New tools for understanding the local asymptotic power of panel unit root tests

Journal of Econometrics 188 (2015) 59–93

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Journal of Econometrics journal homepage: www.elsevier.com/locate/jeconom

New tools for understanding the local asymptotic power of panel unit root tests✩ Joakim Westerlund a,b,∗ , Rolf Larsson c a

Lund University, Sweden

b

Financial Econometrics Group, Centre for Research in Economics and Financial Econometrics, Deakin University, Australia

c

Uppsala University, Sweden

article

info

Article history: Received 24 June 2013 Received in revised form 27 June 2014 Accepted 30 March 2015 Available online 20 April 2015 JEL classification: C22 C23 Keywords: Panel unit root test Local asymptotic power Infinite-order approximation Moment expansion

abstract Motivated by the previously documented discrepancy between actual and predicted power, the present paper provides new tools for analyzing the local asymptotic power of panel unit root tests. These tools are appropriate in general when considering panel data with a dominant autoregressive root of the form ρi = 1 + ci N −κ T −τ , where i = 1, . . . , N indexes the cross-sectional units, T is the number of time periods and ci is a random local-to-unity parameter. A limit theory for the sample moments of such panel data is developed and is shown to involve infinite-order series expansions in the moments of ci , in which existing theories can be seen as mere first-order approximations. The new theory is applied to study the asymptotic local power functions of some known test statistics for a unit root. These functions can be expressed in terms of the expansions in the moments of ci , and include existing local power functions as special cases. Monte Carlo evidence is provided to suggest that the new results go a long way toward bridging the gap between actual and predicted power. © 2015 Elsevier B.V. All rights reserved.

1. Motivation Consider the problem of testing for a unit root in the panel data variable {yi,t }Tt=1 , and assume for simplicity that the data generating process (DGP) is given by yi,t = ρi yi,t −1 + εi,t , where yi,0 = 0 and εi,t ∼ N (0, 1). The analysis of the local power of various unit root test statistics when applied to such variables has attracted much attention in recent years (see Westerlund and Breitung, 2013, Section 2, for a review of this literature). The limit theory makes extensive use of the laws of large numbers and central limit theory, leading to local asymptotic power functions that are stated in terms of moments of various sample quantities. Almost all this theory assumes that ρi = 1 + ci N −κ T −1 , where κ = 1/2 or κ = 1/4, depending on whether the data contain incidental in-

✩ A previous version of the paper was presented at a seminar at University of Barcelona. The authors would like to thank seminar participants, and in particular Cheng Hsiao (Editor) Peter Phillips, Giuseppe Cavaliere, Josep Carrion-i-Silvestre, David Harris, Vasilis Sarafidis, an Associate Editor, and two anonymous referees for many valuable comments and suggestions. ∗ Correspondence to: Department of Economics, Lund University, Box 7082, 220 07 Lund, Sweden. Tel.: +46 46 222 8997; fax: +46 46 222 4613. E-mail address: [email protected] (J. Westerlund).

http://dx.doi.org/10.1016/j.jeconom.2015.03.043 0304-4076/© 2015 Elsevier B.V. All rights reserved.

tercept and trend terms.1 Without such terms, it has been found that in the above DGP with κ = 1/2 local power depends on the mean of ci , but not on the variance, or any other moment for that matter (see, for example, Breitung, 2000; Moon and Perron, 2004; Moon et al., 2007). This means that one can just as well assume that c1 = · · · = cN = c; there are no additional insights to be gained by allowing ci to vary, at least not from a power point of view. The fact that according to theory power should only depend on the mean of ci is somewhat of an anomaly, because in Monte Carlo studies there is also a dependence on higher moments. Indeed, as Moon and Perron (2008, page 91) conclude from their simulation study, ‘‘despite our theoretical results, there is somewhat of a power loss against a heterogeneous alternative in finite samples’’ (see Moon et al., 2007, Section 7, for a similar finding). Let us illustrate this point using as an example the pooled ordinary least

1 Moon and Phillips (1999) show that the maximum likelihood estimator of the local-to-unity parameter in near unit root panels with individual-specific trends is inconsistent. They call this phenomenon, which arises because of the presence of an infinite number of nuisance parameters, an ‘‘incidental trend problem’’, because it is analogous to the well-known incidental parameter problem in dynamic fixed-T panels.

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squares (OLS) t-statistic for a unit root, whose local asymptotic distribution (as N , T → ∞) in the simple DGP considered here is given by

µ1 √c + N (0, 1),

(1)

2

where µ1c = E (ci ) (see, for example, Moon et al., 2007, Section 3). If µ1c < 0 and the test is set up as left-sided, the asymptotic

√

local power implied by (1) is given by Φ (−µ1c / 2 − zα ), where Φ (x) is the cumulative distribution function of N (0, 1) and zα = Φ −1 (1 − α) is the (1 − α)-quantile of that same distribution. In Fig. 1 we plot the 5% power as a function of the variance of ci when ci is drawn from a uniform distribution with mean −5. The variance ranges from 0 to 8, suggesting that the support of ci varies from −5 to [−9.9, −0.1]. There are three curves representing the theoretical local power, and the empirical power for N = 20 and N = 50 when T = 500. Since according to theory power only depends on the mean of ci , the theoretical power curve is flat. We see that when N = 20 the empirical power function is way off the theoretical prediction. It starts at about 7% below (note how the scale of the horizontal axis goes from 75% to 100%) and then the difference just gets larger as the variance of ci increases. Of course, since in this case N is relatively small the asymptotic approximation is not going to be perfect, and some variations are to be expected. However, the same pattern is observed also when N = 50, although the vertical distance to the theoretical power curve is not as large as before. In other words, there seems to be a variance effect at work here that cannot be explained by theory, and that seems to go away, although only very slowly, as N increases. This example suggests that the asymptotic approach commonly used for analyzing the local power of panel unit root tests may not be sharp enough to capture actual behavior. It is therefore necessary to consider alternative frameworks that are sensible not only asymptotically but also in finite samples, and this paper takes a step in this direction. However, unlike existing works, here the rate of shrinking of the local alternative is not pre-specified but is instead set equal to ρi = 1 + ci N −κ T −τ , which includes all previously considered local alternatives as special cases, including the usual time series specification with κ = 0 and τ = 1 (see, for example, Phillips, 1988). Within this framework, terms that would otherwise be negligible are retained, leading to a more detailed asymptotic analysis. Our main contribution is to show how the moments of some commonly encountered sample quantities can be written as infinite-order (IO) series expansions in the moments of ci with coefficients that depend on the rate of shrinking of the local alternative, and hence also on κ and τ . The expansions are up to an error that is of order T −1 , indicating that they should lead to good approximations even for very small values of N, provided that T is large enough. The significance of the new moment expansions is that they can be used to obtain IO local asymptotic power functions of any test statistic that depend on the sample quantities considered. A further contribution of this paper is to provide some detailed illustrations of how such a IO local power analysis can be carried out. As a first illustration we revisit the above example. The results suggest an IO local asymptotic distribution where the meanonly distribution in (1) can be seen as a mere first-order (FO) approximation. Here, the effect of the second moment of ci is of order N −1/2 , with subsequent moments in the expansion decaying according to additional powers of N −1/2 . The results from a small Monte Carlo study show that the new IO theory is very accurate, and that it goes a long way toward explaining the variance effect seen in Fig. 1. As a second illustration, we consider the test statistics of Moon and Perron (2008), whose main difference lies with how they

are adjusted for bias in the presence of incidental intercepts. Our interest in these statistics originates with the fact while it has been observed that the value of κ compatible with non-negligible local asymptotic power depends on the bias adjustment method, little is known as to the working of this dependence (see, for example, Moon and Perron, 2004, 2008; Moon et al., 2006, 2007). The new theory delivers significant insight in this regard. In particular, it shows how different bias adjustment methods cause cancellation of different terms in the moment expansions, and that the observed variation in κ is brought about by a cancellation of the leading term. In a third illustration, we show how the results provided in the present paper can be used to derive IO results also for test statistics that are not stated directly in terms of the moments considered here, but whose asymptotic distributions can still be obtained from these moments. In particular, the uniformly most powerful test of Becheri et al. (2015) is considered, whose FO local asymptotic distribution is equal to that in (1). By deriving the IO version of this distribution we obtain what might be referred to as an ‘‘IO local power envelop’’. In our fourth and final illustration, we show how the provided moment expansions, which are derived under many simplifying assumptions, can be used to derive IO local asymptotic distributions also under more general conditions. As an example, we take one of the test statistics previously considered by Moon and Perron (2004), which is general enough to accommodate heteroskedastic, and serial and cross-section correlated errors. The plan of the paper is as follows. Section 2 lays out DGP considered, which is chosen so as to exclude all distractions but the features that drive local asymptotic power. Section 3 reports the results of the IO expansions of the moments considered, which are put in perspective through a comparison with the corresponding FO moment approximations. Section 4 is concerned with the local power illustrations. Section 5 concludes. 2. Model The DGP is similar to the one considered in Section 1 and is given by yi,t = βi′ dt + ui,t ,

p

(2)

ui,t = ρi ui,t −1 + εi,t ,

(3)

where ui,0 = 0, εi,t is independently and identically distributed (iid) with E (εi,t ) = 0, E (εi2,t ) = σε2 > 0 and E (εi4,t ) < ∞. In the

derivations we assume that σε2 is known (as in, for example, Moon p et al., 2007); hence, we can just as well set σε2 = 1. Also, dt = p ′ (1, . . . , t ) is a (p + 1)-dimensional vector of trends, for which we consider three specifications; (i) no deterministic terms (p = −1), (ii) incidental intercepts (p = 0), and (iii) incidental trends (p = 1). Since in practice incidental intercepts are always included, (ii) and (iii) are the empirically most relevant specifications; however, (i) is relevant too, for its simplicity, and we are going to use it here as an illustrative example. We further assume that

ρi = exp(N −κ T −τ ci ) = exp(T −1 λNT ci ),

(4)

where λNT = N −κ T 1−τ < ∞. The drift parameter ci is assumed to be iid and independent of εj,t for all i, j and t. All the moments of ci exist, and in what follows it is going to be convenient to denote j j these as µc = E (ci ) for j ≥ 1 and µ0c = 0. The null hypothesis of interest is that of a unit root (c1 = · · · = cN = 0), which can be formulated in terms of the moments of ci as H0 : µ2c = 0. The relevant alternative hypothesis is given by H1 : µ2c > 0 (corresponding to ci ̸= 0 for some i). Unlike existing studies where τ = 1 and κ = 1/4 or κ = 1/2 is assumed from the outset (see, for example, Breitung, 2000;

J. Westerlund, R. Larsson / Journal of Econometrics 188 (2015) 59–93

61

Fig. 1. Local power as a function of the variance of ci .

Moon and Perron, 2008; Moon et al., 2006, 2007), except for the requirement that λNT should be bounded, in the present study we make no assumptions regarding the values taken by (τ , κ), which are instead considered a part of the analysis. One of the advantages of this is that it points to a previously unexplored relationship between (τ , κ) and the relative expansion rate of N and T . Let us therefore define

θ=

ln(T ) ln(N )

,

(5)

such that T = N θ (see Moon and Perron, 2004, Assumption 10, for a similar parametrization). Hence, by setting the value of θ we can control the relative expansion rate of N and T . At this point, the only requirement is that θ > 0. One implication of (5) is that λNT can be written as a function of N only;

λNT = λN = N θ(1−τ )−κ ,

(6)

suggesting that the condition that λNT < ∞ can be reformulated as θ(1 − τ ) − κ ≤ 0. Remark. Many of the above assumptions are restrictive, but can be relaxed at the expense of added expositional and technical complexity. However, this seems unnecessary in the present case, because moment expansions are unlikely to be affected by the removal of nuisance parameters. In fact, having their genesis in invariance principles, which apply under very general conditions when it comes to the heteroskedasticity and serial correlation of the errors, the asymptotic expansions developed here are widely applicable (as opposed to, for example, classical Edgeworth expansions). Cross-section dependence in the form of common factors can be accommodated as suggested in, for example, Bai and Ng (2004, 2010), Moon and Perron (2004), and Phillips and Sul (2003) by using ‘‘defactored’’ data. Moreover, while restrictive when p = −1, the assumption that ui,0 = 0 is irrelevant when p ≥ 0, which is also the only empirically relevant case. This is illustrated in Section 4.4 where we allow for nonzero initialization, and also heteroskedastic and serially and cross-sectionally correlated errors. The assumption that ci has all its moments is also not necessary, but is made here in order to keep with the rest of

the local asymptotic power literature.2 That the finite moment restriction is stronger than necessary is particularly obvious under the unit root null, in which case ci and all its moments are zero. The same applies to the requirement that λNT < ∞ (or θ (1 − τ ) − κ ≤ 0). The reason for why it is needed under the alternative is to ensure that the rate of shrinking is at least T −1 . If the rate of shrinking is lower than this, then the expansions provided herein are no longer valid. The assumption in (5) is only for ease of exposition in the illustration of Section 4 not a restriction.3 The statement in (6) is a direct implication of (5), and is hence not an assumption. 3. Main results Let us introduce the OLS detrending operator Dp , which is such that T 

Dp yi,t = yi,t −

p

yi,k ak,t ,

k=1 p′

p ak,t

p p′

p

where = dk ( n=2 dn dn )−1 dt . Define the following sample quantities based on the OLS detrended data: AiT ,p =

BiT ,p =

T 1

T t =2

T

Dp yi,t −1 Dp ∆yi,t ,

T 1 

T 2 t =2

(Dp yi,t −1 )2 . N

Let ANT ,p = N −1 i=1 AiT ,p with a similar definition of BNT ,p . Almost all (within type) panel unit root tests statistics considered in the literature can be written in terms of ANT ,p and

2 Most studies assume that the support of c is bounded (see, for example, Moon i and Perron, 2004, 2008; Moon et al., 2007), which obviously implies finite moments. 3 The only change needed in order to relax (5) is to replace all instances of λ N

in the reported moment expansions (see Theorem 1) by λNT = N −κ T 1−τ . However, since the conclusions are qualitatively the same, and since assuming T = N θ greatly simplifies the discussion of the results, we opt for the less general specification.

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J. Westerlund, R. Larsson / Journal of Econometrics 188 (2015) 59–93

BNT ,p . The most common statistic by far is the t-ratio, which in its most basic √ form and under the conditions of Section 2 can −1/2 be written as NANT ,p BNT ,p . Examples of works involving test statistics of this type include Bai and Ng (2010), Levin et al. (2002), Moon and Perron (2004, 2008), Moon et al. (2007), Westerlund and Larsson (2009), and Westerlund (2015). Another possibility that was recently considered by Westerlund and Larsson (2012) is to use the Lagrange multiplier (LM) principle, leading to test 2

−1

statistics that are of the form NANT ,p BNT ,p . Bai and Ng (2010), and Westerlund (2015) take another route and consider a modified Sargan–Bhargava (MSB) type test statistic, which can be written √ as NBNT ,p . Of course, the actual test statistics considered in these works are more involved than this (and may involve, for example, correction for bias, heteroskedasticity, and serial and cross-section dependence); however, the sample quantities from which the asymptotic distributions derive are still given by ANT ,p and BNT ,p . In Section 4 we elaborate on this. Key elements in the derivation of asymptotic distributions of functions of ANT ,p and BNT ,p are the first and second moments of AiT ,p and BiT ,p . Theorem 1, which is our main result, provides the IO representations of these moments in the empirically relevant case when p > −1.

∞ ∞   192 · 2j + 12 j+1 j+1 192 · 2j j j λ N µc + λN µc (j + 5)! (j + 5)! j=0 j=0 ∞  64 · 2j + 4 j+2 j+2 − λ µ (j + 5)! N c j=0 ∞  7 1 8 · 2j j+3 j+3 5 + λN µc + − λN µ1c + λ2N µ2c , 3 6 6 j + 5 ( )! j=0

µ1B1 = −

µ2A1 =

∞  108 · 4j+8 − 432 · 3j+8 + 504 · 2j+8 − 144

(j + 8)!

j=0

+

∞  −216 · 4j+8 + 432 · 3j+8 + 288 · 2j+8 − 720

(j + 8)!

j=0

+

∞  180 · 4j+8 − 72 · 3j+8 − 456 · 2j+8 − 24

(j + 8)!

j=0

+

(j + 8)!

∞  21 · 4j+8 + 24 · 3j+8 + 14 · 2j+8 + 20

(j + 8)!

j=0

+

∞  −3 · 4j+8 − 3 · 3j+8 − 6 · 2j+8 − 13

(j + 8)!

j=0

Theorem 1. Under the conditions laid out in Section 2,

+

m −1 E (Am ), iT ,p ) = µAp + O(T

+

where p ∈ {0, 1}, m ∈ {1, 2} signifies the power of AiT ,p and BiT ,p , the O(T −1 ) remainder terms are independent of λN and the moments of ci , and

−

∞  2j+1 j + 1 j+2 j+2 λ µ , =− + 2 (j + 4)! N c j=0 ∞  2j (j − 1) + 2

(j + 3)!

j=0

µ

2 A0

=

µ2B1 =

(j + 4)!

λ µ

j c

∞  −192 · 4j + 243/2 · 3j + 8 · 2j − 3/2 j+1 j+1 λ N µc (j + 4)! j=0 ∞ ∞  48 · 4j + 2 · 2j − 1 j+2 j+2  −4 · 2j j+3 j+3 λN µc + λ µ + (j + 4)! (j + 4)! N c j=0 j=0

µ

=

1 6

3

1

2

3

+ λN µ1c − λ2N µ2c ,

∞  3 · 4j+5 − 6 · 3j+6 + 33 · 2j+5 − 18

(j + 6)!

j=0

λjN µjc

∞  −3 · 4j+5 + 3j+7 − 7 · 2j+5 + 1 j+1 j+1 λ N µc (j + 6)! j=0 ∞  3 · 4j+4 + 19 · 2j+3 − 6 j+2 j+2 + λN µc (j + 6)! j=0 ∞  5 · 2j+4 j+3 j+3 5 27 1 − λ N µc + + λN µ1c − λ2N µ2c , j + 6 12 20 3 ( )! j=0

+

µ1A1 =

∞ 1  (2λN )j+1

µcj+1 + 8

∞  (2j+1 − 1)λj

N

µjc

2 j=0 (j + 2)! (j + 2)! j=0 ∞ ∞ j   −16 · 2 + 2 j j 16 · 2j j+1 j+1 +6 λN µc + 6 λ µ (j + 4)! (j + 4)! N c j=0 j=0

+6

∞  −16 · 2j + 1 j=0

1680 2113 4 210

+

(j + 4)!

15681

λjN+2 µjc+2 − 1 − 7λN µ1c ,

λjN+4 µcj+4

λjN+5 µjc+5

λjN+6 µjc+6

λN µ4c +

214139

λN µ1c −

70 317

504

λ5N µ5c −

(j + 10)!

1260 2 6 315

λ2N µ2c +

37969 630

λ3N µ3c

λN µ6c ,

λjN µjc

∞  −216 · 4j+10 − 432 · 3j+10 + 432 · 2j+10 + 432

(j + 10)!

j=0 j N

+

+

181267

∞  108 · 4j+10 − 216 · 2j+10

+

λjN+3 µjc+3

∞  −2j+6 j+7 j+7 λ µ (j + 8)! N c j=0

j=0

λjN µjc ,

∞  192 · 4j − 243 · 3j + 56 · 2j − 1 j=0

2 B0

−

1

µ1B0 =

(j + 8)!

j=0

m −1 E (Bm ), iT ,p ) = µBp + O(T

µ1A0

∞  3 · 4j+6 + 13 · 2j+5 + 2

λjN+1 µjc+1

λjN+2 µjc+2

∞  −81 · 4j+8 − 54 · 3j+8 + 90 · 2j+8 + 102 j=0

+

λjN µjc

× λjN+1 µjc+1 ∞  180 · 4j+10 + 576 · 3j+10 + 408 · 2j+10 − 672 j+2 j+2 + λN µc (j + 10)! j=0 ∞  −81 · 4j+10 − 288 · 3j+10 − 498 · 2j+10 − 192 j+3 j+3 + λ N µc (j + 10)! j=0 ∞  21 · 4j+10 + 66 · 3j+10 + 168 · 2j+10 + 126 j+4 j+4 + λ N µc (j + 10)! j=0 ∞  −3 · 4j+10 − 6 · 3j+10 − 40 · 2j+10 − 30 j+5 j+5 + λN µc (j + 10)! j=0 ∞  3 · 4j+8 + 75 · 2j+7 + 12 j+6 j+6 + λ N µc (j + 10)! j=0 ∞  −5 · 2j+8 j+7 j+7 λ µ + (j + 10)! N c j=0 − −

13079 420 40213 18144

+

4059 70

λ4N µ4c +

λN µ1c − 473 3360

1709 42

λ5N µ5c −

λ2N µ2c + 1 567

51481 3780

λ3N µ3c

λ6N µ6c .

The expressions provided in Theorem 1 are quite different from existing ones based on FO theory (see, for example, Breitung, 2000; Moon and Perron, 2008; Moon et al., 2006). They may look daunting; however, the idea is quite straightforward. As an illustration, consider the simple case when p = −1, such that Dp yi,t = yi,t = ui,t . Because in practice incidental intercepts

J. Westerlund, R. Larsson / Journal of Econometrics 188 (2015) 59–93

are always allowed for, this case is not covered by Theorem 1; however, it is useful as an illustration. We start t by considering t −s E (BiT ,−1 ). Because ui,0 = 0, we have ui,t = εi,s , and s=1 ρi therefore E(

u2i,t −1

t −1  t −1 

|ci ) =

t −1 

ρ

(εi,s εi,k )

ρi2(t −1−s) E (εi2,s ) =

s=1

t −1 

ρi2(t −1−s) ,

=

s=1

in t = 1, . . . , T and γ = (t − 1)T −1 , . . . , tT −1 , from which we obtain exp(tT −1 ) = exp(γ )[1 + O(T −1 )]. By using this result, ρi = exp(T −1 λN ci ), and the fact that λN < ∞, letting t = ⌊γ T ⌋ and s = ⌊ηT ⌋, where ⌊x⌋ is the integer part of x, we obtain T

E(

u2i,t −1

|ci ) =

t −1 1

T s =1



γ

= η=0

exp[2T

−1

(t − 1 − s)λN ci ]

γ

η=0

exp[2(γ − η)λN ci ]dη + O(T −1 ),

exp[2(γ − η)λN ci ]dη =

γ



η=0

1



γ =0

γ η=0

=

1



γ =0

1 2

γ dγ + O(λN )

+ O(λN ),

(7)

and so, by the continuous mapping theorem, E (BiT ,−1 ) =

T 1  −1 T E (u2i,t −1 ) T t =2 1





γ

= γ =0

=

1 2

η=0

exp[2(γ − η)λN ci ]dηdγ + O(T −1 )

+ O(λN ) + O(T −1 ),

(8)

which, in view of O(λN ) = O(N −1/2 ) = o(1), has 1/2 as its asymptote. This is the FO representation of E (BiT ,−1 ). Specifically, since the O(T −1 ) remainder is just due to the continuous time approximation, which is independent of λN , what we refer to here as the ‘‘FO approximation’’ is the one given in (7). To the best of our knowledge, this is the only known representation of E (BiT ,−1 ) in the literature (see, for example, Breitung, 2000; Moon and Perron, 2004; Moon et al., 2007; Levin et al., 2002). One of the points of the present paper is to show how one can actually work without FO approximations. Indeed, since



γ

η=0

[exp(2γ λN ci ) − 1] + O(T −1 ),

exp[2(γ − η)λN ci ]dη =

1 2λN ci

[exp(2γ λN ci ) − 1],

1

2λN ci

=

γ =0



1 2λN ci

[exp(2γ λN ci ) − 1]dγ + O(T −1 ) 

1

[exp(2λN ci ) − 1] − 1 + O(T −1 ),

2λN ci

∞

j=0

xj /j! and j! = (j − 1)!j,



1

2λN ci

1



1

[exp(2λN ci ) − 1] − 1    ∞  1 (2λN ci )j −1 −1 2λN ci j=0 j!

2λN ci



1 2λN ci

∞  (2λN ci )j

(j + 2)!

.

It follows that if we let

[1 + O(λN )]dη = γ + O(λN ).

exp[2(γ − η)λN ci ]dηdγ =



j =0

Since this result is independent of ci , it is also the (limiting) unconditional expectation of T −1 u2i,t −1 . It follows that



2λN ci

where, via exp(x) =

=

which holds uniformly in (γ , η) ∈ [0, 1] × [0, 1]. Existing works on the local power of panel unit root tests are based on first assuming particular values for (τ , κ), and then evaluating the local power for that particular combination only. Suppose therefore, as in Section 1, that (τ , κ)  = (1, 1/2). This ∞ j implies that λN = N −1/2 , which, via exp(x) = j=0 x /j!, yields the following FO approximation of the conditional expectation of T −1 u2i,t −1 :



1

T −1 E (u2i,t −1 |ci ) =

E (BiT ,−1 |ci ) =

where the last two equalities hold because E (εi,s εi,k ) = 0 for all s ̸= k and E (εi2,t ) = 1. Also, |tT −1 − w| = O(T −1 ) uniformly

−1

we have

which in turn implies 2(t −1)−s−k E i

s=1 k=1

=

63

µ1B−1 =

∞  E [(2λN ci )j ]

(j + 2)!

j =0

=

1 2

=

λN µ1c

+

3

+

∞  (2λN )j µjc j =0

λ2N µ2c

(j + 2)!

+ ···,

6

(9)

then E (BiT ,−1 ) = µ1B−1 + O(T −1 ).

(10)

This is the IO representation of E (BiT ,−1 ), in which the FO approximation in (7) appears naturally as the first term in the expansion of µ1B−1 . The representation in (9) and (10) shows not only how E (BiT ,−1 ) depends on the moments of ci , but also how the effect of these moments depends on λN = N θ (1−τ )−κ , and hence on (θ , τ , κ). In Section 4 we illustrate the importance of these dependencies in the context of panel unit root testing. However, already here we see how the accuracy of the FO approximation depends critically on λN tending to zero, and, in finite samples, also on the rate at which this happens. In the extreme case when θ (1 − τ ) − κ = 0, λN = 1, and therefore the higher order terms in (9) are of the same order of magnitude as the FO term, thereby invalidating any finite-order approximation. This would be the case when τ = 1 and κ = 0, in which ρi = exp(T −1 τN ci ) = exp(T −1 ci ), so that now we are considering alternatives of the usual time series type (see, for example, Phillips, 1988). Hence, the slower the rate of shrinking of the local alternative, the poorer the approximation. t t −s Let us now consider E (AiT ,−1 ). Substitution of ui,t = s=1 ρi εi,s yields E (ui,t −1 ∆ui,t |ci )

=E

 t −1 

ρ

ε

t −1−s i,s i

 t 

t −1  t 

ε

v=2

s=1

=

ρ

t −v i,v i

ρi2t −1−s−v E (εi,s εi,v ) −

s=1 v=1

=

t −1  s=1

=

t −1  s=1

ρi2(t −s)−1 −

−

t −1 

ρi2(t −1−s)

s =1

ρit −1−s (ρit −s − ρit −1−s ).

ρ

ε

 |ci

w=1 t −1  t −1  s=1 w=1

t −1 

 t −1−w i,w i

ρi2(t −1)−s−w E (εi,s εi,w )

64

J. Westerlund, R. Larsson / Journal of Econometrics 188 (2015) 59–93

The derivative of ρit = exp(tT −1 λN ci ) with respect to t is given by T −1 λN ci exp(tT −1 λN ci ). This implies



ρit −1−s (ρit −s − ρit −1−s )

and E (B2iT ,0 ) (see in particular (A.16) and (A.20)), it is possible to show that

s=1

= λN ci =

1

γ



η=0

exp[2(γ − η)λN ci ]dη + O(T −1 )

[exp(2γ λN ci ) − 1] + O(T −1 ),

2 which we can use to obtain E (AiT ,−1 |ci ) =

T 1

T t =2  1 1

E (ui,t −1 ∆ui,t |ci )

∞ 1  E [(2λN ci )j+1 ]

(j + 2)!

2 j =0

λN µ1c

λ2N µ2c

=

j +1 ∞ 1  (2λN )j+1 µc

2 j =0

(j + 2)!

λ3N µ3c

+ ···, (11) 2 3 6 where the FO approximation is again given by the first term, we can show that =

+

+

E (AiT ,−1 ) = µ1A−1 + O(T −1 ).

(12)

Note how (12) is stated directly in terms of the expected value of the normalized sum of ui,t −1 ∆ui,t . An alternative way of arriving at the same result, and that is perhaps more common (see Moon and Perron, 2008; Moon et al., 2006), involves using (3) to substitute for ∆ui,t , giving E (ui,t −1 ∆ui,t |ci ) = (ρi − 1)E (u2i,t −1 |ci ) + E (ui,t −1 εi,t |ci ) = (ρi − 1)E (u2i,t −1 |ci ), where T −1 E (u2i,t −1 |ci ) is known from before. Consequently, since ρi = exp(T −1 λN ci ) = 1 + T −1 λN ci + Op (T −2 λ2N ) and Op (T −1 λ2N ) ≤ Op (T −1 ), we have

E (AiT ,−1 |ci ) =

E (B2iT ,−1 |ci ) = µ2B−1 + O(T −1 ),

(14)

=

∞ 3  2j (2j+1 − 1)

+ λN µc +

2

µ2B−1 = 6

7 12

+

13λN µc 15

∞  (2λN ci )j j =0

(j + 2)!

∞ 1  (2λN ci )j+1

2 j=0

(j + 2)!

leading to the result in (12).

+ O(T −1 ) + O(T −1 ),

λjN µjc − 10

(15) ∞  (2λN )j µjc j =0

23λ µ 2 N

30

2 c

1 4

(j + 3)!

+ ···.

(16)

The purpose of this section is to illustrate how the results reported in Section 3 can be used in deriving IO asymptotic distributions of panel unit root test statistics, and also to show how these compare to the corresponding FO distributions reported in the literature. The test statistics that we will consider are all based on ANT ,p and BNT ,p . In this section we therefore begin by deriving the asymptotic distributions of some useful transformations of these quantities. We then show how the test statistics can be written as functions of these transformed quantities, leading to a straightforward asymptotic analysis. We start by considering the probability limit of BNT ,p , and in so doing we apply the law of large numbers given in Corollary 1 of Phillips and Moon (1999). In the current case this requires verifying that BiT ,p is uniformly integrable over T , which can be done by using the same steps as in, for example, Moon and Phillips (2000, page 975).4 It follows that BNT ,p = µ1Bp + op (1),

(17)

which, via Taylor expansion, gives 1

(BNT ,p )−1/2 = (µ1Bp )−1/2 − (µ1Bp )−3/2 (BNT ,p − µ1Bp ) 2

+ Op ((BNT ,p − µ1Bp )2 ). ANT ,p = µ1Ap + op (1).

= λN ci

+ ···,

+

4. Illustrations

T t =2

T 1  −1 T E (u2i,t −1 |ci ) + Op (T −1 λ2N ) T t =2

+

2 j=0 (j + 1)!

Again, the results reported in the literature only refer to the FO terms in (15) and (16) (see, for example, Breitung, 2000; Moon and Perron, 2004; Moon et al., 2007; Levin et al., 2002).

We similarly have

(ρi − 1)E (u2i,t −1 |ci )

12

(j + 4)!

j ∞ 1  (2λN )j µc

λjN µjc −

17λ2N µ2c

∞  2j (2j+3 + 1) j =0

=

(j + 2)!

2 j=0 1

T 1

= λN ci

=

(13)

µ2A−1 =

Hence, letting

µ1A−1 =

E (A2iT ,−1 |ci ) = µ2A−1 + O(T −1 ),

where

[exp(2γ λN ci ) − 1]dγ + O(T −1 ) 2 γ =0   1 1 1 = exp(2λN ci ) − − 1 + O(T −1 ), 2 2λN ci 2λN ci ∞ j where, via substitution of exp(x) = j=0 x /j!,   ∞ 1 1 1  (2λN ci )j − −1 2 2λN ci j=0 j! 2λN ci     ∞  1 1 (2λN ci )j = −1 −1 2 2λN ci j=0 j!   ∞ 1 1  (2λN ci )j+1 = −1 2 2λN ci j=0 (j + 1)!   ∞ ∞ 1  (2λN ci )j 1  (2λN ci )j+1 = −1 = . 2 j=0 (j + 1)! 2 j=0 (j + 2)! =

E (B2iT ,−1 ) are more involved, and we therefore omit them. However, by following the same steps used in Appendix for obtaining E (A2iT ,0 )

t −1

E (ui,t −1 ∆ui,t |ci ) =

Naturally, the derivations of the expansions for E (A2iT ,−1 ) and

(18)

(19)

√

Consider N (ANT ,p − µ1Ap ). We now use the same steps as in Larsson et al. (2001, page 140) to verify that ZiT ,p = [AiT ,p −E (AiT ,p )] [E (A2iT ,p ) − [E (AiT ,p )]2 ]−1/2 satisfies the Lindeberg condition of

4 In the notation of Phillips and Moon (1999), we have Y = C Q , where C = 1 i,T i i ,T i and Qi,T = BiT ,p . Since Qi,T is iid, the conditions for Corollary 1 are that (i) |Qi,T | is uniformly integrable over T for all i and (ii) supi∈[1,N ] |Ci | ≤ ∞, where the latter is obviously satisfied in view of the fact that Ci = 1.

J. Westerlund, R. Larsson / Journal of Econometrics 188 (2015) 59–93

where the op (1) remainder term is due to

Theorem 2 of Phillips and Moon (1999), which is given by N 1 

E [ZiT2 ,p 1(|ZiT ,p | >

(BNT ,−1 − µ

√

N ϵ)] → 0

N i =1 as N , T → ∞ for all ϵ > 0. However, since in our case ZiT ,p is iid in i, the condition simplifies to E [

(|Z1T ,p | >

2 Z1T ,p 1

N −1/2

ZiT ,p →d N (0, 1)

as N , T → ∞, where →d signifies √convergence in distribution. In order to see the implications for N (ANT ,p − µ1Ap ), note first that by using the results reported in Section 3, E (A2iT ,p ) − [E (AiT ,p )]2 =

σA2p + O(T −1 ), where σA2p = µ2Ap − (µ1Ap )2 . Similarly, E (AiT ,p ) = µ1Ap + O(T −1 ), and therefore

N 1 

[E (AiT ,p ) − µ1Ap ]

N i=1 N

N i=1

(20)

√

as N , T → ∞ with θ > 1/2 such that N /T = N The same arguments can be used to show that

1/2−θ

= o(1).

√

N (BNT ,p − µ1Bp ) →d σBp N (0, 1),

where σB2p = µ2Bp − (µ1Bp )2 . Moreover,

√

N µ1A−1

t−1 ∼ 

√

N (ANT ,p + µ1Ap ) and

√

µ

N

4.1. The Section 1 example continued Under the assumptions of Section 2, the OLS t-statistic for a unit root is given

1 B−1

1/

N

T



NANT ,p

(Dp yi,t −1 )

2

√

t−1 = 

−

BNT ,−1

N µ1A−1

=  + µ1B−1

√ µ1A−1 N (BNT ,−1 − µ1B−1 ) 2(µ1B−1 )3/2

(23)

(24)

(µ1A−1 )2 σB2−1 4(µ1B−1 )3

Nµ

∞ √ 

N

1 A−1

= 1 + O(λN ),

(25)

(2λN )j+1 µjc+1 /(j + 2)!

j =0

 = µ1B−1



∞ 

(2λN )j µjc /(j + 2)!

j=0

=

BNT ,p

√

N (0, 1)

showing how the part of power that comes from affecting the variance of t−1 is negligible. Let us now consider the drift. According to (9) and (11),

(22)

where αˆ p = T −1 ANT ,p /BNT ,p is the OLS slope estimator in a pooled panel regression of Dp ∆yi,t onto Dp yi,t −1 . In this section we provide an explanation of the graphical evidence reported in Fig. 1, which is based on applying the above t-statistic to the raw data, that is, Fig. 1 is based on t−1 . The asymptotic distribution of this test statistic follows easily from the p = −1 results provided in Section We √ 3, and (20)–(22). √ √ begin by noting that, by substitution of NANT ,−1 = N µ1A−1 + N (ANT ,−1 −µ1A−1 ) in the numerator of t−1 (see (22)), and then application of the Taylor expansion in (18) to the denominator, NANT ,−1

4(µ1B−1 )3

√ ,

i=1 t =2

√

+

√ = 

+

1/2

µ1A−1 = O(λN ), where under (τ , κ) = (1, 1/2), λN = N −1/2 = o(1), we have that

√



µ1B−1

µ1B−1

(µ1A−1 )2 σB2−1

The presence of the moments of ci under H1 affects both the drift and variance of the asymptotic distribution of t−1 . As for the variance effect, since µ1B−1 = σA2−1 = µ2A−1 = 1/2 + O(λN ) and

σA2−1

Levin et al., 2002; Moon and Perron, 2008), and hence independent by normality.

tp =

+

σA2−1

as N , T → ∞ with θ > 1/2. This is the IO local asymptotic distribution of t−1 . We begin with a discussion of the interpretation of (23); then we also make some general remarks. As (23) makes clear, the asymptotic distribution of t−1 is comprised of two terms; (i) a drift term, and (ii) a normal variable with zero mean but potentially nonunit variance. If H0 is true, so that ci and all its moments are zero, according to (11), µ1A−1 = 0, and therefore the drift is zero, that is, t−1 is asymptotically mean zero. As for the variance under the same null, according to (11) and (15), σA2−1 = µ2A−1 −(µ1A−1 )2 =

(21)

(BNT ,p + µ1Bp ) are asymptotically uncorrelated (see, for example,

αˆ p



t−1 →d N (0, 1).

√ [AiT ,p − E (AiT ,p )] + Op ( NT −1 )

→d σAp N (0, 1)

)

is therefore given by

1 

= √

)/2(µ

N (ANT ,−1 − µ1A−1 )

µ2A−1 = 1/2. By using this, the fact that according to (9), µ1B−1 = 1/2 and µ1A−1 = 0, we have that under H0 the variance in (23) reduces to σA2−1 /µ1B−1 + (µ1A−1 )2 σB2−1 /4(µ1B−1 )3 = σA2−1 /µ1B−1 = 1. The asymptotic null distribution of t−1 as N , T → ∞ with θ > 1/2

N 1  N (ANT ,p − µ1Ap ) = √ [AiT ,p − E (AiT ,p )] N i=1

√

+√

1 B−1

are asymptotically independent, we obtain

N ϵ)] → 0,

i=1

1 3/2 B−1

= Op (1)op (1) = op (1). In view of this, √ √ (20), (21), and the fact that N (ANT ,p + µ1Ap ) and N (BNT ,p + µ1Bp )

√

2 which is implied by the uniform integrability of Z1T ,p over T (details are available upon requests). Hence, by Theorem 2 of Phillips and Moon (1999), N 

65

√

N (ANT ,−1 − µ1A−1 )

 µ1B−1 + op (1),

N (λN µ1c /2 + λ2N µ2c /3 + λ3N µ3c /6 + · · ·)



1/2 + λN µ1c /3 + λ2N µ2c /6 + · · ·

,

(26)

suggesting that while the drift, and hence also power, is affected by all the moments of ci , the effect of µ1c dominates. In order to infer the order of this effect we make use of (6) to substitute for λN , giving

√

N λN = N θ (1−τ )+1/2−κ .

If θ (1 − τ ) + 1/2 − κ > 0, then

√

N λN µ1c diverges and therefore so

√

does t−1 , whereas if θ (1 − τ ) + 1/2 − κ < 0, then N λN µ1c = o(1) and therefore the part of power that comes from shifting the mean of the test statistic is negligible. Hence, only if θ (1 −τ )+ 1/2 −κ = √ 0, such that N λN = 1, is power non-negligible in the usual nonincreasing sense. One possibility is to set (τ , κ) = (1, 1/2) (as in, for example, Moon et al., 2007; Moon and Perron, 2004, 2008), which implies that λN = N −1/2 , and therefore (26) reduces to

√

N µ1A µ1  −1 = √c + O(N −1/2 ), 2 µ1B−1

(27)

66

J. Westerlund, R. Larsson / Journal of Econometrics 188 (2015) 59–93

Fig. 2. Local power as a function of the variance of ci .

√

where the first term is identically the FO drift term reported in (1) in Section 1. Together, (26) and (27) go a long way toward explaining the observed test behavior in Fig. 1, and why the FO local power function is unable to capture it. We begin by noting that, while according to (27) power should be unaffected, as (26) makes clear, there is a second-order variance effect working through µ2c . Moreover, since in the DGP used for creating the figure, µ1c = −5, according to (26) increased variance should lead to a less negative drift term and hence also lower power. The new IO theory should therefore be able to capture the type of behavior seen in Fig. 1. This is illustrated in Fig. 2, which is Fig. 1 with the IO local power function superimposed. As the figure makes clear the IO power functions not only manage to capture the effect of the variance of ci , but are also more accurate in terms of the level of power (note again the scaling of the horizontal axis). We also see that accuracy is increasing in N, and that with N = 50 the curves representing the empirical and IO theoretical powers almost coincide. The above discussion supposes that (τ , κ) = (1, 1/2). However, it is important to realize that the results obtained are not unique to (τ , κ) = (1, 1/2), but apply to all triples (θ , τ , κ) such that θ (1 − τ ) + 1/2 − κ = 0. In other words, in contrast to the impression given by the bulk of existing studies where (τ , κ) = (1, 1/2) is assumed from the outset, there is actually nothing ‘‘special’’ about this specification. This illustrates how the usual practice in the time series literature of setting the rate of shrinking of the local alternative equal to the rate of consistency of the OLS estimator of ρi under H0 need not be very informative in panels. Remark. In the above example with (τ , κ) = (1, 1/2), the effect j of µc for j ≥ 2 is negligible. However, this need not be the case. There are two possibilities. First, the FO term might be zero. As an example, consider θ = τ = 1. √ when √ the drift term in (26) N λ2N = N 1/2−2κ for all Since in this case N λN = N 1/2−κ > κ > 0, power is driven by µ1c . However, this does not mean that power is negligible if µ1c = 0, a natural conclusion from FO theory. Indeed, while certainly negligible for √ κ ≥ 1/2, power is still nonnegligible for κ ≤ 1/4 (such that N λ2N > o(1)). Note also how in this case power is driven by µ2c rather than by µ1c . Similarly, if µ2c = 0, then there is still power, but now only for κ ≤ 1/6 (such

that N λ3N > o(1)) with µ3c taking the role of µ2c (and µ1c ) as the driver of power. The second possibility is that θ (1 − τ ) − κ = 0, such that λN = N θ (1−τ )−κ = 1, in which case all the moments of ci affect power, although µ1c still dominates. Remark. An important lesson from the above discussion is that FO theory can be deceptive in its simplicity, in the sense that power is not just a function of µ1c . In particular, given the heterogeneity of many panel data sets, the common practice of always setting up panel unit root tests as left-sided seems highly questionable. Indeed, it follows from (26) that if the test is set up as left-tailed and µ2c > 0 but µ1c > 0 or µ1c = 0, then one is likely to incorrectly conclude that yi,t has a unit root. That is, the test cannot discriminate between on the one hand a perfect unit root, and on the other hand explosive behavior (µ1c > 0) and/or non-unit roots that average to zero (µ1c = 0 and µ2c > 0). Another example of a situation in which a one-sided test is likely to run into difficulties is when µ1c < 0 but higher-order moments are large enough to outweigh the negative mean, leading to an overall positive drift. Remark. Another lesson of the above results is that the rate of shrinking of the local alternative is not really determined by (τ , κ) but rather by (θ , τ , κ). As already mentioned, the condition for non-negligible and non-increasing power is given by θ (1 − τ ) + 1/2 −κ = 0. By solving this equation for τ or κ the rate of shrinking of the local alternative can be written as N −κ T −τ = N −(θ τ +κ) = N −(θ +1/2) . Thus, the fact that (τ , κ) have been set so as to ensure non-negligible and non-increasing power, as is typically done in the previous literature, does not mean that the rate of shrinking is fixed. In fact, the rate of shrinking can be made arbitrarily high by just setting θ large enough, and this without affecting the level of power. One interpretation of this is that the time series information is relatively more important than the cross-sectional. Therefore, by increasing the value of θ we can be closer to the null and still enjoy the same level of power. 4.2. Incidental intercept and trend terms In this section we consider as a second illustration the t # and t statistics of Moon and Perron (2008), who provide FO results for +

J. Westerlund, R. Larsson / Journal of Econometrics 188 (2015) 59–93

the same DGP considered here with incidental intercepts (p = 0). We make three extensions. First, IO results are provided for the case when p = 0, which are then compared by means of Monte Carlo simulation with the corresponding FO results. Second, where Moon and Perron (2008) only consider p = 0, in this section we also consider incidental trends (p = 1), which has received much interest in the recent literature (see, for example, Moon and Perron, 2004; Moon et al., 2006, 2007). Because of this, in what follows t # and t + will be subscripted by p. Third, by providing explicit expressions for the local power functions of both tp# and tp+ , we complement and extend the results of Moon and Perron (2004), who consider three test statistics, ta∗ , tb∗ and t # . All three test statistics allow quite unrestricted forms of error heteroskedasticity, and serial and cross-section correlation, but differ in the way that the deterministic terms are treated. In particular, while t # allows for unrestricted intercept and trend terms, ta∗ and tb∗ only allow for a restricted intercept that can be ignored in the analysis.5 In this sense, ta∗ and tb∗ are very similar to t−1 , which also does not involve any treatment of deterministic terms. It is therefore not surprising to find that under the conditions of Section 2, the asymptotic distributions reported by Moon and Perron (2004, Theorem 2) for ta∗ and tb∗ are nothing but the FO part of the asymptotic distribution of t−1 . For t # , however, which is empirically more relevant, they only show how the power for p = 0 (p = 1) is negligible when κ > 1/4 (κ > 1/6) (see their Theorem 4). Hence, with this test, not only is the value of κ compatible with non-negligible local power unknown, but so is the actual local asymptotic distribution associated with this value. The current section derives this distribution. Interestingly, under the conditions of Section 2, t # is identically tp# . The IO results provided in this section can therefore be seen as extensions of the results of Moon and Perron (2004), but under the relatively restrictive DGP of Section 2. Define tpk =

αˆ pk  1/

T N  

,

(Dp yi,t −1 )2

i=1 t =2

where k ∈ {#, +} and

αˆ = αˆ p + # p

NT /2 N  T 

here for tp# and tp+ therefore have implications also for other test statistics. We begin by considering tp# . From T αˆ p = ANT ,p /BNT ,p , we have

√

that

√

NT αˆ p# can be written as

NT αˆ =

√

= 

NT αˆ p#

αˆ p+ = αˆ p +

T

with b0 = 3 and b1 = 15/2. The difference between αˆ p# and αˆ p+ originates with the fact that while D−1 yi,t = yi,t is a martingale, D0 yi,t and D1 yi,t are not. The implication is that while αˆ −1 is unbiased, αˆ 0 and αˆ 1 are biased, which calls for some kind of correction. In αˆ p# , the numerator of αˆ p is adjusted to ensure that the mean is asymptotically zero, whereas in αˆ p+ , the bias of the entire estimator is subtracted. As Westerlund and Breitung (2013, Section 3) show, the choice of bias correction method can have a substantial impact on the power of panel unit root tests. The methods considered here are the most common ones by far, suggesting that tp# and tp+ can be viewed as representing many of the existing panel unit root test statistics (see, for example, Bai and Ng, 2010; Levin et al., 2002; Moon and Perron, 2004, 2008; Moon et al., 2007; Westerlund and Larsson, 2009, 2012; Westerlund, 2015). The results provided

,

√ =

N (ANT ,p + 1/2)



.

BNT ,p

√

√

N (µ1Ap + 1/2) + N (ANT ,p − µ1Ap ), the expansion in 1/2) = (18), and then (20) and (21), we obtain

√

tp#

N (ANT ,p + 1/2)

=

 √

N (µ

=

BNT ,p

1 Ap

+ 1/2)

 µ1Bp

√ +

N (ANT ,p − µ1Ap )



µ1Bp

√ (µ1Ap + 1/2) N (BNT ,p − µ1Bp )

−

2(µ1Bp )3/2

√

+ op (1)

N (µ1Ap + 1/2)

∼

 µ1Bp  +

σA2p

+

µ1Bp

(µ1Ap + 1/2)2 σB2p

1/2

4(µ1Bp )3

N (0, 1)

(28)

as N , T → ∞ with θ > 1/2. This establishes the asymptotic distribution of tp# . Let us now consider tp+ . The appropriate value of bp depends on whether p = 0 or p = 1. Suppose that p = 0, such that b0 = 3. By adding and subtracting terms, NT αˆ 0+ =

√

√

√

NT αˆ 0 + 3 N =

N (ANT ,0 + 1/2) BNT ,0

√

+

3 N (BNT ,0 − 1/6) BNT ,0

giving

√

NT αˆ 0+ t0+ =  = 1/BNT ,0

√

,

√

N (ANT ,0 + 1/2)



In view of this, the fact that

√

+

3 N (BNT ,0 − 1/6)

BNT ,0

√

N (BNT ,0 − 1/6) =



.

BNT ,0

√

N (µ

N (BNT ,0 −µ1A0 )+

− 1/6), and the results in (18), (20) and (21), we have √ N (µ1A0 + 3µ1B0 ) N [(ANT ,0 − µ1A0 ) + 3(BNT ,0 − µ1B0 )] +   t0 = + µ1B0 µ1B0 √ (µ1A0 + 3µ1B0 ) N (BNT ,0 − µ1B0 ) − + op (1) 2(µ1B0 )3/2 √ N (µ1A0 + 3µ1B0 )  ∼ µ1B0 1 B0

√

 5 The required assumption for β , . . . , β to be ignorable when p = 0 is given by 1 N supi=1,...,N E (βi2 ) < ∞ (see Moon and Perron, 2004, Assumption 9).

BNT ,p

The asymptotic distribution can now be obtained by using √ the same steps leading up to (23). Indeed, by substitution of N (ANT ,p +

(Dp yi,t −1 )2

,

N (ANT ,p + 1/2)

=

BNT ,p

1/BNT ,p

i=1 t =2

bp

N /2

NT αˆ p +

suggesting that tp#

√

√

√

# p

√ ,

67

+

σA20 µ1B0

+

[36(µ1B0 )2 + (µ1A0 + 3µ1B0 )2 ]σB20 4(µ1B0 )3

1/2 N (0, 1) (29)

68

J. Westerlund, R. Larsson / Journal of Econometrics 188 (2015) 59–93

as N , T → ∞ with θ > 1/2. The derivations for the case when p = 1 are almost identical and are therefore omitted. The following statement covers both cases:

√

N (µ1Ap + bp µ1Bp )

+

tp ∼

 µ1Bp

 +

σA2p µ1Bp

+

[4b2p (µ1Bp )2 + (µ1Ap + bp µ1Bp )2 ]σB2p 4(µ1Bp )3

1/2 N (0, 1). (30)

is materially different when compared to tp# . In particular, while in tp#

+

Analogous to t−1 , the asymptotic distributions of and tp can be divided into two terms: (i) a drift term, and (ii) a normal variable with zero mean but nonunit variance. If H0 is true, according to Theorem 1, µ1A0 = µ1A1 = −1/2, µ1B0 = 1/6 and µ1B1 = 1/15, and therefore

tp#

71/240, σ = 1/45 and σ = 11/6300. The variance in (28) under H0 when p = 0 and p = 1 is therefore given by 1/2 and 71/16, respectively. The corresponding values for the variance in (30) are given by 17/10 and 193/112, respectively. Hence, under H0 and provided that θ > 1/2, the tp# and tp+ statistics scaled by the inverse square root of these variances are asymptotically N (0, 1). As with t−1 , the moments of ci affect both the drift and variance of the asymptotic distributions of tp# and tp+ ; however, the variance effect is negligible, and we therefore focus on the drift. We begin by considering t0# . According to Theorem 1, 2 B1

√

√ =

6N λ2N µ2c 24

µ1B0

√ + O( N λ3N ),

√



µ

√ =

1 B0

6µ2c

24

+ O(N −1/4 ),

(32)

which is the same result as the one reported by Moon and Perron (2008, Theorem 4.1).6 Hence, again, the existing FO results only account for the leading term in the drift. Let us now consider t1# . According to Theorem 1,

√

N (µ

1 A1



+ 1/2)

µ1B1

N (µ1A0 + b0 µ1B0 )

 µ1B0

√ =

15N λ µ 4 N

1440

4 c

√ + O( N λ5N ),

√ =

6N λN µ1c 4

√ + O( N λ2N ),

(34)

in case of t1+ it is given by

√

N (µ1A1 + b1 µ1B1 )

 µ1B1

√ =−

15N λ2N µ2c 56

√ + O( N λ3N ).

(35)

Hence, assuming again that τ = 1, in contrast to tp# , where the introduction of incidental trends caused the value of κ compatible with non-negligible and non-increasing power to increase from κ = 1/4 to κ = 1/8, with tp+ the increase is from κ = 1/2 to κ = 1/4, suggesting that tp+ should be more powerful than tp# .7 Moreover, while the powers of t0# and t1# are driven by µ2c

(31)

suggesting that, unlike t−1 where the drift term is driven by µ1c , the drift of the asymptotic distribution of t0# is driven by µ2c . For this drift to be non-negligible √ and non-increasing, we need θ (1 − τ ) + 1/4 − κ = 0, such that N λ2N = N 2(θ(1−τ )+1/4−κ) = 1. One possibility is to set (τ , κ) = (1, 1/4), such that λN = N −1/4 , in which case the drift in (31) reduces to N (µ1A0 + 1/2)

√

and tp are asymptotically mean zero. As for the

2 B0



case of t0+ the appropriate drift term is given by

+

variance under the same null, by Theorem 1, σA20 = 1/12, σA21 =

N (µ1A0 + 1/2)

N −1/4 T −1 to N −1/8 T −1 , which is suggestive of relatively low power for t1# . As already mentioned, Moon and Perron (2004) consider a version of t1# and show how the local power of this test statistic is negligible for κ > 1/6. The result in (28) is more general in this sense and shows not only how the local power of t1# is negligible for κ > 1/8, but also what the local asymptotic distribution looks like when κ = 1/8. The presence of incidental trends has a similar power-reducing effect on tp+ , although the way in which this effect manifests itself

and µ4c , respectively, those of t0+ and t1+ are driven by µ1c and µ2c , respectively. A small-scale simulation study was conducted to assess the accuracy of the above IO results in small samples. The DGP is given by a restricted version of (2)–(6) that sets εi,t ∼ N (0, 1), βi = 0 (although we do not assume knowledge of this in the testing), θ = 2, τ = 1, and ci ∼ U (a, b), suggesting that  µjc = (j + 1)−1 jk=0 ak bj−k . The reported rejection frequencies are for a double-sided 5% level test when p = 0; the results for the case when p = 1 were, in terms of the quality of the theoretical approximations, almost identical and are therefore omitted.8 The IO local asymptotic distributions in (28) and (30) are truncated such that they only include the first 100 moments of ci .9 The number of replications is 3000. The results reported in Table 1 are generally in agreement with theory and can be summarized as follows:

• Unreported results of size of the test statistics (available upon request) suggest that size accuracy is almost perfect.

(33)

suggesting that the appropriate condition for non-negligible and non-increasing √power is no longer given by θ (1 −τ )+ 1/2 −κ = 0, but rather by N λ4N = N 4(θ(1−τ )+1/8−κ) = 1, or θ (1 − τ ) + 1/8 − κ = 0. Hence, in contrast to the case when p = 0 and τ = 1, if p = τ = 1, then θ (1 − τ ) + 1/8 − κ = 1/8 − κ < 0 for all κ > 1/8, including κ = 1/4. Thus, the allowance for incidental trends that need to be estimated leads to an increase in the neighborhood around unity for which power is negligible, from

• Power is generally very close to what is predicted by our (truncated) IO power functions. In fact, accuracy is surprisingly good, even for N as small as 20. By contrast, the predictions based on the associated FO power functions are way off target in most cases. To take an extreme example, consider the cases when a = b = −4 and a = −8, b = 0, in which the predicted powers for t0# based on FO theory are more then 40 times as large as actual power.

7 This is true when τ = 1; however, the dominance of t + holds for all τ > 0 (and p

θ > 0). 6 Note that in Moon and Perron (2008) t # is scaled to have unit variance. The asymptotic distribution reported in their Theorem 4.1 is therefore not identical to the one given here. However, accounting for the difference in scaling, the distributions are identical.

8 The one-sided rejection frequencies are available from the corresponding author upon request, as are the results for the case when p = 1. 9 In all experiments considered the effect of higher-order moments was negligible. We also re-ran the simulations using only the first 50 moments. The results were unaffected by this.

J. Westerlund, R. Larsson / Journal of Econometrics 188 (2015) 59–93

69

Table 1 5% local power.

κ

N

t0#

FO theory

IO theory

t0+

FO theory

IO theory

a = b = −2 1/4 1/2

20 40 80 20 40 80

1.9 2.4 2.7 3.5 3.1 3.6

8.1 8.6 8.9 5.3 4.8 5.4

1.5 2.1 3.4 2.9 3.1 4.4

24.6 39.4 57.2 10.7 12.9 13.2

49.9 65.8 79.7 14.8 16.1 15.4

18.1 35.0 55.0 7.7 10.1 11.1

2.4 2.5 3.1 3.4 2.9 3.9

10.9 11.6 11.9 5.4 4.8 5.4

2.0 2.5 3.7 3.1 3.2 4.6

21.7 32.6 47.6 10.1 11.8 12.6

49.9 65.8 79.7 14.8 16.1 15.4

15.4 29.1 44.8 7.5 9.2 10.4

1.5 2.9 5.3 1.9 2.3 2.9

64.3 63.2 64.2 7.3 6.1 6.0

1.1 2.7 5.6 1.6 2.3 3.8

57.3 82.1 95.7 23.1 29.0 33.4

97.9 99.7 100.0 45.8 47.7 46.0

55.8 85.1 97.7 16.5 25.3 29.2

2.0 2.8 5.6 2.5 2.5 3.3

87.7 86.8 87.0 9.8 7.3 6.5

1.8 3.0 5.2 2.1 2.6 4.1

48.0 69.5 88.7 20.3 24.8 29.2

97.9 99.7 100.0 45.8 47.7 46.0

42.1 69.2 90.8 14.4 21.4 25.2

a = −4, b = 0 1/4 1/2

20 40 80 20 40 80

a = b = −4 1/4 1/2

20 40 80 20 40 80

a = −8, b = 0 1/4 1/2

20 40 80 20 40 80

Notes: t0# and t0+ refer to the bias-corrected OLS t-statistics for a unit root in the case of incidental intercepts. While t0# is based on correcting only the numerator, in t0+ the whole test statistic is corrected. ‘‘FO theory’’ and ‘‘IO theory’’ refer to the theoretical power according to the first- and infinite-order approaches, respectively. κ , a and b are such that ρi = exp(ci N −κ T −1 ), where ci ∼ U (a, b).

• As expected, power is mainly driven by µ1c . However, there is also a second-order effect working through variance of ci . In particular, while FO theory completely misses this, both the empirical and IO theoretical powers seem to be decreasing in |a − b|. • When√κ = 1/4, since θ = 2 and τ = 1 in the simulations, we have N λN = N 1/4 , suggesting that in this case the power of t0+ should be increasing in N. The results reported in Table 1 are quite suggestive of this. 4.3. A uniformly most powerful test In a very influential paper, Moon et al. (2007) derive the asymptotic local power envelope of panel unit root tests when applied to the DGP of Section 2. They also propose a point-optimal test that depends on a sequence of drift parameters that has to be specified by the researcher, and can be seen as a likelihood ratio test, where the alternative is determined by the specified drifts. If the true drifts, c1 , . . . , cN , are known, then the point-optimal test attains the power envelope and is therefore asymptotically optimal. As Becheri et al. (2015) show, however, with c1 , . . . , cN unknown, which is always the case in practice, the power envelope is not attainable, and in such cases the power of the point-optimal test is in fact equal to envelope under c1 = · · · = cN . They also derive a new ˆ , which is asymptotically uniformly most powerful test, denoted ∆ when c1 , . . . , cN are not observed. The purpose of this section is to ˆ , which can be interpreted derive the IO local power function of ∆ as a ‘‘IO local power envelope’’ under c1 = · · · = cN . The DGP used for this purpose, which is the same as in Becheri et al. (2015), is the one described in Section 2 with p = 0.

According to Theorem 4.3 of Becheri et al. (2015),

√ N T t −1 2  ˆ = √ ∆ ∆yi,s ∆yi,t

NT i=1 t =3 s=2 is asymptotically optimal under c1 = · · · = cN . We now make use of the results of Section 2 to derive the IO local asymptotic distribution of this test. We begin by noting that under p = 0, t −1 s=2 ∆yi,s = yi,t −1 − yi,1 = ui,t −1 − ui,1 and ∆yi,t = ∆ui,t . By

T

using this, t =3 ∆ui,t = ui,T − ui,2 and ui,t = can show that

t

s=1

ρit −s εi,s , we

√ N T 2  ˆ ∆= √ (ui,t −1 − ui,1 )∆ui,t NT i=1 t =3

√

√

N 2  ui,t −1 ∆ui,t − √ ui,1 (ui,T − ui,2 ) NT i=1 t =3 NT i=1

= √

N T 2 

√

N 2 

√

2NANT ,−1 − √ εi ,1 NT i=1

=

√

−√ √ =



T 

 ρ

ε + (ρ

T −s i ,s i

T −2 i

− 1)εi,2

s =3

N 2 

NT i=1

(ρiT −1 − ρi )εi2,1 √

2NANT ,−1 + Op (T −1/2 ) + Op ( NT −1 ),

where both remainder terms are op (1) under θ > 1/2. Hence,

ˆ is in itself not a function of ANT ,−1 , asymptotically ∆ ˆ is just while ∆ √ 2NANT ,−1 , whose distribution follows directly from (20); ˆ = ∆

√

2N µ1A−1 +

√

+

√

2N (ANT ,−1 − µ1A−1 ) + op (1) ∼

2σA−1 N (0, 1).

√

2N µ1A−1 (36)

70

J. Westerlund, R. Larsson / Journal of Econometrics 188 (2015) 59–93

√

Under (τ , κ) = (1, 1/2), this distribution simplifies to µ1c / 2 + N (0, 1), which coincides with the FO local asymptotic distribution reported by Becheri et al. (2015, Proposition 4.3). The FO power √ envelope is therefore given by Φ (−µ1c / 2 − zα ), where Φ (x) and

√

zα are as in Section 1. Since under (τ , κ) = (1, 1/2), 2σA−1 = 1 + O(N −1/2 ), the √ IO version of this envelope may be written as limN , T →∞ Φ (− 2N µ1A−1 − zα ).

Remark. It is interesting to compare the local asymptotic power ˆ with those of t0# and t0+ . Clearly, since under τ = 1 the local of ∆ power of t0# is negligible for all κ > 1/4, including κ = 1/2, the

ˆ . The power of t0+ power of this test is not even close to that of ∆

ˆ. is non-negligible√within the same shrinking neighborhood as ∆ √ However, since 6/4 ≈ 0.61 < 1/ 2 ≈ 0.71, the drift in the ˆ, asymptotic distribution of t0+ (see (34)) is smaller than that of ∆ ˆ . In other words, the powers and therefore t0+ is not as powerful as ∆ of t0# and t0+ both lie below the envelope. 4.4. An example involving a more general DGP In this section we show how the analysis provided so far can be extended to cover test statistics that are suitable under more realistic DGP settings than the one given in Section 2. As already mentioned in Section 4.2, Moon and Perron (2004) consider a heteroskedasticity, and serial and cross-section correlation robust version of tp# , henceforth denoted tR#,p , whose local power for p = 0 (p = 1) is shown to be negligible when κ > 1/4 (κ > 1/6). Our contribution here is to provide IO results for this test statistic. In essence, the analysis provided in Section 4.2 for tp# is in the current tR#,p .

section extended to cover also The DGP is still characterized by (2)–(6); however, the conditions placed on εi,t are different from those of Section 2. In particular, the following dynamic common factor representation is assumed to apply:

εi,t = βi′ ft + ei,t , (37) where ft is a r × 1 vector of unobserved common factors, βi is a vector of factor loadings, and ei,t is an idiosyncratic error term. The rest of the assumptions can be summarized as follows:

• ei,t = φi (L)ϵi,t , where ϵi,t is iid with E (ϵi,t ) = 0, E (ϵi2,t ) = 1, ∞ j E (ϵi8,t ) < ∞, φi (L) = j=0 φi,j L , L is the lag operator, φi (1) > 0 ∞ m and j=0 j |φi,j | < ∞ for m > 1; • ft = Φ (L)ηt , where  ηt is iid across ∞ tm with E (ηt ) = 0r ×1 , ∞ j E (ηt ηt′ ) = Ir , Φ (L) = j=0 Φj L , j=0 j ∥Φj ∥ < ∞ for m > 1 √ and ∥A∥ = tr(A′ A) denotes the Frobenius (Euclidean) norm of the matrix  A;  • limN →∞ N −1 Ni=1 βi βi′ and limN →∞ T −1 Tt=1 ft ft′ are positive definite; • E (|ui,0 |) < ∞ and E (|ei,0 |) < ∞ for all i, and E (∥f0 ∥) < ∞; • cj , ϵi,t and ηs are mutually independent for all i, j, t and s;  • σe2 = limN →∞ N −1 Ni=1 σe2,i ∈ (0, ∞), ωe2 = limN →∞ N −1

ω ∈ (0, ∞) and φ = limN →∞ N i=1 ω  ∞ 2 2 (0, ∞), where σe2,i = ∞ φ and ω = ( φ ) e,i j =0 j=0 i,j . N

i=1

2 e,i

4 e 2 i,j

N −1

4 e,i

∈

Consider the (T − 1) × N matrix εˆ = (ˆε1 , . . . , εˆ N ), where εˆ i = (ˆεi,2 , . . . , εˆ i,T )′ , εˆ i,t = xi,t − αˆ p Xi,t −1 , xi,t = Dp ∆yi,t , Xi,t −1 = Dp yi,t −1 and αˆ p is as before. The idea of Moon and Perron (2004) is to estimate βi by the method of principal components, and to project xi,t and Xi,t −1 on this estimate, which should then be asymptotically cross-section correlation free. Let us therefore ˆ βˆ ′ β) ˆ −1 βˆ ′ , define the N × N projection matrix Mβˆ = IN − β( where βˆ = T −1 εˆ ′ fˆ is √ the principal components estimator of β = (β1 , . . . , βN )′ and fˆ is T times the eigenvectors corresponding to

the r largest eigenvalues of the (T − 1) × (T − 1) matrix εˆ εˆ ′ .10 Let us further introduce the (T − 1) × N matrices x = (x1 , . . . , xN ) and X−1 = (X1,−1 , . . . , XN ,−1 ), where xi = (xi,2 , . . . , xi,T )′ and Xi,−1 = (Xi,1 , . . . , Xi,T −1 )′ are (T − 1)× 1. In this notation, robust versions of ANT ,p and BNT ,p may be constructed as ANTR,p = (NT )−1 tr(X−1 Mβˆ x′ ) ′ and BNTR,p = N −1 T −2 tr(X−1 Mβˆ X− 1 ), respectively. The following estimator can be seen as a robust version of αˆ p :

ANTR,p + σˆ e2 /2

αˆ R#,p =

T BNTR,p

,

where σˆ e2 is any consistent estimator of σe2 such that (σˆ e2 − σe2 ) = op (N −1/2 ). We also assume the existence of estimators ω ˆ e2 and φˆ e4

satisfying (ω ˆ e2 −ωe2 ) = op (1) and (φˆ e4 −φe4 ) = op (1), respectively.11 The Moon and Perron (2004) heteroskedasticity, and serial and cross-section correlation robust version of tp# can now be written as

√

tR#,p

= 

NT αˆ R#,p

√

N (ANTR,p + σˆ e2 /2)

=

 ωˆ e BNTR,p

ωˆ e2 /BNTR,p

.

By using the same arguments as in Bai and Ng (2010, Section 4), and Moon and Perron (2004), we can show that under p > −1 the projection onto βˆ is successful in eliminating the effect of the common factor on BNTR,p (that is, Mβˆ β is asymptotically zero); N T 1 

BNTR,p =



 Dp

NT 2 i=1 t =2

t −1 

2 ρ

t −1−s ei,s i

+ op (1)

s =1

= ωe2 µ1Bp + op (1), where ω µ 2 e

1 Bp

(38)

is just ω times the limit of BNT ,p in (17). The asymp2 e

√

totic distribution of N (BNTR,p − ωe2 µ1Bp ) as N , T → ∞ with θ > 1/2 is also free of this effect, and is given by

√

N (BNTR,p − ωe2 µ1Bp ) →d

 φe4 σBp N (0, 1),

(39)

√

which is φe4 times the asymptotic distribution of N (BNT ,p −µ1Bp ) in as long as p > −1, the asymptotic distribution of √ (21). Similarly, N (ANTR,p − σe2 µ1Ap ) is φe4 times the asymptotic distribution of



√ √

N (ANT ,p − µ1Ap ) in (20); N (ANTR,p − σe2 µ1Ap )



√ =

N

N T 1 

NT i=1 t =2 + op (1)

→d



 Dp

t −1 

 ρ

t −1 −s ei,s i

 Dp ei,t − σ µ 2 e

1 Ap

s=1

φe4 σAp N (0, 1).

(40)

Making use of these results,

√

tR#,p =

N (ANTR,p + σˆ e2 /2)

√ =

 ωˆ e BNTR,p

N (ANTR,p + σe2 /2)



ωe BNTR,p

+

√ (ωˆ e−1 − ωe−1 ) N (ANTR,p + σe2 /2)  BNTR,p

10 Moon and Perron (2004, Equation (11)) suggest using β( ˆ N −1 βˆ ′ β) ˆ 1/2 instead of βˆ . However, since rotations of βˆ do not affect the results (a result that was verified using Monte Carlo simulation), in the present paper we take βˆ as it is. 11 See Moon and Perron (2004, Section 2.2.2) for a detailed description of how σˆ 2 , e ωˆ e2 and φˆ e4 may be constructed.

J. Westerlund, R. Larsson / Journal of Econometrics 188 (2015) 59–93

71

√

N (σˆ e2 − σe2 )

+

2ω ˆe

√ 2



5. Conclusion

BNTR,p

√

+ 1/2) N (ANTR,p − σ µ )   + 1 ωe2 µBp ωe2 µ1Bp √ σe2 (µ1Ap + 1/2) N (BNTR,p − ωe2 µ1Bp ) − + op (1) 2ωe4 (µ1Bp )3/2 √  σe2 N (µ1Ap + 1/2) φ4  ∼ + 2e ωe ω 2 µ1

=

σe

N (µ

1 Ap

e

 ×

σA2p µ1Bp

2 e

1 Ap

Bp

+

σe4 (µ1Ap + 1/2)2 σB2p

1/2

4ωe4 (µ1Bp )3

N (0, 1),

(41)

√

where the third equality follows from the fact that N (σˆ e2 −σe2 ) = op (1), and, by Taylor expansion, we also have (ω ˆ e−1 −ωe−1 ) = op (1). # The asymptotic variance of tR,p depends in a complicated fash-

ion on σe2 , ωe2 and φe4 , which are nuisance parameters that reflect the heteroskedasticity and serial correlation of ei,t . As in Section 4.2, however, because in this case µ1A0 = µ1A1 = −1/2, under H0 the asymptotic variance simplifies to φe4 σAp /ωe4



µ1Bp , which is

just φe4 /ωe4 times the corresponding variance of tp# . This result, together with the consistency of φˆ e4 and ω ˆ e2 , suggests that under H0 and θ > 1/2,

√





2ω ˆ e2 tR#,0 / φˆ e4 and 4ωˆ e2 tR#,1 / 71φˆ e4 are asymptoti-

cally N (0, 1). The asymptotic null distributions of these scaled test statistics are therefore free of nuisance parameters. In order to get a feeling for the effect of σe2 , ωe2 and φe4 under H1 , consider as an ex-

√

ample the local asymptotic distribution of



2ω ˆ e2 tR#,0 / φˆ e4 , which,

Recently, much effort has been directed toward the analysis of the local power of panel unit root tests. The main thrust of this paper is that the conventional FO asymptotic analysis, in which only the leading term in the local power function is considered, can be a rather unreliable guide to what happens in practice. While this observation is in itself nothing new but has been made in several Monte Carlo studies (see, for example, Moon and Perron, 2008; Moon et al., 2007), as far as we are aware, so far there have been no attempts to explain the observation theoretically, and this paper can be seen as a step in this direction. The way we do this is by developing IO expansions of moments of certain sample quantities, which can be used to obtain expansions of any test statistic depending on those quantities. The new results are shown to deliver significant insight when compared to existing ones. First, they show how the local power of panel unit root tests can be written as an IO series expansion in the moments of ci , in which existing results can be seen as mere FO approximations. The results should therefore provide better approximations of actual test behavior, a result that is confirmed by Monte Carlo simulation. Second, the fact that the FO term is zero does not mean that local power is absent altogether, a natural conclusion from FO theory. It just means that local power is zero in the direction of the particular moment being zero. Looking closer to the null there will be power coming from higher-order moments (provided of course that they are different from zero). Third, unless it is known a priori that the FO term is non-zero and non-dominated by the effect of higher-order moments, the appropriate critical region need not be given by the left tail of N (0, 1). Thus, in contrast to the current common practice, panel unit root tests should in general be set up as double-sided. Fourth, unless one is precise about θ , just fixing (τ , κ), as is commonly done in the existing literature, is not enough to infer the rate of shrinking of the local alternative.

in view of the above result for tR#,p , is easily seen to be given by

√

2ω ˆ e2

 φˆ e4

tR#,0

√ σe2 2N (µ1Ap + 1/2)  ∼ φe4 µ1Bp  1/2 σe4 (µ1Ap + 1/2)2 σB2p √ σA2p + 2 + N (0, 1). µ1Bp 4ωe4 (µ1Bp )3

Hence, unlike in the iid case considered in Sections 4.1–4.3, under the above more general conditions, the local power of

Appendix. Proof of Theorem 1 By the definition of Dp , Dp yi,t = Dp ui,t = ρi Dp ui,t −1 + Dp εi,t , and similarly, Dp ∆yi,t = Dp ∆ui,t . By using these results, AiT ,p =

T 1

T t =2

√

2ω ˆ e2 tR#,0 / φˆ e4 is driven partly by nuisance parameters. The only exception is if ei,t is homoskedastic and serially uncorrelated, in  which case σe2 = ωe2 = φe4 , and therefore the nuisance parameters cancel out. If ei,t is serially uncorrelated but heteroskedastic, by  the Jensen inequality, φe4 ≥ ωe2 = σe2 , suggesting that while the

√

variance is unaffected,

2N (µ

1 Ap

 √ + 1/2)/ µ1Bp ≥ σe2 2N (µ1Ap +

BiT ,p =

leads to reduced power, which is in agreement with the results of Moon and Perron (2004), and Westerlund (2015). The main lesson from this section is that the moment expansions of Section 2 are not restricted to test statistics that assume iid errors, but that they are in fact applicable also when analyzing more general test statistics. In this section we focused on the statistic of Moon and Perron (2004); however, intuition suggests that the same steps can be used when analyzing also other test statistics that are similar in construction, such as those considered by Bai and Ng (2010).

T 1

T t =2

T 1 

T 1 

T 2 t =2

T 2 t =2

(Dp yi,t −1 )2 =

Dp ui,t −1 Dd ∆ui,t ,

(Dp ui,t −1 )2 .

Consider AiT ,p . From the definition of Dp , TAiT ,p =

T 

Dp ui,t −1 Dp ∆ui,t

t =2



1/2)/ φe4 µ1Bp . In other words, the presence of heteroskedasticity

Dp yi,t −1 Dd ∆yi,t =

=

T 

 ui,t −1 −

t =2

=

T 

T 

ui,t −1 ∆ui,t −

∆ui,t

T

T 

T 

 p at , s ∆ ui , s

s=2 T 

ui,t −1

t =2

T  t =2

+

∆ui,t −

s=2

t =2

−

 p at ,s ui,s−1

T 

p

at ,s ∆ui,s

s=2

p

at ,s ui,s−1

s =2 T

 t =2 s=2

p

at ,s ui,s−1

T 

v=1

p

at ,v ∆ui,v

(A.1)

72

J. Westerlund, R. Larsson / Journal of Econometrics 188 (2015) 59–93

where, since ui,0 = 0, ui,t =

t 

= λN ci

t 

ρit −s εi,s =

s=1

ri,t −s εi,s

with ri,t = exp(tT −1 λN ci ). We similarly have T

T 2 BiT ,p =



Dp ui,t −1

1

2

T

=

=

 ui , t − 1 −

T 

t =2

s=2

T 

T 

u2i,t −1

−2

t =2

ui , t − 1

t =2

= ∼

p at ,s ui,s−1

s=2

p at ,s ui,s−1

= .

(A.2)

s =2

The expansions in (A.1) and (A.2) will be used throughout this appendix. Moreover, while in the paper (and elsewhere) A ∼ B denotes A = B(1 + o(1)), for notational simplicity, here it will be used to denote A = B(1 + O(T −1 )). p We start by considering E (AiT ,0 ). If p = 0, then dt = d0t = 1, p 0 −1 and so at ,s = at ,s ∼ T . Making use of this (A.1) reduces to TAiT ,0 ∼

T 

T 1

ui,t −1 ∆ui,t −

T t =2

t =2

−

∼

T 1

T t =2

T 

T 

∆ui,t

ui , t − 1

T2

1 T

t =2

ui,T

T 

ui,t −1 ∆ui,t |ci

T t −1 1 

T 2 t =2 s=1 2 τ =0



1

{exp (2τ λN ci ) − 1} dτ

1

E (ui,t −1 ∆ui,t |ci ) = E

ui,t −1

t =2

 

E ui,T ui,t −1 |ci



=E

−

T 

 ri,t −1−w εi,w

u =1

=

=

t −1 

T −1 E ui,T ui,t −1 |ci





=

t −1 

T s =1



τ



γ =0 τ

∼ = γ =0

ri,t −v εi,v

=



T

ri,t −1−s ri,t −1−w E εi,s εi,w

s=1





E 2

ui,T

T 

ui,t −1 |ci

∼

t =2

=



=



E AiT ,0 |ci

ri,t −1−s ri,t −s − ri,t −1−s .

t −1 1

T s=1



τ

∼ γ =0

2λN ci

τ =0

[exp {(1 + τ ) λN ci }

1 2(λN ci )2

{exp (2λN ci )





∼



ri (τ − γ )ri′ (τ − γ )dγ



1



1

1

exp (2λN ci ) −

2

2λN ci

−

2(λN ci )2

1

2λN ci

 −1

{exp (2λN ci ) − 2 exp (λN ci ) + 1} .

Consider the first term on the right-hand side. By substitution of ∞ j exp (x) = j=0 x /j!, 1

ri,t −1−s T ri,t −s − ri,t −1−s

(A.5)

This, together with (A.4) and (A.3), gives

s=1



1



1

− 2 exp (λN ci ) + 1} . ri2,t −1−s

When t = ⌊τ T ⌋ we have |tT −1 − τ | = O(T −1 ), which holds uniformly in τ ∈ [0, 1]. It follows that exp(tT −1 ) ∼ exp(τ ), and so, conditional on ci , we have ri,t ∼ ri (τ ) = exp(τ λN ci ). Denote by ri′ (τ ) = λN ci ri (τ ) the derivative of ri (τ ). Then, E ui,t −1 ∆ui,t |ci

exp {(1 + τ − 2γ ) λN ci } dγ



s =1



r (1 − γ ) r (τ − γ ) dγ

1 [exp {(1 + τ ) λN ci } 2λN ci − exp {(1 − τ ) λN ci }] ,



t −1

=

ri,1−s ri,t −s−1

− exp {(1 − τ ) λN ci }] dτ



t −1 



and therefore

|ci

  ri,t −1−s ri,t −v E εi,s εi,v

ri,t −1−u ri,t −s −



t −1 1

s=1 w=1

=

ri,T −v ri,t −1−u E εi,v εi,u

ri,T −u ri,t −1−u ,

s=1 v=1

−

ri,t −1−u εi,u |ci

v=1 u=1

1

t −1  t −1 



u=1

T  t −1 

(A.3)

t



ri,T −v εi,v

t −1 

implying that if t = ⌊τ T ⌋, then

∆ui,u

w=1 t −1

T 

v=1

v=2

s=1 t −1



t 

(A.4)

We similarly have

t =2

ri,t −1−s εi,s

2λN ci

u=1

ui,t −1 .



 −1 .

1

exp (2λN ci ) −

2λN ci

2



1



1



ri,t −1−s T ri,t −s − ri,t −1−s

Consider the first term in this expansion. Substitution of ui,t = t s=1 ri,t −s εi,s yields

 t −1 

[exp(2τ λN ci ) − 1]



=

t =2

ui,t −1 +

T 

∆ui,t

T (T − 1) 

t =2

ui,t −1 ∆ui,t −

T 

exp[2(τ − γ )λN ci ]dγ

t =2

2

T T   t =2

E

2



+



p at ,s ui,s−1 T 

2

γ =0

and therefore

t =2 T 

1

=

s=1

τ





1

∞  (2λN ci )j

2λN ci j=0

2

=

1 2



1 2λN ci

j!



−

2λN ci

∞  (2λN ci )j j =0

j!



1

−1 



−1 −1

J. Westerlund, R. Larsson / Journal of Econometrics 188 (2015) 59–93

=

=

1



1

∞  (2λN ci )j+1

1

2λN ci j=0

2



2

∞  (2λN ci )j j =0





1 ∞ 1  (2λN ci )j+1

−1 =

(j + 1)!

implying

−1

(j + 1)! 

(j + 2)!

2 j =0

T

.

E 2

T 

 u2i,t −1

|ci

∼

t =2

(A.6)

=

Similarly,



1

1

2λN ci

1



1 2λN ci

τ =0



1

exp (2λN ci ) −

2λN ci

{exp (2τ λN ci ) − 1} dτ

2λN ci

−1 .

(A.10)

Similarly,

1

{exp (2λN ci ) − 2 exp (λN ci ) + 1} 2(λN ci )   ∞ ∞   1 (2λN ci )j (λN ci )j = −2 +1 2(λN ci )2 j=0 j! j! j =0   ∞ ∞   (λN ci )j+1 1 (2λN ci )j+1 = −2 2(λN ci )2 j=0 (j + 1)! (j + 1)! j=0   ∞ ∞   (2λN ci )j+2 (λN ci )j+2 1 −2 = 2(λN ci )2 j=0 (j + 2)! (j + 2)! j=0   ∞  2j+1 − 1 (λN ci )j = , (j + 2)! j=0 2

E

 T   



E AiT ,0 |ci



∼

=

=

=

=

j ci

=

= (A.7)

=

Hence, since E ( ) = µ we obtain



E AiT ,0 ∼ −

1 2

+

(j + 4)!

j =0

λjN+2 µjc+2 ,

(A.8)

as required.   Next, consider E BiT ,0 . From (A.2), T 2 BiT ,0 ∼

T 

u2i,t −1 −

t =2

 T 1  T

T −1 E (u2i,t −1 |ci ) =

=

=

T

E



,

(A.9)

T u=1 v=1 t −1 1

T u=1 τ γ =0

=

E

t −1 q−1 1 

T u=1 v=1 t ∧ q −1 1 

T

τ ∧ξ

ri2,t −1−u ∼



τ γ =0

2λN ci



r (τ − γ )2 dγ

exp {2 (τ − γ ) λN ci } dγ

1

ri,t −1−u ri,q−1−v E ϵu εi,v



{exp (2τ λN ci ) − 1} ,



ri,t −1−u ri,q−1−u

exp {(τ + ξ − 2γ ) λN ci } dγ

= γ =0

1 2λN ci

[exp {(τ + ξ ) λN ci } − exp {|τ − ξ | λN ci }] ,

(A.11)

which yields 1 T3

E

∼

 T   

2 |ci

ui,t −1



t =2 1



1 2λN ci

 

1



τ =0

ξ =0

[exp {(τ + ξ ) λN ci }

− exp (|τ − ξ | λN ci )] dξ dτ  1  1 1 = exp (τ λN ci ) exp (ξ λN ci ) dξ dτ 2λN ci τ =0 ξ =0   1  τ − 2 exp (τ λN ci ) exp (−ξ λN ci ) dξ dτ =

ξ =0

1



1 2(λN ci )2

exp (τ λN ci ) {exp (λN ci ) − 1} dτ 0

1

exp (τ λN ci ) {1 − exp (−τ λN ci )} dτ



   1 } { exp c − 3 exp λ c d τ + 2 (λ ) (τ ) N i N i 2(λN ci )2 0   1 1 {exp (λN ci ) − 3} {exp (λN ci ) − 1} + 2 = 2(λN ci )2 λN ci  1 1 4 = exp (2λN ci ) − exp (λN ci ) 2(λN ci )2 λN ci λN ci  3 + +2 . (A.12) λN ci

=

 

|ci

v=1

0

2  

ri,t −1−u ri,t −1−v E εi,u εi,v

ri,q−1−v εi,v



r (τ − γ ) r (ξ − γ ) dγ

γ =0  τ ∧ξ

=

ri,t −1−u εi,u



u=1



ri,t −1−u εi,u |ci

E (ui,t −1 ui,q−1 |ci ),

 q−1 

u =1

− 2

u =1

t −1 t −1 1 



T

 t −1 

t =2

 t −1  

=

1



where, with t = ⌊τ T ⌋ and τ ∈ [0, 1], 1

T  T  t =2 q=2

τ =0

2 ui,t −1

=



t =2

∼

−

j c,

∞  2j+1 j + 1

|ci

ui,t −1

T −1 E (ui,t −1 ui,q−1 |ci )

 j+1  ∞  2 − 1 (λN ci )j 2 j=0 (j + 2)! (j + 2)! j=0  j+2  ∞ ∞  2 − 1 (λN ci )j+1 1  (2λN ci )j+1 1 − − + 2 2 j=0 (j + 2)! (j + 3)! j= 0  j+3  ∞ ∞  2j+1 (λN ci )j+2  2 − 1 (λN ci )j+2 1 − + − 2 (j + 3)! (j + 4)! j=0 j=0  j+1  ∞ j + 3  2 (j + 4) − 2 + 1 (λN ci )j+2 1 − + 2 (j + 4)! j=0  j+1  ∞  2 j + 1 (λN ci )j+2 1 . − + 2 (j + 4)! j=0 ∞ 1  (2λN ci )j+1

 

2

where, letting q = ⌊ξ T ⌋ and ξ ∈ [0, 1],

suggesting



73

1

This, together with (A.9) and (A.10), implies E (BiT ,0 |ci ) ∼

1 2λN ci



1 2λN ci

exp (2λN ci ) −

1 2λN ci

 −1

74

J. Westerlund, R. Larsson / Journal of Econometrics 188 (2015) 59–93

−

3

λN ci



1

1 2λN ci

−

−

2(λN ci )3 5

4(λN ci )2 1

1 2λN ci

−

∞  

We have

exp (λN ci )

E (ui,t −1 ∆ui,t ui,q−1 ∆ui,q |ci )

2λN ci ∞  

−

−

−



=

=

2 1 6

+2



2

j

2 +

(λN ci )3

(λN ci )j j!

+2

+2 =

=

1

2

2j+1 +

(λN ci )3

(λN ci ) (j + 1)!

j +1

6

−

(λN ci )2 

1

−4

(j + 3)!

2

∞  2j (λN ci )j

∞  (λN ci )j +2 (j + 3)! (j + 3)! j =0

j=0

−4

(j + 3)!



q −1  q t −1  t −1  

q −1  q −1 t −1  t −1  



I1 = 3

(j + 4)!

ri,t −1−u ri,t −u ri,q−1−u ri,q−u

q −1 t −1  

ui,t −1 ∆ui,t T 

− 

ui,t −1

t =2

=

T  T 

 

+

(A.13)

=

1)∧q t ∧( q−1) (t −  

q −1 t −1  

+

2



+

T

T 

ui,t −1 ∆ui,t

+

t =2

1 T

u2 2 i,T



T 

1)∧q t ∧( q−1) (t −   u =1

2

ri,t −1−u ri,t −v ri,q−1−u ri,q−v

v=1

ri,t −1−u ri,t −v ri,q−1−v ri,q−u ,

v=1

and

ui,t −1

t ∧q−1

t =2



ri,t −1−u ri,t −u ri2,q−1−u +

u =1

q−1 t −1  

ri,t −1−u ri,t −u ri2,q−1−a

u=1 a=1 a̸=u

t ∧q−1 t ∧(q−1)

T t =2 q =2

T

ri,t −1−u ri,t −u ri,q−1−a ri,q−a

  u =1



ui,t −1 ∆ui,t ui,q−1 ∆ui,q

u2 2 i ,T

ri,t −1−u ri,t −v ri,q−1−v ri,q−u

v=1 v̸=u

t ∧q−1 t ∧q

+

I2 = 3

T T 2 

1

ri,t −1−u ri,t −v ri,q−1−u ri,q−v

v=1 v̸=u

u =1

t =2 q =2

−

ri,t −1−u ri,t −u ri,q−1−a ri,q−a

u=1 a=1

2

ui,T

ri,t −1−u ri,t −1−w ri,q−1−a ri,q−1−c

t ∧q−1 t ∧q

+

λjN µjc ,

t =2

×



ri,t −1−u ri,t −1−w ri,q−1−a ri,q−b

u=1 a=1 a̸=u

(j + 3)!





u =1

+

∞  2j (j − 1) + 2

T 

ri,t −1−u ri,t −v ri,q−1−a ri,q−1−c E εi,u εi,v εi,a εi,c

where I1 , I2 , I3 and I4 are implicitly defined (with the dependence on i, t and q suppressed for simplicity). I1 and I2 can be expanded in the following way:

which was to be shown. Consider E (A2iT ,0 ). The result in (A.3) implies that T 2 A2iT ,0 ∼



= I1 − I2 − I3 + I4 ,

u =1





  × E εi,u εi,w εi,a εi,c

(j + 4)!

j=0

|ci

c =1

u=1 w=1 a=1 c =1

∞  (λN ci )j , [2j (j − 1) + 2] (j + 3)! j =0



ri,q−1−c εi,c



ri,t −1−u ri,t −v ri,q−1−a ri,q−b E εi,u εi,v εi,a εi,b

q−1  q−1 t −1  t  

giving





t ∧q−1

∞  2j+1 (λN ci )j+1 j=0

+

∞  (λN ci )j+1 [2j+1 (j + 4) − 2j+3 + 2] (j + 4)! j =0

E BiT ,0 ∼

ri,q−b εi,b −

b=1

q −1 

u=1 w=1 a=1 b=1

∞  (λN ci )j+1

+

ri,q−1−a εi,a

q 

ri,t −1−w εi,w

w=1

  × E εi,u εi,w εi,a εi,b

∞  2j (λN ci )j+1

j=0





u=1 v=1 a=1 c =1



1

∞  2j (λN ci )j+1

j =0

ri,t −v εi,v −

t −1 

u=1 v=1 a=1 b=1

2(λN ci )3



t 

v=1

q −1  q t −1  t  

=

−

2(λN ci )3

1

j =0

q −1 

×

3 1

ri,t −1−u εi,u

u =1

2(λN ci )3

−



a=1

(λN ci )j+2 j +2 = − 2 + 4(λN ci )2 2(λN ci )3 (λN ci )3 (j + 2)! j =0    ∞  1 1 1 2 (λN ci )j+3 j +3 = + − 2 + 2 4(λN ci )2 2(λN ci )3 (λN ci )3 (j + 3)! j =0 1

 t −1 

3

2(λN ci )3

4(λN ci )2 1

1

−

j =0

1

5

4(λN ci )2

j =0

−

4

λN ci

∞  2 (λN ci )j (2λN ci )j + j! (λN ci )3 j=0 j!

 ∞

1

4(λN ci )2

j =0

=

exp (2λN ci ) −

=E

∞  

−

1

λN ci 

+2

4(λN ci )2

− =

1 2(λN ci )2

+ =



+

ui,t −1 ∆ui,t ui,T ui,q−1

T  T  t =2 q =2

  u =1

ri,t −1−u ri,t −v ri,q−1−u ri,q−1−v

v=1 v̸=u

t ∧q−1 t ∧(q−1)

ui,t −1 ui,q−1 .

(A.14)

+

  u =1

v=1 v̸=u

ri,t −1−u ri,t −v ri,q−1−v ri,q−1−u

J. Westerlund, R. Larsson / Journal of Econometrics 188 (2015) 59–93

=

q −1 t −1  

ri,t −1−u ri,t −u ri2,q−1−a

+

1)∧q t ∧ q−1 (t −  

u=1 a=1

 

ri,t −1−u ri,t −v ri,q−1−u ri,q−1−v

v=1

u =1

 

+

I1 − I2 =

+

q −1 t −1  

and t ∧q−1

ri,t −1−u ri,t −u ri2,q−1−a



I4 = 3

 

ri,t −1−u ri,t −v ri,q−1−u ri,q−v

+

 

+

ri,t −1−u ri,t −v ri,q−1−u ri,q−1−v

1)∧q t ∧( q−1) (t −  

 

−

ri,t −1−u ri,t −v ri,q−1−v ri,q−u

=

= ri,t −1−u ri,t −v ri,q−1−v ri,q−1−u

 

from which it follows that ri,t −1−u ri,t −v ri,q−1−u ∆ri,q−v 1 {t < q}

I3 − I4 =

q−1 t −1  

u=1 v=1

ri2,t −1−u ri,t −v ∆ri,t −v

+

t −1 



× 1 {t = q} +

q −1  q −1 

+

−

 1 {t > q}

+

ri,t −1−u ri,t −1−v ri,t −v ∆ri,t −u 1 {t = q}

+

−

=

q−1 t −1  

ri,t −1−u ri,t −v ri,q−1−v ∆ri,q−u

+

u=1 v=1 q −1 

ri,t −v ri,q−1−v

1 {t > q} ,

t ∧q−1

ri2,t −1−u ri,q−1−u ri,q−u +

u =1

+

q −1 t −1  

+

u =1

w=1 w̸=u

ri,t −1−u ri,t −1−w ri,q−1−u ∆ri,q−w 1 {t < q}

t −1  t −1 

ri2,t −1−u ri,t −1−w ∆ri,t −w 1 {t = q}

q−1  q−1 

ri,t −1−u ri,t −1−w ri,q−1−u ∆ri,q−w

u=1 w=1

ri2,t −1−u ri,q−1−a ri,q−a

u=1 a=1 a̸=u

 

t −1  t −1 



+ ri,t −1−q

q−1 

 ri,t −1−u ri,q−1−u

1 {t > q}

u=1

t ∧q−1 (t −1)∧q

+

ri2,t −1−u ri,q−1−a ∆ri,q−a

u=1 w=1

where 1 {A} is the indicator function for the event A. We similarly have I3 = 3

ri,t −1−u ri,t −1−w ri,q−1−w ri,q−1−u

w=1

u=1 w=1



v=1



 

u=1 a=1

q −1  q −1 

+ ri,t −1−q

ri,t −1−u ri,t −1−w ri,q−1−w ri,q−u

w=1

t ∧q−1 t ∧q−1

ri,t −1−u ri,t −v ri,q−1−v ∆ri,q−u 1 {t < q}

u=1 v=1



1)∧q t ∧ q −1 (t −  

u=1

t −1  t −1 

ri,t −1−u ri,t −1−w ri,q−1−u ri,q−1−w

w=1

u =1

u=1 v=1

+

  u=1

u=1 t −1  t 

ri,t −1−u ri,t −1−w ri,q−1−u ri,q−w

w=1

t ∧q−1 t ∧q−1

ri,t −1−u ri,t −v ri,q−1−u ∆ri,q−v

ri,t −1−u ri,q−1−u

ri2,t −1−u ri2,q−1−a

u=1 a=1

  u=1

u=1 v=1 q−1 

q −1 t −1  

t ∧q−1 (t −1)∧q

ri2,t −1−u

u =1



+

ri2,t −1−u ri,q−1−a ri,q−a −

u=1 a=1

t −1  t −1 

+ ri,t −q

ri,t −1−u ri,t −1−w ri,q−1−u ri,q−1−w

w=1

u =1

u=1 v=1

+

ri2,t −1−u ri2,q−1−a

t ∧q−1 t ∧q−1

+2

ri,t −1−u ri,t −u ri,q−1−a ∆ri,q−a

t −1  t 



ri,t −1−u ri,t −1−w ri,q−1−w ri,q−1−u

w=1 w̸=u

u=1 a=1

u=1 a=1

+

q−1 t −1  

v=1

u =1 q −1 t −1  

  u=1

t ∧q−1 t ∧(q−1)

 

ri,t −1−u ri,t −1−w ri,q−1−u ri,q−1−w

w=1 w̸=u

t ∧q−1 t ∧q−1

+

v=1

u=1

ri2,t −1−u ri2,q−1−a

u=1 a=1 a̸=u

u=1

v=1

u =1

q −1 t −1  

t ∧q−1 t ∧q−1

v=1

t ∧q−1 t ∧(q−1)

−

ri2,t −1−u ri2,q−1−u +

u=1

t ∧q−1 t ∧q

u=1

ri,t −1−u ri,t −1−w ri,q−1−w ri,q−u ,

w=1

u=1

u =1 a =1

+

ri,t −1−u ri,t −1−w ri,q−1−u ri,q−w

w=1

1)∧q t ∧ q−1 (t −  

ri,t −1−u ri,t −u ri,q−1−a ri,q−a

u =1 a =1

−

  u=1

suggesting that q −1 t −1  

ri2,t −1−u ri,q−1−a ri,q−a

t ∧q−1 (t −1)∧q

ri,t −1−u ri,t −v ri,q−1−v ri,q−1−u ,

v=1

u =1

q−1 t −1   u =1 a =1

t ∧q−1 t ∧(q−1)

+

=

ri,t −1−u ri,t −1−w ri,q−1−w ri,q−u

w=1 w̸=u

u=1

t ∧q−1 t ∧(q−1)

+

75

ri,t −1−u ri,t −1−w ri,q−1−u ri,q−w

+

t −1  t −1  u=1 w=1

ri,t −1−u ri,t −1−w ri,q−1−w ∆ri,q−u 1 {t < q}

76

J. Westerlund, R. Larsson / Journal of Econometrics 188 (2015) 59–93 t −1  t −1 

+

ri,t −1−u ri2,t −1−w ∆ri,t −u 1

u=1 w=1

 +

+ (λN ci )

{t = q}

ri,t −1−u ri,t −1−w ri,q−1−w ∆ri,q−u



q −1 

ri,t −1−w ri,q−1−w

ξ =0

Thus, by adding the results, E (ui,t −1 ∆ui,t ui,q−1 ∆ui,q |ci ) = (I1 − I2 ) + (I4 − I3 )

× dξ 1 {τ < γ }  τ  2 + (λN ci )

ri,t −1−u ri,q−1−a ∆ri,t −u ∆ri,q−a

u=1 a=1 t −1 t −1



ξ =0

ri,t −1−u ri,q−1−u ∆ri,t −v ∆ri,q−v 1 {t < q}

t −1



ri,t −1−u ri,q−1−u 1 {t < q}

ξ =0

u =1

+

t −1  t −1  u=1 v=1

+

q−1  q −1 

ri2,t −1−u ∆ri,t −v



2

1 {t = q} +

t −1 

ri2,t −1−u 1 {t = q}

ri,t −1−u ri,q−1−u ∆ri,t −v ∆ri,q−v 1 {t > q}

+

ξ =0

ri,t −1−u ri,q−1−u 1 {t > q}

+T ξ =0

ri,t −1−u ri,q−1−v ∆ri,t −v ∆ri,q−u 1 {t < q}

t −1 

+

+ 2(λN ci )

+

ri,t −1−u ri,q−1−v ∆ri,t −v ∆ri,q−u 1 {t > q}

∆ri,t −v ri,q−1−v 1 {t > q} ,

+ (λN ci )



γ





ξ =0

τ η=0

ξ =0

η=0

+ λN ci exp {(γ − τ ) λN ci } × dξ 1 {τ < γ }  τ  + (λN ci )2 ξ =0

τ η=0



τ

ξ =0

exp {(τ + γ − 2ξ ) λN ci }

exp {2 (2τ − ξ − η) λN ci }

× dηdξ 1 {τ = γ }  τ +T exp {2 (τ − ξ ) λN ci } dξ 1 {τ = γ }

exp {(τ + γ − 2η) λN ci }

τ η=0 γ η=0



τ ξ =0

exp {(τ + γ − 2ξ ) λN ci }

exp {2 (2τ − ξ − η) λN ci }

exp {2 (τ − ξ + γ − η) λN ci }



γ ξ =0

exp {(τ + γ − 2ξ ) λN ci }

× dξ 1 {τ > γ } =

1 4

+

× dηdξ 1 {τ < γ }

η=0

exp {2 (τ − ξ + γ − η) λN ci }

+ 2λN ci exp {(τ − γ ) λN ci }

exp {2 (τ − ξ + γ − η) λN ci }

γ

× dηdξ 1 {τ > γ }

exp {2 (τ − ξ + γ − η) λN ci } dηdξ

ξ =0 η=0  τ  τ

2

ξ =0

τ

ξ =0

E (ui,t −1 ∆ui,t ui,q−1 ∆ui,q |ci )

∼ (λN ci )2



× dηdξ 1 {τ = γ }  γ  2 + 2(λN ci )

and so, with t = ⌊τ T ⌋, q = ⌊γ T ⌋ and (τ , γ ) ∈ [0, 1] × [0, 1], τ

exp {2 (2τ − ξ − η) λN ci }

η=0

ξ =0

v=1



exp {(τ + γ − 2ξ ) λN ci }

η=0

× dξ 1 {τ < γ }  τ  + 2(λN ci )2

u=1 v=1

+ ri,t −1−q

ξ =0

+ 2λN ci exp {(γ − τ ) λN ci }

ri,t −1−u ri,t −1−v ∆ri,t −v ∆ri,t −u 1 {t = q}

q −1 

τ

× dηdξ 1 {τ < γ }

ri,t −1−u ∆ri,q−u 1 {t < q}

u=1 v=1 q −1 q −1  



exp {2 (τ − ξ ) λN ci } dξ 1 {τ = γ } 2

u =1 t −1  t −1 

τ



u=1 v=1

+ ri,q−1−t

η=0

exp {(τ + γ − 2ξ ) λN ci }

× dη1 {τ > γ }  τ  γ 2 = (λN ci ) exp {2 (τ − ξ + γ − η) λN ci } dηdξ

u =1 t −1  t −1 

τ

ξ =0

exp {2 (τ − ξ + γ − η) λN ci } 1 {τ < γ }

+ λN ci exp {(τ − γ ) λN ci }

u =1

q −1 

η=0

γ

× dηdξ 1 {τ > γ }

u=1 v=1

+ ∆ri,t −q

τ



× dηdξ 1 {τ = γ }  γ  γ exp {2 (τ − ξ + γ − η) λN ci } + (λN ci )2

u=1 v=1

+ ∆ri,q−t

exp {2 (τ − ξ + γ − η) λN ci }

+ λN ci exp {(γ − τ ) λN ci }

t −1 q −1

+

η=0

ξ =0

× dξ 1 {τ > γ }  τ  + (λN ci )2

1 {t > q} .

w=1



γ



+ λN ci exp {(τ − γ ) λN ci }

u=1 w=1

=

γ

× dηdξ 1 {τ > γ }

q−1  q−1 

+ ri,t −1−q



2

(exp{2τ λN ci } − 1)(exp{2γ λN ci } − 1) T 2λN ci

(exp{2τ λN ci } − 1)1 {τ = γ }

1

exp{2(γ − τ )λN ci }(exp{2τ λN ci } − 1)2 1 {τ < γ } 2 + exp{2 (γ − τ ) λN ci }(exp{2τ λN ci } − 1)1 {τ < γ }

+

+ +

1 2 1 2

(exp{2τ λN ci } − 1)2 1 {τ = γ } exp{2 (τ − γ ) λN ci } (exp{2γ λN ci } − 1)2 1 {τ > γ }

+ exp{2 (τ − γ ) λN ci } (exp{2γ λN ci } − 1) 1 {τ > γ } , (A.15)

J. Westerlund, R. Larsson / Journal of Econometrics 188 (2015) 59–93

where the impact of the fifth term on the right-hand side is negligible and may be ignored. Hence, 1 T2

 T  

E

 1

∼

2 ui,t −1 ∆ui,t

t =2



1

+

2λN ci



(exp{2τ λN ci } − 1)(exp{2γ λN ci } − 1)dγ dτ

γ =0

τ =0



1

=

=

ri,t −1−u ri,t −1−w ri,T −w ri,q−1−u

w=1 w̸=u

q −1 t −1  

exp{2 (τ − γ ) λN ci } (exp{2γ λN ci } − 1) dγ dτ

+

ri2,t −1−u ri,T −a ri,q−1−a

[3 + 4λN ci + 4(λN ci ) − 2 (3 + 2λN ci )

× exp{2λN ci } + 3 exp{4λN ci }].

q−1 t −1 t ∧  

ri,t −1−u ri,t −1−w ri,T −u ri,q−1−w

u=1 w=1

2

16(λN ci )2

 

u=1 a=1

γ =0

1

ri,t −1−u ri,t −1−w ri,T −u ri,q−1−w

w=1 w̸=u

u =1

exp{2 (τ − γ ) λN ci } (exp{2γ λN ci } − 1)2 dγ dτ

+2 τ =0

t ∧q−1 t −1

 

+

(A.16)

ri,t −1−u ri,t −1−w ri,T −w ri,q−1−u .

u=1 w=1

Furthermore,

Hence,

E (ui,t −1 ∆ui,t ui,T ui,q−1 |ci )    t −1 t t −1    =E ri,t −1−u εi,u ri,t −v εi,v − ri,t −1−w εi,w

E (ui,t −1 ∆ui,t ui,T ui,q−1 |ci )

v=1

u =1 T 

×

ri,T −a εi,a

q −1 

a=1

=

w=1

q−1 t −1  

 ri,q−1−b εi,b |ci

−

ri,t −1−u ri,t −v ri,T −a ri,q−1−b E εi,u εi,v εi,a εi,b

q−1 T  t −1  t −1  

q −1 t −1  

ri2,t −1−u ri,T −a ri,q−1−a

u=1 a=1





+

u=1 v=1 a=1 b=1

−

ri,t −1−u ri,t −u ri,T −a ri,q−1−a

u =1 a =1

b =1

q −1 t −1  t  T  

=

ri2,t −1−u ri,T −a ri,q−1−a

u=1 a=1 a̸=u

q −1 t −1 t ∧  

+

(exp{2τ λN ci } − 1)dτ

τ



q −1 t −1  

t ∧q−1 t −1

τ =0 τ

+

ri2,t −1−u ri,T −u ri,q−1−u +

u =1

1



1

 u=1

1



1



t ∧ q −1



4 τ =0 γ =0

+

|ci

and J2 = 3

 

77

q−1) t −1 t ∧(  

−

ri,t −1−u ri,t −1−w ri,T −a ri,q−1−b

ri,t −1−u ri,t −v ri,T −u ri,q−1−v

v=1

u=1

q −1 t −1 t ∧  

ri,t −1−u ri,t −1−w ri,T −u ri,q−1−w

u=1 w=1

u=1 w=1 a=1 b=1

  × E εi,u εi,w εi,a εi,b

t ∧q−1

+

= J1 − J2 ,

t  

ri,t −1−u ri,t −v ri,T −v ri,q−1−u

v=1

u =1

t ∧q−1 t −1

where

−

t ∧q−1

J1 = 3



+

q −1 t −1  

= ri,t −1−u ri,t −u ri,T −a ri,q−1−a

u=1 a=1 a̸=u

+

u =1

q−1 t −1  

ri,t −1−u ri,T −a ri,q−1−a ∆ri,t −u

u =1 a =1

 +

q−1) t −1 t ∧(  

ri,t −1−u ri,t −1−w ri,T −w ri,q−1−u

u=1 w=1

ri,t −1−u ri,t −u ri,T −u ri,q−1−u

u =1

 

t −1  t −1 

ri,t −1−u ri,T −u ri,q−1−v ∆ri,t −v

u=1 v=1

ri,t −1−u ri,t −v ri,T −u ri,q−1−v

v=1 v̸=u

+ ri,q−1−t

t −1 

 ri,t −1−u ri,T −u

1 {t < q}

u =1 t ∧q−1

+

t   u =1

=

q −1 t −1  

ri,t −1−u ri,t −v ri,T −v ri,q−1−u

v=1 v̸=u

+

ri,t −1−u ri,t −u ri,T −a ri,q−1−a

+

q−1 t −1  

t ∧q−1 t −1

ri,t −1−u ri,t −v ri,T −u ri,q−1−v

v=1

+

  u=1

t ∧q−1

+

t   u =1

ri,t −1−u ri,q−1−u ri,T −v ∆ri,t −v 1 {t > q}

u=1 v=1

q−1) t −1 t ∧(   u =1

ri,t −1−u ri,T −u ri,t −1−v ∆ri,t −v 1 {t = q}

u=1 v=1

u=1 a=1

+

t −1  t −1 

v=1

ri,t −1−u ri,T −v ri,q−1−u ∆ri,t −v

v=1 t ∧q−1

ri,t −1−u ri,t −v ri,T −v ri,q−1−u

+ ri,T −t

 u =1

ri,t −1−u ri,q−1−u ,

78

J. Westerlund, R. Larsson / Journal of Econometrics 188 (2015) 59–93

showing

which yields

T −1 E (ui,t −1 ∆ui,t ui,T ui,q−1 |ci )

∼ λN ci

τ





γ

exp {(2τ − 2ξ + 1 + γ − 2η) λN ci } dηdξ

ξ =0 η=0  τ  τ

+ λN ci

ξ =0

T −2 E (u2i,T ui,t −1 ui,q−1 |ci )

η=0



τ ξ =0



T4

η=0

ξ =0

T3

E

 ui,t −1 ∆ui,t ui,T ui,q−1 |ci

∼

t =2 q =2

1 8(λN ci )3

× (exp{λN ci } − 1)2 (3 exp{2λN ci } − 3 − 2λN ci ) .

(A.17)

Moreover,

=E

T 

ri,T −u εi,u

×

ri,T −v εi,v

v=1

u =1 q−1 

T 

t −1 



ri2,T −u ri,t −1−u ri,q−1−u +

u =1

+

q −1 t −1  

+

E(

=

u=1 w=1

q −1 T t∧   w=1 w̸=u

+

u=1 v=1

16(λN ci )4 ∞ 

3(2 − λN ci )2

(A.18)

∞  (4λN ci )j

j!

j =0

(3λN ci )j j!

+ 2(28 + 4λN ci + (λN ci )2 − 2(λN ci )3 ) − 8(2 + 3λN ci + 2(λN ci )2 )

∞  j=0

∞  (2λN ci )j

j!

(λN ci )j j! 

ri,T −u ri,T −v ri,t −1−v ri,q−1−u

ri,T −u ri,T −v ri,t −1−u ri,q−1−v

u=1 v=1 q −1  t −1 

|ci ) ∼



1

j=0

ri2,T −u ri,t −1−w ri,q−1−w

ri,T −u ri,T −v ri,t −1−u ri,q−1−v

ri2,T −u ri,t −1−w ri,q−1−w



(exp{λN ci } − 1) [3 exp{3λN ci } − 9 exp{2λN ci }

j=0

t −1 q −1

+

1 4(λN ci )4

+ 24(−2 + λN ci )

− 4 + 4λN ci + 11(λN ci ) + 4(λN ci ) + 4(λN ci ) 2

u=1 v=1 v̸=u q −1 T t∧  

ui,t −1 ui,q−1 |ci

+ 2(28 + 4λN ci + (λN ci )2 − 2(λN ci )3 ) exp{2λN ci } − 8(2 + 3λN ci + 2(λN ci )2 ) exp{λN ci } − 4 + 4λN ci + 11(λN ci )2 + 4(λN ci )3 + 4(λN ci )4 ].

u=1 v=1 v̸=u q −1  t −1 



t =2 q=2

A2iT ,0

u=1

T  T 

1 [3 + 4λN ci + (λN ci )2 − 2(3 + 2λN ci ) 16(λN ci )2 × exp{2λN ci } + 3 exp{4λN ci }] 1 −2 (exp{λN ci } − 1)2 (3 exp{2λN ci } − 3 − 2λN ci ) 8(λN ci )3 1 + (exp{λN ci } − 1)[3 exp{3λN ci } − 9 exp{2λN ci } 4(λN ci )4 + (5 + 2λN ci ) exp{λN ci } + 1 + 2λN ci ] 1 [3(2 − λN ci )2 exp{4λN ci } = 16(λN ci )4 + 24(−2 + λN ci ) exp{3λN ci }

  ri,T −u ri,T −v ri,t −1−w ri,q−1−a E εi,u εi,v εi,w εi,a

t ∧q−1

exp {(2 + τ + γ − 2ξ − 2η) λN ci } dηdξ ,

E (A2iT ,0 |ci ) ∼

u=1 v=1 w=1 a=1

=3

η=0

Further expansions and simplifications yield

ri,q−1−a εi,a |ci

a =1

=

ξ =0

This, together with (A.14), (A.16) and (A.17), implies that E (A2iT ,0 | ci ) can be expanded in the following manner:

ri,t −1−w εi,w

w=1



q −1 T  T  t −1  

exp {(2 + τ + γ − 2ξ − 2η) λN ci } dηdξ

+ (5 + 2λN ci ) exp{λN ci } + 1 + 2λN ci ].

E (u2i,T ui,t −1 ui,q−1 |ci )



η=0  τ

u2i,T

E

∼

where the fourth term is negligible. It follows that T T  

ξ =0  γ



1

× dηdξ 1 {τ > γ }  τ ∧γ  τ exp {(2τ − 2ξ + 1 + γ − 2η) λN ci } dηdξ + λN ci ξ =0 η=0  τ ∧γ + exp {(1 − τ ) λN ci } exp {(τ + γ − 2ξ ) λN ci } dξ ξ =0  τ + exp {(1 − τ ) λN ci } exp {(τ + γ − 2ξ ) λN ci }



γ

and therefore

× dηdξ 1 {τ = γ }  τ  γ exp {(2τ − 2ξ + 1 + γ − 2η) λN ci } + λN ci

1



+

exp {(τ + 1 − 2ξ ) λN ci }

η=0

× dξ 1 {τ < γ } ,

τ

η=0

+

× dξ 1 {τ < γ }  τ  τ exp {(3τ + 1 − 2ξ − 2η) λN ci } + λN ci

ξ =0

exp {(2 + τ + γ − 2ξ − 2η) λN ci } dηdξ

∼

exp {(2τ − 2ξ + 1 + γ − 2η) λN ci }

+ exp {(γ − τ ) λN ci }

τ ∧γ



ξ =0

× dηdξ 1 {τ < γ }

ξ =0

1



=

1 16(λN ci )4

 3(2 − λN ci )2

+ 24(−2 + λN ci )

4

∞  (4λN ci )j+4

(j + 4)!

j=0

∞  j=0

ri,T −u ri,T −v ri,t −1−v ri,q−1−u ,

3

(3λN ci ) (j + 4)!

j+4

+ 2(28 + 4λN ci + (λN ci )2 − 2(λN ci )3 )

∞  (2λN ci )j+4 j=0

(j + 4)!

J. Westerlund, R. Larsson / Journal of Econometrics 188 (2015) 59–93

− 8(2 + 3λN ci + 2(λN ci )2 )

∞  (λN ci )j+4 j =0

1

+ =

6

3

1

2

3



(j + 4)!

a =1

+

(j + 4)!

1

+

6

3

a =1

(λN ci )

= (λN ci )j+1

a=1

192 · 4j − 243 · 3j + 56 · 2j − 1

j =0

(j + 4)!

E (u2i,t −1 ui,u−1 ui,v−1 |ci )

λjN µjc

t ∧u−1 v∧t −1

=

−192 · 4j + 243/2 · 3j + 8 · 2j − 3/2 j+1 j+1 λ N µc (j + 4)!

∞ 

This completes the proof for E (A2iT ,0 ).

t ∧v−1 u∧t −1

T t =2

+

=



T 

T2

b =1

and

a=1

T T T T 1 

+ T T T 2 

T t =2 u=2 v=2

u2i,t −1 ui,u−1 ui,v−1

a =1

w− 1

b=1

ui,t −1 ui,u−1 ui,v−1 ui,w−1 ,

τ ∧γ



(A.19)

ri,u−1−b εi,b

β=0 v−1 

ri,v−1−c εi,c

c =1

ri,w−1−d εi,d |ci



1 T4

E

ri,t −1−a ri,u−1−b ri,v−1−c ri,w−1−d

t ∧u∧v∧w−1



 

1 16(λN ci )4

c =1 c ̸=a

|ci

[−9 − 4λN ci + 4(λN ci )2 (A.20)

T −2 E (u2i,t −1 ui,u−1 ui,v−1 |ci )



τ ∧γ

β=0

ri,t −1−a ri,u−1−a ri,v−1−c ri,w−1−c

 u2i,t −1 u2i,u−1

t =2 u =2



∼

t ∧u−1 v∧w−1

a=1

T  T 

Similarly,

ri,t −1−a ri,u−1−a ri,v−1−a ri,w−1−a

a =1

exp {2 (τ + γ − β − δ) λN ci } dδ dβ,

+ (6 − 20λN ci ) exp{2λN ci } + 3 exp{4λN ci }].

a=1 b=1 c =1 d=1



δ=0

exp {2 (τ + γ − β − δ) λN ci } dδ dβ

implying

∼

× E εi,a εi,b εi,c εi,d

δ=0 γ



+

d=1



τ ∧γ



∼2 β=0 τ



t −1  u −1  v−1 w−  1

ri2,t −1−a ri2,u−1−b ,

T −2 E (u2i,t −1 u2i,u−1 |ci )

E (ui,t −1 ui,u−1 ui,v−1 ui,w−1 |ci ) u −1 

u −1 t −1  

and therefore, letting t = ⌊τ T ⌋, u = ⌊γ T ⌋ and (τ , γ ) ∈ [0, 1] × [0, 1],

t =2 u=2 v=2 w=2

ri,t −1−a εi,a

ri,t −1−a ri,u−1−b ri,u−1−a ri,t −1−b

b =1

a=1 b=1

where

t −1 

  a=1

t =2

ui,t −1

u2i,t −1 u2i,u−1 −

ri,t −1−a ri,u−1−a ri,u−1−c ri,t −1−c

c =1

t ∧u−1 u∧t −1

+

 

 

4

t =2 u=2

T2

a=1

ri2,t −1−a ri,u−1−b ri,v−1−b

t =2

T  T 

+

t −1 u ∧v−1 

+

ri,t −1−a ri,u−1−b ri,v−1−a ri,t −1−b

b=1

t ∧u−1 u∧t −1

). From (A.9),  2  2 T T T   2 2 2 ∼ ui , t − 1 − ui,t −1 ui,t −1 1

 

+

E (u2i,t −1 u2i,u−1 |ci ) =

B2iT ,0

t =2

ri,t −1−a ri,u−1−a ri,v−1−c ri,t −1−c

c =1

a =1

∞  −4 · 2 j j + 3 j + 3 1 3 1 λN µc + + λN µ1c − λ2N µ2c . + 6 2 3 (j + 4)! j =0

Consider E (

  a=1

48 · 4j + 2 · 2j − 1 j+2 j+2 λ N µc + (j + 4)! j =0

+

ri,t −1−a ri,u−1−b ri,v−1−b ri,w−1−a .

b =1

In particular, we have

∞ 

=3

 

3

j =0

=

ri,t −1−a ri,u−1−b ri,v−1−a ri,w−1−b

b=1

t ∧w−1 u∧v−1

+

∞ 

+

×

  a =1

+ λN ci − (λN ci )2

E (A2iT ,0 ) ∼

=E

ri,t −1−a ri,u−1−a ri,v−1−c ri,w−1−c

c =1

t ∧v−1 u∧w−1

+

∞  −4 · 2j (λN ci )j+2 + (λN ci )j+3 j + 4 ( )! j =0

and so,

T 4 B2iT ,0

  a=1

1

2

ri,t −1−a ri,u−1−b ri,v−1−b ri,w−1−a

b=1 b̸=a

t ∧u−1 v∧w−1

(j + 4)!

j =0

 

j

∞  −192 · 4j + 243/2 · 3j + 8 · 2j − 3/2

∞  48 · 4j + 2 · 2j − 1

ri,t −1−a ri,u−1−b ri,v−1−a ri,w−1−b

b=1 b̸=a

t ∧w−1 u∧v−1

(j + 4)!

j =0

 

+

∞  192 · 4j − 243 · 3j + 56 · 2j − 1

+

t ∧v−1 u∧w−1

+

+ λN ci − (λN ci )2

j =0

79



τ ∧ξ

exp {(2τ + γ + ξ − 2β − 2δ) λN ci } dδ dβ

δ=0 τ ∧ξ  τ ∧γ

+ β=0

δ=0

exp {(2τ + γ + ξ − 2β − 2δ) λN ci } dδ dβ

80

J. Westerlund, R. Larsson / Journal of Econometrics 188 (2015) 59–93

τ



γ ∧ξ



β=0 δ=0 τ ∧γ  τ ∧ξ



β=0 τ

δ=0 γ ∧ξ



δ=0

1 T5

E

t =2 u=2 v=2

∼

+

u2i,t −1 ui,u−1 ui,v−1 |ci

+

1

[35 + 8λN ci − 4(λN ci )2 8(λN ci )5 + 4 (−13 + 6λN ci ) exp{λN ci }

=

5

τ ∧γ



ξ ∧η



+

β=0



δ=0 τ ∧ξ  γ ∧η

+ β=0  τ ∧η

δ=0  γ ∧ξ

+



β=0 τ ∧γ



δ=0 ξ ∧η

=3 β=0

δ=0

18

(λN ci )6 27 20



− +

T6

E

∼

4(λN ci )6

5

E(

|ci ) ∼



E (B2iT ,0 ) ∼

27 20

+ + × + + +

27 4(λN ci ) 1

6

4(λN ci )2

−

+

4(λN ci )5

1 4(λN ci )

5

+

7 16(λN ci )

E(

 ∼

4(λN ci )6

−

3

(j + 6)! (j + 6)! (j + 6)!

4

+

4(λN ci )

,

3

λjN µjc

λjN+1 µjc+1

λjN+2 µcj+2

∞  5 · 2j+4 j+3 j+3 λ µ − (j + 6)! N c j=0

5 12

27

+

20

1

λN µ1c − λ2N µ2c . 3

which establishes the required result for E (B2iT ,0 ). p

Let us now consider E (AiT ,1 ). With p = 1, dt = d1t = (1, t )′ . Thus, letting FT = diag (1, T ), we have FT−1

T 

d1t d1t ′ FT−1 =

t =2

T   1 t /T

t /T (t /T )2

t =2



 ∼

1 1/2

1/2 , 1/3



which is invertible. It follows that

 a1t ,u

=

d1t ′ FT−1

−1

FT

T 

−1 d1t d1t ′ FT−1

FT−1 d1u

t =2

3

∼ (1

tT −1 )



1 1/2

 −1 

1/2 1/3

1 uT −1



= 4 − 6tT −1 − 6uT −1 + 12tuT −2 . Substitution into (A.1) yields

|ci ) 3

1

j=0

(A.24)



(λN ci )j+1

(λN ci )j+2

∞  3 · 4j+4 + 19 · 2j+3 − 6

+ 3

(λN ci )j

λN ci − (λN ci )2 ,

j=0

which simplifies to B2iT ,0

3

∞  −3 · 4j+5 + 3j+7 − 7 · 2j+5 + 1

+

+ exp{4λN ci } 16(λN ci )4   6 3 − + exp{3λN ci } (λN ci )6 (λN ci )5   33 7 19 5 − + − 2(λN ci )6 2(λN ci )5 8(λN ci )4 4(λN ci )3 exp{2λN ci }   18 1 6 − exp{λN ci } + − (λN ci )6 (λN ci )5 (λN ci )4

4(λN ci )6

1

λN ci − (λN ci )2

∞  3 · 4j+5 − 6 · 3j+6 + 33 · 2j+5 − 18

+

ui,t −1 ui,u−1 ui,v−1 ui,w−1 |ci

3

j =0

(λN ci )j+6 (j + 6)!

and so we obtain



3

(λN ci )4

(j + 6)!

+

12

Insertion into (A.19) now yields B2iT ,0

(λN ci )5

j=0

(3 + 2λN ci − 4 exp{λN ci } + exp{2λN ci })2 .

−

∞  5 · 2j+4 (λN ci )j+3 (j + 6)! j=0

t =2 u=2 v=2 w=2

3

4(λN ci )3

 ∞

6

∞  3 · 4j+4 + 19 · 2j+3 − 6

(A.23)

T T  T  T  

+

1

(j + 6)!

giving 1

−

8(λN ci )4

(j + 6)!

exp {(τ + γ + ξ + η − 2β − 2δ)

× λN ci } dδ dβ,

+

2(λN ci )5



5

(2λN ci ) (j + 6)!

j=0

exp {(τ + γ + ξ + η − 2β − 2δ) λN ci } dδ dβ

19

∞  −3 · 4j+5 + 3j+7 − 7 · 2j+5 + 1

(A.22)

exp {(τ + γ + ξ + η − 2β − 2δ) λN ci } dδ dβ

+

−

j=0

exp {(τ + γ + ξ + η − 2β − 2δ) λN ci } dδ dβ

j =0

7

∞  3 · 4j+5 − 6 · 3j+6 + 33 · 2j+5 − 18

E (ui,t −1 ui,u−1 ui,v−1 ui,w−1 |ci )

∼

(λN ci )5

(3λN ci )j+6 (j + 6)!

j +6

+

12

 ∞

3

+

j =0

and, letting w = ⌊ηT ⌋ with η ∈ [0, 1], T

−



+ (26 − 8λN ci ) exp{2λN ci } − 12 exp{3λN ci } + 3 exp{4λN ci }], −2

∞ 



where v = ⌊ξ T ⌋ and ξ ∈ [0, 1], suggesting that T  T  T 

(λN ci )6

2(λN ci )6

j =0

(A.21)

6

33

+ ×

× dδ dβ, 

− 

exp {(2τ + γ + ξ − 2β − 2δ) λN ci }

+ β=0

+

exp {(2τ + γ + ξ − 2β − 2δ) λN ci } dδ dβ

=2 



exp {(2τ + γ + ξ − 2β − 2δ) λN ci } dδ dβ

+

3 4(λN ci )5

+

3 16(λN ci )4

 ∞ j=0

(4λN ci )j+6 (j + 6)!

TAiT ,1 ∼

T  t =2

ui,t −1 ∆ui,t −

T T 2 

T t =2 u=2

+ 12tuT −2 )ui,t −1 ∆ui,u

(4 − 6tT −1 − 6uT −1

J. Westerlund, R. Larsson / Journal of Econometrics 188 (2015) 59–93 T T T 1 

+

T 2 t =2 u=2 v=2

(4 − 6tT −1 − 6uT −1 + 12tuT −2 )

∼

ui,t −1 ∆ui,t +

t =2

8 T

T 

ui,T

PiT =

T 2 t =2 u=2

 +

ui,t −1 + 12TPiT ,

t =1

1



1

1





u 

ri,u−w εi,w −

w=1

u −1 

 ri,u−1−w εi,w

 |ci

1

1



 ξ , γ = g2 (τ , γ , ξ ). Furthermore, by symmetry,  1  1 2E (PiT |ci ) ∼ (2τ − 2τ γ ) exp {(τ + γ ) λN ci } dγ dτ τ =0



ri,t −1−v ri,u−v −

v=1



(t −1 )∧(u−1)

 ri,t −1−v ri,u−1−v

v=1

u −1 



x exp (ax) dx =

1 {u < t }

v=1

=h η=0

2E (PiT |ci ) ∼ − ′

 +

exp {(τ + γ − 2η) λN ci } dη

+ exp {(τ − γ ) λN ci } 1 {γ < τ } [exp {(τ + γ ) λN ci } − exp (|τ − γ | λN ci )]

−

1 2

1

[exp {(τ + γ ) λN ci } + exp {(τ − γ ) λN ci } 1 {γ < τ }

− exp {(γ − τ ) λN ci } 1 {τ ≤ γ }], where t = ⌊τ T ⌋, u = ⌊γ T ⌋ and (τ , γ ) ∈ [0, 1] × [0, 1].

(λN ci ) 

3

−

1

(λN ci )4 

+



1

2

(λN ci )

4

(λN ci )

(A.27)

2E (PiT |ci ) ∼

1

τ =0



1

γ =0

g1 (τ , γ ) exp {(τ + γ ) λN ci } dγ dτ

 ∞

1

+

4

3

2

(2λN ci )j j!

1

(λN ci )2 −

+

1

+

(λN ci )3 (λN ci )

(λN ci )2

2

= −(1 − λN ci + (λN ci )2 ) + (2 + (λN ci )2 )

1

 ∞

1

Consequently,



1

−

(λN ci ) (λN ci ) j =0  ∞ j 2 1 (λN ci ) 1 + − (λN ci )4 (λN ci )2 j=0 j! (λN ci )4 1

=−

+ exp {(τ − γ ) λN ci } 1 {γ < τ } =

exp (ax) ,

+ exp{2λN ci }(1 − λN ci + (λN ci )2 )],

r (τ − η) r (γ − η) dη + r (τ − γ ) 1 {γ < τ }

η=0  τ ∧γ

2

a3

and by further expansions and simplifications,

E (ui,t −1 ∆ui,u |ci )

1

a2 x2 − 2ax + 2

[1 + λN ci + (λN ci )2 (λN ci )4 − exp{λN ci }(2 + (λN ci )2 )

implying

=

(−4τ − 4γ + 3τ 2 + 14τ γ + 3γ ξ

exp (ax) ,

a2

2E (PiT |ci ) ∼ −

ri,t −1−v ∆ri,u−v + ri,t −1−u 1 {u < t } ,

∼

ax − 1

x2 exp (ax) dx =

v=1

τ ∧γ

ξ =0

now yields, after simplification,

 ri,t −1−v ∆ri,u−v + ri,t −1−u ri,0

γ =0

Substitution of

t ∧u−1



1



− 12τ 2 γ − 12τ γ ξ + 12τ 2 γ ξ ) × exp {(γ + ξ ) λN ci } dξ dγ dτ .

v=1

=

1



τ =0

ri,t −1−v ∆ri,u−v 1 {t ≤ u}



γ =0

1

 +

ri,t −1−v ri,u−1−w E εi,v εi,w

(A.26)



w=1

v=1 w=1

+

g2 (τ , γ , ξ )

× exp {(γ + ξ ) λN ci } dξ dγ dτ ,

  ri,t −1−v ri,u−w E εi,v εi,w





ξ =0

where the equality holds, because g1 (γ , τ ) = g1 (τ , γ ) and g2 τ ,

t −1 u −1

=

1



γ =0

τ =0

u



t −1 

g2 (τ , γ , ξ ) exp {(ξ − γ ) λN ci } dξ dγ dτ

g1 (τ , γ ) exp {(τ + γ ) λN ci } dγ dτ

γ =0

v=1 w=1

=

ξ =γ

γ =0 1





E (ui,t −1 ∆ui,u |ci )

1)∧u (t − 

1



+

to show that

−

g1 (τ , γ ) exp {(γ − τ ) λN ci } dγ dτ

γ =τ

= τ =0

g2 (τ , γ , ξ ) exp {(γ − ξ ) λN ci } dξ dγ dτ

1



τ =0

g2 (tT −1 , uT −1 , v T −1 )ui,u−1 ∆ui,v ,

− 6τ 2 γ − 6τ 2 ξ − 12τ γ ξ + 12τ 2 γ ξ . By following the same steps as in the proof of E (AiT ,0 ) it is possible

=

ξ =0

−

g2 (τ , γ , ξ ) = −4τ − 2γ − 2ξ + 3τ + 7τ γ + 7τ ξ + 3γ ξ

ri,t −1−v εi,v

γ



γ =0

τ =0

g2 (τ , γ , ξ ) exp {(γ + ξ ) λN ci } dξ dγ dτ

g1 (τ , γ ) exp {(τ − γ ) λN ci } dγ dτ

−

2

t −1

1



v=1

1



τ =0

g1 (τ , γ ) = τ + γ − 2τ γ ,

=E

ξ =0

+

with

 t −1 

1



(A.25)

g1 (tT −1 , uT −1 )ui,t −1 ∆ui,u

T 3 t =2 u=2 v=2

γ =0

τ =0



T T T 1 

+

γ =0 1  τ

81

1



τ =0

where T T 1 

1



+

× (4 − 6tT −1 − 6v T −1 + 12t v T −2 )ui,u−1 ∆ui,v T 

1



∞  j =0

j =0

 ∞ j =0

(2λN ci )j+4 (j + 4)!

(λN ci ) 1 7 − − λN ci 6 6 (j + 4)! j+4

∞ 

2j+4 (λN ci )j

j =0

(j + 4)!

(λN ci ) 1 7 − − λN ci (j + 4)! 6 6 j

82

J. Westerlund, R. Larsson / Journal of Econometrics 188 (2015) 59–93

=

∞ 

−16 · 2j + 2 (λN ci )j + (j + 4)!

j =0

∞  −16 · 2j + 1

+

(j + 4)!

j =0

∞ 

16 · 2j

j=0

(j + 4)! 1

(λN ci )j+2 −

6

(λN ci )j+1

E (AiT ,1 |ci ) ∼

=

T

 E

T 

6

 ui,t −1 ∆ui,t |ci

+

t =1

8 T2

 E

ui,T

T 

 ui,t −1 |ci

t =1

+ 12E (PiT |ci ) ∞ 1  (2λN ci )j+1

+6

 ∞  −16 · 2j + 2 (j + 4)!

j =0

∞ 

+

j =0

j



∞  16 · 2j (λN ci ) + (λN ci )j+1 j + 4 ( )! j =0 

2 j=0 (j + 2)!

µj + 1 + 8

(j + 4)!

∞  −16 · 2j + 1

(j + 4)!

j =0

=



j N

λjN+2 µjc+2 −

1 6

∼



6

+

t =2

8



RiT =



T

∼−

ui , t − 1

+ 12T

−2

−

=−

T 3 t =2 u =2

RiT ,

g1 (tT −1 , uT −1 )ui,t −1 ui,u−1

T T T 1 

T 4 t =2 u=2 v=2

g2 (tT −1 , uT −1 , v T −1 )ui,u−1 ui,v−1 .

Making use of (A.11), E (RiT |ci )



1

γ =0

ξ =0

g2 (τ , γ , ξ ) exp {(γ − ξ ) λN ci } dξ dγ dτ

[−6 + 6(λN ci )2 + 5(λN ci )3



1 2(λN ci )5 1 6(λN ci )5

(A.29)

1 6(λN ci )5

g1 (τ , γ ) [exp {(τ + γ ) λN ci }

2λN ci τ =0 γ =0 − exp (|τ − γ | λN ci )] dγ dτ  1  1  1 1 + g2 (τ , γ , ξ ) [exp {(γ + ξ ) λN ci } 2λN ci τ =0 γ =0 ξ =0 − exp (|γ − ξ | λN ci )] dξ dγ dτ  1  1 1 = g1 (τ , γ ) exp {(τ + γ ) λN ci } dγ dτ 2λN ci τ =0 γ =0

[−6 + 6(λN ci )2 + 5(λN ci )3

[−3 + 3λN ci + 9(λN ci )2 + 5(λN ci )3

Insertion of this, (A.10) and (A.12) into (A.29) gives E (BiT ,1 |ci ) ∼

+

4

(λN ci )2



1 2λN ci



1

λN ci

1 2λN ci

exp (2λN ci ) −

exp (2λN ci ) −

2

4

λN ci

1 2λN ci

 −1

exp (λN ci ) +

3

λN ci

 +2

[−3 + 3λN ci + 9(λN ci ) + 5(λN ci ) (λN ci )5 + (−6λN ci − 6(λN ci )2 ) exp{λN ci } + (3 − 3λN ci + 3(λN ci )2 ) exp{2λN ci }]   6 2 1 6 + − + exp{2λN ci } = − (λN ci )5 (λN ci )4 (λN ci )3 4(λN ci )2   12 4 6 6 + − exp{λN ci } + − (λN ci )4 (λN ci )3 (λN ci )5 (λN ci )4 6

1

[1 + λN ci + (λN ci )2 − exp{λN ci }(2 + (λN ci )2 )

+ (−6λN ci − 6(λN ci )2 ) exp{λN ci } + (3 − 3λN ci + 3(λN ci )2 ) exp{2λN ci }].

t =2

T

1 

1

γ



+ exp{2λN ci }(1 − λN ci + (λN ci )2 )]

−

+

∼

6(λN ci )4

2

T

where T

1



− 3 exp{λN ci }(−2 + 2λN ci + (λN ci )2 )]

u=2

u2i,t −1

1

1

7

− λN µ1c . (A.28)

+ 12tuT −2 )ui,t −1 ui,u−1  2 T T 1   −1 −1 −2 + 2 (4 − 6tT − 6uT + 12tuT )ui,t −1 T

g1 (τ , γ ) exp {(τ − γ ) λN ci } dγ dτ

E (RiT |ci )

∞  16 · 2j j+1 j+1 λjN µjc + λ µ (j + 4)! N c j =0 

T t =2 u =2

t =2

g2 (τ , γ , ξ ) exp {(γ + ξ ) λN ci }

suggesting that, via (A.27),

T T 2  u2i,t −1 − (4 − 6tT −1 − 6uT −1

t =2

T

γ =0

τ =0

This completes the proof for E (AiT ,1 ). Consider E (BiT ,1 ). With p = 1 (A.2) becomes T 2 BiT ,1 ∼

τ





∞  2j+1 − 1 λ µjc ( j + 2 )! j =0



 ∞  −16 · 2j + 2 j =0

T 

ξ =0

− 3 exp{λN ci }(−2 + 2λN ci + (λN ci )2 )],

∞ 1  (2λN )j+1

+

γ =0

τ =0

1



+

−16 · 2 + 1 1 7 (λN ci )j+2 − − λN ci , 6 6 (j + 4)!

+6

1

τ =0

j

and so we obtain E (AiT ,1 ) ∼

1

We have already calculated the first two integrals. The sum of the two last ones is given by

 j+1  ∞  2 − 1 (λN ci )j +8 (j + 2)! (j + 2)! j =0

2 j =0

2λN ci



× dξ dγ dτ  1  τ 1 − g1 (τ , γ ) exp {(τ − γ ) λN ci } dγ dτ λN ci τ =0 γ =0  1  1  γ 1 g2 (τ , γ , ξ ) exp {(γ − ξ ) λN ci } − λN ci τ =0 γ =0 ξ =0 × dξ dγ dτ .

7

− λN ci .

This, together with (A.6) and (A.7), implies 1

1



1

+

2

9

1

3

− − − (λ c )3 4(λN ci )2 2λN ci  N i  6 6 2 1 = − + − + (λN ci )5 (λN ci )4 (λN ci )3 4(λN ci )2 ∞ j + 5  (2λN ci ) × (j + 5)! j =0   ∞ 12 4 (λN ci )j+5 5 + − + 4 3 (λN ci ) (λN ci ) 3 (j + 5)! j =0

J. Westerlund, R. Larsson / Journal of Econometrics 188 (2015) 59–93

7

1

× ui,T ui,t −1 ui,v−1 ∆ui,w T T T T T 288      + 3 g1 (tT −1 , uT −1 )

− λN ci + (λN ci )2 6 6   ∞ 2j+5 (λN ci )j 1 = −6 + 6λN ci − 2(λN ci )2 + (λN ci )3 4 (j + 5)! j =0 + (12λN ci − 4(λN ci )2 )

T

+

∞  64 · 2j + 4 j =0

(j + 5)!

∞ 

8 · 2j

(j + 5)!

j =0

T

E ui,T ui,t −1 ui,u−1 ∆ui,v |ci



5

7 6

192 · 2

j =0

(j + 5)!

j

6

λj µjc +

∞  64 · 2j + 4

−

(j + 5)!

j =0



192 · 2 + 12 j+1 j+1 λ N µc (j + 5)! j =0

(A.30)

3 6 6 as required. Consider E (A2iT ,1 ). From (A.25), T 2 A2iT ,1 ∼

ui,t −1 ∆ui,t +

T

t =2 T T 12  

+

T

+ 



=

ui,t −1 ∆ui,t



ui,u−1 +

u=2

T UiT =

T

g2 (tT

64 T2

+

T 

T T T 192   

T2

−1

16

+

, uT



−1

, vT

−1

+

T

)ui,u−1 ∆ui,v +

 ui,t −1 ∆ui,t

u2i,T

T 

2 ui , t − 1

+ T 2 UiT , (A.31)



, vT

−1

ri,T −a ri,t −1−a ri,u−1−a ri,v−a

a =1

+

)ui,t −1 ∆ui,t ui,u−1 ∆ui,v

ri,T −a ri,t −1−b ri,u−1−a ri,v−b

ri,T −a ri,t −1−b ri,u−1−b ri,v−a

ri,T −a ri,t −1−a ri,u−1−c ri,v−c

c =1

1)∧v u−1 (t −   a =1

+

ri,T −a ri,t −1−a ri,u−1−c ri,v−c

v t∧ u −1   a =1

ri,T −a ri,t −1−b ri,u−1−a ri,v−b

b=1

ri,T −a ri,t −1−b ri,u−1−b ri,v−a ,

b =1

and g1 (uT

−1

, vT

−1

)ui,T ui,t −1 ui,u−1 ∆ui,v

t ∧u∧v−1

K2 = 3

g1 (tT −1 , uT −1 )g1 (v T −1 , w T −1 )

+

ri,T −a ri,t −1−a ri,u−1−a ri,v−1−a

t −1 u ∧v−1  a =1

+

t =2 u=2 v=2 w=2

t =2 u=2 v=2 w=2

 a=1

T

t =2 u=2 v=2 w=2



ri,T −a ri,t −1−b ri,u−1−c ri,v−1−e E εi,a εi,b εi,c εi,e

b=1 b̸=a

−1)∧v t −1 (u 

t =2

× ui,t −1 ∆ui,t ui,v−1 ∆ui,w T T T T 192     + 3 g2 (uT −1 , v T −1 , w T −1 ) T

ri,T −a ri,t −1−b ri,u−1−c ri,v−d E εi,a εi,b εi,c εi,d

b=1 b̸=a

v t∧ u −1   a =1

=

× ui,t −1 ∆ui,u ui,v−1 ∆ui,w T T T T 24     + 2 g2 (uT −1 , v T −1 , w T −1 ) T

|ci

c =1 c ̸=a

1)∧v u−1 (t −   a =1

t =2



−1)∧v t −1 (u  a =1

T



T

−1

144     T2

u−1)∧v (t −1)∧( 

ui,t −1

t =2 u=2 v=2 T

ri,v−1−e εi,e



a=1

t =2 u=2 v=2

T

+

g1 (uT



e=1

T −1  t −1  u −1  v−1 

K1 = 3

where the expectations of the first three terms have already been computed. The remaining term is T T T 24   

v−1 

= K1 − K2 ,

t =2

t =2

× ui,T

−

2

+

T

ri,v−d εi,d −

ri,u−1−c εi,c

c =1

where

ui,T

2

T

v 

u −1 

a=1 b=1 c =1 d=1

g1 (tT −1 , uT −1 )ui,t −1 ∆ui,u

T 2 t =2 u=2 v=2

ri,t −1−b εi,b

b =1

T −1  t −1  u −1  v 

t =2 u=2

T T T 12   



a=1 b=1 c =1 e=1

1

8

=

∞  8 · 2j j+3 j+3 λ µ (j + 5)! N c j =0

− λN µ1c + λ2N µ2c ,

 T 

t −1 

d=1

j

λj+2 µjc+2 +

7

5

+

∞ 

ri,T −a εi,a

a =1

×

E (BiT ,1 ) ∼ −

2

 T −1 

=E

1

− λN ci + (λN ci )2 ,

3

(A.32)

where g1 and g2 are as above. Here, in the same manner as above,

implying ∞ 

t =2 u=2 v=2 w=2 x=2 y=2

× g2 (v T −1 , xT −1 , yT −1 )ui,u−1 ∆ui,v ui,x−1 ∆ui,y ,

(λN ci )j+2

(λN ci )j+3 +

t =2 u=2 v=2 w=2 x=2

× g2 (v T −1 , wT −1 , xT −1 )ui,t −1 ∆ui,u ui,w−1 ∆ui,x T T T T T T 144       + 4 g2 (tT −1 , uT −1 , v T −1 )

∞  5 (λN ci )j 7 1 + − λN ci + (λN ci )2 6 (j + 5)! 3 6 j =0

∞ ∞   192 · 2j 192 · 2j + 12 =− (λN ci )j + (λN ci )j+1 j + 5 j + 5 ( )! ( )! j=0 j=0

−

83

u−1 t ∧v−  1 a=1

+

ri,T −a ri,t −1−b ri,u−1−a ri,v−1−b

b=1 b̸=a

v−1 t ∧ u −1   a=1

ri,T −a ri,t −1−a ri,u−1−c ri,v−1−c

c =1 c ̸=a

b=1 b̸=a

ri,T −a ri,t −1−b ri,u−1−b ri,v−1−a





84

J. Westerlund, R. Larsson / Journal of Econometrics 188 (2015) 59–93

=

t −1 u ∧v−1  a =1

+

c =1

u−1 t ∧v−  1 a =1

+

× dβ 1 {γ > ξ }  γ  τ exp {(1 + τ + γ + ξ − 2β − 2δ) λN ci } + λN ci

ri,T −a ri,t −1−a ri,u−1−c ri,v−1−c

β=0

b=1

v−1 t ∧ u −1   a =1

ri,T −a ri,t −1−b ri,u−1−b ri,v−1−a ,

β=0

implying E (ui,T ui,t −1 ui,u−1 ∆ui,v |ci ) =

a=1

−

t −1 u ∧v−1  a =1

+

a =1

−

u−1 t ∧v−  1

a =1

−

ri,T −a ri,t −1−a ri,u−1−c ri,v−1−c

β=0

+ exp {(1 − ξ ) λN ci }

ri,T −a ri,t −1−b ri,u−1−a ri,v−b

1 T4

E

ri,T −a ri,t −1−a ri,u−1−c ∆ri,v−c 1 {u ≤ v}

t −1 v−1

ri,T −a ri,t −1−a ri,u−1−c ∆ri,v−c

a =1 c =1

+

1

+

E

ri,T −a ri,t −1−b ri,u−1−a ∆ri,v−b

 ri,T −a ri,u−1−a

+

a =1

1 8(λN ci )6

1 {t > v}

− 2 − 2λN ci + 2(λN ci )2 + 5(λN ci )3 ].

ri,T −a ri,t −1−b ri,u−1−b ∆ri,v−a

b=1



ri,t −1−b ri,u−1−b .

Next, consider

=E

b =1

×

T −1 E (ui,T ui,t −1 ui,u−1 ∆ui,v |ci )



τ

β=0

γ



δ=0

× dδ dβ 1 {γ ≤ ξ }  τ  ξ exp {(1 + τ + γ + ξ − 2β − 2δ) λN ci } + λN ci β=0

δ=0

+ exp {(γ − ξ ) λN ci }



τ β=0

 ri,t −1−a εi,a

v−1 

 ri,v−1−d εi,d

d=1

exp {(1 + τ + γ + ξ − 2β − 2δ) λN ci }

× dδ dβ 1 {γ > ξ }

 t −1  a =1

It follows that, with t = ⌊τ T ⌋, u = ⌊γ T ⌋, v = ⌊ξ T ⌋ and (τ , γ , ξ ) ∈ [0, 1] × [0, 1] × [0, 1],

∼ λN ci

exp {(1 + τ − 2β) λN ci }

(A.34)

E (ui,t −1 ∆ui,u ui,v−1 ∆ui,w |ci )

t ∧u−1

+ ri,T −v

[6(1 − λN ci − (λN ci )2 ) exp{4λN ci }

+ 6(−4 + 2λN ci + 5(λN ci )2 ) exp{3λN ci } + (28 + 8λN ci − 40(λN ci )2 − 5(λN ci )3 ) exp{2λN ci } + 2(−4 − 6λN ci + 3(λN ci )2 ) exp{λN ci }

a=1

v−1 t ∧ u −1  

g2 (uT −1 , v T −1 , w T −1 )



a=1 b=1

+ ri,t −1−v

(A.33)

t =2 u=2 v=2 w=2

∼

u −1 

)ui,T ui,t −1 ui,u−1 ∆ui,v |ci

[6(−1 + (λN ci )) exp{4λN ci }

 T  T  T  T 

ri,T −a ri,t −1−b ri,u−1−a ∆ri,v−b 1 {t ≤ v}

u−1  v−1 

, vT

× ui,T ui,t −1 ui,v−1 ∆ui,w |ci

a=1 b=1



g1 (uT

−1

and

a =1 u−1  t −1 

1 4(λN ci )6

1 {u > v}

ri,T −a ri,t −1−a

 −1

+ 2 + 2λN ci − (λN ci )3 ],

T5



t −1 

+ ri,u−1−v

exp {(τ + γ − 2β) λN ci } dβ,

− 6(−4 + 2λN ci + (λN ci )2 ) exp{3λN ci } + (−28 − 8λN ci + 12(λN ci )2 + (λN ci )3 ) exp{2λN ci } + 2(4 + 6λN ci + (λN ci )2 ) exp{λN ci }

ri,T −a ri,t −1−b ri,u−1−b ri,v−1−a



+

 T  T  T 

ri,T −a ri,t −1−b ri,u−1−b ri,v−a

a =1 c =1



β=0

t =2 u=2 v=2

b=1

t −1  u−1 

τ ∧γ



giving ri,T −a ri,t −1−b ri,u−1−a ri,v−1−b

∼

v−1 t ∧ u −1  

exp {(1 + γ − 2β) λN ci }

δ=0

× dδ dβ

b=1

a =1

=

β=0

× dβ 1 {τ > ξ }  ξ  τ ∧γ exp {(1 + τ + γ + ξ − 2β − 2δ) λN ci } + λN ci

ri,T −a ri,t −1−a ri,u−1−c ri,v−c

c =1

b =1

v t∧ u −1  

γ



b =1

a =1

+

+ exp {(τ − ξ ) λN ci }

c =1

1)∧v u−1 (t −  

δ=0

× dδ dβ 1 {τ > ξ }

b =1

−1)∧v t −1 (u 

δ=0

× dδ dβ 1 {τ ≤ ξ }  γ  ξ exp {(1 + τ + γ + ξ − 2β − 2δ) λN ci } + λN ci

ri,T −a ri,t −1−b ri,u−1−a ri,v−1−b

=

t −1  u  v−1  w 

u 

ri,u−b εi,b −

u −1 

b=1

c =1

w 

w− 1

e=1

ri,w−e εi,e −

 ri,u−1−c εi,c

 ri,w−1−f εi,f

|ci

f =1

ri,t −1−a ri,u−b ri,v−1−d ri,w−e E (εi,a εi,b εi,d εi,e )

a=1 b=1 d=1 e=1

−



t −1  u  v−1 w−  1

ri,t −1−a ri,u−b ri,v−1−d ri,w−1−f

a=1 b=1 d=1 f =1

× E (εi,a εi,b εi,d εi,f ) t −1  u −1  v−1  w  − ri,t −1−a ri,u−1−c ri,v−1−d ri,w−e a=1 c =1 d=1 e=1

J. Westerlund, R. Larsson / Journal of Econometrics 188 (2015) 59–93

× E (εi,a εi,c εi,d εi,e ) t −1  u−1  v−1 w−  1 + ri,t −1−a ri,u−1−c ri,v−1−d ri,w−1−f

=

a =1



a=1 c =1 d=1 f =1

+

× E (εi,a εi,c εi,d εi,f )

ri,t −1−a ri,u−a ri,v−1−d ∆ri,w−d

d=1

+ ri,v−1−w

(t − 1)∧u 

 ri,t −1−a ri,u−a + 1 {v > w}

a=1

(t − 1)∧u (v− 1)∧w  

t ∧v−1

+

ri,t −1−a ri,u−a ri,v−1−d ri,w−d

 

+







a=1

b=1

(t − 1)∧u v∧w−  1

+

a =1

b=1

ri,t −1−a ri,u−b ri,v−1−b ri,w−a ,

+ ri,u−w +

a=1

a=1

b =1

+

ri,t −1−a ri,u−1−a ri,v−1−d ri,w−d





a=1

c =1





a=1

d=1

ri,t −1−a ri,u−1−c ri,v−1−c ri,w−a ,

−

ri,t −1−a ri,u−1−a ri,v−1−d ri,w−1−d





a =1

c =1

+

ri,t −1−a ri,u−1−c ri,v−1−a ri,w−1−c

(t − 1)∧w u ∧v−1 

ri,t −1−a ri,u−1−c ri,v−1−c ri,w−1−a .

  a=1

  a=1

(t − 1)∧u (v− 1)∧w  

 ri,t −1−a ri,u−a ri,v−1−d ri,w−d

ri,t −1−a ri,u−a ri,v−1−d ri,w−1−d

+

+ ri,v−1−w

1 {v > w}

ri,t −1−a ri,u−1−a

a=1

ri,t −1−a ri,u−b ri,v−1−a ri,w−b

+

b=1

  a =1





a =1

b=1

 ri,t −1−a ri,u−b ri,v−1−a ri,w−1−b

+

(t − 1)∧w u∧(v−  1)

t ∧v−1 w−1

 



a=1

b =1

ri,t −1−a ri,u−1−c ri,v−1−a ∆ri,w−c

c =1



t ∧v−1

ri,t −1−a ri,u−b ri,v−1−b ri,w−a

+ ri,u−1−w

b=1



ri,t −1−a ri,u−1−c ri,v−1−a ∆ri,w−c 1 {u ≤ w}

c =1

a =1



ri,t −1−a ri,v−1−a

1 {u > w}

a =1

t ∧w−1 u∧(v−1)

−



t ∧v−1 u−1

 

a=1



d=1

t ∧v−1 u∧(w−1)

−

ri,t −1−a ri,u−1−a ri,v−1−d ∆ri,w−d

d=1 t ∧u−1

t ∧v−1 u∧w

a =1

t ∧u−1 w−1

a=1

d=1

(t − 1)∧u v∧w−  1

ri,t −1−a ri,u−1−a ri,v−1−d ∆ri,w−d 1 {v ≤ w}

d=1

 

+

ri,t −1−a ri,u−1−c ri,v−1−c ri,w−1−a

c =1

t ∧u−1 v−1

Hence,

a =1

ri,t −1−a ri,u−1−c ri,v−1−c ri,w−a

c =1

t ∧w−1 u∧v−1

−

c =1

a =1

ri,t −1−a ri,u−1−c ri,v−1−a ri,w−1−c

c =1

a =1

=

+

ri,t −1−a ri,u−1−c ri,v−1−a ri,w−c

  a =1

c =1

 

−

ri,t −1−a ri,u−1−a ri,v−1−d ri,w−1−d

d=1

t ∧v−1 (u−1)∧w

+

t ∧w−1 u∧v−1

L1 − L2 =

(A.36)

t ∧v−1 u∧w−1

 

a=1

ri,u−b ri,v−1−b 1 {t > w} ,

ri,t −1−a ri,u−1−a ri,v−1−d ri,w−d

  a =1

t ∧v−1 u∧w−1

+



t ∧u−1 v∧w−1

−

d=1

a =1



t ∧u−1 (v−1)∧w

L3 − L4 =

t ∧u−1 v∧w−1

+

ri,t −1−a ri,u−b ri,v−1−b ∆ri,w−a

b=1

and

and

a=1

w− 1 u∧(v− 1)

+ ri,t −1−w

ri,t −1−a ri,u−1−c ri,v−1−a ri,w−c



 

ri,t −1−a ri,u−b ri,v−1−b ∆ri,w−a 1 {t ≤ w}

b =1

a =1 c =1 (t −1)∧w u∧v−1

+

1 {u ≥ w}

ri,t −1−a ri,v−1−a

u∧(v−1)

ri,t −1−a ri,u−b ri,v−1−b ri,w−1−a ,

a=1 d=1 t ∧v−1 (u−1)∧w





b =1

a=1





+

ri,t −1−a ri,u−b ri,v−1−a ri,w−1−b





t −1 u∧(v−  1)



t ∧u−1 (v−1)∧w

L3 =

 a =1

t ∧w−1 u∧(v−1)

+

ri,t −1−a ri,u−b ri,v−1−a ∆ri,w−b

b=1 t ∧v−1

a=1 d=1 t ∧v−1 u∧(w−1)



t ∧v−1 w−1

a =1

ri,t −1−a ri,u−a ri,v−1−d ri,w−1−d



ri,t −1−a ri,u−b ri,v−1−a ∆ri,w−b 1 {u < w}

b=1

 

+

ri,t −1−a ri,u−b ri,v−1−a ri,w−b

a=1 b=1 (t −1)∧w u∧(v−1)

+

u   a =1

a=1 d=1 t ∧v−1 u∧w

L4 =

(t − 1)∧u w−  1

(A.35)

where

L2 =

ri,t −1−a ri,u−a ri,v−1−d ∆ri,w−d 1 {v ≤ w}

d=1

a =1

= L1 − L2 − L3 + L4 ,

L1 =

(t − 1)∧u  v−1 

85

ri,t −1−a ri,u−b ri,v−1−b ri,w−1−a

+

t −1 u ∧v−1  a=1

c =1

ri,t −1−a ri,u−1−c ri,v−1−c ∆ri,w−a 1 {t ≤ w}

86

J. Westerlund, R. Larsson / Journal of Econometrics 188 (2015) 59–93

w− ∧v−1 1 u

w− 1

 +

a=1

ri,t −1−a ri,u−1−c ri,v−1−c ∆ri,w−a

+ ri,v−1−u

c =1

+ ri,t −1−w

 ri,u−1−c ri,v−1−c 1 {t > w} ,

(A.37)

c =1

t −1  v−1 

ri,t −1−a ri,v−1−d ∆ri,u−a ∆ri,w−d 1 {v ≤ w, t ≤ u}

u−1  v−1 

ri,t −1−a ri,v−1−d ∆ri,u−a ∆ri,w−d

a=1 d=1

+ ri,t −1−u

v−1 

 ri,v−1−d ∆ri,w−d 1 {v ≤ w, t > u}

d=1 t −1 w−1

+



+

a=1 d=1 u−1 w−  1

ri,t −1−a ri,v−1−d ∆ri,u−a ∆ri,w−d 1 {v > w, t ≤ u} ri,t −1−a ri,v−1−d ∆ri,u−a ∆ri,w−d

+ ri,t −1−u

w− 1

 ri,v−1−d ∆ri,w−d 1 {v > w, t > u}

d=1 t −1



+ ri,v−1−w

ri,t −1−a ∆ri,u−a 1 {v > w, t ≤ u}

a=1

 + ri,v−1−w

u −1 

 ri,t −1−a ∆ri,u−a + ri,t −1−u

a=1

× 1 {v > w, t > u}  t ∧v− u−1 1  + ri,t −1−a ri,v−1−a ∆ri,u−b ∆ri,w−b a=1

b =1 t ∧v−1



+ ∆ri,w−u

 ri,t −1−a ri,v−1−a 1 {u < w}

a=1

a=1

ri,t −1−a ri,u−1−c ri,v−1−a ∆ri,u−c 1 {u = w}

c =1

t ∧v−1 u−1

+

  a=1





+ ∆ri,u−w

ri,t −1−a ri,v−1−a 1 {u > w}

a=1 t −1  u−1 

ri,t −1−a ri,v−1−b ∆ri,u−b ∆ri,w−a

a =1 b =1

+ ri,v−1−u

t −1 

 ri,t −1−a ∆ri,w−a 1 {t ≤ w, u ≤ v}

a=1

+

t −1  v−1 

ri,t −1−a ri,v−1−b ∆ri,u−b ∆ri,w−a 1 {t ≤ w, u > v}

a=1 b=1

 +

It follows that, with t = ⌊τ T ⌋, u = ⌊γ T ⌋, v = ⌊ξ T ⌋, w = ⌊ηT ⌋ and (τ , γ , ξ , η) ∈ [0, 1] × [0, 1] × [0, 1] × [0, 1], E (ui,t −1 ∆ui,u ui,v−1 ∆ui,w |ci )

∼ (λN ci )2



τ

β=0

ξ



δ=0

exp {(τ + γ + ξ + η − 2β − 2δ)λN ci }

× dδ dβ 1 {ξ < η, τ < γ }  γ  ξ 2 + (λN ci ) exp {(τ + γ + ξ + η − 2β − 2δ)λN ci } δ=0

× dδ dβ 1 {ξ < η, τ > γ } + λN ci exp {(τ − γ )λN ci }



ξ δ=0

exp {(ξ + η − 2δ)λN ci }

× dδ 1 {ξ < η, τ > γ }  τ  η + (λN ci )2 exp {(τ + γ + ξ + η − 2β − 2δ)λN ci } β=0

δ=0

β=0

δ=0

× dδ dβ 1 {ξ > η, τ < γ }  γ  η + (λN ci )2 exp {(τ + γ + ξ + η − 2β − 2δ)λN ci } × dδ dβ 1 {ξ > η, τ > γ } + λN ci exp {(τ − γ )λN ci }



δ=0

× dδ 1 {ξ > η, τ > γ }

× dβ 1 {ξ > η, τ < γ } + λN ci exp {(ξ − η)λN ci }

η

τ



β=0 γ



β=0

exp {(ξ + η − 2δ)λN ci }

exp {(τ + γ − 2β)λN ci }

exp {(τ + γ − 2β)λN ci }

× dδ dβ 1 {γ < η}

t ∧v−1

+

(A.38)

b=1

ri,t −1−a ri,v−1−a 1 {u = w}

ri,t −1−a ri,v−1−a ∆ri,u−b ∆ri,w−b 1 {u > w}

a=1



× 1 {t > w, u < v} v−1  + ri,t −1−w ri,v−1−b ∆ri,u−b 1 {t > w, u ≥ v} .

b =1

t ∧v−1

+

ri,v−1−b ∆ri,u−b + ri,v−1−u

× dβ 1 {ξ > η, τ > γ } + exp {(τ − γ + ξ − η)λN ci } 1 {ξ > η, τ > γ }  τ ∧ξ  γ + (λN ci )2 exp {(τ + γ + ξ + η − 2β − 2δ)λN ci }

  a=1

ri,t −1−a ri,u−b ri,v−1−a ∆ri,u−b 1 {u = w}

b =1

t ∧v−1 w−1

+



+ λN ci exp {(ξ − η)λN ci }

t ∧v−1 u−1

 

u −1 

β=0

a=1 d=1

−

ri,t −1−a ri,v−1−b ∆ri,u−b ∆ri,w−a 1 {t > w, u ≥ v}

a=1 b=1

b=1

a=1 d=1

+

w− v−1 1 

+ ri,t −1−w

E (ui,t −1 ∆ui,u ui,v−1 ∆ui,w |ci )



+



implying

=

ri,t −1−a ∆ri,w−a 1 {t > w, u < v}

a=1

u∧v−1





w− u−1 1  a =1 b =1

ri,t −1−a ri,v−1−b ∆ri,u−b ∆ri,w−a

β=0

δ=0

+ λN ci exp {(η − γ )λN ci } × dβ 1 {γ < η}  τ ∧ξ  + (λN ci )2 β=0

η δ=0



τ ∧ξ

β=0

exp {(τ + ξ − 2β)λN ci }

exp {(τ + γ + ξ + η − 2β − 2δ)λN ci }

× dδ dβ 1 {γ > η}  τ ∧ξ + exp {(τ + ξ − 2β)λN ci } dβ 1 {γ = η} β=0  τ ∧ξ + λN ci exp {(γ − η)λN ci } exp {(τ + ξ − 2β)λN ci } × dβ 1 {γ > η}  τ  2 + (λN ci ) β=0

β=0

γ δ=0

exp {(τ + γ + ξ + η − 2β − 2δ)λN ci }

J. Westerlund, R. Larsson / Journal of Econometrics 188 (2015) 59–93

× dδ dβ 1 {τ < η, γ < ξ } + λN ci exp {(ξ − γ )λN ci }

and

τ



β=0

exp {(τ + η − 2β)λN ci }

1

× dβ 1 {τ < η, γ < ξ }  τ  ξ 2 exp {(τ + γ + ξ + η − 2β − 2δ)λN ci } + (λN ci ) β=0

× dδ dβ 1 {τ < η, γ > ξ }  η  γ exp {(τ + γ + ξ + η − 2β − 2δ)λN ci } + (λN ci )2 β=0

T6

E

+ λN ci exp {(ξ − γ )λN ci }

η

β=0

∼

β=0

exp {(τ + η − 2β)λN ci }

δ=0

+ λN ci exp {(τ − η)λN ci }

γ



δ=0

1 24(λN ci )8

, xT

−1

, yT

−1

)ui,u−1 ∆ui,v ui,x−1 ∆ui,y |ci

[18(1 − λN ci − (λN ci )2 )2 exp{4λN ci }

− 5(λN ci )4 − 19(λN ci )5 ]. exp {(γ + ξ − 2δ)λN ci }

(A.42)

Moreover, by putting t = u in (A.35),

× dδ 1 {τ > η, γ < ξ } + exp {(τ − η + ξ − γ )λN ci } 1 {τ > η, γ < ξ }  ξ + λN ci exp {(τ − η)λN ci } exp {(γ + ξ − 2δ)λN ci } × dδ 1 {τ > η, γ > ξ } ,

−1

− 36(1 − λN ci − (λN ci )2 )(2 − 3(λN ci )2 ) exp{3λN ci } + (84 + 48λN ci − 300(λN ci )2 − 47(λN ci )3 + 173(λN ci )4 + 19(λN ci )5 ) exp{2λN ci } − 4(6 + 30λN ci − 15(λN ci )2 − 29(λN ci )3 + (λN ci )4 ) exp{λN ci } − 6 + 36λN ci + 54(λN ci )2 + 27(λN ci )3

× dβ 1 {τ > η, γ < ξ }  η  ξ 2 exp {(τ + γ + ξ + η − 2β − 2δ)λN ci } + (λN ci ) × dδ dβ 1 {τ > η, γ > ξ }

g2 (tT −1 , uT −1 , v T −1 )

t =2 u=2 v=2 w=2 x=2 y=2

× g2 (w T

δ=0



 T  T  T  T  T  T 



δ=0

× dδ dβ 1 {τ > η, γ < ξ }

87

E ui,u−1 ∆ui,u ui,v−1 ∆ui,w |ci



=

u−1  u  v−1  w 



ri,u−1−a ri,u−b ri,v−1−d ri,w−e E εi,a εb εi,d εi,e





a=1 b=1 d=1 e=1

δ=0

(A.39)

−

u−1  u  v−1 w−  1

ri,u−1−a ri,u−b ri,v−1−d ri,w−1−f E εa εi,b εi,d εi,f



ri,u−1−a ri,u−1−c ri,v−1−d ri,w−e E εa εi,c εi,d εi,e





a=1 b=1 d=1 f =1

which in turn implies

−

u −1  u −1  v−1  w 



a=1 c =1 d=1 e=1

1 T4

E

 T  T  T  T 

g1 (tT −1 , uT −1 )g1 (v T −1 , w T −1 )

+

t =2 u=2 v=2 w=2

  × E εa εi,c εi,d εi,f = M1 − M2 − M3 + M4 ,

× ui,t −1 ∆ui,u ui,v−1 ∆ui,w |ci 1 6(λN ci )8

where

[18(1 − λN ci )2 exp{4λN ci } − 36(2 − 2λN ci

− (λN ci )2 + (λN ci )3 ) exp{3λN ci } + (84 + 48λN ci − 132(λN ci )2 + (λN ci )3 + 17(λN ci )4 + (λN ci )5 ) exp{2λN ci } + 4(−6 − 30λN ci + 3(λN ci )2 + 11(λN ci )3 + 2(λN ci )4 ) exp{λN ci } − 6 + 36λN ci + 42(λN ci )2 + 15(λN ci )3 4 5 +  (λN ci ) − (λN ci ) ], T  T  T  T  T  1

T5

E

M1 = 3

(u−1)∧(v−  1)∧w

+

(A.40)

+

=

1)∧w u∧(v− (u−  1)

ri,u−1−a ri,u−b ri,v−1−b ri,w−a

b=1 b̸=a

1)∧w u−1 (v−  

ri,u−1−a ri,u−a ri,v−1−d ri,w−d

a=1 d=1 u∧v−1 (u−1)∧w

[−18(1 − 2λN ci + (λN ci ) ) exp{4λN ci }

+

+ 18(4 − 4λN ci − 6(λN ci ) + 4(λN ci ) + (λN ci )4 ) exp{3λN ci } − (84 + 48λN ci − 216(λN ci )2 − 23(λN ci )3 + 53(λN ci )4 + 4(λN ci )5 ) exp{2λN ci } − 2(−12 − 60λN ci + 18(λN ci )2 + 40(λN ci )3 + 7(λN ci )4 ) exp{λN ci } + 6 − 36λN ci − 48(λN ci )2 − 21(λN ci )3 − (λN ci )4 + 4(λN ci ) ],

ri,u−1−a ri,u−b ri,v−1−a ri,w−b

b=1 b̸=a

a=1

3

5

  a =1

g1 (tT −1 , uT −1 )g2 (v T −1 , w T −1 , xT −1 )

2

ri,u−1−a ri,u−a ri,v−1−d ri,w−d

d=1 d̸=a

u∧v−1 u∧w

+

× ui,t −1 ∆ui,u ui,w−1 ∆ui,x |ci 12(λN ci )8

1)∧w u−1 (v−   a=1



∼

ri,u−1−a ri,u−a ri,v−1−a ri,w−a

a =1

t =2 u=2 v=2 w=2 x=2

1

ri,u−1−a ri,u−1−c ri,v−1−d ri,w−1−f

a=1 c =1 d=1 f =1



∼

u −1  u −1  v−1 w−  1

3





ri,u−1−a ri,u−b ri,v−1−a ri,w−b

a=1 b =1 (u−1)∧w u∧v−1

+





a=1

b =1 u∧v−1

+ ri,w−u



ri,u−1−a ri,u−b ri,v−1−b ri,w−a

ri,u−1−a ri,v−1−a 1 {u ≤ w}

a =1 (u−1)∧w

(A.41)

+ ri,v−1−u

 a =1

ri,u−1−a ri,w−a 1 {u < v} ,

88

J. Westerlund, R. Larsson / Journal of Econometrics 188 (2015) 59–93 u∧v−1 (u−1)∧w

u∧v∧w−1



M2 = 3

ri,u−1−a ri,u−a ri,v−1−a ri,w−1−a

+

a =1 u−1 v∧w−1

 

+

a =1

−

d=1 d̸=a

=





a =1

b=1 b̸=a

ri,u−1−a ri,u−b ri,v−1−a ri,w−1−b

b =1

+



a=1

b =1

1)∧w u (u− ∧v−1 

  a =1

u∧v−1

ri,u−1−a ri,u−a ri,v−1−d ri,w−1−d



+ ri,w−u

u∧v−1

ri,u−1−a ri,u−b ri,v−1−a ri,w−1−b



− ri,w−1−u

ri,u−1−a ri,u−b ri,v−1−b ri,w−1−a

+ ri,v−1−u

b =1 u∧v−1



1)∧w (u− 

ri,u−1−a ri,v−1−a 1 {u < w}

u∧w−1



− ri,v−1−u

u∧w−1



M3 = 3

= ri2,u−1−a ri,v−1−a ri,w−a

1)∧w u−1 (v−  

+

a=1

+

a =1

c =1 c ̸=a

a=1

  a=1

ri,u−1−a ri,u−1−c ri,v−1−c ri,w−a

ri2,u−1−a ri,v−1−d ri,w−d

+



+





a=1

c =1



M4 =

u∧v−1 w−1

+

u−1 u ∧v−1  a =1

 +

ri2,u−1−a ri,v−1−d ri,w−1−d



ri,u−1−a ri,u−1−c ri,v−1−a ri,w−1−c ri,u−1−a ri,u−1−c ri,v−1−c ri,w−1−a .

+

1 {u > w}



ri,u−1−a ri,v−1−a 1 {u < w}

a=1 u∧v−1

1)∧w u−1 (v−  

ri,u−1−a ri,v−1−a 1 {u = w}

a=1 u −1 

ri,u−1−a ∆ri,w−a 1 {u < v, u ≤ w}

a=1

ri,u−1−a ri,u−a ri,v−1−d ri,w−d

d=1

b=1



+ ri,v−1−u

u−1 v∧w−  1 a =1

ri,u−b ri,v−1−b

u∧v−1

+ ∆ri,w−u

Thus,

−



b=1

c =1

a =1

ri,u−1−a ri,u−b ri,v−1−b ∆ri,w−a

b=1

+ ri,u−1−w

u∧w−1 u∧v−1

M1 − M2 =

ri,u−1−a ri,u−b ri,v−1−b ∆ri,w−a 1 {u ≤ w}

w− ∧v−1 1 u

c =1

 

1 {u > w}

ri,u−1−a ri,v−1−a

u∧v−1

 

a=1



b=1

d=1

a =1

+



+ ri,u−w

ri,u−1−a ri,u−1−c ri,v−1−c ri,w−a ,

u∧v−1 u∧w−1

+

ri,u−1−a ri,u−b ri,v−1−a ∆ri,w−b

b=1 u∧v−1

a =1

a=1

1 {v > w}

ri,u−1−a ri,u−b ri,v−1−a ∆ri,w−b 1 {u ≤ w}

 

+

and u−1 v∧w−  1

ri,u−1−a ri,u−a

a =1

ri,u−1−a ri,u−1−c ri,v−1−a ri,w−c

a=1 c =1 (u−1)∧w u∧v−1



b =1

a=1

a=1 d=1 u∧v−1 (u−1)∧w



u−1 

u∧v−1 u−1

+

c =1 c ̸=a

 

ri,u−1−a ri,u−a ri,v−1−d ∆ri,w−d

a =1

u−1 (v−1)∧w

=

u−1 w−  1 a=1 d=1

ri,u−1−a ri,u−1−c ri,v−1−a ri,w−c

1)∧w u (u− ∧v−1 

+



+ ri,v−1−w

u∧v−1 (u−1)∧w



ri,u−1−a ri,u−a ri,v−1−d ∆ri,w−d 1 {v ≤ w}

a=1 d=1

ri2,u−1−a ri,v−1−d ri,w−d

d=1 d̸=a



u−1  v−1 

+

a =1

ri,u−1−a ri,w−1−a 1 {u < v}

a=1

ri,u−1−a ri,w−1−a 1 {u < v} ,

a =1

(u−1)∧(v−  1)∧w

ri,u−1−a ri,w−a 1 {u < v}

a=1

a=1

+ ri,v−1−u

ri,u−1−a ri,v−1−a 1 {u < w}

a =1

 

+ ri,w−1−u

ri,u−1−a ri,v−1−a 1 {u ≤ w}

a=1

b =1

a=1

ri,u−1−a ri,u−b ri,v−1−b ri,w−1−a

b=1

u∧w−1 u∧v−1

+

ri,u−1−a ri,u−b ri,v−1−b ri,w−a

b=1

u∧w−1 u∧v−1

−

ri,u−1−a ri,u−b ri,v−1−b ri,w−1−a

  a =1

ri,u−1−a ri,u−b ri,v−1−a ri,w−1−b

b=1

a =1 b =1 u∧v−1 u∧w−1

+

ri,u−1−a ri,u−b ri,v−1−a ri,w−b

 

a =1



u−1 v∧w− 1 

a=1

a=1

u∧w−1 u∧(v−1)

+



u∧v−1 u∧w−1

ri,u−1−a ri,u−a ri,v−1−d ri,w−1−d

u∧v−1 u∧(w−1)

+



 +

ri,v−1−u

w− 1

 ri,u−1−a ∆ri,w−a + ri,v−1−u ri,u−1−w

a =1

ri,u−1−a ri,u−a ri,v−1−d ri,w−1−d

× 1 {w < u < v}

J. Westerlund, R. Larsson / Journal of Econometrics 188 (2015) 59–93



and, as in (A.37), M3 − M4 =

+

u −1  v−1 

+

u−1 w−  1

ri2,u−1−a ri,v−1−d ∆ri,w−d 1 {v ≤ w}

ri2,u−1−a ri,v−1−d ∆ri,w−d + ri,v−1−w

u −1 

a=1 d=1

+

a=1

ri2,u−1−a

a=1

It follows that E ui,u−1 ∆ui,u ui,v−1 ∆ui,w |ci



∼ (λN ci )

2

ri,u−1−a ri,u−1−c ri,v−1−a ∆ri,w−c



+ ri,u−1−w +

a=1

 +

1 {u > w}

ri,u−1−a ri,u−1−c ri,v−1−c ∆ri,w−a 1 {u ≤ w}



E ui,u−1 ∆ui,u ui,v−1 ∆ui,w |ci

=



× dβ 1 {γ > η}  γ  2 + (λN ci )

ri,u−1−a ri,v−1−d ∆ri,u−a ∆ri,w−d 1 {v ≤ w}

β=0

a=1 d=1

 +

u−1 w−  1

+ ri,v−1−w

u−1 

u∧v−1 u−1

  

ri,u−1−a ri,v−1−a ∆ri,u−b ∆ri,w−b 1 {u ≤ w}

b =1

u∧v−1 w−1

 

+

a=1



1 {u > w}

ri,u−1−a ri,v−1−a

+

ri,u−1−a ri,v−1−b ∆ri,u−b ∆ri,w−a 1 {u ≤ w}

b=1

w− ∧v−1 1 u a =1

ri,u−1−a ri,v−1−b ∆ri,u−b ∆ri,w−a

+ ri,u−1−w



 T

−3

1 {u > w}

ri,u−1−a ri,v−1−a 1 {u < w}

a=1

+

ri,u−1−a ri,v−1−a 1 {u = w}

a=1

+ ri,v−1−u

u −1  a=1

exp {(2γ + ξ + η − 2β − 2δ) λN ci }



γ ∧ξ β=0

exp {(γ + ξ − 2β) λN ci }



η β=0

exp {(γ + η − 2β) λN ci }

T  T  T 

 g1 (uT

−1

, vT

−1

)ui,t −1 ∆ui,t ui,u−1 ∆ui,v |ci

1 4(λN ci )5

[1 − (1 − λN ci ) exp{λN ci }]

× (3 + 2λN ci − 3 exp{2λN ci }) (1 + λN ci − exp{λN ci }) (A.43) and

u∧v−1



E

∼

u∧v−1



exp {(γ + ξ − 2β) λN ci }

t =2 u=2 v=2

b=1

+ ∆ri,w−u

β=0

from which we obtain

 ri,v−1−b ∆ri,u−b

δ=0

γ ∧ξ



× dβ 1 {η < γ < ξ } + exp {(ξ − η) λN ci } 1 {η < γ < ξ } ,

b=1 u∧v−1

γ ∧ξ

+ λN ci exp {(ξ − γ ) λN ci }



a =1 u−1 u ∧v−1 



exp {(2γ + ξ + η − 2β − 2δ) λN ci }

× dβ 1 {γ < ξ , γ < η}

b=1

+ ∆ri,u−w

a=1

δ=0

β=0

ri,u−1−a ri,v−1−a ∆ri,u−b ∆ri,w−b

u∧v−1

+

η

× dβ 1 {γ < η}  γ ∧ξ + exp {(γ + ξ − 2β) λN ci } dβ 1 {γ = η} β=0  γ + λN ci exp {(ξ − γ ) λN ci } exp {(γ + η − 2β) λN ci }

1 {v > w}

a =1

a=1

exp {(2γ + ξ + η − 2β − 2δ) λN ci }

+ λN ci exp {(η − γ ) λN ci }

 ri,u−1−a ∆ri,u−a

exp {(2γ − 2β) λN ci }

× dδ dβ 1 {γ < η}

ri,u−1−a ri,v−1−d ∆ri,u−a ∆ri,w−d

a=1 d=1

+

β=0

+ 2λN ci exp {(γ − η) λN ci }

Consequently,



γ



× dδ dβ 1 {γ > η}

c =1

u−1  v−1 

δ=0

β=0

1 {u > w} .

ri,u−1−c ri,v−1−c

γ

× dδ dβ 1 {γ < η}  γ ∧ξ  + 2(λN ci )2

ri,u−1−a ri,u−1−c ri,v−1−c ∆ri,w−a



exp {(2γ + ξ + η − 2β − 2δ) λN ci }

δ=0

β=0

u∧v−1

+ ri,u−1−w

δ=0

× dβ 1 {ξ > η}  γ ∧ξ  2 + (λN ci )

c =1

a =1

β=0

+ λN ci exp {(ξ − η) λN ci }

c =1

∧v−1 w− 1 u

ξ



β=0

a =1 u−1 u ∧v−1 

γ

× dδ dβ 1 {ξ > η}

 ri,u−1−a ri,v−1−a





× dδ dβ 1 {ξ < η}  γ  η exp {(2γ + ξ + η − 2β − 2δ) λN ci } + (λN ci )2

c =1 u∧v−1

ri,u−1−a ∆ri,w−a + ri,v−1−u ri,u−1−w

× 1 {w < u < v} . 

× 1 {v > w} u ∧v−1  u−1 + ri,u−1−a ri,u−1−c ri,v−1−a ∆ri,w−c 1 {u ≤ w} a=1 c =1 u ∧v−1 w− 1

w− 1 a=1

a=1 d=1



ri,v−1−u

89



T

−4

E

 T  T  T  T 

g2 (uT −1 , v T −1 , w T −1 )

t =2 u=2 v=2 w=2

 ri,u−1−a ∆ri,w−a 1 {u < v, u ≤ w}

× ui,t −1 ∆ui,t ui,v−1 ∆ui,w |ci

90

J. Westerlund, R. Larsson / Journal of Econometrics 188 (2015) 59–93

∼

1

[3(1 − 9λN ci − (λN ci )2 + 2(λN ci )3 ) 24(λN ci )5 + (18 + 36λN ci − 43(λN ci )2 − 18(λN ci )3 ) exp{λN ci } + 2(−6 + 12λN ci + 2(λN ci )2 + 3(λN ci )3 ) exp{2λN ci } + 9(−2 + 3(λN ci )2 ) exp{3λN ci } + 9(1 − λN ci − (λN ci )2 ) × exp{4λN ci }]. (A.44)

+ =

+

+

+

∞  180 · 4j+8 − 72 · 3j+8 − 456 · 2j+8 − 24

(j + 8)!

+

∞  21 · 4j+8 + 24 · 3j+8 + 14 · 2j+8 + 20

(j + 8)!

∞  −3 · 4j+8 − 3 · 3j+8 − 6 · 2j+8 − 13

(j + 8)!

∞  3 · 4j+6 + 13 · 2j+5 + 2

(j + 8)!

j=0

E(

×

|ci ) ∼

1

+ =

181267

16(λN ci )8

3 (2 − λN ci ) (12 − 6λN ci + (λN ci ) )

240

(j + 8)!

− 48(2 − λN ci )(12 − 6λN ci + (λN ci )2 t )(6 − (λN ci )2 ) ∞  (3λN ci )j+8 × (j + 8)! j=0

(λN ci )j+5

(λN ci )j+6

+

15681λN ci

−

214139(λN ci )2

630 1

−

+ λN ci + 3

+

210

797 6300

(λN ci )2 +

1 30

−

504

2(λN ci )6 315

437

(λN ci )3 +

75 600

(λN ci )4 ,

implying E (A2iT ,1 ) ∼

∞  108 · 4j+8 − 432 · 3j+8 + 504 · 2j+8 − 144

(j + 8)!

j =0

+

∞  −216 · 4j+8 + 432 · 3j+8 + 288 · 2j+8 − 720

(j + 8)!

j=0

+

∞  180 · 4j+8 − 72 · 3j+8 − 456 · 2j+8 − 24

(j + 8)! (j + 8)!

+

∞  21 · 4j+8 + 24 · 3j+8 + 14 · 2j+8 + 20

(j + 8)!

j=0

+

∞  −3 · 4j+8 − 3 · 3j+8 − 6 · 2j+8 − 13

(j + 8)!

j=0

+ 2(4032 + 2304λN ci − 3648(λN ci )2 + 720(λN ci )3 + 112(λN ci )4 − 48(λN ci )5 + 13(λN ci )6 − 2(λN ci )7 ) ∞  (2λN ci )j+8 × (j + 8)! j=0

+

+ 16(−144 − 720λN ci − 24(λN ci )2 + 102(λN ci )3 + 20(λN ci )4 − 13(λN ci )5 + 2(λN ci )6 )  ∞  (λN ci )j+8 181267 15681h 214139(λN ci )2 × − + − 1680 70 1260 (j + 8)! j =0

−

∞  3 · 4j+6 + 13 · 2j+5 + 2

(j + 8)!

j=0

λjN µjc

λjN+1 µjc+1

λjN+2 µjc+2

∞  −81 · 4j+8 − 54 · 3j+8 + 90 · 2j+8 + 102 j=0

∞  (4λN ci )j+8 j=0

(λN ci )j+4

1680 70 1260 37969(λN ci )3 2113(λN ci )4 317(λN ci )5

131

 2 2

(λN ci )j+3

∞  2j+6 (λN ci )j+7 − j + 8 ( )! j=0

+ 2

(λN ci )j+1

(λN ci )j+2

(j + 8)!

j=0

+

315

(λN ci )j

∞  −81 · 4j+8 − 54 · 3j+8 + 90 · 2j+8 + 102

j=0

A2iT ,1

504

2(λN ci )6

(j + 8)!

j=0

−

Simplifications now yield

−

∞  −216 · 4j+8 + 432 · 3j+8 + 288 · 2j+8 − 720

j=0

[3(2 − λN ci )2 (12 − 6λN ci

+ (λN ci )2 )2 exp{4λN ci } − 48(2 − λN ci )(12 − 6λN ci + (λN ci )2 ) × (6 − (λN ci )2 ) exp{3λN ci } + 2(4032 + 2304λN ci − 3648(λN ci )2 + 720(λN ci )3 + 112(λN ci )4 − 48(λN ci )5 + 13(λN ci )6 − 2(λN ci )7 ) exp{2λN ci } + 16(−144 − 720λN ci − 24(λN ci )2 + 102(λN ci )3 + 20(λN ci )4 − 13(λN ci )5 + 2(λN ci )6 ) × exp{λN ci } − 576 + 3456λN ci + 3648(λN ci )2 + 1392(λN ci )3 + 80(λN ci )4 + 16(λN ci )5 + 35(λN ci )6 + 4(λN ci )7 + (λN ci )8 ].

317(λN ci )5

(j + 8)!

j=0

and so, via (A.16)–(A.18) and (A.31), 1

+

210

j=0

1

16(λN ci )8

2113(λN ci )4

∞  108 · 4j+8 − 432 · 3j+8 + 504 · 2j+8 − 144

+

[−9(−12 + 24λN ci − 20(λN ci )2 + 9(λN ci )3 (λN ci )8 + 3(λN ci )4 + (λN ci )5 ) exp{4λN ci } + 9(−48 + 48λN ci − 8(λN ci )2 − 6(λN ci )3 + 24(λN ci )4 + (λN ci )5 ) exp{3λN ci } + (504 + 288λN ci − 456(λN ci )2 + 90(λN ci )3 − 210(λN ci )4 − 38(λN ci )5 + 6(λN ci )6 ) exp{2λN ci } − (144 + 720λN ci + 24(λN ci )2 − 102(λN ci )3 − 84(λN ci )4 + 25(λN ci )5 + 6(λN ci )6 ) exp{λN ci } + 3(−12 + 72λN ci + 76(λN ci )2 + 29(λN ci )3 + 7(λN ci )4 + 13(λN ci )5 + 2(λN ci )6 )],

E (A2iT ,1 |ci ) ∼

630

−

j =0

Substitution of (A.33), (A.34) and (A.40)–(A.44) into (A.32) yields, after simplification, E (UiT |ci ) ∼

37969(λN ci )3

λjN+3 µjc+3

λjN+4 µjc+4

λjN+5 µcj+5

λjN+6 µcj+6

∞  −2j+6 j+7 j+7 λ µ + (j + 8)! N c j=0

+

181267 1680 37969 630

as required.

+

15681

λN µ1c −

70 2113

λ3N µ3c −

210

214139 1260 317

λ4N µ4c +

504

λ2N µ2c

λ5N µ5c −

2 315

λ6N µ6c ,

J. Westerlund, R. Larsson / Journal of Econometrics 188 (2015) 59–93

Finally, consider E (B2iT ,1 ). From (A.29), we get T 4 B2iT ,1 ∼

 T 

u2i,t −1 +

 t =2 T T 12  

+

T

∼

Similarly, making use of (A.23) we have 1

2

T6

ui,t −1

 × ui,t −1 ui,u−1 ui,v−1 ui,w−1 |ci

t =2 u =2

T 

2 u2i,t −1

+

T 64 

T2

g2 (tT

−1

T 16 

, uT

−1

 u2i,t −1

t =2

, vT

T 

−1

)ui,u−1 ui,v−1

ui , t − 1

(A.45)

where we have already calculated the expectations of the first three terms. The fourth term is

T

+

t =2 u=2 v=2

T2

T6

E

g1 (v T

, wT

−1

× ui,t −1 ui,u−1 ui,v−1 ui,w−1 |ci

)

t =2 u=2 v=2 w=2

× ui,t −1 ui,u−1 ui,v−1 ui,w−1 T T T T 24     + 2 g2 (uT −1 , v T −1 , w T −1 ) T

+

T

1 T

T3

T6

T

192     

E



t =2 u=2 v=2 w=2 x=2

× ∼

t =2 u=2 v=2 w=2 x=2

T5

E

∼

 T  T  T  t =2 u=2 v=2

1

(A.46)

1

[9(1 − λN ci − (λN ci )2 ) exp{4λN ci } 24(λN ci )7 − 18λN ci (1 − 3λN ci ) exp{3λN ci }

T7

g1 (uT −1 , v T −1 )u2i,t −1 ui,u−1 ui,v−1 |ci

(A.50)

and using (A.23), 1



E

 T  T  T  T  T 

g2 (v T −1 , w T −1 , xT −1 )

t =2 u=2 v=2 w=2 x=2

 × ui,t −1 ui,u−1 ui,w−1 ui,x−1 |ci

[−9(1 − λN ci ) 7

12(λN ci ) × exp{4λN ci } + 18λN ci (1 − λN ci ) exp{3λN ci } + (−42 + 72λN ci − 30(λN ci )2 + (λN ci )3 ) exp{2λN ci } + 6λN ci (13 − 19λN ci + 6(λN ci )2 ) × exp{λN ci } + 51 − 57λN ci + 6(λN ci )2

+ 7(λN ci )3 − 2(λN ci )4 ].

|ci

− 31(λN ci )3 + 14(λN ci )4 ],

where we can use (A.21) to show that 1

u2i,t −1 ui,v−1 ui,w−1

+ (42 − 72λN ci − 48(λN ci )2 + 23(λN ci )3 ) exp{2λN ci } − 6λN ci (13 − 45λN ci + 18(λN ci )2 ) × exp{λN ci } − 51 + 57λN ci − 111(λN ci )2

t =2 u=2 v=2 w=2 x=2 y=2

× g2 (w T −1 , xT −1 , yT −1 ) × ui,u−1 ui,v−1 ui,x−1 ui,y−1 ,

g2 (uT −1 , v T −1 , w T −1 )

t =2 u=2 v=2 w=2

× g2 (v T −1 , wT −1 , xT −1 )ui,t −1 ui,u−1 ui,w−1 ui,x−1 T T T T T T 144       + 4 g2 (tT −1 , uT −1 , v T −1 ) T

 T  T  T  T 

g2 (v T −1 , w T −1 , xT −1 )

× ui,t −1 ui,u−1 ui,w−1 ui,x−1 T T T T T 288      + 3 g1 (tT −1 , uT −1 ) T

(A.49)

Moreover, from (A.21),

× u2i,t −1 ui,v−1 ui,w−1 T

[9(1 − λN ci )2 exp{4λN ci }

+ 6(λN ci )4 + 6(λN ci )5 + 2(λN ci )6 ].

t =2 u=2 v=2 w=2

T

1 3(λN ci )10

− 36λN ci (1 − λN ci )2 exp{3λN ci } + 2(−9 + 18λN ci + 3(λN ci )2 − 27(λN ci )3 + 15(λN ci )4 + (λN ci )5 ) exp{2λN ci } − 4λN ci (−9 + 12λN ci − 3(λN ci )2 + 2(λN ci )4 ) exp{λN ci } + 9 − 18λN ci − 3(λN ci )2 + 6(λN ci )3

t =2 u=2 v=2 w=2

× ui,t −1 ui,u−1 ui,v−1 ui,w−1 T T T T 144     g1 (tT −1 , uT −1 )g1 (v T −1 , w T −1 ) + 2 T

g1 (tT −1 , uT −1 )g1 (v T −1 , w T −1 )



∼ −1

 T  T  T  T  t =2 u=2 v=2 w=2

g1 (uT −1 , v T −1 )u2i,t −1 ui,u−1 ui,v−1

T T T T 192    

(A.48)

and 1

t =2

T T T 24   

[−9(1 − λN ci ) exp{4λN ci }

+ 3 − 9λN ci + 6(λN ci )2 + 21(λN ci )3 + 8(λN ci )4 ],

t =2

+ T 4 QiT ,

ui,t −1

1 6(λN ci )8

− 18(−2 + λN ci + (λN ci )2 ) exp{3λN ci } + (−42 − 24λN ci + 66(λN ci )2 + 7(λN ci )3 ) exp{2λN ci } + (12 + 42λN ci − 42(λN ci )2 − 40(λN ci )3 ) exp{λN ci }

2

4



∼

2

T

g1 (v T −1 , w T −1 )

t =2 u=2 v=2 w=2

g1 (tT −1 , uT −1 )ui,t −1 ui,u−1

t =2

T 4 QiT =

E

 T  T  T  T 

t =2

T 2 t =2 u=2 v=2

+

T 

T

T T T 12   

+ 

8



91

∼

1 12(λN ci )8

[9(1 − λN ci − (λN ci )2 ) exp{4λN ci }

− 18(2 − λN ci − 5(λN ci )2 ) exp{3λN ci } + (42 + 24λN ci − 264(λN ci )2 − 43(λN ci )3 ) exp{2λN ci } + 2(−6 − 21λN ci + 129(λN ci )2 + 92(λN ci )3 ) exp{λN ci } (A.47)

− 3 + 9λN ci − 87(λN ci )2 − 129(λN ci )3 − 44(λN ci )4 ], (A.51)

92

J. Westerlund, R. Larsson / Journal of Econometrics 188 (2015) 59–93



1 T7

E

T  T  T  T  T 

+ 1728 − 3456λN ci − 960(λN ci )2 + 3120(λN ci )3 + 1200(λN ci )4 + 304(λN ci )5 + 103(λN ci )6 + 28(λN ci )7 + 4(λN ci )8 ],

g1 (tT −1 , uT −1 )g2 (v T −1 , w T −1 , xT −1 )

t =2 u=2 v=2 w=2 x=2

 × ui,t −1 ui,u−1 ui,w−1 ui,x−1 |ci ∼

which can be simplified in the following manner:

[−9(1 − 2λN ci + (λN ci )3 ) exp{4λN ci } 6(λN ci )10 + 18λN ci (2 − 6λN ci + 3(λN ci )2 + (λN ci )3 ) exp{3λN ci } + (18 − 36λN ci + 36(λN ci )2 + 78(λN ci )3 − 96(λN ci )4 − 9(λN ci )5 ) exp{2λN ci } + 6λN ci (−6 + 6λN ci − 9(λN ci )2 + 7(λN ci )3 + 8(λN ci )4 ) exp{λN ci } − 9 + 18λN ci + 3(λN ci )3 − 12(λN ci )4 − 27(λN ci ) − 10(λN ci ) ], 5

6

×



T

E 8

T  T  T  T  T  T 

× g2 (w T ∼

, xT

, yT

−1

(A.52)

g2 (tT −1 , uT −1 , v T −1 )

+ 96λN ci (72 − 112λN ci − 32(λN ci )2 + 21(λN ci )3  ∞  (λN ci )j+10 4 5 − 5(λN ci ) + 2(λN ci ) ) (j + 10)! j=0

)ui,u−1 ui,v−1 ui,x−1 ui,y−1 |ci

−

1

[9(1 − λN ci − (λN ci )2 )2 exp{4λN ci } 12(λN ci )10 − 36λN ci (1 − 4λN ci + 2(λN ci )2 + 3(λN ci )3 ) exp{3λN ci } + 2(−9 + 18λN ci − 39(λN ci )2 − 51(λN ci )3 + 180(λN ci )4 + 26(λN ci )5 ) exp{2λN ci } − 4λN ci (−9 + 6λN ci − 24(λN ci )2 + 75(λN ci )3 + 58(λN ci )4 ) exp{λN ci } + 3(3 − 6λN ci + (λN ci )2 − 4(λN ci )3 + 33(λN ci )4 + 52(λN ci )5 + 18(λN ci )6 )].

(A.53)

− =

1

[−9(−12 + 24λN ci − 20(λN ci )2

(λN ci ) + 9(λN ci )3 + 3(λN ci )4 + (λN ci )5 ) exp{4λN ci } + 18λN ci (−24 + 32λN ci − 16(λN ci )2 + 25(λN ci )3 + (λN ci )4 ) exp{3λN ci } + (−216 + 432λN ci + 408(λN ci )2 − 498(λN ci )3 − 888(λN ci )4 − 284(λN ci )5 + 25(λN ci )6 ) exp{2λN ci } − 2λN ci (−216 + 336λN ci + 96(λN ci )2 − 639(λN ci )3 − 421(λN ci )4 + 18(λN ci )5 ) exp{λN ci } + 108 − 216λN ci − 60(λN ci )2 + 195(λN ci )3 − 357(λN ci )4 − 627(λN ci )5 − 201(λN ci )6 + 10(λN ci )7 ]. 10

This, together with (A.20), (A.22), (A.24) and (A.45), yields E (B2iT ,1 |ci ) ∼

1

[3(−24 + 24λN ci 16(λN ci )10 − 8(λN ci )2 + (λN ci )3 )2 exp{4λN ci } − 96λN ci (72 − 96λN ci + 48(λN ci )2 − 11(λN ci )3 + (λN ci )4 ) exp{3λN ci } + (−3456 + 6912λN ci + 6528(λN ci )2 − 7968(λN ci )3 + 2688(λN ci )4 − 640(λN ci )5 + 150(λN ci )6 − 20(λN ci )7 ) exp{2λN ci } + 96λN ci (72 − 112λN ci − 32(λN ci )2 + 21(λN ci )3 − 5(λN ci )4 + 2(λN ci )5 ) exp{λN ci }

13079 420 40213

+

4059 70

1709

λN ci −

(λN ci )4 +

473

42

(j + 10)!

j= 0

+

(j + 10)!

∞  21 · 4j+10 + 66 · 3j+10 + 168 · 2j+10 + 126

(j + 10)!

∞  −3 · 4j+10 − 6 · 3j+10 − 40 · 2j+10 − 30

(j + 10)!

∞  3 · 4j+8 + 75 · 2j+7 + 12

(j + 10)!

j=0

+

∞  −5 · 2j+8 j=0

− −

(λN ci )j

(j + 10)!

13079 420 40213 18144

+

(λN ci )j+3

(λN ci )j+4

(λN ci )j+5

(λN ci )j+6

(λN ci )j+7

4059 70

(λN ci )j+1

(λN ci )j+2

∞  −81 · 4j+10 − 288 · 3j+10 − 498 · 2j+10 − 192

j=0

+

(λN ci )6

(j + 10)!

j=0

+

(λN ci )3

∞  180 · 4j+10 + 576 · 3j+10 + 408 · 2j+10 − 672

j=0

+

567

3780

(j + 10)!

j=0

+

1

51481

∞  −216 · 4j+10 − 432 · 3j+10 + 432 · 2j+10 + 432 j=0

+

(λN ci )2 +

(λN ci )5 −

18144 3360 ∞  108 · 4j+10 − 216 · 2j+10

Insertion of (A.47)–(A.53) into (A.46) now yields E (QiT |ci ) ∼

(j + 10)!

+ (−3456 + 6912λN ci + 6528(λN ci )2 − 7968(λN ci )3 + 2688(λN ci )4 − 640(λN ci )5 + 150(λN ci )6 ∞  (2λN ci )j+10 − 20(λN ci )7 ) (j + 10)! j= 0

 −1

16(λN ci )

− 96λN ci (72 − 96λN ci + 48(λN ci )2 − 11(λN ci )3 + (λN ci )4 ) ∞  (3λN ci )j+10 × (j + 10)! j= 0

t =2 u=2 v=2 w=2 x=2 y=2

−1

3(−24 + 24λN ci − 8(λN ci )2 + (λN ci )3 )2



10

∞  (4λN ci )j+10 j= 0

and 1

1

E (B2iT ,1 |ci ) ∼

1

λN ci −

(λN ci )4 +

473 3360

1709 42

(λN ci )2 +

(λN ci )5 −

1 567

51481 3780

(λN ci )3

(λN ci )6 .

Hence, E (B2iT ,1 ) ∼

∞  108 · 4j+10 − 216 · 2j+10 j=0

+

λjN µjc

∞  −216 · 4j+10 − 432 · 3j+10 + 432 · 2j+10 + 432 j=0

+

(j + 10)! (j + 10)!

∞  180 · 4j+10 + 576 · 3j+10 + 408 · 2j+10 − 672 j=0

(j + 10)!

λjN+1 µjc+1

λjN+2 µjc+2

J. Westerlund, R. Larsson / Journal of Econometrics 188 (2015) 59–93

+

∞ 

j+10

−81 · 4

j= 0

+

j+10

− 288 · 3 − 498 · 2 (j + 10)! (j + 10)!

∞  −3 · 4j+10 − 6 · 3j+10 − 40 · 2j+10 − 30

(j + 10)!

j= 0

+

∞  3 · 4j+8 + 75 · 2j+7 + 12

(j + 10)!

j= 0

+

∞  −5 · 2j+8 j= 0

− −

− 192

∞  21 · 4j+10 + 66 · 3j+10 + 168 · 2j+10 + 126 j= 0

+

j+10

(j + 10)!

13079 420 40213 18144

+

λjN+3 µjc+3

λjN+4 µjc+4

λjN+5 µcj+5

λjN+6 µjc+6

λjN+7 µjc+7

4059 70

λ4N µ4c +

λN µ1c − 473 3360

1709 42

λ5N µ5c −

λ2N µ1c + 1 567

51481 3780

λ3N µ3c

λ6N µ6c .

This establishes the required result for E (B2iT ,1 ), and hence the proof of the theorem is complete.  References Bai, J., Ng, S., 2004. A panic attack on unit roots and cointegration. Econometrica 72, 1127–1177. Bai, J., Ng, S., 2010. Panel unit root tests with cross-section dependence: A further investigation. Econometric Theory 26, 1088–1114.

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