where Aka1 denotes the set of all y which are not in Aka1, and P
C
K`1 Z
k0 Aka1
D
K`1 +
k0 P[Aka1]K2a1,
hence P
C
K`1 Y
k0 Aka1
D
1K2 a1
and choosing a1 such that a1aK2, we get
P
C
K`1 Y
k0 Aka1
D
1 a.
But 5
K`1 k0
Aka1My: 5K`1 k0
Cky, a1U jN. This shows that Cy, a, 5
K`1 k0
Ck a K`2
is a 1 a condence region for j.
5.4. Exact tests of joint hypotheses It is also possible to use 5.5 to derive an exact test of a linear hypothesis on
the vector
j, b m{, where bm is an m]1 subvector of b. Consider the null
hypothesis
H0: jj0 and Rbmr0, where R is a known q]m matrix with rank q, r is a known q]1 vector and
b
mbk
1
, bk
2
,2, bk
m
. The following equivalences hold: jj0
Rb mr0
H
Q
G
h2kj0h1k0, k3Km
Rb mbrb0
H
Q
G
Imhm 2
j0Imhm 1
, R
hm 2
r
b0, where Km,Mk1,k2,2,kmN, hm
i ,
hik
1
, hik
2
,2, hik
m
, i1, 2. Dening Q,
A
Im j0Im 0 R
r
B
, dm,
A
h m
1 hm
2 b
B
, we see that H0 is equivalent to Qdm0. Finally, H0 appears as a linear
hypothesis on the parameters of 5.5: H0: RKdH0 with RK,Q, 0, dH, d
m, dm, dm,h 1,k
h2k, kNKm. Once again, the standard Fisher procedure solves the problem.
J.-M. Dufour, O. Torre s Journal of Econometrics 99 2000 255}289
273
5.5. Linear regression models with AR 1 errors
We now show that model 5.1 with p1 includes as an important special case the linear regression model with AR1 errors. This model is given by
t mtut, utut~1et, t1, 2,2, ¹
with et
... N0,
p 2e and u0 given. An alternative form of this model is
t mtut~1et, t1, 2,2, ¹.
Since uttmt, t1, 2,2,¹, we have t
mHtt~1et, t2, 3,2,¹, 5.6
where mHt,mtmt~1. It is now clear that this model is a special case of 5.1. The procedures developed in the previous sections therefore apply to 5.6. In
particular, exact inference in integrated AR1 models is available. 5.6. A test on the order of an autoregression
We now turn to another kind of inference problem. We are no longer interested in inference on the components of the mean vector or autocovariance
matrix, but rather on the order of the autoregression in ARp models. There is a situation in which Theorem 3.6 and its corollary are of special interest.
Consider MXt :t3TN, a stochastic process for which we know that one of the
following representations is true: U
BXtet, where Uz11z2z22p
1
z p
1
, W
BXtlt, where Wz1t1zt2z22tp
2
z p
2
, where
et and lt are both Gaussian white noises and p1Op2 we set p1p2. Suppose we wish to test H0: MXt: t3TNARp1 against H1: MXt: t3TNAR
p2. If H0 is true, then MXt: t3TN is Markovian of order p1, and we know from Corollary 3.7 that the coecient of Xq in the ane regression of Xt on p2 leads
and p2 lags will be zero for any q such that DqtDp11,2, p2. Since the ane regression is a classical linear regression model, standard inference pro-
cedures apply. From the exposition of the procedures, it is clear that splitting the sample entails an information loss. We may then suspect the tests to lack power.
We investigate this issue in the next section.
6. Combination of tests One of the purposes of this paper is to improve the Ogawara}Hannan testing
procedure. In the previous sections, we showed that Ogawaras results can be
274 J.-M. Dufour, O. Torre
s Journal of Econometrics 99 2000 255}289
extended to a much wider class of processes than those considered in Ogawara 1951 and Hannan 1955a, b, 1956. We also showed one can use these results to
obtain nite sample inference procedures for a wide variety of econometric models. However, when we apply those, we are led to leave one half of the
sample apart, at least. In this section, we discuss methods that allow one to make use of the full sample. We also present simulation results which show our
method performs better than that of Ogawara and Hannan.
6.1. Theoretical results Consider a statistical model characterized by a family of probability laws,
parameterized by h: PMPh, h3HN. Suppose we wish to test H0: P3P0
against H1: P3PCP0. If the model is identied, which will be assumed, this amounts to test H0: h3H0 against H1: h3H1, where h3H08Ph3P0. Assume
we have m statistics ¹i, i3J,M1, 2,2,mN, that can be used for testing H0. Further assume that under H0, Ph[My: ¹iytN] is known, for all t3R, i3J.
The relation between these statistics is typically unknown or dicult to estab- lish. We wish to combine the information provided by each of those m statistics
on the true probability distribution of the model. A natural way of doing this is to proceed as follows. Using the m statistics ¹i,
we build m critical regions =iai,¹~1 i
tiai,R, where the tiais are chosen so that Ph[=iai]ai. We reject H0 with a test based an the ith statistic if y is
in =iai, or equivalently if the observed value ti of ¹i is in tiai,R. Consider the decision rule which consists in rejecting H0 when it has been rejected by at
least one of the tests based on a ¹i statistic. The rejection region corresponding to this decision rule is 6i|J=iai. This test is called an induced test of H0 see
Savin, 1984. Its size is impossible or dicult to determine since the distribution of the vector ¹1,¹2,2, ¹m is generally unknown or intractable. It is however
possible to choose the ais so that the induced test has level a. We have
Ph
C
Z i|J
= i
ai
D
+ i|J
Ph[=iai]+ i|J
ai and so we only need to choose the
ais so that they sum to a. To our knowledge, there is no criterion for choosing the
ais in a way that could be optimal in some sense. Without such a rule, we will set
aia0am for all i3J. It is dicult to compare the power of an
a level test based on a single statistic ¹
i with that of a
a level-induced test. The latter uses the information provided by the whole sample, but is obtained by combining m tests of level
am only, whereas the former has level
aam, but only exploits a subsample. In other words, with respect to power, what can be gained from the larger sample size on
which is based the induced test could be lost because the levels of the individual
J.-M. Dufour, O. Torre s Journal of Econometrics 99 2000 255}289
275
tests combined are lower e.g., am instead of a. We now present simulations
that reveal the power increase associated with combining tests. 6.2. Power simulations for AR
1 processes Let
Mt: t3¹MN, where ¹MM1, 2,2,¹N, a random process admitting an AR1 representation
t jt~1et, et
... N0, IT, t3¹M
6.1 with 0 given. For the sake of simplicity, we assume that ¹ is even with ¹2n.
Since Mt: t3¹MN is a Markov process of order 1, the results of Section 2 apply
and we know that: 1 2t, t1, 2,2, n1, are mutually independent, condi- tionally to 1, 3,2, 2n~1; 2 2t`1, t1, 2,2,n1, are mutually inde-
pendent, conditionally to 2, 4,2, 2n. If we dene two subsets of
T
, J1M2, 4,2, 2n2N and J2M3, 5,2, 2n1N, we obtain two trans- formed models of type 4.2:
t j
1 j2
t`1t~1git, t3Ji, giN0, p2gIn
i
, 6.2
where gi,git,t3Ji, i1, 2, and n1n1, n2n. In each of these two
models it is possible to test H0: jj0 at level a2, as shown in Section 4. We combine these two tests according to the procedure described in Section 6.1.
In our simulations, we proceed as follows. We consider j00, 0.5, 1 and
¹ 100. For a set
j0 of S values of j in a neighborhood of j0, we simulate a sample of size ¹ from the AR1 process 6.1. Then we form the two
subsamples yt: t3Ji, i1, 2, from which we test H0b0: bb0 in the trans- formed model 6.2, with
b0j01j20. For purposes of comparison, these tests are performed at levels 5 and 2.5. The two 2.5 level tests are
combined to give a 5 level induced test. These computations are repeated 1000 times, for each value of
j in j0. The number of rejections of H0b0 gives an estimation of the performance of the test. Results are shown in Figs. 1}6 where
the solid line represents the 5 induced test and the dashed lines } } and represent the 5 subsample-based tests.
Figs. 1}3 display the estimated power function for j0, 0.5, 1, respectively,
whereas the last three Figs. 4}6 show the dierences of rejection frequencies for j0, 0.5, 1, respectively. More precisely these dierences are computed as:
Number of rejections of H0b0 with the induced test } Number of rejections of
H0 with the test based on subsample yt: t3Ji: i1, 2. Apart from the case where
j00, the combination method leads to a power increase, relative to a 5 level test based on a subsample. When
j00, the power loss from combining is about 8 at most, which appears small. For
276 J.-M. Dufour, O. Torre
s Journal of Econometrics 99 2000 255}289
Fig. 1. Rejection frequencies of H0: j0.
Fig. 2. Rejection frequencies of H0: j0.5. J.-M. Dufour, O. Torre
s Journal of Econometrics 99 2000 255}289 277
Fig. 3. Rejection frequencies of H0: j1.
Fig. 4. Dierences of rejection frequencies for H0: j0. 278
J.-M. Dufour, O. Torre s Journal of Econometrics 99 2000 255}289
Fig. 5. Dierences of rejection frequencies for H0: j0.5.
Fig. 6. Dierences of rejection frequencies for H0: j1. J.-M. Dufour, O. Torre
s Journal of Econometrics 99 2000 255}289 279
j0O0, it is important to note that the values j and j~1 yield the same value of b in 6.2. For example j0.5 and j2.0 both yield b0.4. In other words,
unless we impose restrictions such as DjD1 or DjD1, the value of b does not
completely identify j. This explains the presence of the mirror peak at j2
Fig. 2. 7. Conclusion
In this paper we proposed a method allowing one to make nite-sample inference on the parameters of autoregressive models. This was made possible
by special properties of Markov processes. The conditions under which such results hold are very mild since their demonstrations only require the existence
of density functions. In particular, they are general enough to be applied to multivariate and possibly non stationary andor non-Gaussian processes. How-
ever, with the addition of conditional stationarity and normality assumptions, we were able to use these properties to derive exact tests and condence regions
on the parameters of AR1 models. In order to apply our procedure, it is necessary to split the sample as two subsets of observations. Our simulations in
the case of a pure AR1 model showed that a combination of separate inference results based on these subsamples generally leads to an improvement in the
performance of the procedure.
Our method displays several attractive features. First, since it is exact, it controls the probability of making a type I error. Second, it is readily applicable
to a wide range of econometric specications of AR1 models. In particular, it can be used to deal with random walk models, models with a deterministic mean
expressed as a linear combination of exogenous variables, including polynomial deterministic trends, etc. Third, the critical regions are built from standard
distributions which, unlike most asymptotic procedures, do not change with the sample size andor model specication. Finally, Monte Carlo experiments show
that it has good power properties. For those reasons, we think that our procedure should be considered as a good alternative to asymptotic inference
methods.
In Section 6, we argued that simulations of power functions were necessary because we could not say a priori whether the combination method yields more
power. Indeed, on the one side we make use of the whole sample when combining, but on the other side we must lower the bound on the probability of
making a type I error the level in each of the tests we combine. The former should increase the performance of the procedure whereas the latter
should decrease it. The method is easily transposable to higher-order autoreg- ressive models and it appears quite plausible the same eect will take place in
more general processes. It would certainly be of interest to study this issue further.
280 J.-M. Dufour, O. Torre
s Journal of Econometrics 99 2000 255}289
Of course, the nite-sample validity of the t- and F-type tests described in Sections 4 and 5 remain limited to models with Gaussian errors. As usual, these
procedures will however be asymptotically valid under weaker distributional assumptions. Further, it is of interest to remember that the general theorems on
Markovian processes given in Section 3 hold without parametric distributional assumptions. In particular, the conditional independence and truncation prop-
erties do not at all require the Gaussian distributional assumption, hence opening the way to distribution-free procedures. Similarly, the test combination
technique described in Section 6, which is based on the Boole}Bonferroni inequality, is by no way restricted to parametric models. For example, the latter
might be applied to combine distribution-free tests or bootstrap tests see Nankervis and Savin, 1996 which accommodate more easily non-Gaussian
distributions. Such extensions go however beyond the scope of the present paper.
Acknowledgements The authors thank FreHdeHric Jouneau, Pierre Perron, Eugene Savin, an anony-
mous referee and the Editor Peter Robinson for several useful comments. An earlier version of this paper was presented at the Econometric Society European
Meeting Toulouse, August 1997. This work was supported by the Canadian Network of Centres of Excellence [program on Mathematics of Information
Technology and Complex Systems
MITACS], the Canada Council for the Arts Killam Fellowship, the Natural Sciences and Engineering Research Council of
Canada, the Social Sciences and Humanities Research Council of Canada, and the Fonds FCAR Government of QueHbec.
Appendix A. Proofs A.1. Proof of Theorem 3.1
We must show that fX
p`1
,
2p`12
,X
np`1
A
1,p
n t1
fX
tp`1
A
1,p
. The following equality is always true:
fX
p`1
,
2p`12
,X
np`1
A
1,p
fX
p`1
A
1,p
n t2
fX
tp`1
A
1,p
,X
p`1
,
2p`12
,X
t~1p`1
. A.1
J.-M. Dufour, O. Torre s Journal of Econometrics 99 2000 255}289
281
Consider the tth term of the product in A.1 for t2: fX
tp`1
A
1,p
,X
p`1
,
2p`12
,X
t~1p`1
fX
tp`1
X
1
,
212
,X
tp`1~1
,A
t`1,p
fX
tp`1
,A
t`1,p
X
1
,
212
,X
tp`1~1
fA
t`1,p
X
1
,
212
,X
tp`1~1
. A.2
The numerator in A.2 can be written fX
tp`1
,A
t`1,p
X
1
,
21 2
,X
tp`1~1
P
2
P
fX
tp`1
,
212
,X
np`1`p
X
1
,
212
,X
tp`1~1
dxt`1p`1,2p`12,xnp`1
P
2
P
np`1`p stp`1
fX
s
X
1
,
212
,X
s~1
dxt`1p`1,2p`12,xnp`1
P
2
P
np`1`p stp`1
fX
s
X
s~p
,
212
,X
s~1
dxt`1p`1,2p`12, xnp`1, where the last identity follows from the Markovian property Mp. Set
g1at`1,p,xtp`1,fX
tp`1
,A
t`1,p
X
1
,
21 2
,X
tp`1~1
at`1,p,xtp`1. Similarly, we can write the denominator of A.2 as
fA
t`1,p
X
1
,
212
,X
tp`1~1
P
2
P
np`1`p stp`1
fX
s
X
s~p
,
212
,X
s~1
dxtp`1,2p`12, xnp`1 and we denote g2at`1,p,fA
t`1,p
X
1
,
212
,X
tp`1~1
at`1,p. Clearly, neither g1at`1,p, xtp`1 nor g2at`1,p depends on Xp`1,2p`12, Xt~1p`1.
Therefore these variables do not enter the ratio A.2 and we may write the tth term of product A.1 for t2 as
fX
tp`1
A
1,p
,X
p`1
,
2p`12
,X
t~1p`1
fX
tp`1
A
1,p
. Since this is true for any t1, 2,2, n, we can factor the conditional density as
fX
p`1 2p`1 2
,X
np`1
A
1,p
n t1
fX
tp`1
A
1,p
which yields the result to be proved. Q.E.D
A.2. Proof of Theorem 3.2 From Theorem 3.1, Xp`1,X2p`1,2, Xnp`1 are mutually independent con-
ditionally on A1,p, hence fX
tp`1
A
1,p
fX
tp`1
X
1
,
21 2
,X
tp`1~1
,X
tp`1`1
,
212
,X
n`1p`1
282 J.-M. Dufour, O. Torre
s Journal of Econometrics 99 2000 255}289
fX
tp`1
,
212
,X
n`1p`1
X
1
,
21 2
,X
tp`1~1
fX
tp`1`1
,
21 2
,X
n`1p`1
X
1
,
212
,X
tp`1~1
fX
tp`1
,
21 2
,X
n`1p`1
X
1
,
212
,X
tp`1~1
: fX
tp`1
,
21 2
,X
n`1p`1
X
1
,
212
,X
tp`1~1
dxtp`1 n`1p`1
stp`1 fX
s
X
1
,
212
,X
s~1
: n`1p`1
stp`1 fX
s
X
1
,
21 2
,X
s~1
dxtp`1 n`1p`1
stp`1 fX
s
X
s~p
,
21 2
,X
s~1
:n`1p`1 stp`1
fX
s
X
s~p
,
212
,X
s~1
dxtp`1 p,
A.3 where the last equality is derived using the Markovian property Mp. The
product of conditional densities in the numerator of A.3 can be splitted as n`1p`1
stp`1 fX
s
B
s,p
G1]G2, where G1,
t`1p`1~1 stp`1
fX
s
X
s~p
,
21 2
,X
s~1
, G2,
n`1p`1 st`1p`1
fX
s
X
s~p
,
212
,X
s~1
. Clearly, G2 does not depend on Xtp`1. Therefore, the ratio A.3 simplies as
fX
tp`1
A
1,p
G1 :G1 dxtp`1
. A.4
Now, due to the Markovian property Mp, any of the conditional densities in
the product
G1 can be written as fX
s
X
s~p
,
212
,X
s~1
fX
s
X
tp`1~p
,
212
,X
s~1
, stp1, tp11,2,t1p1. Therefore, it is easy to see that
G1 t`1p`1~1
stp`1 fX
s
X
tp`1~p
,
21 2
,X
s~1
fX
tp`1
,
212
,X
t`1p`1~1
X
tp`1~p
,
212
,X
tp`1~1
. Hence,
P
G1 dxtp`1fX
tp`1`1
,
21 2
,X
t`1p`1~1
X
tp`1~p
,
21 2
,X
tp`1~1
and fX
tp`1
A
1,p
G1 :G1 dxtp`1
fX
tp`1
X
tp`1~p
,
21 2
,X
tp`1~1
,X
tp`1`1
,
212
,X
t`1p`1~1
.
J.-M. Dufour, O. Torre s Journal of Econometrics 99 2000 255}289
283
Since Xtp`1`1Xt`1p`1~p, we can use the notation of Section 3.1 to write fX
tp`1
A
1,p
fX
tp`1
B
tp`1,p
,B
t`1p`1
which is the desired result. Q.E.D
A.3. Proof of Theorem 3.6 We need to show that fX
t
B
t,q
,B
t`q`1,q
does not depend on Xt~q and Xt`q, for qp1, p2,2, q. We have:
fX
t
B
t`q`1,q
,B
t,q
fX
t
,B
t`q`1,q
B
t,q
fB
t`q`1,q
B
t,q
fX
t
,B
t`q`1,q
B
t,q
: fX
t
,B
t`q`1,q
B
t,q
dxt .
Now, using the fact that MXt: t3TN is Markovian of order p, the numerator of
this last term can be written fX
t
,B
t`q`1,q
B
t,q
fX
t
,
21 2
,X
t`q
B
t,q
t`q st
fX
s
B
s,p
so that fX
t
B
t`q`1,q
,B
t,q
t`q st
fX
s
B
s,p
: t`q
st fX
s
B
s,p
dxt t`q
st fX
s
B
s,p
t`q st`p`1
fX
s
B
s,p
: t`p
qt fX
q
B
q,p
dxt t`p
st fX
s
B
s,p
: t`p
st fX
s
B
s,p
dxt .
It is easy to see that the variables Xs with tqstp1 and tp
1stq do not appear in the latter expression. Q.E.D
A.4. Proof of Theorem 3.8 Let
Mt: t3TN be a Gaussian process having the same rst and second order moments as
MXt: t3TN. Then Mt: t3TN must also satisfy the condition in the theorem
to 1,212, t~p~1Dt~p,212,t~1, ∀tp1,
which is
equivalent to
the Markovian
condition fY
t
Y
1
,
21 2
,Y
t~1
fY
t
Y
t~p
,
21 2
,Y
t~1
, ∀tp1, since Mt: t3TN is Gaussian. From Theorem 3.1, p`1
,2p`12,np`1 are mutually independent, conditional on AY1,p, where A
Y 1,p
is dened like A1,p in Section 3.1 with X replaced by . Using the normality of
Mt: t3TN, this is equivalent to tp`1o
sp`1D A
Y 1,p
, ∀t, s such that 1t, sn, tOs.
This is a condition on the rst and second order moments of Mt: t3TN, which
must also be satised by the rst and second order moments of MXt: t3TN.
Hence, if A1,p denotes the set of X variables as dened in Section 3.1, Xtp`1oXsp`1DA1,p, ∀t, s such that 1t, sn, tOs.
Q.E.D
284 J.-M. Dufour, O. Torre
s Journal of Econometrics 99 2000 255}289
A.5. Proof of Theorem 3.9 Let
Mt: t3TN be a Gaussian process having the same rst and second order moments as
MXt: t3TN. From the proof of Theorem 3.8, we know that Mt: t3TN must also satisfy
fY
t
Y
1
,
21 2
,Y
t~1
fY
t
Y
t~p
,
21 2
,Y
t~1
, ∀tp1.
Then, from Theorem 3.2, we have fY
tp`1
A
Y 1,p
fY
tp`1
B
Y t`1p`1,p
,B
Y tp`1,p
, ∀t such that 1tn,
where for any s, B Ys,p,s~p,212,s~1. Since Mt: t3TN is Gaussian, this
condition is equivalent to tp`1o
B Y
sp`1,p D BYt`1p`1,p,BYtp`1,p
for all t1 and s1 such that sOt and sOt1. Since this is condition on the rst and second order moments of
Mt: t3TN, it must also be satised by those of
MXt: t3TN. Q.E.D A.6. Proof of Theorem 4.1
EL[Xt DBt`p`1,p, Bt,p]EL[Xt DBt`p`1,p, BHt,p] is the ane regression of Xt on Bt`p`1,p, Bt,p, where BHl,p,Xl~p, Xl~p`1,2, Xl~1. The matrix of
the coecients of this regression is given by W12W~1 22
, where W12,cov[Xt, Bt`p`1,p, BHt,p],
W22,V[Bt`p`1,p, BHt,p]. W22 is non-singular by the regularity assumption. We partition these matrices in
the following way: W12,C1 C2,
W22,
A
A11 A12 A21 A22
B
, where
A11,VBt`p`1,p, A22,VBHt,p, A21A12,cov Bt`p`1,p, BHt,p, C1,covXt, Bt`p`1,p, C2,covXt, BHt,p.
Since MXt: t3TN is weakly stationary, C1C2,C and A11A22,A1. We
next show that A12A21, i.e., A12 is symmetric. The i, jth element of this matrix is
covXt`p`1~i, Xt~p`j~1ct`p`1~i~t`p~j`1c2p`1~i`j,
J.-M. Dufour, O. Torre s Journal of Econometrics 99 2000 255}289
285
where cs~t,covXs, Xt, and its j, ith element is
covXt`p`1~j, Xt~p`i~1ct`p`1~j~t`p~i`1c2p`1~j`i. These two terms are identical and consequently A12A12A21,A2. The
vector P whose components are the coecients of Xt`k and Xt~k, 1kp, in the ane regression of Xt on Bt`p`1,p, BHt,p is given by
P C
C
A
A1 A2 A2 A1
B
~1 .
Dene P1 and P2, the two 1]p subvectors of P whose elements are the coecients of the variables in Bt`p`1,p and in BHt,p, respectively. Then
CP1A1P2A2 CP1A2P2A1
H
N
G
A1P1P2A2P2P10, A2P1P2A1P2P10
which is equivalent to W22
A
P1P2 P2P1
B
0. Since the variance}covariance matrix W22 is non-singular, we must have
P1P2. Q.E.D. Appendix B. Coe
7cients of two-sided autoregressions for AR 1 processes
The model is t
t~1ut, t1, 2,2, n, u
u1,2, unN0, p2uIn, with 0 given. Rewriting tt0+t~1
i0 iut~i and taking expectations, we
get Ett0. The mean deviation process MXt,tEt: t1, 2,2, nN satises the autoregression XtXt~1ut.
B.1. Computation of xrst-order moments Dene
W12,cov[2t, 2t`1, 2t~1] and W22,V[2t`1, 2t~1]. From the denition of
MXt: t1, 2,2,nN, we have Xt+t~1 i0
iut~i and
286 J.-M. Dufour, O. Torre
s Journal of Econometrics 99 2000 255}289
EXt0, EX2tp2u+t~1 i0
2i. Furthermore, the autocovariances are cov2t`1, 2tEX2t`1X2tp2u
2t~1 +
i0 2i,
cov2t, 2t~1EX2tX2t~1p2u 2t~2
+ i0
2i, cov2t`1, 2t~1EX2t`1X2t~1p2u2
2t~2 +
i0 2i,
hence W12p2u
A
2t~1 +
i0 2i,
2t~2 +
i0 2i
B
, W22p2u
A
2t +
i0 2i
2 2t~2
+ i0
2i 2
2t~2 +
i0 2i
2t~2 +
i0 2i
B
.
B.2. The azne regression of 2t on 2t`1, 2t~1 when DDO1 In general we have:
EL[2t D 2t`1 2t~1]E2tW12W~1 22
A
2t`1 E2t`1
2t~1 E2t~1
B
. Using the fact that, for
DDO1, k
+ i0
2i 1
2k`1 1
2 ,
we obtain the following expressions: W12
p 2
u 1
2 1
4t, 14t~2, W22
p 2
u 1
2
A
1 4t`2
214t~2 214t~2 14t~2
B
, hence
EL[2t D 2t`1, Y2t~1]E2t 1
2 1 1
A
2t`1 E2t`1
2t~1 E2t~1
B
a b2t`12t~1,
J.-M. Dufour, O. Torre s Journal of Econometrics 99 2000 255}289
287
where aE2tb[E2t`1E2t~1] and b12. Since for all t
0, EtkEt~k, k0, 1, 2, t, we have a0. B.3. The azne regression of 2t on 2t`1, 2t~1 when DD1
When DD1, we have
W12p2u2t, 2t1, W22p2u
A
2t1 2t1
2t1 2t1
B
, hence
EL[2t D 2t`1, 2t~1]2 2t`12t~1 [1
2] 2t`1
2t~1 .
Note that from the derivations in the case where DDO1, a0 irrespective to
the value of . In any case, the residual variance is V[2tEL[2t D 2t`1 2t~1]]V2tW12W~1
22 W
12 p
2 u
1 2
, 3
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