Minimum LM Unit Root Test with Two Structural Breaks

Junsoo LeeMark Strazicich

Department of EconomicsUniversity of Central Florida

July 1999

Abstract

The two-break unit root test of Lumsdaine and Papell (1997) is examined and found tosuffer from bias and spurious rejections in the presence of structural breaks under thenull. A two-break minimum LM unit root test is proposed as a remedy. The two-breakLM test does not suffer from bias and spurious rejections and is mostly invariant to thesize, location, and misspecification of the breaks. We test the Nelson and Plosser (1982)data and find fewer rejections of the unit root than Lumsdaine and Papell.

JEL classification: C12, C15, and C22

Key words: Lagrange Multiplier, Unit Root Test, Structural Break, and EndogenousBreak

Corresponding author: Junsoo Lee, Associate Professor, Department of Economics,University of Central Florida, Orlando, FL, 32816-1400, USA. Telephone: 407-823-2070. Fax: 407-823-3269. E-mail: Junsoo.Lee@bus.ucf.edu.

We thank John List for helpful comments.

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1. Introduction

Since the influential paper of Perron (1989), researchers have noted the

importance of allowing for a structural break when testing for a unit root. Perron showed

that the ability to reject a unit root decreases when the stationary alternative is true and a

structural break is ignored. Perron used a modified Dickey-Fuller (hereafter DF) unit

root test including dummy variables to account for one known, or “exogenous,” structural

break. Subsequent papers further modify the test to allow for one unknown break point

that is determined “endogenously” from the data. One widely used endogenous

procedure is the “minimum test” of Zivot and Andrews (1992, hereafter ZA), which

selects the break point where the t-statistic testing the null of a unit root is at a minimum.

Given a loss of power when ignoring one structural break in standard unit root

tests, it is logical to expect a similar loss of power when ignoring two, or more, breaks in

the one-break tests. Recent research indicates that many economic time series might

contain more than one structural break.1 Therefore, it may be necessary to allow for

more than one break when testing for a unit root. Recently, Lumsdaine and Papell (1997,

hereafter LP) make a contribution in this direction by extending the ZA test to two

structural breaks.

One critical issue common to these minimum unit root tests is that they typically

assume no breaks under the null, and derive their critical values under this assumption.

Despite their popularity, these tests are invalid if structural breaks occur under the null; as

rejection of the null would not necessarily imply rejection of a unit root per se, but would

1 For example, Ben-David, Lumsdaine, and Papell (1999), Ben-David and Papell (1998), and Papell,Murray, and Ghiblawi (1999) find evidence of more than one structural break in real GDP, per capita realGDP, and unemployment rates among OECD countries.

2

instead imply rejection of a unit root without break. Perron initially allows for a break

under both the null and alternative hypotheses. This is important for his exogenous break

test; otherwise the test statistic will diverge under the null as the size of the break

increases. The same result is true for the endogenous minimum tests. Nunes et al. (1997)

and Lee et al. (1998) provide evidence that assuming no structural break under the null in

the ZA test makes the associated test statistic diverge and leads to spurious rejections

when the data generating process (DGP) contains a break. Lee and Strazicich (1999a)

further investigate the source of these spurious rejections and find that the ZA test most

often selects the break point where bias is maximized. In this paper, we find the same

problems of bias and spurious rejections for the two-break LP unit root test.

To provide a remedy, we propose a “two-break minimum LM test.” The test is

based on the Lagrange Multiplier (LM) unit root test suggested by Schmidt and Phillips

(1992, hereafter SP), and can also be seen as an extension of the one-break minimum LM

test developed in Lee and Strazicich (1999b). The two-break LM test solves the

problems entailed in the LP test: the LM test does not diverge as breaks under the null

increase in size, and is free of bias and spurious rejections. Further, there is no need to

exclude breaks under the null. Whereas, for the two-break LP test it might be necessary

to exclude breaks under the null to make the test statistic invariant to nuisance

parameters, a similar assumption is not required for the two-break LM test. Even with

breaks under the null, the distribution of the two-break LM test statistic is unaffected,

since the test is invariant to break point nuisance parameters. The two-break LM test is

3

also robust to misspecification of the number of breaks under the null; thus providing a

solution to the suggestion raised in LP (p. 212).

The paper proceeds as follows. Section 2 presents the two-break minimum LM

unit root test and compares it to the LP test. Asymptotic properties of the two-break

minimum LM test are discussed. Section 3 compares properties of each test in simulation

experiments. Section 4 examines the Nelson and Plosser (1982) data using the two-break

minimum LM test. Section 5 summarizes and concludes. Throughout the paper, the

symbol “→” denotes weak convergence of the associated probability measure.

2. Test Statistics and Breaks under the Null

Perron previously considered three structural break models as follows. The “crash”

Model A allows for a one-time change in level; the “changing growth” Model B

considers a sudden change in slope of the trend function; and Model C allows for change

in level and trend. We consider the following DGP:

yt = δ'Zt + Xt , Xt = βXt-1 + εt , (1)

where Zt is a vector of exogenous variables, A(L)εt = B(L)ut, and A(L) and B(L) are finite

order polynomials with ut ~ iid (0,σ2). Two structural breaks can be considered from the

above DGP as follows.2 Model A allows for two changes in level and is described by Zt

= [1, t, D1t, D2t]', where Djt = 1 for t ≥ TBj + 1, j=1,2, and zero otherwise. Model C

includes two changes in level and trend, and is described by Zt = [1, t, D1t, D2t, DT1t*,

DT2t*]', where DTjt

* = t for t ≥ TBj + 1, j=1,2, and zero otherwise. The DGP includes

2 Model B is omitted from the discussion that follows, as it is commonly held that most economic timeseries can be described adequately by Model A or C.

4

both the null (β = 1) and alternative (β < 1) models in a consistent manner. For instance,

consider Model A (a similar argument can be applied to Model C); depending on the

value of β, we have:

Null yt = µ0 + d1B1t + d2B 2t + yt-1 + vt , (2a)

Alternative yt = µ1 + γt + d1*D1t + d2*D2t + vt , (2b)

where vt is a stationary error term, and Bjt = 1 for t = TBj + 1, j=1,2, and zero otherwise.

We let d = (d1, d2)′. For Model C, Djt terms are added to (2a) and DTjt* to (2b),

respectively. Note that the null model in (2a) includes dummy variables (Bjt) to allow for

two possible breaks. Nesting both the null and alternative models from (2a) and (2b), we

may consider the two-break augmented unit root test equation of LP as follows:

yt = α0 + α1t + α2B1t + α3B2t + α4D1t + α5D2t + φ yt-1 + ∑j=1

k

cj∆yt-j+ et . (3)

In the exogenous test, the break points are known, and TBj/T → λj as T → ∞, where λ =

(λ1, λ2)′. LP provide the asymptotic distribution of the exogenous t-statistic testing φ = 1

when omitting Bjt terms from (3). We denote the LP exogenous test statistic omitting Bjt

terms as “τ̂*” and when including Bjt terms as “τ̂.” The asymptotic distribution of the LP

exogenous test statistic is found to depend on λ in either case. One critical limitation of

the LP test is that while the test may be valid if the size of breaks under the null is zero,

(i.e., d = 0 in (2a)), it is invalid if d ≠ 0. The asymptotic distribution of the unit root test

statistic is not invariant to d under the null, and the associated t-statistic diverges as d

increases. In this case, it is necessary to include Bjt terms in (3) to insure that the

asymptotic distribution of the test statistic will be invariant to d. This is quite important,

5

as Perron (1989, p. 1393) noted this potential for divergence in his exogenous break test

and therefore included Bt.3

The same divergence problem is found for the endogenous break unit root test of

LP. The authors omit Bjt terms from their testing equation (3), and derive their critical

values assuming d1 = d2 = 0 under the null hypothesis (2a). While this assumption might

be necessary for this type of endogenous test not to depend on the location of the

break(s), this introduces a different problem: the test statistic diverges as the magnitude

of breaks under the null increase, the same as in the exogenous test. Further, unlike in the

exogenous break test, the LP test statistic is shown to diverge even if Bjt terms are

included in the testing equation (3). In sum, ignoring structural breaks under the null can

lead to serious problems.4

Fortunately, the divergence problem of the LP test is not found for the two-break

minimum LM test. Test statistics for the LM unit root test can be obtained according to

the LM (score) principle from the following regression:

∆yt = δ '∆Zt + φ S∼t-1 + ut , (4)

where S∼

t = yt - ψ∼

x - Ztδ∼

, t=2,..,T, δ∼ are coefficients in the regression of ∆yt on ∆Zt, and ψ∼x

is the restricted MLE of ψx (≡ ψ + X0) given by y1 - Z

1δ∼ (see SP). To correct for

3 Perron (1993) and Perron and Vogelsang (1992) rectify their tests by adding the Bt term in order toeliminate dependency of their test statistics on the nuisance parameter d in the additive outlier model.

4 A different, but related, question is what happens in the standard unit root test (without breaks) if the nullhypothesis is true and there is a structural break? This question was initially addressed in Amsler and Lee(1995), who showed that, unlike under the alternative, the standard unit root tests are unaffected byignoring a break under the null. Recently, Leybourne, Mills, and Newbold (1998) show that using astandard Dickey-Fuller test (without break) can lead to spurious rejection of the null if a structural breakoccurs early in the series. Contrary to this, Lee (1999) shows that this spurious rejection problem does notoccur for the LM unit root test of Schmidt and Phillips (1992).

6

autocorrelated errors, one can include augmented terms ∆S∼

t-j, j=1,..,k, in (4), as in the

augmented DF type test. The unit root null hypothesis is described by φ = 0, and the LM

test statistics are given by:

ρ∼ = T· %φ , (5a)

τ∼ = t-statistic testing the null hypothesis φ = 0 . (5b)

Theorem 1. Assume (i) the data are generated according to (1), with Zt = (1, t, D1t, D2t)'

for Model A, and Zt = [1, t, D1t, D2t, DT1t*, DT2t

*]' for Model C; (ii) the innovations εt

satisfy the regularity conditions of Phillips and Perron (1988, p. 336); and (iii) TBj /T →

λj as T → ∞. Then, under the null hypothesis that β = 1:

ρ∼ → - 1 2

σe2

σ2 [⌡⌠0

1 V_(m) 2(r)dr] , (6a)

τ∼ → - 1 2

σ σe

[⌡⌠0

1 V_(m) 2(r)dr]-1/2 , (6b)

where V_(m)(r) is defined for m = A or C, V_(A)(r) is a demeaned Brownian bridge, and

V_(C)(r) = V_(C)(r, λ) is a demeaned and de-breaked Brownian bridge.

Proof is given in the Appendix.

An important implication of Theorem 1 is that the asymptotic null distribution of

the LM statistics in (6) for Model A do not depend on location of the breaks (λj = TBj/T).

Thus, the LM test can allow for two breaks under the null without depending on nuisance

parameters. In addition, the asymptotic distribution of the two-break LM test is the same

as that of the SP test (without breaks), implying that critical values from the SP test can

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be used for the two-break LM test. This invariance result, initially shown in Amsler and

Lee, holds for a finite number of breaks. Regardless of the presence or absence of

structural breaks, the asymptotic null distribution of the two-break LM test statistic is

unaffected, thereby making the test robust to misspecification of break points under the

null. This result, however, does not hold for Model C. The asymptotic distribution of the

two-break LM test statistic in Model C depends on λ, but unlike in the LP test, the two-

break LM test remains free of spurious rejections.

The minimum LM test uses a grid search to determine the location of two breaks

(TBj) as follows:

LMρ = Inf λ

ρ∼(λ) , (7a)

LMτ = Inf λ

τ∼(λ) . (7b)

The break point estimation scheme is similar to that of the LP test. Yet, in spite of

similar estimation schemes, we shall see that the performance of these tests is quite

different. The asymptotic distribution of the two-break minimum LM test can be

described as follows:

Corollary 1. Under the null hypothesis that β = 1:

LMρ → Inf λ

[ - 1 2

σe2

σ2 {⌡⌠0

1 V_(m )2(r) dr} ] , (8a)

LMτ → Inf λ

[- 1 2

σ σe

⌡⌠0

1 V_(m)2(r) dr]-1/2 . (8b)

To derive critical values for the two-break LM unit root test we generate pseudo-

iid N(0,1) random numbers using the Gauss (version 3.2.12) RNDNS procedure. Critical

values are derived using 50,000 replications for the exogenous break tests, and 5,000

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replications for the endogenous break tests in samples of T = 100. Critical values are

shown in Table 1 and 2.

3. Simulation Results

This section examines simulation results to compare performance of the minimum

LM test with that of the LP test. While LP consider only testing equations without B1t

and B2t, we additionally consider their test with these terms included. In the discussion

that follows, we denote “LPt*” as the LP two-break test with Bjt terms included in the

testing equation, and “LPt” omitting these terms. Since performance of the LMρ test is

similar, we discuss only the LMτ test. We first examine the exogenous two-break unit

root test, assuming the break points are known, and then proceed to the endogenous break

test. Separate examination of these tests might be useful to investigate more specifically

the effect of using incorrect break points, since using incorrect break points in the LP test

is shown to introduce a bias in estimating the regression parameters. Simulations are

performed for the LM and LP tests using 20,000 replications for the exogenous tests and

2,000 replications for the endogenous tests in samples of T = 100. Throughout, R denotes

the number of structural breaks, λ is a vector denoting location of the breaks, and d is a

vector denoting magnitude of the breaks in the DGP. Re and λe denote the values used in

the tests. All measures of size and power are reported using 5% critical values.

Exogenous Tests

We first examine the exogenous two-break unit root tests in Table 3. In

Experiment A and B, we investigate the effect of different λ and d in Model A. The LMτ

test is clearly invariant to λ and d, thus confirming the invariance results of Theorem 1.

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As expected, the LPt test shows significant size distortions, which increase with the

magnitude of the breaks. This provides evidence that omitting the Bjt terms from the

exogenous break test introduces a serious problem. While the LPt test appears to have

greater power than the LMτ test in the presence of structural breaks, this result is spurious

and due to the large size distortions. In sum, we can say that the exogenous break LPt

test is simply invalid. While the LPt* test is invariant to d, and therefore free of the

divergence problem, it is still not invariant to λ. In all cases, the LMτ test has greater

power to reject the null than the LPt* test.

In Experiment C, we examine effects of under-specifying the number of breaks

(Re < R). As expected, the LMτ test is mostly invariant to assuming an incorrect number

of breaks under the null, while the LPt test is more seriously affected. Both the LMτ and

LPt* tests lose power under the alternative. This result can be seen as a generalization of

the finding of Perron and Amsler and Lee, indicating that unit root tests lose power when

the number of breaks is underestimated. In Experiment D, we examine effects of

assuming incorrect break points. The LMτ test is again mostly invariant to using incorrect

break points under the null, while other effects are similar to under-specifying the number

of breaks.

Results for Model C are similar to those for Model A, except that the LMτ test is

no longer strictly invariant to λ under the null, but remains invariant to d. The LPt test is

again invalid due to large size distortions and spurious rejections. In Experiment C′ and

D′, we see that the three test statistics all have (mostly negative) size distortions when

break locations are incorrectly estimated or the number of breaks is underestimated.

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Endogenous Minimum Tests

The two-break endogenous unit root test is examined in Table 4. Experiment E

compares rejection rates for Model A at different break locations and magnitudes.

Overall, the LMτ test performs well with no serious size distortions. Contrary to this, the

LPt and LPt* test suffer significant size distortions and spurious rejections, which

increase with the magnitude of the breaks. Unlike for the exogenous test, the LPt* test is

now as invalid as the LPt test. Thus, regardless of including or excluding Bjt terms in the

testing regression (3), the endogenous two-break LP test exhibits spurious rejections

when d ≠ 0.

The above unexpected result for the LPt* test is closely related to estimating

incorrect break points. Table 4 reports the frequency of estimating break points over a

specified range. For the LPt* test, the frequency of estimating the break points correctly

at TB is virtually zero. Instead, the LPt* test most frequently selects break points

incorrectly at TB-1. This problem becomes more serious as the breaks increase in size.

The reason is that the LPt* test statistic is generally smaller when break points are

misspecified, reaching a minimum at TB-1. Therefore, when the LPt* test searches for the

minimum t-statistic, it most frequently selects the break points incorrectly at TB-1. This

causes the LPt* test t-statistic to diverge and become smaller as |d| increases. This

problem is critical, and is associated with bias in estimating the crucial parameter φ in (3),

corresponding to β in (1). Similar to results in Lee and Strazicich (1999a) for the ZA

one-break test, we shall see below (Table 5) that the bias in estimating β using the LPt*

test is maximized at TB LP test in Table 4 appears to select correct break

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points most frequently at TB, this outcome is misleading, as spurious rejections are

maximized.

The endogenous break test in Model C is examined in Panel (2) of Table 4. The

minimum LMτ test has somewhat greater size distortions than in Model A, but rejections

are still near 5%. This is not the case for the LP tests. Again, the LPt and LPt* test each

have significant size distortions and spurious rejections when the DGP contains structural

breaks. Apparent success in estimating the break points with the LPt test is again

misleading, as spurious rejections are maximized at TB. Fortunately, the LMτ test remains

free of spurious rejections. Therefore, the LMτ test may still be used in Model C, as long

as critical values are employed corresponding to the break points estimated.5

Bias Effects

In order to see further why using incorrect break points leads to spurious

rejections, we examine possible bias and mean squared error (MSE) in estimating β (or φ)

and σ for Model A in Table 5. An important advantage of the LM test is revealed; bias in

estimating β under the null is small and unaffected by incorrect break points. A similar

result can be seen examining empirical critical values. At the break points selected most

often with the LMτ test (TB), the empirical critical values are invariant to the magnitude of

the breaks and mostly unaffected by incorrect estimation of their location. This is not the

case for the LP tests. Empirical critical values for the LPt and LPt* test depend both on

the magnitude and location of the breaks. The source of the size distortions is revealed;

5 Critical values for the two-break LM test in Model C are provided in Table 2 for a variety of two breakpoint combinations.

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the LPt* test selects break points where the t-statistic testing for a unit root is minimized

and the bias in estimating β is maximized. Excluding Bjt terms from the testing

regression, the LPt test also selects break points where bias is maximized. Regardless of

including or excluding Bjt terms in the testing regression, the t-statistic in the LPt and

LPt* test is minimized where the bias in estimating β is maximized. Omitting the Bjt

terms in the LPt test only transfers the bias one period from TB-1 to TB. Just as the

exogenous test omitting Bjt terms diverges, the endogenous LPt test also diverges when

selecting break points where bias is maximized, thus both the LPt and LPt* tests are

equally invalid. Except for the break points estimated, the LPt and LPt* tests are quite

similar, since both tend to select break points where bias and spurious rejections are the

greatest as |d| increases. Results for the LPt and LPt* test under the alternative, in terms

of power and break point estimation, are therefore misleading. MSE tells a similar story

for β. Results for σ in terms of bias and MSE are similar to those for β.

4. Empirical Tests

In this section, the two-break minimum LM and LP tests are applied to the Nelson

and Plosser (1982) data. The data comprise fourteen annual time series ranging from

1860 (or later) to 1970 and have the advantage of being examined often in the literature.

All series are in logs except the interest rate. For each test, we determine the number of

augmentation terms, ∆S∼

t-i, i = 1,..,k, in (4) for the LMτ test, and ∆yt-i in (3) for the LPt and

LPt* test, by following the procedure in Perron and LP. Starting from a maximum of k =

8 lagged terms, we examine each combination of two break points over the time interval

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[.1T, .9T].6 After determining the “optimal k,” we determine the break points where the t-

statistic is a minimum. Throughout, we follow Perron and ZA and assume Model A for

all series except real wages and the S & P 500 stock index, in which we assume Model C.

Overall, we find stronger rejections of the unit root null using the LPt and LPt*

tests than with the LMτ test. At the 5% significance level, the null is rejected for six

series by the LPt* and LPt test, and for four series by the LMτ test.7 For example, while

the null is rejected at the 5% significance level for Real GNP, Nominal GNP, Per-capita

Real GNP, and Employment in the LPt and LPt* tests, the null is rejected at only higher

significance levels using the LMτ test.8 Compared with results in ZA (finite sample

critical values) for one-break, we observe the same number of unit root rejections. This

result may indicate that power to reject the null diminishes if the number of breaks is

under-specified.

To investigate the potential for spurious rejections, we also estimate the size of

structural breaks under the null. The null model is estimated in (2a) using the first

differenced series. Briefly, for each possible combination of TB1 and TB2 in the interval

[.1T, .9T], we again determine the k-augmented terms by using the general to specific

procedure. We then determine the break points where the Schwarz Bayesian Criterion is

6 This “general to specific” procedure looks for significance of the last augmented term. We use the 10%asymptotic normal value of 1.645 on the t-statistic of the last lagged term. The procedure has been shownto perform better than other data-dependant procedures (see, e.g., Ng and Perron, 1995). The trimming ofend points does not affect estimation of the test statistics, but critical values are affected by the trimming.Lumsdaine and Papell (1997) use 1% trimming.

7 Throughout the empirical section, we use the critical values from Table 2 (Model A) and Table 3 (ModelC) in Lumsdaine and Papell (1997) for the LPt and LPt* test statistics (asymptotically equivalent). Forcomparison, LM test critical values were derived using the same sample size and trimming as in Lumsdaineand Papell (T = 125 and 1%). LM test critical values are -4.571, -3.937, and -3.564 for Model A, and -6.281, -5.620, and -5.247 for Model C, at the 1%, 5%, and 10% significance levels, respectively.

14

minimized. Estimated break coefficients in standardized units are shown, along with

other results, in Table 6. Break terms appear significant under the null in most series,

with magnitudes ranging from near 2 to 8. This suggests that even modest size breaks

can lead to different test results, or at least move significance levels in that direction.

Similar to the simulation results in Section 3, we observe that the LPt* test often

selects break points one period before the LPt and LMτ test. For example, the LPt* test

estimates break points for Nominal GNP at 1919 and 1928, while the LPt and LMτ test

estimate break points at 1920 and 1929, and 1920 and 1948, respectively. There is no

evidence that structural breaks occurred in either 1919 or 1928; on the contrary, it is more

reasonable to argue that the correct break points are 1920 and 1929, since Nominal GNP

fell by 24% in 1921 and 12% in 1930. A similar pattern is frequently observed in the

other series. Throughout, the LPt* test tends to select incorrect break points in a

consistent manner when there are significant structural breaks under the null. Omitting

the Btj terms in the LPt test only moves the estimated break points to one period later.

Other than this, the LPt* and LPt tests have similar results throughout. We note that

estimation of the break points can be imprecise and different tests produce different

results. This difficulty may not pose a serious problem for the LM unit root test, since

simulation results indicated the LM test is robust to break point misspecification.

5. Summary and Concluding Remarks

In many time series, allowing for one structural break may be too restrictive. A

unit root test allowing for more than one structural break could, therefore, lead to greater

8 For real wages and the money stock the opposite is the case.

power to reject the null. This paper proposes a two-break minimum LM unit root test as

a remedy to the spurious rejections found for the two-break minimum test of Lumsdaine

critical issue in these Dickey-Fuller type minimum tests is their assumption of no

break(s) under the null. This assumption was found to be detrimental in the presence of

different inference results in unit root tests. On the contrary, the asymptotic null

distribution of the two-break minimum LM unit root test was shown to be invariant to

test, our findings serve to caution researchers inclined to apply endogenous break unit

root tests that assume no breaks under the null.

References

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Lee, J., J. List, and M. Strazicich. (1998). “Spurious Rejections with the Minimum UnitRoot Test in the Presence of a Structural Break under the Null,” Working Paper,

Lee, J. and M. Strazicich. (1999a). “Break Point Estimation with Minimum Unit RootTests and Spurious Rejections of the Null,” Working Paper, University of Central

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Table 1

Critical Values of Exogenous LM Tests (T=100)

(1) Model A

1% 5% 10%

τ∼ -3.63 -3.06 -2.77

ρ∼ -23.8 -17.5 -14.6

Note: Critical values for Model A are the same as those in Schmidt and Phillips(1992).

(2) Model C

(i) τ∼

λ2

λ1 .4 .6 .8.2 -4.83, -4.19, -3.89 -4.93, -4.31, -4.00 -4.76, -4.19, -3.88.4 - -4.91, -4.33, -4.03 -4.88, -4.32, -4.03.6 - - -4.84, -4.19, -3.89

(ii) ρρ∼∼

λ2

λ1 .4 .6 .8.2 -38.1, -30.2, -26.4 -39.4, -31.6, -27.9 -37.2, -30.1, -26.3.4 - -39.2, -31.8, -28.1 -38.7, -31.7, -28.1.6 - - -38.3, -30.2, -26.4

Note: Critical values are at the 1%, 5%, and 10% levels, respectively.

19

Table 2

Critical Values of the Endogenous Two-Break Minimum Tests (T = 100)

(1) Model A

1% 5% 10%LMτ -4.545 -3.842 -3.504LMρ -35.73 -26.89 -22.89LPt -6.420 -5.913 -5.587

LPt* -6.400 -5.853 -5.560

Note: The DGP in the simulation does not include breaks. The LP tests areaffected by breaks, while LM tests are invariant to breaks in Model A.

(2) Model C (I)

1% 5% 10%LMτ -5.825 -5.286 -4.989LMρ -52.551 -45.532 -41.664LPt -6.936 -6.386 -6.108

LPt* -6.945 -6.344 -6.064

Note: The DGP in the simulation does not include breaks. Both LP and LM testsare affected by breaks in Model C.

(3) Model C (II)

(i) LMττλ2

λ1 .4 .6 .8.2 -6.16, -5.59, -5.28 -6.40, -5.74, -5.32 -6.33, -5.71, -5.33.4 - -6.46, -5.67, -5.31 -6.42, -5.65, -5.32.6 - - -6.32, -5.73, -5.32

(ii) LMρρλ2

λ1 .4 .6 .8.2 -55.5, -47.9, -44.0 -58.6, -50.0, -44.4 -57.6, -49.6, -44.6.4 -59.3, -49.0, -44.3 -58.8, -48.7, -44.5.6 -57.5, -49.8, -44.4

Note: Critical values are at the 1%, 5%, and 10% levels, respectively.

20

Table 3Rejection Rates of the Exogenous Two-Break Tests (T = 100)

(1) Model A

Exp DGP EstimationUnder the Null

(β = 1.0)Under the Alternative

(β = 0.9)R λ′ d′ Re λe′ τ

∼ τ^ τ^* τ∼ τ^ τ^*

A 0 - - 2 .25, .50 .047 .040 .042 .243 .114 .1132 .25, .75 .047 .040 .040 .239 .110 .1082 .50, .75 .047 .039 .039 .240 .115 .114

B 2 .25, .50 5, 5 2 .25, .50 .047 .487 .042 .243 .763 .11310, 10 2 .25, .50 .047 .955 .042 .243 .998 .113

2 .25, .75 5, 5 2 .25, .75 .047 .485 .040 .239 .757 .10810, 10 2 .25, .75 .047 .956 .040 .239 .997 .108

C 2 .25, .50 5, 5 0 - .053 .005 .005 .125 .010 .010.25, .50 5, 5 1 .25 .045 .109 .013 .151 .201 .033.25, .50 5, 5 1 .50 .045 .092 .012 .137 .127 .023

2 .25, .50 10, 10 0 - .037 .001 .001 .019 .000 .000.25, .50 10, 10 1 .25 .032 .245 .005 .040 .305 .004.25, .50 10, 10 1 .50 .031 .156 .003 .026 .085 .001

D 2 .25, .50 5, 5 2 .25, .75 .046 .151 .028 .147 .250 .05810, 10 2 .25, .75 .033 .292 .010 .038 .359 .010

(2) Model C

Exp DGP EstimationUnder the Null

(β = 1.0)Under the Alternative

(β = 0.9)R λ′ d′ Re λe′ τ

∼ τ^ τ^* τ∼ τ^ τ^*

A′ 0 - - 2 .25, .50 .052 .050 .050 .118 .101 .0972 .25, .75 .048 .052 .050 .115 .101 .0992 .50, .75 .051 .055 .054 .119 .105 .101

B′ 2 .25, .50 5, 5 2 .25, .50 .052 .625 .050 .118 .773 .09710, 10 2 .25, .50 .052 .986 .050 .118 .998 .097

2 .25, .75 5, 5 2 .25, .75 .048 .627 .051 .115 .773 .09910, 10 2 .25, .75 .048 .986 .051 .115 .999 .099

C′ 2 .25, .50 5, 5 0 - .000 .000 .000 .000 .000 .000.25, .50 5, 5 1 .25 .002 .011 .004 .003 .007 .006.25, .50 5, 5 1 .50 .004 .080 .005 .004 .102 .008

2 .25, .50 10, 10 0 - .000 .000 .000 .000 .000 .000.25, .50 10, 10 1 .25 .000 .001 .000 .000 .000 .000.25, .50 10, 10 1 .50 .000 .103 .000 .000 .096 .000

D′ 2 .25, .50 5, 5 2 .25, .75 .015 .045 .015 .014 .031 .01610, 10 2 .25, .75 .000 .006 .000 .000 .001 .000

21

Table 4Rejection Rates and Estimated

Break Points of the Endogenous Break Tests (T = 100)

(1) Model A

Frequency of Estimated Break Points in theRangeExp λ′ d′ Test 5% Rej.

TB-1 TB TB± 10 TB± 30

Under the null (β = 1)

E - 0, 0 LMτ .044 - - - -LPt .046 - - - -

LPt* .038 - - - -.25, .5 5, 5 LMτ .064 .000 .126 .254 .766

LPt .192 .000 .176 .402 .804LPt* .180 .206 .000 .426 .816

.25, .5 10, 10 LMτ .034 .002 .252 .500 .804LPt .748 .000 .818 .882 .976

LPt* .748 .826 .000 .890 .978.25,.75 5, 5 LMτ .052 .000 .028 .152 .646

LPt .170 .008 .162 .392 .752LPt* .166 .190 .000 .408 .760

.2, .3 5, 5 LMτ .040 .000 .266 .438 .668LPt .206 .000 .136 .318 .598

LPt* .174 .148 .000 .286 .586

Under the alternative (β = .9)

F .25, .5 0, 0 LMτ .282 - - - -LPt .098 - - - -

LPt* .078 - - - -.25, .5 5, 5 LMτ .172 .000 .276 .480 .808

LPt .318 .002 .332 .600 .912LPt* .292 .356 .000 .624 .910

.25, .5 10, 10 LMτ .042 .000 .564 .778 .894LPt .954 .000 .972 .990 .998

LPt* .944 .974 .000 .986 .996.25,.75 5, 5 LMτ .204 .004 .132 .304 .734

LPt .298 .016 .304 .594 .820LPt* .284 .318 .000 .616 .824

.2, .3 5, 5 LMτ .130 .000 .364 .580 .738LPt .336 .000 .248 .462 .698

LPt* .302 .262 .000 .434 .702

22

(2) Model C

Frequency of Estimated Break Points in theRangeExp λ′ d′ Test 5% Rej.

TB-1 TB TB± 10 TB± 30

Under the null (β = 1)

E′ - 0, 0 LMτ .080 - - - -LPt .052 - - - -

LPt* .054 - - - -.25, .5 5, 5 LMτ .022 .006 .020 .462 .876

LPt .272 .000 .400 .630 .930LPt* .286 .418 .000 .612 .934

.25, .5 10, 10 LMτ .018 .002 .020 .738 .990LPt .882 .000 .966 .988 .996

LPt* .886 .968 .000 .984 .998.25,.75 5, 5 LMτ .042 .004 .018 .550 .948

LPt .262 .004 .318 .576 .954LPt* .296 .338 .000 .582 .956

.2, .3 5, 5 LMτ .056 .002 .000 .152 .486LPt .146 .000 .152 .248 .528

LPt* .158 .136 .000 .250 .534

Under the alternative (β = .9)

F′ .25, .5 0, 0 LMτ .124 - - - -LPt .098 - - - -

LPt* .096 - - - -.25, .5 5, 5 LMτ .050 .006 .040 .534 .942

LPt .346 .004 .514 .728 .956LPt* .370 .560 .000 .738 .964

.25, .5 10, 10 LMτ .040 .000 .042 .726 1.00LPt .968 .000 .996 1.00 1.00

LPt* .968 .998 .000 1.00 1.00.25,.75 5, 5 LMτ .064 .006 .026 .590 .970

LPt .348 .002 .412 .664 .966LPt* .372 .454 .000 .664 .968

.2, .3 5, 5 LMτ .130 .002 .002 .198 .542LPt .246 .000 .248 .362 .620

LPt* .258 .238 .000 .358 .626

Note: Critical values of the model without breaks are used.

23

Table 5Bias Effects of Using Incorrect Break Points (T = 100)

(Model A)

(a) Under the Null (β = 1)

β σd Test BreakPoint

5%Rej.

Emp.Crit. Bias MSE Bias MSE

5, 5 τ∼

TB -2 .065 -3.16 -.086 .010 .181 .039TB -1 .065 -3.14 -.086 .010 .181 .039

TB .063 -3.14 -.084 .010 -.038 .006TB+1 .065 -3.14 -.086 .010 .181 .039

τ^TB -2 .239 -4.86 -.189 .040 .120 .021TB -1 .376 -5.49 -.210 .050 .107 .018

TB .514 -6.50 -.241 .068 .087 .015TB+1 .027 -3.85 -.141 .028 .159 .032

τ^*TB –2 .380 -5.48 -.215 .052 .097 .016TB –1 .523 -6.56 -248 .072 .075 .013

TB .049 -4.10 -.145 .026 -.070 .010TB+1 .026 -3.84 -.144 .029 .150 .029

10, 10 τ∼

TB –2 .042 -3.00 -.083 .009 .678 .468TB –1 .043 -2.98 -.083 .009 .678 .469

TB .063 -3.14 -.084 .010 -.038 .006TB+1 .042 -3.00 -.083 .009 .678 .468

τ^TB –2 .745 -5.57 -.250 .066 .522 .280TB –1 .903 -7.01 -.312 .103 .460 .221

TB .960 -10.9 -.430 .201 .335 .132TB+1 .009 -3.48 -.141 .037 .651 .432

τ^*TB -2 .907 -6.99 -.318 .107 .450 .212TB -1 .962 -10.99 -.440 .210 .319 .122

TB .049 -4.10 -.145 .026 -.070 .010TB+1 .009 -3.48 -.143 .039 .644 .423

24

(b) Under the Alternative (β = .9)

β σd Test BreakPoint

5%Rej.

Emp.Crit. Bias MSE Bias MSE

5, 5 τ∼

TB -2 .141 -3.45 -.017 .003 .194 .045TB -1 .138 -3.46 -.017 .003 .195 .045

TB .281 -3.86 -.045 .006 -.026 .006TB+1 .141 -3.48 -.017 .003 .194 .045

τ^TB -2 .477 -5.20 -.149 .027 .127 .022TB -1 .661 -5.90 -.182 .039 .108 .018

TB .794 -7.24 -.232 .064 .077 .014TB+1 .061 -4.19 -.111 .021 .176 .038

τ^*TB –2 .659 -5.91 -.186 .040 .098 .016TB –1 .797 -7.27 -.239 .067 .066 .012

TB .137 -4.58 -.114 .019 -.051 .008TB+1 .060 -4.18 -.113 .022 .167 .035

10, 10 τ∼

TB –2 .023 -2.84 .010 .001 .692 .487TB –1 .024 -2.84 .010 .001 .392 .487

TB .281 -3.86 -.045 .006 -.026 .006TB+1 .025 -2.86 .010 .001 .691 .487

τ^TB –2 .949 -5.78 -.191 .039 .515 .273TB –1 .995 -7.37 -.274 .079 .436 .198

TB .999 -12.05 -.440 .206 .260 .085TB+1 .016 -3.67 -.110 .031 .666 .452

τ^*TB -2 .995 -7.33 -.278 .081 .427 .190TB -1 .999 -12.13 -.449 .213 .246 .077

TB .137 -4.58 -.114 .019 -.051 .008TB+1 .015 -3.66 -.113 .032 .659 .443

25

Table 6Empirical Results

LPt* LPt LMτ Null ModelSERIES Model

k̂ T̂BStat.

k̂ T̂BStat.

k̂ T̂BStat.

d̂1*, d̂2* a, b T̂B

Real GNP A 2 19281937

-7.00* 1 19291940

-6.65* 7 19201941

-3.62 3.09, -2.67(2.97, -2.65)

19211929

Nominal GNP A 8 19191928

-7.50* 8 19201929

-7.42* 8 19201948

-3.65 -4.84, -3.461(-4.80, -3.27)

19201931

Per-capita real GNP A 2 19281939

-6.88* 2 19291939

-6.67* 7 19201941

-3.68 3.07, -2.60(2.94, -2.57)

19211929

Industrial Production A 8 19171928

-7.67* 8 19181929

-7.78* 8 19201930

-4.32* -3.73, -4.38(-3.63, -4.13)

19201931

Employment A 8 19281955

-6.80* 8 19291956

-6.83* 7 19201945

-3.91 -2.90, 2.51(-2.73, 2.33)

19311941

Unemployment Rate A 7 19281941

-6.31* 7 19291941

-6.63* 7 19261942

-4.47* -3.43, 1.97(-3.38, 1.83)

19171920

GNP Deflator A 8 19161920

-4.74 1 19291945

-4.64 1 19191922

-3.18 3.88, -8.49(3.73, -7.14)

19171921

CPI A 2 19141944

-4.03 5 19151940

-4.04 4 19161941

-3.92 -2.44, -7.78(-7.13, -2.41)

19201930

Nominal Wage A 7 19141929

-5.85 7 19301949

-5.59 7 19211942

-3.84 -3.75, -2.98(-3.62, -2.89)

19201931

Real Wage C 4 19211940

-6.27 4 19221940

-6.63 8 19221939

-6.24* -3.10, -.57(-2.54, -3.01)

19311945

Money Stock A 8 19291958

-6.22 8 19301958

-6.03 7 19271931

-4.31* -3.54, -3.63(-3.50, -3.50)

19201931

Velocity A 1 18831953

-4.62 1 18841949

-4.77 1 18931947

-2.52 2.33, -2.28(2.32, -2.27)

19411944

Interest Rates A 2 19311957

-1.74 2 19321958

-1.74 3 19491958

-1.58 2.67, -2.50(2.64, -2.45)

19171921

SP500 C 1 19251938

-6.37 1 19241937

-6.12 3 19251941

-5.57 3.12, 3.35(4.82, 2.54)

19281932

Note: * denotes significant at 5%. a: Standardized coefficients (d^

i* = (d^

i/σ^) are reported. b: t-statistics for di = 0 are given in parentheses.

26

Appendix

Proof of Theorem 1

(a) Model A

Amsler and Lee (1995) derive the asymptotic distributions of LM test statistics

with one known, or exogenous, structural break. Here, we consider a more general case

with a finite number of, say, m << T structural breaks. Let Zt = (t, Wt′)′, where Wt =

(D1t,..,Dmt)′ and δ = (δ1, δ2′)′. From the regression of ∆yt on ∆Zt, we obtain δ∼ =

(∆Z′∆Z)−1∆Z′∆y, where ∆Z = (∆Z1, ∆Z2,.., ∆ZT)′, and ∆y = (∆y1, ∆y2,.., ∆yT)′. Following

SP, we define S∼

t = yt - (y1 - Z1δ∼) - Ztδ∼

. Letting St = ∑j=2

t εj and [rT] be the integer part of

rT, r ∈ [0,1], we obtain

T -1/2

S∼

[rT] = T

-½S

[rT] - T -1([rT]-1) T ½(δ∼

1-δ

1) – T -1(W

[rT] -W1)′ T ½(δ∼

2-δ

2 ) . (A.1)

The first term on the right hand side of (A.1) follows T-1/2S[rT] → σW(r). For the second

term, we note T(δ∼1-δ1) = (

1 T i′Μ∆W i)

-1 1 T

i′Μ∆Wε, where Μ∆W=I-∆W(∆W′∆W)-1∆W′.

Here, 1 T i′Μ∆W i → 1, since i′∆W = im′ (1 × m vector of ones), and i′Μ∆W i = T-m-1.

Then,

1 T

i′Μ∆Wε = 1

T ∑j=2

T

εj - 1

T ∑i=1

m

εTbi+1 → σW(1) ,

T -1

([rT]-1) T ½(δ∼

1-δ

1) → σ rW(1) .

We can show that the third term vanishes asymptotically. Since W[rT]

-W1 → im,

T(δ∼2-δ

2) = (

1 T ∆W′Μ1 ∆W)-1 1

T ∆W′Μ1ε = op(1) ,

27

whereas ∆W′Μ1∆W = Im - ImT-1→ Im, and ∆W′Μ1ε = (εTB1+1,..,εTBm+1)′ - imε_. Thus,

combining results, we can show that the terms in (A.1) follow

T -½

S∼

[rT] → σ[W(r) - rW(1)] = σV(r) , (A.2)

where V(r) is a Brownian bridge. This is the same expression as obtained from the usual

SP test ignoring a break (see the equation before (A3.1) in SP, 1992, p. 283). We also

define a demeaned Brownian bridge V_(A)(r) = V(r) –⌡⌠0

1V(r)dr as in SP, and it follows that

ρ∼ and τ∼ have the same asymptotic distributions as the usual SP tests not allowing for

structural breaks.

(b) Model C

We let Zt = (t, Wt*′)′, and Wt* = (D1t,D2t, DT1t*,D2t*)′. We define u

∼1 = 0 and u

∼t = OLS

residuals, t = 2,..,T, from the regression

∆yt = ∆Ztδ + ut . (A.3)

Note that the invariance result does not hold in Model C, and the test statistics in (5)

depend on λ, the parameter indicating the break points. Thus, we define a de-break

Brownian bridge V(C)(λ,r) as a residual process projected onto the subspace generated by

dz(λ,r) = [1, d1(λ1,r), d2(λ2,r)], where dj(λj,r) = 1 if r > λj, for j=1,2, and 0 otherwise.

Here, this expression is free of the effect of bj(λj,r) asymptotically, where bj(λj,r) = 1 if r

= λj, j=1,2, and 0 otherwise, but depends on λ. Then, the residuals from (A.3) follow

1 T

∑j=1

[rT]

u∼

j → V(C)(λ,r) .

Further, as in Perron (1997), we consider the following regression

28

∆yt = δ(λ)'∆Zt(λ) + φ(λ) S∼t-1(λ) + et , t = 2,..,T, (A.4)

where S∼

t(λ) = ∑j=2

t εj - (δ∼(λ)' - δ(λ)')(Zt(λ)- Z1(λ)). We let M∆Ζ(λ) = I - P∆Ζ(λ), where

P∆Ζ(λ) = ∆zT(λ)[∆zT(λ)′∆zT(λ)]-1∆zT(λ), and where ∆zT(λ) = (∆z1,T(λ),..,∆zT,T(λ))′. Pre-

multiplying (A.4) by M∆Ζ(λ), we obtain

M∆Ζ(λ)∆Y = φ(λ) M∆Ζ(λ)S∼

1(λ) + M∆Ζ(λ) e , (A.5)

where ∆Y = (∆y2,.., ∆yT)′, S∼1(λ) = (S∼1(λ),..,S∼T-1(λ))′ and e = (e2,..,eT)′. Then, the τ∼ statistic

in (5a) can be written as

τ∼ = [T -2S∼

1(λ)′ M∆Ζ(λ) S∼

1(λ)] -1/2[T -1S∼

1(λ)′ M∆Ζ(λ) e] / sT(λ) ,

where sT(λ) is the corresponding standard error of the regression. We obtain

T -2S∼

1(λ)′ M∆Ζ(λ) S∼

1(λ) = σ2

⌡⌠0

1[ST(r) - P∆Ζ(λ) ST(r)]2 dr , (A.6)

T -1S∼

1(λ)′ M∆Ζ(λ) e = σ2

⌡⌠0

1ST(r)dST(r) - σ2

⌡⌠0

1P∆Ζ(λ) ST(r)dST(r) . (A.7)

The effect of applying M∆Ζ(λ) or P∆Ζ(λ) to the above expressions is twofold; one is to

demean the process, and the other is to de-trend the structural dummy effect. Then, it is

given that

⌡⌠0

1[ST(r) - P∆Ζ(λ) ST(r)]2 dr = σ2

⌡⌠0

1V_(C)(λ,r)2 dr ,

where V_(C)(λ,r) is a demeaned and de-breaked Brownian bridge. The rest of the proof

follows that of SP, except that V_(C)(λ,r) replaces V_(r) in the expressions in the asymptotic

distribution of the SP statistics.

Proof of Corrolary1

29

The main procedure of the proof is to show continuity of a composite function.

We simply utilize the result of Zivot and Andrews (1992) on continuity of the composite

functional and make a note on corresponding notations. The minimum LMτ statistic can

be expressed as

Inf τ∼(λ∼) = g[ST(r), V_T(λ,r), ⌡⌠0

1ST(r)dST(r), ⌡⌠0

1P∆Ζ(λ) ST(r)dST(r), s2] + op(1) ,

where g = h*[h[H1(•), H2(•), sT(λ)]], with h*(m) = Inf m(•) for any real function m(•),

and h[m1, m2, m3] = m1-1/2m2/m3. The functionals H1 and H2 are defined by (A.6) and

(A.7) for Model C, while the term λ is absent in these expressions for Model A.

Continuity of h* and h is proved in ZA.