E¢cient Unit Root Tests and Structural Change
when the Initial Observation is Drawn from its
Unconditional Distribution¤
Hui Liu
University of Ottawa
Gabriel Rodríguezy
University of Ottawa
This version: April 30 2004
(Still Preliminary)
Abstract
Following Perron and Rodríguez (2003) and Elliott (1999), we develop
e¢cient unit root tests in the context of structural change using GLS
detrended data (Elliott, Rothenberg and Stock, 1996) when the initial
observation is drawn from its unconditional distribution. We derive
the limiting distributions of the M-tests (Stock, 1999; Perron, and Ng,
1996), the ADF statistic and optimal point test. We simulate the …nite
sample size and power under various error processes, using di¤erent lag
selection methods and two di¤erent methods to select the break point.
Empirical applications are also provided.
Keywords: Unit Root, GLS Detrending, Optimal Point Test, Initial
Condition, Power Envelope.
JEL: C2, C5.
¤
This pap er is drawn from the second chapter of the PhD dissertation of Hui Lui at
the University of Ottawa.
y
Address for Correspondence: Department of Economics, University of Ottawa,
P.O. Box 450, Station A, Ottawa, Ontario, Canada, K1N 6N5. E-mail address:
gabrielr@uottawa.ca.
1
Introduction
There has been a large literature on unit root tests since the seminal work of
Nelson and Plosser (1982), where they applied augmented Dickey-Fuller test
(Dickey and Fuller, 1979; Said and Dickey, 1984) to 14 US macroeconomic
time series and found unit root in 13 of them.
Perron (1989) challenged this …nding by introducing a structural break
associated with major events under the null and alternative hypothesis. He
considered shift in level and/or slope a priori and concluded that most
of Nelson-Plosser series are stationary about a broken trend except CPI,
velocity of money and interest rate.
However Christiano (1992) argued that treating structural breaks as
known by pre-examining the data would give rise to data mining problems. To circumvent this shortcoming, two data-dependent methods were
proposed to estimate the break point endogenously by Zivot and Andrews
(1992) and Perron (1997). One is to select the break point that gives the
least favorable result for the null hypothesis (Zivot and Andrews, 1992),
and their empirical application to Nelson-Plosser (1982) data set found less
evidence against the null of a unit root than Perron (1989) for many of the
series, but stronger evidence against the null than Perron (1989) for some
time series (nominal GNP, real GNP, and industrial production). Another
one is to choose the break point that is related to the maximum of the absolute value of the t-statistic on the parameter associated with the change
in slope (Perron, 1997), and it found a rejection of unit root hypothesis for
most of Nelson-Plosser series except for Consumer Price Index, Velocity and
Interest Rate.
Perron and Rodríguez (2003) applied GLS detrending (ERS, 1996) to
the so-called M-tests (Stock, 1999) and the point optimal test (ERS, 1996)
and extended them to the case of unknown structural break. Their methods
are applied to two time series of Nelson-Plosser data - real wages and stock
prices, and a rejection is found for most of the test statistics analyzed
Another group of studies believe that the performance of unit root tests
also has dependence on the various methods of selecting the order of augmented autoregression. Campell and Perron (1991) proposed the so-called
sequential t-statistc method based on sequential testing on the signi…cance
of the coe¢cient on the last included lag. Ng and Perron (1995) showed that
data-dependent methods have a better performance than a …xed truncation
lag proportionate to the sample size, and when comparing with information
criteria based methods, they found that the sequential t-statistic method has
less size distortions and comparable power. In order to circumvent the size
1
problem of information criteria methods, Ng and Perron (2001) proposed
modi…ed AIC=BIC by including a penalty factor to under…tting. Using
these methods, they were able to have more acceptable size distortions when
there is a negative moving average error process.
But few papers explored the impact of initial observations on unit root
testing in large samples. Elliott (1999) addressed the loss of power when the
initial observation is drawn from its unconditional distribution under the
alternative hypothesis. He derived point optimal test under the new initial
condition and showed that power envelope shifted down from the one corresponding to the …xed initial value. On the other hand, Müller and Elliott
(2003) treated a variety of initial values under the alternative hypothesis
as nuisance parameters and derived a family of point optimal tests over a
weighting function of di¤erent initial conditions: They also related unit root
test statistics that don’t have optimality properties to this family of optimal tests, in order to understand what implicit assumptions these statistics
make on the initial condition. They found that many test statistics can be
closely related to the optimal test families but with very di¤erent weightings
for the initial condition.
This paper follows the research lines of Elliott (1999) and Perron and
Rodríguez (2003). Under the alternative hypothesis of stationarity, we consider an unknown break point and assume the initial value is drawn from its
unconditional distribution. We evaluate the performance of unit root tests
in both large and small samples. By doing so, we achieve a deeper understanding on the impact of the initial condition in unit root testing when
there is a structural break under the alternative hypothesis.
The rest of this paper is organized as follows. Next section derives the
limiting distributions of the MGLS , ADF GLS , and PTGLS statistics under the
new initial condition assumption. Section 4 calculates the asymptotic critical
values, the power envelope and the asymptotic power functions. Section 5
presents evidence on size and power in …nite sample. Section 6 shows an
empirical application and the last section concludes. All the proofs are in
the Appendix.
2
2.1
The Models and Asymptotic Theory
The models
For the purpose of comparison, this paper considers the same models and
tests as Perron and Rodríguez (2003) except that the initial condition is
2
di¤erent. Hence, the data generating process is,
yt = dt + ut
(1)
ut = ®ut¡1 + vt
® = 1 + cT ¡1
(2)
(3)
where dt = '0 zt includes the deterministic components. The so-called model
I contains a break in slope, therefore the deterministic components are:
zt = f1; t; 1(t ¸ TB ) (t ¡ TB )g
where 1(¢) is the indicator function, TB is³ the break
´0 point, and the set of
estimate of coe¢cients is denoted as '
^= ¹
^ 1; ¯^ 1; ¯^ 2 . Model II contains a
break in both intercept and slope, then the deterministic components are,
zt = f1; 1(t ¸ TB ) ; t; 1 (t ¸ TB ) (t ¡ TB )g
³
´0
and the set of estimate of coe¢cients is denoted as '
^ = ¹
^ 1; ¹
^ 2 ; ¯^ 1; ¯^ 2 :
Equation (3) represents the local to unity framework examined in Phillips
(1987) and Chan and Wei (1987), the parameter c measures the deviation
from unity. When c = 0, we are under null hypothesis, when c < 0; we are
under the alternative hypothesis. The innovations fvt g satis…es the mixingtype conditions (see Davidson 1994, for a general treatment) such that the
functional central limiting theorem
¡ 2can
¢ be applied to the partial sums St =
Pt
¡1=2
S[T r] ) N 0; ¾ r ´ ¾W (r) ; where ) signi…es weak
j=1 vj and T
convergence, W¡ (r)¢is a standard wiener process on the interval [0, 1], ¾2 =
limT !1 T ¡1E ST2 is the non-normalized spectral density at frequency zero.
Using the Continuous Mapping Theorem, we have the following limiting
distributions which hold throughout the paper,
R1
P
1. T ¡3=2 Tt=1 ut ) ¾ 0 Wc (r) dr;
R
P
2. T ¡2 Tt=1 u2t ) ¾2 01 Wc2 (r) dr;
nR
o
¡
¢
P
3. T ¡1 Tt=1 ut¡1 vt ) ¾2 01 Wc (r) dW (r) + ¸ with ¸ = ¾2 ¡ ¾2v =2¾2;
4. T ¡5=2
PT
t=1 tut¡1
) ¾2
R1
0
rWc (r) dr;
where Wc (r) is the Ornstein-Uhlenbeck process de…ned by the stochastic
di¤erential equation Wc (r) = cWc (r)+dW (r). In stead of assuming u0 = 0
as the initial condition; we have the following assumption adopted from
Elliott (1999),
3
Assumption 1. We assume u0 is zero when ® = 1; so u1 = v1; and u1
¾2
has mean zero and variance (1¡®
2) when ® < 1:
Under this assumption, the initial observation doesn’t disappear at the convergence rate of T 1=2 under the alternative hypothesis (see Lemma 1 in
Appendix), and it will have e¤ect on the limiting distribution. In order to
isolate the e¤ect of the initial condition, we also keep the GLS detrending
approach as Perron and Rodríguez (2003) and we use the same notation.
That is, we …rst transform the data into,
h¡
i
¢1=2
yt®¹ =
1 ¡®
¹2
y1; (1 ¡ ®
¹ L)yt
(4)
h¡
i
¢1=2
zt®¹ =
1 ¡®
¹2
z1; (1 ¡ ®
¹ L)zt
h¡
i
¢
¹
2 1=2
u®
=
1
¡
®
¹
u
(1
¡
®
¹
L)u
1;
t
t
for t = 2; ¢¢¢; T and ®
¹ = 1+¹cT ¡1: Then we calculate the set of coe¢cients related to the deterministic component by estimating the following regression
using OLS:
yt®¹ = '
^ 0z ¹® + u¹®
(5)
¡
¢
The limiting distribution of ¤ ¡1 (^
' ¡ ') with ¤ = diag T 1=2 ; T ¡1=2; T ¡1=2
is stated in Lemma 2 in the Appendix, and the new times series after eliminating the deterministic component is
y~t = yt ¡ '
^ 0 zt
2.2
(6)
ADF, M-Tests and their Limiting Distribution
We include two types of tests. One is the widely used ADF statistic (Dickey
and Fuller, 1979; Said and Dickey, 1984) which tests if ¯ = 0 in the following
augmented regression:
¢~
yt = ¯^ 0y~t¡1 +
k
X
¯^ j ¢~
yt¡j + e^tk
(7)
j=1
where the laged …rst di¤erences are used to account for the serial correlations.
The other tests are the so-called M-tests
by Stock (1999) based
¡ proposed
¢
on the idea that an I(1) process is Op T 1=2 whereas an I(0) process is
Op (1) : This class of tests include a modi…ed version of Phillips-Perron’s
4
(1988) Z® test; Sargan and Bhargava’s (1983) uniformly most powerful test
and Bhargava’s (1986) locally most powerful invariant tests; and PhillipsPerron’s (1988) Zt® test. Using the above de…ned y~t series, the M-tests
are:
MZ®GLS =
T
X
¡ 2
¢
2
y~T ¡ T ¡1 s2 (2T ¡2
y~t¡1
)¡1
(8)
t=1
MSB GLS = (T ¡2
T
X
y~2t¡1=s2)1=2
(9)
t=1
MZtGLS
T
¡ ¡1 2
¢ 2 ¡2 X
2
2
= T y~T ¡ s (4s T
y~t¡1
)¡1=2 :
(10)
t=1
Perron and Ng (1996) showed that the main advantage of the M¡tests
is that they have less size distortions when error term contains negative
moving average dynamics and in other cases they also have acceptable size
distortions. In Equations (8) - (10), an estimator of the spectral density at
the frequency zero, s2, is required. Following Perron and Ng (1996), we use
the following autoregressive estimate s2 :
³
´2
2
2
^
s = sek = 1 ¡ ¯(1)
P
P
^
where s2ek = (T ¡ k)¡1 Tt=k+1 e^2tk ; ¯(1)
= kj=1 ¯^ j ; f^
etk g and ¯^ j are obtained from the augmented regression (7)
The following theorem speci…es the limiting distribution of the unit root
statistics with initial condition as in Assumption 1.
Theorem 1 Suppose fyt g is generated by (1) to (3), the initial value is
given by Assumption 1, GLS-detrending is applied according to (4) to (6),
± = TB =T is the break point, then the M GLS and ADF GLS statistics for
Model I and II have the following limiting distribution
MZ®GLS (±) )
0:5g1 (c; c; ±)
GLS
´ J MZ® (c; c; ±)
g2(c; c; ±)
GLS
MSB GLS (±) ) (g2(c; c; ±))1=2 ´ J MSB
(c; c; ±)
0:5g1 (c; c; ±)
GLS
MZtGLS (±) )
´ J MZt (c; c; ±)
1=2
(g2 (c; c; ±))
0:5g1 (c; c; ±)
GLS
ADF GLS (±) )
´ J ADF
(c; c; ±)
1=2
(g2 (c; c; ±))
5
where
g1(c; c; ±) = Vcc(1) (1; ±)2 ¡ 2Vcc(2) (1; ±) ¡ 1;
Z1
Z1
g2(c; c; ±) =
Vcc(1) (r; ±)dr ¡ 2
Vcc(2) (r; ±)dr;
0
(1)
Vcc (r; ±)
(2)
Vcc (r; ±)
±
= Wc (r) ¡ b4 ¡ rb5;
= b6 (r ¡ ±) [Wc (r) ¡ rb5 ¡ b 4 ¡ (1=2) (r ¡ ±) b6] :
The elements b4 ; b5 ; b6 are calculated using
2 2
3 0
1
1 2
¹c ¡ 2¹c
c ¡ c¹
¡¹c (1 ¡ ±) + 12 c¹2 (1 ¡ ±)2
¾b1
2¹
4 1 ¹c2 ¡ ¹c
5 ¢ @ ¾b2 A
1 + 13 ¹c2 ¡ c¹ m
2
2
1 2
¾b3
¡¹
c (1 ¡ ±) + 2 ¹c (1 ¡ ±) m
d
0
1
b4
= ¾ @ b5 A
b6
where
b1 = ¡2¹c» ¡ ¹c (c ¡ ¹c)
b2 =
b3 =
m =
d =
» »
3
Z
1
wc (r) ¡ ¹cW (1) ;
Z 1
Z1
Z 1
(1 ¡ ¹c) W (1) + (c ¡ ¹c)
wc (r) ¡ ¹c (c ¡ c¹)
rwc (r) + c¹
wc (r) ;
0
0
0
R1
R1
(1 ¡ ¹c + ±¹c) W (1) + ¹c ± Wc (r) ¡ W (±) ¡ ¹c (c ¡ c¹) ± rWc (r)
R1
R1
;
+±¹c (c ¡ ¹c) ± Wc(r) + (c ¡ ¹c) ± Wc (r)
1
1
1
1 ¡ ± ¡ c¹ + c¹± ¡ ¹c2± + ¹c2 + ¹c2±3;
2
3
6
1 2
2
3
1 ¡ ± ¡ c¹ (1 ¡ ±) + ¹c (1 ¡ ±) ;
3
µ
¶
1
N 0;
:
¡2c
0
The Feasible Point Optimal Test and its Asymptotic Distribution
ERS (1996) showed that no uniformly optimal tests exist for unit root testing. Based on Dufour and King (1981), they developed a feasible point
6
optimal test, which has power function tangent to the power envelope at
one point of the alternative hypothesis. The statistic is de…ned as
PTGLS = fS (¹
®) ¡ ®
¹ S (1)g =s2
(11)
¡
¢
¡
¢
0
0
where S (¹
®) = (y ®¹ ¡ '0 z®¹ ) (y ®¹ ¡ '0 z®¹ ) and S (1) = y1 ¡ '0 z1 y 1 ¡ '0 z1 ;
the squared sum of residuals under the alternative and the null hypothesis
respectively.
Perron and Rodríguez (2003) extended PTGLS test to the case of an unknown structural break. We derive the limiting distribution considering the
e¤ect of the new initial condition and the results are summarized in the
following theorem.
Theorem 2 Suppose fy tg is generated by (1) to (3), the initial condition is
given by Assumption 1, GLS-detrending is applied according to (4) to (6),
the break point is ± = TB =T; then the PTGLS test for Model I and II have the
following limiting distribution
PTGLS
c; ±) ) ¡2¹c»2 ¡ 2¹
c
» (c; ¹
Z
0
1
Wc (r)dW(r) + (¹c2 ¡ 2¹cc)
¹ 0; ±) ¡ M(c;
¹ ¹c; ±) ¡ ¹c
+M(c;
Z
0
1
Wc2 (r)dr
GLS
´ J PT » (c; ¹c; ±)
¹ ¹c; ±) = A(c;
¹ ¹c; ±)0 B
¹ (¹c; ±) A(c;
¹ ¹c; ±), and A(c;
¹ c¹; ±), B
¹ (¹c; ±) are dewhere M(c;
…ned in the appendix. Unlike Perron and Rodríguez (2003), the term »
enters the limiting distribution of the statistic.
4
4.1
Selecting the Break Point and Asymptotic Results
Selecting the Break Point
We use two data-dependent methods to estimate the break point endogenously. One is the so-called in…mum method proposed by Zivot and Andrews
(1992), the other is the so-called supremum method suggested by Perron
(1997). These two methods were proposed to circumvent the data mining
problem caused by pre-examination of the data.
Zivot and Andrews (1992) proposed selecting the break point that gives
the strongest rejection against the null hypothesis of ® = 1: If smaller values
7
of the statistic lead to rejection of the null, then the break point ± can be
selected by
J (c; c¹) = inf J (c; ¹c; ±)
±2[0;1]
±2[0;1]
where J is the M GLS ; ADF GLS statistics derived above. The selection of
± for PTGLS
is slightly di¤erent. According to Perron and Rodríguez (2003),
»
we select ± using
PTGLS
c) = f inf
» (c; ¹
±2[";1¡"]
S(¹
®; ±) ¡
inf
±2[";1¡"]
®
¹ S(1; ±)g=s2
(13)
where a truncation ² is needed for critical values to be bounded and ² = 0:15
is used throughout the paper. Applying (13) to Theorem 2, we get
Z 1
Z 1
GLS
2
PT » (c; ¹c) ) ¡2¹
c
Wc (r)dW (r) + (¹c ¡ 2¹cc)
Wc2 (r)dr (14)
0
´
0
¹ 0; ±) ¡ sup M(c;
¹ ¹c; ±) ¡ ¹c ¡ 2¹c»2
+sup M(c;
GLS
J PT » (c; c¹)
In the supremum method, Perron (1997) recommends to choose the ±¤
that is related to the largest absolute value of t¡statistic related to the parameter of break on slope; since we don’t have information on the sign of the
shocks. After selecting ±¤ , we calculate the M GLS (±¤) ; and ADF GLS (±¤)
statistics: There is no feasible optimal point test available using this method
to select break point.
4.2
Asymptotic Critical Values, Power Envelope, and Asymptotic Power Functions
Under the null hypothesis c = 0; we use T=1000, 10,000 replications to
simulate asymptotic critical values for ¹c = ¡1 to -70 (¹
® = 0:999 to 0.93),
then we let c = c¹ to calculate power at each c. As suggested by ERS
(1996), we choose the value of ¹c that gives 50% power as the one used for
GLS-detrending and we select ¹c = ¡24. Intuitively, lower c¹ in this paper
(compared to ¹c = ¡22:5 in Perron and Rodríguez, 2003) tells us that it
may take longer to reach the same percentage of power. In other words, the
power envelope shifts down from the previous one in Perron and Rodríguez
(2003), and the loss of power is caused by the relaxation of assumption for
the initial observation. To be more precise, we graph both power envelopes
in Figure 1.
8
Next we use the critical values when ¹c = ¡24; T = 1000 and 10,000
replications to calculate the asymptotic power for each test using ¹c = ¡24
to detrend data and the results are graphed in Figure 1 and 2. We can see
that the power curve for each test lies under the power envelope, but not
far from it. Using the in…mum method to choose break point sometimes
gives a slightly higher power than supremum method, although it does not
necessarily give a consistent estimate of the true break point (Vogelsang
and Perron, 1998). The results from Perron and Rodríguez (2003) are also
graphed in the same …gure as a comparison. We can see that the power of
each test has dropped due to the change of assumption for initial condition.
5
Finite Sample Results
In practice, many time series have small sample size. Therefore it is necessary to simulate …nite sample critical values and to evaluate the performance
in terms of size and power. When doing this, there is always a question of
how many lags should be chosen to account for the serial correlation and
to maintain certain power. Now it is generally agreed that data dependent
methods give a better test performance than choosing lag k a priori (see Ng
and Perron, 1995). In the following, we use …ve data dependent methods to
choose k:
5.1
The Selection of k
We …rst use Akaike and Bayesian Information Criteria (AIC and BIC hereafter). They take the form of
© ¡ 2 ¢
ª
AIC(k) = arg min
ln sek + 2k=T ¤
k2[0;kmax ]
© ¡ 2 ¢
ª
BIC(k) = arg min
ln sek + ln T ¤k=T ¤
k2[0;kmax ]
here T ¤ = T ¡ kmax; and kmax should be large enough to account for serial
correlations. We use kmax = int[12(T=100)1=4 ] as recommended by Perron
and Ng (2001). The shortcoming of these information criteria is that when
there is strong negative MA component in the error term, they tend to
select a smaller k than that is necessary for unit root tests to have good
size. Perron and Ng (2001) proposed modi…ed AIC and BIC (MAIC and
MBIC) to account for this problem. The idea is to use a penalty factor
^¿ T (k) to correct under…tting. The MAIC and MBIC are de…ned as:
© ¡ 2 ¢
ª
MAIC(k) = arg min
ln sek + 2 (^
¿ T (k) + k) =T ¤
k2[0;kmax ]
9
MBIC(k) = arg
min
k2[0;kmax ]
© ¡ 2 ¢
ª
ln sek + ln T ¤(^¿ T (k) + k)=T ¤
¡ ¢¡1 2 PT
^¯ 0
where ¿^T (k) = s2ek
^2t¡1, ¯^ 0 is estimated using augmented
t=kmax +1 y
P
autoregression (7) and s2ek = (T ¡ kmax )¡1 Tt=kmax +1 e^2tk : Ng and Perron
(2001) showed that as a strong negative MA error exists, ^¿ T (k) increases as
k decreases. Therefore ¿^T (k) is used as a penalty factor for small k:
The last method is the so-called sequential t¡test or recursive method
proposed by Campell and Perron (1991). To apply, we start from augmented
regression (7) with kmax = int(4(T =100)1=4): If the t-statistic associated with
the kmax th lag is signi…cant (p¡value less than 0.1), then k = kmax is chosen.
Otherwise we redo the regression with kmax ¡ 1 lags, and so on, until we
…nd the lag that has a signi…cant t-statistic. Note that if k = 0 and no
rejection is found, we select k = 0: This method has less size distortion than
AIC and BIC when there is a strong negative MA error, but it tends to
overparameterize in the other cases.
5.2
Size and Power
The critical values for model I and II, using k selected by four data-dependent
methods and 1000 replications, are tabulated in Table 1 to 4. The performance of AIC is very poor and hence not included. Table 1 and 2 give the
critical values using in…mum method to choose break point, whereas Table
3 and 4 calculate the critical values using supremum method. Based on
these critical values, we calculate …nite sample size and size adjusted power
using 1000 replications, T=100, 200, iid; MA(1); and AR(1) errors. The
results are in Table 5 to 12. We summarize the following two characteristics from these results. First, MAIC and MBIC have much acceptable
size distortions than AIC and BIC when there are strong negative moving
average errors: For example, when µ = ¡0:8; T = 200 in Table 11, the size
distortions for MAIC and MBIC are 0.1050, 0.1040, 0.1050, 0.1380 and
0.1110, 0.1100, 0.1120, 0.1430, respectively. Using BIC they are 0.8610,
0.8620, 0.8590, 0.9000: Second, all tests except the ADF GLS test have low
power when there is strong negative autoregressive errors. For example,
when ½ = ¡0:8; T = 200 in Table 8 and 12, the power for M GLS and PTGLS
tests are 6% to the most.
10
6
Empirical Application
In a similar way as Perron and Rodríguez (2003), two time series from the
Nelson-Plosser data set are examined. They are real wages (1900-1970) and
common stock prices (1870-1970). A common characteristic is that they
both exhibit a change in level and slope. Therefore the model II is used
to test whether the null hypothesis of a unit root is rejected or not. The
test results using information criteria to select lag k are tabulated in Tables
13a,b (AIC is not included for its low power). We …nd that the break points
selected are the same as those in Perron and Rodríguez (2003) and they are
associated with major events. The real wages series with a break at 1938
is graphed in Figure 3, and the stock prices series with a break at 1931 is
graphed in Figure 4.
When using information criteria BIC, MAIC and MBIC, most test
statistics can reject the null at least at the 5% level. Comparing the results
from those of Perron and Rodríguez (2003), we can see that in some cases
there are less evidences against the unit root null hypothesis. For example,
the real wages are rejected at 1% level in Perron and Rodríguez (2003)
but at 2.5% level here when using supremum method and MAIC: When
using in…mum method, the real wages are rejected at 5% for BIC and 2.5%
for MAIC in Perron and Rodríguez (2003), here the null of a unit root is
not rejected for BIC and is rejected at 10% for MAIC: These evidences
indicate that the power of these unit root tests has decreased due to the
new assumption on initial value.
The results for sequential t-statistic method are summarized in Tables
14a,b. When using sequential t-statistics, the null hypothesis is rejected in
all cases except when using supremum method to select break point for real
wages. But combing all the results, we can still conclude with a rejection of
the null.
7
Concluding Remarks
According to Elliott (1999) and Müller and Elliott (2003), changing the initial condition in the DGP has di¤erent impact under the null and alternative
hypothesis in unit root testing. Under the null, the initial value change is
equivalent to a mean shift. Therefore invariance method can be applied for
various initial conditions under the null hypothesis. But this is not true
under the alternative hypothesis, where invariant tests will have a di¤erent
distribution as the initial condition changes, and hence impacts on power
11
performance are expected.
This paper examines M GLS ; ADF GLS ; PTGLS tests in the context of
structural change when the initial observation is drawn from its unconditional distribution, in comparison with zero or …xed initial values as dealt
in Perron and Rodríguez (2003). We …nd asymptotic power loss and one
should be cautious when using unit root tests for the time series believed to
start from an unconditional distribution. The …nite sample size and power
performance are also studied for processes with iid errors, MA(1) and AR
(1) errors. The performances are quite di¤erent when di¤erent procedures
are used to select the order of autoregressions. But they are consistent with
what standard literatures predict.
References
[1] Bhargava, A. (1986), “On the Theory of Testing for Unit Root in Observed Time Series,” Review of Economic Studies 53, 369-384.
[2] Banerjee, A., R. Lumsdaine and J. H. Stock (1992), “Recursive and
Sequential Tests of the Unit Root and Trend Break Hypothesis: Theory and International Evidence,” Journal of Business and Economics
Statistics 10, 271-287.
[3] Campbell, J. Y. and P. Perron (1991), “Pitfalls and Opportunities:
What Macroeconomists Should Know About Unit Roots” in Blanchard,
O.J., Fischer, S. eds., NBER Macroeconomics Annual 6, 141-201.
[4] Chan, N. H. and C. Z. Wei (1987), “Asymptotic Inference for Nearly
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[5] Christiano, L. J. (1992), “Searching for a Break in GNP,” Journal of
Business and Economics Statistics 10, 237-250.
[6] Davidson J. (1994), “Stochastic Limit Theory,” Oxford University
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[7] Dickey, D. A. and W. A. Fuller (1979), “Distribution of the Estimator for Autoregressive Time Series with a Unit Root,” Journal of the
American Statistical Association 74, 427-431.
12
[8] Dufour, J. -M. and King, M. (1991), “Optimal Invariant Tests for the
Autocorrelation Coe¢cient in Linear Regressions with Stationary or
Nonstationary Errors,” Journal of Econometrics 47, 115-143.
[9] Elliott, G., T. Rothenberg and J. H. Stock (1996), “E¢cient Tests for
an Autoregressive Unit Root,” Econometrica 64, 813-839.
[10] Elliott, G. (1999), “E¢cient Tests for a Unit Root When the Initial Observations is Drawn from its Unconditional Distribution,” International
Economic Review 40, 767-783.
[11] Müler, U. K. and G. Elliott (2003), “Tests for Unit Roots and the Initial
Condition,” Econometrica 71, 1269-1286.
[12] Nelson, C. R. and C. I. Plosser (1982), “Trends and Random Walks
in Macroeconomics Time Series: Some Evidence and Implications,”
Journal of Monetary Economics 10, 139-162.
[13] Ng, S. and P. Perron (1995), “Unit Root Tests in ARMA Models with
Data Dependent Methods for the Selection of the Truncation Lag,”
Journal of the American Statistical Association 90, 268-281.
[14] Ng, S. and Perron, P. (2001), “ Lag Length Selection and the Construction of Unit Root Tests with Good Size and Power,” Econometrica 69,
1519-1554.
[15] Perron, P. (1989), “The Great Crash, the Oil Price Shock and the Unit
Root Hypothesis,” Econometrica 57, 1361-1401
[16] Perron, P. (1997), “Further Evidence of Breaking Trend Functions in
Macroeconomics Variables,” Journal of Econometrics 80, 355-385.
[17] Perron, P. and S. Ng (1996), “Useful Modi…cations to Some Unit Root
Tests with Dependent Errors and Their Local Asymptotic Properties,”
Review of Economics Studies 63, 435-463.
[18] Perron, P. and G. Rodríguez (2003), “GLS Detrending, E¢cient Unit
Root Tests and Structural Change,” Journal of Econometrics 115, 127.
[19] Phillips, P. C. B. (1987), “Time Series Regression with Unit Roots,”
Econometrica 55, 277-302.
13
[20] Said, S. E. and Dickey, D. A. (1984), “Testing for Unit Roots in
Autoregressive-Moving Average Models of Unknown Order. Biometrika
71, 599-608.
[21] Sargan, J. D. and Bhargava, A. (1983), “Testing Residuals from Least
Squares Regression for being Generated by the Gaussian Random
Walk” Econometrica 51, 153-174.
[22] Stock, J. H. (1999), “A Class of Tests for Integration and Cointegration,” in Engle, R.F. and H. White (eds.), Cointegration, Causality and
Forecasting. A Festschrift in Honour of Clive W.J. Granger, Oxford
University Press, 137-167.
[23] Zivot, E. and D. W. K. Andrews (1992), “Further Evidence on the Great
Crash, The Oil-Price Shock and the Unit Root Hypothesis,” Journal of
Business and Economics Statistics 10, 251-270.
14
8
Appendix
Lemma 3 Under the initial condition speci…ed in Assumption 1, we have
T ¡1= 2u1
)
¾N(0;
´
¾»:
1
)
¡2c
Using the fact that ® = 1 + Tc , we can show that
¾ 2 ), we have the above result.
Applying to u1 » N(0; 1¡®
2
Proof:
¡
¢
T 1 ¡ ®2 ) ¡2c:
Lemma 4 Suppose yt is generated by (1) to (3) and the deterministic components given by
Model I, the initial condition is de…ned by Assumption 1, ± = TB =T is the break point, then we
have,
T ¡1=2 (b
¹1 ¡ ¹1 )
)
¾b4;
¡ ¯1 )
)
¾b5;
(b̄ 2 ¡ ¯ 2 )
)
¾b6:
T 1=2 (b̄
T
1=2
1
where the de…nitions of b4, b5, b6 are given in the following proof.
Proof: In matrix notation, we have:
£
¤ ¡1 £ ®¹ ®0
¤
¹
^
¤¡1(Ã(±)
¡ Ã) = ¤z ¹® z®0
¤
¤z u ¹
where
z ¹®
=
u¹®
=
¤
=
(A.1)
2 ¡
3
¢1=2
¡2¹
c=T ¡ c¹2 =T 2
¡¹
cT ¡1 ¢ ¢ ¢
¡¹
cT ¡1
6 ¡
7
¢1=2
6 ¡2¹
7;
c=T ¡ c¹2 =T 2
¢ ¢¢
¡¹
cT ¡1 (t ¡ 1) + 1
¢¢¢
4
5
¡1
0
¢ ¢¢
¡¹
cT (t ¡ 1) + ±¹
c+1 ¢ ¢ ¢
h¡
i
¢1=2
¡2¹
c=T ¡ ¹
c2 =T 2
u1 ; (c ¡ ¹
c) T ¡1 u1 + v2; ¢ ¢ ¢ (c ¡ ¹
c) T ¡1uT ¡1 + vT ;
diag(T 1=2 ; T ¡1=2 ; T ¡1=2):
Let D be a 3 £3 matrix which is the limiting distribution of ¤z ®¹ z ®¹0 ¤: We know
that D11; D12; D22 are the same as those in Elliott (1999) and D23 ; D33 are the
same as the terms ¡23; ¡33 in Perron and Rodríguez (2003). That is,
D11
=
D12
=
D22
=
D23
=
´
D33
=
´
c¹2 ¡ 2¹
c;
1 2
c ¡¹
¹
c;
2
1 2
1+ ¹
c ¡ c¹;
3
1
1
1 2 3
1 ¡ ± ¡ c¹ + ¹
c± ¡ c¹2± + c¹2 + ¹
c ±
2
3
6
m;
1 2
1 ¡ ± ¡ c¹(1 ¡ ±) 2 + ¹
c (1 ¡ ±)3
3
d:
15
Therefore we only need to calculate D13 :
D13
h ¡
i
¢1=2
¡2¹
c=T ¡ ¹
c2 =T 2
¡¹
c=T ¢ ¢ ¢ ¡¹
c=T ¡¹
c=T ¢ ¢ ¢ ¡¹
c=T ¢
h
i0
0 ¢ ¢¢ 1 1¡¹
c=T 1 ¡ 2¹
c=T ¢ ¢ ¢ 1 ¡ (T ¡ TB ¡ 1) c¹=T
2
3
·
¸
T
c 4 X
¹
c
¹
¡
1 ¡ ¡ (t ¡ TB ¡ 1) 5
T t=T +1
T
=
=
B
¢
1 ¡
¡¹
c (1 ¡ ±) + c¹2 1 ¡ ±2 ¡ ¹
c2± (1 ¡ ±)
2
1 2
¡¹
c (1 ¡ ±) + ¹
c (1 ¡ ±) 2 :
2
)
=
¹ ;
Next we calculate the limiting distribution of ¤z ®¹ u®0
¤z ®¹ u
=
®0
¹
2
T 1= 2
6
6
=4 0
0
0
0
T ¡1=2
0
3
7
7¢
5
0
T ¡1=2
2 ¡
3
¢
1= 2
¡2¹
c=T ¡ ¹
c2=T 2
¡¹
cT ¡1 ¢ ¢ ¢
¡¹
cT ¡1
6 ¡
7
¢
1= 2
7¢
¢6
c=T ¡ ¹
c2=T 2
¢¢¢
¡¹
cT ¡1 (t ¡ 1) + 1
¢¢ ¢
4 ¡2¹
5
0
¢¢¢
¡¹
cT ¡1 (t ¡ 1) + ±¹
c +1 ¢¢ ¢
h¡
i0
¢1=2
¡2¹
c=T ¡ ¹
c2=T 2
u1; (c ¡ c¹) T ¡1 u1 + v2 ; ¢ ¢ ¢ (c ¡ ¹
c) T ¡1 uT ¡1 + vT ;
2
³
´
1
1 P
¡
¢£
¤
c ¡ ¹
c2
T 2 ¡ 2¹
u1 + T 2 T
cT ¡1 (c ¡ ¹
c) T ¡1ut¡1 + vt
2
t=2 ¡¹
T
T
6
³
´
¤£
¤
PT £
6 ¡ 12
c
c¹2
¡1
2
6 T
¡ 2¹
cT ¡1 (t ¡ 1) + 1 (c ¡ ¹
c) T ¡1ut¡1 + vt
t=2 ¡¹
T ¡ T 2 u1 + T
4
1 PT
£
¤
£
¤
T ¡ 2 t=TB +1 ¡¹
cT ¡1 (t ¡ 1) + ±¹
c + 1 (c ¡ ¹
c) T ¡1 ut¡1 + vt
where the …rst element of this 3 £ 1 vector may be expressed as:
=
=
)
´
¡2¹
cT ¡1=2 u1 ¡ ¹
c2T ¡3=2 u1 + T 1= 2
t=2
t=2
0
1
Wc (r) ¡ c¹W (1)
¾b1;
7
7
7;
5
T
T
X
X
¡
¢
¡
¢
¡¹
cT ¡1 (c ¡ c¹) T ¡1 ut¡1 + T 1=2
¡¹
cT ¡1 vt
¡2¹
cT ¡1=2 u1 ¡ ¹
c2T ¡3=2 u1 ¡ T ¡3=2 ¹
c (c ¡ ¹
c)
½
Z
¾ ¡2¹
c» ¡ ¹
c (c ¡ ¹
c)
3
¾
T
X
t=2
ut¡1 ¡ c¹T ¡1=2
T
X
vt
t=2
and the second element may be written as
=
¡2T ¡3=2 ¹
cu1 ¡ ¹
c2 T ¡5=2u1 +
T ¡1=2
T ©
X
t=2
=
¡¹
cT ¡1 (t ¡ 1) (c ¡ ¹
c) T ¡1 ut¡1 ¡ ¹
cT ¡1 (t ¡ 1) vt + (c ¡ ¹
c) T ¡1 ut¡1 + vt
¡2T ¡3=2 ¹
cu1 ¡ ¹
c2 T ¡5=2u1 ¡ ¹
c (c ¡ ¹
c) T ¡5=2
16
T
X
t=2
(t ¡ 1) ut¡1 ¡ ¹
cT ¡3=2
T
X
t=2
(t ¡ 1) vt
ª
+ (c ¡ ¹
c) T ¡3=2
)
´
´
½
Z
¾ ¡¹
c (c ¡ ¹
c)
0
T
X
ut¡1 + T ¡1=2
t=2
1
T
X
vt
t=2
·
Z
rWc (r) ¡ ¹
c W (1) ¡
½
Z
¾ (1 ¡ ¹
c) W (1) + (c ¡ ¹
c)
0
0
1
1
¸
Z
Wc (r) + (c ¡ ¹
c)
Wc (r) ¡ ¹
c (c ¡ ¹
c)
¾b2;
Z
1
0
1
rWc (r) + ¹
c
0
¾
Wc (r) + W (1)
Z
0
1
¾
Wc (r)
and the third element
=
=
)
´
8
< ¡¹
cT ¡1 (t ¡ 1) (c ¡ ¹
c) T ¡1ut¡1 ¡ ¹
cT ¡1 (t ¡ 1) vt + ±¹
c (c ¡ ¹
c) T ¡1ut¡1
:
+±¹
cvt + (c ¡ c¹) T ¡1 ut¡1 + vt
t=TB +1
8
T
< ¡¹
X
cT ¡1 (c ¡ ¹
c) T ¡1tut¡1 + ¹
cT ¡1 (c ¡ ¹
c) T ¡1ut¡1 ¡ ¹
cT ¡1 tvt + ¹
cT ¡1 vt
T ¡1=2
:
¡1
¡1
+±¹
c (c ¡ ¹
c) T ut¡1 + ±¹
cvt + (c ¡ ¹
c) T ut¡1 + vt
t=TB +1
8
9
R
R
< (1 ¡ ¹
c + ±¹
c) W (1) + c¹ ±1 Wc (r) ¡ W (±) ¡ ¹
c (c ¡ ¹
c) ±1 rWc (r) =
¾
R
R
:
;
+±¹
c (c ¡ ¹
c) 1 Wc (r) + (c ¡ ¹
c) 1 Wc (r)
T ¡1=2
T
X
±
9
=
;
9
=
±
¾b3 :
Therefore
0
=
)
´
1
¹
^1 ¡ ¹1
B
C
C
^
¤¡1 B
@ ¯1 ¡ ¯1 A
^
¯2 ¡ ¯2
¡ ®¹ ®0
¢
¹
¹
¤z z ¤ ¡1 ¤z ®¹ u®0
2
c2 ¡ 2¹
¹
c
6
6 1¹
2
c
4 2c ¡ ¹
¡¹
c (1 ¡ ±) + 12 ¹
c2 (1 ¡ ±)2
0
1
b4
B
C
B
¾ @ b5 C
A¥
b6
1 2
c
2¹
¡¹
c
1 + 13 c¹2 ¡ c¹
m
¡¹
c (1 ¡ ±) + 12 c¹2 (1 ¡ ±) 2
m
d
3¡1 0
7
7
5
¾b1
1
B
C
C
¢B
@ ¾b2 A
¾b3
Lemma 5 Suppose fytg is generated by (1) to (3), the deterministic component is given by that
^ (±) be the
of Model II, the initial condition is de…ned in Assumption 1, ± is the break point. Let Ã
estimates of the coe¢cients of (5), then the results of lemma 2 stil l hold with the addition that
¹2 ¡ ¹2 ) limT !1 ·(¹
^
c; ±)vTB ´ v¤ :
Proof :
We have
0
¹
^ ¡ ¹1
B 1
B ¹
B ^ ¡ ¹2
¤¡1 B 2
B ¯
^
@ 1 ¡ ¯1
^ ¡¯
¯
2
2
1
C
C ¡
¢
C
¹ ¤ ¡1 ¤z®
¹ u®0
¹
C = ¤z ®¹ z ®0
C
A
17
;
¡
¢
¤ = diag T 1=2 ; 1; T ¡1=2 ; T ¡1= 2 ;
2 ¡
¢ 1= 2
¡2¹
c=T ¡ ¹
c2=T 2
¡ T¹c
6
6 0
¢¢¢
6
z®¹ = 6 ¡
¢
2=T 2 1= 2
6 ¡2¹
c
=T
¡
c
¹
1 ¡ T¹c
4
0
¢¢¢
where
and
¢
¢ ¢¢
¢¢¢
1¡
¡ T¹c
1
¢¢¢
1¡
(t¡1)c¹
T
(t¡1)c¹
T
¢
¡ T¹c
¢¢ ¢
1¡
¢¢ ¢
+ ±¹
c
³
¢¢ ¢
¡¹
c; ª14 ) ¡¹
c (1 ¡ ±) +
¹2
c
2
¡ T¹c
1¡
(T ¡1)¹
c
T
(T ¡1)¹
c
T
´ ¡1
0
¤z ¹® z®¹ ¤
;
We …rst calculate the limiting distribution of
¹ : From the proof of Lemma 2, we know that
then ¤z ®¹ u®0
c¹2
2
3
+ ±¹
c
7
7
7
7:
7
5
denoted as
ª11 )
c2
¹
(1 ¡ ±) 2 ; ª33 ) 1 + 13 ¹
c2 ¡ ¹
c ´ a; ª34 ) m; ª44 ) d:
ª;
¡ 2¹
c; ª13 )
Therefore
we only need to calculate the terms
ª12
=
ª22
=
ª23
=
=
ª12; ª22; ª23 ; ª24 :
½
¾
½
¾
1
c
¹
c2
¹
c2
¹
c
¹
c2
¹
T 1=2 ¡ + 2 + ¢ ¢ ¢ + 2 = T 2 ¡ + 2 (T ¡ TB ) ) 0;
T
T
T
T
T
c¹2
c¹2
c2
¹
1 + 2 + ¢ ¢ ¢ + 2 = 1 + 2 (T ¡ TB ) ) 1;
T 8
T
T
9
·
¸
T
<
c
¹
c X
¹
c =
¹
¡1=2
T
1 ¡ (TB ¡ 1) ¡
1 ¡ (t ¡ 1)
:
T
T t=T +1
T ;
B
8
9
T
<
=
2
X
c
¹
c
¹
c
¹
T ¡1=2 1 ¡ (TB ¡ 1) ¡ (T ¡ TB ) + 2
(t ¡ 1)
:
;
T
T
T
t=TB +1
)
ª24
=
=
0;
8
<
9
·
¸=
T
X
¹
c
c
¹
c
¹
T ¡1=2 1 ¡ (TB ¡ 1) + ±¹
c¡
1 ¡ (t ¡ 1) + ±¹
c
:
;
T
T t=T +1
T
B
8
9
T
<
=
2
2
X
c
¹
c
¹
c
¹
±¹
c
T ¡1=2 1 + ¡ (T ¡ TB ) + 2
(t ¡ 1) ¡
(T ¡ TB )
:
;
T
T
T t=T +1
T
B
)
0:
¹ ; the …rst, third and fourth elements
For the limiting distribution of ¤z ¹® u®0
are already calculated in the proof for Lemma 2. We only need to calculate
the second element which is:
h
=
0
¢¢¢
1
¡ T¹c
¢¢¢
(c ¡ c¹) T ¡1 uTB ¡1 + vTB ¡
c¹
T
Therefore, we have:
=
i
¡ T¹c
³
´ ¡1
0
¹
¤z ®¹ z ®¹ ¤
¤z ¹® u®0
2
c2 ¡ 2¹
¹
c
6
6 0
6
6 c¹2
6
4 2 ¡ c¹
2
¡¹
c (1 ¡ ±) + c¹2 (1 ¡ ±)2
¢
"µ
¡
T
X
t=TB +1
£
2¹
c
c¹2
¡ 2
T
T
¶ 1=2
#0
u1 ; ¢ ¢ ¢ (c ¡ c¹) T ¡1 ut¡1 + vt; ¢ ¢ ¢
¤
(c ¡ ¹
c) T ¡1ut¡1 + vt = vTB + o p (1) ) vTB :
0
¹2
c
2
1
0
0
a
m
0
m
d
¡¹
c
18
¡¹
c (1 ¡ ±) +
0
c¹2
2
(1 ¡ ±)2
3 ¡1 2
7
7
7
7
7
5
¾b1
6
6 v
6 TB
¢6
6 ¾b2
4
¾b3
3
7
7
7
7;
7
5
where according to matrix algebra, the second element of the resulting matrix is equal to:
· (¹
c; ±) vTB =
·¤ (¹
c; ±) vTB
³ 2
´2 ;
(¹
c2 ¡ 2¹
c) ad ¡ ¹c2 ¡ ¹
c d
i h
¢
¡ 2
¢h
2
c¹2 ¡ 2¹
c ad+2m ¹
c ¡ 2¹
c ¡¹
c (1 ¡ ±) + c¹2 (1 ¡ ±)2 ¡ ¡¹
c (1 ¡ ±) +
³ 2
´2
¡
¢
m2 c¹2 ¡ 2¹
c ¡ d ¹c2 ¡ ¹
c : That is,
where ·¤ (¹c; ±) =
¡
¹2
c
2
(1 ¡ ±)2
¹
^2 ¡ ¹2 ) lim · (¹
c; ±) vTB ´ v¤¥
t!1
Proof of Theorem 1. We only show the proof in detail for Model I and
the statistic MZ®GLS . The proof for Model II and other statistics follow
analogously. Firstly, consider the limiting distribution of T ¡1y~2T :
T ¡1 ~
yT2
=
n
h
³
´
³
´
io 2
T ¡1 uT ¡ (b
¹1 ¡ ¹1) + b̄1 ¡ ¯1 T + b̄2 ¡ ¯ 2 1 (¢) (T ¡ T ±)
T ¡1 fu2T + (b
¹1 ¡ ¹1) 2 + ( b̄1 ¡ ¯ 1)2 T 2
+( b̄2 ¡ ¯ 2) 21 (¢) (T ¡ T ±) 2 ¡ 2uT (b
¹1 ¡ ¹1 )
+2(b̄ 1 ¡ ¯ 1 )T ( b̄ 2 ¡ ¯ 2)1 (¢) (T ¡ T ±)
¡2uT ( b̄1 ¡ ¯ 1)T ¡ 2uT ( b̄ 2 ¡ ¯ 2)1 (¢) (T ¡ T ±)
+2(b
¹1 ¡ ¹1 )( b̄1 ¡ ¯1 )T
+2(b
¹1 ¡ ¹1 )( b̄2 ¡ ¯2 )1 (¢) (T ¡ T ±)g
The limiting distributions of some terms are calculated as follows. The other
terms are the same as those in Perron and Rodríguez (2003).
1.
T ¡1(b
¹1 ¡ ¹1) 2 = [T ¡1=2 (b
¹1 ¡ ¹1 )]2 ) ¾ 2 b24 :
2.
¡2T ¡1 uT (b
¹1 ¡ ¹1) = ¡2(T ¡1=2 uT )[T ¡1=2 (b
¹1 ¡ ¹1 )] ) ¡2¾ 2 Wc (1)b4:
3.
2T ¡1(b
¹1 ¡ ¹1 )( b̄ 1 ¡ ¯ 1)T = 2[T ¡1= 2(b
¹1 ¡ ¹1 )][T 1=2( b̄1 ¡ ¯ 1)] ) 2¾ 2b4 b5:
4.
2T ¡1(b
¹1 ¡ ¹1 )( b̄ 2 ¡ ¯ 2)(T ¡ T ±) = 2(b
¹1 ¡ ¹1 )( b̄2 ¡ ¯2 ) ¡ 2(b
¹1 ¡ ¹1 )( b̄2 ¡ ¯ 2)±
= 2[T ¡1= 2(b
¹1 ¡¹1 )][T 1=2 ( b̄2 ¡ ¯ 2)] ¡2T ¡1=2(b
¹1 ¡¹1 )T 1=2 (b̄ 2 ¡ ¯ 2)±] ) 2¾2 b4b6 (1 ¡ ±):
Therefore,
19
i2
a¡
T ¡1y~2T
)
¾2 Wc2 (1) + ¾ 2b24 + ¾ 2 b25 + ¾ 2b26 (1 ¡ ±)2
¡2¾ 2Wc (1)b4
¡2¾ 2b
=
=
´
(1)
Vcc (1; ±)
=
(2)
Vcc (1; ±)
=
2
¾
+
2¾ 2 b
6 Wc (1)(1 ¡
f[Wc2(1) ¡
±)
(A.2)
5b6(1 ¡ ±)
¡ 2¾2 b
5Wc (1)
+ 2¾ 2b
+ 2¾2 b
4b6 (1 ¡ ±)
2b5 Wc (1) +
b25
4 b5
+ b24
¡ 2b4Wc (1) + 2b4b5]
¡[2b6 (1 ¡ ±)Wc (1) ¡ 2b5b6 (1 ¡ ±) ¡ b26(1 ¡ ±)2 ¡ 2b4b6 (1 ¡ ±)]
1
¾2 f[Wc (1) ¡ b4 ¡ b5]2 ¡ 2[b6 (1 ¡ ±)][Wc (1) ¡ b4 ¡ b5 ¡ (1 ¡ ±)b6 ]
2
n
o
(1)
(2)
¾2 Vcc (1; ±) 2 ¡ 2Vcc (1; ±)
Wc (1) ¡ b4 ¡ b5
1
[b6(1 ¡ ±)][Wc (1) ¡ b4 ¡ b5 ¡ (1 ¡ ±)b6]:
2
P
Next we calculate the limiting distribution of term T ¡2 Tt=1 y~2t¡1 : Using the
above results and by the Continuous Mapping Theorem (CMT), we have
2T ¡2
T
X
t=1
Z
~t2 ) 2¾ 2 f
y
1
0
(1)
Vcc (r; ±)2 dr ¡ 2
Z
1
±
(2)
(A.3)
Vcc (r; ±)drg
then substitute (A.2), (A.3) into (8) to (10) and using the fact that
consistent estimate of ¾2 , the proof is complete ¥
s2
is a
Proof of Theorem 2. Here we only give the proof for Model I. De…ning
³
´³
´
0
0 ¡1
0
¹ ) ; we have S (¹
¹
¹ T (c; 0; ±) and
MT (c; ¹
c; ±) = u¹® z®¹
z ®¹ z ®¹
(z®¹ u®0
®; ±) = u¹® u®¹ ¡ M
£
¤
1
10
2
1=2
¹ T (0; 0; ±) . By de…nition, u = (1 ¡ ¹
S (1) = u u ¡ M
® ) u1;(1 ¡ ®
¹L)u2 ¢ ¢ ¢ (1 ¡ ®
¹L)T ;
u1 = [0;(1 ¡ L)u2 ¢ ¢ ¢ (1 ¡ L) T ] ;
)
and some algebra gives
¹ ¡ u1 u10
u¹® u®0
¡
¢Z
¡2¹
c» 2 + c¹2 ¡ 2c¹
c
1
0
Wc2 (r) dr ¡ 2¹
c
We have calculated each element of M¹ T (c; ¹c; ±) =
ie,
³
Z
1
Wc (r) dr
0
´³
´¡1
0
0
¹ )above ;
u®¹ z ®¹ ¤ ¤z®¹ z ®¹ ¤
(¤z®¹ u®0
¹ T (c; ¹
M
c; ±)
=
h
¾b1
¾b2
¾b3
2
2
c
i 6 c¹ ¡ 2¹
6 1 c¹2 ¡ ¹
c
4 2
¡¹
c (1 ¡ ±) + 12 c¹2 (1 ¡ ±)2
1¹
2
2c
¡ c¹
1 + 13 c¹2 ¡ ¹
c
m
¡¹
c (1 ¡ ±) + 12 ¹
c2 (1 ¡ ±)2
m
d
The calculation of M¹T (0; 0; ±) follows that of Perron and Rodríguez (2003)
since the initial value is not changed under the null. The feasible point
optimal test in the case of an unknown break is
PTGLS
c) = f
» (c; ¹
inf
±2[";1¡"]
S(¹
®; ±) ¡
20
inf
±2[";1¡"]
®
¹ S(1; ±)g=s2 :
3 ¡1 2
7
7
5
3
¾b1
6
7
7
¢6
4 ¾b2 5
¾b3
therefore the limiting distribution is:
PTGLS
c)
» (c; ¹
)
sup
±2[";1¡"]
¹ 0; ±) ¡
M(c;
¡2¹
c» 2 ¡ 2¹
c
´
JPGLS
(c; ¹
c):
T»
and this completes the proof
Z
1
0
sup
¹ c¹; ±)
M(c;
±2[";1¡"]
Wc (r)dW(r) + (¹
c2 ¡ 2¹
cc)
¥
21
Z
1
0
Wc (r)2 dr ¡ ¹
c
Table 1. Critical Values for P TG»LS test, M GL S and ADF G LS tests choosing T B minimizing the statistics; Model I
(¹c = ¡24 when constructing the tests and s2 )
Test
Size
T =1
T = 100
k=0
BIC
MAIC
MBIC
t-sig
BIC
MAIC
MBIC
t-sig
T = 100
T = 200
M Z®
.01
.025
.05
.10
.20
-43.2104
-37.2836
-33.3663
-28.7889
-24.1055
-30.4229
-27.5891
-25.5075
-22.9946
-20.0160
-43.9198
-35.7359
-30.2096
-25.8172
-21.4082
-30.8763
-27.9567
-25.3798
-22.4927
-19.3680
-30.8763
-28.3175
-25.5268
-22.5960
-19.3878
-261.0613
-146.4954
-75.5349
-50.7655
-34.7101
-39.1318
-33.6974
-30.1851
-27.0805
-22.0681
-35.5371
-31.1744
-28.4615
-24.5458
-20.8158
-35.5371
-31.1744
-28.7532
-24.6024
-21.1245
-60.4129
-51.0969
-43.0684
-32.5375
-26.7672
M SB
.01
.025
.05
.10
.20
.01
.025
.05
.10
.20
.01
.025
.05
.10
.20
0.1071
0.1151
0.1214
0.1308
0.1426
-4.6211
-4.3005
-4.0643
-3.7666
-3.4499
-4.6211
-4.3005
-4.0643
-3.7666
-3.4499
0.1271
0.1333
0.1387
0.1459
0.1559
-3.8741
-3.6914
-3.5459
-3.3673
-3.1325
-4.9195
-4.5628
-4.3040
-3.9977
-3.6734
0.1064
0.1181
0.1272
0.1381
0.1508
-4.6743
-4.2170
-3.8632
-3.5776
-3.2408
-5.1228
-4.8046
-4.4888
-4.1761
-3.7558
0.1258
0.1323
0.1392
0.1475
0.1592
-3.8948
-3.7012
-3.5410
-3.3401
-3.0758
-4.9805
-4.5386
-4.2308
-3.8658
-3.5327
0.1258
0.1323
0.1390
0.1472
0.1590
-3.8948
-3.7092
-3.5430
-3.3453
-3.0779
-4.9805
-4.5762
-4.2393
-3.9013
-3.5471
0.0437
0.0582
0.0813
0.0981
0.1185
-11.4080
-8.5255
-6.1433
-5.0339
-4.1298
-5.2971
-4.9842
-4.6555
-4.2925
-3.9294
0.1130
0.1200
0.1275
0.1351
0.1490
-4.4217
-4.0678
-3.8768
-3.6576
-3.2992
-4.8890
-4.5307
-4.2567
-4.0460
-3.6256
0.1184
0.1262
0.1314
0.1420
0.1528
-4.2138
-3.9230
-3.7696
-3.4551
-3.2059
-4.7908
-4.3966
-4.1723
-3.8195
-3.4797
0.1184
0.1262
0.1313
0.1414
0.1524
-4.2138
-3.9230
-3.7776
-3.4762
-3.2121
-4.7908
-4.4103
-4.1971
-3.8514
-3.5062
0.0901
0.0980
0.1075
0.1230
0.1352
-5.4683
-5.0145
-4.6016
-4.0202
-3.6072
-5.0672
-4.6754
-4.4324
-4.1493
-3.7462
.01
.025
.05
.10
.20
6.9674
8.0650
9.3401
10.8664
13.1103
7.6005
9.1496
10.4705
12.1358
14.4013
7.3799
8.9509
10.4300
12.0165
14.4134
9.9487
11.3400
12.1186
13.4793
16.0046
9.9487
11.2643
12.0880
13.3318
15.9125
1.6626
2.9359
4.5099
7.1297
9.9347
8.0541
9.2601
10.1610
11.7554
14.0881
9.0214
10.0080
10.8464
12.9627
15.1298
9.0214
10.0039
10.7962
12.8079
14.7528
5.3409
6.3991
7.6818
10.0039
11.9629
M Zt
ADF
PT »
Table 2. Critical Values for PTGLS
test, M G LS and ADF GLS tests choosing T B minimizing the statistics; Model II
»
(¹c = ¡24 when constructing the tests and s2 )
Test
Size
T =1
T = 100
k=0
BIC
MAIC
MBIC
t-sig
BIC
MAIC
MBIC
t-sig
T = 100
T = 200
M Z®
.01
.025
.05
.10
.20
-43.2104
-37.2836
-33.3663
-28.7889
-24.1055
-32.0496
-29.4361
-27.2141
-24.5661
-21.6006
-92.0178
-43.7198
-36.6878
-30.0820
-24.6100
-33.0652
-30.3964
-27.0392
-24.6868
-21.2387
-32.4640
-29.6516
-26.8044
-24.2571
-21.0856
-529.1307
-220.1458
-119.0889
-73.0580
-47.7979
-44.6410
-34.8546
-32.0666
-28.5460
-23.5335
-36.0791
-32.2178
-29.7736
-26.2192
-21.8524
-36.0791
-32.2178
-29.9471
-26.2477
-21.9167
-70.3351
-58.0172
-49.1422
-36.8659
-30.1404
M SB
.01
.025
.05
.10
.20
.01
.025
.05
.10
.20
.01
.025
.05
.10
.20
0.1071
0.1151
0.1214
0.1308
0.1426
-4.6211
-4.3005
-4.0643
-3.7666
-3.4499
-4.6211
-4.3005
-4.0643
-3.7666
-3.4499
0.1235
0.1294
0.1342
0.1414
0.1502
-3.9772
-3.8055
-3.6603
-3.4759
-3.2552
-5.1099
-4.7587
-4.4921
-4.1833
-3.8451
0.0735
0.1066
0.1163
0.1276
0.1416
-6.7634
-4.6535
-4.2543
-3.8364
-3.4654
-5.2810
-5.0688
-4.7399
-4.4020
-3.9873
0.1220
0.1276
0.1339
0.1413
0.1515
-4.0345
-3.8782
-3.6539
-3.4805
-3.2169
-5.0726
-4.6853
-4.3813
-4.1038
-3.7163
0.1235
0.1289
0.1350
0.1422
0.1523
-4.0100
-3.8096
-3.6360
-3.4634
-3.2130
-5.0726
-4.7196
-4.4125
-4.1183
-3.7479
0.0307
0.0476
0.0647
0.0821
0.1018
-16.2653
-10.4897
-7.7107
-6.0148
-4.8320
-5.5183
-5.2270
-4.9458
-4.5767
-4.1634
0.1058
0.1179
0.1245
0.1314
0.1444
-4.7166
-4.1642
-3.9633
-3.7493
-3.4033
-5.0656
-4.6614
-4.3853
-4.1211
-3.7543
0.1177
0.1245
0.1292
0.1369
0.1487
-4.2334
-3.9714
-3.8540
-3.5961
-3.2770
-4.8118
-4.5091
-4.2433
-3.9649
-3.5933
0.1177
0.1245
0.1286
0.1369
0.1488
-4.2334
-3.9714
-3.8540
-3.5961
-3.2831
-4.8118
-4.5255
-4.2493
-3.9834
-3.6135
0.0842
0.0925
0.1006
0.1151
0.1277
-5.9142
-5.3678
-4.9135
-4.2746
-3.8486
-5.1875
-4.8412
-4.5706
-4.2608
-3.8944
.01
.025
.05
.10
.20
6.9674
8.0650
9.3401
10.8664
13.1103
9.1361
10.4098
11.6000
13.2085
15.2320
7.8244
9.3537
11.0876
13.0612
15.2557
10.7618
12.1748
13.2501
14.5721
16.7392
10.7148
11.9795
13.0746
14.3445
16.6628
2.2765
3.5404
4.8532
7.4151
10.3941
8.9647
10.3576
11.3883
12.6174
14.8423
10.1956
11.1091
11.9117
13.7618
15.9041
10.1584
10.9864
11.8585
13.4543
15.7512
5.3073
6.9266
8.3451
10.4570
12.7655
M Zt
ADF
PT »
Table 3. Critical Values for M GL S and ADF G LS tests choosing T B maximizing jt b̄ j; Model I
2
(¹c = ¡24 when constructing the tests and s2 )
Test
Size
T =1
T = 100
k=0
BIC
MAIC
MBIC
t-sig
BIC
MAIC
MBIC
t-sig
T = 100
T = 200
M Z®
.01
.025
.05
.10
.20
-42.4320
-36.8108
-32.6893
-28.2664
-23.7486
-30.8763
-28.6782
-25.7177
-23.2047
-19.9864
-40.0494
-33.9881
-29.4984
-24.9678
-20.7516
-30.4244
-27.5856
-24.9826
-21.9668
-18.8297
-30.4244
-27.7928
-25.1308
-22.1559
-18.8921
-187.4052
-99.5411
-66.2327
-42.8809
-30.3537
-38.1112
-33.5067
-29.8513
-26.7054
-21.7640
-34.8049
-30.6945
-27.9640
-24.0488
-20.4058
-34.8049
-30.6945
-28.0354
-24.4379
-20.7038
-57.5728
-50.1020
-40.3330
-31.5249
-25.7182
M SB
.01
.025
.05
.10
.20
.01
.025
.05
.10
.20
.01
.025
.05
.10
.20
0.1078
0.1159
0.1226
0.1319
0.1439
-4.5798
-4.2635
-4.0192
-3.7372
-3.4234
-4.5798
-4.2635
-4.0192
-3.7372
-3.4234
0.1263
0.1318
0.1383
0.1459
0.1560
-3.9287
-3.7501
-3.5755
-3.3817
-3.1243
-4.9815
-4.6151
-4.3667
-3.9605
-3.6216
0.1109
0.1213
0.1294
0.1395
0.1530
-4.4408
-4.1007
-3.7841
-3.5064
-3.8364
-5.1030
-4.7136
-4.4272
-4.0767
-3.6817
0.1272
0.1329
0.1396
0.1493
0.1607
-3.8627
-3.6755
-3.5148
-3.2910
-3.0085
-4.8971
-4.4359
-4.1639
-3.8143
-3.4586
0.1272
0.1327
0.1396
0.1487
0.1603
-3.8627
-3.6864
-3.5194
-3.3084
-3.0197
-4.8971
-4.4405
-4.1669
-3.8245
-3.4842
0.0515
0.0704
0.0868
0.1078
0.1263
-9.6530
-7.0082
-5.7510
-4.6004
-3.8753
-5.2051
-4.9284
-4.5439
-4.2269
-3.8407
0.1141
0.1215
0.1293
0.1361
0.1499
-4.3555
-4.0592
-3.8455
-3.6347
-3.2746
-4.7880
-4.5064
-4.2266
-3.9895
-3.5678
0.1187
0.1267
0.1332
0.1430
0.1545
-4.1715
-3.8991
-3.7349
-3.4150
-3.1622
-4.6062
-4.3241
-4.0815
-3.7570
-3.4199
0.1187
0.1267
0.1324
0.1424
0.1535
-4.1715
-3.8991
-3.7391
-3.4525
-3.1902
-4.6380
-4.3408
-4.1178
-3.7976
-3.4708
0.0926
0.0992
0.1112
0.1254
0.1377
-5.3427
-4.9700
-4.4739
-3.9263
-3.5607
-4.9411
-4.5973
-4.3888
-4.0797
-3.6891
M Zt
ADF
Table 4. Critical Value for M GL S and ADF G LS tests choosing T B maximizing jtb̄ j; Model II
2
(¹c = ¡24 when constructing the tests and s2 )
Test
Size
T =1
T = 100
k=0
BIC
MAIC
MBIC
t-sig
BIC
MAIC
MBIC
t-sig
T = 100
T = 200
M Z®
.01
.025
.05
.10
.20
-42.4320
-36.8108
-32.6893
-28.2664
-23.7486
-28.6517
-26.6342
-23.6385
-20.8656
-17.9730
-38.5839
-32.0902
-27.1841
-22.6054
-18.6641
-27.3671
-24.4422
-21.7802
-19.5608
-16.9754
-27.3671
-24.4422
-21.9509
-19.5899
-17.1633
-108.6946
-78.0695
-49.4972
-36.5664
-25.3750
-34.7313
-31.1557
-26.7210
-22.8812
-19.3138
-30.7842
-27.0364
-24.6923
-21.1674
-17.8371
-31.1557
-28.3081
-24.9101
-21.5344
-18.3070
-54.6617
-40.4134
-33.1226
-27.7477
-21.7951
M SB
.01
.025
.05
.10
.20
.01
.025
.05
.10
.20
.01
.025
.05
.10
.20
0.1078
0.1159
0.1226
0.1319
0.1439
-4.5798
-4.2635
-4.0192
-3.7372
-3.4234
-4.5798
-4.2635
-4.0192
-3.7372
-3.4234
0.1307
0.1360
0.1435
0.1534
0.1650
-3.7630
-3.6088
-3.4210
-3.2081
-2.9632
-4.6747
-4.4033
-4.0625
-3.7509
-3.4684
0.1135
0.1238
0.1340
0.1474
0.1624
-4.3851
-3.9773
-3.6605
-3.3268
-3.0288
-4.8235
-4.4831
-4.2202
-3.8450
-3.5125
0.1335
0.1422
0.1492
0.1588
0.1697
-3.6842
-3.4835
-3.2907
-3.1029
-2.8826
-4.5290
-4.1219
-3.7843
-3.5486
-3.2752
0.1335
0.1422
0.1487
0.1586
0.1692
-3.6842
-3.4835
-3.2992
-3.1108
-2.8958
-4.5290
-4.1219
-3.8195
-3.5552
-3.2988
0.0678
0.0799
0.1004
0.1157
0.1388
-7.3678
-6.2401
-4.9722
-4.2532
-3.5555
-4.9873
-4.6006
-4.2892
-3.9873
-3.6278
0.1197
0.1267
0.1368
0.1463
0.1595
-4.1587
-3.9098
-3.6402
-3.3516
-3.0969
-4.6317
-4.2861
-4.0300
-3.7243
-3.3811
0.1270
0.1353
0.1417
0.1520
0.1649
-3.9098
-3.6756
-3.4910
-3.2190
-2.9577
-4.3521
-4.1305
-3.8031
-3.5201
-3.2239
0.1267
0.1322
0.1409
0.1514
0.1632
-3.9220
-3.7436
-3.5205
-3.2579
-2.9959
-4.3774
-4.1573
-3.8472
-3.5745
-3.2763
0.0951
0.1109
0.1223
0.1340
0.1502
-5.2068
-4.4949
-4.0509
-3.6874
-3.2821
-4.6815
-4.4347
-4.1630
-3.8013
-3.4767
MZt
ADF
Table 5. Size and Power when using In¯mum Method for Model I; T=100
Size
i:i:d:
µ = ¡0:8
µ = ¡0:4
µ = 0:4
µ = 0:8
Power
Criteria
BIC
MAIC
MBIC
t-sig
M Z®
M SB
M Zt
ADF
PT
M Z®
M SB
M Zt
ADF
PT
0.0510
0.0510
0.0510
0.0500
0.050
0.0510
0.0490
0.0500
0.0510
0.050
0.0500
0.0510
0.050
0.050
0.0510
0.0510
0.0510
0.051
0.0500
0.0510
0.3180
0.5080
0.5100
0.1800
0.2990
0.4990
0.5050
0.1810
0.3240
0.5020
0.5140
0.1800
0.5100
0.4480
0.4700
0.4330
0.3810
0.4950
0.5160
0.1640
BIC
MAIC
MBIC
t-sig
BIC
MAIC
MBIC
t-sig
BIC
MAIC
MBIC
t-sig
BIC
MAIC
MBIC
t-sig
0.9370
0.3990
0.4070
0.1070
0.4010
0.1350
0.1350
0.0400
0.2140
0.1000
0.0010
0.0660
0.5260
0.2230
0.0150
0.0890
0.9350
0.3970
0.4070
0.1080
0.3960
0.1350
0.1320
0.0400
0.2120
0.1050
0.0010
0.0660
0.5260
0.2310
0.0170
0.0900
0.9380
0.4000
0.4090
0.1070
0.3950
0.1360
0.1380
0.0400
0.2130
0.0940
0.0000
0.0660
0.5210
0.2200
0.0130
0.0870
MA(1) Errors
0.9760
0.9320
0.3780
0.3390
0.3900
0.3530
0.7980
0.0880
0.4870
0.4020
0.1000
0.1110
0.1000
0.1170
0.2530
0.0380
0.0950
0.2010
0.0060
0.0480
0.0000
0.0020
0.0610
0.0640
0.1180
0.4480
0.0080
0.1050
0.0000
0.0060
0.0490
0.0910
0.9960
0.8270
0.8330
0.2970
0.8870
0.5560
0.5600
0.1360
0.7290
0.1340
0.0590
0.1890
0.7290
0.3380
0.0220
0.2820
0.9960
0.8290
0.8350
0.2990
0.8840
0.5520
0.5600
0.1360
0.7200
0.1290
0.0560
0.1900
0.7280
0.3360
0.0220
0.2830
0.9960
0.8250
0.8310
0.2970
0.8860
0.5500
0.5600
0.1360
0.7320
0.1370
0.0660
0.1870
0.7280
0.3350
0.0270
0.2810
1.0000
0.8290
0.8370
0.9610
0.9130
0.4570
0.4680
0.7370
0.5240
0.0650
0.0610
0.4040
0.4070
0.0460
0.0120
0.2270
0.9950
0.7070
0.7160
0.2630
0.8840
0.4880
0.5090
0.1210
0.7350
0.1010
0.0780
0.1910
0.7070
0.2650
0.0260
0.2990
Table 6. Size and Power when using In¯mum Method for Model I; T=100
Size
i:i:d:
½ = ¡0:8
½ = ¡0:4
½ = 0:4
½ = 0:8
Power
Criteria
BIC
MAIC
MBIC
t-sig
M Z®
M SB
M Zt
ADF
PT
M Z®
M SB
M Zt
ADF
PT
0.0510
0.0510
0.0510
0.0500
0.050
0.0510
0.0490
0.0500
0.0510
0.050
0.0500
0.0510
0.050
0.050
0.0510
0.0510
0.0510
0.051
0.0500
0.0510
0.3180
0.5080
0.5100
0.1800
0.2990
0.4990
0.5050
0.1810
0.3240
0.5020
0.5140
0.1800
0.5100
0.4480
0.4700
0.4330
0.3810
0.4950
0.5160
0.1640
BIC
MAIC
MBIC
t-sig
BIC
MAIC
MBIC
t-sig
BIC
MAIC
MBIC
t-sig
BIC
MAIC
MBIC
t-sig
0.0180
0.0000
0.0000
0.0200
0.1420
0.0620
0.0600
0.0480
0.1550
0.1030
0.0020
0.0760
0.2930
0.3260
0.2960
0.1450
0.0170
0.0000
0.0000
0.0200
0.1380
0.0640
0.0620
0.0490
0.1520
0.1100
0.0030
0.0790
0.3030
0.3390
0.3120
0.1490
0.0180
0.0000
0.0000
0.0190
0.1450
0.0640
0.0620
0.0480
0.1490
0.0980
0.0010
0.0750
0.2820
0.3080
0.2800
0.1430
AR(1) Errors
0.0470
0.0130
0.0430
0.0000
0.0430
0.0000
0.0410
0.0150
0.1680
0.1360
0.0500
0.0580
0.0490
0.0560
0.0850
0.0380
0.0530
0.1310
0.0130
0.0550
0.0000
0.0000
0.0460
0.0590
0.0600
0.1860
0.0490
0.1890
0.0320
0.1560
0.0610
0.0790
0.0400
0.0080
0.0070
0.0720
0.5100
0.3530
0.3480
0.1500
0.5380
0.0740
0.0140
0.1700
0.3500
0.3830
0.1700
0.1170
0.0390
0.0070
0.0050
0.0730
0.5100
0.3390
0.3490
0.1500
0.5270
0.0730
0.0140
0.1700
0.3470
0.3900
0.1800
0.1170
0.0400
0.0080
0.0080
0.0720
0.5110
0.3560
0.3630
0.1500
0.5430
0.0750
0.0150
0.1670
0.3510
0.3870
0.1700
0.1140
0.4560
0.3300
0.3350
0.3860
0.5960
0.3350
0.3420
0.4340
0.2960
0.0210
0.0130
0.2620
0.1310
0.1040
0.0150
0.1320
0.0330
0.0100
0.0100
0.0570
0.5100
0.3350
0.3390
0.1380
0.5620
0.0480
0.0190
0.1650
0.3460
0.3380
0.0710
0.1000
Table 7. Size and Power when using In¯mum Method for Model I; T=200
Size
i:i:d:
µ = ¡0:8
µ = ¡0:4
µ = 0:4
µ = 0:8
Power
Criteria
BIC
MAIC
MBIC
t-sig
M Z®
M SB
M Zt
ADF
PT
M Z®
M SB
M Zt
ADF
PT
0.0500
0.0510
0.0500
0.0500
0.0500
0.0500
0.0500
0.0510
0.0500
0.0510
0.0500
0.0510
0.0500
0.0500
0.0510
0.0510
0.0500
0.0510
0.0510
0.0510
0.5060
0.4310
0.4420
0.2000
0.4970
0.4230
0.4470
0.2040
0.4950
0.4200
0.4380
0.2130
0.5170
0.3720
0.3830
0.4320
0.4860
0.4230
0.4420
0.2420
BIC
MAIC
MBIC
t-sig
BIC
MAIC
MBIC
t-sig
BIC
MAIC
MBIC
t-sig
BIC
MAIC
MBIC
t-sig
0.8740
0.1400
0.1450
0.4250
0.2360
0.0970
0.0970
0.0560
0.1770
0.1030
0.0070
0.0640
0.3380
0.1890
0.0330
0.0530
0.8730
0.1370
0.1430
0.4270
0.2380
0.0950
0.0990
0.0580
0.1750
0.0990
0.0090
0.0660
0.3380
0.1950
0.0380
0.0530
0.8710
0.1400
0.1450
0.4320
0.2280
0.0880
0.0920
0.0580
0.1730
0.0990
0.0060
0.0680
0.3280
0.1880
0.0310
0.0540
MA(1) Errors
0.9110 0.8490
0.1670 0.1030
0.1700 0.1100
0.7060 0.4180
0.2410 0.2130
0.0580 0.0760
0.0640 0.0860
0.1230 0.0610
0.1040 0.1560
0.0370 0.0800
0.0020 0.0040
0.0600 0.0640
0.0970 0.2640
0.0060 0.1140
0.0030 0.0220
0.0300 0.0550
0.9960
0.3940
0.3970
0.8130
0.8490
0.4160
0.4270
0.3370
0.7190
0.4500
0.0530
0.3380
0.7300
0.4300
0.2200
0.2670
0.9960
0.3900
0.3970
0.8130
0.8420
0.4100
0.4310
0.3380
0.7080
0.4430
0.0620
0.3420
0.7260
0.4230
0.2220
0.2690
0.9960
0.3910
0.3950
0.8160
0.8450
0.4060
0.4250
0.3460
0.7110
0.4410
0.0510
0.3470
0.7230
0.4260
0.2190
0.2790
0.9990
0.4210
0.4280
0.9730
0.8610
0.3260
0.3460
0.6330
0.5670
0.2190
0.0320
0.3780
0.3930
0.0830
0.0640
0.1900
0.9880
0.2810
0.2860
0.7550
0.8090
0.3630
0.3800
0.3840
0.6990
0.3780
0.0500
0.3730
0.6950
0.3660
0.1810
0.3100
Table 8. Size and Power when using In¯mum Method for Model I; T=200
Size
i:i:d:
½ = ¡0:8
½ = ¡0:4
½ = 0:4
½ = 0:8
Power
Criteria
BIC
MAIC
MBIC
t-sig
M Z®
M SB
M Zt
ADF
PT
M Z®
M SB
M Zt
ADF
PT
0.0500
0.0510
0.0500
0.0500
0.0500
0.0500
0.0500
0.0510
0.0500
0.0510
0.0500
0.0510
0.0500
0.0500
0.0510
0.0510
0.0500
0.0510
0.0510
0.0510
0.5060
0.4310
0.4420
0.2000
0.4970
0.4230
0.4470
0.2040
0.4950
0.4200
0.4380
0.2130
0.5170
0.3720
0.3830
0.4320
0.4860
0.4230
0.4420
0.2420
BIC
MAIC
MBIC
t-sig
BIC
MAIC
MBIC
t-sig
BIC
MAIC
MBIC
t-sig
BIC
MAIC
MBIC
t-sig
0.0050
0.0010
0.0010
0.0110
0.0530
0.0440
0.0450
0.0340
0.1150
0.1090
0.0090
0.0540
0.1450
0.1530
0.1390
0.0910
0.0050
0.0010
0.0010
0.0120
0.0510
0.0430
0.0450
0.0360
0.1110
0.1080
0.0120
0.0520
0.1490
0.1540
0.1510
0.0960
0.0050
0.0010
0.0010
0.0120
0.0500
0.0440
0.0460
0.0340
0.1080
0.1040
0.0060
0.0520
0.1380
0.1360
0.1300
0.0920
AR(1) Errors
0.0530 0.0040
0.0400 0.0000
0.0400 0.0000
0.0470 0.0100
0.0630 0.0470
0.0410 0.0410
0.0450 0.0430
0.0490 0.0330
0.0570 0.0970
0.0410 0.0900
0.0000 0.0020
0.0470 0.0520
0.0660 0.1020
0.0420 0.0930
0.0460 0.0960
0.0510 0.0610
0.0270
0.0130
0.0140
0.0550
0.4380
0.3460
0.3550
0.1610
0.5710
0.4540
0.0040
0.2690
0.4030
0.3650
0.3700
0.2000
0.0250
0.0130
0.0140
0.0550
0.4290
0.3390
0.3600
0.1660
0.5580
0.4480
0.0060
0.2730
0.3980
0.3710
0.3810
0.2000
0.0290
0.0130
0.0140
0.0550
0.4270
0.3360
0.3510
0.1670
0.5620
0.4390
0.0040
0.2760
0.3930
0.3570
0.3690
0.2040
0.4750
0.2760
0.2760
0.3730
0.4730
0.3100
0.3210
0.3850
0.4010
0.2460
0.0040
0.3500
0.2340
0.1620
0.1730
0.1920
0.0210
0.0100
0.0090
0.0510
0.4020
0.3020
0.3300
0.1830
0.5510
0.4040
0.0040
0.3150
0.3800
0.3450
0.3570
0.2150
Table 9. Size and Power when using Supremum Method for Model I; T=100
Size
i:i:d:
µ = ¡0:8
µ = ¡0:4
µ = 0:4
µ = 0:8
Power
Criteria
BIC
MAIC
MBIC
t-sig
M Z®
M SB
M Zt
ADF
M Z®
M SB
M Zt
ADF
0.0500
0.0510
0.0500
0.0500
0.0500
0.0490
0.0500
0.0500
0.0500
0.0500
0.0510
0.0500
0.0500
0.0510
0.0510
0.0510
0.3350
0.4950
0.5010
0.1690
0.3360
0.4650
0.4840
0.1700
0.3810
0.4900
0.5060
0.1690
0.5100
0.4450
0.4680
0.4510
BIC
MAIC
MBIC
t-sig
BIC
MAIC
MBIC
t-sig
BIC
MAIC
MBIC
t-sig
BIC
MAIC
MBIC
t-sig
0.9310
0.3330
0.3420
0.0800
0.3950
0.1090
0.1150
0.0380
0.1900
0.0470
0.0010
0.0660
0.4400
0.1030
0.0040
0.0920
0.9310
0.3280
0.3390
0.0800
0.4010
0.1020
0.1100
0.0390
0.1920
0.0480
0.0010
0.0680
0.4440
0.1010
0.0050
0.0920
MA(1) Errors
0.9350 0.9690
0.3340 0.3290
0.3450 0.3400
0.0800 0.7700
0.4010 0.4660
0.1110 0.0870
0.1180 0.0950
0.0380 0.2340
0.2020 0.0880
0.0470 0.0050
0.0010 0.0010
0.0640 0.0600
0.4510 0.0980
0.1020 0.0030
0.0040 0.0000
0.0910 0.0500
0.9930
0.7070
0.7150
0.2660
0.8760
0.4900
0.5090
0.1210
0.7180
0.0980
0.0680
0.1940
0.7030
0.2380
0.0180
0.2990
0.9930
0.7060
0.7150
0.2660
0.8750
0.4810
0.5010
0.1210
0.7140
0.0820
0.0590
0.1940
0.7050
0.2240
0.0170
0.3020
0.9940
0.7060
0.7150
0.2650
0.8830
0.4890
0.5080
0.1210
0.7340
0.0970
0.0720
0.1930
0.7080
0.2390
0.0190
0.2990
1.0000
0.7110
0.7210
0.9340
0.9060
0.4120
0.4320
0.7040
0.5190
0.0710
0.0700
0.4230
0.3970
0.0350
0.0160
0.2230
Table 10. Size and Power when using Supremum Method for Model I; T=100
Size
i:i:d:
½ = ¡0:8
½ = ¡0:4
½ = 0:4
½ = 0:8
Power
Criteria
BIC
MAIC
MBIC
t-sig
M Z®
M SB
M Zt
ADF
M Z®
M SB
M Zt
ADF
0.0500
0.0510
0.0500
0.0500
0.0500
0.0490
0.0500
0.0500
0.0500
0.0500
0.0510
0.0500
0.0500
0.0510
0.0510
0.0510
0.3350
0.4950
0.5010
0.1690
0.3360
0.4650
0.4840
0.1700
0.3810
0.4900
0.5060
0.1690
0.5100
0.4450
0.4680
0.4510
BIC
MAIC
MBIC
t-sig
BIC
MAIC
MBIC
t-sig
BIC
MAIC
MBIC
t-sig
BIC
MAIC
MBIC
t-sig
0.0120
0.0000
0.0000
0.0140
0.1350
0.0570
0.0530
0.0410
0.1340
0.0530
0.0010
0.0650
0.1930
0.2050
0.1690
0.0870
0.0110
0.0000
0.0000
0.0140
0.1330
0.0520
0.0520
0.0420
0.1370
0.0530
0.0020
0.0650
0.2030
0.2110
0.1770
0.0880
AR(1) Errors
0.0120 0.0420
0.0000 0.0380
0.0000 0.0380
0.0140 0.0420
0.1400 0.1580
0.0560 0.0500
0.0560 0.0490
0.0410 0.0880
0.1340 0.0470
0.0480 0.0060
0.0000 0.0000
0.0640 0.0420
0.1960 0.0360
0.1980 0.0340
0.1620 0.0230
0.0850 0.0380
0.0300
0.0060
0.0050
0.0570
0.4970
0.3220
0.3190
0.1370
0.5370
0.0440
0.0160
0.1650
0.3300
0.3270
0.0660
0.1030
0.0290
0.0050
0.0050
0.0570
0.4940
0.2950
0.3030
0.1380
0.5350
0.0360
0.0130
0.1650
0.3300
0.3160
0.0620
0.1040
0.0300
0.0080
0.0070
0.0570
0.5040
0.3240
0.3290
0.1360
0.5580
0.0460
0.0170
0.1640
0.3420
0.3260
0.0650
0.1030
0.4170
0.2790
0.2820
0.3620
0.5820
0.3260
0.3330
0.4250
0.2960
0.0190
0.0170
0.2850
0.1230
0.0950
0.0060
0.1320
Table 11. Size and Power when using Supremum Method for Model I; T=200
Size
i:i:d:
µ = ¡0:8
µ = ¡0:4
µ = 0:4
µ = 0:8
Power
Criteria
BIC
MAIC
MBIC
t-sig
M Z®
M SB
M Zt
ADF
M Z®
M SB
M Zt
ADF
0.0490
0.0480
0.0500
0.0500
0.0520
0.0500
0.0510
0.0510
0.0510
0.0510
0.0510
0.0500
0.0510
0.0500
0.0500
0.0510
0.5060
0.4230
0.4520
0.2400
0.5230
0.4210
0.4380
0.2410
0.5050
0.4100
0.4360
0.2370
0.5100
0.3660
0.3790
0.4220
BIC
MAIC
MBIC
t-sig
BIC
MAIC
MBIC
t-sig
BIC
MAIC
MBIC
t-sig
BIC
MAIC
MBIC
t-sig
0.8610
0.1050
0.1110
0.4140
0.2280
0.0840
0.0930
0.0600
0.1660
0.0960
0.0050
0.0660
0.2890
0.1180
0.0250
0.0540
0.8620
0.1040
0.1100
0.4150
0.2380
0.0860
0.0920
0.0610
0.1710
0.0980
0.0050
0.0690
0.2930
0.1260
0.0230
0.0570
MA(1) Errors
0.8590 0.9000
0.1050 0.1380
0.1120 0.1430
0.4160 0.6720
0.2290 0.2320
0.0790 0.0620
0.0870 0.0670
0.0590 0.1130
0.1630 0.0940
0.0910 0.0350
0.0030 0.0020
0.0660 0.0520
0.2840 0.0880
0.1150 0.0070
0.0240 0.0060
0.0530 0.0300
0.9890
0.2830
0.2880
0.7560
0.8260
0.3760
0.3960
0.3860
0.7110
0.3860
0.0590
0.3650
0.6990
0.3640
0.1810
0.2890
0.9890
0.2830
0.2850
0.7560
0.8270
0.3770
0.3860
0.3870
0.7160
0.3870
0.0560
0.3660
0.7050
0.3590
0.1730
0.2910
0.9890
0.2830
0.2870
0.7560
0.8230
0.3700
0.3910
0.3800
0.7080
0.3740
0.0560
0.3630
0.6970
0.3600
0.1760
0.2890
0.9940
0.3290
0.3310
0.9450
0.8320
0.3080
0.3180
0.5990
0.5610
0.2070
0.0420
0.3730
0.3740
0.0910
0.0590
0.1790
Table 12. Size and Power when using Supremum Method for Model I; T=200
Size
i:i:d:
½ = ¡0:8
½ = ¡0:4
½ = 0:4
½ = 0:8
Power
Criteria
BIC
MAIC
MBIC
t-sig
M Z®
M SB
M Zt
ADF
M Z®
M SB
M Zt
ADF
0.0490
0.0480
0.0500
0.0500
0.0520
0.0500
0.0510
0.0510
0.0510
0.0510
0.0510
0.0500
0.0510
0.0500
0.0500
0.0510
0.5060
0.4230
0.4520
0.2400
0.5230
0.4210
0.4380
0.2410
0.5050
0.4100
0.4360
0.2370
0.5100
0.3660
0.3790
0.4220
BIC
MAIC
MBIC
t-sig
BIC
MAIC
MBIC
t-sig
BIC
MAIC
MBIC
t-sig
BIC
MAIC
MBIC
t-sig
0.0050
0.0000
0.0000
0.0110
0.0500
0.0430
0.0450
0.0330
0.1030
0.0960
0.0020
0.0510
0.1130
0.1050
0.1070
0.0670
0.0050
0.0000
0.0000
0.0110
0.0510
0.0440
0.0410
0.0340
0.1080
0.0970
0.0020
0.0530
0.1260
0.1130
0.1090
0.0730
AR(1) Errors
0.0050 0.0450
0.0000 0.0400
0.0000 0.0400
0.0110 0.0370
0.0490 0.0530
0.0410 0.0400
0.0430 0.0410
0.0330 0.0370
0.1020 0.0490
0.0890 0.0380
0.0020 0.0000
0.0520 0.0440
0.1090 0.0520
0.0990 0.0390
0.1010 0.0400
0.0630 0.0360
0.0180
0.0130
0.0120
0.0480
0.4280
0.3240
0.3550
0.1820
0.5650
0.4020
0.0060
0.3010
0.3770
0.3420
0.3590
0.2010
0.0200
0.0120
0.0110
0.0480
0.4410
0.3270
0.3410
0.1870
0.5780
0.4050
0.0060
0.3060
0.3850
0.3470
0.3550
0.2050
0.0190
0.0120
0.0110
0.0480
0.4270
0.3110
0.3440
0.1820
0.5650
0.3900
0.0050
0.2990
0.3760
0.3380
0.3550
0.2000
0.4270
0.2750
0.2720
0.3410
0.4520
0.3030
0.3200
0.3630
0.3900
0.2270
0.0050
0.3490
0.2200
0.1700
0.1730
0.1860
Table 13a. Empirical Results using Informatiioin Criteria to select lag k and In¯mum Method to choose Break Point T B
Series
T
Stock Prices
100
Real Wages
71
Criterial
M Z®
BIC
M AIC
M BIC
BIC
M AIC
M BIC
b
-49.89
-49.22a
-49.22a
-39.12c
-39.12a
-39.12a
k
1
1
1
1
1
1
TB
1941
1937
1837
1938
1938
1938
M Zt
k
b
-4.95
-4.93a
-4.93a
-4.37c
-4.37a
-4.37a
1
1
1
1
1
1
TB
1941
1937
1937
1938
1938
1938
ADF
k
b
-5.25
-5.25a
-5.25a
-4.69
-4.69d
-4.69c
1
1
1
1
1
1
TB
1937
1937
1937
1938
1938
1938
PT
b
8.92
13.26d
13.26d
9.43c
11.28b
11.28b
k
TB
®^
1
2
2
1
1
1
1931
1931
1931
1940
1940
1940
0.65
0.65
0.65
0.61
0.61
0.61
We use a, b, c, d to represent rejection at 1%, 2.5%, 5%, 10% signi¯cance level.
Table 14b. Empirical Results using Information Criteria to select Lag k and Supremum Method to choose Break Point T B
Series
T
Stock Prices
100
Real Wages
71
Criterial
BIC
M AIC
M BIC
BIC
M AIC
M BIC
M Z®
b
-33.10
-20.25d
-20.25c
-27.94c
-27.94a
-27.94a
M Zt
ADF
k
TB
®
^
b
c
1
2
2
1
1
1
1931
1931
1931
1933
1933
1933
0.73
0.77
0.77
0.69
0.69
0.69
-4.05
-3.21d
-3.21c
-3.67c
-3.67b
-3.67b
-4.32
-3.38
-3.38
-3.86d
-3.86c
-3.86c
We use a, b, c, d to represent rejection at 1%, 2.5%, 5%, 10% signi¯cance level.
Table 15a. Empirical Results using Recursive Method to select Lag k and In¯mum Method to choose T B
Series
Stock Prices
Real Wages
T
100
71
M Z®
k
c
-143.10
-11628.50a
3
4
TB
1948
1941
M Zt
k
c
-8.43
-76.24a
3
4
TB
ADF
1948
1941
b
-5.25
-4.69d
k
1
1
TB
PT
1936
1938
d
5.94
4.05c
k
TB
®
^
3
3
1930
1940
0.65
0.61
We use a, b, c, d to represent rejection at 1%, 2.5%, 5%, 10% signi¯cance level.
Table 16b. Empirical Results using Recursive Method to select Lag k and supremum Method to choose TB
Series
T
M Z®
M Zt
ADF
k
TB
®
^
Stock Prices
Real Wages
100
71
-49.87c
-27.94
-4.97c
-3.67
-3.97c
-3.86
3
1
1930
1933
0.73
0.69
We use a, b, c, d to represent rejection at 1%, 2.5%, 5%, 10% signi¯cance level.
Figure 1. Gaussian Power Envelope and Asymptotic Power Functions; In…mum Method and
Fixed and Random Initial Condition.
36
Figure 2. Gaussian Power Envelope and Asymptotic Power Functions; Supremum Method
and Fixed and Random Initial Condition.
37
Figure 3. Logarithm of Real Wages with a broken time trend; 1900-1970.
38
Figure 4. Logarithm of Stock Prices with a broken time trend; 1871-1971.
39