The generalized fluctuation test : a unifying view

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The Generalized Fluctuation Test:
A Unifying View

Chung-Ming Kuan

Department of Economics

University of Illinois

Kurt Hornik

Institut fUr Statistik und
Wahrscheinlichkeitstheorie
Techniscbe Universitat Wien

Bureau of Economic and Business Research

College of Commerce and Business Administration

University of Illinois at Urbana-Champaign

BEBR

FACGLTY WORKING PAPER NO. 93-0154

College of Commerce and Business Administration

University of Illinois at Grbana-Champaign

August 1993

The Generalized Fluctuation Test:
A Unifying View

Chung-Ming Kuan
Kurt Hornik

THE GENERALIZED FLUCTUATION TEST:
A UNIFYING VIEW

Chung-Ming Kuan

Department of Economics
University of Illinois at Urbana- Champaign

Kurt Hornik

Institut fiir Statistik und Wahrscheinlichkeitstheorie
Technische Universitat Wien

August 17, 1993

t Chung-Ming Kuan thanks the College of Commerce and Business Administration of the University of
Illinois for research support. This paper was presented in the 1993 North American Summer Meeting of
the Econometric Society.

Abstract

In this paper a general principle of constructing tests for parameter constancy without
assuming a specific alternative is introduced. A unified asymptotic result is established
to analyze this class of tests. As applications, tests based on the range of recursive and
moving estimates are also considered, and their asymptotic distributions are characterized
analytically. Our simulations show that different tests have quite different behavior under
various alternatives and that no test uniformly dominates the other tests.

JEL Classification Number: 211

Keywords: CUSUM, MOSUM, Brownian bridge, functional central limit theorem, gen-
eralized fluctuation test, moving estimate, moving-estimates test, range test, recursive
estimate, recursive-estimates test, structural change, Wiener process.

1 Introduction

The topic of testing the goodness-of-fit of a probability model has a long history in the sta-
tistical literature, of which tests for the constancy of a mean function are a special case. In
the linear regression context, this type of tests reduces to tests for constant regression co-
efficients. It is quite typical to construct tests against certain specific alternatives ba^ed on
a prior belief. A popular aJternative is a one-time structural change at known or unknown
change point, e.g.. Chow (1960), Quandt (1960), Hawkins (1987), and Andrews (1993).
This alternative is convenient for deriving tests but may not describe many interesting
phenomena, however. In the study of business cycle, for example, it is not uncommon to
believe that a downswing of major aggregates takes place suddenly (Hicks (1950)), but
there do not exist similar abrupt changes when the economy moves to a upturn period
(e.g., Neftci (1979)). Another popular alternative is that parameters foUow a random
walk (or a martingale), e.g., Cooley & Prescott (1976), Lamotte & McWhorter (1978),
Leybourne & McCabe (1989), and Nyblom (1989). This alternative is also somewhat re-
strictive. For example, suppose that a policy causes the economy shifting to a new regime,
either suddenly or gradually, it is quite likely that, when rational expectation prevails, the
economy will be returning to, instead of drifting away from, a "normal" level.

The specific tests can be extended in different ways. Andrews & Ploberger (1992)
introduce a clztss of optimal tests against multiple structural changes. Another strategy is
to construct tests without bearing any specific alternatives in mind. As one rarely knows
how regression coefficients evolve over time, it would be desirable to construct tests with
power against all possible mean functions. This class of tests is our primary interest in this
paper, which includes estimates-based tests, such as the recursive-estimates (RE) test, also
known as the fluctuation test, of Sen (1980) and Ploberger, Kramer, & Kontrus (1989),
and the class of moving-estimates (ME) tests of Chu, Hornik, & Kuan (1992a), as spe-
cial cases. The well known residual-batsed tests, such as the CUSUM tests of Brown,
Durbin, Sz Evans (1975) and Ploberger & Kramer (1992) and the class of MOSUM tests of
Bauer & Hackl (1978) and Chu, Hornik & Kuan (1992b), also belong to this class. Note,
however, that the class of ME (MOSUM) tests differs from the RE (CUSUM) test in an
important respect. Moving estimates (or moving sums of residuals) can be interpreted as
non-parametric estimates of corresponding mean functions, whereas recursive estimates
(or cumulated sums of residuals) do not have similar interpretation.

On the other hand, we observe that a common feature of the above "general" tests
is that they are based on empirical processes consisting of two additive components, one

satisfying a functional central limit theorem and one that is roughly a "straight line" under
the null hypothesis. By suitable construction, this straight line component can be elimi-
nated, for example, by applying a linear operator annihilating the straight line, so that the
resulting empirical process under the null is essentially governed by the functional central
limit theorem. Under the alternative, however, this empirical process will "fluctuate", in
the sense that its behavior is not completely characterized by the functional central limit
theorem. A test for parameter constancy can then be obtained by assigning an appropri-
ate functional to measure the "fluctuation" of the empirical process; the nuU hypothesis
is rejected if this process fluctuates too much. This class of tests will be referred to as the
generalized fluctuation test. It includes the RE, ME, CUSUM, and MOSUM tests as spe-
cial cases. Clearly, numerous tests can be constructed according to this general principle.
As their power properties under different alternatives are far from obvious, it is extremely
interesting to find out, by simulations, which combination of functional and operator can
deliver "better" power results.

In this paper we first establish an asymptotic result for the generalized fluctuation test
that can be written as A(£7'yj), where A is a functional and Ct is an operator annihilating
the straight line component of an empirical process Yt, from which many known results can
be derived ats corollaries. Our result greatly facilitates the analysis of these tests under the
null and alternatives. In particular, we also consider tests based on the range functional,
instead of the majcimal functional typically adopted in existing tests. Specifically, the
range of recursive-estimates (RR) and moving-estimates (RM) tests are investigated. The
asymptotic null distribution of the RR test is well known in literature, but that of the
RM tests is unknown. For certain bandwidths of moving windows, we derive a formula
representing the asymptotic distribution of the RM test, from which critical values can
be easily calculated; for other bandwidths of moving windows, critical values of the RM
tests are obtained by simulations. Power simulations are also conducted to compare the
performance of different tests.

This paper is organized as follows. We introduce the generalized fluctuation test in
a simple location model and provide a unified asymptotic result in section 2. We then
introduce range tests and derive their asymptotic null distributions in section 3. These
results are extended to multiple regression in section 4. Power performance and simulation
results are reported in section 5. Section 6 concludes the paper. Applications of the general
result to known tests and mathematical proofs are summarized in the Appendix.

2 The Generalized Fluctuation Test

To illustrate the idea of a general class of tests for parameter constancy, first consider the
data generating process (DGP):

Vi = Z^t + e,-, i = 1,...,T, (1)

where {c,} is a sequence of i.i.d. random variables with mean zero and variance one. It is
well known that €, satisfy a functional central limit theorem (FCLT):

as T ^^ GO, where [Tt] is the integer part ofTt, => denotes weak convergence of associated
probability measures, and ly is a standard Wiener process. The null hypothesis of interest
is fii = /zo for all i. In what follows, a function / is either in C[0, r], the space of continuous
functions on [0,r], or in Z)[0,r], the space of functions that are right continuous with left-
hand limits on [0,r]. We always assume that the space C is endowed with the uniform
topology and that the space D is endowed with the Skorohod topology. For more details
about the spaces C and D we refer to Billingsley (1968). We also let -^^ denote convergence
in probability, and ='^ denote equality in distribution.

Consider the piecewise constant process Yt on [0, 1] with jump points

yr{f) = ^Zy- (3)

Under the null hypothesis,

Yrit) = %/T/xoIp + ET{t\ (4)

where Et is also a piecewise constant process with jump points

Observe that, apart from the factor T'/^, the first term in (4) is roughly a "straight line"
passing through the origin and that the second term satisfies the FCLT (2). When the
straight line component is removed, the resulting empirical process is well behaved by the
FCLT under the null hypothesis. If the null hypothesis is false, this empirical process
will fluctuate, in the sense that its behavior is not completely characterized by the FCLT.

Hence, a test can be constructed by evaluating the fluctuation of an empirical process.
This is the underlying idea of the generalized fluctuation (GF) test.

To fix the idea, consider the GF test that can be written as X{CTyT)i where Ct is a
linear operator in D which annihilates the straight line of (4), i.e., CtYt = J^tEt, and A
is a functional measuring the fluctuation of CtYt- If CtEt = CEj + Op(l), then under
the null, Cxyr => CW. When the null hypothesis is false, the deterministic component of
Yt is not a straight line so that CtYt = CtEt + something. For example, the operator
Ct such that for / in D[Q, 1]

CTfit) = /(0-^/(i)

eliminates the straight line component of Yr under the nuU. It follows that
CtYt = CtEt => CW,

where Cf{t) = f{t) — tf{l)- This class of tests includes many well known tests as special
CcLses, as the examples below show.

In what follows, for functions / in D[0, 1], let

max(/; r) = max fit), min(/;r) = min f(t),

be the maximum and minimum of / on [0, r], and let

range(/;r) = max(/; r) - min(/; r). (5)

be the range of / on [0,r]. Finally, we write p for the function f{i) - f{t) - //(I) such
that W° is the familiar Brownian bridge ("tied-down Wiener process").

Example I. Estimates-Based Tests:

1. The RE test: Sen (1980), Ploberger, Kramer, k Kontrus (1989).

Let recursive estimates of fxo be fik = k~^ J2t-\ 2/t, ^ = 1,...,T. The RE test is
based on the fluctuation of recursive estimates in terms of the deviations p-k — p-T-

^^ = .=T^t;^i^'=-^^' = k^i^jif

t=l t=i

(6)

Hence, REt = maxd^TVrl; 1) with

2. The ME test: Chu, Hornik, k Kuan (1992a).

Let moving estimates of fio be /ijt./i = [Th]~^ ^i=k+i Vi, k = 0,. . .,T - [Th], where
[Th] is the bandwidth of moving windows and 0 < h < 1. The ME test is ba^ed on
the fluctuation of moving estimates in terms of the deviations /ijt,/i — P'T'

MET,h = rmx

k=0,-,T-[Th] y/T

1

max —p=
k=0,-,T-[Th] vT

\fik,h - AtI
k+[Th]

i=k+l

I^E.

t=i

max

0<t<l-/lT

^,(M^)_>.,(M)_M^,(i

(8)

T J '' \ T J T
Straightforward rescaling shows that MEj^h — maxd^T^/iVrl; 1 — /i) with

>Ct,/./(0 = /(KT(0 + M-/(MO)-/^r/(i)

= /O(kt(0 + /^t)-/°(kt(0), (9)

where hr = [Th]/T, kt(0 = [NTt]/T, Nt = {T - [T/i])/(l - h).

Example II. Residual- Ba^ed Test:

1. The Recursive-CUSUM test: Brown, Durbin, & Evans (1975).

The recursive residuals are u, = y, — /i,_i, t = 2, . . .,T. The Recursive-CUSUM test

is ba^ed on the fluctuation of cumulated sums of recursive residuals:

k

E-.

QSr = max —7=

^ k=2,-,T y/T

1

= max —==

k=2,-,T y/T

t=2
k

t-1

It is readily seen that QSj- = maxd^rVVl; 1) with

'^rm = m-[^dr. (H)

2. The OLS-CUSUM test: Ploberger k Kramer (1992).

Let e, = T/, — Atj ^ = Ij-'-j^i be OLS residuals. Analogous to the Recursive-
CUSUM test, the OLS-CUSUM test is based on the fluctuation of cumulated sums
of OLS residuals:

1

QSr = max _

-* k=\,...,T y/T

»=i

1

= max — F=
k=\-J y/T

Y^V^-tY. 2/'

t=i

1=1

(12)

Clearly, QS°t = REt, cf. (6).

3. The Recursive-MOSUM test: Bauer & Hackl (1978), Chu, Hornik, & Kuan (1992b).
In contrast with the CUSUM-type of test, the Recursive-MOSUM test is based on
moving sums (with bandwidth [Th], 0 < /i < 1) of recursive residuals. Letting
T' = T — 1, the statistic is

MSt k = max — =

k=o,-,T'-[T'h] ^/T

1

max —=
k=0,-,T'-[T'h] VT

k+l+[T'h]

t = A:+2
k+l + [T'h] / ^ ,_i

(13)

t=Jt-|-2 \ ■ " j = l

In view of (8)-(ll), we can write MSj-f^ = m3ix(\ CT,hyT\'i I — h) with

(0 iTVJTr

CT,km = f{KT'{t) + hT')-fiKT'it))- H'

dr.

(14)

4. The OLS-MOSUM test: Chu, Hornik & Kuan (1992b).

Analogous to the Recursive-MOSUM test, the OLS-MOSUM test is based on moving
sums of OLS residuals:

MS J- f^ = max

1

k+[Th]

k=0,-,T-[Th] y/T

Clearly, MS^^^ = MET,k, cf. (8) and (9)

(15)

The tests above apply different operators to remove the straight line component but
adopt the same maximal functional to evaluate the fluctuation of empirical processes.
It is clear that numerous tests can be constructed by choosing different combinations of
functional and annihilators. For example, by applying the functional max(/; r) we obtain
one-sided tests in the above examples, and by applying the range functional range(/; 1) we
obtain range tests which will be discussed in details in next section. Therefore, a unified
asymptotic result can facilitate the analysis of this class of tests.

More precisely, we assume the following conditions.
[Gl] Ct and £ are linear operators from /^[0, 1] to £)[0,r] such that Cti-t = Oi where

LT{t) = [Tt]/T.

[G2] A is a positively homogeneous functional on Z)[0,r] which is continuous with respect
to the Skorohod topology, i.e., /t -* / in the Skorohod metric implies A(/r) — >• A(/).

In what follows the function J/, the anti-derivative of /, is defined as

J fit) = f'f{u)du,
Jo

and the function Ahf is defined by A/i/(f) = f{t + h) - f(t) (for h = 1 we simply write
Ai = A). We then have the following.

Theorem 2.1 Given the DGP (1), suppose that

tii = fio + T-^gii/T), (16)

where 6 < 1/2 and g is a function of bounded variation on [0, 1]. // [Gl] and [G2] hold
with CtYt = CYt + Op(l), then for S = 1/2,

\{CTYT)=>X{C{W + Jg));

for 8 < 1/2,

T'-'/^'XiCTYT) -" KC{Jg)).

Under the null hypothesis, g is identically zero so that this class of tests converges in
distribution to X(CW). The first result indicates that under local alternatives of order
T"^/^, A(£7'y7') has non-trivial local power, provided that CJg ^ 0; the second conclusion
says that the OF test diverges whenever \{CJg) > 0, hence are consistent against the claiss
of alternatives (16) with 6 < 1/2. Note that the term Jg characterizes the deviation of
the limiting process under the alternative from the limiting process under the null. Note
also that negative values of S are allowed. Applying this theorem to tests discussed above
we immediately obtain many known results in literature as special cases; these results are
summarized in the Appendix.

3 Range Tests

We have noted that a typical choice in the existing OF tests is the maximal functional.
Other choices are possible; for example, the integral functional is used in the Cramer-
von Mises test, and the weighted integral functional is used in the Anderson-Darling test.
Following Feller (1951), we consider the range functional (5). Specifically, we consider the
RR (range of recursive estimates) test:

k t

RRt = ^majc^--^(/i^ - /ir) - ^jnin^ -^(/i^ - fij), (17)

and the RM (range of moving estimates) test:

[Th],. . , . [Th],.

RMt = max — ^=(/za:,a - /^t) - mm —p^i^irh-^iT)- (18)

That is, the RR and RM tests are based on the largest possible difference between the
deviations /i^ — /xt a^nd /ijt,/v — /tTi respectively. Intuitively, the range functional can better
pick up smaller fluctuations of a process which changes its signs, e.g., if g{t) = sin(27r<),
max(l^l) = 1, but range(^) = 2. Note that there is little problem of constructing tests
with correct asymptotic size based on either the range or maximal functional. What
matters is the behavior of tests under various alternatives. Comparison of tests is done
by simulations and will be discussed in section 5.

It is ea^y to see from (17) and (18) that

RRt = range(£TVT;l) = range(y;^; 1),
RMT,h = r^nge{CT,hY^\l-h) = range(A/,^y:^; 1 - /i),

where Ct and CT,h are defined in (7) and (9), respectively. We then obtain from Theo-
rem 2.1 that:

Theorem 3.1 Given the DGP (1) with (16), suppose that the FCLT (2) hold. Then for
S = 1/2, we have

RRt => T3inge{C{W + Jg);l),
RMT,h => range(£,(W^o + Jy); 1 - /i);
for6< 1/2,

T^-^I''RRt -p range(/:(Jy);l),
T^-'l''RMT,k -" range(£;,( Jy); 1 - /i).

where C and Ch are such that Cf{t) = f{t) - tf{l) and Chf{t) = Akfit).

Under the null hypothesis, we thus have

RRt => range(VF<';l),
RMT,h => range(A/,Vy"; 1 - h).

It is noted in Chu, Hornik, & Kuan (1992a) that, if g is periodic with period h and if l/h
is an integer, then jChJg = 0. Consequently, the RM test has only trivial power (or is

inconsistent) for local (or non-locaJ) alternatives with this type of g function. As far as
the asymptotic null distribution is concerned, it is well known that (see e.g., Shorack &
WeUner (1986, p. 142)),

IP{range(W^; 1) < 5} = 1 - 2 J^iAkh"^ - l)e-^^'\ (19)

k=i

which is the distribution of the Kuiper (1960) statistic. A detailed table of this distribution
can be found in Shorack & Wellner (1986, p. 144). We note that this distribution can be
easily derived from Equation (4.3) of Feller (1951), cf. Dudley (1976). The distribution of
the range of A/iVF*^ on [0, 1 — /i] is unknown, but for 1/2 < /i < 1 it can be represented in
terms of the range of a Wiener process on [0, 1], as shown in the following theorem.

Theorem 3.2 For 1/2 < h < I,

Ta.nge{AhW°; I - h) ='^ y^2(l - h) range(iy; 1).

Let (f> and $ denote the density and distribution functions of the standard normal
random variable, respectively. Feller (1951) shows that the density of range(iy; 1) at

ly > 0 is

00
Sj^i-l)''-^ k^(f>(kw).
Jk=i

It follows that

IP{range(iy;l) < 5}

/oo °°
J2{-l)''-^k^<f){kw)dw

- k=\
00

= \-^Y^{-\f-'^k^{-ks).
k=\

With a little more effort we obtain an equivalent series representation of this probability.
Corollary 3.3 Under the null hypothesis, for 5 > 0 and h > 1/2,

lim JP{RMT,h < \/2(l - h)s}
T— ►00 ' *

00
= l-8^(-l)'=-^A:4»(-A:5)

k=i

00

,ttV(2;-l)

The asymptotic critical values for the RR and RM tests with h > 1/2 can then be solved
from the formulae above. Table 1 summarizes some of these critical values; the critical
values of RMT,h with h > 1/2 are not included because they are those of RM t,i/2 times
(2(1 — h)y/^. Asymptotic critical values of the RM test with h < 1/2 can be obtained
by simulating the behavior of A^W^ on [0, 1 — h]. Simulated critical values for various h
based on a sample of 2000 are summarized in Table 2. Note that the simulated critical
values for the RM test with h = 1/2 are quite close to those in Table 1. Using a larger
sample of 3000 or 5000 only results in a slight improvement, however.

4 Extension to Multiple Regression

The general approach of Section 2 can be extended to multiple regression models. Consider
now the DGP:

y, = x\0, + €„ i=l,---,T, (20)

where x, is the n x 1 vector of explanatory variables. The null hypothesis is /?, = Po for
all i. Following Kramer, Ploberger, & Alt (1988), we assume:

[Ml] {(i) is a martingale difference sequence with respect to {^*}, the cr-algebra gener-
ated by {{xt+uU),t < i} such that IE(€^|/"'-') = a^.

[M2] {x.} is such that limsupj^^T-^ ^J^^ IE|x.f+^ < oo, and

1 [Tt]

Q[Tt] = {ttiE^.^: --' ^' (21)

uniformly inc<<< l,c>0, where Q is a non-stochastic, positive definite matrix.
Under these conditions, if a^ is a consistent estimator for cr^, we have

/ 1 [Tt] \

l—i—Q-'/'J^x,u, 0<t<l\=>W, (22)

where W is an n-dimensional, standard Wiener process. We also let W^ denote the
n-dimensional Brownian bridge.

Define now the piecewise constant process Yt on [0, 1] with jump points:

10

so that under the null hypothesis

\Tt]

The first term on the right-hand side is the "straight line" component to be removed by
an operator Ct\ the second term is the component satisfying the FCLT. In the present
context, <T^ and Q must be estimated suitably to ensure proper FCLT effect. Now Ct and
C in [Gl] are linear operators from D[0, 1]" to D[0,r]'*, and A in [G2] is a functional in
D[0, 1]''. For / in D[0, l]"" with elements /,, define

range(/; r) = max (max(/,; r) - min(/,; r)),

t=l,---,n

and let ||.|| denote the maximal norm.
Let the recursive OLS estimates be

\t=i / t=i
and the moving OLS estimates be

/h+[Th] \ -^ k+[Th]

Pk,h=l E Xix'A ^ x,y., k = Q,---,T -[Th].

It can be easily verified that

REt = m^x -^\\Q}/\0,-M\\ = max(||£TrT||;l)

k=n,-,T arVT

with Ct defined in (7) and that

MET,k = ^^^max^^^^H||D^^/^4A-/3T)|| = max(||£T,/.>T||; 1 - /i)
with CT,h defined in (9). We also have

RRt = max f max^-^[Z)-'/'(/3fc - /3t)]. -

t=l, •••,n \k=n,--,T \J^

,™n^^[6;"'(A-/3r)].)
= range(£Tl'r;l) (24)

^^•* = .=r.'!„ (*=o,"'.f-tr., 7f 1^t"\/^M - ih)\, -

= range(£T,/.>T; 1 - /i). (25)

11

Let CTj- = T ^ J2i=i{yi ~ ^[Pt)^ be the estimate of a^. Then under the alternative

A = (3o + T-'g{i/T), (26)

where S < 1/2 and ^ is a vector-valued function of bounded variation on [0, 1], we have
a J' — >P aj, where

al = h'' , , , , 0<<5<i,

\ <^^ + /o (di^) - /o ^(") ^^) Q {aiu) - Jo g{v) dv^ du, 6 = 0;
see e.g., Chu, Hornik, & Kuan (1992a). The result below is an extension of Theorem 2.1:

Theorem 4.1 Given the DGP (20) with (26), suppose that [Ml] and [M2] hold. If CtYt =
CYt + Op(l) for some C, then for 6 = 1/2,

X{CtYt) =^ X{C{W + a-^Q^I'^Jg));

forS < 1/2,

T'-'/^X{CtYt) -" X(C{a^'Q'/'Jg)).

It is now straightforward to verify that Theorem 2 of Ploberger, Kramer, & Kontrus (1989)
and Corollary 4.4 of Chu, Hornik, & Kuan (1992a) can be obtained from this theorem.
For range tests we have, analogous to Theorem 3.1:

Corollary 4.2 Given the DGP (20) with (26), suppose that the conditions [Ml] and [M2]
hold. Then for 6 = 1/2, we have

RRt => range(£(Vy + <7-^Q^/V^);l),
RMT,h ^ range(A(W° + (T-iQi/V^);l-/i);

for 8 <II2,

T'-'/^RRt -" range(£(a,-igi/2j^).l)^
T'-'/'RMT,h -" range(A(a7^g^/V5);l-/i),
where C and Ch are such that Cf{t) = f{t) - tf{\) and Chf{t) = Ahf°{t).
Corollary 4.2 implies that under the null hypothesis,

RRt =^ range(VyO;l),
IiMT,h => range(A;,VyO; 1 _ /i).

Then by (19) and Corollary 3.3, we have the following distributions.

12

Corollary 4.3 Under the null hypothesis, for 5 > 0,

lim W{RRt <s} = 1-2 y2{4k^s^ - l)e
and for h > 1/2,

-2k

2,2

lim JP {RMT,h < \/2(l -h)s}
"—►00 ' "

T— ►00

= (l-SY^i-l)^-'^k^{-ks)

k=i

Simulated critical values of the RM test with various h and n up to 5 are summarized in
Table 2. Other critical values for n = 6, . . ., 10 are available upon request.

For residual-based tests, consider the empirical process Yt with jump points:

Yrik/T) = T^J^y.

It is readily seen that the straight line component of VV can be removed exactly if x-/3fc_i or
x[Pt is subtracted from y,. Hence, the CUSUM- and MOSUM-type of tests are GF tests.
Additional structures are needed to incorporate residual-based tests into the functional-
operator framework, however. To reduce excessive notations, we do not pursue this pos-
sibility here.

5 Simulations

In this section we evaluate finite-sample performance of different tests by simulations. Size
simulations are based on the location model

Vt = 2 + U,

where tj are i.i.d. A''(0, 1). We consider the RR test and RM tests with h = 0.1,..., 0.5
and samples T = 100, 200, 300, and 500. The number of replications is 10000. These
results are summarized in Table 3. It can be seen that all tests are conservative but not
very different from nominal sizes; in particular, the RR test has the largest size distortion
in different finite samples, and the RM tests with smaller window bandwidth h has larger
size distortion.

In power simulations competing tests we consider are the ME, RE, MAX-F (An-
drews (1993)), AVG-F and EXP-F (Andrews k Ploberger (1992) and Andrews, Lee, k

13

Ploberger (1992)) tests. Note that the AVG-F and EXP-F tests are optimal in the sense
of Andrews & Ploberger (1992). For moving-estimates based tests, we compute tests
with h = 0.1, 0.2 and 0.5. All power results are based on empirical critical values simu-
lated from a sample of 100 observations with 10000 replications. In what follows we shall
write moving-estimates based tests as ME(/i) or RM(/i). The empirical critical values
are RM(0.1) = 1.602, RM(0.2) = 2.005, RM(0.5)=2.065, ME(0.1)=0.910, ME(0.2)=1.149,
ME(0.5)=1.289, RR=1.472, RE=1.176. The MAX-F, AVG-F and EXP-F tests are com-
puted specifically for the alternative in simulations. For the alternative of a single struc-
tural change:

\ 2 + A + u, i = [rA]+l,---,T,

empirical critical values are MAX-F=7.328, AVG-F=2.157, EXP-F=1.60, which are com-
puted for treating each observation [Ts], s G [0.1,0.9], as a hypothetical change point. For
the alternative of double structural changes:

' 2 + f„ i= l,---,[TAi],

y^ = I 2 + A, + e„ z = [TAi] + 1, • • •, [TA^], (28)

^ 2-h A2 + 6., t = [TA2]+l,---,T,

empirical critical values are MAX-F=5.718, AVG-F= 1.861, EXP-F=2.756, which are
computed by treating each pair of observations ([T^i], [T52]), ^1 € [0.1,0.85] and 52 =
S\ + 0.05, .. .,0.9, as a pair of two change points. Note that the trimming of observa-
tions is arbitrary; see Andrews (1993), Andrews & Ploberger (1992), and Andrews, Lee &
Ploberger (1992).

For the alternative of a single structural change (27), we consider two cases: A =
0.5 and 0.25. The number of replications is 5000. Because these tests have symmetric
performance, we only report results for A = 0.1, . . .,0.5 in Table 4. We can ignore the ME
tests in this case because Chu, Hornik, & Kuan (1992a) have shown that under a single
change the RE test dominates the ME test for every possible change point. We observe
from Table 4 A that:

1. A = 0.1, the MAX-F test is the best;

2. A = 0.2, the AVG-F and EXP-F tests are the best;

3. A = 0.3, the RE, AVG-F and EXP-F tests are the best;

4. A = 0.4, the RM(l/2), RE, AVG-F and EXP-F tests are the best;

14

5. A = 0.5, the RM(l/2) test is the best.

When the parameter changes becomes smaller, the differences between these tests are less
significant. It is interesting to note that it is possible to find some tests outperforming the
AVG-F and EXP-F tests which are optimal.

For the alternative of double structural changes (28), we consider four cases: Ai = 0.5
with A2 = 0.75, 0.25, 0, -0.25. The first change points Aj are 0.2, 0.4, 0.6 and 0.8, the
second change points are Ai + 0.1, . . .,0.9, and the number of replications is 5000. These
results are summarized in Table 5. The results are quite mixed; for example:

1. Ai = 0.2 and A2 = 0.5: the best tests are AVG-F in Table 5A, RR in Table 5B,
RM(0.5) and RR in Table 5C, and RM(0.5) in Table 5D. In this case, the RM(0.2)
test performs similarly to the AVG-F or EXP-F test in Tables 5B, 5C and 5D.

2. Ai = 0.4 and A2 = 0.9: the best tests are RE and AVG-F in Table 5A, RM(0.5) in
Table 5B, and ME(0.5) and RR in Tables 5C and 5D.

3. Ai = 0.6 and A2 = 0.9: the best tests are RE and AVG-F in Table 5A, RM(0.5) in
Tables 5B and 5C, and RR in Table 5D.

In particular, there is no test uniformly better than the other tests.

6 Conclusions

In this paper we provide a unifying view of the tests for parameter constancy which are
determined by the fluctuation of empirical processes. We establish a unified asymptotic
result which allows us to analyze the behavior of these tests quite easily. As applications
we also consider tests based on the range functional, rather than the typical maximal
functional, and characterize their asymptotic null distributions. Our simulation results
show that tests may have very different power performance under different alternatives
and that it is possible to find tests outperforming tests that are optimal in the sense of
Andrews Sz Ploberger (1992). What we want to convey here is that if one is uncertain
about the behavior of parameter changes, it would be better to conduct a family of tests
to safeguard various directions of alternatives. For this purpose, different estimates-based
tests can be easily computed and complement other likelihood-based tests.

15

Appendix

Proof of Theorem 2.1: Let Vg and Mg be the variation of g on [0, 1] and max(|y|; 1),
respectively. Clearly,

where U = i/T. Hence, as \[Tt]/T - t\ = \[Tt] - Tt\/T < \/T and
T^^aiU)- / 9{s)ds

l/T [^'1

ds

= I ^{g{U)-g{U-i+s))ds- f g{s)

Jo fr{ J[Tl]/T

< {Vg + Mg)/T,

we have

Yt = T^/'iT/xo + T^'''-^Jg + ^T + Rt.
where \RT{t)\ < T-^l'^~\Vg + M^). As Ct annihilates ij,

A(£r>V) = XiT'l^'-^Ug + ££t + CRt + Op(l)).

We immediately conclude that for 6 = 1/2, A(£7'y7') => X{CJg + £VF), and that for
(!) < 1/2, T*-i/2A(£^y^) -^P A(£J^) as asserted. D

Applications of Theorem 2.1: It is easily verified that for Ct in the RE test, the
corresponding C is such that

cm = fit) - tfii).

For CT,h in the ME test, the corresponding Ch is such that

jCkfit) = fit + h)- fit) - hfil) = A,/(0 - /i/(l);

see also Chu, Hornik, & Kuan (1992a). For Ct in the Recursive-CUSUM test, the corre-
sponding C is such that

■' fir)

Cfit) = fit)- f ^dr-

Jo T

16

for CT,h in the Recursive-MOSUM test, the corresponding Ch is such that

Chfit) = f{t + h)-m- / ^^dr;

Jt "T

= AhM-Ah f^dr.

Jo T

Given the DGP (1), the results of the RE, ME, CUSUM, and MOSUM tests now follow
straightforwardly from Theorem 2.1. For the Recursive-CUSUM test, note that

^' W{t)

Z{t) := W{t)- f -^dr

Jo T

is a Gaussian process with continuous sample paths, mean zero, and covariance function
min(t, 5), hence a Wiener process. □

Proof of Theorem 3.1: Straightforward application of Theorem 2.1. □

To prove Theorem 3.2, we utilize the following two lemmas.
Lemma A.l For 0 < h < I,

range(A,iW^°; I - h) ='^ Arange(Aiy; (1 - h)/h).

Proof: Note that

Ts.nge{AhW°; 1 - h) = max \AhW°{t) - AhW°{s)\

0<s,t<l—h

max \AhW{t) - AkW{s)\

0<3,t<l—h

= range(A/iVF; 1 - h).

As Wh{u) = h~^^^W{hu) is a Wiener process,

{h-'^/^AhW{t),0 <t<l-h)
='^ {h-^^^AhWihu),Q<u<{l-h)/h)
='^ {AWk{u),Q<u<{\-h)lh)
='^ (AV^(t/),0<u< (l-/i)//i). □

Lemma A. 2 For 0 < r < 1,

range(AVy;r) ='^ \/2rrange(V^; 1).

17

Proof: Let Ct be the space of continuous functions on [0,r], and let fix, fix and fiw be
the measures on Cr induced by AW conditional on Aiy(O) = x, by x + y/2W, and by W,
respectively. By (16.11) of Shepp (1966),

dfjLj

if) = (2/(2 - r))i/V'/2g-(^+^(^))'/^(2-^);

dfla

hence, as under fix, the functions g{t) = {f{t) — x)/\/2 are distributed according to fiw-,
we have

P{range(AVy; r) < s\AW{Q) = x)

= I (2/(2-r))i/V'/2e-(^+^(^))'/''<2-)rf^^(/)

= / (2/(2-r))^/V'/2e-(2-+^^^(-))V4(2-r)^^^(^)

Aange(5;T)<a/N/2
= / (2/(2 - ^))l/2e-V2g-(2x+v/2v))V4(2-r)

X dP{range(W^; r) < 5/^2, W{t) < y}
and thus

IP{range(AVF; r) < 5}

= / IP{range( AW^; r) < s\AWiO) = x} <^(x) dx

= 11 (7r(2 - ^))-l/2e-(2x+y2v)V4(2-r)

X rfIP{range(iy; r) < s/y/2, W{t) < y} dx
= I rfP{range(Vr; r) < 5/v/2, W{r) < y}

= IP{range(Py;r) < 5/v/2}
= IP{range(Py; 1) < s/V^},

where the last equation again follows by rescaling. □

Proof of Theorem 3.2: By successively putting together the previous lemmas, we have

RMT;h ^ range(A,,H^°; 1 - h)

='^ Vh range( AVF ; ( 1 - h)/h)

='^ y/hy/2il - h)/h Ta.nge{W]l). □

Proof of Corollary 3.3: It remains to show that two expressions of the distribution of
the RM test are the same. By the extended Poisson summation formula (see e.g. Feller,

18

1971) applied to the standard norma] density </> with characteristic function ^(a) = e~°- Z'^,
we find that for w 7^ 0,

, w .^-^ \ w J

k=-—oo j=— 00

Differentiating this identity twice with respect to z, we obtain
00 1 00

A;=-oo j = -oo

which upon letting z = -k then gives

cx> 1 00

Y, {-lf-^k'<j>{kw) = —^ Y. ((2j + l)V-Ti;2)e-(2j+^)'^'/2u.^.

k=—oo j=—oo

Thus, by substituting u = 1/w, we have

F{range(H^; 1) < 5}

rs °°

j = -oo

00 ^00

= 4 Y ((2; + l)Vu3-u)e-(2j+^)'''"'/2rfu

j = — 00
00

j = — 00
00

= ^.E-("^ + 72TTTW

\£»V.2 + (2J+ 1)2^2 j

1/5

,-(2j + l)2,r2uV2

,-(2j + l)2;rV252

1/,

as asserted. D

Proof of Theorem 4.1: The proof is essentially the same of that of Theorem 2.1. Here,
Yrit) =

(Tj

1 / 1 ^^'^

[Tt]

Qt%t\]E^^'^]-

t=l

19

We observe that for some M*,

j,^Xix'ig{U)-Q j g{s)ds

1=1

<

[Tt\

^Y,{x,x\-Q)g{U)

t=i
< M*IT.

+

/ 1 [^'1 ft '

Hence,

where Ili^HOII < T-^/^-^M'. Thus, for 6 = 1/2, a^ ^p a^ and

X{CtYt) => XiCW + a-'CQ^/^Jg);
and for <!) < 1/2, aj ->p a^ and T^-'^/^\{CtYt) ^p X{Cct6Q^/^ Jg). O

Proof of Corollary 4.2: Straightforv.ard application of Theorem 4.1. D

Proof of Corollary 4.3: It is easy to see that

IP{range(Vy°;l) < s]
= IP{range(W°;l) < 5 for all i = 1, • • -,71}
= (P{range(vy?;l)<5})".

The first assertion follows from (19). Similarly, the second assertion follows from Corol-
lary 3.3. n

20

References

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Andrews, D. W. K., I. Lee, h W. Ploberger (1992) Optimal changepoint tests for normal
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Bauer, P., h P. Hackl (1978). The use of MOSUMS for quality control, Technometrics ,
20, 431-436.

Billingsley, P. (1968). Convergence of Probability Measures, New York: Wiley.

Brown, R. L., J. Durbin, & J. M. Evans (1975). Techniques for testing the constancy of
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Chu, C.-S., K. Hornik, & C.-M. Kuan (1992b). MOSUM tests for parameter constancy,
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variables, Annals of Mathematical Statistics, 22, 427-432.

Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Volume II,
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21

Hawkins, D. L. (1987). A test for a change point in a parametric model based on a
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Lamotte, L. R., & A. McWhorter (1978). An exact test for the presence of random
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Leybourne, S. J., h B. P. M. McCabe (1989). On the distribution of some test statistics
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Neftci, S. N. (1979). Lead-lag relations, exogeneity and prediction of economic time series,
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Nyblom, J. (1989). Testing for the constancy of parameters over time, Journal of the
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Quandt, R. E. (1960). Tests of the hypothesis that a linear regression system obeys two
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Shepp, L. (1966). Radon-Nikodym derivatives of Gaussian process. Annals of Mathemat-
ical Statistics, 37, 312-354.

Shorack, G. R., & J. A. Wellner (1986). Empirical Processes with Applications to Statis-
tics, New York: Wiley.

22

Table 1: Asymptotic Critical Values of Range Tests.

Tests

n

Tail Pre

)bability

0.20

0.15

0.10

0.05

0.025

0.01

1

1.47337

1.53692

1.61960

1.74726

1.86243

2.00090

2

1.60894

1.66698

1.74272

1.86040

1.96747

2.09740

3

1.68277

1.73796

1.81017

1.92280

2.02578

2.15134

4

1.73294

1.78629

1.85620

1.96558

2.06590

2.18862

RR

5

1.77069

1.82270

1.89096

1.99797

2.09637

2.21700

6

1.80083

1.85179

1.91877

2.02396

2.12085

2.23985

7

1.82583

1.87596

1.94190

2.04560

2.14128

2.25896

8

1.84715

1.89658

1.96166

2.06413

2.15878

2.27535

9

1.86571

1.91454

1.97889

2.08029

2.17407

2.28969

10

1.88211

1.93043

1.99413

2.09462

2.18764

3.30242

1

1.95843

2.07958

2.24117

2.49767

2.73436

3.02334

2

2.22011

2.33550

2.48844

2.73017

2.95322

3.22650

3

2.36716

2.47875

2.62645

2.85987

3.07558

3.34055

4

2.46855

2.57741

2.72147

2.94926

3.16006

3.41950

RMi/2

5

2.54549

2.65226

2.79357

3.01716

3.22433

3.47968

6

2.60725

2.71235

2.85147

3.07175

3.27605

3.52819

7

2.65871

2.76242

2.89974

3.11729

3.31926

3.56875

8

2.70274

2.80527

2.94106

3.15631

3.35630

3.60357

9

2.74116

2.84266

2.97713

3.19040

3.38869

3.63404

10

2.77519

2.87579

3.00910

3.22064

3.41743

3.66110

Notes: Critical values are solved from the formulae in Corollary 4.3 with 5 terms in the summation;
n is the number of parameters in a linear regression model.

23

Table 2: Simulated Asymptotic Critical Values of RM Tests.

n

h

TaU Probability |

0.20

0.15

0.10

0.05

0.025

0.01

0.05

1.2758

1.3101

1.3533

1.4208

1.4811

1.5569

0.10

1.6224

1.6752

1.7418

1.8433

1.9398

2.0514

0.15

1.8300

1.8986

1.9866

2.1199

2.2409

2.3788

0.20

1.9600

2.0409

2.1472

2.3100

2.4577

2.6354

1

0.25

2.0421

2.1354

2.2604

2.4450

2.6068

2.8068

0.30

2.0816

2.1877

2.3230

2.5329

2.7171

2.9385

0.35

2.0826

2.1976

2.3468

2.5781

2.7922

3.0376

0.40

2.0648

2.1830

2.3401

2.5885

2.8140

3.0834

0.45

2.0074

2.1298

2.2941

2.5553

2.7808

3.0715

0.50

1.9193

2.0407

2.2037

2.4628

2.7023

3.0016

0.05

1.3464

1.3775

1.4161

1.4784

1.5350

1.6026

0.10

1.7330

1.7805

1.8410

1.9368

2.0260

2.1282

0.15

1.9754

2.0372

2.1181

2.2416

2.3495

2.4836

0.20

2.1324

2.2058

2.3015

2.4501

2.5840

2.7467

2

0.25

2.2432

2.3277

2.4384

2.6072

2.7571

2.9458

0.30

2.3077

2.4018

2.5285

2.7216

2.8974

3.1107

0.35

2.3307

2.4357

2.5695

2.7816

2.9755

3.2081

0.40

2.3207

2.4343

2.5808

2.8103

3.0191

3.2791

0.45

2.2704

2.3857

2.5437

2.7842

3.0050

3.2872

0.50

2.1839

2.2997

2.4533

2.6961

2.9243

3.2103

0.05

1.3857

1.4143

1.4524

1.5128

1.5690

1.6350

0.10

1.7933

1.8387

1.8974

1.9897

2.0736

2.1748

0.15

2.0516

2.1098

2.1874

2.3043

2.4080

2.5411

0.20

2.2313

2.3021

2.3928

2.5326

2.6571

2.8086

3

0.25

2.3506

2.4296

2.5337

2.6996

2.8463

3.0344

0.30

2.4253

2.5166

2.6345

2.8162

2.9843

3.1839

0.35

2.4657

2.5648

2.6985

2.9015

3.0861

3.3173

0.40

2.4620

2.5730

2.7142

2.9326

3.1412

3.3894

0.45

2.4239

2.5369

2.6831

2.9120

3.1310

3.3938

0.50

2.3283

2.4406

2.5889

2.8243

3.0355

3.2950

0.05

1.4115

1.4395

1.4763

1.5345

1.5894

1.6543

0.10

1.8333

1.8765

1.9341

2.0242

2.1058

2.2074

0.15

2.1048

2.1605

2.2342

2.3460

2.4534

2.5785

0.20

2.2919

2.3600

2.4466

2.5871

2.7098

2.8665

4

0.25

2.4232

2.5000

2.6024

2.7609

2.9012

3.0802

0.30

2.5084

2.5956

2.7105

2.8896

3.0495

3.2457

0.35

2.5592

2.6527

2.7778

2.9761

3.1576

3.3762

0.40

2.5610

2.6649

2.8007

3.0145

3.2079

3.4414

0.45

2.5257

2.6379

2.7805

3.0030

3.2107

3.4561

0.50

2.4286

2.5387

2.6850

2.9099

3.1216

3.3740

0.05

1.4322

1.4594

1.4956

1.5536

1.6063

1.6707

0.10

1.8646

1.9082

1.9629

2.0497

2.1273

2.2281

0.15

2.1452

2.2004

2.2728

2.3829

2.4855

2.6057

0.20

2.3430

2.4074

2.4917

2.6258

2.7462

2.8959

5

0.25

2.4800

2.5541

2.6530

2.8074

2.9483

3.1270

0.30

2.5717

2.6568

2.7705

2.9455

3.1049

3.3065

0.35

2.6230

2.7165

2.8371

3.0300

3.2068

3.4197

0.40

2.6367

2.7399

2.8725

3.0892

3.2801

3.5177

0.45

2.5979

2.7069

2.8463

3.0632

3.2653

3.5153

0.50

2.5099

2.6158

2.7573

2.9813

3.1916

3.4545

24

Table 3: Size Simulation at 10% Level.

Sample
Size

RM Tests

RR

Test

h = 0.5*

/» = 0.5

h = OA

/i = 0.3

h = 0.2

h = 0.l

100

6.5

7.4

7.3

6.7

5.3

4.0

4.2

200

7.6

8.3

8.6

8.0

7.2

5.8

5.9

300

8.0

8.9

8.3

7.9

7.8

6.6

6.1

500

8.4

9.3

9.4

9.0

8.6

7.5

6.7

Notes: All numbers are in percentage. The first column of the RM test size is based on asymptotic
critical value from Table 1; other columns of the RM tests are based on simulated asymptotic critical
values from Table 2.

Table 4A: Power Simulation under a Single Structural Change: A = 0.5.

A

RM Tests

ME Tests

RR

Test

RE

Test

Tests for a Single

Change

^ = ^

/'=i

^ = Tfi

^ = ^

f^ = i

^ = To

MAX-F

AVG-F

EXP-F

0.1

14.3

14.2

16.6

14.5

12.7

16.0

16.0

16.4

26.0

19.7

23.0

0.2

26.9

36.0

27.4

28.4

32.5

25.5

33.5

39.1

45.2

42.6

45.4

0.3

47.3

46.3

32.9

42.7

42.4

29.7

47.2

57.3

55.9

58.0

58.2

0.4

65.5

50.8

35.2

55.0

45.0

30.6

55.5

67.2

61.1

65.9

64.4

0.5

76.4

54.0

37.3

62.7

46.6

31.5

60.0

70.9

64.2

69.4

67.6

Table 4B: Power Simulation under a Single Structural Change: A = 0.25.

A

RM Tests

ME Tests

RR

Test

RE

Test

Tests for a Single Change

^ = ^

^ = '.

''^To

^ = '.

^^ = i

^ = rn

MAX-F

AVG-F

EXP-F

0.1

11.0

11.2

11.4

10.8

10.8

11.5

11.3

11.7

13.9

12.3

12.8

0.2

14.3

15.6

13.8

15.1

14.3

13.3

16.2

16.4

18.4

18.0

18.4

0.3

21.9

19.4

16.1

19.8

18.4

14.4

21.1

24.1

22.8

25.0

24.4

0.4

27.2

21.8

17.0

22.3

19.9

15.4

22.4

26.6

24.2

26.8

25.5

0.5

31.8

21.7

16.0

23.5

19.8

14.7

22.1

27.3

24.4

28.2

26.8

25

Table 5A: Power Simulation under Double Structural Changes: Ai = 0.5 and A2 = 0.75.

Ai

A2

RM Tests

ME Tests

RR

Test

RE

Test

Tests for Double Changes

h = -^

h = i

^ = 1^

h = -^

h = i

f^ = Tn

MAX-F

AVG-F

EXP-F

0.3

58.4

68.9

51.5

56.0

65.1

47.5

65.5

76.6

69.0

80.9

78.3

0.4

64.5

65.1

48.9

57.0

60.5

44.1

63.5

75.9

65.8

80.9

76.1

0.5

69.3

62.4

46.6

57.1

58.1

41.6

60.3

75.1

63.1

78.6

73.9

0.2

0.6

60.2

58.6

43.1

47.1

54.4

38.8

54.7

72.2

60.1

77.1

71.4

0.7

49.6

53.6

39.2

37.5

49.3

35.6

46.3

66.2

55.3

71.6

66.3

0.8

39.6

47.4

36.5

29.5

43.7

33.6

36.7

57.3

48.8

64.5

59.1

0.9

30.6

38.9

31.4

24.4

36.6

28.8

31.1

47.0

42.1

53.3

50.2

0.5

94.5

83.5

63.7

88.7

76.3

53.2

88.7

94.5

85.1

94.0

91.5

0.6

91.0

78.5

57.5

83.0

71.1

48.4

84.1

92.1

81.5

92.1

88.3

0.4

0.7

85.7

73.4

53.4

75.3

66.4

45.0

76.9

88.6

74.9

88.7

83.9

0.8

78.0

65.8

47.3

65.7

58.3

39.4

67.5

83.0

68.4

83.7

78.6

0.9

70.0

55.8

41.9

56.5

49.4

35.9

58.8

75.3

60.4

76.1

70.7

0.7

88.3

79.5

60.1

82.0

73.5

51.5

84.3

92.4

82.0

91.7

89.0

0.6

0.8

82.0

72.1

53.4

73.9

65.7

45.7

75.7

86.4

75.3

87.4

83.3

0.9

73.2

58.2

44.3

63.9

50.8

37.6

64.4

78.8

64.5

79.2

74.7

0.8

0.9

36.3

49.2

41.0

36.5

45.4

38.2

41.7

55.6

53.6

64.6

62.9

Table 5B: Power Simulation under Double Structural Changes: Ai = 0.5 and A2 = 0.25.

Ai

A2

RM Tests

ME Tests

RR

Test

RE

Test

Tests for Double Chcinges

h = ^

^ = i

^ = r.

^ = '2

^=i

^ = r,

MAX-F

AVG-F

EXP-F

0.3

12.3

16.2

16.0

13.4

15.4

14.8

15.4

13.3

18.4

17.0

18.7

0.4

13.4

23.6

19.9

17.7

22.6

18.8

21.6

15.0

24.7

19.2

23.9

0.5

15.1

26.9

21.3

24.2

24.6

19.8

29.0

16.9

27.1

23.1

27.8

0.2

0.6

18.0

31.4

24.7

32.5

28.0

23.1

36.3

21.1

32.6

29.3

34.3

0.7

21.7

32.8

26.1

39.6

29.5

22.9

40.1

22.9

34.2

34.1

36.5

0.8

23.1

34.0

26.9

39.0

29.5

24.0

42.5

26.7

35.9

37.0

39.5

0.9

26.0

34.0

26.9

34.1

31.3

23.8

39.3

30.9

36.4

40.4

41.4

0.5

23.0

24.6

19.7

23.4

22.0

19.1

25.0

27.5

24.6

27.9

27.1

0.6

30.3

34.4

25.6

31.9

31.2

22.5

34.0

33.2

32.1

33.1

34.0

0.4

0.7

40.5

40.2

28.6

41.1

36.9

26.2

44.1

41.0

38.6

41.2

42.0

0.8

52.3

44.5

31.4

51.0

40.0

27.2

52.2

49.3

43.6

49.5

48.8

0.9

61.7

49.1

34.4

57.5

44.2

30.0

57.2

58.0

49.0

57.0

55.3

0.7

35.9

26.7

21.2

27.6

24.4

19.8

27.1

33.1

26.3

31.4

30.1

0.6

0.8

47.0

39.1

28.6

37.2

35.8

26.0

39.6

43.9

35.7

41.3

39.7

0.9

58.0

47.9

33.3

47.7

43.0

29.1

50.9

55.7

44.7

54.1

50.9

0.8

0.9

21.7

25.5

21.1

20.3

23.0

20.3

22.3

25.9

24.6

28.8

27.5

26

Table 5C: Power Simulation under Double Structural Changes: Ai = 0.5 and A2 = 0.

Ai

A2

RM Tests

ME Tests

RR

Test

RE

Test

Tests fo

r Double Chcinges

h = ^.

f^ = i

f^ = ia

f^ = i

f^=i

^ = 1^

MAX-F

AVG-F

EXP-F

0.3

14.9

17.7

18.3

15.1

16.6

17.6

15.9

14.3

19.4

14.3

17.4

0.4

32.3

36.4

28.7

27.8

35.9

26.1

32.8

24.3

32.1

25.8

31.0

0.5

47.3

44.4

31.9

42.7

42.7

28.9

48.2

32.2

42.3

37.6

42.7

0.2

0.6

43.1

48.3

35.7

55.2

42.9

31.3

57.4

35.5

47.6

45.9

49.4

0.7

38.4

48.6

35.8

63.3

42.3

29.8

61.7

33.0

50.6

49.7

52.9

0.8

29.0

42.5

32.8

55.7

36.7

27.8

59.1

27.3

47.6

48.2

50.5

0.9

25.2

34.2

29.4

42.5

30.4

26.0

49.6

25.6

41.9

42.1

45.1

0.5

14.4

18.5

18.1

14.8

18.1

18.2

16.5

12.4

18.8

13.3

16.5

0.6

16.1

36.7

28.6

28.3

36.3

27.2

33.3

20.1

32.1

23.5

30.1

0.4

0.7

26.9

46.0

32.6

43.8

43.6

29.4

50.2

27.7

43.3

37.2

43.5

0.8

42.6

47.9

35.6

56.0

44.6

31.3

59.1

35.1

48.0

45.5

50.1

0.9

59.0

48.0

34.3

62.8

42.3

28.8

62.0

48.9

50.8

54.3

55.1

0.7

15.8

18.6

18.5

15.0

17.4

18.3

17.3

13.0

19.2

13.1

16.9

0.6

0.8

31.7

35.4

27.2

27.6

35.8

25.8

32.1

23.0

31.4

25.4

.30.4

0.9

51.7

44.9

33.0

43.0

42.9

30.4

48.0

43.5

43.4

44.1

45.5

0.8

0.9

15.8

18.9

18.7

15.3

17.1

18.7

16.2

15.4

18.4

16.8

17.5

Table 5D: Power Simulation under Double Structural Changes: Aj = 0.5 and A2 = —0.25.

Ai

A2

RM Tests

ME Tests

RR

Test

RE

Test

Tests for Double Chcinges

h = -^

^ = i

^ = To

^-J

/^-i

^-11)

MAX-F

AVG-F

EXP-F

0.3

39.5

40.2

35.6

33.8

36.6

33.8

37.6

42.8

40.2

40.5

42.6

0.4

68.3

67.1

50.1

57.8

64.6

45.7

62.8

63.9

61.7

61.4

64.9

0.5

86.4

74.6

54.9

77.1

70.7

47.9

79.9

74.4

73.5

74.6

76.7

0.2

0.6

79.4

75.7

57.1

83.2

70.5

47.9

84.2

74.7

76.8

79.2

80.3

0.7

69.9

73.6

54.9

85.0

66.4

45.2

83.6

67.9

75.9

78.9

79.6

0.8

44.6

64.2

49.1

72.7

56.2

41.8

77.3

47.8

69.4

69.2

72.5

0.9

28.0

36.6

38.0

52.3

31.6

32.3

60.8

25.2

54.3

50.9

57.2

0.5

48.6

36.5

33.0

36.6

32.9

30.5

37.4

39.6

39.1

36.3

39.8

0.6

35.6

58.3

44.1

48.1

55.7

40.2

54.6

44.9

54.7

47.7

54.8

0.4

0.7

29.9

63.4

48.3

58.0

60.7

42.1

67.2

44.1

63.6

56.1

64.4

0.8

39.0

61.3

46.5

66.9

56.2

38.6

72.4

38.3

64.9

59.2

66.0

0.9

58.1

47.3

39.7

68.9

43.7

33.2

68.6

40.8

59.2

55.4

60.9

0.7

14.1

26.9

26.9

20.6

25.1

25.5

24.1

21.6

30.9

24.6

30.2

0.6

0.8

23.3

43.3

35.1

28.0

43.2

33.0

37.3

18.2

42.9

28.0

40.6

0.9

46.4

43.3

34.6

40.9

43.9

30.9

51.1

32.2

47.8

38.1

47.4

0.8

0.9

13.1

15.1

18.7

12.7

15.0

18.4

14.8

10.4

20.5

12.4

18.1

27

HECKMAN IXI
BINDERY INC. |s|

JUN95

u , T Pi,.,..P N. MANCHESTER
Bound -To -Pkas. i^di^nA 46962