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orKing raper beries
Testing for Cointegration When Some of
the Cointegrating Vectors are Known
Michael T. K. Horvath and Mark W. Watson
Working Papers Series
Macroeconomic Issues
Research Department
Federal Reserve Bank of Chicago
December 1993 (WP-93-15)
FEDERAL RESERVE BANK
OF CHICAGO
Testing for Cointegration When Some of the Cointegrating Vectors are Known
by
Michael T.K. Horvath
Department of Economics, Northwestern University
and
Mark W. Watson
Department of Economics, Northwestern University
and Federal Reserve Bank of Chicago
First Draft: October 5, 1993
This Draft: December 6 , 1993
This paper has benefited from comments by Neil Ericsson, Gordon Kemp, Andrew Levin, Soren
Johansen, John McDermott, Pierre Perron and James Stock, and was supported by grants from
the National Science Foundation (SES-91-22463) and the Sloan Foundation.
1. Introduction
Economic models often imply that variables are cointegrated with simple known
cointegrating vectors. Examples include the neoclassical growth model, which implies that
income, consumption, investment and. the capital stock will grow in a balanced way, so that any
stochastic growth in one of the series must be matched by corresponding growth in the others.
Asset pricing models with stable risk premia imply corresponding stable differences in spot and
forward prices, long- and short-term interest rates, and the logarithms of stock prices and
dividends. Most theories of international trade imply long run purchasing power parity, so that
long-run movements in nominal exchange rates are matched by countrys’ relative price levels.
Certain monetarist propositions are centered around the stability of velocity, implying
cointegration among the logarithms of money, prices and income. Each of these theories have
two distinct implications for the properties of economic time series under study: first, the series
are cointegrated, and second, the cointegrating vector takes on a specific value. For example,
balanced growth implies that the logarithms of income and consumption are cointegrated, and
that the cointegrating vector takes on the value of [1 - 1].
The most widely used approach to testing these cointegration propositions is articulated and
implemented in Johansen and Juselius (1992), who investigate the empirical support for longrun purchasing power parity. They implement a two-stage testing procedure. In the first
stage, the null hypothesis of no cointegration is tested against the alternative that the data are
cointegrated with an unknown cointegrating vector using Johansen’s (1988) test for
cointegration. If the null hypothesis is rejected, a second stage test is implemented with
cointegration maintained under both the null and alternative. The null hypothesis is that the
data are cointegrated with the specific cointegrating vector implied by the relevant economic
theory ([1 - 1] in the consumption-income example), and the alternative is that data are
cointegrated with another unspecified cointegrating vector. Since a consistent test for
cointegration is used in the first stage, potential cointegration in the data is found with
probability approaching 1 in large samples. Thus, the probability of rejecting the cointegration
constraints on the data imposed by the economic model are given by the size of the test in the
second step, at least in large samples. An important strength of this procedure is that it can
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uncover cointegration in the data with a cointegrating vector different from the cointegrating
vector imposed by the theory. The disadvantage is that the sample sizes used in economics are
often relatively small, so that the first stage tests have low power.
This paper discusses an alternative procedure in which the null of no cointegration is tested,
against the composite alternative of cointegration with a prespecified cointegrating vector. As
we will demonstrate, this approach has two advantages. First, and most important, the resulting
test for cointegration is significantly more powerful than the test that does not impose the
cointegrating vector. For example, in the bivariate example analyzed in Section 3 these power
gains correspond to sample size increases ranging from 40%-70% for a test with power equal to
50%. The second advantage is that the test statistic is very easy to calculate: it is the standard
Wald test for the presence of the candidate error correction terms in the first difference vector
autoregression. The countervailing disadvantage of the testing approach is that it does not
separate the two components of the alternative hypothesis, and so may fail to reject the null of
no cointegration when the data are cointegrated with a cointegrating vector different from that
used to construct the test. We investigate this in Section 3, where it is shown that in situations
with weak cointegration (represented by a local-to-unity error correction term), even inexact
information on the value of the cointegrating vector often leads to power improvements over
the test that uses no information.
The plan of this paper is as follows. In Section 2, we consider the general problem of
testing for cointegration in a model in which some of the potential cointegrating vectors are
known, and some are unknown, under both the null and the alternative. In particular we
present Wald and Likelihood Ratio tests for the hypothesis that the data are cointegrated with
r^ known and r^ unknown cointegrating vectors under the null. Under the alternative there
are r^ and r^ additional known and unknown cointegrating vectors respectively. The tests are
constructed in the context of a finite order Gaussian vector error correction model (VECM),
and generalize the procedures of Johansen (1988) who considered the hypothesis testing problem
with rOk= rak=0. In Section 2 we also derive the asymptotic null distributions of the test
statistics and tabulate critical values. Section 3 focuses on the power properties of the test.
First, we present comparisons of the power of likelihood based tests that do and do not use
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information about the value of the cointegrating vector. Next, since information about the
potential cointegrating vector might be inexact, we investigate the power loss associated with
using an incorrect value of the cointegrating vector. Finally, when there are no cointegrating
vectors under the null and only one cointegrating vector under the alternative, simple univariate
unit root tests provide an alternative to the multivariate VECM-based tests. Section 3 compares
the power of these univariate unit root tests to the multivariate VECM-based tests. Section 4
contains an empirical application which investigates the forward premia in foreign exchange
markets by examining the cointegration properties of forward and spot prices. Section 5
contains some concluding remarks.
2.
Testing for Cointegration in the Gaussian VAR Model
As in Johansen (1988) we derive tests for cointegration in the context of the reduced rank
Gaussian VAR:
(2 . 1a)
Y t
=
dt
+
X {
(2 .ib) xt = rP=1nixt.i + e,
where Yt is an n x 1 data vector from a sample of size T, dj represents deterministic drift in Yt,
Xt is an n x 1 random vector generated by (2. lb), et is NUD(0,E6), and for convenience, the
initial conditions X.j, i=0,...,p are assumed to equal zero. To focus attention on the long-run
behavior of the process, it is useful to rewrite (2 . lb) as:
(2.1c) AXt =
U X tA
+ E f l ^ A X ^ + et,
where Et=-In+
Our interest is focused on r=rank(II), and we consider tests of the hypotheses:
H0: rank(II)=r=r0
H&: rank(II)=r=r0 +ra, with ra>0.
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The alternative is written so that ra represents the number of additional cointegrating vectors
that are present under the alternative. We assume that r0 =rok+rOu’ where r0k *s
number of
cointegrating vectors that are known under the null and r^ represents the number of
cointegrating vectors that are unknown (or alternatively, unrestricted) under the null. Similarly
ra=rak+ rau»where the subscripts "k" and "u" denote known and unknown, respectively. The r^
prespecified vectors are thought to be cointegrating vectors under the alternative; under the null
they do not cointegrate the series. In spite of this, for expositional ease, they will be referred
to as cointegrating vectors.
As in Engle and Granger (1987), Johansen (1988), and Ahn and Reinsel (1990), it is
convenient to write the model in vector error correction form by factoring the matrix II as
n=5a’, where 5 and a are nxr matrices of full column rank, and the columns of a denote the
cointegrating vectors. The columns of a are partitioned as a=(aQ<*a), where aQis an n x rQ
matrix whose columns are the cointegrating vectors present under the null, aa is an nxra matrix
whose columns are the additional cointegrating vectors present under the alternative. The
matrix 5 is partitioned conformably as $ = ($ 0 5a), where d Q is nXrQand 5a is nxr&. It is also
useful to partition aa to isolate the known and unknown cointegrating vectors. Thus,
aa= (aaka m
)’
w^ere the rak columns of
the alternative, and the r^ columns of
are the additional cointegrating vectors known under
are the additional cointegrating vectors that are
present but unrestricted under the alternative. The matrix 5a is partitioned conformably as
5a=(5ak
Using this notation, IIXt_j = 50 (o!^Xt. j) + 5a(aaXt_^), and the competing
hypotheses are: HQ:5a=0 vs. Ha:5a£0, with rank(5aaa)=ra.
We develop tests for HQvs. Ha in three steps. First, we abstract from deterministic
components and derive the likelihood ratio statistic and some useful asymptotically equivalent
statistics under the maintained assumption that dt=0. Second, we discuss how these statistics
can be modified for nonzero values of dt. Finally, the asymptotic null distributions of the
resulting statistics are derived and critical values based on these asymptotic distributions are
tabulated.
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C a l c u l a t in g
th e L R
an d
W a ld
T e s t S t a t is t ic s w h e n d ^ = 0 :
The likelihood ratio statistic for testing HQ:r=r0 k+r0u vs. Ha:r=r0 k+rak+r0 u+r3u will depend
on r-.Ok’, r_ , r_
Ou , r„
a^, and the values of ou
Ok and
LRj.
x
(.<Xq ^
at.
We write the statistic as
«aic)- The values of r^ and ra>_appear implicitly as the ranks of aQk and aafc
respectively. When ^ = 0 , the statistic is written as LR^ r^(0, a^), and as LRj. r (a^.O) when
w =0-
To derive the LR statistic, we limit attention to the problem with ro=rok=rOu:=0- F°r the
purposes of deriving the computational formula for the LR statistic, this is without loss of
generality since, in the general case, the LR statistic is identically
LRr0,r.(<V aak) " LR0,ro+r.<0’f°fok aaJ)‘LR0,
With rQ=0, and ignoring the deterministic components, dt, the model can be written as::
(2.3) AY, = S ^ Y , . , ) +
where /3=(4>j
$2
+ B Z l
••• ^p-l)> ^
+
ef
Zt=(AY^_i AY|_2 ... AYJ_p+ j)\ In the context of (2.3)
the null hypothesis HQ: r=0 can be written as the composite null HQ: <5^=0, 5^=0.* It is
convenient to discuss each part of this null separately, and thus we first consider testing <5 ^ = 0
maintaining 5^ = 0 , then the converse, and finally the joint hypothesis.
The test statistic for HQ: r=Q vs. Ha: r=r^ : When ^ = 0 , equation (2.3) simplifies to:
(2.4) AYt = i ^ Y ^ j ) +
fiZ t
+ et.
Since a^Yt_^ does not depend on unknown parameters, (2.4) is a standard multivariate linear
regression, so that the LR, Wald and LM statistics have their standard regression form. Letting
Y=[Yj Y2 ... Yt ] \ Y_1 =[Yq Yj ... Yt . j]’, AY=Y-Y_1, Z =
e=[ci
«2
[ Z {
... ej]’, and M£=p-Z(Z’Z)"*Z)], the OLS estimator of
A
5^=(AY’
Y_^<2^X0^.Y_j
1
is
Y. ^aak)' which corresponds to the Gaussian MLE. The
corresponding Wald test statistic for H0 vs. Ha is:
2^ ... Z ^ ’,
-5-
(2.5)
W = [v e c ty i’K ^Y.j'M zY .
, ^ - 1
® E /W s ^ )]
= [vec(AY'Mz Y ^ 3 ^'K a^Y.fM zY . ! ^
)-1
® E^H vec^Y'M zY.,^)].
where Ef is the usual estimator value of Ef, i.e., E€=T*Ae’€, and where e is the matrix of
OLS residuals from (2.4). For values of 6^ that are T‘* local to 5^=0, the LR and LM
statistics are asymptotically equal to W.
The test statistic for HQ: r=0 vs. Ha: r=r^ : The model simplifies to (2.4) with <5^ and
replacing 5 ^ and a ^ . However, the analogue of the Wald statistic in (2.5) cannot be calculated
since the regressor
depends on unknown parameters. However, the LR statistic can
be calculated, and useful formulae for the LR statistic are developed in Anderson (1951) and
Johansen (1988). Unknown cointegrating vectors complicate the testing problem because under
the null hypothesis 5^=0, so that the cointegrating vectors are unidentified. The problem can
be avoided when ra=n, since in this case n is unrestricted under the alternative and the null and
alternative become HQ: n=0 vs. Ha: n£0. The problem cannot be avoided when the rank(II) <n
under the alternative. Indeed, in the standard classical reduced rank regression, the general
form of the asymptotic distribution of the LR statistic has only been derived for the case in
which the matrix of regression coefficients has full rank under the alternative. In this case,
9
Anderson (1951) shows that the LR statistic has an asymptotic x null distribution. When the
matrix of regression coefficients has reduced rank under the alternative, the asymptotic
distribution of the LR statistic depends on the distribution of the regressors. Still, the special
structure of the regressors in the cointegrated VAR allows Johansen (1988) to circumvent this
problem and derive the asymptotic distribution of the LR test even when n has reduced rank
under the alternative.
As pointed out by Hansen (1990), when some parameters are unidentified under the null,
the LR statistic can be interpreted as a maximized version of the the Wald statistic. This
interpretation is useful here, because it suggests a simple way to compute the statistic. Since
this form of the statistic appears as one component in the test statistic for the general r=r&k+rau
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alternative, we derive it here.
Let LR denote the likelihood ratio statistic for testing HQversus Ha, and let LR (Ef) denote
the (infeasible) LR statistic that would be calculated if E e were known. As usual,
**
1
LR=LR (E€)+Op(l) under HQand local alternatives (here, T ). Let L(5au,aau,E6) denote the
log likelihood written as a function of 5^, a^, and
with /? concentrated out, and let
A
5au(aau) denote the MLE of 5^ for fixed a^. Then the well known relation between the
Wald and LR statistic in the linear model implies that
(2.6)
W (o^
= 2 rU V a a,),«a!1ip-L (0,av
where W(a^) is the Wald statistic in (2.5) written as a function of a^, and the second equality
follows since aQ
does not enter the likelihood when $_“u=0. Thus:
*ki
S“Pa.„
= SuPa.„
“ LR‘ < ^
where the Sup is taken over all nxra matrices aa.
To calculate Sup„
W(a„ ), rewrite (2.5) as:
Cxau
(2.7)
W foJ = [vec(AY’Mz Y
®^ W
a Y-Mz Y. , 0^ )]
= TR[^'A(AY’MZY iaa.)(aa.Y .1,Mz Y^aao)-1(aiiMz Y:1AY)^14-]
= TR[Eg1/i(AY’Mz Y.1) DD’O ^ Y ^ A Y )^ ’], where D=a^(a^Y:1Mz Y.1aau) ' 1/4
= TR[D’(Y:1Mz AY)E^1(AY’Mz Y.1)D]
= TR[F’CC’F],
where F-fYljM zY. i)‘V < ° i Y’-lMZY-l‘Ia / A- and C=(Y;,MZY p ’V l
1MZAY)E^V4’.
Notice that F’F=Irau and Sup
W(aa)=SuppF=I TR[F’(CC’)F]. Letting
Xj(CC’) denote the eigenvalues of (CC’) ordered so that Xj ^ X2 ^
-7-
^
Xn. Then
(2.8)
Supau WCa^) = SuppT =I TR[F’(CC’)F] =
Xj(CC’)
= LR*(E6) = LR + op(l),
where the final equality holds under the null and local (T‘*) alternatives. Since Xj(CC’)=Aj(C’C),
the likelihood ratio statistic can then be calculated (up to a term that vanishes in probability) as
the largest r^ eigenvalues of C,C=[E‘^(AY,Mz Y.1)(Y:1Mz Y.1)*1(Y:1Mz AY)E;1A’].
To see the relationship between the expression in (2.8) and the well known formula for the
LR statistic developed in Anderson (1951) and Johansen (1988), note that their formula can be
written as LR= -T £ [su_ ^ln[ 1-7 -], where yj are the ordered squared canonical correlations
between AYt and Yt_j, after controlling for AYt_j,...,AYt_p+ j. Since y-=X-(S’S), where
S’SKAY’M ^ Y r^ A Y ’M ^.jXYIjN^Y.jrtYljN^AYXAY’N^AY)"^’ (Brillinger
(1980), Ch.10), LR=-T£[‘“= 1ln[l-Xi(S’S)]=T E p = 1Xi(S’S)+op(l) = £ f“ = 1Xi(TS’S)+op(l).
Finally, since T(S’S) =
(AY’Mz Y_!)(YIjMz Y_1) ' 1(Y11MZAY)E^’, where
Ee=T'^(AY’Mz AY), this expression is identical to (2.7), except that Efi is estimated under the
null.
The test statistic for HQ: r=0 vs. Ha: r=ra^+ra^ : The model now has the general form of (2.3).
As above, the LR statistic can be approximated up to an op(l) term by maximizing the Wald
statistic over the unknown parameters in
Let
Mzk=[I-(MzY_iaak)(a^kY’iMzY_iaak)"^(a^kMzY’i)]Mz denote the matrix that partials
both Z and Y^a^ out of the regression (2.8). The Wald statistic (as a function of
and a^)
can be written as:
(2.9)
= [vec(AY'MzY.[aai)]'[(o^Y., ’MzY. , aak) - 1 ® E;1][vec(AY’MzY.r a i)]
+ [vec(AY'MzkY
, ’MzkY iaaii)-> ® E;>][vec(AY’MzkY [C^)]
The first term is identical to equation (2.5) above, and the second term is the same as (2.6),
except that MZAY and MzY_j are replaced with MzjcAY and M^Y.^.
When maximizing W ta^a^) over the unknown cointegrating vectors in a^, we can restrict
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attention to cointegrating vectors that are linearly independent of a^, so that the LR statistic is
obtained by maximizing (2.9) over all nxr^ matrices au satisfying a^aat=0. Let G denote an
(arbitrary) n x (n-r^) matrix whose columns span the null space of the columns of
Then
can be written as a linear combination of the columns of G, so that (x^ G a^ , where
(n-r^) xr^ matrix, so that aauaak=aauG’
is an
for all a^. Substituting Ga^ into (2.9)
and carrying out the maximization yields:
(2.10) Sup^ W(V a u ) = [vec(AY’MzY 1aat)]’[(a^Y.1’MzY
® ^ [v e c ^ Y ’M ^ o ^ ]
+
= LR + op(l),
where H’H = £^1>4 (AY,MzkY.1G)(G,Y:1MzkY.1G)'1(G,Y:1MzkAY)^^:
Before proceeding, we make three computational notes about (2.10). First, when r^=0, the
statistic is just the standard Wald statistic testing for the presence of the error correction terms
a^Yt_i that is calculated by most econometric software packages. Second, any consistent
A.
estimator of
can be used as
A particularly easy estimator, consistent under the most
general hypothesis considered here, is the residual covariance matrix from the regression of Yt
onto p lagged levels of Yt. Third, the columns of the matrix G (appearing in the definition of
H) can be formed in a number of ways, for example using the Gram-Schmidt orthogonalization
procedure.
M o d ifica tio n s R eq u ired F o r N on zero D rift C om ponent:
When dt£0 in (2.1a), Yt is not directly observed, and the procedures outlined above require
modification. The necessary modification depends on the precise form of drift function. Here
we assume that d^/XQ+jijt, and thus allow Yt to have a nonzero mean and, when ^ £ 0 , a non
zero trend. While more general drift functions are certainly possible, this formulation of dt has
proved to be adequate for most applications.
(2.11)
In this case the VECM for yt becomes:
Ayt = 6 + 7 t + 5(a’yt. 1) + E P= l^i^Yt-i + €t
-9-
where 0 = (I-£f=
^
y = - b a 'p ^ .
There are three complications that arise when /xqor /xj are nonzero. First, as discussed in
Johansen (1991),(1992a),(1992b) and Johansen and Juselius (1990), relationships between /
xq
, /x j
and the cointegrating vectors can lead to different interpretations of the drift parameters. For
example, some linear combinations of /
xq
are related to initial conditions in the Yt process, while
other are related to means of the "error-correction" terms a ’Y^. Second, these different
interpretations can imply different trend properties of the data and this leads to changes in the
asymptotic distribution of test statistics. Third, in the context of the univariate unit root model,
Elliott, Rothenberg and Stock (1992) show that different methods for detrending Yt (associated
with different estimators of /
xq
and p ^ ) can lead to large differences in the power of unit root
test statistics, and Elliott (1993) shows has the tests’ power depends on assumptions concerning
initial conditions of the process.
Rather than investigate all of the possible methods here, we present results for what are
arguably the three most important cases. The first is simply the baseline case with /xq=/xj =0; in
this case 0 = 7 = 0 in (2.11). In the second case, /x^=0 so that the data are not "trending", but /xq=£0
and is unrestricted. This is appropriate when there are no restrictions on the initial conditions of
the Xt process or the means of the error correction terms, a’Yt. Hence 7 = 0 in (2.11), but 6 ¥ = 0 ,
but is constrained because in captures only the non-zero mean of the error correction terms a ’yt.
Imposing the constraint, leads to:
(2.12)
Ay, = «((*>,.!-» +
+ 't
where j8 = a Vo* I® die third case, /xq£ 0 and is unrestricted and /ij ¥=0 , but is restricted by the
requirement that a’/xj= 0 ; in this case 7 = 0 in (2 . 11) and 0 is unrestricted.
A s y m p t o t ic D i s t r i b u t i o n
o f t h e S t a t is t ic s :
Above, the Gaussian likelihood ratio statistic for testing H0 :r=r0 k+r0u vs.
Ha:r=rOi+rat "t"rou+rau was ^ e d as L R ^ /a ^ , a^). Let W ^ fa ^ , «at) define the
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corresponding Wald statistic constructed by maximizing over all values of the unknown
cointegrating vectors. In particular, defining Wq ^(0 ,0^ ) sSupa^ Wfa^or^) from (2.10), then
Wr0.r /a<V aak^= W 0 ,r0+ ra^°’^Ot %J)"W0 ,ro(°’aOi)'
To derive the asymptotic distribution of Wro
aak) we make four sets of assumptions:
A. The data are generated by (2. la)-(2. lc) with:
(A. 1) E(et |eM , ... , ^ = 0 ,
E(et€t’ I €t-l»
’ eP =Se»
E(e| t) < k < oo for all i and t.
(A.2) Letting <i>(z) = I-4>jz-...-^>p_^z^"*, then the roots of | $>(z) | are all outside the unit circle.
(A.3) X_i=0, i= 0,...,p -l.
(A.4) Three alternative assumptions are made about dt:
(A.4.i) dt=0 for all t;
(A.4.ii) dt=/XQ for all t;
(A.4.iii) dt=piQ+/ijt for all t, with a^tj=0 and a^/ij=0.
Note that under assumption (A.4.iii) we assume that
annihilates the deterministic drift in the
series under both the null and the alternative.
The test statistic will be formed as described above, when dt=0. When dt£0, the VECM is
augmented with a constant, and the statistic is calculated as above with Zt in (2.3) augmented by
a constant. Since, under assumption (A.4.iii), the constant term in the VECM (2.11) is
unrestricted, augmenting Zt with a constant and carrying out least squares produces the Gaussian
maximum likelihood estimator. However, under assumption (A.4.ii), the constant term in the
VECM (2.11) is constrained (see (2.12)), and thus the least squares estimator does not correspond
to the Gaussian MLE. We nevertheless, consider test statistics based on this formulation for two
reasons. First, when some columns of a are known, the unconstrained estimator and test
statistics are much easier to calculate than the constrained estimator; the required calculations
when a is known are discussed in Johansen and Juselius (1990) and Johansen (1991). Second, we
show that when a is unknown, the test based on the unconstrained estimator has somewhat better
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local power than the test based on the constrained estimator.
Convenient representations for the asymptotic null distribution can be derived using the
following notation. Let: B(s)=(Bj(s) I^Cs) ... Bn(s))’ denote an n x l dimensional standard Wiener
process; J QF(s)ds= f F and J QF(s)dB(s) = f FdB, for arbitrary function F(s); B^(s)=B(s)- J B
denote the corresponding "demeaned" process; s^=s- J s=s - ‘/2 denote the demeaned time trend; and
B- j(s)=(B-(s), ... Bj(s))’ denote a (j-i+1)x 1 subvector of B(s), and B^ j be defined analogously.
T h e o re m
1:
The asymptotic null distribution of
Wr„,r.<“W
=>
i FldBl,k>'< i
EfelXiK ! F2 dBl k’)'( J F2 F2’)-1( J F2 dBl k')]
^(0^ , 0^ ) can be represented as:
! FldBl,k» +
where k=n-rQu> F2 (s)=F 3 (s)-7 3 jF j(s) with 73 j = [ J FjFj’]'^ j F3 F1 ’, \[ .] is the i’th largest
eigenvalue of the matrix in brackets, and the definition of Fj(s) and F3 (s) depends on the
particular assumptions employed. In particular:
Case (1). Suppose that (A.1)-(A.3) and (A.4.i) hold, and the statistic is calculated with
Zt=(AY[ . 1 AY[_2 ... AY|.p+1)’, then F 1(s)=B 1>m(s) with m=rafc and F3 (s)=Bij(s) with
i=rat+ l a n d j - n - r ^ .
Case (21. Suppose that (A.1)-(A.2) and (A.4.ii) hold, and the statistic is calculated with
Zt= (l AY^4 AY’ _2 ... AY’.p+1)’, then F 1(s)=B<(>m(s) with m=rak and
F 3(s)= B f,j(s) with i=rat+ 1 andi= n - V rok-
Case (31. Suppose that (A.1)-(A.2) and (A.4.iii) hold, and the statistic is calculated with
Zt=(l
A Y [
a
AYj_2 ... AY{_p+1) \ then F1(s)=B^ m(s) with m=rak, and
F3 (s)=(s/i(s)’ B^’j(s)) with i=rafc+ l and j=n-r0 k-r0 u*l.
Proof: See Appendix
We make six remarks about these results. First, Theorem 1 is a generalization of the results
in Johansen [(1988)(1991)] who considered the problem with rok=rak=0. Second, when a
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constant included in Zt, the test statistic is invariant to the initial conditions for Xt, t=0,...,-p+l
under the null hypothesis. Thus, assumption (A.3) is not necessary under Cases (2) and (3) in
Theorem 1. Third, when ro^= ra^= 0, the limiting distributions in Cases (2) and (3) are the same.
Fourth, under Cases (1) and (3), the
r,(aOk’aatP statistic is asymptotically equivalent to the
LR statistics; this equivalence fails to obtain in Case (2) because the constraint on the constant
term in the VECM (2.11) and (2.12) is imposed when the LR statistic is calculated, but the W
statistic is calculated using an unconstrained estimator. Fifth, while the case with dt=^Q+/x^t for
all t, with a ^ i j = 0 and
^0
is not covered by the theorem, the limiting distribution of the
test statistic is readily deduced in this case as well. Since we did not tabulate critical values for
this case, we did not include the limiting distribution in the theorem. As a practical matter, our
calculations indicated that the critical values for the test statistic under the assumption that
“ak^l =0 ^
than those under the assumption a^/x^0, and so using the Case (3)
distribution results in conservative inference. Finally, it is also straightforward to generalize the
theorem to accommodate linear restrictions on the cointegrating vector of the form Ha_<Ml=h,
where H and h are known ( xn and t xr^ matrices respectively.
Critical values for n-r^ < 5 are provided in Table 1. These critical values were calculated by
simulation using 10,000 replications and T = 1000. Extended critical values of n-r_ <9 are
tabulated in Horvath and Watson (1993). When rOk= raj^= 0 these correspond to the critical values
tabulated in Johansen (1988), Johansen and Juselius (1990) and Osterwald-Lenum (1992).^
3. Comparison of Testing Procedures
In this section we carry out three power comparisons. First, we compare the local power of
the W/LR tests that impose the value of the cointegrating vector under the alternative to the
corresponding tests that do not use this information. Second, since a
p r io r i
information about
the cointegrating vector may only be approximately correct, we investigate the power
implications of imposing an incorrect value of the cointegrating vector. Finally, for the special
case with ^ = ^ = 0 and ^ = 1, we compare the power of the VECM-based tests to univariate unit
root tests applied to the error correction terms.
For tractability, most of the discussion will focus on a bivariate version of (2.11), with
- 13-
=0:
*1=*2
(3.1)
Ayi,t
ei,t
-Ay2,t-
-e2 ,t-
Since the likelihcxxi based procedures are invariant to nonsingular transformations of Yt, we set
a = ( 0 1)’ and 5 ^=0 , so that the model becomes:
(3.2a) Ayl t =
(3.2b) Ay2>t = 62
+ e1>t
+
^
2 , 1-1 + e2,t*
To investigate the local power of the tests, we suppose that 52 is local to zero; specifically we set
62=62
j=-c/T . This allows us to study local power using local-to-unity asymptotics familiar
from the work of Bobkowsky (1983), Cavanagh (1985), Chan and Wei (1987), Chan (1988),
Phillips (1987b, 1988) and Stock (1991). To rule out drift in the error correction term we set
02 =O, and for simplicity we set E€=I2.
The local power results are conveniently stated in terms of a two dimensional
Wiener/diffusion process, Bc(s)=(Bj c(s) B2 c(s))\ Let B(s)=(Bj(s) B2 (s))’ denote a two
dimensional standardized Wiener process, let Bj c(s)=Bj(s), and let B2 c(s) evolve as
dB2] c(s)=- cB2 c(s)+dB2 (s). Thus, the first element of Bc(s) is a random walk, and the second
element is generated by a diffusion process with parameter c. Let B^(s)=Bc(s)- $ Bc denote the
demeaned version of this bivariate process, and let Dc(s)=(s^(s) B^ c(s)) denote the bivariate
process composed of the demeaned values of the time trend and B2 c. Corresponding to the
three cases in Theorem 1, it is straightforward to derive limiting representations for the
cointegration test statistics under local departures from the null. Let 7 =(yj
y 2Y
denote an
arbitrary 2x1 vector, and let a=(0 1) denote the true value of the cointegrating vector. Using the
notation introduced above Wq j(0,y) denotes the test statistic for Hg: r=0 vs. Ha: r=rak= 1
constructed using 7 as the cointegrating vector under the alternative; similarly Wq j(0,0) denotes
the test statistic for Hg: r=0 vs. Ha: r=r3u= l. The limiting distribution of these statistic is given
by:
-14-
(Case 1): Suppose that the data are generated by (3.2a)-(3.2b) with 0 j = 0 2 = 0 , 5 2 =-c/T, and et
satisfies assumption (A. 1) with Ee=I. If the test statistic is calculated without including a
constant in Zt, then:
W0>1(0,7) - > (7’ I BcdB')’(7’ ! BcBc,7 )'1(7’ J BcdB’);
W0 1 (0,0i = > XjK i BcdB’)’( j BCBC')'‘( ( BcdB')].
(Case 2): Suppose that the data are generated by (3.2a)-(3.2b) with 0j = 0 2 = 0 , 5 2 =-c/T, and et
satisfies assumption (A. 1) with E€=I. If the test statistic is calculated including a constant in
Zt, then:
W0 ,i(°,7) = > (V f BgdB’)’(7’ J BjBj’7 )-1(7’ J BgdB’);
WQ) 1(0,0) = > Xj[(
l
BgdB’)’( J BqB^’)_1( \ BgdB’)].
(Case 3): Suppose that the data are generated by (3.2a)-(3.2b) with 0j £0,
0 2 = 0 , 52 =-c/T,
and
et satisfies assumption (A. 1) with Ee=I. If the test statistic is calculated including a constant
in Zt, then:
w 0 ,l(°. 7 ) = > (7’ l DcdB’) ’ (7 I DcDc’7T 1(7’ I DcdB’), for7 l = 0 ;
w 0 ,i(°. 7 ) = > U sMdB’)’( j (s^ )2 ) ' h f s^dB’), for 7 ^
0;
W0,i(°,0) = > Xj[( l DcdB’)’( f DCDC’)_1( l DcdB’)].
In Case 3, when 0^ £0 and 7 j £0, the regressor 7 ’yt_i is dominated by the linear trend 7 ^0 jt. In
contrast, 7 *yt_j is linear function of diffusion process in Cases (1) and (2) for all values of 7 ^,
and in Case (3) when 7 j =0. This difference leads to the two possible limiting representations for
WQ j(0 ,7 ) in Case (3).
In Figure 3.1, we plot the local power curves associated with these limiting random variables
for a
= y ?
Thus, the Wq ^(0 ,0^ ) plot shows the power of the test that imposes the true value of
the cointegrating vector, while the Wq j(0,0) plots shows the power of the test that does not use
this information. The power gains from incorporating the true value of the cointegrating vector
are substantial: at 50% power they correspond to sample size increases of approximately 70%,
-15-
50%, and 40% for cases 1-3 respectively. Panel B of the figure also shows the local power of the
LR analogue of W
q
j(0,0)
that imposes the constraint on the constant term shown in (2.12). As
discussed in Johansen and Juselius (1990) and Johansen (1991), this statistic is calculated by
augmenting the matrix Y_i in (2 . 10) by a column of l ’s and excluding the constant from Z^.
Letting Fc(s) denote (1 Bc(s)), this statistic has a limiting distribution given by
Xj[( J FcdB’)’( | FCFC’)'*( f FcdB’)]. Interestingly, the power curve lies below the corresponding
W
q
^(0,0) power curve that does not impose this constraint on the constant term, and of course
both curves lie below their case 1 analogue. The reduction in power for the case 2 LR statistic
relative to case 1 arises because, under the null that 5=0, the constant term /3 in (2.12) is
unidentified. The LR statistic maximizes over this parameter, leading to an increase in critical
value relative to case 1. The reduction in power for the W
q
j(0,0)
statistic in case 2 relative to
case 1 arises because the data are demeaned in case 2 , leading to a reduction in the variance of
the regressor. Apparently, the reduction in power from demeaning the data is less than the
reduction from maximizing over the parameter /3.
Since the a
p r io r i
knowledge of the cointegrating vector may be inexact, it is also of interest
to consider the behavior of the statistics constructed from incorrect values of the cointegrating
vector. Asymptotic results for fixed values of 82 <0, imply that using the correct value of the
cointegrating vector is critical to the power gains apparent in Figure 3.1. For fixed alternatives,
the W
q
1(0,0)
and corresponding LR statistics are consistent. On the other hand, since 7 ’yt is
1(1) when 7 is not proportional to a, the test based on W
q
1(0 ,7 )
for y £
a ,
will not be consistent.
Thus, imposing the incorrect value of the cointegrating vector would seem to have disasterous
effects on the power of the test.
However, this drawback is somewhat artificial, since it applies in a situation when the power
of the W
q
i (0,0)
test is unity. An arguably more meaningful comparison obtains from the local-
to-unity results where cointegration is weak. Figure 3.2 shows the power results for W
test for a variety of values of 7 =(7 i 1); also plotted are the power results for W
q
q
1(0 ,7 )
i (0,0).
Results
are presented for the non-trending data Cases 1 and 2; results for Case 3 will be discussed
shortly. It is apparent from Figure 3.2 that for values of 71 reasonably close to the true value of
0, the W
q
1(0 ,7 )
test continues to dominate the W
- 16-
q
i (0,0)
test. For example, for the entire range
of values of c considered, the Wq j(0,y) test dominates the Wq j test for 7 ^< .1. On the other
hand, when 7 ^=0.5, the Wq j(0,0) test dominates for most values of c.
The results are quite different in Case 3. These results are not shown because the rejection
probability for the test constructed from incorrect values of 7 ^ for the Wq j (0 ,7 > test are very
small for all values of c. The reason for this can be seen from the results for Case 3 given
above. When 7 ^ £ 0 and c= 0 the Wq 1 (0 ,7 ) statistic converges to ( j s^dB’)’( j (s**)^)"*( f s^dB’) which
9
has a X2 distribution. From Table 1, the 5% critical value for the Wq 1 (0 ,7 ) test is 10.18, so
that the corresponding rejection probability for the Wq 1 (0 ,7 ) test using the incorrect value of 7
9
and assuming c= 0 is P(x2 > 10-18) =0.6%. The rejection probabilities for other values of c are
also close to this value.
Arguably, these results for Case 3 have little relevance. After all, when 0 1 =£0, 7 ’yt will be
trending when 7 1 £0. This behavior would be obvious in a large sample, and so the hypothesis
that 7 ’yt is 1(0) could easily be dismissed. This suggests that the comparison should be made, for
example, with 0^ or 7 1 local to zero, say 01 =c^[/T
XL
\L
or 71 =c /T . Since these power functions
depend critically on the assumed values of the constant c ^ and c ^ , and since reasonable values
of these parameters will differ from application to application, we do not report these functions.
Instead we carry out an experiment for a fixed sample size and Gaussian errors, using values for
the parameters in (3.2a)-(3.2b) and values of 7 ^ that are relevant for a typical application: the
analysis of postwar U.S. quarterly data on income and consumption. Letting yj t denote the
logarithm of per capita consumption, and y2 t denote the logarithm of the consumption/income
ratio, then 0} = .004,
= .006, cr2 = . 0 1 1 , cor(ej ^2 t) =0.21 and T=174.^ Results are shown for
values of 7 j ranging from 0 to . 10. For comparison with previous graphs ^ is written as -c/T,
and the power is plotted against c. For this example, the Wq j (0,7) dominates the Wq j (0,0)
statistic for all values of c considered when the error in the postulated cointegrating vector is 5 %
or less.
When there is only one cointegrating vector under the alternative, simple univariate tests
provide an alternative to the likelihood based tests. Thus, if the cointegrating vector is assumed
to be known, then the error correction term a ’yt can be formed, and cointegration tested by
employing a standard unit root test. The final task of this section is to compare the VECM
- 17-
likelihood based test to standard univariate tests.
There are three distinct differences between the multivariate tests considered in this paper
and standard univariate unit root tests. To highlight these differences, consider the VECM
(3.3)
AYt = 5a’YM +
and let Wj=a’Y^ and Vj=u’Y^ where v is chosen so that P=[a v] has full rank. The elements of wt
are the 1(0 ) error correction terms and the elements of vt are the remaining 1( 1) non-cointegrated
linear combinations of the data. After multiplying (3.3) by P \ the system can be partitioned as:
(3.4a)
Awt = pwt_j + E f= i^ Ww,i^wt-i + ^P= l^wv,i^vt-i + ew,t
(3.4a)
Avt = ?wt.j 4- Sf=i^vw ,i^wt-i "** E f=l^ w ,i^ vt-i + ev,t
where p=a’5, £=u’5, ew>t= a’et, ev>t=u’et and <£ww j,
j, etc., are the appropriate elements of
P’$j. The multivariate tests considered in this paper test HQ:5=0 in (3.3), or equivalently HQ:p=0
and |= 0 in (3.4a) and (3.4b). In contrast, univariate tests concentrate their attention on the
equation for Awt, written as:
(3.5) Awt = pwt.j + ut,
and test the null hypothesis HQ: p=0.
Thus, the first major difference in the univariate and multivariate tests is the null hypothesis:
univariate tests consider the simple null, p= 0 , while multivariate tests consider the composite null,
p =0 and £ =0. Since wt is 1(1) under the null, the lagged level of wt will not help predict any 1(0)
variable after controlling for lagged first differences of wt. The univariate tests look for
predictive power for future values of Awt, while the multivariate tests look for predictive power
for future values of all of the 1(0) variables, Awt and Avt. The second difference between the
univariate and multivariate tests is that the univariate tests typically use a one-sided alternative
(p < 0), while the multivariate tests consider two-sided alternatives. The third major difference is
-18-
the conditioning set used to estimate p in (3.5) or (3.4a). The multivariate tests include lagged
values of Awt and Avt in the regression; univariate procedures, such as augmented Dickey-Fuller
regression, include only lags of Awt. Thus, when lags of Avt help predict Awt, the error term in
the multivariate regression will have smaller error than the error term in the univariate
regression. This leads to a more efficient estimator of p and a more powerful test.
This last point is the subject of recent papers by Kremers, Ericsson and Dolado (1992) and
Hansen (1993). These papers carefully document the power gains associated with augmenting
standard Dickey-Fuller regressions with additional 1(0) regressors, and allow us to focus instead
on the the potential power gains and losses associated with the first two differences in the
univariate and multivariate procedures. We do this in the context of the bivariate example
introduced in (3.1) and (3.2) above, except now <5^0, so that (3.2a) becomes:
(3.2a’)
Ay1>t = 5 ^ ^
As before,
+
e1>t.
t is generated by (3.2b) with h j =-c/T, and the errors are mutually independent
iid(0,1). Since lagged first differences do not appear in the model and since the errors are
mutually uncorrelated, E(e2 1 1{Ay^ t}) =0, so that the standard univariate Dickey-Fuller test for
(3.2b) and the generalization of this procedure suggested by Hansen (1993) coincide.
We consider the local power of the one-sided Dickey-Fuller univariate t-test, the
corresponding two-sided test based on the squared Dickey-Fuller t-test statistic, and the W(0,a)
statistics. The power functions for W(0,a) are calculated for various values of
parameterized as 5j =( kc )/T. (When
is not T’* local to zero, e.g., <5^=
kc
local to zero,
for fixed non-zero
values of of k and c, then the asymptotic power of the W(0,a) test is unity.) Figure 3.4 plots the
power functions. We make two remarks. First, the power functions of the one-sided DickeyFuller t-test and the two-sided test based on the squared t-statistic are nearly identical. This is a
reflection of the skewed distribution of the Dickey-Fuller t-statistic. Thus, the two-sided nature
of the W statistics has little impact on the power relative to the one-sided univariate test.
Second, the relative performance of the W(0,a) statistic depends critically on the value of k .
When
k
=
0, the power loss in the W(0,a) statistic relative to the univariate test corresponds to a
- 19-
sample size reduction of 10% at 50% power. On the other hand, when k is large, the W(0,a)
statistic dominates the Dickey-Fuller test. For example, when x=30, the power gain in the
W(0,a) statistic relative to the univariate test corresponds to a sample size increase of
approximately 35% at 50% power.
4. Stability of the Forward-Spot Foreign Exchange Premium
In this section we examine forward and spot exchange rates, focusing on whether the
forward-spot premium, defined as the forward exchange rate minus the spot exchange rate, is
1(0). The data come from Citicorp Database Services, are sampled weekly for the period January
1975 through December 1989 (for a total of 778 observations), and are adjusted for transactions
costs induced by bid-ask spreads and for the two-day/non-holiday delivery lag for spot market
exchange orders as described in Bekaert and Hodrick (1993). The forward-spot premia for the
British Pound, Swiss Franc, German Mark, and Japanese Yen, the currencies used in our analysis,
are shown in Figure 4.1.
The tests for cointegration are performed on bivariate systems of forward and spot rates in
levels, currency-by-currency. In each case, the number of lagged first differences in the VECM
was determined by step-down testing, beginning with a lag length of 18 and using a 5% test for
each lag length. (See Ng and Perron (1993) for an analysis of step-down testing in the context
of testing for unit roots.) Results for testing for cointegration between forward and spot rates
are presented in Table 4.1. For each currency we report the test statistic for the case where we
impose a = (l - 1)’ (denoted by Wq j(0 ,a&k)), the test statistic for the case where a is unspecified
(denoted by W
q
j(0,0)),
the cointegrating vector estimated in this case (denoted by a^), and the
ADF statistic calculated from the forward premium. All statistics are reported for the optimal
lag length chosen via the step-down procedure. Constant terms were included in all regressions,
and so the p-values for the W
q
^(O.a^) statistic are from the Case (3) asymptotic null distribution
(equivalently Case (2)) . Since nominal exchange rates exhibit some trending behavior over the
sample period, the p-values for the W
q
j(0,0)
statistic are reported for the Case (3) asymptotic
null distribution.
Looking first at the W
q
^(0 ,0^ ) column, the null of no cointegration is rejected for all
-20-
currencies at the 5% level. The W
q
^(0,0) statistics, which can be interpreted as W
maximized over all values of a, differ little from the W
q
q
j(0,a)
statistics. Their p-values are
much greater however, since their null distribution must account for the fact that they are
maximized versions of W
q
i(0,aak). The next column shows why the two statistics are so similar:
the estimated values of the cointegrating vector are equal to (1 -1), at least to two decimal
places.
O
The final column shows the ADF test statistic applied directly to the forward-spot
premium. Like the Wq ^(0,0^) statistic, the ADF tests reject the null at the 5% level for all of
the currencies. This application clearly shows the power advantage of testing for cointegration
using a prespecified value of the cointegrating vector. Using the Wq j (0,0) statistic, the null of
no cointegration is rejected at the 5 % level for only two of the four currencies.
Concluding Remarks
In this paper we have generalized VECM-based tests for cointegration to allow for known
cointegrating vectors under both the null and alternative hypotheses. The results presented in
Section 3 suggest that the power gains associated with these new methods can be substantial.
These power gains were evident in the tests for cointegration involving forward and spot
exchange rates. Cointegration was found in all currencies using tests that imposed a
cointegrating vector of (1 -1), but the null of cointegration was rejected in only one half of the
cases when this information was not used. Yet, in these bivariate exchange rate models, the
univariate ADF test applied to the forward premium (Ft-St), yielded roughly the same inference
as the multivariate VECM-based tests that imposed the cointegrating vector. Arguably, a more
interesting application of the new procedures will be in larger systems with some known and
some unknown cointegrating vectors.
The tests developed here rely on simple methods for eliminating trends in the data —
incorporating unrestricted constants in the VECM. In the unit root context, the work in Elliott,
Rothenberg and Stock (1992) suggests that large power gains can be achieved using alternative
detrending methods. Hence, one extension of the current research will be a thorough
investigation of alternative methods of detrending and their effects on tests for cointegration.
-21 -
Appendix
P r o o f o f T heorem 1:
To prove the theorem, it useful to introduce two alternative representations
for the model. The first is a triangular simultaneous equations model used by Park (1988); the
second is Phillips’ (1991) triangular moving average representation. The first representation is
useful because it allows the test statistic to be written in a particularly simple form; the second
representation is useful because it neatly separates the regressors into 1 (0) and 1 ( 1 ) components.
We begin by defining some additional notation. First, partition Yt as
Yt=<Yi,t Y2,t Y3,t Y4,t) \ where Y, t is r ^ x l , Yj ( i s r ^ x l , Y3>t i s r ^ x l and Y4 t is
(n -r^ -r^ -r^ ) x 1. Since the cointegration test statistic is invariant to nonsingular
transformations on Yt, we set a Ok=[0 1 ^ 0 0]’ and <*^=[0 0 Ir 0]’, where these matrices are
partitioned conformably with Yt. Thus, a^Y t =Y 2 t and c^_Yt =Y 3 t. Without loss
generality, we write
= [ 1 ^ ^ w3 “ 4!
columns of a = [ a Qu
a ^ = [ 0 0 0 ajjj, which insures that the
a ^ ] are linearly independent. Finally, we assume that the true (but
unknown) values of
and
are zero. These normalizations imply that
ut =(Yj t Y2 t)’ denotes the 1(0) components of Yt and vt =(Y^ t Y4 t)’ denotes the 1(1),
non-cointegrated components.
Using this notation, the VECM in equation (2.3) can be reparameterized as the simultaneous
equations models:
(A-1)
AYM =
(A.2)
AQt = «a(HvM ) + 7 ’St + et
<TYtA
+
0 f t
+ eu ,
where Qt = (Y ^t Y ^t Y ^ ) ’, St =(AYJt
H -
1
*ak
Z’)’, and
0
These equations follow from writing the first r ^ equations in (2.3) as:
(A.3) AYj t - S i^ o ^ Y j.j +
0kY2>t_1 + ^ ^ 3 ^
-22-
+ ^ ( S J Y 4 , t - i ) + ^ l Zt + €l,t»
and the last (n-r^) equations as:
(A.4) AQt = $Q>0uai Yt-l + 5Q>0kY2,t-l + 5Q>akY3jt. 1 + 8Q,att(54IY4>t. 1) +
0 Q Z t + eQ v
In equation (A. 1), the term 0’Yt^ captures the effect of all of the error correction terms on
AYj t. Since o^,
and
are unknown,
8
is unrestricted. To obtain (A.2), equation (A.3) is
solved for a ^ Y ^ as a function of AYj t, the other error correction terms, Zt, and
expression is then substituted into (A.4). Thus for example, et = eq t-5q
t; this
i
in
(A.2). In terms of the reparameterized model (A.1)-(A.2), the only constraints on the
parameters are those imposed by the null hypothesis: HQ: $a =0.
Equations (A.l) and (A.2) are useful because, for given
a
the parameters in (A.2) can be
efficiently estimated by 2SLS using Ct =(Uj_^, v ^ , Z |)’ as instruments. Thus, letting
Q=[Ql Q2 - O r]’. V-I= tv 0 v ! ...
S = [Sj s 2 ... ST] \ C =[C j c 2 ... c T] \
e= [e i e2 ... &p]\ S=C(C’C)-1C’S, and Mg=I-S(S’S)"*S’, the Wald statistic for testing
Ho:5a =0 using a fixed a ^ , is:
(A.5)
W(aau) = [vec(AQ’M$V .1H’)],[(HV.1,M$V .1H’) '1 ® Sg1][vec(AQ’MgV.1H’)]
=[vec(e’Ms V .1H’)]’[(HV:1M$V .1H’) '1 ® E^lIvecfe’M gV.jH’)],
where the second equality holds under HQ.
To asymptotic distribution of su p ^ W fa ^ ) depends on the behavior of the regressors and
instruments, which is readily deduced from the triangular moving average representation of the
model:
(A. 6)
ut = Du(L)at +
(A .l)
Avt = Dv(L)at +
nu
fiy
\L
where at =E‘ et, where n u = Q in Case 1 and
-23-
=0 in Case 1 and Case 2. Since the variables are
generated by a finite order VAR, the matrix coefficients in the lag polynomials DU(L) and Dy(L)
eventually decay at an exponential rate. Since vt is 1(1) and not cointegrated, Dy(l) has full row
rank. Furthermore, the error term et in (A.2) can be written as et=Dat, and Dy(l)D’ has full
row rank since only the first differences of Yj t enter (A.2).
The theorem now follows from applying standard results from the analysis of integrated
regressors to the components Wfa^). (For example, see Chan and Wei (1988), Park and Phillips
(1988), Phillips (1988), Sims, Stock and Watson (1990), Tsay and Tiao (1990), or the
comprehensive summary in Phillips and Solo (1992).) We now consider the theorem’s three cases
in turn.
Case 1: In this case, n u = 0 and n y
= 0
in (A.6 ) and (A.7), and it is readily verified that
(A.8 .i) r V j M s V . j = r V j V . j + op(l)
(A.8 .ii) T^V^Mge =
Y
l V 'A e
+
op(l),
(A.8 .iii) plim(Ee) = Ee = FF’
so that
w (Q'au) ==[vec(T"1e’V_1H’)]’[(T'2HV_1’V_1H’) '1®(D,D)"1][vec(T‘1e’V .1H’)] + op(l).
From the partitioned inverse formula:
(A.9)
[vec(T’ 1e’V_1H’)]’[(T"2 HV_1 ’V_1H’)"1 ®(D’D)‘ 1][vec(T'1e’V_1H’)] =
[vecCr^e’V ^ . p i ’t C r ^ V i ^ V ^ . p - ^ f D ’D r ^ v e c C r ^ e ’V!.!)] +
[vec(T-1e’M v V 2}.1aau)]’[(T-2a ^ ).1M v V 2>.1aau)-1® ( D ’D)-1][vec(T-1e’M v V 2>.1aau)]
where Vj
denotes the first r^ columns of V.j, and V2
denotes the remaining n-r^-r^-
rafc columns. Letting Dj denote the first r^ rows of Dy(l),
(A. 10)
[vecCT^e’V j.p i’tC T ^V i^V ^.^-^fD ’Dr^tvecCr^e’V !.!)) =
-24-
T r a c e [ ( D ’ D ) - V 4( T 1 e ’ V 1> . 1 ) ( ( r 2 V i >. 1 V 1>. 1 ) - 1 ( r 1 V i >. 1 e ) ( D ’D ) - I /4 ’]
= > T r a c e [ ( D ’ D ) _1/^ ( D j J B d B ’ F T C D j J B B ’ D J ) " 1 ^ ! J B d B ’ F O f D ’ D ) " ^ ’ ]
= T r a c e [( J F ^ B ^ J X
\F j F j r t J F j d B 1 > n . r J ]
where B(s) denotes an nx 1 standard Brownian motion process, F^(s)=Bj rak(s) (the first raj_
elements of B(s)), and the last equality denotes equality in distribution.
As shown in equation (2.7)), maximizing the second terms in (A.9) over all values of a du
.
yields:
(A. 11)
SupaJvecfl-'e'M v V z^ S ^ rK T -^ V ^ .jM v V2 ,. 1o aa) - 1 SCD’D)'1]
[vecC r'e'M vV j^a^)]
= E fS l^W
where,
(A. 12) R = (D’D)"1/^[T '1e’MViV2 _1][T'2V2>. 1MV iV2 _1] '1[T '1e,MViV2 . j l ’fD’D)-1^2.
Using notation borrowed from Phillips and Hansen (1990), R is readily seen to converge to:
(A. 13) R = > ( J F2dBj>n.rJ ’( ! F ^ ) - 1! ! F2dBi n.rJ
where F2 (s)=F 3 (s)"rF[(s), with 7 =[ J F1Fj]*1[ ( FjFj] where F3 =Brak+|in^(s). Case (1) of
the Theorem follows from (A. 10) and (A. 13).
Case 2: In Case (2),
n u £ 0
but /*v=0. Letting V_j =T~^ £ vt.j, the proof follows as in Case (1)
with (V.j-V_j) replacing V_j in (A.8)-(A.12) and /3^(s) replacing B(s) in the limiting
representation (A. 10) and (A. 13).
Case 3: In Case (3), both /*u and /xy£0. However, since E(a^Yt)=0 is assumed in Case 3, the
first r^ elements of /iy=0. Thus the first term of the statistic (the analogue of (A. 10)) is
-25-
identical to the corresponding term in Case 2. The last n -r^ -r^ -r^ elements of vt contain a
linear trend, and so, appropriately transformed, this set of regressors behaves like a single time
trend and n -r^-r^ -r^-1 martingale components. With this modification, the result for Case (3)
follows as in Case (2).
-26-
Footnotes
1. Formally, the restriction rank(5ac*a)= ra should added to the alternative. Since this
constraint is satisfied almost surely by the estimators under the alternative, it can be ignored
when constructing the likelihood ratio test statistics.
2. The formulation used here is not as general as that used in Johansen (1992b), who considers
a model of the form: AYt ==/?Q+/3jt+nYt.^ +
Johansen’s formulation allows
for the possibility of quadratic trends in Yt, which are ruled out in our formulation of dt. See
Johansen (1992b) for more discussion.
3. For example, when h=0, the statistic is formed as in (2.10) where now the matrix G is nx(nrafc- 0 with columns spanning the null space of the columns of («afc H’); the asymptotic
distribution Theorem 1 continues to hold except that the index j in the definition of F-j(s)
becomes j =n-r_ -r~ -L Similar results obtain when h=£0.
4. There are many repeated entries in Table 1. For example, as noted above, when r^ = 0 , the
Case (2) and Case (3) critical values are identical. Furthermore, within each case, the critical
values are the same for all combinations of rafc and r ^ with rak+ r2u=n-r0u. In this situation
when ^ = 0 , these hypotheses all correspond to HQ: n = 0 in equation (2.2). There are a number
of other examples of identical critical values that are not listed here.
5. These power curves were computed using 10,000 replications and T = 1000.
6. These parameter values were calculated using consumption and output from the Citibase
database, spanning the quarters 1947:1 through 1992:4, and are in constant (1987) dollar, per
capita terms. The consumption series is the sum of consumption expenditures on nondurables
and services. The output series corresponds to gross, private sector, nonresidential, domestic
product and is constructed as gross domestic product minus farm, nonfarm housing, and
government production.
7. We thank Robert Hodrick for making the data available to us.
8. Evans and Lewis (1993) using monthly data over the 1975-1989 period also find estimates of
cointegrating vectors very close to (1 -1). While their estimated standard errors suggest that the
-27-
cointegrating vectors may be different from (1 -1), Evans and Lewis argue that this arises from
large outliers or "regime shifts" in the data.
-28-
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Bobkowsky, M.J. (1983), Hypothesis Testing in Nonstationary Time Series, Ph.D. thesis,
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Chan, N.H. (1988), "On Parameter Inference for Nearly Nonstationary Time Series," Journal o f
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Davies, R. B. (1987), "Hypothesis Testing When a Parameter is Present Only Under the
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Engle, R.F. and C.W.J. Granger (1987), "Cointegration and Error Correction: Representation,
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R o c h e ste r .
Horvath, Michael and Mark Watson (1993), "Critical Values for Likelihood Based Tests for
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University.
Johansen, S., (1988), "Statistical Analysis of Cointegrating Vectors," Journal o f Economic
Dynamics and Control, 12, pp. 231-54. Reprinted in Long-Run Economic Relations:
Readings in Cointegration, edited by R.F. Engle and C.W.J. Granger, Oxford University
Press, New York, 1991.
Johansen, S. (1991), "Estimation and Hypothesis Testing of Cointegrating Vectors in Gaussian
Vector Autoregression Models," Econometrica, 59, 1551-1580.
Johansen, S. (1992a), "The Role of the Constant Term in Cointegration Analysis of Nonstationary
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Johansen, S. (1992b), "Determination of Cointegration Rank in the Presence of a Linear Trend,"
Oxford Bulletin o f Economics and Statistics, 54, pp. 383-397.
Johansen, S. and K. Juselius (1990), "Maximum Likelihood Estimation and Inference on
Cointegration —with Applications to the Demand for Money," Oxford Bulletin o f
Economics and Statistics, 52, no. 2, pp. 169-210.
Johansen, S. and K. Juselius (1992), "Testing Structural Hypotheses in a Multivariate
Cointegration Analysis of the PPP and UIP of UK," Journal o f Econometrics, 53, pp. 21144.
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MacKinnon, James G. (1991), "Critical Values for Cointegration Tests," in R.F. Engle and
C.W.J. Granger (eds) Long-Run Economic Relations, Readings in Cointegration, Oxford
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Maximum Likelihood Cointegration Rank Test Statistics," Oxford Bulletin o f Economics and
Statistics, 54, pp. 461-71.
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Methods for the Selection of the Truncation Lag," manuscript, University of Montreal.
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-30-
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pp. 971-1001.
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Some Unit Roots," Econometrica, Vol. 58, No. 1, pp. 113-44.
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Macroeconomic Time Series," Journal o f Monetary Economics 28, pp. 435-60.
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Tsay, R.S and G.C Tiao (1990), "Asymptotic Properties of Multivariate Nonstationary Processes
with Applications to Autoregressions," Annals o f Statistics, 18. pp. 220-50.
-31 -
T a b le 2 .1
C r i t i c a l V a lu e s f o r T e s ts fo r C o in te g r a tio n
•- Case 1
n-r
r
r
r
ok «k au
2.95
12.18
8.47
4.12
‘2.95
12.18
8.47
0
0
0
0
0
0
0
0
0
1
1
1
1
1
2
2
0 1
0 2
0 3
1 0
1 1
1 2
2 0
2 1
3
0
0 1
0 2
1 0
1 1
2 0
0 1
1 0
22.25
28.02
29.31
11.44
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
0 1
0 2
0 3
0 4
1 0
1 1
1 2
1 3
2 0
2 1
2 2
3
0
3
1
4
0
0 1
0 2
0 3
1 0
1 1
3
3
3
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
—
10Z
4.12
3
—
5Z
7.26
0 0 1
0 0 2
0 1 0
0 1 1
0 2 0
1 0 1
1 1 0
3
Case 2
7.26
2
2
2
2
2
2
2
3
—
1Z
5Z
0
0
3
1
0
—
10Z
1Z
1
1
3
0
1
—
—
Case 3 ....
1Z
5Z
10Z
6.63
6.84
3.98
2.73
6.63
12.18
8.47
6.63
14.83
11.03
9.35
19.14
14.93
13.01
18.13
14.18
12.36
16.10
12.21
10.45
22.43
18.17
15.87
19.66
15.41
13.54
9.43
6.28
4.73
13.73
10.18
8.30
13.73
10.18
8.30
16.10
10.45
22.43
18.17
15.87
19.66
15.41
13.54
16.10
12.21
12.21
10.45
22.43
18.17
15.87
22.43
18.17
15.87
9.43
6.28
4.73
13.73
10.18
8.30
8.94
6.02
4.64
9.43
6.28
4.73
13.73
10.18
8.30
13.73
10.18
8.30
17.51
15.42
25.93
21.19
19.12
26.17
21.14
18.62
23.28
20.81
35.98
29.46
26.79
34.84
28.75
26.08
23.91
21.52
37.72
31.66
28.82
35.83
29.62
27.05
7.94
6.43
15.41
11.62
9.72
15.41
11.62
9.72
24.91
20.30
18.05
31.42
26.08
23.67
30.67
25.70
23.04
29.31
23.91
21.52
37.72
31.66
28.82
35.83
29.62
27.05
19.75
15.20
13.04
25.35
20.74
18.51
25.35
20.74
18.51
29.31
23.91
21.52
37.72
31.66
28.82
35.83
29.62
27.05
29.31
23.91
21.52
37.72
31.66
28.82
37.72
31.66
28.82
16.84
12.89
11.03
21.62
16.65
14.51
20.36
15.93
13.93
19.75
15.20
13.04
25.35
20.74
18.51
22.90
18.18
16.25
11.44
7.94
6.43
15.41
11.62
9.72
15.41
11.62
9.72
19.75
15.20
13.04
25.35
20.74
18.51
22.90
18.18
16.25
19.75
15.20
13.04
25.35
20.74
18.51
25.35
20.74
18.51
11.44
7.94
6.43
15.41
11.62
9.72
11.39
7.87
6.36
11.44
7.94
6.43
15.41
11.62
9.72
15.41
11.62
9.72
28.33
23.82
21.51
32.35
27.40
24.94
32.19
27.07
24.84
40.14
34.35
31.63
47.03
40.50
37.78
46.00
40.27
37.17
44.62
45.66
39.17
35.90
47.31
36.58
49.16
44.03
45.61
53.14
54.34
46.30
39.91
54.25
56.17
43.32
44.09
13.60
9.73
7.93
17.16
13.20
11.16
17.16
13.20
11.16
32.75
27.86
25.43
39.55
33.55
30.73
39.47
33.22
30.45
42.47
36.93
33.81
51.82
44.98
41.45
50.96
43.78
40.94
45.66
39.91
36.58
56.17
49.16
45.61
54.34
47.33
44.09
22.85
17.92
15.81
28.62
23.41
21.10
28.62
23.41
21.10
47.33
38.43
33.36
30.69
47.26
40.98
38.11
46.82
40.76
37.50
45.66
39.91
36.58
56.17
49.16
45.61
54.34
47.33
44.09
33.53
27.80
25.24
41.08
35.33
32.33
41.08
35.33
32.33
45.66
39.91
36.58
56.17
49.16
45.61
54.34
47.33
44.09
45.66
39.91
36.58
56.17
49.16
45.61
56.17
49.16
45.61
24.15
19.28
17.30
27.09
22.73
20.61
28.06
22.74
20.36
31.30
26.19
23.82
37.76
32.45
29.49
38.01
31.74
28.65
33.53
27.80
25.24
41.08
35.33
32.33
40.07
33.57
30.41
17.16
13.20
11.16
33.45
28.25
25.73
13.60
28.04
9.73
23.19
7.93
20.82
17.16
13.20
33.83
28.87
11.16
26.10
T a b le 2 .1 (C o n tin u ed )
C r i t i c a l V a lu es fo r T e s ts fo r C o in te g r a tio n
2 ..... -
1 .....
r
r
r
ak au
A
1
1
1
1
2
2
2
2
2
A
3
A
3
1 2
2 0
2‘ 1
0
3
0 1
0 2
1 0
1 1
2 0
0 1
1 0
5
0 0 1
0 0 2
0 0 3
0 0 A
0 0 5
0 1 0
0 1 1
0 1 2
0 1 3
0 1 A
0 2 0
0 2 1
0 2 2
0 2 3
0 3 0
0 3 1
0 3 2
0 A 0
0 A 1
0 5 0
1 0 1
1 0 2
1 0 3
1 0 A
1 1 0
1 1 1
1 1 2
1 1 3
1 2 0
1 2 1
1 2 2
1 3 0
1 3 1
1 A 0
2 0 1
n-r
°u
4
A
A
A
A
A
A
A
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
5
1z
5Z
10Z
1Z
5Z
10Z
3 --1Z
5Z
10Z
33.53
27.80
25.2A
41.08
35.33
32.33
40.07
33.57
30.41
22.85
17.92
15.81
28.62
23.41
21.10
28.62
23.41
21.10
33.53
27.80
25.2A
41.08
35.33
32.33
AO.07
33.57
30.41
33.53
27.80
25.2A
41.08
35.33
32.33
41.08
35.33
32.33
18.59
1A.60
12.78
23.09
18.37
16.12
21.92
17.52
15.51
22.85
17.92
15.81
28.62
23.41
21.10
25.82
21.00
18.74
13.60
9.73
7.93
17.16
13.20
11.16
17.16
13.20
11.16
22.85
17.92
15.81
28.62
23.41
25.82
21.00
18.74
22.85
17.92
15.81
28.62
23.41
21.10
21.10
28.62
23.41
21.10
13.60
9.73
7.93
17.16
13.20
11.16
12.81
9.54
7.85
13.60
9.73
7.93
17.16
13.20
11.16
17.16
13.20
11.16
35.29
30.51
27.76
39.10
33.87
31.08
38.95
33.51
30.89
51.50
A5.8A
A2.75
59.27
52.05
A8.77
57.99
51.53
48.24
61.05
5A.A2
51.22
70.75
63.29
59. AA
70.30
62.45
58.82
65.5A
58.65
55.23
77.A6
69.37
65.20
75.6A
67.89
64.37
66.00
59.39
55.80
78.85
70.93
66.58
76.36
68.62
65.15
15.32
11.A1
9.A6
19.00
1A.53
12.49
19.00
14.53
12.49
A1.09
35.77
32.98
A7.18
41.36
38.AA
A6.58
40.78
38.15
56.00
A9.75
A6.A1
6A.19
57.55
53.98
63.59
56.60
53.43
63.52
56.83
53.56
74.61
66.88
63.00
73.49
65.73
62.31
66.00
59.39
55.80
78.85
70.93
66.58
76.36
68.62
65.15
26.01
20.92
18.55
31.26
26.15
23.51
31.26
26.15
23.51
46.51
A8.36
A2.5A
39.5A
56.90
50.15
A6.93
56.23
49.55
60.5A
5A.27
50.93
71.63
64.20
60.46
70.31
62.86
59.64
66.00
59.39
55.80
78.85
70.93
66.58
76.36
68.62
65.15
37.35
31.75
28.9A
AA.87
39.03
36.03
AA.87
39.03
36.03
57.01
50.AA
A7.36
67.41
60.14
56.68
66.72
59.62
55.85
66.00
59.39
55.80
78.85
70.93
66.58
76.36
68.62
65.15
50.02
AA. A2
A1.A3
61.0A
53.88
50.1A
61.OA
53.88
50.14
66.00
66.00
59.39
55.80
78.85
70.93
66.58
76.36
68.62
65.15
59.39
55.80
78.85
70.93
66.58
78.85
70.93
66.58
30.10
25.62
23.21
34.36
29.09
26.61
33.87
28.72
26.37
A2.91
A8.63
37.30
3A.70
AO.60
49.21
42.92
40.02
A0.13
50.23
58.91
A3.52
A2.91
51.22
47.80
57.59
50.31
47.15
50.02
AA.A2
A1.A3
61.04
53.88
50.1A
59.39
51.95
48.67
15.32
11.41
9.A6
19.00
14.53
12. A9
19.00
14.53
12.49
36.01
A6.5A
30.7A
28.25
41.68
36.30
33.62
41.37
35.94
33.11
A0.78
37.76
55.99
48.54
A5.25
54.54
47.42
44.73
50.02
AA.42
A1.A3
61.0A
53.88
50.14
59.39
51.95
48.67
26.01
20.92
18.55
31.26
26.15
23.51
31.26
26.15
23.51
A2.58
37.A0
3A.60
50.71
AA.76
A1.71
50.25
44.34
41.27
50.02
AA.A2
41.43
61.04
53.88
50.14
59.39
51.95
48.67
37.35
31.75
28.9A
AA.87
39.03
36.03
44.87
39.03
36.03
50.02
AA.A2
41.43
61.0A
53.88
50.1A
59.39
51.95
48.67
50.02
AA.A2
41. A3
61.04
53.88
50.14
61.04
53.88
50.14
25.AA
20.91
18.95
28.77
24.48
22.09
29.62
24.41
21.83
Table 2.1 (Continued)
Critical Values for Tests for Cointegration
Case 1 ----r
r
r
ak au
5
2
2
2
2
2
2
2
2
5
3
5
3
5
3
5
3
0 2
0 3
1 0
1 1
1 2
2 0
2 1
3
0
0 1
0 2
1 0
1 1
2 0
0 1
1 0
n-r
°u
5
5
5
5
5
5
5
5
3
5
4
5
4
12
52
102
------- Case 2 -------
12
52
102
------- Case 3
12
52
102
34.64
29.41
26.66
40.57
35.03
32.20
40.73
34.50
31.42
37.35
31.75
28.94
44.87
39.03
36.03
43.65
37.21
34.13
15.32
11.41
9.46
19.00
14.53
12.49
19.00
14.53
12.49
31.01
25.99
23.64
36.35
31.39
28.72
36.34
30.99
28.34
37.35
31.75
28.94
44.87
39.03
36.03
43.65
37.21
34.13
26.01
20.92
18.55
31.26
26.15
23.51
31.26
26.15
23.51
37.35
31.75
28.94
44.87
39.03
36.03
43.65
37.21
34.13
37.35
31.75
28.94
44.87
39.03
36.03
44.87
39.03
36.03
20.52
16.39
14.39
24.46
19.95
17.70
23.82
19.16
16.94
26.01
20.92
18.55
31.26
26.15
23.51
28.71
23.83
21.25
15.32
11.41
9.46
19.00
14.53
12.49
19.00
14.53
12.49
26.01
20.92
18.55
31.26
26.15
23.51
28.71
23.83
21.25
26.01
20.92
18.55
31.26
26.15
23.51
31.26
26.15
23.51
15.32
11.41
9.46
19.00
14.53
12.49
15.02
11.23
9.31
15.32
11.41
9.46
19.00
14.53
12.49
19.00
14.53
12.49
Table 4.1
Tests for Cointegration
Between Spot and Forward Exchange Rates
(Weekly Data, January 1975 - December 1989)
A
Currency
_Ho iJl<Ls*
■— a*-11-
wn ,(0 .0)
British Pound
10.95 (0.04)
10.97 (0.21)
[1
-1.001 (.004)]
-3.12 (0.03)
Swiss Franc
12.73 (0.02)
13.67 (0.08)
[1
-0.998 (.003)]
-3.33 (0.02)
German Mark
23.38 (<•01)
25.00 (<.01)
tl
-0.999 (.002)]
-3.58 «.01)
Japanese Yen
15.00 «.01)
15.02 (0.05)
[1
-1.001 (.003)]
-2.99 (0.04)
Notes:
The statistics W q ^(0>aak) were calculated using a^-Cl -1)'.
ADF
The
numbers in parentheses next to the test statistics are p-values. The estimated
A
cointegrating vector a
is normalized as (1 0 ), and the numbers in
au
A
parentheses are the standard errors for fi computed under the maintained
hypothesis that the' data are cointegrated.
Figure 3.1
Local Asymptotic Power
A. C a s e
1
o
B. C a s e 2
Power
o
c
C. Case 3
Power
o
c
c
Figure 3.2
Local Asymptotic Power
Incorrectly Specified Cointegrating Vector
A. C a s e
o
B. C a s e 2
1
o
Figure 3.3
Power in the Income/Consumtion System
Incorrectly Specified Cointegrating Vector
Figure 3.4
Local Asymptotic Power
C o n su m p tio n /ln co m e System
o
o
U n i v a r i a t e and M u l t i v a r i a t e T e s t s
c
c
- 0 .0 2 0
- 0 .0 1 6
- 0 .0 1 2
- 0 .0 0 8
-0 .0 0 4
0 .000
0 .0 0 4
0 .0 0 8
0.012
0.016
Figure 4.1
Forward-Spot Premia
@FedFRASER