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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 à 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:
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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
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of Econometrics. Amsterdam: North-Holland.
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Method-of-Moments Estimators, Econometrica, 64, 891-916.
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[16] Neyman, J., and E.L. Scott, (1948), Consistent Estimates Based on Partially Consistent Ob-
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1426.
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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
α²i ·Z∂Ui∂αi
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 ]
à PTt=1 Vit√TE [V 2i ]
!2nXi=1
α²i ·Z∂Ui∂αi
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 ]
à 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
E [Ki (xit; θ0,αi0)]
¯¯
≤ |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
E [Ki (xit; θ0,αi0)]
¯¯
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 .
Lemma 9 Suppose that Condition 4 holds. Then, we have
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¢.
Lemma 10 Suppose that Condition 4 holds. Then, we have
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
¢for some constant C > 0 and υ such that 0 ≤ υ < 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.
Lemma 11 Suppose that Condition 4 holds. Then, we have
Pr
"max
0≤²≤ 1√T
|θ²² (²)| > C³T
110−υ
´2#= o
¡T−1
¢for some constant C > 0 and υ such that 0 ≤ υ < 1
160 .
Proof. The conclusion follows by using the characterization of θ²² (²) in Appendix C.3, and Lemmas
7, 8, 9, and 10.
Lemma 12 Suppose that Condition 4 holds. Then, we have
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
¢for some constant C > 0 and υ such that 0 ≤ υ < 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
+
µZ∂2Vi (·, θ, ²)∂θr∂αi
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 +
µZ∂2Vi (·, θ, ²)∂θr∂αi
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 =
µZ∂2Vi (·, θ, ²)∂θr∂αi
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
µZ∂2Vi (·, θ, ²)∂θr∂αi
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
¢for some constant C > 0 and υ such that 0 ≤ υ < 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
à 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
k1 (xi,t) k3 (xi,t) k4 (xi,t)
!
− T 1/2
(T − 1)2³T 1/2ki,3
´Ã 1T
TXt=1
k1 (xi,t) k2 (xi,t) k4 (xi,t)
!
− T 1/2
(T − 1)2³T 1/2ki,4
´Ã 1T
TXt=1
k1 (xi,t) k2 (xi,t) k3 (xi,t)
!
+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
For m = 3, it suffices to note that
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
k1 (xi,t) k2 (xi,t) k3 (xi,t)
For m = 4, it suffices to note that
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
k2 (xi,t) k3 (xi,t) + ki,2ki,3T
(T − 1)4TXt=1
k1 (xi,t) k4 (xi,t)
+ki,2ki,4T
(T − 1)4TXt=1
k1 (xi,t) k3 (xi,t) + ki,3ki,4T
(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
k1 (xi,t) k3 (xi,t) k4 (xi,t)
− 1
(T − 1)4 ki,3TXt=1
k1 (xi,t) k2 (xi,t) k4 (xi,t)− 1
(T − 1)4 ki,4TXt=1
k1 (xi,t) k2 (xi,t) k3 (xi,t)
+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
à 1√n
nXi1=1
k[1,1] (xi1,t)
!2Ã1√n
nXi2=1
k[2,1] (xi2,t)
!2≤ E
à 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)
2ià 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
à 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
¶
Lemma 18 Suppose that Condition 5 holds. WhenP`m` = 4, L = 2,m1 = 1,m2 = 3, 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 = 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
óT 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
óT 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
óT 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
óT 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
óT 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
óT 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
óT 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
óT 1/2ki2,[2,1]
´Ã 1T
TXt=1
k[2,2] (xi2,t) k[2,3] (xi2,t)
!!= Op (1)
Because¯¯ 1n
nXi2=1
óT 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
óT 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¸¶1/2Eà 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
à 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
à 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¸¶1/2Eà 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
from which the conclusion follows.
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) ,
from which the conclusion follows.
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) ,
from which the conclusion follows.
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
¶
Lemma 20 Suppose that Condition 5 holds. WhenP`m` = 4, L = 2,m1 = 2,m2 = 2, we have
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
óT 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
óT 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
óT 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
óT 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
óT 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
óT 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
óT 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
óT 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
óT 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
óT 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
óT 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]
¯M (xi1,t)M (xi2,t)
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]
¯M (xi1,t)M (xi2,t)
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
k[1,1] (xi1,t) k[1,2] (xi1,t) k[2,1] (xi2,t) k[2,2] (xi2,t)
!= op
µ1√T
¶(43)
This follows easily from
E
"¯¯ 1n2
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)
!¯¯#
≤ 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
from which we obtain
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
E [Uαiαii ]¡αθi¢2!
(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
RV αii d∆iTE [V 2i ]
= 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
E [Uαiαii ]¡αθi¢2!
= 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
A.3.6 Coefficient of (θ²)3
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
A.4.2 Coefficient of θ²²²
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
A.4.4 Coefficient of θ²²
6Uθαiαii (α²i)2+ 6Uθαii α²²i + 6U
αii α
θ²²i + 12Uαiαii αθ²i α
²i + 6U
αiαiαii αθi (α
²i)2+ 6Uαiαii αθiα
²²i
A.4.5 Coefficient of θ²²θ²
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
A.4.8 Coefficient of θ²
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
A.4.9 Coefficient of (θ²)2
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
A.4.10 Coefficient of (θ²)3
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
θαiαii αθ²i α
θi + 12U
θαiαiαii
¡αθi¢2α²i + 12U
θαiαii αθθi α
²i
A.4.11 Coefficient of (θ²)4
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
¡αθi¢2+ 5Uαii α
θθi + 10U
θαii αθi + 5U
θθi
A.5.4 Coefficient of θ²²²
10Uαiαii αθiα²²i + 20U
αiαii αθ²i α
²i + 10U
αiαiαii αθi (α
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
¡αθi¢2+ 10Uαii α
θθi + 20U
θαii αθi + 10U
θθi
3
A.5.8 Coefficient of θ²²
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
A.5.9 Coefficient of θ²²θ²
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
A.5.10 Coefficient of θ²² (θ²)2
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
θαiαii αθ²i α
θi + 90U
θαiαiαii
¡αθi¢2α²i
+30Uαiαiαiαii
¡αθi¢3α²i + 90U
θθαiαii αθiα
²i + 30U
αii α
θθθ²i + 30Uθθθαii α²i + 90U
θθαii αθ²i
+90Uθαii αθθ²i
A.5.11 Coefficient of θ²² (θ²)3
40Uαiαii αθθθi αθi + 30Uαiαii
¡αθθi¢2+ 120Uθαiαii αθθi α
θi + 60U
αiαiαii αθθi
¡αθi¢2
+10Uαiαiαiαii
¡αθi¢4+ 10Uαii α
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
¡αθi¢2α²i + 30U
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
¡αθi¢3+ 15Uαii α
θθθi + 45Uθαii αθθi + 45U
θαiαii
¡αθi¢2
+45Uθθαii αθi + 15Uθθθi
A.5.14 Coefficient of θ²
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
A.5.15 Coefficient of (θ²)2
10Uαii αθθ²²²i + 60Uαiαiαii αθ²²i αθiα
²i + 30U
αiαii αθθ²²i α²i + 20U
αiαii αθ²²²i αθi + 60U
αiαiαii
¡αθ²i¢2α²i
+60Uαiαiαii αθ²i α²²i α
θi + 30U
αiαiαii αθθi α
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
A.5.16 Coefficient of (θ²)3
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
αiαiαii αθθi α
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
A.5.17 Coefficient of (θ²)4
5Uαiαii αθθθθi α²i + 60Uαiαiαii αθθi α
θ²i α
θi + 5U
αiαiαiαiαii
¡αθi¢4α²i + 15U
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
¡αθi¢2α²i + 60U
θθαiαii αθ²i α
θi
+60Uθαiαii αθθi αθ²i + 60U
θαiαii αθθ²i αθi + 60U
θαiαiαii αθ²i
¡αθi¢2+ 20Uθθθαiαii αθiα
²i
+20Uθαiαiαiαii
¡αθi¢3α²i + 20U
θαiαii αθθθi α²i + 30U
θθαii αθθ²i + 20Uθαii αθθθ²i + 5Uθθθθαii α²i
+20Uθθθαii αθ²i
A.5.18 Coefficient of (θ²)5
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 ])
from which 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) ,
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) ,
and
α²²i = (α²i)2 E [V
αiαii ]
E [V 2i ]+ 2α²i
RV αii d∆iTE [V 2i ]
= 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
Evaluating at ² = 0, we obtain
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
RV αii d∆iTE [V 2i ]
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
RV θαii d∆iTE [V 2i ]
+ 2α²iαθi
RV αiαii d∆iTE [V 2i ]
+ 2αθ²i
RV αii d∆iTE [V 2i ]
α²²²i
ZV αii dF (²) = (α²i)
3 E [Vαiαiαii ]
E [V 2i ]+ 3α²iα
²²i
E [V αiαii ]
E [V 2i ]
+3 (α²i)2
RV αiαii d∆iTE [V 2i ]
+ 3α²²i
RV αii d∆iTE [V 2i ]
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
ZV θαiαii dF (²) + 4α²iα
θ²i
ZV θαiαii dF (²)
+ (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
ZV αiαiαii dF (²)
+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
ZV αiαii d∆iT + 2α
²iαθθi
ZV αiαii d∆iT + 4α
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
ZV αiαiαii dF (²)
+α²²²i αθi
ZV αiαii dF (²) + 3α²²i α
θ²i
ZV αiαii dF (²) + 3α²iα
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
ZV αiαii d∆iT + 6α
²iαθ²i
ZV αiαii d∆iT + 3α
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
ZV αiαii d∆iT + 4α
²²²i
ZV αii d∆iT
Evaluating at ² = 0, we obtain
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
RV θαii d∆iTE [V 2i ]
+¡αθi¢3 R V αiαiαii d∆iT
E [V 2i ]+ 3αθiα
θθi
RV αiαii d∆iTE [V 2i ]
+ αθθθi
RV αii d∆iTE [V 2i ]
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
RV θθαii d∆iTE [V 2i ]
+ 4αθ²i
RV θαii d∆iTE [V 2i ]
+ 2αθθ²i
RV αii d∆iTE [V 2i ]
+ 2α²i¡αθi¢2 R V αiαiαii d∆iT
E [V 2i ]
+4αθiαθ²i
RV αiαii d∆iTE [V 2i ]
+ 2α²iαθθi
RV αiαii d∆iTE [V 2i ]
+ 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
RV θαiαii d∆iTE [V 2i ]
+ 3α²²i
RV θαii d∆iTE [V 2i ]
+ 3 (α²i)2 αθi
RV αiαiαii d∆iTE [V 2i ]
+ 3α²²i αθi
RV αiαii d∆iTE [V 2i ]
+6α²iαθ²i
RV αiαii d∆iTE [V 2i ]
+ 3αθ²²i
RV αii d∆iTE [V 2i ]
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
RV αiαiαii d∆iTE [V 2i ]
+ 12α²iα²²i
RV αiαii d∆iTE [V 2i ]
+ 4α²²²i
RV αii d∆iTE [V 2i ]
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
ZV θθαiαii dF (²)
+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
ZV θαiαii dF (²)
+4αθiαθθθ²i
ZV αiαii dF (²) + α²i
¡αθi¢4 Z
V αiαiαiαiαii dF (²) + 4¡αθi¢3αθ²i
ZV αiαiαiαii dF (²)
+6¡αθi¢2αθθ²i
ZV αiαiαii dF (²) + 4αθ²i α
θθθi
ZV αiαii dF (²) + 3α²i
¡αθθi¢2 Z
V αiαiαii dF (²)
+6αθθi αθθ²i
ZV αiαii dF (²) + 6α²iα
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
ZV αiαiαii dF (²) + 12αθiα
θ²i α
θθi
ZV αiαiαii dF (²) + 6
¡αθi¢2α²iα
θθi
ZV αiαiαiαii dF (²)
+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
ZV αii d∆iT + 4α
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
ZV θθαiαii dF (²)
+3 (α²i)2αθθi
ZV θαiαiαii dF (²) + (α²i)
2αθθθi
ZV αiαiαii dF (²) + 3α²²i α
θθi
ZV θαiαii dF (²)
+α²²i αθθθi
ZV αiαii dF (²) + 6α²iα
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
V αiαiαii dF (²)
+3αθ²²i αθθi
ZV αiαii dF (²) + 3αθθi α
²²i α
θi
ZV αiαiαii dF (²) + 3αθθi (α
²i)2αθi
ZV αiαiαiαii dF (²)
+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
ZV αiαiαii d∆iT
+6αθiαθθ²i
ZV αiαii d∆iT + 6α
θ²i α
θθi
ZV αiαii d∆iT + 6α
²iαθi
ZV θθαiαii d∆iT
+6α²iαθθi
ZV θαiαii d∆iT + 2α
²iαθθθi
ZV αiαii d∆iT + 6α
²i
¡αθi¢2 Z
V θαiαiαii d∆iT
+6αθiα²iαθθi
ZV αiαiαii d∆iT + 6α
θ²i
ZV θθαii d∆iT + 6α
θθ²i
ZV θαii d∆iT
+2αθθθ²i
ZV αii d∆iT + 2α
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
ZV αiαiαiαii dF (²)
+2α²²²i αθi
ZV θαiαii dF (²) + 3α²iα
θθ²²i
ZV αiαii dF (²) + 3 (α²i)
2αθθ²i
ZV αiαiαii dF (²)
+6αθ²²i α²iαθi
ZV αiαiαii dF (²) + 3α²iα
θθi α
²²i
ZV αiαiαii dF (²) + 6αθiα
²²i α
θ²i
ZV αiαiαii dF (²)
+6α²iαθiα
²²i
ZV θαiαiαii dF (²) + 6αθi (α
²i)2αθ²i
ZV αiαiαiαii dF (²) + 3α²i
¡αθi¢2α²²i
ZV αiαiαiαii dF (²)
+ (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
ZV θαiαii dF (²) + 3α²iα
²²i
ZV θθαiαii dF (²)
+6 (α²i)2 αθi
ZV θαiαiαii d∆iT + 6α
θiα
θ²²i
ZV αiαii d∆iT + 12α
²iαθ²i
ZV θαiαii d∆iT + 6α
²iαθθ²i
ZV αiαii d∆iT
+3α²²i αθθi
ZV αiαii d∆iT + 3 (α
²i)2αθθi
ZV αiαiαii d∆iT + 3α
²²i
¡αθi¢2 Z
V αiαiαii d∆iT
+3 (α²i)2 ¡αθi¢2 Z
V αiαiαiαii d∆iT + 6α²²i α
θi
ZV θαiαii d∆iT + 12α
θiα
²iαθ²i
ZV αiαiαii d∆iT
+3αθθ²²i
ZV αii d∆iT + 3α
²²i
ZV θθαii d∆iT + 6α
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
ZV αiαiαii dF (²)
+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
ZV αiαii dF (²) + 12α²iα
θ²i α
²²i
ZV αiαiαii dF (²) + α²²²²i
ZV θαii dF (²)
+αθ²²²²i
ZV αii dF (²) + 4α²iα
²²²i
ZV θαiαii dF (²) + 4α²iα
θiα
²²²i
ZV αiαiαii dF (²)
+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
ZV αiαii d∆iT + 12α
²iαθiα
²²i
ZV αiαiαii d∆iT + 4 (α
²i)3ZV θαiαiαii d∆iT
+4α²²²i
ZV θαii d∆iT + 4α
θ²²²i
ZV αii d∆iT + 12α
²²i α
θ²i
ZV αiαii d∆iT + 12α
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
ZV αiαii dF (²) + 5α²iα
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
ZV αiαiαii d∆iT + 20α
²iα²²²i
ZV αiαii d∆iT
Evaluating at ² = 0, we obtain
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 αiαiαiαii ]
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 ]
E [V 2i ]+ 12αθiα
²iαθθi
EhV θαiαiαii
iE [V 2i ]
+4αθiα²iαθθθi
E [V αiαiαii ]
E [V 2i ]+ 12αθiα
θ²i α
θθi
E [V αiαiαii ]
E [V 2i ]+ 6
¡αθi¢2α²iα
θθi
E [V αiαiαiαii ]
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
RV αiαii d∆iTE [V 2i ]
+ 6¡αθi¢2αθθi
RV αiαiαii d∆iTE [V 2i ]
+ 12αθiαθθi
RV θαiαii d∆iTE [V 2i ]
+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
RV θαii d∆iTE [V 2i ]
+ αθθθθi
RV αii d∆iTE [V 2i ]
+ 4αθi
RV θθθαii d∆iTE [V 2i ]
+ 6αθθi
RV θθαii d∆iTE [V 2i ]
+¡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
RV θαiαii d∆iTE [V 2i ]
+ 2α²i¡αθi¢3 R V αiαiαiαii d∆iT
E [V 2i ]+ 6
¡αθi¢2αθ²i
RV αiαiαii d∆iTE [V 2i ]
+6αθiαθθ²i
RV αiαii d∆iTE [V 2i ]
+ 6αθ²i αθθi
RV αiαii d∆iTE [V 2i ]
+ 6α²iαθi
RV θθαiαii d∆iTE [V 2i ]
+6α²iαθθi
RV θαiαii d∆iTE [V 2i ]
+ 2α²iαθθθi
RV αiαii d∆iTE [V 2i ]
+ 6α²i¡αθi¢2 R V θαiαiαii d∆iT
E [V 2i ]
+6αθiα²iαθθi
RV αiαiαii d∆iTE [V 2i ]
+ 6αθ²i
RV θθαii d∆iTE [V 2i ]
+ 6αθθ²i
RV θαii d∆iTE [V 2i ]
+2αθθθ²i
RV αii d∆iTE [V 2i ]
+ 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 αiαiαiαii ]
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 αiαiαiαii ]
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
RV αiαii d∆iTE [V 2i ]
+ 12α²iαθ²i
RV θαiαii d∆iTE [V 2i ]
+6α²iαθθ²i
RV αiαii d∆iTE [V 2i ]
+ 3α²²i αθθi
RV αiαii d∆iTE [V 2i ]
+ 3 (α²i)2αθθi
RV αiαiαii d∆iTE [V 2i ]
+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
RV θαiαii d∆iTE [V 2i ]
+12αθiα²iαθ²i
RV αiαiαii d∆iTE [V 2i ]
+ 3αθθ²²i
RV αii d∆iTE [V 2i ]
+ 3α²²i
RV θθαii d∆iTE [V 2i ]
+ 6αθ²²i
RV θαii d∆iTE [V 2i ]
+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 αiαiαiαii ]
E [V 2i ]+ 4 (α²i)
3αθ²i
E [V αiαiαiαii ]
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
RV θαiαii d∆iTE [V 2i ]
+ 12 (α²i)2αθ²i
RV αiαiαii d∆iTE [V 2i ]
+ 4 (α²i)3αθi
RV αiαiαiαii d∆iTE [V 2i ]
+4α²²²i αθi
RV αiαii d∆iTE [V 2i ]
+ 12α²iαθiα
²²i
RV αiαiαii d∆iTE [V 2i ]
+ 4 (α²i)3
RV θαiαiαii d∆iTE [V 2i ]
+4α²²²i
RV θαii d∆iTE [V 2i ]
+ 4αθ²²²i
RV αii d∆iTE [V 2i ]
+ 12α²²i αθ²i
RV αiαii d∆iTE [V 2i ]
+12α²iαθ²²i
RV αiαii d∆iTE [V 2i ]
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
RV αiαii d∆iTE [V 2i ]
+ 5α²²²²i
RV αii d∆iTE [V 2i ]
+30 (α²i)2α²²i
RV αiαiαii d∆iTE [V 2i ]
+ 20α²iα²²²i
RV αiαii d∆iTE [V 2i ]
18