text: 39 (cid:88) (cid:0) (cid:1) ∞ − ∗ −4 G ∂ ( 1)l ∗ ∗ − · ∗ I2 (τ + t ) = c3(cid:16)σjPik ∂(cid:17)ξk (cid:16) l! (cid:17)M˙ (cid:104)AjL(cid:18)−1(cid:105) (τ + t(cid:19)) ∂(cid:104)L−1(cid:105) ln rAN (τ + t ) σ rNA (τ + t ) l=1 O v O O v2 + A M˙ A + M¨ A + A MA . (F2) c L L c2 L Finally, by means of relation (D5), we just obtain the following integrals, each of which will be considered sep- expression in the last line of Eq. (111). arately. Appendix G: Integrals of second integration of 1. Integral I mass-multipole terms 3 By inserting (110) and (111) into (117) we obtain the The integral I , using relation (D5), reads 3 (cid:90) (cid:88) (cid:2) (cid:3) ∞ − τ ∗ ∗ −2 G ∂ ( 1)l (cid:48) (cid:48) ∗ (cid:48) (cid:48) ∗ − · (cid:48) ∗ I (τ + t , τ + t ) = Pij dcτ M(cid:104)A (cid:105) (τ + t ) ∂(cid:104) (cid:105) ln rN (τ + t ) σ rN (τ + t ) .(G1) 3 0 c2 ∂ξj l! L L A A l=0 τ0 We insert relation (D11) into (G1) and obtain: ∗ ∗ (cid:32) (cid:33) I (τ + t , τ + t ) = 3 0 (cid:88) (cid:2) (cid:3) ∞ − −2 G ∂ ( 1)l ∗ ∗ · ∗ ∗ − · ∗ Pij M(cid:104)A (cid:105) (τ + t ) ∂(cid:104) (cid:105) (cid:32)rN (τ + t ) + σ rN (τ + t ) ln rN (τ + t ) σ rN (τ + t ) (cid:33) c2 ∂ξj (cid:88) l! L L A A A(cid:2) A (cid:3) l=0 ∞ − 2 G ∂ ( 1)l ∗ ∗ · ∗ ∗ − · ∗ + Pij M(cid:104)A (cid:105) (τ + t ) ∂(cid:104) (cid:105) rN (τ + t ) + σ rN (τ + t ) ln rN (τ + t ) σ rN (τ + t ) c2 (cid:88) ∂ξj (cid:90) l! L 0 L A 0 A 0 A 0 A 0 l=0 ∞ − τ (cid:48) ∗ 2 G ( 1)l (cid:48) (cid:48) ∗ (cid:48) di (τ + t ) + dcτ M˙ (cid:104)A (cid:105) (τ + t ) ∂(cid:104) (cid:105) (cid:48) ∗A − · (cid:48) ∗ c3 l! (cid:90) L L rN (τ + t ) σ rN (τ + t ) l=1 (cid:88)τ0 A A (cid:2) (cid:3) ∞ − τ −2 G ∂ ( 1)l (cid:48) (cid:48) ∗ (cid:48) · (cid:48) ∗ (cid:48) ∗ − · (cid:48) ∗ Pij dcτ M(cid:104)A (cid:105) (τ + t ) ∂(cid:104) (cid:105) σ v (τ + t ) ln rN (τ + t ) σ rN (τ + t ) c3 ∂ξj l! (cid:90) L L A A A (cid:88)l=0 τ0 ∞ − τ (cid:48) ∗ · (cid:48) ∗ −2 G ∂ ( 1)l (cid:48) (cid:48) ∗ (cid:48) d (τ + t ) v (τ + t ) Pij dcτ M(cid:104)A (cid:105) (τ + t ) ∂(cid:104) (cid:105) A(cid:48) ∗ − ·A (cid:48) ∗ . (G2) c3 ∂ξj l! L L rN (τ + t ) σ rN (τ + t ) l=0 τ0 A A In order to get the expression in (G2), we have performed sequently, (G2) represents an exact expression for the an integration by parts which results in the expressions integral in (G1). in the ﬁrst and second line. Furthermore, for the expres- Now we are going to proceed with the consideration sion in the third line we have used relation (D6). Con- of the remaining three integrals in the third, fourth, and ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: paragraph

label: equation

text: −4

text: G

text: (τ

text: t

text: l!

text: (τ

text: ln

text: rAN

text: (τ

text: t

text: σ

text: rNA

text: (τ

text: t

text: O

text: v

text: O

text: O

text: A

text: M˙

text: A

text: M¨

text: A

text: A

text: MA

text: c

text: L

text: L

text: L

text: Finally,

text: by

text: means

text: of

text: relation

text: we

text: just

text: obtain

text: the

text: following

text: integrals,

text: each

text: of

text: which

text: will

text: be

text: considered

text: sep-

text: expression

text: in

text: the

text: last

text: line

text: of

text: Eq.

text: arately.

text: Appendix

label: section

text: G:

text: Integrals

text: of

text: second

text: integration

text: of

text: Integral

text: I

text: mass-multipole

text: terms

text: By

text: inserting

text: and

text: into

text: we

text: obtain

text: the

text: The

text: integral

text: I

text: using

text: relation

text: reads

text: τ

text: −2

text: G

text: I

text: (τ

text: t

text: τ

text: t

text: Pij

text: dcτ

text: (τ

text: t

text: ln

text: rN

text: (τ

text: t

text: σ

text: rN

text: (τ

text: t

text: ∂ξj

text: l!

text: L

text: L

text: A

text: A

text: τ0

text: We

text: insert

text: relation

text: into

text: and

text: obtain:

text: I

text: (τ

text: t

text: τ

text: t

text: −2

text: G

text: Pij

text: (τ

text: t

text: (τ

text: t

text: σ

text: rN

text: (τ

text: t

text: ln

text: rN

text: (τ

text: t

text: σ

text: rN

text: (τ

text: t

text: ∂ξj

text: l!

text: L

text: L

text: A

text: A

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text: G

text: Pij

text: (τ

text: t

text: rN

text: (τ

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text: σ

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text: rN

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text: σ

text: rN

text: (τ

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text: ∂ξj

text: l!

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text: L

text: A

text: A

text: A

text: A

text: τ

text: G

text: di

text: (τ

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text: dcτ

text: M˙

text: (τ

text: t

text: ∗A

text: l!

text: L

text: L

text: rN

text: (τ

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text: σ

text: rN

text: (τ

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text: A

text: A

text: τ

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text: G

text: Pij

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text: ∂ξj

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text: (τ

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text: Pij

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text: ﬁrst

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text: the

text: third

text: line

text: we

text: have

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text: remaining

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text: third,

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text: ##LTLine##


text: 6 some time-moment t . However, such an approach is are having arbitrary shape and inner structure and rota- A problematic mainly for two reasons: tional motion, given in terms of time-dependent intrin- (i) all terms of the inﬁnite series expansion (16) con- sic mass-multipoles MA (t) and spin-multipoles SA (t), in L L tribute on 1PN or 1.5PN level, because the expan- accordance with the IAU recommendations [19, 20] and sion in (16) is not performed with respect to the the theory of relativistic reference systems [18, 55–62]. inverse powers of the speed of light. The given solution for the light trajectory is considered as a further step towards a consistent model of general- (ii) the time-moment tA remains an open parameter relativistic theory of light propagation in the gravita- as long as no additional arguments are introduced, tional ﬁeld of the Solar system, which ﬁnally aims at which would uniquely allow to identify that param- accuracies on sub-µas-level and even on nas-level. eter with the time of closest approach or with the The article is organized as follows: In section the retarded time. geodesic equation in 1.5PN approximation is considered. A compendium of the DSX framework is presented in sec- These both aspects (1) and (2) enforce to determine the tion . The transformation of geodesic equation in terms light trajectory x (t) of a light-signal from the celestial of new variables, which are more eﬃcient than the stan- light source toward the observer as function of intrinsic dard parametrization, is given in section . The ﬁrst and multipoles MA and SA as well as function of the arbitrary L L second integration of geodesic equation is determined in worldlines x (t) of these massive Solar system bodies. A section and section , respectively. The important case In a previous investigation [26] a solution for the light of light-propagation in the gravitational ﬁeld of moving trajectory in 1PN approximation in the gravitational spin-dipoles is investigated in section . Finally, the ex- ﬁeld of N massive bodies in arbitrary motion and ex- pressions for the observables of time delay and light de- pressed in terms of their intrinsic multipoles has been ﬂection are obtained in section and . Especially, numer- obtained: (cid:0) (cid:1) ical values for the impact of the leading mass-multipoles and spin-multipoles on time delay and light deﬂection are − O − given in Table II and Table III, respectively. A summary x (t) = x + c (t t ) σ + ∆x + c 3 . (17) 0 0 1PN and outlook can be found in section . The used notations and conventions and further details and several checks of the calculations are shifted into appendix. However, as outlined in more detail in [26], such 1PN so- lution is not suﬃcient for astrometry on sub-µas-level of accuracy. For instance, the rotational motion of the mas- II. GEODESIC EQUATION IN 1.5PN sive bodies cannot be taken into account in 1PN approx- APPROXIMATION imation. However, the impact of the spin-dipole struc- ture of the massive bodies on light deﬂection amounts The Solar system is composed of N arbitrarily shaped, to be about 0.7 µas , 0.2 µas , and 0.04 µas for a graz- rotating and deformable massive bodies which move un- ing lightray at Sun, Jupiter, and Saturn, respectively der the inﬂuence of their mutual gravitational interaction [28, 36]. Moreover, also higher spin-multipoles have a among their common barycenter. It is clear, that the signiﬁcant impact on sub-µas-level [26, 64]. Furthermore, metric of such a highly complicated N-body system is in 1PN approximation there are no terms proportional v not known in its exact form and can only be determined to A M where M is the mass-quadrupole term. Al- c ab ab within an approximative scheme. In view of the weak ready a straightforward estimate reveals that such terms gravitational ﬁelds and slow motions of the bodies, the become relevant on sub-µas-level of accuracy [26], see metric tensor of the Solar system in the BCRS coordinate also Table III. In order to scrutinize the impact of such system xµ = (ct, x) can be expanded in terms of inverse terms one is necessarily enforced to determine the 1.5PN powers in the light-velocity, called post-Newtonian ex- solution for the light trajectory. pansion [17]: (cid:0) (cid:1) In view of these facts, the primary goal of this investi- (cid:0) (cid:1) gation is to determine a solution for the light trajectory O − in 1.5PN aOppro−ximation, which includes all terms up to gαβ (t, x) = ηαβ + h(α2β) (t, x) + h(α3β) (t, x) + c 4 , the order c 4 , where both of the important aspects (19) (1) and (2) addressed above are fully taken into account: (cid:0) (cid:1) − O − wh(cid:0)ere η(cid:1) is the metr(cid:0)ic te(cid:1)nsor of ﬂat Minkowski space- x (t) = x + c (t t ) σ + ∆x + ∆x + c 4 . αβ 0 0 1PN 1.5PN time and the metric perturbations are of the order h(2) = (18) O − O − αβ c 2 and h(3) = c 3 , cf. Eq. (10). Inserting (19) αβ into (6a) yields the geodesic equation in 1.5PN approx- Especially, the massive bodies of the Solar system are al- imation, which in terms of global coordinate time reads lowed to move along arbitrary worldlines x (t) and they [18, 24, 32, 35, 39, 40]: A ##LTLine##

text: some

text: time-moment

text: t

text: However,

text: such

text: an

text: approach

text: is

text: are

text: having

text: arbitrary

text: shape

text: and

text: inner

text: structure

text: and

text: rota-

text: A

text: problematic

text: mainly

text: for

text: two

text: reasons:

text: tional

text: motion,

text: given

text: in

text: terms

text: of

text: time-dependent

text: intrin-

text: (i)

label: list

text: all

text: terms

text: of

text: the

text: inﬁnite

text: series

text: expansion

text: con-

text: sic

text: mass-multipoles

text: MA

text: (t)

text: and

text: spin-multipoles

text: SA

text: (t),

text: in

text: L

text: L

text: tribute

text: on

text: or

text: level,

text: because

text: the

text: expan-

text: accordance

text: with

text: the

text: IAU

text: recommendations

text: and

text: sion

text: in

text: is

text: not

text: performed

text: with

text: respect

text: to

text: the

text: the

text: theory

text: of

text: relativistic

text: reference

text: systems

text: inverse

text: powers

text: of

text: the

text: speed

text: of

text: light.

text: The

text: given

text: solution

text: for

text: the

text: light

text: trajectory

text: is

text: considered

text: as

text: a

text: further

text: step

text: towards

text: a

text: consistent

text: model

text: of

text: general-

text: (ii)

text: the

text: time-moment

text: tA

text: remains

text: an

text: open

text: parameter

text: relativistic

text: theory

text: of

text: light

text: propagation

text: in

text: the

text: gravita-

text: as

text: long

text: as

text: no

text: additional

text: arguments

text: are

text: introduced,

text: tional

text: ﬁeld

text: of

text: the

text: Solar

text: system,

text: which

text: ﬁnally

text: aims

text: at

text: which

text: would

text: uniquely

text: allow

text: to

text: identify

text: that

text: param-

text: accuracies

text: on

text: sub-µas-level

text: and

text: even

text: on

text: nas-level.

text: eter

text: with

text: the

text: time

text: of

text: closest

text: approach

text: or

text: with

text: the

text: The

text: article

text: is

text: organized

text: as

text: follows:

text: In

text: section

text: the

text: retarded

text: time.

text: geodesic

text: equation

text: in

text: approximation

text: is

text: considered.

text: A

text: compendium

text: of

text: the

text: DSX

text: framework

text: is

text: presented

text: in

text: sec-

text: These

text: both

text: aspects

text: and

text: enforce

text: to

text: determine

text: the

text: tion

text: The

text: transformation

text: of

text: geodesic

text: equation

text: in

text: terms

text: light

text: trajectory

text: x

text: (t)

text: of

text: a

text: light-signal

text: from

text: the

text: celestial

text: of

text: new

text: variables,

text: which

text: are

text: more

text: eﬃcient

text: than

text: the

text: stan-

text: light

text: source

text: toward

text: the

text: observer

text: as

text: function

text: of

text: intrinsic

text: dard

text: parametrization,

text: is

text: given

text: in

text: section

text: The

text: ﬁrst

text: and

text: multipoles

text: MA

text: and

text: SA

text: as

text: well

text: as

text: function

text: of

text: the

text: arbitrary

text: L

text: L

text: second

text: integration

text: of

text: geodesic

text: equation

text: is

text: determined

text: in

text: worldlines

text: x

text: (t)

text: of

text: these

text: massive

text: Solar

text: system

text: bodies.

text: A

text: section

text: and

text: section

text: respectively.

text: The

text: important

text: case

text: In

text: a

text: previous

text: investigation

text: a

text: solution

text: for

text: the

text: light

text: of

text: light-propagation

text: in

text: the

text: gravitational

text: ﬁeld

text: of

text: moving

text: trajectory

text: in

text: approximation

text: in

text: the

text: gravitational

text: spin-dipoles

text: is

text: investigated

text: in

text: section

text: Finally,

text: the

text: ex-

text: ﬁeld

text: of

text: N

text: massive

text: bodies

text: in

text: arbitrary

text: motion

text: and

text: ex-

text: pressions

text: for

text: the

text: observables

text: of

text: time

text: delay

text: and

text: light

text: de-

text: pressed

text: in

text: terms

text: of

text: their

text: intrinsic

text: multipoles

text: has

text: been

text: ﬂection

text: are

text: obtained

text: in

text: section

text: and

text: Especially,

text: numer-

text: obtained:

text: ical

text: values

text: for

text: the

text: impact

text: of

text: the

text: leading

text: mass-multipoles

text: and

text: spin-multipoles

text: on

text: time

text: delay

text: and

text: light

text: deﬂection

text: are

text: O

text: given

text: in

text: Table

text: II

text: and

text: Table

text: III,

text: respectively.

text: A

text: summary

text: x

text: (t)

text: x

text: c

text: (t

text: t

text: σ

text: ∆x

text: c

text: and

text: outlook

text: can

text: be

text: found

text: in

text: section

text: The

text: used

text: notations

text: and

text: conventions

text: and

text: further

text: details

text: and

text: several

text: checks

text: of

text: the

text: calculations

text: are

text: shifted

text: into

text: appendix.

text: However,

text: as

text: outlined

text: in

text: more

text: detail

text: in

text: such

text: so-

text: lution

text: is

text: not

text: suﬃcient

text: for

text: astrometry

text: on

text: sub-µas-level

text: of

text: accuracy.

text: For

text: instance,

text: the

text: rotational

text: motion

text: of

text: the

text: mas-

text: II.

text: GEODESIC

text: EQUATION

text: IN

text: sive

text: bodies

text: cannot

text: be

text: taken

text: into

text: account

text: in

text: approx-

text: APPROXIMATION

text: imation.

text: However,

text: the

text: impact

text: of

text: the

text: spin-dipole

text: struc-

text: ture

text: of

text: the

text: massive

text: bodies

text: on

text: light

text: deﬂection

text: amounts

text: The

text: Solar

text: system

text: is

text: composed

text: of

text: N

text: arbitrarily

text: shaped,

text: to

text: be

text: about

text: µas

text: µas

text: and

text: µas

text: for

text: a

text: graz-

text: rotating

text: and

text: deformable

text: massive

text: bodies

text: which

text: move

text: un-

text: ing

text: lightray

text: at

text: Sun,

text: Jupiter,

text: and

text: Saturn,

text: respectively

text: der

text: the

text: inﬂuence

text: of

text: their

text: mutual

text: gravitational

text: interaction

text: Moreover,

text: also

text: higher

text: spin-multipoles

text: have

text: a

text: among

text: their

text: common

text: barycenter.

text: It

text: is

text: clear,

text: that

text: the

text: signiﬁcant

text: impact

text: on

text: sub-µas-level

text: Furthermore,

text: metric

text: of

text: such

text: a

text: highly

text: complicated

text: N-body

text: system

text: is

text: in

text: approximation

text: there

text: are

text: no

text: terms

text: proportional

text: v

text: not

text: known

text: in

text: its

text: exact

text: form

text: and

text: can

text: only

text: be

text: determined

text: to

text: A

text: M

text: where

text: M

text: is

text: the

text: mass-quadrupole

text: term.

text: Al-

text: c

text: ab

text: ab

text: within

text: an

text: approximative

text: scheme.

text: In

text: view

text: of

text: the

text: weak

text: ready

text: a

text: straightforward

text: estimate

text: reveals

text: that

text: such

text: terms

text: gravitational

text: ﬁelds

text: and

text: slow

text: motions

text: of

text: the

text: bodies,

text: the

text: become

text: relevant

text: on

text: sub-µas-level

text: of

text: accuracy

text: see

text: metric

text: tensor

text: of

text: the

text: Solar

text: system

text: in

text: the

text: BCRS

text: coordinate

text: also

text: Table

text: III.

text: In

text: order

text: to

text: scrutinize

text: the

text: impact

text: of

text: such

text: system

text: xµ

text: (ct,

text: x)

text: can

text: be

text: expanded

text: in

text: terms

text: of

text: inverse

text: terms

text: one

text: is

text: necessarily

text: enforced

text: to

text: determine

text: the

text: powers

text: in

text: the

text: light-velocity,

text: called

text: post-Newtonian

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text: solution

text: for

text: the

text: light

text: trajectory.

text: pansion

text: In

text: view

text: of

text: these

text: facts,

text: the

text: primary

text: goal

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text: this

text: investi-

text: gation

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text: determine

text: a

text: solution

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text: the

text: light

text: trajectory

text: O

text: in

text: aOppro−ximation,

text: which

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text: all

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text: up

text: to

text: gαβ

text: (t,

text: x)

text: ηαβ

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text: x)

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text: x)

text: c

text: the

text: order

text: c

text: where

text: both

text: of

text: the

text: important

text: aspects

text: and

text: addressed

text: above

text: are

text: fully

text: taken

text: into

text: account:

text: O

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text: the

text: of

text: ﬂat

text: Minkowski

text: space-

text: x

text: (t)

text: x

text: c

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text: ∆x

text: ∆x

text: c

text: αβ

text: time

text: and

text: the

text: metric

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text: are

text: of

text: the

text: order

text: O

text: O

text: αβ

text: c

text: and

text: c

text: cf.

text: Eq.

text: Inserting

text: αβ

text: into

text: yields

text: the

text: geodesic

text: equation

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text: Especially,

text: the

text: massive

text: bodies

text: of

text: the

text: Solar

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text: al-

text: imation,

text: which

text: in

text: terms

text: of

text: global

text: coordinate

text: time

text: reads

text: lowed

text: to

text: move

text: along

text: arbitrary

text: worldlines

text: x

text: (t)

text: and

text: they

text: A

text: ##LTLine##


text: 51 [60] S.M. Kopeikin, Celestial coordinate reference systems in [73] S.M. Kopeikin, V.V. Makarov, Gravitational bending curved space-time, Cel. Mech. 44 (1988) 87. of light by planetary multipoles and its measurement [61] S.M. Kopeikin, Theory of relativity in observational radio with micro-arcsecond astronomical interferometers, Phys. astronomy, Sov. Astron. 34 (1990) 5. Rev. D 75 (2007) 062002. [62] S.A. Klioner, A.V. Voinov, Relativistic Theory of Refer- [74] S. Zschocke, M.H. Soﬀel, Gravitational ﬁeld of one uni- ence Systems in Closed Form, Phys. Rev. D 48 (1993) formly moving extended body and N arbitrarily mov- 1451. ing pointlike bodies in post-Minkowskian approximation, [63] Highly accurate ephemerides for solar system objects Class. Quantum Grav. 31 (2014) 175001. provided by Jet Propulsion Laboratory (JPL) at [75] A. Fienga, H. Manche, J. Laskar, M. Gastineau, IN- http://ssd.jpl.nasa.gov/ POP06: a new numerical planetary ephemeris, Astron. [64] J. Meichsner, Physics in gravity ﬁelds with higher spin Astrophys. 477 (2008) 315. multipole moments, Diploma Thesis, Technical Univer- [76] I. de Pater, J.J. Lissauer, Planetary Science, 2.Ed., Cam- sity Dresden, 2015. bridge University Press, 2015. [65] S.A. Klioner, The problem of clock synchronization: a [77] W. B. Hubbard, B. Militzer, A preliminary Jupiter relativistic approach, Celestial Mechanics 53 (1992) 81. model, The Astronomical Journal 820 (2016) 80. [66] E.F. Arias, P. Charlot, M. Feissel, J.-F. Lestrade, [78] J.D. Anderson, G. Schubert, Saturn’s gravitational ﬁeld, The Extragalactic Reference System of the International internal rotation, and interior structure, Science 317 Earth Rotation Service, ICRS, Astron. Astrophys. 303 (2007) 1384. (1995) 604. [79] S. Zschocke, S.A. Klioner: On the eﬃcient computation [67] S. Zschocke, A detailed proof of the fundamental theorem of the quadrupole light deﬂection, Class. Quantum Grav. of STF multipole expansion in linearized gravity, Int. J. 28 (2011) 015009. Mod. Phys. D 23 (2014) 1450003. [80] C. Xu, X. Wu, Extending the ﬁrst-order post-Newtonian [68] T. Hartmann, M.H. Soﬀel, T. Kioustelidis, On the use of scheme in multiple systems to the second-order contribu- STF-tensors in celestial mechanics, Celestial Mechanics tions to light propagation, Chin. Phys. Lett. 20 (2003) and Dynamical Astronomy 60 (1994) 139. 195. [69] Private communication with Prof. M.A. Soﬀel and Prof. [81] C. Xu, Y. Gong, X. Wu, M. Soﬀel, S.A. Klioner (2005), S.M. Kopeikin (27 June 2012) where a wrong overall sign Second order post-Newtonian Equations of light propaga- for all spin-multipoles in Ref. [31] has been conﬁrmed tion in multiple systems, arXiv:gr-qc/0510074. which were caused by a wrong sign of the vectorial po- [82] O. Minazzolli, B. Chauvineau, Post-Newtonian metric of − tential in Eq. (11) in [31]. general relativity including all the c 4 terms in the conti- [70] I.I. Shapiro, Fourth Test of General Relativity, Phys. Rev. nuity of the IAU2000 resolutions, Phys. Rev. D 79 (2009) Lett. 13 (1964) 789. 084027. [71] I.I. Shapiro, G.H. Pettengill, M.E. Ash, et al., Fourth [83] S.A. Klioner, M.H. Soﬀel, The post-linear Schwarzschild Test of General Relativity: Preliminary Results, Phys. solution in harmonic coordinates: elimination of Rev. Lett. 20 (1968) 1265. structure-dependent terms, Phys. Rev. D 89 (2014) [72] S. Bertone, O. Minazzoli, M. Crosta, C. Le Poncin- 104056. Laﬁtte, A. Vecchiato, M.C. Angonin, Time Transfer [84] S.A. Klioner, Physically adequate reference system of a functions as a way to validate light propagation solutions test observer and relativistic description of the Gaia at- for space astrometry, Class. Quantum Grav. 31 (2014) titude, Phys. Rev. D 69 (2004) 124001. 015021.

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text: Laboratory

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text: Manche,

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text: Laskar,

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text: Gastineau,

text: IN-

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text: numerical

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text: Astron.

text: J.

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text: Physics

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text: Pater,

text: J.J.

text: Lissauer,

text: Planetary

text: Science,

text: Cam-

text: sity

text: Dresden,

text: bridge

text: University

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text: Klioner,

text: The

text: problem

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text: clock

text: synchronization:

text: a

text: W.

text: B.

text: Hubbard,

text: B.

text: Militzer,

text: A

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text: Jupiter

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text: Mechanics

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text: Astronomical

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text: Arias,

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text: S.A.

text: Klioner:

text: On

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text: computation

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text: Zschocke,

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text: proof

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text: quadrupole

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text: Quantum

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text: expansion

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text: gravity,

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text: Mod.

text: Phys.

text: D

text: C.

text: Xu,

text: X.

text: Wu,

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text: Hartmann,

text: M.H.

text: Soﬀel,

text: T.

text: Kioustelidis,

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text: the

text: use

text: of

text: scheme

text: in

text: multiple

text: systems

text: to

text: the

text: second-order

text: contribu-

text: STF-tensors

text: in

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text: mechanics,

text: Celestial

text: Mechanics

text: tions

text: to

text: light

text: propagation,

text: Chin.

text: Phys.

text: Lett.

text: and

text: Dynamical

text: Astronomy

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text: communication

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text: Prof.

text: M.A.

text: Soﬀel

text: and

text: Prof.

text: C.

text: Xu,

text: Y.

text: Gong,

text: X.

text: Wu,

text: M.

text: Soﬀel,

text: S.A.

text: Klioner

text: S.M.

text: Kopeikin

text: June

text: where

text: a

text: wrong

text: overall

text: sign

text: Second

text: order

text: post-Newtonian

text: Equations

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text: light

text: propaga-

text: for

text: all

text: spin-multipoles

text: in

text: Ref.

text: has

text: been

text: conﬁrmed

text: tion

text: in

text: multiple

text: systems,

text: which

text: were

text: caused

text: by

text: a

text: wrong

text: sign

text: of

text: the

text: vectorial

text: po-

text: O.

text: Minazzolli,

text: B.

text: Chauvineau,

text: Post-Newtonian

text: metric

text: of

text: tential

text: in

text: Eq.

text: in

text: general

text: relativity

text: including

text: all

text: the

text: c

text: terms

text: in

text: the

text: conti-

text: I.I.

text: Shapiro,

text: Fourth

text: Test

text: of

text: General

text: Relativity,

text: Phys.

text: Rev.

text: nuity

text: of

text: the

text: resolutions,

text: Phys.

text: Rev.

text: D

text: Lett.

text: I.I.

text: Shapiro,

text: G.H.

text: Pettengill,

text: M.E.

text: Ash,

text: et

text: al.,

text: Fourth

text: S.A.

text: Klioner,

text: M.H.

text: Soﬀel,

text: The

text: post-linear

text: Schwarzschild

text: Test

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text: Physically

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text: reference

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text: a

text: functions

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text: a

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text: 8 at the barycenter of the Solar system, and N local charts with coordinates Xα = (cT , X ), one for each individ- The gravitational potentials in (29) - (31) are given by A A A ual body A = 1, ..., N and having their origins at the the integrals (cid:90) (cid:0) (cid:1) barycenter of these massive bodies and co-moving with them. The local coordinate systems are tied to each indi- (cid:48) G (cid:48) t00 (t, x ) O − vidual massive body and are deﬁned very similar to the w (t, x) = (cid:90) d3x | − (cid:48)| + (cid:0) c 2(cid:1) , (32) Geocentric Celestial Reference System (GCRS) which is c2 x x in use for the Earth and, therefore, they are called GCRS- (cid:48) G (cid:48) t0i (t, x ) O − like reference systems. A central result of the DSX ap- wi (t, x) = d3x | − (cid:48)| + c 2 , (33) c x x proach is the form of the global metric g of BCRS and µν the form of the local metric GA for each GCRS-like sys- αβ tem, and the coordinate transformation among all these where the integrals in (32) and (33) run over the entire ↔ reference systems (ct, x) (cTA, XA). Another central Solar sys(cid:0)tem(cid:1) , and tµν(cid:0)is t(cid:1)he energy(cid:0)-mo(cid:1)mentum tensor of achievement in the DSX formalism is the decomposition the Solar system in global BCRS coordinates; recall the of the global metric in terms of intrinsic mass-multipoles components of energy-momentum tensor scale as follows: O O O MA and intrinsic spin-multipoles SA. In this section we t00 = c2 , t0i = c1 , tij = c0 . L L will present a compendium of the DSX theory, which has The global gravitational potentials in (32) - (33) ad- become a basic part of IAU resolution B1.3 (2000) [19] mit an expansion in terms of global Blanchet-Damour and which are of upmost relevance for our own consid- (BD) mass-multipoles and spin-multipoles, mL and sL, erations aiming at applications of the DSX approach in [32, 51–53], which characterize the multipole structure of the astrometrical science. the Solar system as a whole, (cid:88) (cid:0) (cid:1) A. BCRS (cid:0) (cid:1) ∞ − ( 1)l 1 O − The harmonic BCRS coordinates are denoted by xµ = w (t, x) = G m(cid:104) (cid:105) (t) ∂(cid:104) (cid:105) + c 2 ,(34) ct, xi , where t = TCB is the BCRS coordinate time; (cid:88) l! L L r l=0 about a practical synchronization of a set of clocks dis- ∞ − tributed somewhere in the Solar system we refer to [65]. wi (t, x) = −G ( 1)l m˙ (cid:104) − (cid:105) (t) ∂(cid:104) − (cid:105) 1 The origin of the spatial axes of BCRS is located at the (cid:88) l! iL 1 L 1 r (cid:0) (cid:1) l=2 barycenter of the Solar system and cover the entire three- ∞ − dimensional space and can therefore be used to model − ( 1)l l 1 O − G (cid:15) s(cid:104) − (cid:105) (t) ∂(cid:104) − (cid:105) + c 2 . light trajectories from distant celestial objects to the ob- (l + 1)! iab bL 1 aL 1 r server. The IAU Resolution B2 (2006) [20] recommends l=1 the spatial axes of BCRS to be oriented according to (35) the spatial axes of the International Celestial Reference System (ICRS) [66]. Furthermore, according to IAU res- (cid:0) (cid:1) The global mass-multipoles and global spin-multipoles olution B1.3 (2000) [19] the Solar system is assumed to in (34) - (35) are Cartesian symmetric and trace-free be isolated and the space-time is asymptotically ﬂat, that O − (STF) tensors, and up to order c 2 given by (cf. means the BCRS metric g (t, x) at inﬁnity reads: µν Eqs. (2.34a) and (2.34b(cid:90)) in [52]): rl→im∞ gµν (t, x) = ηµν , (28) m(cid:104) (cid:105) (t) = STF (cid:90) d3x x t00 (t, x) , (36) L L L c2 | | where r = x . The BCRS is completely characterized by t0b (t, x) the form of its metric tensor which, however, is not known s(cid:104)L(cid:105) (t) = STLF d3x (cid:15)abcl xaL−1 c , (37) in its exact form. According to the geodesic equation in (cid:0) (cid:1) 1.5PN approximation (25), for our intentions the metric where m = const. is the mass of the entire Solar system, is required to be known only up to terms of the order 0 O − the mass-dipole m = 0 because the origin of BCRS is c 4 , which are given by [19]: i (cid:0) (cid:1) located at the barycenter of the Solar system, and the spin-dipole s = const describes the spin of the entire i − 2 w (t, x) O − Solar system which safely can be assumed to be time- g00 (t, x) = 1 + c2 +(cid:0) c(cid:1) 4 , (29) independent. The spatial derivative operator in (34) - (35) is deﬁned by g0i (t, x) = −(cid:18) 4 wic(3t, x) + O(cid:19) c−5 , (cid:0) (cid:1) (30) ∂ ∂ ∂(cid:104) (cid:105) = STF . . . , (38) g (t, x) = 1 + 2 w (t, x) δ + O c−4 . (31) L i1...il ∂xi1 ∂xil ij c2 ij ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: at

text: the

text: barycenter

text: of

text: the

text: Solar

text: system,

text: and

text: N

text: local

text: charts

text: with

text: coordinates

text: Xα

text: (cT

text: X

text: one

text: for

text: each

text: individ-

text: The

text: gravitational

text: potentials

text: in

text: are

text: given

text: by

text: A

text: A

text: A

text: ual

text: body

text: A

text: N

text: and

text: having

text: their

text: origins

text: at

text: the

text: the

text: integrals

text: barycenter

text: of

text: these

text: massive

text: bodies

text: and

text: co-moving

text: with

text: them.

text: The

text: local

text: coordinate

text: systems

text: are

text: tied

text: to

text: each

text: indi-

text: G

text: (t,

text: x

text: O

text: vidual

text: massive

text: body

text: and

text: are

text: deﬁned

text: very

text: similar

text: to

text: the

text: w

text: (t,

text: x)

text: c

text: Geocentric

text: Celestial

text: Reference

text: System

text: (GCRS)

text: which

text: is

text: x

text: x

text: in

text: use

text: for

text: the

text: Earth

text: and,

text: therefore,

text: they

text: are

text: called

text: GCRS-

text: G

text: (t,

text: x

text: O

text: like

text: reference

text: systems.

text: A

text: central

text: result

text: of

text: the

text: DSX

text: ap-

text: wi

text: (t,

text: x)

text: c

text: c

text: x

text: x

text: proach

text: is

text: the

text: form

text: of

text: the

text: global

text: metric

text: g

text: of

text: BCRS

text: and

text: µν

text: the

text: form

text: of

text: the

text: local

text: metric

text: GA

text: for

text: each

text: GCRS-like

text: sys-

text: αβ

text: tem,

text: and

text: the

text: coordinate

text: transformation

text: among

text: all

text: these

text: where

text: the

text: integrals

text: in

text: and

text: run

text: over

text: the

text: entire

text: reference

text: systems

text: (ct,

text: x)

text: (cTA,

text: XA).

text: Another

text: central

text: Solar

text: and

text: tensor

text: of

text: achievement

text: in

text: the

text: DSX

text: formalism

text: is

text: the

text: decomposition

text: the

text: Solar

text: system

text: in

text: global

text: BCRS

text: coordinates;

text: recall

text: the

text: of

text: the

text: global

text: metric

text: in

text: terms

text: of

text: intrinsic

text: mass-multipoles

text: components

text: of

text: energy-momentum

text: tensor

text: scale

text: as

text: follows:

text: O

text: O

text: O

text: MA

text: and

text: intrinsic

text: spin-multipoles

text: SA.

text: In

text: this

text: section

text: we

text: tij

text: L

text: L

text: will

text: present

text: a

text: compendium

text: of

text: the

text: DSX

text: theory,

text: which

text: has

text: The

text: global

text: gravitational

text: potentials

text: in

text: ad-

text: become

text: a

text: basic

text: part

text: of

text: IAU

text: resolution

text: mit

text: an

text: expansion

text: in

text: terms

text: of

text: global

text: Blanchet-Damour

text: and

text: which

text: are

text: of

text: upmost

text: relevance

text: for

text: our

text: own

text: consid-

text: (BD)

text: mass-multipoles

text: and

text: spin-multipoles,

text: mL

text: and

text: sL,

text: erations

text: aiming

text: at

text: applications

text: of

text: the

text: DSX

text: approach

text: in

text: which

text: characterize

text: the

text: multipole

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text: ##LTLine##

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text: Alternative asymptotics for cointegration tests in large VARs. Alexei Onatski and Chen Wang Faculty of Economics, University of Cambridge May 29, 2020 Abstract Johansen’s (1988, 1991) likelihood ratio test for cointegration rank of a Gaussian VAR depends only on the squared sample canonical correlations between current changes and past levels of a simple transformation of the data. We study the asymptotic behavior of the empirical distribution of those squared canonical correlations when the number of observations and the dimensionality of the VAR diverge to inﬁnity simultaneously and pro- portionally. We ﬁnd that the distribution almost surely weakly converges to the so-called Wachter distribution. This ﬁnding provides a theoretical explanation for the observed tendency of Johansen’s test to ﬁnd “spurious cointegration”. It also sheds light on the workings and limitations of the Bartlett correction approach to the over-rejection problem. We propose a simple graphical device, similar to the scree plot, for a preliminary assess- ment of cointegration in high-dimensional VARs. 1 Introduction Johansen’s (1988, 1991) likelihood ratio (LR) test for cointegration rank is a very popular econometric technique. However, it is rarely applied to systems of more than three or four variables. On the other hand, there exist many applications involving much larger systems. For example, Davis (2003) discusses a possibility of applying the test to the data on seven aggregated and individual commodity prices to test Lewbel’s (1996) generalization of the Hicks-Leontief composite commodity 1

text: Alternative

label: title

text: asymptotics

text: for

text: cointegration

text: tests

text: in

text: large

text: VARs.

text: Alexei

label: author

text: Onatski

text: and

text: Chen

text: Wang

text: Faculty

text: of

text: Economics,

text: University

text: of

text: Cambridge

text: May

text: Abstract

text: Johansen’s

label: abstract

text: likelihood

text: ratio

text: test

text: for

text: cointegration

text: rank

text: of

text: a

text: Gaussian

text: VAR

text: depends

text: only

text: on

text: the

text: squared

text: sample

text: canonical

text: correlations

text: between

text: current

text: changes

text: and

text: past

text: levels

text: of

text: a

text: simple

text: transformation

text: of

text: the

text: data.

text: We

text: study

text: the

text: asymptotic

text: behavior

text: of

text: the

text: empirical

text: distribution

text: of

text: those

text: squared

text: canonical

text: correlations

text: when

text: the

text: number

text: of

text: observations

text: and

text: the

text: dimensionality

text: of

text: the

text: VAR

text: diverge

text: to

text: inﬁnity

text: simultaneously

text: and

text: pro-

text: portionally.

text: We

text: ﬁnd

text: that

text: the

text: distribution

text: almost

text: surely

text: weakly

text: converges

text: to

text: the

text: so-called

text: Wachter

text: distribution.

text: This

text: ﬁnding

text: provides

text: a

text: theoretical

text: explanation

text: for

text: the

text: observed

text: tendency

text: of

text: Johansen’s

text: test

text: to

text: ﬁnd

text: “spurious

text: cointegration”.

text: It

text: also

text: sheds

text: light

text: on

text: the

text: workings

text: and

text: limitations

text: of

text: the

text: Bartlett

text: correction

text: approach

text: to

text: the

text: over-rejection

text: problem.

text: We

text: propose

text: a

text: simple

text: graphical

text: device,

text: similar

text: to

text: the

text: scree

text: plot,

text: for

text: a

text: preliminary

text: assess-

text: ment

text: of

text: cointegration

text: in

text: high-dimensional

text: VARs.

text: Introduction

text: Johansen’s

text: likelihood

text: ratio

text: (LR)

text: test

text: for

text: cointegration

text: rank

text: is

text: a

text: very

text: popular

text: econometric

text: technique.

text: However,

text: it

text: is

text: rarely

text: applied

text: to

text: systems

text: of

text: more

text: than

text: three

text: or

text: four

text: variables.

text: On

text: the

text: other

text: hand,

text: there

text: exist

text: many

text: applications

text: involving

text: much

text: larger

text: systems.

text: For

text: example,

text: Davis

text: discusses

text: a

text: possibility

text: of

text: applying

text: the

text: test

text: to

text: the

text: data

text: on

text: seven

text: aggregated

text: and

text: individual

text: commodity

text: prices

text: to

text: test

text: Lewbel’s

text: generalization

text: of

text: the

text: Hicks-Leontief

text: composite

text: commodity


text: is provided by the tables of the asymptotic critical values for Johansen’s LR test (see, for example, MacKinnon, Haug and Michelis (1999)). The critical values in such tables become uncomfortably large for p > 4. Of course, the reason for such an unpleasant growth is that those critical values are of order 2p2. The transformation (cid:55)→ − LR LR /p 2p 0,p,T 0,p,T makes the LR statistic ‘well-behaved’ under the sequential asymptotics. The divi- sion by p reduces the ‘second order behavior’ to O (1), while subtracting 2p elim- P inates the remaining explosive ‘highest order term’. We report the corresponding transformed 95% critical values alongside the original ones in Table 1. The transformed critical values resemble 97-99 percentiles of N(0, 1). Since the LR test is one-sided, the resemblance is coincidental. However, we do expect the sequential asymptotic distribution of the transformed LR statistic (as well as its simultaneous asymptotic distribution) to be normal (possibly with non- zero mean and non-unit variance). Our expectation is based on the fact that LR /p behaves as the eigenvalue average (see (22)), which is a special case of 0,p,T the so-called linear spectral statistic. The asymptotic normality of linear spectral statistics for relatively simple classes of high-dimensional random matrices is a well established result in the Large Random Matrix Theory (see Bai and Silverstein (2010)). Extending it to the linear spectral statistics of matrices of form (19) is left as an important direction for future research. 3.2 Over-rejection phenomenon, and the Bartlett correc- tion In this subsection, let us assume that the following conjecture holds. Conjecture 9 The simultaneous and sequential asymptotic lower bounds for the scaled LR statistics derived in Corollaries 4 and 8 represent the corresponding simultaneous and sequential asymptotic limits. Speciﬁcally, for c < 1/2, − − 1 1 + c − 1 c − 1 2c − lim LR = ln (1 + c) ln (1 c) + ln (1 2c) , (24) p,T→c∞ 2p2 0,p,T 2c2 2c2 2c2 17 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: is

text: provided

text: by

text: the

text: tables

text: of

text: the

text: asymptotic

text: critical

text: values

text: for

text: Johansen’s

text: LR

text: test

text: (see,

text: for

text: example,

text: MacKinnon,

text: Haug

text: and

text: Michelis

text: The

text: critical

text: values

text: in

text: such

text: tables

text: become

text: uncomfortably

text: large

text: for

text: p

text: Of

text: course,

text: the

text: reason

text: for

text: such

text: an

text: unpleasant

text: growth

text: is

text: that

text: those

text: critical

text: values

text: are

text: of

text: order

text: The

text: transformation

text: LR

text: LR

text: /p

text: makes

text: the

text: LR

text: statistic

text: ‘well-behaved’

text: under

text: the

text: sequential

text: asymptotics.

text: The

text: divi-

text: sion

text: by

text: p

text: reduces

text: the

text: ‘second

text: order

text: behavior’

text: to

text: O

text: while

text: subtracting

text: elim-

text: P

text: inates

text: the

text: remaining

text: explosive

text: ‘highest

text: order

text: term’.

text: We

text: report

text: the

text: corresponding

text: transformed

text: critical

text: values

text: alongside

text: the

text: original

text: ones

text: in

text: Table

text: The

text: transformed

text: critical

text: values

text: resemble

text: percentiles

text: of

text: Since

text: the

text: LR

text: test

text: is

text: one-sided,

text: the

text: resemblance

text: is

text: coincidental.

text: However,

text: we

text: do

text: expect

text: the

text: sequential

text: asymptotic

text: distribution

text: of

text: the

text: transformed

text: LR

text: statistic

text: (as

text: well

text: as

text: its

text: simultaneous

text: asymptotic

text: distribution)

text: to

text: be

text: normal

text: (possibly

text: with

text: non-

text: zero

text: mean

text: and

text: non-unit

text: variance).

text: Our

text: expectation

text: is

text: based

text: on

text: the

text: fact

text: that

text: LR

text: /p

text: behaves

text: as

text: the

text: eigenvalue

text: average

text: (see

text: which

text: is

text: a

text: special

text: case

text: of

text: the

text: so-called

text: linear

text: spectral

text: statistic.

text: The

text: asymptotic

text: normality

text: of

text: linear

text: spectral

text: statistics

text: for

text: relatively

text: simple

text: classes

text: of

text: high-dimensional

text: random

text: matrices

text: is

text: a

text: well

text: established

text: result

text: in

text: the

text: Large

text: Random

text: Matrix

text: Theory

text: (see

text: Bai

text: and

text: Silverstein

text: Extending

text: it

text: to

text: the

text: linear

text: spectral

text: statistics

text: of

text: matrices

text: of

text: form

text: is

text: left

text: as

text: an

text: important

text: direction

text: for

text: future

text: research.

text: Over-rejection

text: phenomenon,

text: and

text: the

text: Bartlett

text: correc-

text: tion

text: In

text: this

text: subsection,

text: let

text: us

text: assume

text: that

text: the

text: following

text: conjecture

text: holds.

text: Conjecture

text: The

text: simultaneous

text: and

text: sequential

text: asymptotic

text: lower

text: bounds

text: for

text: the

text: scaled

text: LR

text: statistics

text: derived

text: in

text: Corollaries

text: and

text: represent

text: the

text: corresponding

text: simultaneous

text: and

text: sequential

text: asymptotic

text: limits.

text: Speciﬁcally,

text: for

text: c

text: c

text: c

text: lim

text: LR

text: ln

text: c)

text: ln

text: c)

text: ln

text: p,T→c∞

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: No constant in model and DGP Constant in model and DGP 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 Constant in model but not in DGP Constant in DGP but not in model 0.8 1 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 Quantile of Wachter Quantile of Wachter Figure 8: The qq Wachter plots for p = 20 and T = 100. The data generating process (DGP) is (11) with k = 1, Π = 0, and either F = 1 (constant in DGP) or t F = 0 (no constant in GDP). The econometrician’s model is (7) with Π = 0 and t either D = 1 (constant in model) or D = 0 (no constant in model). t t 26 elitnauq laciripmE elitnauq laciripmE ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: No

font: default

text: constant

text: in

text: model

text: and

text: DGP

text: Constant

text: in

text: model

text: and

text: DGP

text: Constant

text: in

text: model

text: but

text: not

text: in

text: DGP

text: Constant

text: in

text: DGP

text: but

text: not

text: in

text: model

text: Quantile

text: of

text: Wachter

text: Quantile

text: of

text: Wachter

text: Figure

text: The

text: qq

text: Wachter

text: plots

text: for

text: p

text: and

text: T

text: The

text: data

text: generating

text: process

text: (DGP)

text: is

text: with

text: k

text: Π

text: and

text: either

text: F

text: (constant

text: in

text: DGP)

text: or

text: t

text: F

text: (no

text: constant

text: in

text: GDP).

text: The

text: econometrician’s

text: model

text: is

text: with

text: Π

text: and

text: t

text: either

text: D

text: (constant

text: in

text: model)

text: or

text: D

text: (no

text: constant

text: in

text: model).

text: t

text: t

text: elitnauq

text: laciripmE

text: elitnauq

text: laciripmE

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##


text: In this paper, we consider the simultaneous, large-p, large-T , asymptotic be- havior of the squared sample canonical correlations between p-dimensional ran- dom walk and its innovations. We ﬁnd that the empirical distribution of these squared sample canonical correlations almost surely weakly converges to the so- called Wachter distribution with parameters that depend only on the limit of p/T → ∞ → ∞ as p, T . In contrast, under the sequential asymptotics, when ﬁrst T c → ∞ and then p , we establish the convergence in probability to the so-called Marchenko-Pastur distribution. The diﬀerences between the limiting distributions allow us to explain from a theoretical point of view the tendency of the LR test for cointegration to severely over-reject the null when the dimensionality of the data is relatively large. Furthermore, we derive a simple analytic formula for the Bartlett-type correction factor in systems with relatively large p/T ratio. We propose a quick graphical method, the Wachter plot, for a preliminary analysis of cointegration in large-dimensional systems. The Monte Carlo analysis shows that the quantiles of the Wachter distribution constitute very good center- ing points for the ﬁnite sample distributions of the corresponding squared sample canonical correlations. The quality of the centering is excellent even for such small p and T as p = 10 and T = 20. However, for such small values of p and T, the empirical distribution of the squared sample canonical correlation can considerably ﬂuctuate around the Wachter limit. As p increases to 100, the ﬂuctuations become numerically very small. Our analysis leaves many open questions. First, it is very important to study the ﬂuctuations of the empirical distribution around the Wachter limit. We con- jecture that linear combinations of reasonably smooth functions of the squared − sample canonical correlations, including the log(1 λ) used by the LR statistic, will be asymptotically Gaussian after appropriate centering and scaling. The cen- tering can be derived from the results obtained in this paper. A proof of the asymptotic Gaussianity would require diﬀerent methods from those used here. We are currently investigating this research direction. Further, it would be important to remove the Gaussianity assumption on the data. We believe that the existence of the ﬁnite fourth moments is a suﬃcient condition for the validity of the Wachter limit. Next, it would be interesting to study the simultaneous asymptotic behavior of a few of the largest sample canonical correlations. This may lead to a modiﬁcation of Johansen’s maximum eigenvalue test. Another interesting direction of research is to study situations where the num- 32

text: In

text: this

text: paper,

text: we

text: consider

text: the

text: simultaneous,

text: large-p,

text: large-T

text: asymptotic

text: be-

text: havior

text: of

text: the

text: squared

text: sample

text: canonical

text: correlations

text: between

text: p-dimensional

text: ran-

text: dom

text: walk

text: and

text: its

text: innovations.

text: We

text: ﬁnd

text: that

text: the

text: empirical

text: distribution

text: of

text: these

text: squared

text: sample

text: canonical

text: correlations

text: almost

text: surely

text: weakly

text: converges

text: to

text: the

text: so-

text: called

text: Wachter

text: distribution

text: with

text: parameters

text: that

text: depend

text: only

text: on

text: the

text: limit

text: of

text: p/T

text: as

text: p,

text: T

text: In

text: contrast,

text: under

text: the

text: sequential

text: asymptotics,

text: when

text: ﬁrst

text: T

text: c

text: and

text: then

text: p

text: we

text: establish

text: the

text: convergence

text: in

text: probability

text: to

text: the

text: so-called

text: Marchenko-Pastur

text: distribution.

text: The

text: diﬀerences

text: between

text: the

text: limiting

text: distributions

text: allow

text: us

text: to

text: explain

text: from

text: a

text: theoretical

text: point

text: of

text: view

text: the

text: tendency

text: of

text: the

text: LR

text: test

text: for

text: cointegration

text: to

text: severely

text: over-reject

text: the

text: when

text: the

text: dimensionality

text: of

text: the

text: data

text: is

text: relatively

text: large.

text: Furthermore,

text: we

text: derive

text: a

text: simple

text: analytic

text: formula

text: for

text: the

text: Bartlett-type

text: correction

text: factor

text: in

text: systems

text: with

text: relatively

text: large

text: p/T

text: ratio.

text: We

text: propose

text: a

text: quick

text: graphical

text: method,

text: the

text: Wachter

text: plot,

text: for

text: a

text: preliminary

text: analysis

text: of

text: cointegration

text: in

text: large-dimensional

text: systems.

text: The

text: Monte

text: Carlo

text: analysis

text: shows

text: that

text: the

text: quantiles

text: of

text: the

text: Wachter

text: distribution

text: constitute

text: very

text: good

text: center-

text: ing

text: points

text: for

text: the

text: ﬁnite

text: sample

text: distributions

text: of

text: the

text: corresponding

text: squared

text: sample

text: canonical

text: correlations.

text: The

text: quality

text: of

text: the

text: centering

text: is

text: excellent

text: even

text: for

text: such

text: small

text: p

text: and

text: T

text: as

text: p

text: and

text: T

text: However,

text: for

text: such

text: small

text: values

text: of

text: p

text: and

text: T,

text: the

text: empirical

text: distribution

text: of

text: the

text: squared

text: sample

text: canonical

text: correlation

text: can

text: considerably

text: ﬂuctuate

text: around

text: the

text: Wachter

text: limit.

text: As

text: p

text: increases

text: to

text: the

text: ﬂuctuations

text: become

text: numerically

text: very

text: small.

text: Our

text: analysis

text: leaves

text: many

text: open

text: questions.

text: First,

text: it

text: is

text: very

text: important

text: to

text: study

text: the

text: ﬂuctuations

text: of

text: the

text: empirical

text: distribution

text: around

text: the

text: Wachter

text: limit.

text: We

text: con-

text: jecture

text: that

text: linear

text: combinations

text: of

text: reasonably

text: smooth

text: functions

text: of

text: the

text: squared

text: sample

text: canonical

text: correlations,

text: including

text: the

text: λ)

text: used

text: by

text: the

text: LR

text: statistic,

text: will

text: be

text: asymptotically

text: Gaussian

text: after

text: appropriate

text: centering

text: and

text: scaling.

text: The

text: cen-

text: tering

text: can

text: be

text: derived

text: from

text: the

text: results

text: obtained

text: in

text: this

text: paper.

text: A

text: proof

text: of

text: the

text: asymptotic

text: Gaussianity

text: would

text: require

text: diﬀerent

text: methods

text: from

text: those

text: used

text: here.

text: We

text: are

text: currently

text: investigating

text: this

text: research

text: direction.

text: Further,

text: it

text: would

text: be

text: important

text: to

text: remove

text: the

text: Gaussianity

text: assumption

text: on

text: the

text: data.

text: We

text: believe

text: that

text: the

text: existence

text: of

text: the

text: ﬁnite

text: fourth

text: moments

text: is

text: a

text: suﬃcient

text: condition

text: for

text: the

text: validity

text: of

text: the

text: Wachter

text: limit.

text: Next,

text: it

text: would

text: be

text: interesting

text: to

text: study

text: the

text: simultaneous

text: asymptotic

text: behavior

text: of

text: a

text: few

text: of

text: the

text: largest

text: sample

text: canonical

text: correlations.

text: This

text: may

text: lead

text: to

text: a

text: modiﬁcation

text: of

text: Johansen’s

text: maximum

text: eigenvalue

text: test.

text: Another

text: interesting

text: direction

text: of

text: research

text: is

text: to

text: study

text: situations

text: where

text: the

text: num-


text: (cid:16) (cid:17) (cid:0) (cid:1) The relationship zv˜ + sˆ = wˆ is obtained as follows (cid:16) (cid:17) − (cid:48) − − −(cid:0) − (cid:1) T (zv˜ + sˆ) = tr D 1 zI C A 1CD 1 zI 1 + I p p p (cid:16) (cid:17) − − (cid:48) − − (cid:48) − − − − = tr D 1 I + C A 1CD 1 C A 1CD 1 zI 1 + I p p p (cid:0) (cid:1) − − − − (cid:48) − 1 = tr D 1 I DC 1A (C ) 1 z p − (cid:48) − (cid:48) − − − = tr D 1C CD 1C Az 1 CD 1 = T wˆ. The identity zvˆ + s˜ = w˜ is obtained similarly to zv˜ + sˆ = wˆ by interchanging the (cid:48) (cid:3) roles of D, C and A, C . The identities (49) imply the following equality − − (1 z) f (ϕ) + f (ϕ) = (1 z) g + g (ϕ) . 1 2 1 2 We denote the reciprocal of the common value of the right and left hand sides of this equality as hˆ (z,(cid:0)ϕ) . A di(cid:0)rect calcu(cid:1)lation shows that (cid:1) − − − − hˆ (z, ϕ) = (1 z) zv˜vˆ uˆ2 + zv˜ + 4 sin2 ϕ (zvˆ + uˆ 1) 1 , (50) and the asymptotic relationships (44) and (48) can be written in the following (cid:40) (cid:82) (cid:0)(cid:0) (cid:1) (cid:1) form (cid:82) − − mˆ = 1 2π hˆ (z, ϕ) zv˜ 4 sin2 ϕ vˆ uˆ2 dϕ + o(1) 2πc 0 − − . (51) mˆ = 1 2π hˆ (z, ϕ) ((zvˆ 1) v˜ uˆ2) dϕ + o(1) 2πc 0 This can be viewed as an asymptotic system of two equations with four un- knowns: mˆ , v˜, vˆ, and uˆ. We shall now complete the system by establishing the other two asymptotic relationships connecting these unknowns. Multiplying both sides of the identity − − (cid:48) − − MA 1 = CD 1C A 1 zI (52) p − by AM 1, taking trace, dividing by p(cid:2), and rearran(cid:3)ging terms, we obtain 1 − (cid:48) − 1 + zmˆ = tr CD 1C M 1 . (53) p 44 ##LTLine## ##LTLine## ##LTLine##

text: The

text: relationship

text: zv˜

text: sˆ

text: wˆ

text: is

text: obtained

text: as

text: follows

text: T

text: (zv˜

text: sˆ)

text: tr

text: D

text: zI

text: C

text: A

text: zI

text: I

text: p

text: p

text: p

text: tr

text: D

text: I

text: C

text: A

text: C

text: A

text: zI

text: I

text: p

text: p

text: p

text: tr

text: D

text: I

text: DC

text: (C

text: z

text: p

text: tr

text: D

text: CD

text: Az

text: CD

text: T

text: wˆ.

text: The

text: identity

text: zvˆ

text: s˜

text: w˜

text: is

text: obtained

text: similarly

text: to

text: zv˜

text: sˆ

text: wˆ

text: by

text: interchanging

text: the

text: roles

text: of

text: D,

text: C

text: and

text: A,

text: C

text: The

text: identities

text: imply

text: the

text: following

text: equality

text: z)

text: f

text: (ϕ)

text: f

text: (ϕ)

text: z)

text: g

text: g

text: (ϕ)

text: We

text: denote

text: the

text: reciprocal

text: of

text: the

text: common

text: value

text: of

text: the

text: right

text: and

text: left

text: hand

text: sides

text: of

text: this

text: equality

text: as

text: hˆ

text: A

text: shows

text: that

text: hˆ

text: (z,

text: ϕ)

text: z)

text: zv˜vˆ

text: zv˜

text: ϕ

text: (zvˆ

text: uˆ

text: and

text: the

text: asymptotic

text: relationships

text: and

text: can

text: be

text: written

text: in

text: the

text: following

text: form

text: mˆ

text: 2π

text: hˆ

text: (z,

text: ϕ)

text: zv˜

text: ϕ

text: vˆ

text: dϕ

text: mˆ

text: 2π

text: hˆ

text: (z,

text: ϕ)

text: ((zvˆ

text: v˜

text: dϕ

text: This

text: can

text: be

text: viewed

text: as

text: an

text: asymptotic

text: system

text: of

text: two

text: equations

text: with

text: four

text: un-

text: knowns:

text: mˆ

text: v˜,

text: vˆ,

text: and

text: uˆ.

text: We

text: shall

text: now

text: complete

text: the

text: system

text: by

text: establishing

text: the

text: other

text: two

text: asymptotic

text: relationships

text: connecting

text: these

text: unknowns.

text: Multiplying

text: both

text: sides

text: of

text: the

text: identity

text: MA

text: CD

text: A

text: zI

text: p

text: by

text: AM

text: taking

text: trace,

text: dividing

text: by

text: and

text: terms,

text: we

text: obtain

text: zmˆ

text: tr

text: CD

text: M

text: p

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:2) (cid:3) (cid:2) (cid:3) 1 (cid:48) − 1 − (cid:48) − tr C M 1 = tr DD 1C M 1 , (58) p p which yiel(cid:88)ds (cid:2) (cid:3) (cid:88) (cid:2) (cid:3) T/2 T/2 1 (cid:48) − 1 (cid:48) − (cid:48) − tr ∆ ε M 1ε = tr ∆ ε D 1C M 1ε . (59) p 2j (j) (j) p 1j (j) (j) j=1 j=1 Then, we proceed as in the above analysis of (54) and (40) to obtain the remaining asymptotic relationship. The proof of the following proposition is given in the Supplementary Appendix. Proposition 19 There exists ζ > 0 such that, for any z with Rz = 0 and Iz > ζ, (cid:90) we have (cid:0) (cid:1) 1 2π 0 = hˆ (z, ϕ) 4vˆ sin2 ϕ + 2uˆ dϕ + o(1), where (60) 2πc 0 → → ∞ a.s o(1) 0, as p, T . c Summing up the results in Propositions 14, 16, 18, and 19, the unknowns  (cid:82) (cid:0)(cid:0) (cid:1) (cid:1) mˆ , vˆ, v˜, and uˆ must satisfy the following system of asymptotic equations   (cid:82)  − − mˆ = 1 2π hˆ(cid:82)(z, ϕ) zv˜ (cid:0) 4 sin2 ϕ vˆ uˆ2 dϕ(cid:1)+ o(1)  2πc 0 − −  mˆ = 1 (cid:82) 2π hˆ (z, ϕ)(cid:0)((zvˆ 1) v˜ uˆ(cid:1)2) dϕ + o(1) 2πc 0 − . (61) 1 + zmˆ = 1 2π hˆ (z, ϕ) 2uˆ sin2 ϕ + zv˜vˆ uˆ2 dϕ + o(1) 2πc 0 0 = 1 2π hˆ (z, ϕ) 4vˆ sin2 ϕ + 2uˆ dϕ + o(1) 2πc 0 6.1.4 Solving the system Recall that the unknowns mˆ , vˆ, v˜, and uˆ in the asymptotic relationships (61) | | − depend on p, T. The deﬁnition (36) of mˆ implies that mˆ is bounded by (Iz) 1 . Further, as shown in the proof of Proposition 14, uˆ and vˆ are a.s. bounded by absolute value, and it can be similarly shown that v˜ is a.s. bounded by absolute value. Therefore, there exist a subsequence of p, T along which mˆ , vˆ, v˜, and uˆ a.s. converge to some limits m, v, y, and u. These limits must satisfy a non-asymptotic system of equations 46 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: tr

text: C

text: M

text: tr

text: DD

text: M

text: p

text: p

text: which

text: tr

text: ε

text: M

text: 1ε

text: tr

text: ε

text: D

text: M

text: 1ε

text: p

text: (j)

text: (j)

text: p

text: (j)

text: (j)

text: Then,

text: we

text: proceed

text: as

text: in

text: the

text: above

text: analysis

text: of

text: and

text: to

text: obtain

text: the

text: remaining

text: asymptotic

text: relationship.

text: The

text: proof

text: of

text: the

text: following

text: proposition

text: is

text: given

text: in

text: the

text: Supplementary

text: Appendix.

text: Proposition

text: There

text: exists

text: ζ

text: such

text: that,

text: for

text: any

text: z

text: with

text: Rz

text: and

text: Iz

text: ζ,

text: we

text: have

text: 2π

text: hˆ

text: (z,

text: ϕ)

text: ϕ

text: dϕ

text: where

text: a.s

text: as

text: p,

text: T

text: c

text: Summing

text: up

text: the

text: results

text: in

text: Propositions

text: and

text: the

text: unknowns

text: mˆ

text: vˆ,

text: v˜,

text: and

text: uˆ

text: must

text: satisfy

text: the

text: following

text: system

text: of

text: asymptotic

text: equations

text: mˆ

text: 2π

text: ϕ)

text: zv˜

text: ϕ

text: vˆ

text: mˆ

text: 2π

text: hˆ

text: (z,

text: v˜

text: dϕ

text: zmˆ

text: 2π

text: hˆ

text: (z,

text: ϕ)

text: ϕ

text: zv˜vˆ

text: dϕ

text: 2π

text: hˆ

text: (z,

text: ϕ)

text: ϕ

text: dϕ

text: Solving

text: the

text: system

text: Recall

text: that

text: the

text: unknowns

text: mˆ

text: vˆ,

text: v˜,

text: and

text: uˆ

text: in

text: the

text: asymptotic

text: relationships

text: depend

text: on

text: p,

text: T.

text: The

text: deﬁnition

text: of

text: mˆ

text: implies

text: that

text: mˆ

text: is

text: bounded

text: by

text: (Iz)

text: Further,

text: as

text: shown

text: in

text: the

text: proof

text: of

text: Proposition

text: uˆ

text: and

text: vˆ

text: are

text: a.s.

text: bounded

text: by

text: absolute

text: value,

text: and

text: it

text: can

text: be

text: similarly

text: shown

text: that

text: v˜

text: is

text: a.s.

text: bounded

text: by

text: absolute

text: value.

text: Therefore,

text: there

text: exist

text: a

text: subsequence

text: of

text: p,

text: T

text: along

text: which

text: mˆ

text: vˆ,

text: v˜,

text: and

text: uˆ

text: a.s.

text: converge

text: to

text: some

text: limits

text: m,

text: v,

text: y,

text: and

text: u.

text: These

text: limits

text: must

text: satisfy

text: a

text: non-asymptotic

text: system

text: of

text: equations

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Finally, elementary calculations given in the Supplementary Appendix show (cid:18) (cid:90) (cid:19) that 1 2π 1 2 1 dϕ = , (71) 2π x + 2 sin2 ϕ x (x + 2) 0 ∈ C\ − where x [ 2, 0]. Using (71), (65), and the deﬁnition of h (z, ϕ), we obtain the following relationship (cid:18) (cid:19) − 2cv (zv + u 1) 2 − − (72) u (2zv + u) (zv v 1) − 4 (zv + u 1)2 = − − − − − − , u ((1 z) ( 2zv u) 2z) ( u + uz + 2) (u + 2vz 2) that holds as long as − − − − u ((1 z) ( 2zv u) 2z) ∈ C\ − − [ 2, 0] . 2 (zv + u 1) The latter inclusion holds because otherwise h (z, ϕ) is not a bounded function of ϕ, which would contradict Lemma 15. Using (69) in (72), and simplifying, we ﬁnd that there exist only three possi- bilities. Either − 1 v = − , (73) 1 z or − − − − 1 (c + cz 1) v + z (1 z) (1 c) v2 = 0, (74) or c − − − − − (c + cz z) v + z (1 z) (1 c) v2 = 0. (75) 1 c 49 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Finally,

text: elementary

text: calculations

text: given

text: in

text: the

text: Supplementary

text: Appendix

text: show

text: that

text: 2π

text: dϕ

text: 2π

text: x

text: ϕ

text: x

text: (x

text: C\

text: where

text: x

text: Using

text: and

text: the

text: deﬁnition

text: of

text: h

text: (z,

text: ϕ),

text: we

text: obtain

text: the

text: following

text: relationship

text: (zv

text: u

text: u

text: u)

text: (zv

text: v

text: (zv

text: u

text: u

text: z)

text: u)

text: u

text: uz

text: (u

text: that

text: holds

text: as

text: long

text: as

text: u

text: z)

text: u)

text: C\

text: (zv

text: u

text: The

text: latter

text: inclusion

text: holds

text: because

text: otherwise

text: h

text: (z,

text: ϕ)

text: is

text: not

text: a

text: bounded

text: function

text: of

text: ϕ,

text: which

text: would

text: contradict

text: Lemma

text: Using

text: in

text: and

text: simplifying,

text: we

text: ﬁnd

text: that

text: there

text: exist

text: only

text: three

text: possi-

text: bilities.

text: Either

text: v

text: z

text: or

text: (c

text: cz

text: v

text: z

text: z)

text: c)

text: or

text: c

text: (c

text: cz

text: z)

text: v

text: z

text: z)

text: c)

text: c

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: → First, let us show that the weak limit F (λ) of F (λ) as γ 0 exists and equals 0 γ the continuous part of the Marchenko-Pastur distribution with density (21). By (cid:104) (cid:105) deﬁnition and Theorem 1, F (λ) is the (scaled) Wachter d.f. W (γλ; γ/ (1 + γ) , 2γ/ (1 + γ)) . γ Therefore, by (9) and (10), the density, f (λ), and the boundaries of the support, γ ˆb−, ˆb , of the distribution F equ(cid:114)al(cid:16) (cid:17) (cid:16) (cid:17) + γ − − ˆb λ λ ˆb− 1 + γ + fγ(λ) = (cid:16) (cid:112) − (cid:17) , and 2π λ (1 γλ) √ − (cid:0) ∓ (cid:1)− 2 ˆb± = 2 1 γ . √ → → ± As γ 0, ˆb± a±, where a± = 1 2 2 as in (20), and f (λ) converges to γ the density given by (21). This implies the weak convergence of F (λ) to F (λ) γ 0 with F supported on [a−, a ] and having density (21). 0 + → ∞ To establish the theorem, it remains to show that, as p , F ∞(λ) weakly p, converges to F (λ), in probability. Recall that the weak convergence is metrized 0 L · · by the L´evy distance ( , ). We need to show that for any δ > 0, there exists p 0 such that (s.t.) for all p > p , 0 L − Pr ( (F , F ∞) < δ) > 1 δ. (78) 0 p, Let γ > 0 be so small that L (F , F ) < δ/4. (79) 0 γ ≤ For any p, let T be the smallest even integer satisfying p/T γ. That is, γ γ { ≤ } T = min T : p/T γ . γ ∈ Z T 2 For any T∞ > T , by the triangle inequality, we have γ (cid:0) (cid:1) (cid:0) (cid:1) L ≤ L L L L (F , F ∞) (F , F ) + F , F + F , F + (F , F ∞) , (80) 0 p, 0 γ γ p,Tγ p,Tγ p,T∞ p,T∞ p, 51 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: First,

text: let

text: us

text: show

text: that

text: the

text: weak

text: limit

text: F

text: (λ)

text: of

text: F

text: (λ)

text: as

text: γ

text: exists

text: and

text: equals

text: γ

text: the

text: continuous

text: part

text: of

text: the

text: Marchenko-Pastur

text: distribution

text: with

text: density

text: By

text: deﬁnition

text: and

text: Theorem

text: F

text: (λ)

text: is

text: the

text: (scaled)

text: Wachter

text: d.f.

text: W

text: (γλ;

text: γ/

text: γ)

text: γ))

text: γ

text: Therefore,

text: by

text: and

text: the

text: density,

text: f

text: (λ),

text: and

text: the

text: boundaries

text: of

text: the

text: support,

text: γ

text: ˆb−,

text: ˆb

text: of

text: the

text: distribution

text: F

text: γ

text: ˆb

text: λ

text: λ

text: ˆb−

text: γ

text: fγ(λ)

text: and

text: 2π

text: λ

text: γλ)

text: ˆb±

text: γ

text: As

text: γ

text: ˆb±

text: a±,

text: where

text: a±

text: as

text: in

text: and

text: f

text: (λ)

text: converges

text: to

text: γ

text: the

text: density

text: given

text: by

text: This

text: implies

text: the

text: weak

text: convergence

text: of

text: F

text: (λ)

text: to

text: F

text: (λ)

text: γ

text: with

text: F

text: supported

text: on

text: [a−,

text: a

text: and

text: having

text: density

text: To

text: establish

text: the

text: theorem,

text: it

text: remains

text: to

text: show

text: that,

text: as

text: p

text: F

text: ∞(λ)

text: weakly

text: p,

text: converges

text: to

text: F

text: (λ),

text: in

text: probability.

text: Recall

text: that

text: the

text: weak

text: convergence

text: is

text: metrized

text: L

text: by

text: the

text: L´evy

text: distance

text: We

text: need

text: to

text: show

text: that

text: for

text: any

text: δ

text: there

text: exists

text: p

text: such

text: that

text: (s.t.)

text: for

text: all

text: p

text: p

text: L

text: Pr

text: (F

text: F

text: δ)

text: δ.

text: p,

text: Let

text: γ

text: be

text: so

text: small

text: that

text: L

text: (F

text: F

text: γ

text: For

text: any

text: p,

text: let

text: T

text: be

text: the

text: smallest

text: even

text: integer

text: satisfying

text: p/T

text: γ.

text: That

text: is,

text: γ

text: γ

text: T

text: min

text: T

text: p/T

text: γ

text: γ

text: Z

text: T

text: For

text: any

text: T∞

text: T

text: by

text: the

text: triangle

text: inequality,

text: we

text: have

text: γ

text: L

text: L

text: L

text: L

text: L

text: (F

text: F

text: (F

text: F

text: F

text: F

text: F

text: F

text: (F

text: F

text: p,

text: γ

text: γ

text: p,Tγ

text: p,Tγ

text: p,T∞

text: p,T∞

text: p,

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: [35] Yang, Y. and G. Pan (2012) “The convergence of the empirical distribution of canonical correlation coeﬃcients,” Electronic Journal of Probability 64, 1-13. 57

text: Yang,

text: Y.

text: and

text: G.

text: Pan

text: “The

text: convergence

text: of

text: the

text: empirical

text: distribution

text: of

text: canonical

text: correlation

text: coeﬃcients,”

text: Electronic

text: Journal

text: of

text: Probability


text: Optimal Rates for Multi-pass Stochastic Gradient Methods ∗ Junhong Lin junhong.lin@iit.it Laboratory for Computational and Statistical Learning Istituto Italiano di Tecnologia and Massachusetts Institute of Technology Bldg. 46-5155, 77 Massachusetts Avenue, Cambridge, MA 02139, USA Lorenzo Rosasco lrosasco@mit.edu DIBRIS, Universit`a di Genova Via Dodecaneso, 35 — 16146 Genova, Italy Laboratory for Computational and Statistical Learning Istituto Italiano di Tecnologia and Massachusetts Institute of Technology Bldg. 46-5155, 77 Massachusetts Avenue, Cambridge, MA 02139, USA Abstract We analyze the learning properties of the stochastic gradient method when multiple passes over the data and mini-batches are allowed. We study how regularization properties are controlled by the step-size, the number of passes and the mini-batch size. In particular, we consider the square loss and show that for a universal step-size choice, the number of passes acts as a regularization parameter, and optimal ﬁnite sample bounds can be achieved by early-stopping. Moreover, we show that larger step-sizes are allowed when considering mini-batches. Our analysis is based on a unifying approach, encompassing both batch and stochastic gradient methods as special cases. As a byproduct, we derive optimal convergence results for batch gradient methods (even in the non-attainable cases). 1. Introduction Modern machine learning applications require computational approaches that are at the same time statistically accurate and numerically eﬃcient (Bousquet and Bottou, 2008). This has motivated a recent interest in stochastic gradient methods (SGM), since on the one hand they enjoy good practical performances, especially in large scale scenarios, and on the other hand they are amenable to theoretical studies. In particular, unlike other learning approaches, such as empirical risk minimization or Tikhonov regularization, theoretical results on SGM naturally integrate statistical and computational aspects. Most generalization studies on SGM consider the case where only one pass over the data is allowed and the step-size is appropriately chosen, see (Cesa-Bianchi et al., 2004; Nemirovski et al., 2009; Ying and Pontil, 2008; Tarres and Yao, 2014; Dieuleveut and Bach, 2016; Orabona, 2014) and references therein, possibly considering averaging (Poljak, 1987). In particular, recent works show how the step-size can be seen to play the role of a regularization parameter whose choice controls the bias and variance properties of the obtained solution (Ying and Pontil, 2008; Tarres and Yao, 2014; Dieuleveut and Bach, 2016; Lin et al., 2016a). These latter works show that balancing these contributions, it is possible to derive a step-size choice leading to optimal learning bounds. Such a choice typically depends on some unknown properties of the data generating distributions and it can be chosen by cross-validation in practice. ∗ . J.L. is now with the Laboratory for Information and Inference Systems, E´cole Polytechnique F´ed´erale de Lausanne, Lausanne 1015, Switzerland. (jhlin5@hotmail.com) 1 ##LTLine##

text: Optimal

text: Rates

text: for

text: Multi-pass

text: Stochastic

text: Gradient

text: Methods

text: Junhong

text: Lin

text: junhong.lin@iit.it

text: Laboratory

text: for

text: Computational

text: and

text: Statistical

text: Learning

text: Istituto

text: Italiano

text: di

text: Tecnologia

text: and

text: Massachusetts

text: Institute

text: of

text: Technology

text: Bldg.

text: Massachusetts

text: Avenue,

text: Cambridge,

text: MA

text: USA

text: Lorenzo

text: Rosasco

text: lrosasco@mit.edu

text: DIBRIS,

text: Universit`a

text: di

text: Genova

text: Via

text: Dodecaneso,

text: Genova,

text: Italy

text: Laboratory

text: for

text: Computational

text: and

text: Statistical

text: Learning

text: Istituto

text: Italiano

text: di

text: Tecnologia

text: and

text: Massachusetts

text: Institute

text: of

text: Technology

text: Bldg.

text: Massachusetts

text: Avenue,

text: Cambridge,

text: MA

text: USA

text: Abstract

text: We

text: analyze

text: the

text: learning

text: properties

text: of

text: the

text: stochastic

text: gradient

text: method

text: when

text: multiple

text: passes

text: over

text: the

text: data

text: and

text: mini-batches

text: are

text: allowed.

text: We

text: study

text: how

text: regularization

text: properties

text: are

text: controlled

text: by

text: the

text: step-size,

text: the

text: number

text: of

text: passes

text: and

text: the

text: mini-batch

text: size.

text: In

text: particular,

text: we

text: consider

text: the

text: square

text: loss

text: and

text: show

text: that

text: for

text: a

text: universal

text: step-size

text: choice,

text: the

text: number

text: of

text: passes

text: acts

text: as

text: a

text: regularization

text: parameter,

text: and

text: optimal

text: ﬁnite

text: sample

text: bounds

text: can

text: be

text: achieved

text: by

text: early-stopping.

text: Moreover,

text: we

text: show

text: that

text: larger

text: step-sizes

text: are

text: allowed

text: when

text: considering

text: mini-batches.

text: Our

text: analysis

text: is

text: based

text: on

text: a

text: unifying

text: approach,

text: encompassing

text: both

text: batch

text: and

text: stochastic

text: gradient

text: methods

text: as

text: special

text: cases.

text: As

text: a

text: byproduct,

text: we

text: derive

text: optimal

text: convergence

text: results

text: for

text: batch

text: gradient

text: methods

text: (even

text: in

text: the

text: non-attainable

text: cases).

text: Introduction

text: Modern

text: machine

text: learning

text: applications

text: require

text: computational

text: approaches

text: that

text: are

text: at

text: the

text: same

text: time

text: statistically

text: accurate

text: and

text: numerically

text: eﬃcient

text: (Bousquet

text: and

text: Bottou,

text: This

text: has

text: motivated

text: a

text: recent

text: interest

text: in

text: stochastic

text: gradient

text: methods

text: (SGM),

text: since

text: on

text: the

text: one

text: hand

text: they

text: enjoy

text: good

text: practical

text: performances,

text: especially

text: in

text: large

text: scale

text: scenarios,

text: and

text: on

text: the

text: other

text: hand

text: they

text: are

text: amenable

text: to

text: theoretical

text: studies.

text: In

text: particular,

text: unlike

text: other

text: learning

text: approaches,

text: such

text: as

text: empirical

text: risk

text: minimization

text: or

text: Tikhonov

text: regularization,

text: theoretical

text: results

text: on

text: SGM

text: naturally

text: integrate

text: statistical

text: and

text: computational

text: aspects.

text: Most

text: generalization

text: studies

text: on

text: SGM

text: consider

text: the

text: case

text: where

text: only

text: one

text: pass

text: over

text: the

text: data

text: is

text: allowed

text: and

text: the

text: step-size

text: is

text: appropriately

text: chosen,

text: see

text: (Cesa-Bianchi

text: et

text: al.,

text: Nemirovski

text: et

text: al.,

text: Ying

text: and

text: Pontil,

text: Tarres

text: and

text: Yao,

text: Dieuleveut

text: and

text: Bach,

text: Orabona,

text: and

text: references

text: therein,

text: possibly

text: considering

text: averaging

text: (Poljak,

text: In

text: particular,

text: recent

text: works

text: show

text: how

text: the

text: step-size

text: can

text: be

text: seen

text: to

text: play

text: the

text: role

text: of

text: a

text: regularization

text: parameter

text: whose

text: choice

text: controls

text: the

text: bias

text: and

text: variance

text: properties

text: of

text: the

text: obtained

text: solution

text: (Ying

text: and

text: Pontil,

text: Tarres

text: and

text: Yao,

text: Dieuleveut

text: and

text: Bach,

text: Lin

text: et

text: al.,

text: These

text: latter

text: works

text: show

text: that

text: balancing

text: these

text: contributions,

text: it

text: is

text: possible

text: to

text: derive

text: a

text: step-size

text: choice

text: leading

text: to

text: optimal

text: learning

text: bounds.

text: Such

text: a

text: choice

text: typically

text: depends

text: on

text: some

text: unknown

text: properties

text: of

text: the

text: data

text: generating

text: distributions

text: and

text: it

text: can

text: be

text: chosen

text: by

text: cross-validation

text: in

text: practice.

text: J.L.

text: is

text: now

text: with

text: the

text: Laboratory

text: for

text: Information

text: and

text: Inference

text: Systems,

text: E´cole

text: Polytechnique

text: F´ed´erale

text: de

text: Lausanne,

text: Lausanne

text: Switzerland.

text: ##LTLine##