Minimum LM Unit Root Test with Two Structural Breaks
Junsoo Lee Mark Strazicich
Department of Economics University of Central Florida
July 1999
Abstract The two-break unit root test of Lumsdaine and Papell (1997) is examined and found to suffer from bias and spurious rejections in the presence of structural breaks under the null. A two-break minimum LM unit root test is proposed as a remedy. The two-break LM test does not suffer from bias and spurious rejections and is mostly invariant to the size, 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 Endogenous Break
Corresponding author: Junsoo Lee, Associate Professor, Department of Economics, University of Central Florida, Orlando, FL, 32816-1400, USA. Telephone: 407-8232070. Fax: 407-823-3269. E-mail:
[email protected]. We thank John List for helpful comments.
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 real GDP, and unemployment rates among OECD countries.
1
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
2
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 time series can be described adequately by Model A or C.
3
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: k
yt = α0 + α1t + α2B1t + α3B2t + α4D1t + α5D2t + φ yt-1 +
∑ cj∆yt-j+ et .
(3)
j=1
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 “τ^*” and when including Bjt terms as “τ^.” 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,
4
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 St-1 + ut ,
(4)
∼ ∼ ∼ ∼ ∼ where St = yt - ψx - Ztδ, t=2,..,T, δ are coefficients in the regression of ∆yt on ∆Zt, and ψx ∼ is the restricted MLE of ψx (≡ ψ + X0) given by y1 - Z1 δ (see SP). To correct for
3
Perron (1993) and Perron and Vogelsang (1992) rectify their tests by adding the Bt term in order to eliminate 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 null hypothesis 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 by ignoring a break under the null. Recently, Leybourne, Mills, and Newbold (1998) show that using a standard Dickey-Fuller test (without break) can lead to spurious rejection of the null if a structural break occurs early in the series. Contrary to this, Lee (1999) shows that this spurious rejection problem does not occur for the LM unit root test of Schmidt and Phillips (1992).
5
∼ autocorrelated errors, one can include augmented terms ∆St-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% , ∼ τ = t-statistic testing the null hypothesis φ = 0 .
(5a) (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: 1 σe2 ∼ ρ→_ (m) 2(r)dr] , [ 1V 2 σ2 ⌠ ⌡0 1 σ ∼ τ→_ (m) 2(r)dr]-1/2 , [ 1V 2 σe ⌠ ⌡0
(6a) (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
6
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 twobreak 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 ρ(λ) ,
(7a)
∼ LMτ = Inf τ (λ) .
(7b)
λ
λ
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: 1 σe2 1 _ (m )2(r) dr} ] , 2 {⌠ V 2 σ ⌡ λ 0 1 σ 1 (m)2 LMτ → Inf [V _ (r) dr]-1/2 . ⌠ 2 σ ⌡ e λ 0
LMρ → Inf [ -
(8a) (8b)
To derive critical values for the two-break LM unit root test we generate pseudoiid 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
7
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. 8
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.
9
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
10
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 break point combinations.
11
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, ∆St-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
12
[.1T, .9T].6 After determining the “optimal k,” we determine the break points where the tstatistic 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 shown to perform better than other data-dependant procedures (see, e.g., Ng and Perron, 1995). The trimming of end 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 (Model C) in Lumsdaine and Papell (1997) for the LPt and LPt* test statistics (asymptotically equivalent). For comparison, LM test critical values were derived using the same sample size and trimming as in Lumsdaine and 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.
13
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.
14
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 Amsler, C. and J. Lee. (1995). “An LM Test for a Unit-Root in the Presence of a Econometric Theory 11, 359-368.
and Long-Run Growth: Evidence from Two Structural Breaks,” Working Paper, University of Houston.
Process Among the G7 Countries,” Working Paper, University of Houston. Lee, J. (1999). “The End-Point Issue and the LM Unit Root Test,” Working Paper,
Lee, J., J. List, and M. Strazicich. (1998). “Spurious Rejections with the Minimum Unit Root 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 Root Tests and Spurious Rejections of the Null,” Working Paper, University of Central
Lee. J. and M. Strazicich. (1999b). “Minimum LM Unit Root Tests,” Working Paper, University of Central Florida.
Fuller Tests in the Presence of a Break Under the Null,” Journal of Econometrics 191-203. Lumsdaine, R. and D. Papell. (1997). “Multiple Trend Breaks and the Unit-Root Review of Economics and Statistics, 212-218.
Time Series,” Journal of Monetary Economics Ng and Perron. (1995). “Unit Root Tests in ARMA Models with Data-Dependent Methods for the Selection of the Truncation Lag,” Association 90, 269-281.
Evidence on the Great Crash and the Unit Root Hypothesis Reconsidered,” Oxford 59, 435-448. Papell, D., C. Murray, and H. Ghiblawi. (1999). “The Structure of Unemployment,”
Perron, P. (1989). “The Great Crash, the Oil Price Shock, and the Unit Root Hypothesis,” Econometrica 57, 1361-1401. Perron, P. (1997). “Further Evidence on Breaking Trend Functions in Macroeconomic Variables,” Journal of Econometrics 80, 355-385. Perron, P. (1993). “Erratum,” Econometrica 61, 248-249. Perron, P. and T.J. Vogelsang. (1992), “Testing for a Unit Root in Time Series with a Changing Mean: Corrections and Extensions,” Journal of Business and Economic Statistics 10, 467-470. Phillips, P.C.B. and P. Perron. (1988). Regression,” Biometrika 75, 335-346.
“Testing for a Unit Root in Time Series
Schmidt, P. and P.C.B. Phillips. (1992) “LM Tests for a Unit Root in the Presence of Deterministic Trends,” Oxford Bulletin of Economics and Statistics 54, 257-287. Zivot, E. and D. W. K. Andrews. (1992). “Further Evidence on the Great Crash, the OilPrice Shock and the Unit Root Hypothesis,” Journal of Business and Economic Statistics 10, 251-270.
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Table 1 Critical Values of Exogenous LM Tests (T=100) (1) Model A ∼ τ ∼ ρ
1%
5%
10%
-3.63
-3.06
-2.77
-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 λ1 .2 .4 .6
.4 -4.83, -4.19, -3.89 -
λ2 .6 -4.93, -4.31, -4.00 -4.91, -4.33, -4.03 -
.8 -4.76, -4.19, -3.88 -4.88, -4.32, -4.03 -4.84, -4.19, -3.89
∼ (ii) ρ λ1 .2 .4 .6
.4 -38.1, -30.2, -26.4 -
λ2 .6 -39.4, -31.6, -27.9 -39.2, -31.8, -28.1 -
.8 -37.2, -30.1, -26.3 -38.7, -31.7, -28.1 -38.3, -30.2, -26.4
Note: Critical values are at the 1%, 5%, and 10% levels, respectively.
18
Table 2 Critical Values of the Endogenous Two-Break Minimum Tests (T = 100) (1) Model A
LMτ LMρ LPt LPt*
1% -4.545 -35.73 -6.420 -6.400
5% -3.842 -26.89 -5.913 -5.853
10% -3.504 -22.89 -5.587 -5.560
Note: The DGP in the simulation does not include breaks. The LP tests are affected by breaks, while LM tests are invariant to breaks in Model A.
(2) Model C (I)
LMτ LMρ LPt LPt*
1% -5.825 -52.551 -6.936 -6.945
5% -5.286 -45.532 -6.386 -6.344
10% -4.989 -41.664 -6.108 -6.064
Note: The DGP in the simulation does not include breaks. Both LP and LM tests are affected by breaks in Model C. (3) Model C (II) (i) LMτ λ1 .2 .4 .6
.4 -6.16, -5.59, -5.28 -
λ2 .6 -6.40, -5.74, -5.32 -6.46, -5.67, -5.31 -
.8 -6.33, -5.71, -5.33 -6.42, -5.65, -5.32 -6.32, -5.73, -5.32
(ii) LMρ λ1 .2 .4 .6
.4 -55.5, -47.9, -44.0
λ2 .6 -58.6, -50.0, -44.4 -59.3, -49.0, -44.3
.8 -57.6, -49.6, -44.6 -58.8, -48.7, -44.5 -57.5, -49.8, -44.4
Note: Critical values are at the 1%, 5%, and 10% levels, respectively.
19
Table 3 Rejection Rates of the Exogenous Two-Break Tests (T = 100) (1) Model A Exp R
DGP λ′
d′
A
0
-
-
B
2
.25, .50
2
.25, .75
2
.25, .50 .25, .50 .25, .50 .25, .50 .25, .50 .25, .50 .25, .50
5, 5 10, 10 5, 5 10, 10 5, 5 5, 5 5, 5 10, 10 10, 10 10, 10 5, 5 10, 10
C
2
D
2
Estimation Re λe′ 2 2 2 2 2 2 2 0 1 1 0 1 1 2 2
.25, .50 .25, .75 .50, .75 .25, .50 .25, .50 .25, .75 .25, .75 .25 .50 .25 .50 .25, .75 .25, .75
Under the Null (β = 1.0) ^ ^ ∼ τ* τ τ .047 .040 .042 .047 .040 .040 .047 .039 .039 .047 .487 .042 .047 .955 .042 .047 .485 .040 .047 .956 .040 .053 .005 .005 .045 .109 .013 .045 .092 .012 .037 .001 .001 .032 .245 .005 .031 .156 .003 .046 .151 .028 .033 .292 .010
Under the Alternative (β = 0.9) ^ ^ ∼ τ* τ τ .243 .114 .113 .239 .110 .108 .240 .115 .114 .243 .763 .113 .243 .998 .113 .239 .757 .108 .239 .997 .108 .125 .010 .010 .151 .201 .033 .137 .127 .023 .019 .000 .000 .040 .305 .004 .026 .085 .001 .147 .250 .058 .038 .359 .010
(2) Model C Exp R
DGP λ′
d′
A′
0
-
-
B′
2
.25, .50
2
.25, .75
2
.25, .50 .25, .50 .25, .50 .25, .50 .25, .50 .25, .50 .25, .50
5, 5 10, 10 5, 5 10, 10 5, 5 5, 5 5, 5 10, 10 10, 10 10, 10 5, 5 10, 10
C′
2
D′
2
Estimation Re λe′ 2 2 2 2 2 2 2 0 1 1 0 1 1 2 2
.25, .50 .25, .75 .50, .75 .25, .50 .25, .50 .25, .75 .25, .75 .25 .50 .25 .50 .25, .75 .25, .75
Under the Null (β = 1.0) ^ ^ ∼ τ* τ τ .052 .050 .050 .048 .052 .050 .051 .055 .054 .052 .625 .050 .052 .986 .050 .048 .627 .051 .048 .986 .051 .000 .000 .000 .002 .011 .004 .004 .080 .005 .000 .000 .000 .000 .001 .000 .000 .103 .000 .015 .045 .015 .000 .006 .000
20
Under the Alternative (β = 0.9) ^ ^ ∼ τ* τ τ .118 .101 .097 .115 .101 .099 .119 .105 .101 .118 .773 .097 .118 .998 .097 .115 .773 .099 .115 .999 .099 .000 .000 .000 .003 .007 .006 .004 .102 .008 .000 .000 .000 .000 .000 .000 .000 .096 .000 .014 .031 .016 .000 .001 .000
Table 4 Rejection Rates and Estimated Break Points of the Endogenous Break Tests (T = 100) (1) Model A Exp
λ′
d′
Test
Frequency of Estimated Break Points in the Range
5% Rej.
TB-1
TB
TB± 10
TB± 30
.126 .176 .000 .252 .818 .000 .028 .162 .000 .266 .136 .000
.254 .402 .426 .500 .882 .890 .152 .392 .408 .438 .318 .286
.766 .804 .816 .804 .976 .978 .646 .752 .760 .668 .598 .586
.480 .600 .624 .778 .990 .986 .304 .594 .616 .580 .462 .434
.808 .912 .910 .894 .998 .996 .734 .820 .824 .738 .698 .702
Under the null (β = 1) E
-
0, 0
.25, .5
5, 5
.25, .5
10, 10
.25,.75
5, 5
.2, .3
5, 5
LMτ LPt LPt* LMτ LPt LPt* LMτ LPt LPt* LMτ LPt LPt* LMτ LPt LPt*
.044 .046 .038 .064 .192 .180 .034 .748 .748 .052 .170 .166 .040 .206 .174
.000 .000 .206 .002 .000 .826 .000 .008 .190 .000 .000 .148
Under the alternative (β = .9) F
.25, .5
0, 0
.25, .5
5, 5
.25, .5
10, 10
.25,.75
5, 5
.2, .3
5, 5
LMτ LPt LPt* LMτ LPt LPt* LMτ LPt LPt* LMτ LPt LPt* LMτ LPt LPt*
.282 .098 .078 .172 .318 .292 .042 .954 .944 .204 .298 .284 .130 .336 .302
.000 .002 .356 .000 .000 .974 .004 .016 .318 .000 .000 .262
21
.276 .332 .000 .564 .972 .000 .132 .304 .000 .364 .248 .000
(2) Model C Exp
λ′
d′
Test
Frequency of Estimated Break Points in the Range
5% Rej.
TB-1
TB
TB± 10
TB± 30
.020 .400 .000 .020 .966 .000 .018 .318 .000 .000 .152 .000
.462 .630 .612 .738 .988 .984 .550 .576 .582 .152 .248 .250
.876 .930 .934 .990 .996 .998 .948 .954 .956 .486 .528 .534
.534 .728 .738 .726 1.00 1.00 .590 .664 .664 .198 .362 .358
.942 .956 .964 1.00 1.00 1.00 .970 .966 .968 .542 .620 .626
Under the null (β = 1) E′
-
0, 0
.25, .5
5, 5
.25, .5
10, 10
.25,.75
5, 5
.2, .3
5, 5
LMτ LPt LPt* LMτ LPt LPt* LMτ LPt LPt* LMτ LPt LPt* LMτ LPt LPt*
.080 .052 .054 .022 .272 .286 .018 .882 .886 .042 .262 .296 .056 .146 .158
.006 .000 .418 .002 .000 .968 .004 .004 .338 .002 .000 .136
Under the alternative (β = .9) F′
.25, .5
0, 0
.25, .5
5, 5
.25, .5
10, 10
.25,.75
5, 5
.2, .3
5, 5
LMτ LPt LPt* LMτ LPt LPt* LMτ LPt LPt* LMτ LPt LPt* LMτ LPt LPt*
.124 .098 .096 .050 .346 .370 .040 .968 .968 .064 .348 .372 .130 .246 .258
.006 .004 .560 .000 .000 .998 .006 .002 .454 .002 .000 .238
.040 .514 .000 .042 .996 .000 .026 .412 .000 .002 .248 .000
Note: Critical values of the model without breaks are used.
22
Table 5 Bias Effects of Using Incorrect Break Points (T = 100) (Model A) (a) Under the Null (β = 1) d
5, 5
Test
∼ τ
^ τ
^ τ*
10, 10
∼ τ
^ τ
^ τ*
β
σ
Break Point
5% Rej.
Emp. Crit.
Bias
MSE
Bias
MSE
TB -2 TB -1 TB TB+1
.065 .065 .063 .065
-3.16 -3.14 -3.14 -3.14
-.086 -.086 -.084 -.086
.010 .010 .010 .010
.181 .181 -.038 .181
.039 .039 .006 .039
TB -2 TB -1 TB TB+1
.239 .376 .514 .027
-4.86 -5.49 -6.50 -3.85
-.189 -.210 -.241 -.141
.040 .050 .068 .028
.120 .107 .087 .159
.021 .018 .015 .032
TB –2 TB –1 TB TB+1
.380 .523 .049 .026
-5.48 -6.56 -4.10 -3.84
-.215 -248 -.145 -.144
.052 .072 .026 .029
.097 .075 -.070 .150
.016 .013 .010 .029
TB –2 TB –1 TB TB+1
.042 .043 .063 .042
-3.00 -2.98 -3.14 -3.00
-.083 -.083 -.084 -.083
.009 .009 .010 .009
.678 .678 -.038 .678
.468 .469 .006 .468
TB –2 TB –1 TB TB+1
.745 .903 .960 .009
-5.57 -7.01 -10.9 -3.48
-.250 -.312 -.430 -.141
.066 .103 .201 .037
.522 .460 .335 .651
.280 .221 .132 .432
TB -2 TB -1 TB TB+1
.907 .962 .049 .009
-6.99 -10.99 -4.10 -3.48
-.318 -.440 -.145 -.143
.107 .210 .026 .039
.450 .319 -.070 .644
.212 .122 .010 .423
23
(b) Under the Alternative (β = .9) d
5, 5
Test
∼ τ
^ τ
^ τ*
10, 10
∼ τ
^ τ
^ τ*
β
σ
Break Point
5% Rej.
Emp. Crit.
Bias
MSE
Bias
MSE
TB -2 TB -1 TB TB+1
.141 .138 .281 .141
-3.45 -3.46 -3.86 -3.48
-.017 -.017 -.045 -.017
.003 .003 .006 .003
.194 .195 -.026 .194
.045 .045 .006 .045
TB -2 TB -1 TB TB+1
.477 .661 .794 .061
-5.20 -5.90 -7.24 -4.19
-.149 -.182 -.232 -.111
.027 .039 .064 .021
.127 .108 .077 .176
.022 .018 .014 .038
TB –2 TB –1 TB TB+1
.659 .797 .137 .060
-5.91 -7.27 -4.58 -4.18
-.186 -.239 -.114 -.113
.040 .067 .019 .022
.098 .066 -.051 .167
.016 .012 .008 .035
TB –2 TB –1 TB TB+1
.023 .024 .281 .025
-2.84 -2.84 -3.86 -2.86
.010 .010 -.045 .010
.001 .001 .006 .001
.692 .392 -.026 .691
.487 .487 .006 .487
TB –2 TB –1 TB TB+1
.949 .995 .999 .016
-5.78 -7.37 -12.05 -3.67
-.191 -.274 -.440 -.110
.039 .079 .206 .031
.515 .436 .260 .666
.273 .198 .085 .452
TB -2 TB -1 TB TB+1
.995 .999 .137 .015
-7.33 -12.13 -4.58 -3.66
-.278 -.449 -.114 -.113
.081 .213 .019 .032
.427 .246 -.051 .659
.190 .077 .008 .443
24
SERIES Real GNP
Model A
^k 2
LPt* Stat. ^ T B
1928 1937 1919 1928 1928 1939 1917 1928 1928 1955 1928 1941 1916 1920 1914 1944 1914 1929 1921 1940 1929 1958 1883 1953 1931 1957 1925 1938
-7.00*
Table 6 Empirical Results LPt Stat. ^k ^ ^k T B
1
LMτ ^ T B
Stat.
Null Model ^d *, ^d * a, b 1
2
1929 7 1920 -3.62 3.09, -2.67 -6.65* 1940 1941 (2.97, -2.65) Nominal GNP A 8 -7.50* 8 1920 8 1920 -3.65 -4.84, -3.461 -7.42* 1929 1948 (-4.80, -3.27) Per-capita real GNP A 2 -6.88* 2 1929 7 1920 -3.68 3.07, -2.60 -6.67* 1939 1941 (2.94, -2.57) Industrial Production A 8 -7.67* 8 1918 8 1920 -4.32* -3.73, -4.38 -7.78* 1929 1930 (-3.63, -4.13) Employment A 8 -6.80* 8 1929 7 1920 -3.91 -2.90, 2.51 -6.83* 1956 1945 (-2.73, 2.33) Unemployment Rate A 7 -6.31* 7 1929 7 1926 -4.47* -3.43, 1.97 -6.63* 1941 1942 (-3.38, 1.83) GNP Deflator A 8 -4.74 1 1929 -4.64 1 1919 -3.18 3.88, -8.49 1945 1922 (3.73, -7.14) CPI A 2 -4.03 5 1915 -4.04 4 1916 -3.92 -2.44, -7.78 1940 1941 (-7.13, -2.41) Nominal Wage A 7 -5.85 7 1930 -5.59 7 1921 -3.84 -3.75, -2.98 1949 1942 (-3.62, -2.89) Real Wage C 4 -6.27 4 1922 -6.63 8 1922 -6.24* -3.10, -.57 1940 1939 (-2.54, -3.01) Money Stock A 8 -6.22 8 1930 -6.03 7 1927 -4.31* -3.54, -3.63 1958 1931 (-3.50, -3.50) Velocity A 1 -4.62 1 1884 -4.77 1 1893 -2.52 2.33, -2.28 1949 1947 (2.32, -2.27) Interest Rates A 2 -1.74 2 1932 -1.74 3 1949 -1.58 2.67, -2.50 1958 1958 (2.64, -2.45) SP500 C 1 -6.37 1 1924 -6.12 3 1925 -5.57 3.12, 3.35 1937 1941 (4.82, 2.54) ^ ^ ^ Note: * denotes significant at 5%. a: Standardized coefficients (di * = (di /σ) are reported. b: t-statistics for di = 0 are given in parentheses.
25
^
TB 1921 1929 1920 1931 1921 1929 1920 1931 1931 1941 1917 1920 1917 1921 1920 1930 1920 1931 1931 1945 1920 1931 1941 1944 1917 1921 1928 1932
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 λj, for j=1,2, and 0 otherwise. Here, this expression is free of the effect of bj(λ j,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
∼ ∑ uj → V(C)(λ,r) .
[rT] j=1
Further, as in Perron (1997), we consider the following regression
27
∼ ∆yt = δ(λ)'∆Zt(λ) + φ(λ) St-1(λ) + et , ∼ where St(λ) =
∑
t j=2
t = 2,..,T,
(A.4)
∼
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))′. Premultiplying (A.4) by M∆Ζ(λ), we obtain ∼ M∆Ζ(λ)∆Y = φ(λ) M∆Ζ(λ)S1(λ) + M∆Ζ(λ) e ,
(A.5)
∼ ∼ ∼ ∼ where ∆Y = (∆y2,.., ∆yT)′, S1(λ) = (S1(λ),..,ST-1(λ))′ and e = (e2,..,eT)′. Then, the τ statistic in (5a) can be written as ∼ ∼ ∼ ∼ τ = [T -2S1(λ)′ M∆Ζ(λ) S1(λ)] -1/2[T -1S1(λ)′ M∆Ζ(λ) e] / sT(λ) , where sT(λ) is the corresponding standard error of the regression. We obtain ∼ ∼ T -2S1(λ)′ M∆Ζ(λ) S1(λ) = σ2⌠1[ST(r) - P∆Ζ(λ) ST(r)]2 dr , ⌡0
(A.6)
∼ T -1S1(λ)′ M∆Ζ(λ) e = σ2⌠1ST(r)dST(r) - σ2⌠1P∆Ζ(λ) ST(r)dST(r) . ⌡0 ⌡0
(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 2 2 1 (C) 1 _ (λ,r)2 dr , ⌠0[ST(r) - P∆Ζ(λ) ST(r)] dr = σ ⌡ ⌠0V ⌡
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 28
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), ⌠1ST(r)dST(r), ⌠1P∆Ζ(λ) ST(r)dST(r), s2] + op(1) , ⌡0 ⌡0 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.
29
