

EÃ 1

nT

TXt=1

nXi1=1

M (xi1,t)M (xi2,t)

!21/2

×µE

·³T 1/2ki2,[2,1]

´4¸¶1/4µE

·³T 1/2ki2,[2,2]

´4¸¶1/4But because

E

Ã 1

nT

TXt=1

nXi1=1

M (xi1,t)M (xi2,t)

!2 ≤ 1

nT

TXt=1

nXi1=1

EhM (xi1,t)

2M (xi2,t)2i

≤ 1

nT

TXt=1

nXi1=1

maxiEhM (xi,t)

4i

= maxiEhM (xi,t)

4i,

E

·³T 1/2ki2,[2,1]

´4¸=

TEhk[2,1] (xi2,t)

4i+ 3T (T − 1)E

hk[2,1] (xi2,t)

2i2

T 2

≤ 3maxiEhM (xi,t)

4i,

and

E

·³T 1/2ki2,[2,2]

´4¸≤ 3max

iEhM (xi,t)

4i,

we should have

E

"¯¯ 1n

nXi2=1

³T 1/2ki2,[2,1]

´³T 1/2ki2,[2,2]

´Ã 1T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t)

!k[2,3] (xi2,t)

!¯¯#= O (1) ,

from which the conclusion follows.

We now show that

T 1/2

(T − 1)21

n

nXi1=1

³T 1/2ki1,[1,1]

´ 1n

nXi2=1

Ã1

T

TXt=1

k[2,1] (xi2,t) k[2,2] (xi2,t) k[2,3] (xi2,t)

!= op

µ1√T

¶(33)

41

For this purpose, it suffices to prove that

1

n

nXi1=1

³T 1/2ki1,[1,1]

´ 1n

nXi2=1

Ã1

T

TXt=1

k[2,1] (xi2,t) k[2,2] (xi2,t) k[2,3] (xi2,t)

!= Op (1)

This follows from

1

n

nXi1=1

³T 1/2ki1,[1,1]

´= Op (1)

and

1

n

nXi2=1

Ã1

T

TXt=1

k[2,1] (xi2,t) k[2,2] (xi2,t) k[2,3] (xi2,t)

!= Op (1)

We now show that

T 1/2

(T − 1)21

n

nXi2=1

³T 1/2ki2,[2,1]

´Ã 1T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t)

!k[2,2] (xi2,t) k[2,3] (xi2,t)

!= op

µ1√T

¶T 1/2

(T − 1)21

n

nXi2=1

³T 1/2ki2,[2,2]

´Ã 1T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t)

!k[2,1] (xi2,t) k[2,3] (xi2,t)

!= op

µ1√T

¶T 1/2

(T − 1)21

n

nXi2=1

³T 1/2ki2,[2,3]

´Ã 1T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t)

!k[2,1] (xi2,t) k[2,2] (xi2,t)

!= op

µ1√T

¶(34)

Only the Þrst claim is proved. Others can be proved similarly. For this purpose, it suffices to prove that

1

n

nXi2=1

³T 1/2ki2,[2,1]

´Ã 1T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t)

!k[2,2] (xi2,t) k[2,3] (xi2,t)

!= Op (1)

Note that ¯¯ 1n

nXi2=1

³T 1/2ki2,[2,1]

´Ã 1T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t)

!k[2,2] (xi2,t) k[2,3] (xi2,t)

!¯¯

≤ 1

n

nXi2=1

¯T 1/2ki2,[2,1]

¯ Ã 1

nT

TXt=1

nXi1=1

M (xi1,t)M (xi2,t)2

!Therefore, we have

E

"¯¯ 1n

nXi2=1

³T 1/2ki2,[2,1]

´Ã 1T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t)

!k[2,2] (xi2,t) k[2,3] (xi2,t)

!¯¯#

≤ 1

n

nXi2=1

µE

·¯T 1/2ki2,[2,1]

¯2¸¶1/2EÃ 1

nT

TXt=1

nXi1=1

M (xi1,t)M (xi2,t)2

!21/2

≤ 1

n

nXi2=1

³EhM (xi2,t)

2i´1/2Ã

E

"1

nT

TXt=1

nXi1=1

M (xi1,t)2M (xi2,t)

4

#!1/2

≤ 1

n

nXi2=1

³EhM (xi2,t)

2i´1/2Ã 1

nT

TXt=1

nXi1=1

³EhM (xi1,t)

4i´1/2 ³

EhM (xi2,t)

8i´1/2!1/2

≤ 1

n

nXi2=1

³maxiEhM (xi,t)

2i´1/2 ³

maxiEhM (xi,t)

4i´1/4 ³

maxiEhM (xi,t)

8i´1/4

= O (1) ,

42


We now show that

1

(T − 1)2Ã1

T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t)

!Ã1

n

nXi2=1

k[2,1] (xi2,t) k[2,2] (xi2,t) k[2,3] (xi2,t)

!!= op

µ1√T

¶(35)

For this purpose, it suffices to prove thatÃ1

T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t)

!Ã1

n

nXi2=1

k[2,1] (xi2,t) k[2,2] (xi2,t) k[2,3] (xi2,t)

!!= Op (1)

Note that¯¯ 1T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t)

!Ã1

n

nXi2=1

k[2,1] (xi2,t) k[2,2] (xi2,t) k[2,3] (xi2,t)

!¯¯ ≤ 1

n2T

TXt=1

nXi1=1

nXi2=1

M (xi1,t)M (xi2,t)3

Therefore, we have

E

"¯¯ 1T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t)

!Ã1

n

nXi2=1

k[2,1] (xi2,t) k[2,2] (xi2,t) k[2,3] (xi2,t)

!¯¯#= O (1) ,


We now get back to the proof of Lemma 18. Using equations (30) and (31), (32), (33), (34), (35), we

conclude that

1

T

TXt=1

WT,(−t) =WT + op

µ1√T

¶

Lemma 19 Suppose that Condition 5 holds. WhenP`m` = 4, L = 3,m1 = 1,m2 = 2,m3 = 1, we have

1

T

TXt=1

WT,(−t) =WT + op

µ1√T

¶.

Proof. We have

W[1],T = T1/2 1

n

nXi=1

ki,[1,1], W[2],T = T1

n

nXi=1

ki,[2,1]ki,[2,2], W[3],T = T1/2 1

n

nXi=1

ki,[3,1]

and therefore,

WT = T2 1

n3

nXi1=1

nXi2=1

nXi3=1

ki1,[1,1]ki2,[2,1]ki3,[3,1]

43

and

1

T

TXt=1

WT,(−t)

=1

T (T − 1)21

n3

nXi1=1

nXi2=1

nXi3=1

TXt=1

¡Tki1,[1,1] − k[1,1] (xi1,t)

¢ ¡Tki2,[2,1] − k[2,1] (xi2,t)

¢× ¡Tki2,[2,2] − k[2,2] (xi2,t)¢ ¡Tki3,[3,1] − k[3,1] (xi3,t)¢

=T 2 − 4T(T − 1)2WT

+T

(T − 1)2Ã1

n

nXi1=1

T 1/2ki1,[1,1]

!Ã1

T

TXt=1

Ã1

n

nXi2=1

³T 1/2ki2,[2,1]

´k[2,2] (xi2,t)

!Ã1

n

nXi3=1

k[3,1] (xi3,t)

!!

+T

(T − 1)2Ã1

n

nXi1=1

T 1/2ki1,[1,1]

!Ã1

T

TXt=1

Ã1

n

nXi2=1

³T 1/2ki2,[2,2]

´k[2,1] (xi2,t)

!Ã1

n

nXi3=1

k[3,1] (xi3,t)

!!

+T

(T − 1)2Ã1

n

nXi1=1

T 1/2ki1,[1,1]

!1

n

nXi2=1

ÃÃ1

T

TXt=1

k[2,1] (xi2,t) k[2,2] (xi2,t)

!!Ã1

n

nXi3=1

³T 1/2ki3,[3,1]

´!

+T

(T − 1)2Ã1

n

nXi2=1

³T 1/2ki2,[2,1]

´³T 1/2ki2,[2,2]

´!Ã 1T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t)

!Ã1

n

nXi3=1

k[3,1] (xi3,t)

!!

+T

(T − 1)21

n

nXi2=1

³T 1/2ki2,[2,1]

´Ã 1T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t)

!k[2,2] (xi2,t)

!Ã1

n

nXi3=1

³T 1/2ki3,[3,1]

´!

+T

(T − 1)21

n

nXi2=1

³T 1/2ki2,[2,2]

´Ã 1T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t)

!k[2,1] (xi2,t)

!Ã1

n

nXi3=1

³T 1/2ki3,[3,1]

´!

− T 1/2

(T − 1)21

n

nXi1=1

³T 1/2ki1,[1,1]

´Ã 1T

TXt=1

Ã1

n

nXi2=1

k[2,1] (xi2,t) k[2,2] (xi2,t)

!Ã1

n

nXi3=1

k[3,1] (xi3,t)

!!

− T 1/2

(T − 1)21

n

nXi2=1

³T 1/2ki2,[2,1]

´Ã 1T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t)

!k[2,2] (xi2,t)

Ã1

n

nXi3=1

k[3,1] (xi3,t)

!!

− T 1/2

(T − 1)21

n

nXi2=1

³T 1/2ki2,[2,2]

´Ã 1T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t)

!k[2,1] (xi2,t)

Ã1

n

nXi3=1

k[3,1] (xi3,t)

!!

− T 1/2

(T − 1)21

n

nXi2=1

Ã1

T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t)

!k[2,1] (xi2,t) k[2,2] (xi2,t)

!Ã1

n

nXi3=1

³T 1/2ki3,[3,1]

´!

+1

(T − 1)2Ã1

T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t)

!Ã1

n

nXi2=1

k[2,1] (xi2,t) k[2,2] (xi2,t)

!Ã1

n

nXi3=1

k[3,1] (xi3,t)

!!(36)

44

We now show that

T

(T − 1)2Ã1

n

nXi1=1

T 1/2ki1,[1,1]

!Ã1

T

TXt=1

Ã1

n

nXi2=1

³T 1/2ki2,[2,1]

´k[2,2] (xi2,t)

!Ã1

n

nXi3=1

k[3,1] (xi3,t)

!!= op

µ1√T

¶T

(T − 1)2Ã1

n

nXi1=1

T 1/2ki1,[1,1]

!Ã1

T

TXt=1

Ã1

n

nXi2=1

³T 1/2ki2,[2,2]

´k[2,1] (xi2,t)

!Ã1

n

nXi3=1

k[3,1] (xi3,t)

!!= op

µ1√T

¶T

(T − 1)2Ã1

n

nXi1=1

T 1/2ki1,[1,1]

!1

n

nXi2=1

ÃÃ1

T

TXt=1

k[2,1] (xi2,t) k[2,2] (xi2,t)

!!Ã1

n

nXi3=1

³T 1/2ki3,[3,1]

´!= op

µ1√T

¶T

(T − 1)2Ã1

n

nXi2=1

³T 1/2ki2,[2,1]

´³T 1/2ki2,[2,2]

´!Ã 1T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t)

!Ã1

n

nXi3=1

k[3,1] (xi3,t)

!!= op

µ1√T

¶T

(T − 1)21

n

nXi2=1

³T 1/2ki2,[2,1]

´Ã 1T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t)

!k[2,2] (xi2,t)

!Ã1

n

nXi3=1

³T 1/2ki3,[3,1]

´!= op

µ1√T

¶T

(T − 1)21

n

nXi2=1

³T 1/2ki2,[2,2]

´Ã 1T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t)

!k[2,1] (xi2,t)

!Ã1

n

nXi3=1

³T 1/2ki3,[3,1]

´!= op

µ1√T

¶(37)

Only the Þrst assertion will be proved. Other can be proved similarly. For this purpose, it suffices to

prove thatÃ1

n

nXi1=1

T 1/2ki1,[1,1]

!Ã1

T

TXt=1

Ã1

n

nXi2=1

³T 1/2ki2,[2,1]

´k[2,2] (xi2,t)

!Ã1

n

nXi3=1

k[3,1] (xi3,t)

!!= Op (1)

Because

1

n

nXi1=1

T 1/2ki1,[1,1] = Op (1) ,

we only need to prove thatÃ1

T

TXt=1

Ã1

n

nXi2=1

³T 1/2ki2,[2,1]

´k[2,2] (xi2,t)

!Ã1

n

nXi3=1

k[3,1] (xi3,t)

!!= Op (1)

Note that

E

"¯¯ 1T

TXt=1

Ã1

n

nXi2=1

³T 1/2ki2,[2,1]

´k[2,2] (xi2,t)

!Ã1

n

nXi3=1

k[3,1] (xi3,t)

!¯¯#

≤ 1

n2T

nXi2=1

nXi3=1

TXt=1

Eh¯T 1/2ki2,[2,1]

¯M (xi2,t)M (xi3,t)

i

≤ 1

n2T

nXi2=1

nXi3=1

TXt=1

µE

·¯T 1/2ki2,[2,1]

¯2¸¶1/2 ³EhM (xi2,t)

2M (xi3,t)

2i´1/2

= O (1) ,


45

We now show that

T 1/2

(T − 1)21

n

nXi1=1

³T 1/2ki1,[1,1]

´Ã 1T

TXt=1

Ã1

n

nXi2=1

k[2,1] (xi2,t) k[2,2] (xi2,t)

!Ã1

n

nXi3=1

k[3,1] (xi3,t)

!!= op

µ1√T

¶T 1/2

(T − 1)21

n

nXi2=1

³T 1/2ki2,[2,1]

´Ã 1T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t)

!k[2,2] (xi2,t)

Ã1

n

nXi3=1

k[3,1] (xi3,t)

!!= op

µ1√T

¶T 1/2

(T − 1)21

n

nXi2=1

³T 1/2ki2,[2,2]

´Ã 1T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t)

!k[2,1] (xi2,t)

Ã1

n

nXi3=1

k[3,1] (xi3,t)

!!= op

µ1√T

¶T 1/2

(T − 1)21

n

nXi2=1

Ã1

T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t)

!k[2,1] (xi2,t) k[2,2] (xi2,t)

!Ã1

n

nXi3=1

³T 1/2ki3,[3,1]

´!= op

µ1√T

¶(38)

Only the Þrst assertion will be proved. Other can be proved similarly. For this purpose, it suffices to

prove that

1

n

nXi1=1

³T 1/2ki1,[1,1]

´Ã 1T

TXt=1

Ã1

n

nXi2=1

k[2,1] (xi2,t) k[2,2] (xi2,t)

!Ã1

n

nXi3=1

k[3,1] (xi3,t)

!!= Op (1)

But it follows from

1

n

nXi1=1

³T 1/2ki1,[1,1]

´= Op (1) ,

and ¯¯ 1T

TXt=1

Ã1

n

nXi2=1

k[2,1] (xi2,t) k[2,2] (xi2,t)

!Ã1

n

nXi3=1

k[3,1] (xi3,t)

!¯¯

≤ 1

n2T

TXt=1

nXi2=1

nXi3=1

M (xi2,t)2M (xi3,t)

= Op (1)

We now show that

1

(T − 1)2Ã1

T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t)

!Ã1

n

nXi2=1

k[2,1] (xi2,t) k[2,2] (xi2,t)

!Ã1

n

nXi3=1

k[3,1] (xi3,t)

!!= op

µ1√T

¶(39)

For this purpose, it suffices to prove that

1

T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t)

!Ã1

n

nXi2=1

k[2,1] (xi2,t) k[2,2] (xi2,t)

!Ã1

n

nXi3=1

k[3,1] (xi3,t)

!= Op (1)

This follows because¯¯ 1T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t)

!Ã1

n

nXi2=1

k[2,1] (xi2,t) k[2,2] (xi2,t)

!Ã1

n

nXi3=1

k[3,1] (xi3,t)

!¯¯

≤ 1

n2T

TXt=1

nXi2=1

nXi3=1

M (xi1,t)M (xi2,t)M (xi3,t)

= Op (1)

46

We now get back to the proof of Lemma 19. Using equations (36) and (37), (38), (39), we conclude

1

T

TXt=1

WT,(−t) =WT + op

µ1√T

¶


1

T

TXt=1

WT,(−t) =WT + op

µ1√T

¶.

Proof. We have

W[1],T = T1

n

nXi=1

ki,[1,1]ki,[1,2], W[2],T = T1

n

nXi=1

ki,[2,1]ki,[2,2]

and therefore,

WT = T2 1

n3

nXi1=1

nXi2=1

ki1,[1,1]ki1,[1,2]ki2,[2,1]ki3,[3,1]

47

and

1

T

TXt=1

WT,(−t)

=1

T (T − 1)21

n2

nXi1=1

nXi2=1

TXt=1

¡Tki1,[1,1] − k[1,1] (xi1,t)

¢ ¡Tki1,[1,2] − k[1,2] (xi1,t)

¢× ¡Tki2,[2,1] − k[2,1] (xi2,t)¢ ¡Tki2,[2,2] − k[2,2] (xi2,t)¢

=T 2 − 4T(T − 1)2WT

+T

(T − 1)2Ã1

n

nXi1=1

³T 1/2ki1,[1,1]

´³T 1/2ki1,[1,2]

´!Ã 1n

nXi2=1

1

T

TXt=1

k[2,1] (xi2,t) k[2,2] (xi2,t)

!

+T

(T − 1)21

n2

nXi1=1

nXi2=1

Ã³T 1/2ki1,[1,1]

´³T 1/2ki2,[2,1]

´Ã 1T

TXt=1

k[1,2] (xi1,t) k[2,2] (xi2,t)

!!

+T

(T − 1)21

n2

nXi1=1

nXi2=1

Ã³T 1/2ki1,[1,1]

´³T 1/2ki2,[2,2]

´Ã 1T

TXt=1

k[1,2] (xi1,t) k[2,1] (xi2,t)

!!

+T

(T − 1)21

n2

nXi1=1

nXi2=1

Ã³T 1/2ki1,[1,2]

´³T 1/2ki2,[2,1]

´Ã 1T

TXt=1

k[1,1] (xi1,t) k[2,2] (xi2,t)

!!

+T

(T − 1)21

n2

nXi1=1

nXi2=1

Ã³T 1/2ki1,[1,2]

´³T 1/2ki2,[2,2]

´Ã 1T

TXt=1

k[1,1] (xi1,t) k[2,1] (xi2,t)

!!

+T

(T − 1)21

n2

nXi1=1

nXi2=1

Ã³T 1/2ki2,[2,1]

´³T 1/2ki2,[2,2]

´Ã 1T

TXt=1

k[1,1] (xi1,t) k[1,2] (xi1,t)

!!

− T 1/2

(T − 1)21

n

nXi1=1

³T 1/2ki1,[1,1]

´Ã 1T

TXt=1

k[1,2] (xi1,t)

Ã1

n

nXi2=1

k[2,1] (xi2,t) k[2,2] (xi2,t)

!!

− T 1/2

(T − 1)21

n

nXi1=1

³T 1/2ki1,[1,2]

´Ã 1T

TXt=1

k[1,1] (xi1,t)

Ã1

n

nXi2=1

k[2,1] (xi2,t) k[2,2] (xi2,t)

!!

− T 1/2

(T − 1)21

n

nXi2=1

³T 1/2ki2,[2,1]

´Ã 1T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t) k[1,2] (xi1,t)

!k[2,2] (xi2,t)

!

− T 1/2

(T − 1)21

n

nXi2=1

³T 1/2ki2,[2,2]

´Ã 1T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t) k[1,2] (xi1,t)

!k[2,1] (xi2,t)

!

+1

(T − 1)21

n2

nXi1=1

nXi2=1

Ã1

T

TXt=1

k[1,1] (xi1,t) k[1,2] (xi1,t) k[2,1] (xi2,t) k[2,2] (xi2,t)

!(40)

48

We now show that

T

(T − 1)2Ã1

n

nXi1=1

³T 1/2ki1,[1,1]

´³T 1/2ki1,[1,2]

´!Ã 1n

nXi2=1

1

T

TXt=1

k[2,1] (xi2,t) k[2,2] (xi2,t)

!= op

µ1√T

¶T

(T − 1)21

n2

nXi1=1

nXi2=1

Ã³T 1/2ki1,[1,1]

´³T 1/2ki2,[2,1]

´Ã 1T

TXt=1

k[1,2] (xi1,t) k[2,2] (xi2,t)

!!= op

µ1√T

¶T

(T − 1)21

n2

nXi1=1

nXi2=1

Ã³T 1/2ki1,[1,1]

´³T 1/2ki2,[2,2]

´Ã 1T

TXt=1

k[1,2] (xi1,t) k[2,1] (xi2,t)

!!= op

µ1√T

¶T

(T − 1)21

n2

nXi1=1

nXi2=1

Ã³T 1/2ki1,[1,2]

´³T 1/2ki2,[2,1]

´Ã 1T

TXt=1

k[1,1] (xi1,t) k[2,2] (xi2,t)

!!= op

µ1√T

¶T

(T − 1)21

n2

nXi1=1

nXi2=1

Ã³T 1/2ki1,[1,2]

´³T 1/2ki2,[2,2]

´Ã 1T

TXt=1

k[1,1] (xi1,t) k[2,1] (xi2,t)

!!= op

µ1√T

¶T

(T − 1)21

n2

nXi1=1

nXi2=1

Ã³T 1/2ki2,[2,1]

´³T 1/2ki2,[2,2]

´Ã 1T

TXt=1

k[1,1] (xi1,t) k[1,2] (xi1,t)

!!= op

µ1√T

¶(41)

The Þrst assertion follows from Lemma 13. For the remaining assertions, only the second will be estab-

lished. Others can be proved similarly. In order to prove the Þrst assertion, it suffices to note that

E

"¯¯ 1n2

nXi1=1

nXi2=1

Ã³T 1/2ki1,[1,1]

´³T 1/2ki2,[2,1]

´Ã 1T

TXt=1

k[1,2] (xi1,t) k[2,2] (xi2,t)

!!¯¯#

≤ 1

n2T

nXi1=1

nXi2=1

TXt=1

Eh¯T 1/2ki1,[1,1]

¯ ¯T 1/2ki2,[2,1]


i

≤ 1

n2T

nXi1=1

nXi2=1

TXt=1

µE

·³¯T 1/2ki1,[1,1]

¯ ¯T 1/2ki2,[2,1]

¯´2¸¶1/2 ³EhM (xi1,t)

2M (xi2,t)

2i´1/2

≤ 1

n2T

nXi1=1

nXi2=1

TXt=1

µE¯T 1/2ki1,[1,1]

¯4¶1/4µE¯T 1/2ki2,[2,1]

¯4¶1/4 ³EhM (xi1,t)

2M (xi2,t)

2i´1/2

≤ 1

n2T

nXi1=1

nXi2=1

TXt=1

³3max

iEhM (xi,t)

4i´1/4 ³

3maxiEhM (xi,t)

4i´1/4 ³

maxiEhM (xi,t)

4i´1/2

= O (1)

We now show that

T 1/2

(T − 1)21

n

nXi1=1

³T 1/2ki1,[1,1]

´Ã 1T

TXt=1

k[1,2] (xi1,t)

Ã1

n

nXi2=1

k[2,1] (xi2,t) k[2,2] (xi2,t)

!!= op

µ1√T

¶T 1/2

(T − 1)21

n

nXi1=1

³T 1/2ki1,[1,2]

´Ã 1T

TXt=1

k[1,1] (xi1,t)

Ã1

n

nXi2=1

k[2,1] (xi2,t) k[2,2] (xi2,t)

!!= op

µ1√T

¶T 1/2

(T − 1)21

n

nXi2=1

³T 1/2ki2,[2,1]

´Ã 1T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t) k[1,2] (xi1,t)

!k[2,2] (xi2,t)

!= op

µ1√T

¶T 1/2

(T − 1)21

n

nXi2=1

³T 1/2ki2,[2,2]

´Ã 1T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t) k[1,2] (xi1,t)

!k[2,1] (xi2,t)

!= op

µ1√T

¶(42)

49

Only the Þrst assertion will be proved. Others can be established similarly. In order to prove the Þrst

assertion, it suffices to note that

E

"¯¯ 1n

nXi1=1

³T 1/2ki1,[1,1]

´Ã 1T

TXt=1

k[1,2] (xi1,t)

Ã1

n

nXi2=1

k[2,1] (xi2,t) k[2,2] (xi2,t)

!!¯¯#

≤ 1

n2T

nXi1=1

nXi2=1

TXt=1

Eh¯T 1/2ki1,[1,1]


2i

=1

n2T

nXi1=1

nXi2=1

TXt=1

µE

·¯T 1/2ki1,[1,1]

¯2¸¶1/2 ³EhM (xi1,t)

2M (xi2,t)4i´1/2

= O (1)

We now show that

1

(T − 1)21

n2

nXi1=1

nXi2=1

Ã1

T

TXt=1


!= op

µ1√T

¶(43)

This follows easily from

E

"¯¯ 1n2

nXi1=1

nXi2=1

Ã1

T

TXt=1


!¯¯#

≤ 1

n2T

nXi1=1

nXi2=1

TXt=1

EhM (xi1,t)

2M (xi2,t)

2i

= O (1)

We now get back to the proof of Lemma 20. Using equations (40) and (41), (42), (43), we conclude

1

T

TXt=1

WT,(−t) =WT + op

µ1√T

¶

Lemma 21 Suppose that Condition 5 holds. WhenP`m` = 4, L = 4,m1 = 1,m2 = 1,m3 = 1,m4 = 1,

we have

1

T

TXt=1

WT,(−t) =WT + op

µ1√T

¶.

Proof. We have

W[1],T = T 1/21

n

nXi=1

ki,[1,1], W[2],T = T1/2 1

n

nXi=1

ki,[2,1],

W[3],T = T 1/21

n

nXi=1

ki,[3,1], W[4],T = T1/2 1

n

nXi=1

ki,[4,1]

and therefore,

WT = T2 1

n4

nXi1=1

nXi2=1

nXi3=1

nXi4=1

ki1,[1,1]ki2,[2,1]ki3,[3,1]ki4,[4,1]

50

and

1

T

TXt=1

WT,(−t)

=1

T (T − 1)21

n4

nXi1=1

nXi2=1

nXi3=1

nXi3=1

TXt=1

¡Tki1,[1,1] − k[1,1] (xi1,t)

¢ ¡Tki2,[2,1] − k[2,1] (xi2,t)

¢× ¡Tki3,[3,1] − k[3,1] (xi3,t)¢ ¡Tki4,[4,1] − k[4,1] (xi4,t)¢

=T 2 − 4T(T − 1)2WT

+T

(T − 1)2Ã1

n

nXi1=1

T 1/2ki1,[1,1]

!Ã1

n

nXi2=1

T 1/2ki2,[2,1]

!Ã1

T

TXt=1

Ã1

n

nXi3=1

k[3,1] (xi3,t)

!Ã1

n

nXi4=1

k[4,1] (xi4,t)

!!

+T

(T − 1)2Ã1

n

nXi1=1

T 1/2ki1,[1,1]

!Ã1

n

nXi3=1

T 1/2ki3,[3,1]

!Ã1

T

TXt=1

Ã1

n

nXi2=1

k[2,1] (xi2,t)

!Ã1

n

nXi4=1

k[4,1] (xi4,t)

!!

+T

(T − 1)2Ã1

n

nXi1=1

T 1/2ki1,[1,1]

!Ã1

n

nXi4=1

T 1/2ki4,[4,1]

!Ã1

T

TXt=1

Ã1

n

nXi2=1

k[2,1] (xi2,t)

!Ã1

n

nXi3=1

k[3,1] (xi3,t)

!!

+T

(T − 1)2Ã1

n

nXi2=1

T 1/2ki2,[2,1]

!Ã1

n

nXi3=1

T 1/2ki3,[3,1]

!Ã1

T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t)

!Ã1

n

nXi4=1

k[4,1] (xi4,t)

!!

+T

(T − 1)2Ã1

n

nXi2=1

T 1/2ki2,[2,1]

!Ã1

n

nXi4=1

T 1/2ki4,[4,1]

!Ã1

T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t)

!Ã1

n

nXi3=1

k[3,1] (xi3,t)

!!

+T

(T − 1)2Ã1

n

nXi3=1

T 1/2ki3,[3,1]

!Ã1

n

nXi4=1

T 1/2ki4,[4,1]

!Ã1

T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t)

!Ã1

n

nXi2=1

k[2,1] (xi2,t)

!!

− T 1/2

(T − 1)2Ã1

n

nXi1=1

T 1/2ki1,[1,1]

!Ã1

T

TXt=1

Ã1

n

nXi2=1

k[2,1] (xi2,t)

!Ã1

n

nXi3=1

k[3,1] (xi3,t)

!Ã1

n

nXi4=1

k[4,1] (xi4,t)

!!

− T 1/2

(T − 1)2Ã1

n

nXi2=1

T 1/2ki2,[2,1]

!Ã1

T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t)

!Ã1

n

nXi3=1

k[3,1] (xi3,t)

!Ã1

n

nXi4=1

k[4,1] (xi4,t)

!!

− T 1/2

(T − 1)2Ã1

n

nXi3=1

T 1/2ki3,[3,1]

!Ã1

T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t)

!Ã1

n

nXi2=1

k[2,1] (xi2,t)

!Ã1

n

nXi4=1

k[4,1] (xi4,t)

!!

− T 1/2

(T − 1)2Ã1

n

nXi4=1

T 1/2ki4,[4,1]

!Ã1

T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t)

!Ã1

n

nXi2=1

k[2,1] (xi2,t)

!Ã1

n

nXi3=1

k[3,1] (xi3,t)

!!

+1

(T − 1)2Ã1

T

TXt=1

Ã1

n

nXi1=1

k[1,1] (xi1,t)

!Ã1

n

nXi2=1

k[2,1] (xi2,t)

!Ã1

n

nXi3=1

k[3,1] (xi3,t)

!Ã1

n

nXi4=1

k[4,1] (xi4,t)

!!(44)

Using equation (44) and lemmas in previous subsections, we can conclude that

1

T

TXt=1

WT,(−t) =WT + op

µ1√T

¶

51

G Derivatives of θ when dim (θ) = 1

Recall that

hi (·, ²) ≡ Ui (·; θ (F (²)) ,αi (θ (F (²)) , Fi (²)))

The Þrst order condition may be written as

0 =1

n

nXi=1

Zhi (·, ²) dFi (²)

Differentiating repeatedly with respect to ², we obtain

0 =1

n

nXi=1

Zdhi (·, ²)d²

dFi (²) +1

n

nXi=1

Zhi (·, ²) d∆iT (45)

0 =1

n

nXi=1

Zd2hi (·, ²)d²2

dFi (²) + 21

n

nXi=1

Zdhi (·, ²)d²

d∆iT (46)

0 =1

n

nXi=1

Zd3hi (·, ²)d²3

dFi (²) + 31

n

nXi=1

Zd2hi (·, ²)d²2

d∆iT (47)

0 =1

n

nXi=1

Zd4hi (·, ²)d²4

dFi (²) + 41

n

nXi=1

Zd3hi (·, ²)d²3

d∆iT (48)

0 =1

n

nXi=1

Zd5hi (·, ²)d²5

dFi (²) + 51

n

nXi=1

Zd4hi (·, ²)d²4

d∆iT (49)

G.1 θ² (0)

Evaluating (45) at ² = 0, and noting that E [Uαii ] = 0, we obtain

θ² (0) =1n

Pni=1

RUid∆iT

1n

Pni=1E [U

2i ]

(50)

We therefore have

√nT

1√Tθ² (0) =

1√n√T

Pni=1

PTt=1 Uit

1n

Pni=1E [U

2i ]

G.2 αθi and α²i

In the ith stratum, αi (θ, Fi (²)) solves the estimating equationZVi [θ,αi (θ, Fi (²))] dFi (²) = 0 (51)

Differentiating the LHS with respect to θ and ², we obtain

0 =

ZV θi dFi (²) + α

θi

ZV αii dFi (²)

0 = α²i

ZV αii dFi (²) +

ZVid∆iT

52

where ∆iT ≡ dd²Fi (²) =

√T³ bFi − Fi´. Equating these equations to zero and solving for derivatives of

αi evaluated at ² = 0 gives

αθi = −E£V θi¤

E [V αii ]=E£V θi¤

E [V 2i ]= O (1) (52)

α²i = −PTt=1 Vit√TE [V αii ]

=T−1/2

PTt=1 Vit

E [V 2i ]= Op (1) (53)

G.3 θ²² (0)

Evaluating each term of (46) at ² = 0, we obtain

1

n

nXi=1

Zd2hi (·, ²)d²2

dFi (²)

¯¯²=0

= θ²² (0)1

n

nXi=1

E£Uθi¤

+2θ² (0)

Ã1

n

nXi=1

EhUθαii

iα²i +

1

n

nXi=1

E [Uαiαii ]αθiα²i

!

+(θ² (0))2Ã1

n

nXi=1

E£Uθθi

¤+ 2

1

n

nXi=1

EhUθαii

iαθi +

1

n

nXi=1

E [Uαiαii ]¡αθi¢2!

+1

n

nXi=1

E [Uαiαii ] (α²i)2

and

21

n

nXi=1

Zdhi (·, ²)d²

d∆iT

¯¯²=0

= 2θ² (0)1

n

nXi=1

ZUθi d∆iT + 2θ

² (0)1

n

nXi=1

αθi

ZUαii d∆iT

+21

n

nXi=1

α²i

ZUαii d∆iT


θ²² (0)1

n

nXi=1

E£U2i¤=1

n

nXi=1

E [Uαiαii ] (α²i)2 + 2

1

n

nXi=1

α²i

ZUαii d∆iT

+ 2θ² (0)

Ã1

n

nXi=1

EhUθαii

iα²i +

1

n

nXi=1

E [Uαiαii ]αθiα²i +

1

n

nXi=1

ZUθi d∆iT +

1

n

nXi=1

αθi

ZUαii d∆iT

!

+ (θ² (0))2Ã1

n

nXi=1

E£Uθθi

¤+ 2

1

n

nXi=1

EhUθαii

iαθi +

1

n

nXi=1


(54)

G.4 αθθi , αθ²i , and α

²²i

Evaluating the second order derivatives³∂2

∂θ2, ∂2

∂θ∂² ,∂2

∂²2

´of (51) at ² = 0, we obtain

αθθi =E£V θθi

¤E [V 2i ]

+ 2αθi

EhV θαii

iE [V 2i ]

+¡αθi¢2 E [V αiαii ]

E [V 2i ]= O (1) , (55)

αθ²i = α²i

EhV θαii

iE [V 2i ]

+ αθiα²i

E [V αiαii ]

E [V 2i ]+

RV θi d∆iTE [V 2i ]

+ αθi

RV αii d∆iTE [V 2i ]

= Op (1) , (56)

53

and

α²²i = (α²i)2 E [V

αiαii ]

E [V 2i ]+ 2α²i


= Op (1) . (57)

G.5 θ²²² (0)

Evaluating each term of (47) at ² = 0, it can be concluded that θ²²² (0) 1nPni=1E

£U2i¤can be expressed

as a sum of normalized V-statistics of order 3.1 Condition 4 along with Lemma 13 imply that |θ²²² (0)| =Op (1).

G.6 θ²²²² (0)

Evaluating each term of (48) at ² = 0, it can be concluded that θ²²²² (0) 1nPni=1E

£U2i¤can be expressed

as a sum of normalized V-statistics of order 4.2 Condition 4 along with Lemma 13 imply that |θ²²²² (0)| =Op (1).

H JackkniÞng bθH.1 T

³θ² (0)−

qTT−1

1T

PTt=1 θ

²(t) (0)

´and

qTT−1

1T

PTt=1 θ

²(t) (0)

Because

θ² (0) =1n

Pni=1

RUid∆iT

1n

Pni=1E [U

2i ]

,

we can use Lemma 14 and obtain

1

T

TXt=1

θ²(t) (0) =

rT − 1T

θ² (0)

It therefore follows that

T

Ã√nθ² (0)−

rT

T − 11

T

TXt=1

√nθ²(t) (0)

!+

rT

T − 11

T

TXt=1

√nθ²(t) (0) =

√nθ² (0) (58)

H.2 T³p

nTθ²² (0)−

qTT−1

1T

PTt=1

pnT−1θ

²²(t) (0)

´and

qTT−1

1T

PTt=1

pnT−1θ

²²(t) (0)

Write

θ²² (0)1

n

nXi=1

E£U2i¤=1

n

nXi=1

W1,i,T +W2,T ,

where

W1,i,T ≡ E [Uαiαii ] (α²i)2 + 2α²i

ZUαii d∆iT

= 2

"1√T

TXt=1

VitE [V 2i ]

#"1√T

TXt=1

µUαii +

E [Uαiαii ]

2E [V 2i ]Vit

¶#1For exact expression of the terms in (47), see Supplementary Appendix, which is available upon request.2For exact expression of the terms in (48), see Supplementary Appendix, which is available upon request.

54

and

W2,T ≡ 2θ² (0)

Ã1

n

nXi=1

EhUθαii

iα²i +

1

n

nXi=1

E [Uαiαii ]αθiα²i +

1

n

nXi=1

ZUθi d∆iT +

1

n

nXi=1

αθi

ZUαii d∆iT

!

+(θ² (0))2Ã1

n

nXi=1

E£Uθθi

¤+ 2

1

n

nXi=1

EhUθαii

iαθi +

1

n

nXi=1


= Op

µ1

n

¶By Lemma 14, we obtain

1

T

TXt=1

Ã1

n

nXi=1

W1,i,T,(−t)

!=T − 2T − 1

1

n

nXi=1

W1,i,T +1

T (T − 1)1

n

nXi=1

TXt=1

2Vit

E [V 2i ]

µUαii +

E [Uαiαii ]

2E [V 2i ]Vit

¶By Lemma 15, we obtain

1

T

TXt=1

W2,T,(−t) =T − 2T − 1W2,T +Op

µ1

nT

¶It therefore follows thatÃ

1

n

nXi=1

E£U2i¤! · 1

2T

Ãrn

Tθ²² (0)−

rT

T − 11

T

TXt=1

rn

T − 1θ²²(t) (0)

!

+

Ã1

n

nXi=1

E£U2i¤! · 1

2

rT

T − 11

T

TXt=1

rn

T − 1θ²²(t) (0)

=

Ã1

n

nXi=1

E£U2i¤! · √nT

2

Ãθ²² (0)− 1

T

TXt=1

θ²²(t) (0)

!

=

√nT

2

Ã1

n

nXi=1

W1,i,T

!µ1− T − 2

T − 1¶

−√nT

2

Ã1

T (T − 1)1

n

nXi=1

TXt=1

2Vit

E [V 2i ]

µUαii +

E [Uαiαii ]

2E [V 2i ]Vit

¶!

+

√nT

2

µ1− T − 2

T − 1¶W2,T +Op

µ1√nT

¶=

1

2

√nT

1

T − 11

n

nXi=1

W1,i,T

−12

√nT

1

T − 11

nT

nXi=1

TXt=1

2Vit

E [V 2i ]

µUαii +

E [Uαiαii ]

2E [V 2i ]Vit

¶+1

2

√nT

1

T − 1W2,T +Op

µ1√nT

¶= op (1) (59)

55

H.3 T³p

nT

1√Tθ²²² (0)−

qTT−1

1T

PTt=1

pnT−1

1√T−1θ

²²²(t) (0)

´and

qTT−1

1T

PTt=1

pnT−1

1√T−1θ

²²²(t) (0)

By Lemmas 14, 16, and 17, we obtainrT

T − 11

T

TXt=1

rn

T − 11√T − 1θ

²²²(t) (0) =

√nTq

(T − 1)3θ²²² (0) +

√nTq

(T − 1)3op

µ1√T

¶= op (1) (60)

and

T

Ãrn

T

1√Tθ²²² (0)−

rT

T − 11

T

TXt=1

rn

T − 11√T − 1θ

²²²(t) (0)

!

= T

r n

T 2−

√nTq

(T − 1)3

θ²²² (0) + T√nTq

(T − 1)3op

µ1√T

¶= op (1) (61)

H.4 T³p

nT1Tθ²²²² (0)−

qTT−1

1T

PTt=1

pnT−1

1T−1θ

²²²²(t) (0)

´and

qTT−1

1T

PTt=1

pnT−1

1T−1θ

²²²²(t) (0)

By Lemmas 14, 18, 19, 20, and 21, we obtainrT

T − 11

T

TXt=1

rn

T − 11

T − 1θ²²²²(t) (0) =

√nTq

(T − 1)4θ²²² (0) +

√nTq

(T − 1)4op

µ1√T

¶= op (1) (62)

and

T

Ãrn

T

1

Tθ²²²² (0)−

rT

T − 11

T

TXt=1

rn

T − 11

T − 1θ²²²²(t) (0)

!

= T

r n

T 3−

√nTq

(T − 1)4

θ²²² (0) + T√nTq

(T − 1)4op

µ1√T

¶= op (1) (63)

I Proof of Theorem 3

Using the same argument as in the proof of Theorem 6, it can be shown that

Pr

·¯1√Tθ²²²²² (e²)¯ ≥ η¸ = oµ 1

T

¶.

Therefore,

T · 1120

rn

T

1

T√Tθ²²²²² (e²) = op (1)

56

and¯¯T · 1T

TXt=1

1

120

rn

T − 11

(T − 1)√T − 1θ²²²²²(t)

¡e²(t)¢¯¯ ≤ 1

120

rn

T − 1T

(T − 1)√T − 1 maxt¯θ²²²²²(t)

¡e²(t)¢¯= O

µmaxt

¯1√Tθ²²²²²(t)

¡e²(t)¢¯¶= op (1) ,

where θ²²²²²(t) denotes the delete-t version of θ²²²²²(t) , and the second equality is based on

Pr

·maxt

¯1√Tθ²²²²²(t)

¡e²(t)¢¯ ≥ η¸ ≤Xt

Pr

·¯1√Tθ²²²²²(t)

¡e²(t)¢¯ ≥ η¸ = T · oµ 1T

¶= o (1) .

We may therefore write

√nT³eθ − θ0´ = T

Ã√nT³bθ − θ0´−r T

T − 11

T

TXt=1

pn (T − 1)

³bθ(t) − θ0´!

+

rT

T − 11

T

TXt=1

pn (T − 1)

³bθ(t) − θ0´or

√nT³eθ − θ0´ = T

Ã√nθ² (0)−

rT

T − 11

T

TXt=1

√nθ²(t) (0)

!

+1

2T

Ãrn

Tθ²² (0)−

rT

T − 11

T

TXt=1

rn

T − 1θ²²(t) (0)

!

+1

6T

Ãrn

T

1√Tθ²²² (0)−

rT

T − 11

T

TXt=1

rn

T − 11√T − 1θ

²²²(t) (0)

!

+1

24T

Ãrn

T

1

Tθ²²²² (0)−

rT

T − 11

T

TXt=1

rn

T − 11

T − 1θ²²²²(t) (0)

!

+

rT

T − 11

T

TXt=1

√nθ²(t) (0)

+1

2

rT

T − 11

T

TXt=1

rn

T − 1θ²²(t) (0)

+1

6

rT

T − 11

T

TXt=1

rn

T − 11√T − 1θ

²²²(t) (0)

+1

24

rT

T − 11

T

TXt=1

rn

T − 11

T − 1θ²²²²(t) (0)

+op (1)

57

Using equations (58), (59), (60), (61), (62), and (63) in Section H, we have

T

Ã√nθ² (0)−

rT

T − 11

T

TXt=1

√nθ²(t) (0)

!+

rT

T − 11

T

TXt=1

√nθ²(t) (0) =

√nθ² (0)

1

2T

Ãrn

Tθ²² (0)−

rT

T − 11

T

TXt=1

rn

T − 1θ²²(t) (0)

!+1

2

rT

T − 11

T

TXt=1

rn

T − 1θ²²(t) (0) = op (1)

T

Ãrn

T

1√Tθ²²² (0)−

rT

T − 11

T

TXt=1

rn

T − 11√T − 1θ

²²²(t) (0)

!+

rT

T − 11

T

TXt=1

rn

T − 11√T − 1θ

²²²(t) (0) = op (1)

T

Ãrn

T

1

Tθ²²²² (0)−

rT

T − 11

T

TXt=1

rn

T − 11

T − 1θ²²²²(t) (0)

!+

rT

T − 11

T

TXt=1

rn

T − 11

T − 1θ²²²²(t) (0) = op (1)

58

Supplementary Appendix for Jackknife and Analytical BiasReduction for Nonlinear Panel Models available upon request

Jinyong Hahn Whitney NeweyBrown University MIT

A Derivatives of h

A.1 First order derivative of hdhi (·, ²)d²

= θ²¡Uθi + U

αii α

θi

¢+ Uαii α

²i

A.2 Second order derivative of h

d2hi (·, ²)d²2

= θ²²¡Uθi + U

αii α

θi

¢+2θ²

³Uθαii α²i + U

αiαii αθiα

²i + U

αii α

θ²i

´+(θ²)2

³Uθθi + 2Uθαii αθi + U

αiαii

¡αθi¢2+ Uαii α

θθi

´+Uαiαii (α²i)

2 + Uαii α²²i

A.3 Third order derivative of h

A.3.1 Coefficient of θ²²²

Uαii αθi + U

θi

A.3.2 Coefficient of θ²²

3Uθαii α²i + 3Uαiαii αθiα

²i + 3U

αii α

θ²i

A.3.3 Coefficient of θ²²θ²

3Uθθi + 3Uαiαii

¡αθi¢2+ 3Uαii α

θθi + 6Uθαii αθi

A.3.4 Coefficient of θ²

3Uαii αθ²²i + 3Uθαiαii (α²i)

2 + 3Uθαii α²²i + 6Uαiαii αθ²i α

²i + 3U

αiαii α²²i α

θi + 3U

αiαiαii αθi (α

²i)2

A.3.5 Coefficient of (θ²)2

3Uαii αθθ²i +3Uθθαii α²i +6U

θαii αθ²i +6U

θαiαii αθiα

²i +3U

αiαiαii

¡αθi¢2α²i +3U

αiαii αθθi α

²i +6U

αiαii αθiα

θ²i


Uαii αθθθi + Uαiαiαii

¡αθi¢3+ 3Uθθαii αθi + 3U

θαiαii

¡αθi¢2+ 3Uθαii αθθi + 3Uαiαii αθθi α

θi + U

θθθi

A.3.7 Constant term

Uαii α²²²i + 3Uαiαii α²²i α

²i + U

αiαiαii (α²i)

3

1

A.4 Fourth order derivative of h

A.4.1 Coefficient of θ²²²²

Uαii αθi + U

θi


4Uαiαii αθiα²i + 4U

θαii α²i + 4U

αii α

θ²i

A.4.3 Coefficient of θ²²²θ²

4Uθθi + 4Uαiαii

¡αθi¢2+ 8Uθαii αθi + 4U

αii α

θθi


6Uθαiαii (α²i)2+ 6Uθαii α²²i + 6U

αii α

θ²²i + 12Uαiαii αθ²i α

²i + 6U


²i)2+ 6Uαiαii αθiα

²²i


12Uαiαii αθθi α²i + 24U

αiαii αθiα

θ²i + 24U

θαiαii αθiα

²i + 12U

αiαiαii

¡αθi¢2α²i + 24U

θαii αθ²i + 12U

θθαii α²i

+12Uαii αθθ²i

A.4.6 Coefficient of θ²² (θ²)2

6Uθθθi + 18Uαiαii αθθi αθi + 6U

αiαiαii

¡αθi¢3+ 18Uθαii αθθi + 18U

θθαii αθi + 18U

θαiαii

¡αθi¢2

+6Uαii αθθθi

A.4.7 Coefficient of (θ²²)2

3Uαii αθθi + 6U

θαii αθi + 3U

αiαii

¡αθi¢2+ 3Uθθi


4Uθαii α²²²i + 4Uθαiαiαii (α²i)3 + 4Uαii α

θ²²²i + 12Uαiαiαii αθiα

²²i α

²i + 12U

αiαii αθ²²i α²i + 4U

αiαiαiαii αθi (α

²i)3

+4Uαiαii αθiα²²²i + 12Uαiαiαii αθ²i (α

²i)2 + 12Uθαiαii α²²i α

²i + 12U

αiαii αθ²i α

²²i


6Uαii αθθ²²i + 6Uθθαiαii (α²i)

2 + 6Uθθαii α²²i + 12Uθαii αθ²²i + 12Uαiαii

¡αθ²i¢2+ 24Uαiαiαii αθiα

θ²i α

²i

+12Uαiαii αθθ²i α²i + 12Uαiαii αθ²²i αθi + 6U

αiαiαii

¡αθi¢2α²²i + 6U

αiαiαiαii

¡αθi¢2(α²i)

2

+6Uαiαiαii αθθi (α²i)2 + 12Uθαiαiαii αθi (α

²i)2 + 12Uθαiαii αθiα

²²i + 24U

θαiαii αθ²i α

²i + 6U

αiαii αθθi α

²²i

2


12Uθθαii αθ²i + 4Uθθθαii α²i + 12U

θαii αθθ²i + 4Uαii α

θθθ²i + 12Uαiαiαii αθθi α

θiα

²i + 4U

αiαiαiαii

¡αθi¢3α²i

+4Uαiαii αθθθi α²i + 12Uαiαii αθθi α

θ²i + 12U

αiαii αθθαii αθi + 12U

αiαiαii

¡αθi¢2αθ²i

+12Uθθαiαii αθiα²i + 24U


θi + 12U

θαiαiαii


θαiαii αθθi α

²i


Uαiαiαiαii

¡αθi¢4+ 6Uθθαii αθθi + 6U

θθαiαii

¡αθi¢2+ 4Uθθθαii αθi + 4U

θαii αθθθi + 4Uθαiαiαii

¡αθi¢3

+Uαii αθθθθi + 3Uαiαii

¡αθθi¢2+ 4Uαiαii αθθθi αθi + 6U

αiαiαii αθθi

¡αθi¢2+ 12Uθαiαii αθθi α

θi + U

θθθθi

A.4.12 Constant term

6Uαiαiαii α²²i (α²i)2 + 4Uαiαii α²²²i α

²i + 3U

αiαii (α²²i )

2 + Uαii α²²²²i + Uαiαiαiαii (α²i)

4

A.5 Fifth Order Derivative of h

A.5.1 Coefficient of θ²²²²²

Uαii αθi + U

θi

A.5.2 Coefficient of θ²²²²

5Uαiαii αθiα²i + 5U

αii α

θ²i + 5U

θαii α²i

A.5.3 Coefficient of θ²²²²θ²

5Uαiαii


θθi + 10U

θαii αθi + 5U

θθi


10Uαiαii αθiα²²i + 20U

αiαii αθ²i α

²i + 10U


²i)2 + 10Uαii α

θ²²i + 10Uθαiαii (α²i)

2 + 10Uθαii α²²i

A.5.5 Coefficient of θ²²²θ²

20Uαiαii αθθi α²i+20U

αiαiαii

¡αθi¢2α²i+40U

αiαii αθ²i α

θi+40U

θαiαii αθiα

²i+20U

αii α

θθ²i +40Uθαii αθ²i +20U

θθαii α²i

A.5.6 Coefficient of θ²²² (θ²)2

10Uαiαiαii

¡αθi¢3+30Uαiαii αθθi α

θi +10U

αii α

θθθi +30Uθθαii αθi +30U

θαii αθθi +30U

θαiαii

¡αθi¢2+10Uθθθi

A.5.7 Coefficient of θ²²²θ²²

10Uαiαii


θθi + 20U

θαii αθi + 10U

θθi

3


30Uαiαii αθ²i α²²i + 30U

αiαiαii αθiα

²²i α

²i + 30U

αiαii αθ²²i α²i + 10U

αiαii α²²²i α

θi + 10U

αiαiαiαii αθi (α

²i)3

+30Uαiαiαii αθ²i (α²i)2 + 10Uαii α

θ²²²i + 30Uθαiαii α²²i α

²i + 10U

θαii α²²²i + 10Uθαiαiαii (α²i)

3


30Uαiαii αθθi α²²i + 60U

αiαii αθθ²i α²i + 60U

αiαii αθ²²i αθi + 30U

αiαiαii

¡αθi¢2α²²i + 120U

αiαiαii αθ²i α

²iαθi

+30Uαiαiαii αθθi (α²i)2 + 60Uαiαii

¡αθ²i¢2+ 120Uθαiαii αθ²i α

²i + 60U

θαiαiαii αθi (α

²i)2 + 60Uθαiαii αθiα

²²i

+30Uαiαiαiαii

¡αθi¢2(α²i)

2 + 30Uαii αθθ²²i + 60Uθαii αθ²²i + 30Uθθαiαii (α²i)

2 + 30Uθθαii α²²i


30Uαiαii αθθθi α²i + 90Uαiαiαii αθ²i

¡αθi¢2+ 90Uαiαii αθθ²i αθi + 90U

αiαiαii αθθi α

θiα

²i

+90Uαiαii αθ²i αθθi + 90U

θαiαii αθθi α

²i + 180U


θi + 90U

θαiαiαii

¡αθi¢2α²i

+30Uαiαiαiαii


θθαiαii αθiα

²i + 30U

αii α

θθθ²i + 30Uθθθαii α²i + 90U

θθαii αθ²i

+90Uθαii αθθ²i


40Uαiαii αθθθi αθi + 30Uαiαii

¡αθθi¢2+ 120Uθαiαii αθθi α

θi + 60U

αiαiαii αθθi

¡αθi¢2

+10Uαiαiαiαii


θθθθi + 60Uθθαii αθθi + 60Uθθαiαii

¡αθi¢2

+40Uθαii αθθθi + 40Uθαiαiαii

¡αθi¢3+ 10Uθθθθi + 40Uθθθαii αθi

A.5.12 Coefficient of (θ²²)2

15Uαii αθθ²i + 15Uαiαii αθθi α

²i + 15U

αiαiαii


αiαii αθ²i α

θi + 30U

θαiαii αθiα

²i

+15Uθθαii α²i + 30Uθαii αθ²i

A.5.13 Coefficient of (θ²²)2 θ²

45Uαiαii αθθi αθi + 15U

αiαiαii


θθθi + 45Uθαii αθθi + 45U

θαiαii

¡αθi¢2

+45Uθθαii αθi + 15Uθθθi


30Uαiαii αθ²²i α²²i + 60Uαiαiαii αθ²i α

²²i α

²i + 20U

αiαii αθ²²²i α²i + 20U

αiαiαii α²²²i α

²iαθi + 20U

αiαii α²²²i α

θ²i

+30Uαiαiαiαii α²²i (α²i)2αθi + 30U

αiαiαii αθ²²i (α²i)

2 + 15Uαiαiαii (α²²i )2αθi + 20U

αiαiαiαii αθ²i (α

²i)3

+5Uαiαii α²²²²i αθi + 5Uαii α

θ²²²²i + 30Uθαiαiαii α²²i (α

²i)2 + 20Uθαiαii α²²²i α

²i + 5U

αiαiαiαiαii αθi (α

²i)4

+5Uθαiαiαiαii (α²i)4 + 5Uθαii α²²²²i + 15Uθαiαii (α²²i )

2

4


10Uαii αθθ²²²i + 60Uαiαiαii αθ²²i αθiα

²i + 30U

αiαii αθθ²²i α²i + 20U

αiαii αθ²²²i αθi + 60U

αiαiαii

¡αθ²i¢2α²i

+60Uαiαiαii αθ²i α²²i α

θi + 30U


²²i α

²i + 10U

αiαiαiαiαii

¡αθi¢2(α²i)

3+ 60Uαiαii αθ²²i αθ²i

+30Uαiαii αθθ²i α²²i + 10Uαiαii α²²²i α

θθi + 60Uθαiαiαii α²²i α

²iαθi + 60U

αiαiαiαii αθ²i α

θi (α

²i)2

+30Uαiαiαiαii α²²i α²i

¡αθi¢2+ 10Uαiαiαiαii αθθi (α

²i)3+ 60Uθαiαii αθ²i α

²²i + 60U

θαiαii αθ²²i α²i

+60Uθαiαiαii αθ²i (α²i)2 + 20Uθαiαii α²²²i α

θi + 20U

θαiαiαiαii (α²i)

3αθi + 10U

αiαiαii α²²²i

¡αθi¢2

+30Uαiαiαii αθθ²i (α²i)2 + 20Uθαii αθ²²²i + 10Uθθαiαiαii (α²i)

3 + 10Uθθαii α²²²i + 30Uθθαiαii α²²i α²i


60Uαiαiαii αθθi αθ²i α

²i + 20U

αiαii αθθθ²i α²i + 10U

αiαiαiαiαii

¡αθi¢3(α²i)

2 + 30Uαiαii αθθ²²i αθi

+30Uαiαiαiαii αθθi αθi (α

²i)2+ 60Uαiαiαii αθθ²i α²iα

θi + 30U


²²i α

θi + 60U

αiαii αθθ²i αθ²i

+30Uαiαii αθ²²i αθθi + 10Uαiαii αθθθi α²²i + 120U

θαiαiαii αθ²i α

²iαθi + 60U

αiαiαiαii αθ²i

¡αθi¢2α²i

+10Uαiαiαiαii α²²i¡αθi¢3+ 60Uαiαiαii

¡αθ²i¢2αθi + 10U

αii α

θθθ²²i + 30Uθθαiαiαii αθi (α

²i)2

+60Uθθαiαii αθ²i α²i + 30U

θθαiαii α²²i α

θi + 30U

θαiαiαii αθθi (α

²i)2 + 30Uθαiαii αθθi α

²²i

+60Uθαiαii αθ²²i αθi + 30Uθαiαiαiαii

¡αθi¢2(α²i)

2 + 30Uθαiαiαii α²²i¡αθi¢2+ 60Uθαiαii αθθ²i α²i

+30Uαiαiαii αθ²²i¡αθi¢2+ 10Uαiαiαii αθθθi (α²i)

2 + 30Uθθαii αθ²²i + 60Uθαiαii

¡αθ²i¢2

+10Uθθθαiαii (α²i)2 + 10Uθθθαii α²²i + 30U

θαii αθθ²²i


5Uαiαii αθθθθi α²i + 60Uαiαiαii αθθi α

θ²i α

θi + 5U

αiαiαiαiαii


αiαiαii

¡αθθi¢2α²i

+30Uαiαiαiαii αθθi¡αθi¢2α²i + 20U

αiαii αθθθi αθ²i + 30U

αiαii αθθ²i αθθi + 60Uθαiαiαii αθθi α

²iαθi

+20Uαiαiαii αθθθi α²iαθi + 20U

αiαiαiαii αθ²i

¡αθi¢3+ 30Uαiαiαii αθθ²i

¡αθi¢2+ 20Uαiαii αθθθ²i αθi

+5Uαii αθθθθ²i + 30Uθθαiαii αθθi α

²i + 30U

θθαiαiαii


θθαiαii αθ²i α

θi

+60Uθαiαii αθθi αθ²i + 60U

θαiαii αθθ²i αθi + 60U

θαiαiαii αθ²i

¡αθi¢2+ 20Uθθθαiαii αθiα

²i

+20Uθαiαiαiαii


θαiαii αθθθi α²i + 30U

θθαii αθθ²i + 20Uθαii αθθθ²i + 5Uθθθθαii α²i

+20Uθθθαii αθ²i


Uθθθθθi + 5Uαiαii αθθθθi αθi + Uαiαiαiαiαii

¡αθi¢5+ 10Uαiαii αθθθi αθθi + 10U

αiαiαiαii αθθi

¡αθi¢3

+15Uαiαiαii

¡αθθi¢2αθi + 30U

θθαiαii αθθi α

θi + 20U

θαiαii αθθθi αθi + 30U

θαiαiαii αθθi

¡αθi¢2

+10Uαiαiαii αθθθi

¡αθi¢2+ Uαii α

θθθθθi + 5Uθθθθαii αθi + 5U

θαii αθθθθi + 10Uθθαii αθθθi

+5Uθαiαiαiαii

¡αθi¢4+ 15Uθαiαii

¡αθθi¢2+ 10Uθθαiαiαii

¡αθi¢3+ 10Uθθθαiαii

¡αθi¢2+ 10Uθθθαii αθθi

5

A.5.19 Constant term

Uαiαiαiαiαii (α²i)5 + 10Uαiαiαiαii α²²i (α

²i)3 + 15Uαiαiαii (α²²i )

2 α²i + 5Uαiαii α²²²²i α²i

+10Uαiαii α²²²i α²²i + 10U

αiαiαii α²²²i (α²i)

2+ Uαii α

²²²²²i

B Derivatives of αi

Recall that, in the ith stratum, αi (θ, Fi (²)) solves the estimating equationZVi [θ,αi (θ, Fi (²))] dFi (²) = 0 (B1)

B.1 αθθi , αθ²i , and α

²²i

Second order differentiation³∂2

∂θ2, ∂2

∂θ∂² ,∂2

∂²2

´of (B1) yields

0 =

ZV θθi dFi (²) + 2α

θi

ZV θαii dFi (²) + α

θθi

ZV αii dFi (²) +

¡αθi¢2 Z

V αiαii dFi (²)

0 = α²i

ZV θαii dFi (²) + α

θ²i

ZV αii dFi (²) + α

θiα

²i

ZV αiαii dFi (²) +

ZV θi d∆iT + α

θi

ZV αii d∆iT

0 = α²²i

ZV αii dFi (²) + (α

²i)2ZV αiαii dFi (²) + 2α

²i

ZV αii d∆iT

Evaluating at ² = 0, we obtain

0 = E£V θθi

¤+ 2

E£V θi¤

E [V 2i ]EhV θαii

i− αθθi E

£V 2i¤+

ÃE£V θi¤

E [V 2i ]

!2E [V αiαii ]

0 =T−1/2

PTt=1 Vit

E [V 2i ]EhV θαii

i− αθ²i E

£V 2i¤+E£V θi¤

E [V 2i ]

T−1/2PTt=1 Vit

E [V 2i ]E [V αiαii ]

+T−1/2TXt=1

¡V θi −E

£V θi¤¢+E£V θi¤

E [V 2i ]T−1/2

TXt=1

(V αii −E [V αii ])

0 = −α²²i E£V 2i¤+

ÃT−1/2

PTt=1 Vit

E [V 2i ]

!2E [V αiαii ] + 2

T−1/2PTt=1 Vit

E [V 2i ]T−1/2

TXt=1

(V αii −E [V αii ])


αθθi =E£V θθi

¤E [V 2i ]

+ 2αθi

EhV θαii

iE [V 2i ]

+¡αθi¢2 E [V αiαii ]

E [V 2i ]= O (1) ,

αθ²i = α²i

EhV θαii

iE [V 2i ]

+ αθiα²i

E [V αiαii ]

E [V 2i ]+

RV θi d∆iTE [V 2i ]

+ αθi


= Op (1) ,

and

α²²i = (α²i)2 E [V

αiαii ]

E [V 2i ]+ 2α²i


= Op (1) .

6

B.2 αθθθi , αθθ²i , αθ²²i , and α

²²²i

Third order differentiation³∂3

∂θ3, ∂3

∂θ2∂², ∂3

∂θ∂²2 ,∂3

∂²3

´of (B1) yields

0 =

ZV θθθi dF (²) + 3αθi

ZV θθαii dF (²) + 3

¡αθi¢2 Z

V θαiαii dF (²) + 3αθθi

ZV θαii dF (²)

+¡αθi¢3 Z

V αiαiαii dF (²) + 3αθiαθθi

ZV αiαii dF (²) + αθθθi

ZV αii dF (²)

0 = α²i

ZV θθαii dF (²) + 2α²iα

θi

ZV θαiαii dF (²) + 2αθ²i

ZV θαii dF (²)

+α²i¡αθi¢2 Z

V αiαiαii dF (²) + 2αθiαθ²i

ZV αiαii dF (²)

+α²iαθθi

ZV αiαii dF (²) + αθθ²i

ZV αii dF (²) +

ZV θθi d∆iT

+2αθi

ZV θαii d∆iT +

¡αθi¢2 Z

V αiαii d∆iT + αθθi

ZV αii d∆iT

0 = (α²i)2ZV θαiαii dF (²) + α²²i

ZV θαii dF (²)

+ (α²i)2αθi

ZV αiαiαii dF (²) + α²²i α

θi

ZV αiαii dF (²)

+2α²iαθ²i

ZV αiαii dF (²) + αθ²²i

ZV αii dF (²) + 2α²i

ZV θαii d∆iT

+2α²iαθi

ZV αiαii d∆iT + 2α

θ²i

ZV αii d∆iT

0 = (α²i)3ZV αiαiαii dF (²) + 3α²iα

²²i

ZV αiαii dF (²) + α²²²i

ZV αii dF (²)

+3 (α²i)2ZV αiαii d∆iT + 3α

²²i

ZV αii d∆iT


αθθθi =E£V θθθi

¤E [V 2i ]

+ 3αθi

EhV θθαii

iE [V 2i ]

+ 3¡αθi¢2 E hV θαiαii

iE [V 2i ]

+ 3αθθi

EhV θαii

iE [V 2i ]

+¡αθi¢3 E [V αiαiαii ]

E [V 2i ]+ 3αθiα

θθi

E [V αiαii ]

E [V 2i ]

αθθ²i = α²i

EhV θθαii

iE [V 2i ]

+ 2α²iαθi

EhV θαiαii

iE [V 2i ]

+ 2αθ²i

EhV θαii

iE [V 2i ]

+α²i¡αθi¢2 E [V αiαiαii ]

E [V 2i ]+ 2αθiα

θ²i

E [V αiαii ]

E [V 2i ]+ α²iα

θθi

E [V αiαii ]

E [V 2i ]

+

RV θθi d∆iTE [V 2i ]

+ 2αθi

RV θαii d∆iTE [V 2i ]

+¡αθi¢2 R V αiαii d∆iT

E [V 2i ]+ αθθi


7

αθ²²i = (α²i)2EhV θαiαii

iE [V 2i ]

+ α²²i

EhV θαii

iE [V 2i ]

+ (α²i)2αθiE [V αiαiαii ]

E [V 2i ]

+α²²i αθi

E [V αiαii ]

E [V 2i ]+ 2α²iα

θ²i

E [V αiαii ]

E [V 2i ]

+2α²i


+ 2α²iαθi

RV αiαii d∆iTE [V 2i ]

+ 2αθ²i


α²²²i

ZV αii dF (²) = (α²i)

3 E [Vαiαiαii ]

E [V 2i ]+ 3α²iα

²²i

E [V αiαii ]

E [V 2i ]

+3 (α²i)2


+ 3α²²i


B.3 αθθθθi , αθθθ²i , αθθ²²i , αθ²²²i , and α²²²²i

Fourth order differentiation³∂4

∂θ4, ∂4

∂θ3∂², ∂4

∂θ2∂²2, ∂4

∂θ∂²3 ,∂4

∂²4

´of (B1) yields

0 = 6¡αθi¢2αθθi

ZV αiαiαii dF (²) + 12αθiα

θθi

ZV θαiαii dF (²) + 4αθiα

θθθi

ZV αiαii dF (²)

+3¡αθθi¢2 Z

V αiαii dF (²) +¡αθi¢4 Z

V αiαiαiαii dF (²) + 4¡αθi¢3 Z

V θαiαiαii dF (²)

+αθθθθi

ZV αii dF (²) + 6

¡αθi¢2 Z

V θθαiαii dF (²) + 6αθθi

ZV θθαii dF (²)

+4αθi

ZV θθθαii dF (²) +

ZV θθθθi dF (²) + 4αθθθi

ZV θαii dF (²)

0 = 3αθiα²iαθθi

ZV αiαiαii dF (²) + 3α²iα

θθi

ZV θαiαii dF (²) + 3α²iα

θi

ZV θθαiαii dF (²)

+3¡αθi¢2αθ²i

ZV αiαiαii dF (²) + α²i

¡αθi¢3 Z

V αiαiαiαii dF (²) + 6αθiαθ²i

ZV θαiαii dF (²)

+3α²i¡αθi¢2 Z

V θαiαiαii dF (²) + α²iαθθθi

ZV αiαii dF (²) + 3αθ²i α

θθi

ZV αiαii dF (²)

+3αθiαθθ²i

ZV αiαii dF (²) + α²i

ZV θθθαii dF (²) + 3αθ²i

ZV θθαii dF (²)

+3αθθ²i

ZV θαii dF (²) + αθθθ²i

ZV αii dF (²)

+

ZV θθθi d∆iT + 3α

θi

ZV θθαii d∆iT + 3

¡αθi¢2 Z

V θαiαii d∆iT + 3αθθi

ZV θαii d∆iT

+¡αθi¢3 Z

V αiαiαii d∆iT + 3αθiα

θθi

ZV αiαii d∆iT + α

θθθi

ZV αii d∆iT

8

0 = 2 (α²i)2αθi

ZV θαiαiαii dF (²) + 2α²²i α

θi


θ²i


+ (α²i)2 ¡αθi¢2 Z

V αiαiαiαii dF (²) + 2αθiαθ²²i

ZV αiαii dF (²) + (α²i)

2αθθi

ZV αiαiαii dF (²)

+α²²i αθθi

ZV αiαii dF (²) + 2α²iα

θθ²i

ZV αiαii dF (²) + α²²i

¡αθi¢2 Z

V αiαiαii dF (²)

+αθθ²²i

ZV αii dF (²) + (α²i)

2ZV θθαiαii dF (²) + α²²i

ZV θθαii dF (²)

+2αθ²²i

ZV θαii dF (²) + 2

¡αθ²i¢2 Z

V αiαii dF (²) + 4αθiα²iαθ²i


+2α²i

ZV θθαii d∆iT + 4α

θ²i

ZV θαii d∆iT + 2α

θθ²i

ZV αii d∆iT + 2α

²i

¡αθi¢2 Z

V αiαiαii d∆iT

+4αθiαθ²i


²iαθθi


²iαθi

ZV θαiαii d∆iT

0 = (α²i)3ZV θαiαiαii dF (²) + 3α²iα

²²i

ZV θαiαii dF (²) + α²²²i

ZV θαii dF (²)

+ (α²i)3αθi

ZV αiαiαiαii dF (²) + 3α²iα

θiα

²²i

ZV αiαiαii dF (²) + 3 (α²i)

2αθ²i


+α²²²i αθi

ZV αiαii dF (²) + 3α²²i α

θ²i


θ²²i

ZV αiαii dF (²)

+αθ²²²i

ZV αii dF (²)

+3 (α²i)2ZV θαiαii d∆iT + 3α

²²i

ZV θαii d∆iT + 3 (α

²i)2αθi

ZV αiαiαii d∆iT

+3α²²i αθi


²iαθ²i


θ²²i

ZV αii d∆iT

0 = (α²i)4ZV αiαiαiαii dF (²) + 6 (α²i)

2α²²i

ZV αiαiαii dF (²) + 3 (α²²i )

2ZV αiαii dF (²)

+4α²iα²²²i

ZV αiαii dF (²) + α²²²²i

ZV αii dF (²)

+4 (α²i)3ZV αiαiαii d∆iT + 12α

²iα²²i


²²²i

ZV αii d∆iT


αθθθθi = 6¡αθi¢2αθθi

E [V αiαiαii ]

E [V 2i ]+ 12αθiα

θθi

EhV θαiαii

iE [V 2i ]

+ 4αθiαθθθi

E [V αiαii ]

E [V 2i ]+ 3

¡αθθi¢2 E [V αiαii ]

E [V 2i ]

+¡αθi¢4 E [V αiαiαiαii ]

E [V 2i ]+ 4

¡αθi¢3 E hV θαiαiαii

iE [V 2i ]

+ 6¡αθi¢2 E hV θθαiαii

iE [V 2i ]

+ 6αθθi

EhV θθαii

iE [V 2i ]

+4αθi

EhV θθθαii

iE [V 2i ]

+E£V θθθθi

¤E [V 2i ]

+ 4αθθθi

EhV θαii

iE [V 2i ]

9

αθθθ²i = 3αθiα²iαθθi

E [V αiαiαii ]

E [V 2i ]+ 3α²iα

θθi

EhV θαiαii

iE [V 2i ]

+ 3α²iαθi

EhV θθαiαii

iE [V 2i ]

+ 3¡αθi¢2αθ²i

E [V αiαiαii ]

E [V 2i ]

+α²i¡αθi¢3 E [V αiαiαiαii ]

E [V 2i ]+ 6αθiα

θ²i

EhV θαiαii

iE [V 2i ]

+ 3α²i¡αθi¢2 E hV θαiαiαii

iE [V 2i ]

+ α²iαθθθi

E [V αiαii ]

E [V 2i ]

+3αθ²i αθθi

E [V αiαii ]

E [V 2i ]+ 3αθiα

θθ²i

E [V αiαii ]

E [V 2i ]+ α²i

EhV θθθαii

iE [V 2i ]

+ 3αθ²i

EhV θθαii

iE [V 2i ]

+ 3αθθ²i

EhV θαii

iE [V 2i ]

+

RV θθθi d∆iTE [V 2i ]

+ 3αθi

RV θθαii d∆iTE [V 2i ]

+ 3¡αθi¢2 R V θαiαii d∆iT

E [V 2i ]+ 3αθθi


+¡αθi¢3 R V αiαiαii d∆iT

E [V 2i ]+ 3αθiα

θθi


+ αθθθi


αθθ²²i = 2 (α²i)2 αθi

EhV θαiαiαii

iE [V 2i ]

+ 2α²²i αθi

EhV θαiαii

iE [V 2i ]

+ 4α²iαθ²i

EhV θαiαii

iE [V 2i ]

+ (α²i)2 ¡αθi ¢2 E [V αiαiαiαii ]

E [V 2i ]

+2αθiαθ²²i

E [V αiαii ]

E [V 2i ]+ (α²i)

2αθθi

E [V αiαiαii ]

E [V 2i ]+ α²²i α

θθi

E [V αiαii ]

E [V 2i ]+ 2α²iα

θθ²i

E [V αiαii ]

E [V 2i ]

+α²²i¡αθi¢2 E [V αiαiαii ]

E [V 2i ]+ (α²i)

2EhV θθαiαii

iE [V 2i ]

+ α²²i

EhV θθαii

iE [V 2i ]

+ 2αθ²²i

EhV θαii

iE [V 2i ]

+2¡αθ²i¢2 E [V αiαii ]

E [V 2i ]+ 4αθiα

²iαθ²i

E [V αiαiαii ]

E [V 2i ]

+2α²i


+ 4αθ²i


+ 2αθθ²i


+ 2α²i¡αθi¢2 R V αiαiαii d∆iT

E [V 2i ]

+4αθiαθ²i


+ 2α²iαθθi


+ 4α²iαθi

RV θαiαii d∆iTE [V 2i ]

αθ²²²i = (α²i)3EhV θαiαiαii

iE [V 2i ]

+ 3α²iα²²i

EhV θαiαii

iE [V 2i ]

+ α²²²i

EhV θαii

iE [V 2i ]

+ (α²i)3αθiE [V αiαiαiαii ]

E [V 2i ]

+3α²iαθiα

²²i

E [V αiαiαii ]

E [V 2i ]+ 3 (α²i)

2 αθ²iE [V αiαiαii ]

E [V 2i ]+ α²²²i α

θi

E [V αiαii ]

E [V 2i ]+ 3α²²i α

θ²i

E [V αiαii ]

E [V 2i ]

+3α²iαθ²²i

E [V αiαii ]

E [V 2i ]

+3 (α²i)2


+ 3α²²i


+ 3 (α²i)2 αθi

RV αiαiαii d∆iTE [V 2i ]

+ 3α²²i αθi


+6α²iαθ²i


+ 3αθ²²i


α²²²²i = (α²i)4 E [V

αiαiαiαii ]

E [V 2i ]+ 6 (α²i)

2α²²iE [V αiαiαii ]

E [V 2i ]+ 3 (α²²i )

2 E [Vαiαii ]

E [V 2i ]+ 4α²iα

²²²i

E [V αiαii ]

E [V 2i ]

+4 (α²i)3


+ 12α²iα²²i


+ 4α²²²i


10

B.4 αθθθθθi , αθθθθ²i , αθθθ²²i , αθθ²²²i , αθ²²²²i , and α²²²²²i

Fifth order differentiation³∂5

∂θ5, ∂5

∂θ4∂², ∂5

∂θ3∂²2, ∂5

∂θ2∂²3, ∂5

∂θ∂²4 ,∂5

∂²5

´of (B1) yields

0 = 10αθθi

ZV θθθαii dF (²) + 5αθi

ZV θθθθαii dF (²) + 30αθiα

θθi


+20αθiαθθθi

ZV θαiαii dF (²) + 30

¡αθi¢2αθθi

ZV θαiαiαii dF (²) + 10αθθi α

θθθi

ZV αiαii dF (²)

+5αθiαθθθθi

ZV αiαii dF (²) + 15αθi

¡αθθi¢2 Z

V αiαiαii dF (²) + 10¡αθi¢3αθθi

ZV αiαiαiαii dF (²)

+10¡αθi¢2αθθθi

ZV αiαiαii dF (²) + 10αθθθi

ZV θθαii dF (²) + 5αθθθθi

ZV θαii dF (²)

+¡αθi¢5 Z

V αiαiαiαiαii dF (²) + 5¡αθi¢4 Z

V θαiαiαiαii dF (²) + 15¡αθθi¢2 Z

V θαiαii dF (²)

+10¡αθi¢3 Z

V θθαiαiαii dF (²) + αθθθθθi

ZV αii dF (²) + 10

¡αθi¢2 Z

V θθθαiαii dF (²) +

ZV θθθθθi dF (²)

0 = 6α²i¡αθi¢2 Z

V θθαiαiαii dF (²) + 12αθiαθ²i

ZV θθαiαii dF (²) + 4α²i

¡αθi¢3 Z

V θαiαiαiαii dF (²)

+12¡αθi¢2αθ²i

ZV θαiαiαii dF (²) + 12αθiα

θθ²i

ZV θαiαii dF (²) + 12αθ²i α

θθi


+4αθiαθθθ²i

ZV αiαii dF (²) + α²i

¡αθi¢4 Z

V αiαiαiαiαii dF (²) + 4¡αθi¢3αθ²i


+6¡αθi¢2αθθ²i

ZV αiαiαii dF (²) + 4αθ²i α

θθθi

ZV αiαii dF (²) + 3α²i

¡αθθi¢2 Z


+6αθθi αθθ²i


θθi

ZV θθαiαii dF (²) + 4α²iα

θi

ZV θθθαiαii dF (²)

+4α²iαθθθi

ZV θαiαii dF (²) + α²iα

θθθθi

ZV αiαii dF (²) + 12αθiα

²iαθθi

ZV θαiαiαii dF (²)

+4αθiα²iαθθθi


θ²i α

θθi

ZV αiαiαii dF (²) + 6

¡αθi¢2α²iα

θθi


+αθθθθ²i

ZV αii dF (²) + 4αθθθ²i

ZV θαii dF (²) + α²i

ZV θθθθαii dF (²)

+4αθ²i

ZV θθθαii dF (²) + 6αθθ²i

ZV θθαii dF (²)

+4αθiαθθθi

ZV αiαii d∆iT + 6

¡αθi¢2αθθi

ZV αiαiαii d∆iT + 12α

θiα

θθi

ZV θαiαii d∆iT

+3¡αθθi¢2 Z

V αiαii d∆iT + 6¡αθi¢2 Z

V θθαiαii d∆iT + 4¡αθi¢3 Z

V θαiαiαii d∆iT

+4αθθθi

ZV θαii d∆iT + α

θθθθi


θi

ZV θθθαii d∆iT + 6α

θθi

ZV θθαii d∆iT

+¡αθi¢4 Z

V αiαiαiαii d∆iT +

ZV θθθθi d∆iT

11

0 = 6αθi¡αθ²i¢2 Z

V αiαiαii dF (²) + 3 (α²i)2αθi

ZV θθαiαiαii dF (²) + 3α²²i α

θi


+3 (α²i)2αθθi

ZV θαiαiαii dF (²) + (α²i)

2αθθθi

ZV αiαiαii dF (²) + 3α²²i α

θθi


+α²²i αθθθi


θ²i

ZV θθαiαii dF (²) + 3αθθ²²i αθi

ZV αiαii dF (²)

+3 (α²i)2 ¡αθi¢2 Z

V θαiαiαiαii dF (²) + 3α²²i¡αθi¢2 Z

V θαiαiαii dF (²) + (α²i)2 ¡αθi¢3 Z

V αiαiαiαiαii dF (²)

+6α²iαθθ²i

ZV θαiαii dF (²) + α²²i

¡αθi¢3 Z

V αiαiαiαii dF (²) + 6αθ²i

Zαθθ²i V αiαii dF (²)

+2α²iαθθθ²i

ZV αiαii dF (²) + 6αθ²²i αθi

ZV θαiαii dF (²) + 3αθ²²i

¡αθi¢2 Z


+3αθ²²i αθθi

ZV αiαii dF (²) + 3αθθi α

²²i α

θi

ZV αiαiαii dF (²) + 3αθθi (α

²i)2αθi


+6αθθi α²iαθ²i

ZV αiαiαii dF (²) + 6αθ²i α

²i

¡αθi¢2 Z

V αiαiαiαii dF (²) + 12αθ²i α²iαθi

ZV θαiαiαii dF (²)

+6αθθ²i α²iαθi

ZV αiαiαii dF (²) + 6

¡αθ²i¢2 Z

V θαiαii dF (²) + (α²i)2ZV θθθαiαii dF (²)

+αθθθ²²i

ZV αii dF (²) + 3αθ²²i

ZV θθαii dF (²) + 3αθθ²²i

ZV θαii dF (²) + α²²i

ZV θθθαii dF (²)

+12αθiαθ²i

ZV θαiαii d∆iT + 2α

²i

¡αθi¢3 Z

V αiαiαiαii d∆iT + 6¡αθi¢2αθ²i


+6αθiαθθ²i


θ²i α

θθi


²iαθi

ZV θθαiαii d∆iT

+6α²iαθθi


²iαθθθi


²i

¡αθi¢2 Z

V θαiαiαii d∆iT

+6αθiα²iαθθi


θ²i


θθ²i

ZV θαii d∆iT

+2αθθθ²i


²i

ZV θθθαii d∆iT

12

0 = 3α²²i αθθ²i

ZV αiαii dF (²) + 2 (α²i)

3αθi

ZV θαiαiαiαii dF (²)

+6 (α²i)2αθ²i

ZV θαiαiαii dF (²) + α²²²i

¡αθi¢2 Z

V αiαiαii dF (²) + (α²i)3 ¡αθi¢2 Z

V αiαiαiαiαii dF (²)

+2αθ²²²i αθi

ZV αiαii dF (²) + α²²²i α

θθi

ZV αiαii dF (²) + (α²i)

3 αθθi


+2α²²²i αθi


θθ²²i

ZV αiαii dF (²) + 3 (α²i)

2αθθ²i


+6αθ²²i α²iαθi

ZV αiαiαii dF (²) + 3α²iα

θθi α

²²i


²²i α

θ²i


+6α²iαθiα

²²i

ZV θαiαiαii dF (²) + 6αθi (α

²i)2αθ²i

ZV αiαiαiαii dF (²) + 3α²i

¡αθi¢2α²²i


+ (α²i)3ZV θθαiαiαii dF (²) + αθθ²²²i

ZV αii dF (²) + 2αθ²²²i

ZV θαii dF (²) + α²²²i

ZV θθαii dF (²)

+6α²²i αθ²i

ZV θαiαii dF (²) + 6α²i

¡αθ²i¢2 Z

V αiαiαii dF (²) + 6αθ²²i αθ²i

ZV αiαii dF (²)

+6α²iαθ²²i


²²i


+6 (α²i)2 αθi

ZV θαiαiαii d∆iT + 6α

θiα

θ²²i


²iαθ²i


²iαθθ²i

ZV αiαii d∆iT

+3α²²i αθθi

ZV αiαii d∆iT + 3 (α

²i)2αθθi


²²i

¡αθi¢2 Z

V αiαiαii d∆iT

+3 (α²i)2 ¡αθi¢2 Z

V αiαiαiαii d∆iT + 6α²²i α

θi


θiα

²iαθ²i


+3αθθ²²i


²²i


θ²²i

ZV θαii d∆iT + 6

¡αθ²i¢2 Z

V αiαii d∆iT

+3 (α²i)2ZV θθαiαii d∆iT

0 = 6 (α²i)2αθiα

²²i

ZV αiαiαiαii dF (²) + 4 (α²i)

3αθ²i

ZV αiαiαiαii dF (²) + 6 (α²i)

2αθ²²i


+6 (α²i)2 α²²i

ZV θαiαiαii dF (²) + 4α²²²i α

θ²i

ZV αiαii dF (²) + 6α²²i α

θ²²i

ZV αiαii dF (²)

+α²²²²i αθi

ZV αiαii dF (²) + 3 (α²²i )

2αθi

ZV αiαiαii dF (²) + (α²i)

4αθi

ZV αiαiαiαiαii dF (²)

+4α²iαθ²²²i


θ²i α

²²i

ZV αiαiαii dF (²) + α²²²²i

ZV θαii dF (²)

+αθ²²²²i

ZV αii dF (²) + 4α²iα

²²²i


θiα

²²²i


+3 (α²²i )2ZV θαiαii dF (²) + (α²i)

4ZV θαiαiαiαii dF (²)

+12α²iα²²i

ZV θαiαii d∆iT + 12 (α

²i)2αθ²i

ZV αiαiαii d∆iT + 4 (α

²i)3αθi

ZV αiαiαiαii d∆iT

+4α²²²i αθi


²iαθiα

²²i

ZV αiαiαii d∆iT + 4 (α

²i)3ZV θαiαiαii d∆iT

+4α²²²i

ZV θαii d∆iT + 4α

θ²²²i


²²i α

θ²i


²iαθ²²i

ZV αiαii d∆iT

13

0 = α²²²²²i

ZV αii dF (²) + 10 (α²i)

2α²²²i

ZV αiαiαii dF (²) + (α²i)

5ZV αiαiαiαiαii dF (²)

+10 (α²i)3α²²i

ZV αiαiαiαii dF (²) + 10α²²i α

²²²i


²²²²i

ZV αiαii dF (²)

+15α²i (α²²i )

2ZV αiαiαii dF (²)

+5 (α²i)4ZV αiαiαiαii d∆iT + 15 (α

²²i )

2ZV αiαii d∆iT + 5α

²²²²i

ZV αii d∆iT

+30 (α²i)2 α²²i


²iα²²²i

ZV αiαii d∆iT


αθθθθθi = 10αθθi

EhV θθθαii

iE [V 2i ]

+ 5αθi

EhV θθθθαii

iE [V 2i ]

+ 30αθiαθθi

EhV θθαiαii

iE [V 2i ]

+ 20αθiαθθθi

EhV θαiαii

iE [V 2i ]

+30¡αθi¢2αθθi

EhV θαiαiαii

iE [V 2i ]

+ 10αθθi αθθθi

E [V αiαii ]

E [V 2i ]+ 5αθiα

θθθθi

E [V αiαii ]

E [V 2i ]

+15αθi¡αθθi¢2 E [V αiαiαii ]

E [V 2i ]+ 10

¡αθi¢3αθθi

E [V αiαiαiαii ]

E [V 2i ]+ 10

¡αθi¢2αθθθi

E [V αiαiαii ]

E [V 2i ]

+10αθθθi

EhV θθαii

iE [V 2i ]

+ 5αθθθθi

EhV θαii

iE [V 2i ]

+¡αθi¢5 E [V αiαiαiαiαii ]

E [V 2i ]+ 5

¡αθi¢4 E hV θαiαiαiαii

iE [V 2i ]

+15¡αθθi¢2 E hV θαiαii

iE [V 2i ]

+ 10¡αθi¢3 E hV θθαiαiαii

iE [V 2i ]

+ 10¡αθi¢2 E hV θθθαiαii

iE [V 2i ]

+E£V θθθθθi

¤E [V 2i ]

14

αθθθθ²i = 6α²i¡αθi¢2 E hV θθαiαiαii

iE [V 2i ]

+ 12αθiαθ²i

EhV θθαiαii

iE [V 2i ]

+ 4α²i¡αθi¢3 E hV θαiαiαiαii

iE [V 2i ]

+12¡αθi¢2αθ²i

EhV θαiαiαii

iE [V 2i ]

+ 12αθiαθθ²i

EhV θαiαii

iE [V 2i ]

+ 12αθ²i αθθi

EhV θαiαii

iE [V 2i ]

+4αθiαθθθ²i

E [V αiαii ]

E [V 2i ]+ α²i

¡αθi¢4 E [V αiαiαiαiαii ]

E [V 2i ]+ 4

¡αθi¢3αθ²i


E [V 2i ]

+6¡αθi¢2αθθ²i

E [V αiαiαii ]

E [V 2i ]+ 4αθ²i α

θθθi

E [V αiαii ]

E [V 2i ]+ 3α²i

¡αθθi¢2 E [V αiαiαii ]

E [V 2i ]

+6αθθi αθθ²i

E [V αiαii ]

E [V 2i ]+ 6α²iα

θθi

EhV θθαiαii

iE [V 2i ]

+ 4α²iαθi

EhV θθθαiαii

iE [V 2i ]

+4α²iαθθθi

EhV θαiαii

iE [V 2i ]

+ α²iαθθθθi

E [V αiαii ]


²iαθθi

EhV θαiαiαii

iE [V 2i ]

+4αθiα²iαθθθi

E [V αiαiαii ]


θ²i α

θθi

E [V αiαiαii ]

E [V 2i ]+ 6

¡αθi¢2α²iα

θθi


E [V 2i ]

+4αθθθ²i

EhV θαii

iE [V 2i ]

+ α²i

EhV θθθθαii

iE [V 2i ]

+ 4αθ²i

EhV θθθαii

iE [V 2i ]

+ 6αθθ²i

EhV θθαii

iE [V 2i ]

+4αθiαθθθi


+ 6¡αθi¢2αθθi


+ 12αθiαθθi


+3¡αθθi¢2 R V αiαii d∆iT

E [V 2i ]+ 6

¡αθi¢2 R V θθαiαii d∆iT

E [V 2i ]+ 4

¡αθi¢3 R V θαiαiαii d∆iT

E [V 2i ]

+4αθθθi


+ αθθθθi


+ 4αθi

RV θθθαii d∆iTE [V 2i ]

+ 6αθθi


+¡αθi¢4 R V αiαiαiαii d∆iT

E [V 2i ]+

RV θθθθi d∆iTE [V 2i ]

15

αθθθ²²i = 6αθi¡αθ²i¢2 E [V αiαiαii ]

E [V 2i ]+ 3 (α²i)

2αθi

EhV θθαiαiαii

iE [V 2i ]

+ 3α²²i αθi

EhV θθαiαii

iE [V 2i ]

+3 (α²i)2αθθi

EhV θαiαiαii

iE [V 2i ]

+ (α²i)2αθθθi

E [V αiαiαii ]

E [V 2i ]+ 3α²²i α

θθi

EhV θαiαii

iE [V 2i ]

+α²²i αθθθi

E [V αiαii ]

E [V 2i ]+ 6α²iα

θ²i

EhV θθαiαii

iE [V 2i ]

+ 3αθθ²²i αθiE [V αiαii ]

E [V 2i ]

+3 (α²i)2 ¡αθi¢2 E hV θαiαiαiαii

iE [V 2i ]

+ 3α²²i¡αθi¢2 E hV θαiαiαii

iE [V 2i ]

+ (α²i)2 ¡αθi¢3 E [V αiαiαiαiαii ]

E [V 2i ]

+6α²iαθθ²i

EhV θαiαii

iE [V 2i ]

+ α²²i¡αθi¢3 E [V αiαiαiαii ]

E [V 2i ]+ 6αθ²i α

θθ²i

E [V αiαii ]

E [V 2i ]

+2α²iαθθθ²i

E [V αiαii ]

E [V 2i ]+ 6αθ²²i αθi

EhV θαiαii

iE [V 2i ]

+ 3αθ²²i¡αθi¢2 E [V αiαiαii ]

E [V 2i ]

+3αθ²²i αθθiE [V αiαii ]

E [V 2i ]+ 3αθθi α

²²i α

θi

E [V αiαiαii ]

E [V 2i ]+ 3αθθi (α

²i)2αθiE [V αiαiαiαii ]

E [V 2i ]

+6αθθi α²iαθ²i

E [V αiαiαii ]

E [V 2i ]+ 6αθ²i α

²i

¡αθi¢2 E [V αiαiαiαii ]

E [V 2i ]+ 12αθ²i α

²iαθi

EhV θαiαiαii

iE [V 2i ]

+6αθθ²i α²iαθi

E [V αiαiαii ]

E [V 2i ]+ 6

¡αθ²i¢2 E hV θαiαii

iE [V 2i ]

+ (α²i)2EhV θθθαiαii

iE [V 2i ]

+3αθ²²i

EhV θθαii

iE [V 2i ]

+ 3αθθ²²i

EhV θαii

iE [V 2i ]

+ α²²i

EhV θθθαii

iE [V 2i ]

+12αθiαθ²i


+ 2α²i¡αθi¢3 R V αiαiαiαii d∆iT

E [V 2i ]+ 6

¡αθi¢2αθ²i


+6αθiαθθ²i


+ 6αθ²i αθθi


+ 6α²iαθi

RV θθαiαii d∆iTE [V 2i ]

+6α²iαθθi


+ 2α²iαθθθi


+ 6α²i¡αθi¢2 R V θαiαiαii d∆iT

E [V 2i ]

+6αθiα²iαθθi


+ 6αθ²i


+ 6αθθ²i


+2αθθθ²i


+ 2α²i

RV θθθαii d∆iTE [V 2i ]

16

αθθ²²²i = 3α²²i αθθ²i

E [V αiαii ]

E [V 2i ]+ 2 (α²i)

3αθi

EhV θαiαiαiαii

iE [V 2i ]

+ 6 (α²i)2αθ²i

EhV θαiαiαii

iE [V 2i ]

+α²²²i¡αθi¢2 E [V αiαiαii ]

E [V 2i ]+ (α²i)

3 ¡αθi¢2 E [V αiαiαiαiαii ]

E [V 2i ]+ 2αθ²²²i αθi

E [V αiαii ]

E [V 2i ]

+α²²²i αθθi

E [V αiαii ]

E [V 2i ]+ (α²i)

3αθθi


E [V 2i ]+ 2α²²²i α

θi

EhV θαiαii

iE [V 2i ]

+3α²iαθθ²²i

E [V αiαii ]

E [V 2i ]+ 3 (α²i)

2 αθθ²i

E [V αiαiαii ]

E [V 2i ]+ 6αθ²²i α²iα

θi

E [V αiαiαii ]

E [V 2i ]

+3α²iαθθi α

²²i

E [V αiαiαii ]

E [V 2i ]+ 6αθiα

²²i α

θ²i

E [V αiαiαii ]

E [V 2i ]+ 6α²iα

θiα

²²i

EhV θαiαiαii

iE [V 2i ]

+6αθi (α²i)2αθ²i


E [V 2i ]+ 3α²i

¡αθi¢2α²²iE [V αiαiαiαii ]

E [V 2i ]+ (α²i)

3EhV θθαiαiαii

iE [V 2i ]

+2αθ²²²i

EhV θαii

iE [V 2i ]

+ α²²²i

EhV θθαii

iE [V 2i ]

+ 6α²²i αθ²i

EhV θαiαii

iE [V 2i ]

+ 6α²i¡αθ²i¢2 E [V αiαiαii ]

E [V 2i ]

+6αθ²²i αθ²iE [V αiαii ]

E [V 2i ]+ 6α²iα

θ²²i

EhV θαiαii

iE [V 2i ]

+ 3α²iα²²i

EhV θθαiαii

iE [V 2i ]

+6 (α²i)2αθi

RV θαiαiαii d∆iTE [V 2i ]

+ 6αθiαθ²²i


+ 12α²iαθ²i


+6α²iαθθ²i


+ 3α²²i αθθi


+ 3 (α²i)2αθθi


+3α²²i¡αθi¢2 R V αiαiαii d∆iT

E [V 2i ]+ 3 (α²i)

2 ¡αθi ¢2 R V αiαiαiαii d∆iTE [V 2i ]

+ 6α²²i αθi


+12αθiα²iαθ²i


+ 3αθθ²²i


+ 3α²²i


+ 6αθ²²i


+6¡αθ²i¢2 R V αiαii d∆iT

E [V 2i ]+ 3 (α²i)

2

RV θθαiαii d∆iTE [V 2i ]

17

αθ²²²²i = 6 (α²i)2αθiα

²²i


E [V 2i ]+ 4 (α²i)

3αθ²i


E [V 2i ]+ 6 (α²i)

2αθ²²i

E [V αiαiαii ]

E [V 2i ]

+6 (α²i)2α²²i

EhV θαiαiαii

iE [V 2i ]

+ 4α²²²i αθ²i

E [V αiαii ]

E [V 2i ]+ 6α²²i α

θ²²i

E [V αiαii ]

E [V 2i ]

+α²²²²i αθiE [V αiαii ]

E [V 2i ]+ 3 (α²²i )

2 αθiE [V αiαiαii ]

E [V 2i ]+ (α²i)

4 αθiE [V αiαiαiαiαii ]

E [V 2i ]

+4α²iαθ²²²i

E [V αiαii ]

E [V 2i ]+ 12α²iα

θ²i α

²²i

E [V αiαiαii ]

E [V 2i ]+ α²²²²i

EhV θαii

iE [V 2i ]

+4α²iα²²²i

EhV θαiαii

iE [V 2i ]

+ 4α²iαθiα

²²²i

E [V αiαiαii ]

E [V 2i ]+ 3 (α²²i )

2EhV θαiαii

iE [V 2i ]

+ (α²i)4EhV θαiαiαiαii

iE [V 2i ]

+12α²iα²²i


+ 12 (α²i)2αθ²i


+ 4 (α²i)3αθi

RV αiαiαiαii d∆iTE [V 2i ]

+4α²²²i αθi


+ 12α²iαθiα

²²i


+ 4 (α²i)3

RV θαiαiαii d∆iTE [V 2i ]

+4α²²²i


+ 4αθ²²²i


+ 12α²²i αθ²i


+12α²iαθ²²i


α²²²²²i = 10 (α²i)2α²²²i

E [V αiαiαii ]

E [V 2i ]+ (α²i)

5 E [Vαiαiαiαiαii ]

E [V 2i ]+ 10 (α²i)

3α²²iE [V αiαiαiαii ]

E [V 2i ]

+10α²²i α²²²i

E [V αiαii ]

E [V 2i ]+ 5α²iα

²²²²i

E [V αiαii ]

E [V 2i ]+ 15α²i (α

²²i )

2 E [Vαiαiαii ]

E [V 2i ]

+5 (α²i)4

RV αiαiαiαii d∆iTE [V 2i ]

+ 15 (α²²i )2


+ 5α²²²²i


+30 (α²i)2α²²i


+ 20α²iα²²²i


18

Jackknife and Analytical Bias Reduction for Nonlinear ...Mcfadden/E242_f02/Hahn.pdfJackknife and...

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