Disserta¸c˜
ao de Mestrado
Comparing Two Multivariate Unit Root Tests
Guilherme N. L. Veiga da Rocha
1EPGE - Funda¸c˜ao Getulio Vargas
Orientador: Renato G. Flˆores Jr. (EPGE / FGV-RJ)
Abstract
In this essay, a method for comparing the asymptotic power of the multivariate unit root tests proposed in Phillips & Durlauf (1986) and Flˆores, Preumont & Szafarz (1996) is proposed. In order to determine the asymptotic power of the tests the asymptotic distributions under the null hypothesis and under the set of alternative hypotheses described in Phillips (1988) are determined.
Contents
1 Introduction 1
2 Functional Limit Results 4
3 Description of the Testing Procedures 6
3.1 The PD Unit Root Test . . . 9 3.2 The RPD Unit Root Test . . . 15 3.3 The FPS Unit Root Test . . . 20
4 Distribution under H1 24
4.1 The PD Test under H1 . . . 27
4.2 The RPD Test under H1 . . . 31
4.3 The FPS Test under H1 . . . 32
5 Conclusion 34
A Estimation of unrestricted VAR(1) coefficients 37
B A Lengthy Algebra Result 38
C Some notes on the results regarding near unit root processes - Phillips
Agradecimentos
Primeiramente, gostaria de agradecer ao meu orientador, professor Renato Flˆores. O seu profundo conhecimento sobre diversas ´areas aliado `a sua grande disposi¸c˜ao para ensinar certamente contribu´ıram para tornar os meus anos na EPGE extremamente proveitosos, al´em de terem sido um enorme incentivo a prosseguir na carreira acadˆemica. Agrade¸co tamb´em ao professor Marcelo Fernandes pela paciˆencia e inab´alavel disposi¸c˜ao para discutir d´uvidas e id´eias, assim como pelas sugest˜oes dadas na realiza¸c˜ao desse trabalho. Agrade¸co tamb´em aos professores Paulo Klinger e Alo´ısio Ara´ujo por terem incrementado tanto os meus conhecimentos como o meu interesse pela Matem´atica. Aos demais professores da EPGE, agrade¸co pela oportunidade de conviver por trˆes anos em um ambiente intelectualmente rico. Esses agradecimentos n˜ao estariam completos sem a presen¸ca dos amigos Frederico Es-trella Carneiro Valladares, Bernardo de S´a Mota e Mark G. Sourdoutovich pelas in´umeras conversas sobre minhas n˜ao raras d´uvidas e pelas discuss˜oes de alto n´ıvel que tornaram esses anos breves.
1
Introduction
The assumption of stationarity required in some statistical models makes them inadequate to analyze many series of economic data. This fact caused a great deal of research to be conducted in order to relax such an annoying assumption.
A simple way to reach this goal consists in modelling a series as a process which is stationary around a deterministic trend. This allows one to model series whose mean is clearly nonstationary and the tools needed to deal with such processes can be easily extended from the techniques employed for stationary processes. The development of these models, along with more recent advances in dealing with hetereogeneously distributed errors by White (1980) and White & Domowitz (1984), rendered applied workers generally satisfied with the tools available for data analysis.
However, the study reported in Nelson & Plosser (1982) greatly influenced to change this situation. Applying the test developed by Dickey & Fuller (1979) to macroeconomic time series, they concluded that some of the series analyzed would be better modelled as integrated processes. The effects of spuriously detrending an integrated time series may be quite harmful, including a bias to accept the hypothesis of a nonzero trend coefficient in a driftless random walk (Nelson & Kang (1981)) and the rise of spurious correlations in the detrended series.
Besides providing guidance to model selection, the ability to detect the presence of a unit root may be relevant to test economic theories. In an integrated process, shocks are persistent and mean reversion does not take place. Therefore, hypotheses such as the Purchasing Power Parity in which reversion to a long run equilibrium is assumed can be tested by unit root tests as in Flˆores, Jorion, Preumont & Szafarz (1999).
More recently, Granger (1981) introduced the concept of cointegration, in which two or more integrated series exhibit a common stochastic trend. This is of great importance to economic theory, since it allows one to detect long run equilibrium relationships among nonstationary series. Since cointegration analysis presumes integrated series, it is necessary to somehow be able to test the presence of a unit root in a time series.
are derived when a driftless random walk is the true data generating process (DGP). It is shown that, under such circunstances, the inclusion of a constant and a trend has influence on the distribution of both test statistics.
An important extension of the Dickey & Fuller test is made in Phillips (1987a) and Phillips & Perron (1988). In those papers milder assumptions about the shocks are made and the asymptotic distributions determined. One important result is the proof that, when a unit root is present, the OLS estimate of the autoregressive parameter is consistent regardless of serial correlation in the shocks. Adjustments in the test statistics are made to account for the error serial correlation and the resulting statistics have the same asymptotic distribution of those presented in Dickey & Fuller (1979).
However popular the DF and PP tests proposed by Dickey & Fuller (1979) and Phillips & Perron (1988) respectively are, they do offer some inconveniences. Firstly, both tests suffer from low power when dealing with alternatives close to the null. Although one may argue that any test proposed will have poor performance for some alternative close enough to the null, this does not mean that no effort should be employed in order to improve on the available ones. Some economic series appear to have long persistence and being able to reliably test them for unit roots would be rather desirable. Hansen (1995) suggested exploring the correlations between the series being tested and a series whose order of integration is known as a way of increasing power. It is important to notice that, despite using more than one series, this test will only deal with the existence of a unit root in one of them, which makes evident another problem suffered by the DF and PP tests: its inability to deal with multivariate series.
Phillips & Durlauf (1986) provided some useful results concerning a unit root test in a multivariate environment. The consistency of the OLS estimator under a unit root in each series is established and the consistency of the covariance matrix estimator follows. These results are derived under mild restrictions on the serial correlation of shocks. The test then proposed can be thought of as a multivariate version of the Phillips & Perron (1988) test.
to the use of information contained in the covariance matrix and partly to the restriction of common dynamics.
The FPS test presented in Flˆores, Preumont & Szafarz (1996) can be thought of as a generalization of the Abuaf & Jorion (1990) test, since the restriction of common dynamic is relaxed. In that paper, asymptotic results are derived which allow a variety of tests to be perfomed. The estimation procedure proposed presents two main differences from that proposed in Phillips & Durlauf (1986): the estimation of parameters is made by using Zellner’s SURE method instead of using the OLS method and the coefficients matrix of the VAR assumed for the data generated process is imposed to be diagonal.
In this essay, interest lies in obtaining results allowing one to determine the gain in power of the FPS test over the PD test and what fraction of it is due to the use of the restriction that the matrix of coefficients is diagonal for a certain class of alternative hypothesis. In order to do so, the results related to testing for the presence of unit roots presented in Phillips & Durlauf (1986) will be adapted to the case where a restriction on the matrix of coefficients is imposed. This allows us to determine how much power is added because of this restriction. The remaining difference in power between the FPS test and the restricted PD test shall be attributed to the use of a possibly more efficient weighting matrix in the estimations.
Since our interest is in the asymptotic power, we must be able to determine the asymptotic behavior of the test statistics under the class of alternative hypotheses being considered. In order to do so, we adopt the class of alternatives presented in Phillips (1988) and adapt those results to both the PD and FPS tests as well as to a new test proposed. The class of alternative hypotheses considered in this paper are a sequence of drifting DGPs which converge to the null in a suitable rate, following the original ideas in Pitman (1949) and Neyman (1937).
2
Functional Limit Results
Before we discuss the unit root tests themselves, a review of the results derived in Phillips & Durlauf (1986) is presented. Those results consist of a multivariate generalization of the univariate results due to McLeish (1975a) used in Phillips (1987a) and will be much helpful in the following sections.
Consider a sequence of randomp-dimensional vectors{ut}defined on a probability space
(Ω, B, P) such that:
E(ut) = 0,∀t (1)
Define the vector of partial sums:
St= t
X
j=1
uj (2)
and the vector random sequence of functions on [0,1]:
XT(r) =
1 √
TΣ
−1 2S
[T r] =
1 √
TΣ
−1 2S
j−1,
(j−1)
T ≤r < j
T, T = 1,2, . . .
XT(1) =
1 √
TΣ
−1 2S
T (3)
where r∈[0,1] and Σ12 is the symmetric positive definite square root of the positive definite
matrix Σ = limT↑∞E(T−1STST′ )
We now describe a result concerning the asymptotic behavior of the processXT(r) which
will be helpful when we turn our attention to the limiting distribution of our test statistics under different conditions. A set of assumptions on the temporal dependence of the process {ut} is made and an appropriate metric chosen in order to establish such a result.
In order to determine the metric under which convergence will be established note that:
XT(•)∈D[0,1]p =D[0,1]×D[0,1]× · · · ×D[0,1] (4)
whereD[0,1] is the space of all real valued functions from [0,1] to the real line that are right continuous at each point of [0,1] and possess finite left limits. The Skorohod metric d is chosen in order to turn the metric space (D[0,1], d) into a complete separable metric space. For the n-dimensional product space, the metric chosen is:
Once an appropriate metric for the space where XT(r) lies is chosen, a way to measure
the temporal dependence of the process {ut} must be determined before any necessary
re-striction can be made. We now describe the measure of dependence employed in Phillips & Durlauf(1986). For the σ-algebras F, G⊆ B , define:
ϕ(F, G) = sup{F ∈F,G∈G,P(F)>0}|P(G|F)−P(G)|
α(F, G) = sup{F ∈F,G∈G}|P(G)P(F)−P(F,G)| (6)
Let a < b and Fb
a be the σ-algebra generated by {ua, . . . , ub} and Rba be the σ-algebra
generated by {Sb−Sa−1,∀a≤b}. The temporal dependence of{ut} is measured by:
ϕm = supnsupj≥n+mϕ(F1n, R
j n+m)
αm = supnsupj≥n+mα(F1n, R
j
n+m) (7)
We say that ϕm (or αm) is of size−p if ϕm (or αm) = O(m−p−ǫ) for some ǫ >0 as m↑ ∞.
Having described the metric for which convergence will be derived and the measures of temporal dependence which will be used, we are now ready to state a result concerning the asymptotic behavior of XT(r).
Theorem 1 Let {ut}∞t=1 be a weakly stationary sequence of random p-dimensional vectors
with E(ut) = 0,∀t. If:
1. E|uit|β <∞,∀i= 1, . . . , p for some2≤β <∞;
2. either P∞n=1ϕ 1−1
β
n <∞ OR
β >2 and P∞n=1α1−
2
β
n <∞
Then:
lim
T→∞E(T −1S
TST′ ) = Σ =E(u1u
′
1) +
∞ X
k=2
[E(u1u′k) +E(uku′1)]
If, furthermore, Σ is positive definite, then
XT(r)⇒W(r)
Proof. See Phillips & Durlauf (1986).
In Phillips & Durlauf (1986), conditions that assure the convergence of XT to W when
{ut}is not identically distributed are also obtained. However, for that convergence to hold we
must have limT→∞E(T−1STST′ ) = Σ as a hypothesis. Under stationarity and the conditions
of Theorem 1, not only XT converges to W, but also T−1STST′ converges to Σ. This will be
important as consistent estimation of the long run covariance matrix Σ will be involved in the test procedures under analysis.
3
Description of the Testing Procedures
In this section, the three tests will be described in detail. We begin by describing the common features and assumptions of the procedures and briefly discussing the relationships between them. The details of each procedure, including the test statistics to be used and their distributions under H0, are then described. The distributions of the test statistics
under H1 are given in section 4.
All testing procedures considered in this paper are based on the following model:
yt = Ayt−1+ut (8)
where
{yt}Tt=0 is a sequence ofp-dimensional random vectors which will be tested for the presence
of unit roots;
A is a p×p matrix of coefficients;
{ut}Tt=0 is a sequence of p-dimensional shocks.
Throughout this work, the assumption below will be made:
Assumption 1 The process {ut} is given by:
ut= Φ(L)εt (9)
where:
2. Φ(1) is a non singular matrix;
3. E(εt) = 0,∀t;
4. E(εtε′t) = Ip, ∀t;
5. E(εtε′t−j) = 0,∀t,∀j 6= 0;
6. εt has finite fourth moments;
7. All eigenvalues of Φ(L) lie inside the unit circle;
8. P∞
k=0ΦkΦ′k is a positive definite matrix;
Under such conditions, it is possible to assure that {ut} is a weakly stationary process
satisfying all conditions under which the results of Theorem 1 are established. The p×p
matrix Σ0 will denote the contemporaneous covariance matrix, while Σ will be used to refer
to the long run covariance matrix:
Σ0 = E(utu′t) =
∞ X
k=0
ΦkΦ′k (10)
Σ = Φ(1)Φ(1)′ (11)
Under the assumptions made, Σ is a positive definite matrix.
Besides the long run and contemporaneous covariance matrices, the autocovariance ma-trices will appear in some asymptotic distributions. We define Γj, the autocovariance matrix
of j-th order, by:
Γj =
∞ X
k=j
ΦkΦ′k−j
, j ≥1 (12)
and we define:
Σ1 =
∞ X
j=1
Γj (13)
It is easy to show that:
The hypothesis we will be interested in is given by:
H0 : A=Ip
H1 : A6=Ip (15)
where Ip is the p-dimensional identity matrix and the class of matrices A must be duly
defined.
Our main goal consists in determining a method in order to compare the asymptotic power of the multivariate unit root test proposed in Flˆores, Preumont & Szafarz (1996) – the FPS procedure – and that described in Phillips & Durlauf (1986) – the PD procedure. Two main differences exist between these two testing procedures:
• The FPS test imposes a restriction a priori that A is diagonal, while in the PD proce-dure A is unrestricted;
• The FPS test estimates the matrix of coefficients A using the seemingly unrelated regressions technique, which can be considered – in their case – a least squares where the weighting matrix is given by Σ−1 whereas the PD test relies on OLS estimation;
Since we are also interested in providing a way to determine what fraction of any difference in power is due to each of these peculiarities in the FPS test, a procedure in between the FPS and the PD is proposed.
Such testing procedure will impose the restriction that the matrix of coefficients A is diagonal as in the FPS test, but its estimation, similarly to what happens in the PD test, will make use of OLS. This can be thought of as a restricted PD test and therefore shall be referred to as the RPD test.
One could be tempted to propose a second testing procedure in which no restriction onA
is made but OLS estimation is replaced by SURE. Although such a test may seem to stand in between the FPS and PD tests, its results will be equivalent to those of the PD test. This happens because the set of regressors will be the same for all pequations. Under these circumstances, the SURE can be efficiently estimated by using OLS. It is worth noting that when one imposes that A is diagonal, the set of regressors for each equation will no longer be the same and therefore the use of Σ−1 as weighting matrix may lead to a more efficient
Before we proceed to the description of the details of each test, a result which will be very helpful to determine the consistency of some of the estimators proposed and the asymptotic distribution of some test statistics under H0 is presented:
Theorem 2 Suppose{yt} is given by (8) with A=Ip and {ut}satisfies the assumptions in
Theorem 1. Then:
1. T−32 PT
t=1yt⇒Σ
1 2 R1
0 W(r)dr
2. T−2PT
t=1yt−1yt′−1 ⇒Σ
1 2 R1
0 W(r)W(r)′drΣ
1 2
3. T−1PT
t=1(yt−1u′t+utyt′−1)⇒Σ
1
2W(1)W(1)′Σ 1 2 −Σ
0
4. T−1PT
t=1yt−1u′t ⇒Σ
1 2 R1
0 W(r)dW(r)′Σ
1 2 + Σ
1
where:
Σ0 = limT→∞T−1
T
X
t=1
E(utu′t)
Σ1 = limT→∞T−1
T
X
t=1
t−1
X
j=1
E(uju′t)
Σ = limT→∞E(T−1STST′ ) = Σ0+ Σ1+ Σ′1
When {ut} satisfies asumption 1, Σ0, Σ and Σ1 are as defined in (10), (11) and (13)
respectively.
Proof. See Phillips & Durlauf (1986).
3.1
The PD Unit Root Test
Define:
Y−1 =
h
y0 y1 · · · yT−1
i
(16)
Y = h y1 y2 · · · yT
i
(17)
Using OLS, two different consistent estimators underH0 for the matrixAcan be obtained
(see Appendix A):
ˆ
AP D1 = (YY−′1)(Y−1Y−′1)−1 (18)
ˆ
AP D2 = (Y−1Y′)(Y−1Y−′1)−1 (19)
Phillips & Durlauf (1986) prove that, under H0, ˜AP D1
p
→ Ip and ˜AP D2
p
→ Ip. A third
estimator can be obtained by combining these two:
˜
AP D =
1 2
ˆ
AP D1+ ˆAP D2
= 1 2
(Y−1Y′) + (YY−′1) Y−1Y−′ 1
−1
(20)
It follows immediately from the consistency of ˜AP D1 and ˜AP D2 and the symmetry of
A under H0 that ˜AP D p
→ Ip in that case. A result proven in the same paper concerns
the consistency of the estimator of the contemporaneous covariance matrix under the same situation:
˜
Σ0P D = T−1Y
IT − Y−′ 1(Y−1Y−′1)−1Y−1
Y′ (21)
Before discussing the test statistics, we define an estimator for the long run covariance matrix Σ. The Newey and West (1987) estimator for Σ will be used. It is given by:
˜
ΣP D = T−1 T
X
t=1
˜
uP Dtu˜′P Dt
+ T−1
l
X
τ=1
υτ l T
X
t=τ+1
(˜uP Dtu˜′P Dt−τ + ˜uP Dt−τu˜′P Dt) (22)
with:
υτ l = 1−
τ l+ 1 ˜
uP Dt = yt−A˜P Dyt−1
The next result assures the consistency of ˜ΣP D
Proposition 1 Suppose {yt} is given by (8) with A=Ip and {ut} satisfies the assumptions
1. supi,tE|uit|2β <∞ for some β >2;
2. limT→∞lT = +∞;
3. l =o(T14);
Then:
˜ ΣP D
p
→Σ
Proof. Let
˜
ST l =T−1 T
X
t=1
utu′t+T
−1Xl
τ=1
l
X
t=τ+1
(utu′t−τ +ut−τu′t)
If l and ut satisfy the conditions above, Theorem 3.6 in Phillips & Durlauf (1986) garantees
that ˜ST l p
→Σ as T ↑ ∞. Choosing the same sequence for l, we have that υτ l →1 as T ↑ ∞
and therefore
Σ˜P D−S˜T l
p
→ 0. Since ˜AP D is consistent, ˜uP Dt converges in probability to
uP Dt,∀t and the proof is complete.
In Phillips & Durlauf (1986) the following test statistic, based on a Wald test, is proposed to test H0:
F0P D = vec( ˜AP D−Ip)′
h
˜
Σ0P D⊗(Y−1Y−′ 1)−1
i−1
vec( ˜AP D−Ip)
= trh( ˜AP D−Ip) ˜Σ−0P D1 ( ˜AP D−Ip)(Y−1Y−′ 1)
i
(23)
If instead of the weight matrix used in (23), we use as weight matrix an estimate of the inverse of hΣ⊗(Y−1Y−′ 1)−1
i
, the test statistics become:
FP D = vec( ˜AP D−Ip)′
h
˜
ΣP D⊗(Y−1Y−′1)−1
i−1
vec( ˜AP D −Ip)
= trh( ˜AP D−Ip) ˜Σ−P D1 ( ˜AP D−Ip)(Y−1Y−′1)
i
(24)
Proposition 2 Suppose that conditions in Proposition 1 are satisfied and F0P D and FP D
are defined as in (23) and (24) respectively. Then:
F0P D ⇒
1 4tr
nh
W(1)W(1)′−Σ−12Σ
0Σ−
1 2
i
Σ12Σ−1
0 Σ
1 2
h
W(1)W(1)′−Σ−12Σ
0Σ−
1 2
i
×
Z 1
0 W(r)W(r)
′dr−1 )
(25)
FP D ⇒
1 4tr
nh
W(1)W(1)′−Σ−12Σ
0Σ−
1 2
i h
W(1)W(1)′−Σ−12Σ
0Σ−
1 2
i
×
Z 1
0 W(r)W(r)
′dr−1 )
(26)
Proof. For the first convergence, see Phillips & Durlauf (1986). For the second one, note that:
FP D = tr
h
( ˜AP D−Ip) ˜Σ−P D1 ( ˜AP D−Ip)(Y−1Y−′1)
i = = 1 4tr T X t=1
yt−1u′t+utyt′−1
! T X
t=1
yt−1yt′−1
!−1
˜ Σ−P D1
T
X
t=1
yt−1u′t+uty′t−1
!
= 1 4tr
"
T−1
T
X
t=1
yt−1u′t+utyt′−1
!
˜
Σ−P D1 T−1
T
X
t=1
yt−1u′t+utyt′−1
!
×
T−2
T
X
t=1
yt−1y′t−1
!−1
where the second equality comes from the fact that ( ˜AP D −Ip) is symmetric. Using the
results in theorem 2, it follows that:
FP D ⇒ tr
1
4
Σ12W(1)W(1)′Σ 1 2 −Σ
0
Σ−1Σ12W(1)W(1)′Σ 1 2 −Σ
0
×
Σ12
Z 1
0 W(r)W(r)
′drΣ1 2
−1#
⇒ 14trhW(1)W(1)′ −Σ−12Σ
0Σ−
1
2 W(1)W(1)′−Σ− 1 2Σ
0Σ−
1 2
×
Z 1
0 W(r)W(r)
′dr−1 #
corrections are:
F0SP D = F0P D −
1 4tr
nh
T−2yTyT′ ( ˜Σ
−1 0P D−Σ˜
−1
P D)yTyT′ + ( ˜Σ0P D−Σ˜P D)
i
× (T−2Y−1Y−′ 1)−1
o
(27)
FSP D = FP D−
1 4tr
nh
T−1yTyT′ Σ˜
−1
P D( ˜ΣP D −Σ˜0P D) + ( ˜ΣP D−Σ˜0P D) ˜Σ−P D1T
−1y
TyT′ +
( ˜ΣP D−Σ˜0P D) ˜Σ−P D1( ˜ΣP D−Σ˜0P D)−2( ˜ΣP D −Σ˜0P D)
i
(T−2Y−1Y−′1)−1
o
(28)
We now turn our attention to determining the distribution underH0 of these test
statis-tics:
Proposition 3 Suppose that conditions in Theorem 1 are satisfied and F0SP D andFSP D are
as defined in (27) and (28). Then:
FS0P D ⇒
1 4tr
(
[W(1)W(1)′−Ip] [W(1)W(1)′−Ip]
Z 1
0 W(r)W(r)
′dr−1 )
(29)
FSP D ⇒
1 4tr
(
[W(1)W(1)′−Ip] [W(1)W(1)′−Ip]
Z 1
0 W(r)W(r)
′dr−1 )
(30)
Proof. For the first convergence, define:
C1 = tr
h
T−2yTyT′ ( ˜Σ−0P D1 −Σ˜P D−1)yTyT′ (T−2Y−1Y−1)−1
i
C2 = tr
( ˜Σ0P D−Σ˜P D)
T−2Y−1Y−1
−1
It follows that:
C1 ⇒ tr
"
W(1)W(1)′Σ12(Σ−1
0 −Σ−1)Σ
1
2W(1)W(1)′
Z 1
0 W(r)W(r)
′dr−1 #
C2 ⇒ tr
"
Σ−12(Σ
0−Σ)Σ−
1 2
Z 1
0 W(r)W(r)
′dr−1 #
In Appendix B it is proved that:
W(1)W(1)′−Σ−12Σ
0Σ−
1 2 Σ
1 2Σ−1
0 Σ
1
2 W(1)W(1)′−Σ− 1 2Σ
0Σ−
1 2
=
(W(1)W(1)′−Ip) (W(1)W(1)′−Ip) +
Σ−12(Σ
0 −Σ)Σ−
1 2
+
W(1)W(1)′Σ12(Σ−1
0 −Σ−1)Σ
1
2W(1)W(1)′
Therefore:
F0P D ⇒
1 4tr
nh
(W(1)W(1)′ −Ip) (W(1)W(1)′−Ip) +
Σ−12(Σ
0−Σ)Σ−
1 2
+
W(1)W(1)′Σ12(Σ−1
0 −Σ−1)Σ
1
2W(1)W(1)′
iZ 1
0 W(r)W(r)
′dr−1 )
It follows that:
F0SP D = F0P D −
1 4C1−
1 4C2
⇒ 14tr
(
[(W(1)W(1)′ −Ip) (W(1)W(1)′−Ip)]
Z 1
0 W(r)W(r)
′dr−1 )
For the second convergence, define:
C1 = tr
T−1yTyT′ Σ˜−P D1( ˜ΣP D−Σ˜0P D) T−2Y−1Y−′ 1
−1
= tr
( ˜ΣP D −Σ˜0P D) ˜Σ−P D1 T−1yTyT′ T−2Y−1Y−′ 1
−1
C2 = tr
( ˜ΣP D −Σ˜0P D) ˜Σ−P D1 ( ˜ΣP D−Σ˜0P D) T−2Y−1Y−′1
−1
C3 = tr
( ˜ΣP D −Σ˜0P D) T−2Y−1Y−′1
−1
It follows that:
C1 ⇒ tr
"
Σ−12(Σ−Σ
0)Σ−
1
2W(1)W(1)′
Z 1
0 W(r)W(r)
′dr−1 #
⇒ tr
"
W(1)W(1)′Σ−12(Σ−Σ
0)Σ−
1 2
Z 1
0 W(r)W(r)
′dr−1 #
C2 ⇒ tr
"
Σ−12(Σ−Σ
0)Σ−
1 2Σ−
1
2(Σ−Σ
0)Σ−
1 2
Z 1
0 W(r)W(r)
′dr−1 #
C3 ⇒ tr
"
Σ−12(Σ−Σ
0)Σ−
1 2
Z 1
0 W(r)W(r)
′dr−1 #
Besides:
h
W(1)W(1)′ −Σ−12Σ
0Σ−
1 2
i h
W(1)W(1)′−Σ−12Σ
0Σ−
1 2
i
=
h
(W(1)W(1)′−Ip) + Σ−
1
2(Σ−Σ
0)Σ−
1 2
i
×
h
(W(1)W(1)′−Ip) + Σ−
1
2(Σ−Σ
0)Σ−
1 2
i
=
+ (W(1)W(1)′−I
p)(Σ−
1
2(Σ−Σ
0)Σ−
1 2)+
+ (Σ−12(Σ−Σ
0)Σ−
1
2)(W(1)W(1)′ −I
p)
+ (Σ−12(Σ−Σ
0)Σ−
1 2)(Σ−
1
2(Σ−Σ
0)Σ−
1 2)
i
=
[(W(1)W(1)′−Ip)(W(1)W(1)′−Ip)+
+ W(1)W(1)′Σ−12(Σ−Σ
0)Σ−
1 2 + Σ−
1
2(Σ−Σ
0)Σ−
1
2W(1)W(1)′+
− 2(Σ−12(Σ−Σ
0)Σ−
1
2) + (Σ− 1
2(Σ−Σ
0)Σ−
1 2)(Σ−
1
2(Σ−Σ
0)Σ−
1 2)
i
Therefore:
FP D ⇒
1
4tr{[(W(1)W(1)
′
−Ip) (W(1)W(1)′−Ip)
+ W(1)W(1)′Σ12Σ−1(Σ
0−Σ)Σ−
1 2
+ Σ−12(Σ
0−Σ)Σ−1Σ
1
2W(1)W(1)′
+ Σ−12(Σ−Σ
0)Σ−
1 2Σ−
1
2(Σ−Σ
0)Σ−
1 2
+
− 2(Σ−12(Σ−Σ
0)Σ−
1 2)
iZ 1
0 W(r)W(r)
′dr−1 )
It results that:
FSP D = FP D−2C1−C2+ 2C3
FSP D ⇒
1 4tr
(
[(W(1)W(1)′−Ip) (W(1)W(1)′−Ip)]
Z 1
0 W(r)W(r)
′dr−1 )
3.2
The RPD Unit Root Test
As mentioned above, in the RPD testing procedure, we will impose a restriction that the matrix of coefficients A is diagonal. By measuring the performance of such a procedure, we may get a picture of how much of any difference in power between the FPS and the PD procedures is due to the restriction that the matrix of coefficients is diagonal used in the former. Taking the restriction into account, we are left with the following model:
y1t
y2t
... ypt =
a1 0 · · · 0
0 a2 · · · 0
... ... ... 0 0 · · · 0 ap
y1t−1
y2t−1
...
ypt−1
+
u1t
u2t
... upt (31) Defining:
α = h a1 a2 · · · ap
i′
, a p−dimensional vector (32)
ιp =
h
1 1 · · · 1 i′, a p−dimensional vector (33)
φt = diag(yt), a p×p matrix (34)
µt = diag(ut)a p×p matrix (35)
Φ−1 =
h
φ0 φ1 · · · φT−1
i′
, a T p×p matrix (36) Φ = h φ1 φ2 · · · φT
i′
, a T p×p matrix (37) the model becomes:
yt = φt−1α+ut (38)
Using the notation just introduced, it is possible to rewrite the OLS estimate of matrix A
for the restricted model in a way that will allow us to derive some results in a more general way, which will be useful when dealing with both the RPD and the FPS test. The RPD estimate ARP D of A is the diagonal matrix whose diagonal is defined by ˜αRP D below:
˜
αRP D = T
X
t=1
φ′t−1φt−1
!−1 T X
t=1
φ′t−1yt
!
=
T
X
t=1
φ′t−1φt−1
!−1 T
X
t=1
φ′t−1phit
!
ιp
= Φ′−1Φ−1
−1
Φ′−1Φ
ιp (39)
As in the PD test, estimates of the contemporaneous and long run covariance matrices will be needed. For the long run covariance matrix, the Newey-West estimator will be once more employed. The estimates of these matrices are given by:
˜
Σ0RP D = T−1 T
X
t=1
˜
˜
ΣRP D = T−1 T
X
t=1
˜
utu˜′t+
T−1
l
X
τ=1
υτ l T
X
t=τ+1
(˜utu˜′t−τ + ˜ut−τu˜′t) (41)
where
˜
ut = yt−A˜RP Dyt−1
υτ l = 1−
τ l+ 1
The results presented in Theorem 2 may be easily adapted to deal with the asymptotics related to the RPD and FPS procedure. The following proposition consists in a generalization of a result presented in Flˆores, Preumont & Szafarz (1996) which will allow us to derive the asymptotic distributions to both tests.
Proposition 4 Let AR=Ip andyt, φt andµtbe given as in (31), (34) and (35) respectively
and Θ be a p×p matrix. Assume that ut satisfies the conditions listed in Assumption 1.
Under these conditions, we have as T ↑ ∞: 1. T−2PT
t=1φ′t−1Θφt−1
⇒Θ⊙Σ12 R1
0 W(t)W(t)′dtΣ
1 2
2. T−1PT
t=1φ′t−1Θµt
⇒Θ⊙Σ12 R1
0 W(t)dW(t)′Σ
1 2 + Σ
1
where Σ12 is as defined in Theorem 2 and ⊙ denotes the Hadamard product.
Proof. From (34) and (35), it follows that:
T−2
T
X
t=1
φ′t−1Θφt−1 = Θ⊙T−2
T
X
t=1
yt−1y′t−1
T−1
T
X
t=1
φ′t−1Θµt = Θ⊙T−1 T
X
t=1
yt−1u′t
Combining these and Theorem 2, the results follow from the continuous mapping theorem.
These results are helpful in establishing the consistency of ˜ARP D, ˜Σ0 and ˜ΣRP D which
Proposition 5 Suppose{yt}is given by (38) with α=ιp and{ut}satisfies the assumptions
in Theorem 1. Suppose further that:
1. supi,tE|uit|2β <∞ for some β >2;
2. limT→∞lT = +∞;
3. lT =o(T
1 4);
Then:
1. α˜RP D p
→ιp
2. Σ˜0
p
→Σ0
3. Σ˜RP D p
→Σ
Proof. Consistency of ˜αRP D under H0 follows from the fact that in such conditions:
T(˜αRP D−ιp) = T−2 T
X
t=1
φ′t−1φt−1
!−1
T−1
T
X
t=1
φ′t−1µt
!
ιp
T(˜αRP D−ιp) ⇒
Ip⊙Σ
1 2(
Z 1
0 W(r)W(r)
′dr)Σ1 2
−1
×
Ip⊙Σ
1 2(
Z 1
0 W(r)dW(r)
′)Σ1 2
ιp
LetZ be a random vector such that Z =Ip⊙Σ
1 2 R1
0 W(r)W(r)′drΣ
1 2
−1
×
Ip⊙Σ
1 2 R1
0 W(r)dW(r)′Σ
1 2
ιp. Since T(˜αRP D−ιp)⇒Z, T(˜αRP D−ιp) isOp(1) (See White
(1999), Lemma 4.5). It follows that (˜αRP D−ιp) = T1Op(1) and since T1 isop(1), we have that
(˜αRP D−ιp) is op(1) and therefore ˜αRP D converges in probability toιp.
The consistency of ˜ΣRP D follows from the consistency of ˜αRP D and an argument similar
to that of Proposition 1.
In order to test test H0, two test statistics based on the Wald test will be proposed. As
matrix Σ0, while the second one (FRP D) makes use of the long run covariance matrix Σ.
They are defined by:
F0RP D = (˜αRP D−ιp)′
Φ′ −1Φ−1
h
Φ′ −1
IT ⊗Σ˜0RP D
Φ−1
i−1
Φ′−1Φ−1
×
(˜αRP D−ιp) (42)
FRP D = (˜αRP D−ιp)′
Φ′−1Φ−1
h
Φ′−1IT ⊗Σ˜RP D
Φ−1
i−1
Φ′−1Φ−1
×
(˜αRP D−ιp) (43)
Finally, we turn our attention to the asymptotic distribution of the just defined test statistics under H0:
Proposition 6 Suppose {yt}is given by (38) with α =ιp and the conditions in Proposition
5 are observed. Then:
F0RP D ⇒ ι′p
Ip ⊙(Σ
1 2
Z 1
0 dW(r)W(r)
′Σ1 2 + Σ′
1)
×
Σ0⊙Σ
1 2
Z 1
0 W(r)W(r)
′Σ1 2
−1
×
Ip⊙(Σ
1 2
Z 1
0 W(r)dW(r)
′Σ1 2 + Σ
1)
ιp (44)
FRP D ⇒ ι′p
Ip ⊙(Σ
1 2
Z 1
0 dW(r)W(r)
′Σ1 2 + Σ′
1)
×
Σ⊙Σ12
Z 1
0 W(r)W(r)
′Σ1 2
−1
×
Ip⊙(Σ
1 2
Z 1
0 W(r)dW(r)
′Σ1 2 + Σ
1)
ιp (45)
Proof. Let ˜∆T be a sequence of p× p positive definite matrices such that ˜∆T p
→∆. Consider the test statistics given by:
F∆˜T = (˜αRP D−ιp)′×
Φ′−1Φ−1
h
Φ′−1IT ⊗∆˜T
Φ−1
i−1
Φ′−1Φ−1
F∆˜T =
"
ι′p
T
X
t=1
µ′tφt−1
!
Φ′−1Φ−1
−1
×
Φ′−1Φ−1 Φ′−1(IT ⊗∆˜T)Φ−1
−1
Φ′−1Φ−1
×
Φ′−1Φ−1
−1 XT
t=1
φ′t−1µt
!
ιp
#
=
ι′p Ip⊙
T
X
t=1
uty′t−1
!
˜ ∆T ⊙
T
X
t=1
yt−1yt′−1
!−1
Ip ⊙ T
X
t=1
yt−1u′t
!
ιp
Using the results in proposition 4 it follows that:
F∆˜T ⇒
ι′p
Ip⊙(Σ
1 2
Z 1
0 dW(r)W(r)
′Σ1 2 + Σ′
1)
×
∆⊙Σ12
Z 1
0 W
′(r)W(r)drΣ1 2
−1
×
Ip ⊙(Σ
1 2
Z 1
0 W(r)dW(r)
′Σ1 2 + Σ
1)
ιp
The results proposed follow from setting ˜∆T = ˜Σ0RP D and ˜∆T = ˜ΣRP D.
3.3
The FPS Unit Root Test
As mentioned above, the FPS test bears some resemblance to the RPD test in that both make use of the restriction that A is diagonal. However, contrary to the RPD test, the FPS test uses a different weighting matrix based on the long run covariance matrix when minimizing the sum of squared residuals. We now describe this procedure in detail.
Under the conditions imposed on ut, Σ is a positive definite matrix and, as a
conse-quence, it is possible to choose P positive definite (and symmetric), such that P P = Σ. Premultiplying (38) by P−1, we get:
P−1yt = P−1φt−1α+P−1ut
yt∗ = φ∗t−1α+u∗t (46) It is easy to notice that (46) is very similar to (38). The restrictions regardingut ensure
As an estimate ofα, we shall use:
˜
αF P S =
"T X
t=1
φ∗t−′1φ∗t−1
#−1" T X
t=1
φ∗t−′ 1φ∗t
#
ιp
=
"T X
t=1
φ′t−1Σ−1φt−1
#−1" T X
t=1
φ′t−1Σ−1φt
#
ιp (47)
One point worth noting is that the estimator ˜αF P S as presented in (47) is unfeasible.
However, a consistent estimate ˜P can be obtained from the spectral decomposition of any consistent estimate ˜Σ of the long run covariance matrix. We could for instance use the estimate given by ˜ΣRP D. Actually, the estimate ˜ΣP D could also be used, since it is also
consistent under H0.
As in the other tests, estimates of the contemporaneous and long run covariance matrices are needed. They will be given by:
˜
Σ0F P S = T−1 T
X
t=1
˜
utu˜′t (48)
˜
ΣF P S = T−1 T
X
t=1
˜
utu˜′t+
T−1
l
X
τ=1
υτ l T
X
t=τ+1
(˜utu˜′t−τ + ˜ut−τu˜′t) (49)
where
υτ l = 1−
τ l+ 1 ˜
ut = yt−A˜F P Syt−1
= yt−φt−1α˜F P S
We must now present a result establishing the consistency of the estimates just defined. This is our next step.
Proposition 7 Suppose{yt}is given by (38) with α=ιp and{ut}satisfies the assumptions
in Theorem 1. Suppose further that:
2. limT→∞l= +∞;
3. l =o(T14);
Then:
1. α˜F P S p
→ιp
2. Σ˜0F P S p
→Σ0
3. Σ˜F P S p
→Σ
Proof. Consistency of ˜αF P S under H0 follows from the fact that in such conditions:
T(˜αF P S −ιp) = T−2 T
X
t=1
φt−1Σ˜−F P S1 φ′t−1
!−1
T−1
T
X
t=1
φt−1Σ˜−F P S1 µ′t
!
ιp
T(˜αRP D−ιp) ⇒
Σ−1⊙Σ12
Z 1
0 W(r)W(r)
′drΣ1 2
−1
×
Σ−1⊙(Σ12
Z 1
0 W(r)dW(r)
′Σ1 2 + Σ
1)
ιp
By an argument similar to that presented in Proposition 5, it is possible to show that this leads us to the consistency of ˜αF P S.
The consistency of ˜ΣF P S follows from the consistency of ˜αRP D and an argument similar
to that of Proposition 1.
As in the previous cases, we shall use two different test statistics to test H0. They are
similar to the ones presented above in that one of them is derived from a Wald test using the contemporaneous covariance matrix as weighting matrix, while the other one makes use of the long run covariance matrix. The two test statistics are presented below:
F0F P S = (˜αF P S −ιp)′
Φ′−1Σ˜−1Φ−1
×
h
Φ′−1IT ⊗P˜−1
′˜
Σ0F P SP˜−1
Φ−1
i−1
×
Φ′−1Σ˜−1Φ−1
(˜αF P S −ιp) (50)
FF P S = (˜αF P S −ιp)′×
Φ′−1Σ˜−1Φ−1
h
Φ′−1Φ−1
i−1
Φ′−1Σ˜−1Φ−1
×
where ˜Σ is any consistent estimate of Σ and ˜P is a consistent estimate of Σ12.
Our next result concerns the asymptotic distributions under H0 of the test statistics
defined above.
Proposition 8 Suppose {yt} is given by (38) with α=ιp and the conditions in proposition
7 are observed. Let F0F P S and FF P S be as defined as in (50) and (51) respectively. Then:
F0F P S ⇒ ι′p
Σ−1⊙(Σ12
Z 1
0 dW(r)W(r)
′Σ1 2 + Σ′
1)
×
Σ−12Σ
0Σ−
1 2 ⊙(Σ
1 2
Z 1
0 W(r)W(r)
′drΣ1 2)
−1
×
Σ−1⊙(Σ12
Z 1
0 W(r)dW(r)
′Σ1 2 + Σ
1)
ιp (52)
FF P S ⇒ ι′p
Σ−1⊙(Σ12
Z 1
0 dW(r)W(r)
′Σ1 2 + Σ′
1)
×
Ip⊙(Σ
1 2
Z 1
0 W(r)W(r)
′drΣ1 2)
−1
×
Σ−1⊙(Σ12
Z 1
0 W(r)dW(r)
′Σ1 2 + Σ
1)
ιp (53)
Proof. Let ˜∆T be a sequence of p×ppositive definite matrices such that ˜∆T p
→∆. Consider the test statistics given by:
F∆˜T = (˜αF P S −ιp)′
Φ′−1Σ˜−1Φ−1
h
Φ′−1P˜−1′IT ⊗∆˜T
˜
P−1Φ−1
i−1
Φ′−1Σ˜−1Φ−1
× (˜αF P S −ιp)
F∆˜T =
"
ι′p
T
X
t=1
µtΣ˜−1φ′t−1
!
Φ′−1Σ˜−1Φ−1
−1
×
Φ′−1Σ˜−1Φ−1 Φ′−1P˜−1
′
(IT ⊗∆˜T) ˜P−1Φ−1
−1
Φ′−1Σ˜−1Φ−1
×
Φ′−1Σ˜−1Φ−1
−1 XT
t=1
φt−1Σ˜−1µ′t
!
ιp
#
=
ι′p
T
X
t=1
µtΣ˜−1φ′t−1
! T X
t=1
φ′t−1P˜−1′∆˜TP˜−1φ′t−1
!−1 T X
t=1
φt−1Σ˜−1µ′t
!
ιp
F∆˜T ⇒
ι′p
Σ−1⊙(Σ12
Z 1
0 dW(r)W(r)
′Σ1 2 + Σ′
1)
×
Σ−12∆Σ− 1 2 ⊙Σ
1 2
Z 1
0 W(r)W
′(r)drΣ1 2
−1
×
Σ−1⊙(Σ12
Z 1
0 W(r)dW(r)
′Σ1 2 + Σ
1)
ιp
The results follows from setting ˜∆T = ˜Σ0RP D and ˜∆T = ˜ΣRP D.
4
Distribution under
H
1In the last section, the four test statistics that will be studied here were described and their asymptotic distribution underH0 were calculated. We now turn our attention to determining
their asymptotic distribution under a sequence of alternatives that converge asymptotically to the null. We will be following the theory of nearly integrated processes described for the univariate case in Phillips (1987b).
The alternative processes will be given by:
zt = ACzt−1+et
AC = exp
1
TC
, a p×p matrix
z0 = 0 (54)
We will use the results presented in Phillips (1988) in order to derive the distribution of our test statistics as T ↑ ∞ when the true process is given by (54). It is important to note that here we have a sequence of alternative hypotheses which approaches H0 asT ↑ ∞.
The following result based on Phillips (1988) is presented here since much of the discussion in this section will be based on it:
Theorem 3 Let {zt}∞t=1 be given by (54) and {et}∞t=1 be a weakly stationary sequence of
p-dimensional vectors. Assume that:
1. E(et) = 0,∀t
3. {et}∞t=1 is strong mixing with mixing numbers αm that satisfy:
∞ X
j=1
α1−
2
β
m <∞
Define:
• Σ is the long run covariance matrix for process et
• J(r;C, P) is the Ornstein-Uhlenbeck process with matrix of coefficients C and covari-ance matrix P P′, that is:
J(r;C, P) =
Z r
0 exp(C(r−s))P dWs, a p dimensional vector
Under these conditions, the following results hold:
1. T−1 2z
[T r]⇒J(r;C,Σ
1 2)
2. T−32 PT
t=1zt ⇒
R1
0 J(r;C,Σ
1 2)dr
3. T−2PT
t=1zt−1zt′−1 ⇒
R1
0 J(r;C,Σ
1
2)J(r;C,Σ 1 2)′dr
4. T−1PT
t=1zt−1e′t⇒
R1
0 J(r;C,Σ
1
2)dW(r)′Σ 1 2 + Σ
1
5. T−1PT
t=1zt−1e′t+etzt′−1 ⇒J(1;C,Σ
1
2)J(1;C,Σ 1 2)−C
R1
0 J(r;C,Σ
1
2)J(r;C,Σ 1 2)′dr
−
R1
0 J(r;C,Σ
1
2)J(r;C,Σ 1 2)′dr
C′−Σ0 Proof. See Phillips(1988).
Having enunciated this result, it is straightforward to notice that if the processetsatisfies
the same assumptions regarding the process ut described in Assumption 1 then Theorem 3
applies to et.
In order to derive the asymptotic distribution of the test statistics described in the previous section, it is useful to establish a result which is an analogue to Proposition 4 under
H1. In order to do that, we define:
λt = diag(zt) (55)
υt = diag(et) (56)
Λ−1 =
h
λ0 λ1 · · · λT−1
i′
(57)
Λ = h λ1 λ2 · · · λT
i′
Having defined these variables, we proceed to state a result which will be helpful in determining the asymptotic distribution of the RP D and F P S tests under the sequence of alternatives described in (54).
Proposition 9 Suppose that et satisfies Assumption 1 and letλt and υt be given as in (55)
and (56) respectively and Θ be a p×p matrix. Under these conditions, we have as T ↑ ∞: 1. T−32
PT
t=1λt−1
⇒Ip⊙
(R1
0 J(r;C,Σ
1
2)dr)⊗ι′
p
2. T−2PT
t=1λ′t−1Θλt−1
⇒Θ⊙R1
0 J(r;C,Σ
1
2)J(r;C,Σ 1 2)′dr
3. T−1PT
t=1λ′t−1Θυt
⇒Θ⊙R1
0 J(r;C,Σ
1
2)dW(r)′Σ 1 2 + Σ
1
where Σ12 is as defined in Theorem 2.
Proof. From (55) and (56), it follows that:
T−32
T
X
t=1
λt = Ip⊙(T−
3 2
T
X
t=1
zt⊗ιp)
T−2
T
X
t=1
λt−1Θλ′t−1 = Θ⊙T−2
T
X
t=1
zt−1z′t−1
T−1
T
X
t=1
λt−1Θυt′ = Θ⊙T
−1XT
t=1
zt−1u′t
Combining these and Theorem 3, the results follow from the continuous mapping theorem.
Having the last two results, the steps involved in determining the asymptotic distribution of the test statistics described in section 3 under the sequence of alternatives considered by Phillips (1987b) are very similar to those employed when determining their asymptotic behavior under H0. In the following subsections, results concerning the distributions under
any alternative of the form described in (54) are obtained. One can then determine, for each test, what is the probability that the test statistic under a certain alternative hypothesis will be in the region of acceptance of H0, that is, the asymptotic power of the test against that
4.1
The PD Test under
H
1The following proposition provides the distributions of the test statistics of the PD procedure under the set of alternative hypotheses considered.
Proposition 10 Let {zt}∞t=1 be given by (54) and consider the test statistics F0P D, FP D,
F0SP D and FSP D defined in (23), (24), (27) and (28) respectively. Suppose further that ut
satisfies assumption 1. Define:
G = C
Z 1
0 J(r;C,Σ
1
2)J(r;C,Σ 1 2)′dr
+
Z 1
0 J(r;C,Σ
1
2)J(r;C,Σ 1 2)′dr
C′
Then:
F0P D ⇒
1 4tr
h
J(1;C,Σ12)J(1;C,Σ 1
2)′−G−Σ
0
Σ−01 ×
J(1;C,Σ12)J(1;C,Σ 1
2)′−G−Σ
0
×
Z 1
0 J(r;C,Σ
1
2)J(r;C,Σ 1 2)′dr
−1#
(59)
FP D ⇒
1 4tr
h
J(1;C,Σ12)J(1;C,Σ 1
2)′−G−Σ
0
Σ−1 ×
J(1;C,Σ12)J(1;C,Σ 1
2)′−G−Σ
0
×
Z 1
0 J(r;C,Σ
1
2)J(r;C,Σ 1 2)′dr
−1#
(60)
F0SP D ⇒
1 4tr
nh
J(1;C,Σ12)J(1;C,Σ 1
2)′−G−Σ
Σ−1×
J(1;C,Σ12)J(1;C,Σ 1
2)′−G−Σ
+ J(1;C,Σ12)J(1;C,Σ 1
2)′+G Σ−1−Σ−1
0
×
J(1;C,Σ12)J(1;C,Σ 1 2)′+G
−J(1;C,Σ12)J(1;C,Σ 1
2)′(Σ−1−Σ−1
0 )J(1;C,Σ
1
2)J(1;C,Σ 1 2)′
i
× (
Z 1
0 J(r;C,Σ
1
2)J(r;C,Σ 1
2)′dr)−1
(61)
FSP D ⇒
1 4tr
nh
J(1;C,Σ12)J(1;C,Σ 1
2)′−G−Σ
Σ−1×
J(1;C,Σ12)J(1;C,Σ 1
2)′−G−Σ
+GΣ−1(Σ0−Σ) + (Σ0−Σ) Σ−1G
i
×
Z 1
0 J(r;C,Σ
−1
2)J(r;C,Σ− 1 2)′dr
−1)
Proof. The first convergence is obtained by following the steps:
F0P D = tr
h
˜
AP D−I
Σ−01A˜P D −I
Y−1Y′−1i = 1
4tr
"
T−1
T
X
t=1
yt−1u′t+utyt′−1
!
Σ−01 T−1
T
X
t=1
yt−1u′t+utyt′−1
!
×
T−2
T
X
t=1
yt−1yt′−1
!−1
⇒ 14trhJ(1;C,Σ12)J(1;C,Σ 1
2)′−G−Σ
0
Σ−01 ×
J(1;C,Σ12)J(1;C,Σ 1
2)′−G−Σ
0
×
Z 1
0 J(s;C,Σ
1
2)J(s;C,Σ 1 2)′ds
−1#
We now prove the second convergence:
FP D = tr
h
˜
AP D−I
Σ−1A˜P D−I
Y−1Y′−1i = 1
4tr
"
T−1
T
X
t=1
yt−1u′t+utyt′−1
!
Σ−1 T−1
T
X
t=1
yt−1u′t+utyt′−1
!
×
T−2
T
X
t=1
yt−1y′t−1
!−1
⇒ 14trhJ(1;C,Σ12)J(1;C,Σ 1
2)−G−Σ
0
Σ−1 ×
J(1;C,Σ12)J(1;C,Σ 1
2)−G−Σ
0
×
Z 1
0 J(s;C,Σ
1
2)J(s;C,Σ 1 2)′ds
−1#
For the third convergence, define:
C1 = tr
h
T−2yTyT′ ( ˜Σ−0P D1 −Σ˜P D−1)yTyT′ (T−2Y−1Y−1)−1
i
C2 = tr
( ˜Σ0P D−Σ˜P D)
T−2Y−1Y−1
−1
Thus
C1 ⇒ tr
h
J(1;C,Σ12)J(1;C,Σ 1 2)′(Σ−1
0 −Σ−1)J(1;C,Σ
1
2)J(1;C,Σ 1 2)′
×
Z 1
0 J(r;C,Σ
1
2)J(r;C,Σ 1 2)′dr
−1#
C2 ⇒ tr
"
(Σ0 −Σ)
Z 1
0 JC(r)JC(r)
In the following, we denoteJ(1;C,Σ12) by J(1) and
R1
0 JC(r)JC(r)′dr
byI(J) for sim-plicity. We then get that F0P D− 14[C1+C2] converges to
1 4tr
h
J(1)J(1)′Σ−1
0 −GΣ−01−I
(J(1)J(1)′−G−Σ
0)
−J(1)J(1)′Σ0−1J(1)J(1)′−J(1)J(1)′Σ−1J(1)J(1)′−(Σ0−Σ)
i
(I(J))−1 = 1
4tr
h
J(1)J(1)′Σ−01J(1)J(1)′−J(1)J(1)′Σ−01G−J(1)J(1)′ −GΣ−01J(1)J(1)′+GΣ−01G+G
−J(1)J(1)′+G+ Σ0)−
J(1)J(1)′Σ−01J(1)J(1)′ −J(1)J(1)′Σ−1J(1)J(1)′ −(Σ0−Σ)] (I(J))−1
= 1 4tr
nh
J(1)J(1)′Σ−1J(1)J(1)′−J(1)J(1)′Σ−1G−J(1)J(1)′ −GΣ−1J(1)J(1)′ +GΣ−1G+G−J(1)J(1)′+G+ Σi
+hJ(1)J(1)′(Σ−1−Σ0−1)G+G(Σ−1 −Σ0−1)J(1)J(1)′+G(Σ−1−Σ−01)Gio(I(J))−1 = 1
4tr
nh
(J(1)J(1)′−G−Σ) Σ−1(J(1)J(1)′−G−Σ)i
+hJ(1)J(1)′(Σ−1 −Σ−01)J(1)J(1)′+J(1)J(1)′(Σ−1−Σ−01)G
+G(Σ−1−Σ−01)J(1)J(1)′+G(Σ−1−Σ−01)Gi
−J(1)J(1)′(Σ−1 −Σ−01)J(1)J(1)′o(I(J))−1 = 1
4tr
nh
(J(1)J(1)′−G−Σ) Σ−1(J(1)J(1)′−G−Σ)i +h(J(1)J(1)′+G)Σ−1−Σ−01(J(1)J(1)′+G)i −J(1)J(1)′(Σ−1 −Σ−01)J(1)J(1)′o(I(J))−1 Thus, we can write
F0SP D ⇒
1 4tr
nh
J(1;C,Σ12)J(1;C,Σ 1
2)′−G−Σ
Σ−1×
J(1;C,Σ12)J(1;C,Σ 1
2)′−G−Σ
+J(1;C,Σ12)J(1;C,Σ 1
2)′+G Σ−1−Σ−1
0 J(1;C,Σ
1
2)J(1;C,Σ 1 2)′+G
−J(1;C,Σ12)J(1;C,Σ 1
2)′(Σ−1 −Σ−1
0 )J(1;C,Σ
1
2)J(1;C,Σ 1 2)′
i
(
Z 1
0 J(r;C,Σ
1
2)J(r;C,Σ 1
2)′dr)−1
Finally, we prove the fourth convergence. For that, define:
C1 = tr
T−1yTyT′ Σ˜−P D1( ˜ΣP D−Σ˜0P D) T−2Y−1Y−′ 1
= tr
( ˜ΣP D −Σ˜0P D) ˜Σ−P D1 T−1yTyT′ T−2Y−1Y−′ 1
−1
C2 = tr
( ˜ΣP D −Σ˜0P D) ˜Σ−P D1 ( ˜ΣP D−Σ˜0P D) T−2Y−1Y−′1
−1
C3 = tr
˜
ΣP D −Σ˜0P D T−2Y−1Y−′1
−1
It follows that:
C1 ⇒ tr
h
J(1)J(1)′Σ−1(Σ−Σ0)
(I(J))−1i
C2 ⇒ tr
h
(Σ−Σ0)Σ−1(Σ−Σ0)
(I(J))−1i
C3 ⇒ tr
h
(Σ−Σ0) (I(J))−1
i
We then note that, by definingK1 =J(1)J(1)Σ−1(Σ−Σ0) and K2 =I(J) and, sinceK2
is symmetric, tr(K1K2) =tr(K2′K1′) = tr(K2K1′) = tr(K1′K2). Hence:
C1 ⇒ tr
h
(Σ−Σ0) Σ−1J(1)J(1)′
(I(J))−1i
We then note that FSP D =FP D− 24C1− 14C2+ 24C3 converges to:
1 4tr
h
J(1)J(1)′Σ−1−GΣ−1−Σ0Σ−1
(J(1)J(1)′−G−Σ0)
−J(1)J(1)′Σ−1(Σ−Σ0)−(Σ−Σ0)Σ−1J(1)J(1)′
−(Σ−Σ0)Σ−1(Σ−Σ0)
+ 2 ((Σ−Σ0))
i
(I(J))−1 = 1
4tr
h
J(1)J(1)′Σ−1J(1)J(1)′ −J(1)J(1)′Σ−1G−J(1)J(1)′Σ−1Σ0
−GΣ−1J(1)J(1)′+GΣ−1G+GΣ−1Σ0
−Σ0Σ−1J(1)J(1)′+ Σ0Σ−1G+ Σ0Σ−1Σ0
−J(1)J(1)′Σ−1Σ +J(1)J(1)′Σ−1Σ0−ΣΣ−1J(1)J(1)′+ Σ0Σ−1J(1)J(1)′
−ΣΣ−1Σ + ΣΣ−1Σ + ΣΣ−1Σ−Σ0Σ−1Σ0)
i
(I(J))−1 = 1
4tr
h
J(1)J(1)′Σ−1J(1)J(1)′ −J(1)J(1)′Σ−1G−J(1)J(1)′Σ−1Σ
−GΣ−1J(1)J(1)′+GΣ−1G+GΣ−1Σ −ΣΣ−1J(1)J(1)′+ ΣΣ−1G+ ΣΣ−1Σ
+GΣ−1(Σ0−Σ) + (Σ0−Σ)Σ−1G
i
= 1 4tr
h
(J(1)J(1)′−G−Σ) Σ−1(J(1)J(1)′−G−Σ) +GΣ−1(Σ0−Σ)
+(Σ0−Σ)Σ−1G
i
(I(J))−1 Therefore, we can write:
FSP D ⇒
1 4tr
nh
J(1;C,Σ−12)J(1;C,Σ− 1
2)′ −G−Σ
Σ−1×
J(1;C,Σ−12)J(1;C,Σ− 1
2)′−G−Σ
+GΣ−1(Σ0−Σ) + (Σ0−Σ) Σ−1G
i
×
Z 1
0 J(r;C,Σ
−1
2)J(r;C,Σ− 1 2)′dr
−1)
One important fact given these asymptotic distributions is that, under the set of alterna-tive hypotheses considered, theF0SP DandFSP Dare not free of nuisance parameters contrary
to what happens under the null. This suggests that the power of the test may be affected by the nuisance parameters, despite the corrections made in order to free its distributions under H0 from them.
4.2
The RPD Test under
H
1The following proposition provides the distributions of the test statistics of the RPD proce-dure under the set of alternative hypotheses considered.
Proposition 11 Let {zt}∞t=1 be given by (54) and consider the test statistics F0RP D and
FRP D defined in (42) and (43) respectively. Suppose further that ut satisfies assumption 1.
Then:
F0RP D ⇒
ι′p
Ip⊙(Σ
1 2
Z 1
0 dW(r)J(r;C,Σ
1
2)′+ Σ′
1)
×
Σ0⊙
Z 1
0 J(r;C,Σ
1
2)J(r;C,Σ 1 2)′
−1
×
Ip⊙(
Z 1
0 J(r;C,Σ
1
2)dW(r)′Σ 1 2 + Σ
1)
ιp
FRP D ⇒
ι′p
Ip⊙(Σ
1 2
Z 1
0 dW(r)J(r;C,Σ
1
2)′+ Σ′
1) × Σ⊙ Z 1
0 J(r;C,Σ
1
2)J(r;C,Σ 1 2)′
−1
×
Ip ⊙(
Z 1
0 J(r;C,Σ
1
2)dW(r)′Σ 1 2 + Σ
1)
ιp
(64)
Proof. Let ˜∆T be a sequence of p×ppositive definite matrices such that ˜∆T p
→∆. Consider the test statistics given by:
F∆˜T = (˜αRP D−ιp)′
Φ′−1Φ−1
h
Φ′−1IT ⊗∆˜T
Φ−1
i−1
Φ′−1Φ−1
(˜αRP D−ιp)
=
"
ι′p
T
X
t=1
µtφ′t−1
!
Φ′ −1Φ−1
−1
×
Φ′−1Φ−1 Φ′−1(IT ⊗∆˜T)Φ−1
−1
Φ′−1Φ−1
×
Φ′−1Φ−1
−1 XT
t=1
φt−1µ′t
!
ιp
#
=
ι′p Ip ⊙ T
X
t=1
utyt′−1
!
˜ ∆T ⊙
T
X
t=1
yt−1yt′−1
!−1
Ip⊙ T
X
t=1
yt−1u′t
!
ιp
Using Proposition 9 it follows that:
F∆˜T ⇒
ι′p
Ip ⊙(Σ
1 2
Z 1
0 dW(r)J(r;C,Σ
1
2)′+ Σ′
1) × ∆⊙ Z 1
0 J(r;C,Σ
1
2)J(r;C,Σ 1 2)′dr
−1
×
Ip⊙(
Z 1
0 J(r;C,Σ
1
2)dW(r)′Σ 1 2 + Σ
1)
ιp
The results follow from setting ˜∆T = ˜Σ0RP D and ˜∆T = ˜ΣRP D.
4.3
The FPS Test under
H
1Proposition 12 Let {zt}∞t=1 be given by (54) and consider the test statistics F0F P S and
FF P S defined in (50) and (51) respectively. Suppose further that ut satisfies assumption 1.
Then:
F0F P S ⇒
ι′p
Σ−1⊙(Σ12
Z 1
0 dW(r)J(r;C,Σ
1
2)′ + Σ′
1)
Σ−12Σ
0Σ−
1 2 ⊙
Z 1
0 J(r;C,Σ
1
2)J(r;C,Σ 1 2)′
−1
×
Σ−1⊙(
Z 1
0 J(r;C,Σ
1
2)dW(r)′Σ 1 2 + Σ
1)
ιp
(65)
FF P S ⇒
ι′p
Σ−1⊙(Σ12
Z 1
0 dW(r)J(r;C,Σ
1
2)′ + Σ′
1)
×
Ip⊙
Z 1
0 J(r;C,Σ
1
2)J(r;C,Σ 1 2)′
−1
×
Σ−1⊙(Z 1
0 J(r;C,Σ
1
2)dW(r)′Σ 1 2 + Σ
1)
ιp
(66)
Proof. Let ˜∆T be a sequence of p×ppositive definite matrices such that ˜∆T p
→∆ and ˜P be a positive definite consistent estimator of Σ12. Consider the test statistics given by:
F∆˜T = (˜αF P S −ιp)′
Φ′−1Φ−1
h
Φ′−1P˜−1′IT ⊗∆˜T
˜
P−1Φ−1
i−1
Φ′−1Φ−1
(˜αF P S −ιp)
=
"
ι′p
T
X
t=1
µ′tΣ˜−1φ
t−1
!
Φ′
−1Σ˜−1Φ−1
−1
×
Φ′−1Σ˜−1Φ−1 Φ′−1P˜−1
′
(IT ⊗∆˜T) ˜P−1Φ−1
−1
Φ′−1Σ˜−1Φ−1
×
Φ′−1Σ˜−1Φ−1
−1 XT
t=1
φ′t−1Σ˜−1µt
!
ιp
#
=
ι′p
T
X
t=1
µ′tΣ˜−1φt−1
! T X
t=1
φ′t−1P˜−1′∆˜TP˜−1φ′t−1
!−1 T
X
t=1
φ′t−1Σ˜−1µt
!
ιp
Using Proposition 9 it follows that:
F∆˜T ⇒
ι′p
Σ−1⊙(Σ12
Z 1
0 dW(r)J(r;C,Σ
1
2)′+ Σ′
1)
×
Σ−12∆Σ− 1 2 ⊙
Z 1
0 J(r;C,Σ
1
2)J(r;C,Σ 1 2)′dr
−1
×
Σ−1⊙(
Z 1
0 J(r;C,Σ
1
2)dW(r)′Σ 1 2 + Σ
1)
ιp
5
Conclusion
In this essay, the distributions under a certain set of alternative hypotheses of the multivariate unit roots tests proposed in Phillips & Durlauf (1986) and Flˆores, Preumont & Szafarz (1996) were determined. In addition to that, a new test is proposed in which the restriction employed in the Flˆores, Preumont & Szafarz test is combined with the OLS estimation used in the Phillips & Durlauf test and its distributions both under the null of unit roots and the same set of alternative hypotheses are determined.
The family of alternative hypotheses considered may be regarded as the multivariate ver-sion of the set of alternatives considered in Phillips (1987b). One interesting result emerging from the analysis made is the fact that the corrections made in order to free the asymptotic distribution of the Phillips & Durlauf test under the null of nuisance parameters are not able to do the same under the set of alternative hypotheses considered. This strongly sug-gests that the power of the test against alternatives in this set is affected by some nuisance parameters, including the long run covariance matrix.
Knowing the asymptotic distribution under the null hypothesis and a set of alternatives allows us to determine the asymptotic power of the tests in order to compare them. With the new test proposed, one may be able not only to compare the power of each test but to identify what causes the difference in performance.
References
[1] Abuaf, N., and Jorion, P. (1990), “Purchasing Power Parity in the Long Run”, The Journal of Finance, 45, 157-174.
[2] Bucchianico, A.D., Einmahl, J.H.J., and Mushkudiani, N.A. (2001), “Smallest Non-parametric Tolerance Regions”, The Annals of Statistics, 29, 1320-1343.
[3] Davidson, R., and Mackinnon, J.G. (1993), Estimation and Inference in Econometrics, Oxford University Press.
[4] Dickey, D.A., and Fuller, W.A. (1979), “Distribution of the Estimators of Autoregressive Time Series with a Unit Root”, Journal of the American Statistical Association, 74, 427-431.
[5] Durlauf, S.N., and Phillips, P.C.B. (1988), “Trends versus Random Walks in Time Series Analysis”, Econometrica,56, 1333-1354.
[6] Flˆores, R.G., Jorion, P., Preumont, P.Y., and Szafarz, A. (1999), “Multivariate Unit Root Tests of the PPP Hypothesis”, Journal of Empirical Finance, 6, 335-353.
[7] Flˆores, R.G., Preumont, P.Y., and Szafarz, A. (1996), “Multivariate Unit Root Tests”, CEME-ULB, Discussion Paper 9601.
[8] Granger, C.W. (1981), “Some Properties of Time Series Data and Their Use in Econo-metric Model Specification”, Journal of Econometrics,16, 121-130.
[9] Hamilton, J.D. (1994) Time Series Analysis, Princeton University Press.
[10] Hansen, B.E. (1995) “Rethinking the Univariate Approach to Unit Root Testing: Using Covariates to Increase Power”, Econometric Theory,11, 1148-1171.
[11] Magnus, J.R., and Neudecker, H. (1988) Matrix Differential Calculus with Applications in Statistics and Econometrics, John Wiley & Sons.
