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Econometrica, Vol. 80, No. 2 (March, 2012), 821–861
STRICT STATIONARITY TESTING AND ESTIMATION OF
EXPLOSIVE AND STATIONARY GENERALIZED
AUTOREGRESSIVE CONDITIONAL
HETEROSCEDASTICITY MODELS
CHRISTIAN FRANCQ
University Lille 3 (EQUIPPE), 59653 Villeneuve d’Ascq cedex, France and
CREST
JEAN-MICHEL ZAKOÏAN
CREST, 92245 Malakoff Cedex, France and University Lille 3 (EQUIPPE)
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Econometrica, Vol. 80, No. 2 (March, 2012), 821–861
STRICT STATIONARITY TESTING AND ESTIMATION OF
EXPLOSIVE AND STATIONARY GENERALIZED
AUTOREGRESSIVE CONDITIONAL
HETEROSCEDASTICITY MODELS
BY CHRISTIAN FRANCQ AND JEAN-MICHEL ZAKOÏAN1
This paper studies the asymptotic properties of the quasi-maximum likelihood estimator of (generalized autoregressive conditional heteroscedasticity) GARCH(1 1)
models without strict stationarity constraints and considers applications to testing problems. The estimator is unrestricted in the sense that the value of the intercept, which
cannot be consistently estimated in the explosive case, is not fixed. A specific behavior
of the estimator of the GARCH coefficients is obtained at the boundary of the stationarity region, but, except for the intercept, this estimator remains consistent and
asymptotically normal in every situation. The asymptotic variance is different in the
stationary and nonstationary situations, but is consistently estimated with the same estimator in both cases. Tests of strict stationarity and nonstationarity are proposed. The
tests developed for the classical GARCH(1 1) model are able to detect nonstationarity
in more general GARCH models. A numerical illustration based on stock indices and
individual stock returns is proposed.
KEYWORDS: GARCH model, inconsistency of estimators, nonstationarity, quasimaximum likelihood estimation.
1. INTRODUCTION
TESTING FOR STRICT STATIONARITY is an important issue in the context of financial time series. A standard assumption is that the prices are nonstationary
while the returns (or log returns) are stationary. Numerous econometric tools,
such as the unit root tests, have been introduced for testing the nonstationarity
of prices. For the log returns, the most widely used models are arguably generalized autoregressive conditional heteroscedasticity (GARCH) models introduced by Engle (1982) and Bollerslev (1986). No econometric tools are available for testing strict stationarity in the GARCH framework. The main aim
of this paper is to develop such tools. The problem is nonstandard because,
contrary to stationarity in linear time series models, which solely depends on
the lag polynomials, the strict stationarity condition for GARCH models has a
nonexplicit form, involving the distribution of the underlying independent and
identically distributed (i.i.d.) sequence.
The asymptotic properties of the quasi-maximum likelihood estimator
(QMLE) for classical GARCH models have been extensively studied. Lumsdaine (1996) proved that the local QMLE is consistent and asymptotically
1
We are most thankful to a co-editor and three referees for their constructive comments and
suggestions. We would like to thank Youri A. Davydov, Siegfried Hörmann, Lajos Horváth,
and several seminar and conference audiences for helpful discussions. We are also grateful
to the Agence Nationale de la Recherche (ANR), which supported this work via the project
ECONOM&RISK (ANR 2010 blanc 1804 03).
© 2012 The Econometric Society
DOI: 10.3982/ECTA9405
822
C. FRANCQ AND J.-M. ZAKOÏAN
normal (CAN) in the GARCH(1 1) case. These results were extended to the
GARCH(p q) model, under less stringent conditions, by Berkes, Horváth,
and Kokoszka (2003) and Francq and Zakoïan (2004) (see also the references
therein). For valid inference based on those results, strict stationarity must
hold. Thus, from the point of view of the validity of the asymptotic results for
the QMLE, strict stationarity testing in GARCH models is also an important
issue. Surprisingly, this issue has not been addressed in the literature, to the
best of our knowledge.
1.1. Modes of Divergence in the Nonstationary Case
The complexity of the statistical problem arises from the specificities of the
probabilistic framework, even for the simplest GARCH model. To fix ideas,
consider the GARCH(1 1) model given by
t = ht ηt t = 1 2 (1.1)
ht = ω0 + α0 2t−1 + β0 ht−1 with initial values 0 and h0 ≥ 0, where ω0 > 0, α0 β0 ≥ 0, and (ηt ) is a sequence of independent and identically distributed (i.i.d.) variables such that
Eη1 = 0, Eη21 = 1, and P(η21 = 1) < 1. The top Lyapunov exponent associated
to this model (see Bougerol and Picard (1992)) is given by
γ0 = E log a0 (η1 )
a0 (x) = α0 x2 + β0 The necessary and sufficient condition for the existence of a strictly stationary
solution to (1.1) is (by Nelson (1990))
(1.2)
γ0 < 0
More precisely, if (1.2) holds, we have
(1.3)
ht − ht∞ → 0
almost surely (a.s.)
where
(1.4)
ht∞ = lim ↑ htn n→∞
htn = ω0 1 +
as t → ∞
n−1
a0 (ηt−1 ) · · · a0 (ηt−k ) k=1
In particular, the integrated GARCH model, obtained when α0 + β0 = 1, satisfies the condition (1.2).2 Let us now turn to the nonstationary case, for which
it is necessary to consider γ0 > 0 and γ0 = 0 separately. Under the assumption
(1.5)
γ0 > 0
2
Berkes, Horváth, and Kokoszka (2005) studied the asymptotic behavior of the GARCH(1 1)
process when α0 + β0 approaches 1 as the sample size increases.
TESTING AND ESTIMATION OF GARCH MODELS
823
ht → ∞ almost surely as t → ∞, as shown by Nelson (1990). In this case, the
increasing sequence htn goes to infinity almost surely as n → ∞, by the Cauchy
root test. The case γ0 = 0 is much more intricate. By the Chung–Fuchs theorem, it can be seen that htn goes to infinity almost surely as n → ∞. However, the a.s. convergence of ht to infinity may not hold when γ0 = 0. Actually,
Klüppelberg, Lindner, and Maller (2004) (see also Goldie and Maller (2000))
showed that
(1.6)
when
γ0 = 0
ht → ∞ in probability
instead of almost surely in the case γ0 > 0.3 The astonishing difficulties encountered in the case γ0 = 0 are related to the fact that the sequence ht =
htt + a0 (ηt−1 ) · · · a0 (η0 )h0 does not increase with t.
1.2. The Econometric Problem
We consider the QMLE, which is the commonly used estimator for autoregressive conditional heteroscedasticity (ARCH) models. Denote by θ =
(ω α β) the GARCH(1 1) parameter and define the QMLE as any measurable solution of
1
t (θ)
n t=1
n
(1.7)
θ̂n = (ω̂n α̂n β̂n ) = arg min
θ∈Θ
t (θ) =
2t
+log σt2 (θ)
σ (θ)
2
t
where Θ is a compact subset of (0 ∞)3 that contains the true value θ0 =
2
(ω0 α0 β0 ) and where σt2 (θ) = ω + α2t−1 + βσt−1
(θ) for t = 1 n (with ini2
2
tial values for 0 and σ0 (θ)). The rescaled residuals are defined by η̂t = ηt (θ̂n ),
where ηt (θ) = t /σt (θ) for t = 1 n.
To construct a test of the strict stationarity assumption, we establish the
asymptotic distribution of the statistic
1
γ̂n =
log(α̂n η̂2t + β̂n )
n t=1
n
(1.8)
To study the asymptotic properties of the test, it is necessary to analyze the
asymptotic behavior of the QMLE when γ0 ≥ 0 Jensen and Rahbek (2004a,
2004b) were the first researchers to establish an asymptotic theory for estimators of nonstationary GARCH.4 However, they only considered a constrained
QMLE of (α0 β0 ) (in the sense that the value of the intercept is fixed) that is
consistent in the nonstationary case, but is inconsistent in the stationary case.
3
Klüppelberg, Lindner, and Maller (2004) noted that the arguments given by Nelson (1990)
for the a.s. convergence fail when γ0 = 0.
4
See Linton, Pan, and Wang (2010) for extensions in the case of non-i.i.d. errors.
824
C. FRANCQ AND J.-M. ZAKOÏAN
Instead, we use the standard (unconstrained) QMLE. We complete the well
known results in the case γ0 < 0 by establishing the consistency and asymptotic
normality of the QMLE of (α0 β0 ), the only components that matter for our
testing problem, in the cases γ0 > 0 and γ0 = 0. When γ0 > 0, the estimator
(α̂n β̂n ) is shown to be strongly consistent and it turns out that its asymptotic
distribution is simpler than in the strict stationarity case, and is given by
√
d
n(α̂n − α0 β̂n − β0 ) → N {0 (κη − 1)I −1 } as n → ∞
d
where → stands for the convergence in distribution, κη = Eη41 , and I is a matrix which has an explicit form and does not depend on ω0 . When γ0 = 0, the
QMLE of (α0 β0 ) is shown to be weakly consistent with the same asymptotic
normal distribution as in the case γ0 > 0. The asymptotic variances of (α̂n β̂n )
when γ0 ≥ 0 and when γ0 < 0 do not coincide, but we propose an estimator
which is consistent in both situations. This is in accordance with similar results for autoregressive models with random coefficients derived by Aue and
Horváth (2011).
Even if the QMLE of (α0 β0 ) is consistent in every situation, we show that
the QMLE of ω0 is only consistent in the stationary case. For this reason, it is
important to test the sign of γ0 .
The rest of the paper is organized as follows. Section 2 is devoted to the
asymptotic properties of the QMLE. In Section 3, we first consider the problem of testing the value of (α0 β0 ) without any stationarity restriction. Then
we consider strict stationarity testing. The asymptotic distributions of two tests
are studied when the null assumption is either the stationarity or the nonstationarity. We also consider testing stationarity in more general GARCH-type
models. Numerical illustrations are provided in Section 5. In particular, the stationarity of 11 major stock returns is analyzed. Section 6 concludes. Proofs and
complementary results are collected in the Appendix. Replication files may be
found in the online supplement (Francq and Zakoïan (2012)).
2. ASYMPTOTIC PROPERTIES OF THE QMLE
In this paper, we consider the standard QMLE, which is the commonly used
estimator for GARCH models.
2.1. Consistency and Asymptotic Normality of α̂n and β̂n
The following result completes those already established in the stationary
case, which we recall for convenience. The asymptotic distribution in the case
γ0 = 0 is treated separately. To handle initial values, we introduce the following notation. For any asymptotically stationary process (Xt )t≥0 , let E∞ (Xt ) =
limt→∞ E(Xt ) provided this limit exists.5
5
For instance, for the process (t ), we have E∞ (2t ) = ω0 /(1 − α0 − β0 ) when α0 + β0 < 1.
825
TESTING AND ESTIMATION OF GARCH MODELS
THEOREM 2.1: For the GARCH(1 1) model (1.1), the QMLE defined in (1.7)
satisfies the following properties.
(i) When γ0 < 0 for Θ such that ∀θ ∈ Θ, β < 1, then
α̂n → α0 β̂n → β0
and ω̂n → ω0
a.s.
as n → ∞
(ii) When γ0 > 0, if P(η1 = 0) = 0, then
α̂n → α0
and β̂n → β0
a.s. as n → ∞
(iii) When γ0 = 0, if P(η1 = 0) = 0, for Θ such that ∀θ ∈ Θ, β < 1/a0 (η1 )
−1
p
for some p > 1, then
α̂n → α0
and β̂n → β0
in probability
as
n → ∞
◦
(iv) When γ0 < 0, κη = Eη41 ∈ (1 ∞), θ0 belongs to the interior Θ of Θ, and
for Θ such that ∀θ ∈ Θ, β < 1, then
(2.1)
√
d
n(θ̂n − θ0 ) → N {0 (κη − 1)J −1 }
and
(2.2)
J = E∞
as n → ∞
1 ∂σt2 ∂σt2
(θ0 ) h2t ∂θ ∂θ
◦
(v) When γ0 > 0, κη ∈ (1 ∞), E| log η21 | < ∞, and θ0 ∈ Θ, then
(2.3)
where
√
d
n(α̂n − α0 β̂n − β0 ) → N {0 (κη − 1)I −1 }
as n → ∞
⎞
1
ν1
⎜
α0 β0 (1 − ν1 ) ⎟
α20
⎟
I =⎜
⎝
⎠
(1 + ν1 )ν2
ν1
α0 β0 (1 − ν1 ) β20 (1 − ν1 )(1 − ν2 )
i
β0
with νi = E
α0 η21 + β0
⎛
To obtain the asymptotic distribution of (α̂n β̂n ) in the case γ0 = 0, we need
an additional assumption on the distribution of η2t . Let Zt = α0 η2t + β0 Note
that γ0 = E log Zt = 0 entails EZt ≥ 1 by Jensen’s inequality and, thus, in view
of the independence, E(1 + Zt−1 + Zt−1 Zt−2 + · · · + Zt−1 · · · Z1 ) ≥ t. We introduce the following assumption.
826
C. FRANCQ AND J.-M. ZAKOÏAN
ASSUMPTION A: When t tends to infinity,
1
1
=o √ E
1 + Z1 + Z1 Z2 + · · · + Z1 · · · Zt−1
t
Note that Assumption A is obviously satisfied when ηt = ±1 with equal probabilities and for α0 + β0 = 1, since the expectation is then equal to 1/t. It can
√
be shown that the expectation involved in Assumption A is of order (log t)/ t
for any distribution such that E| log Z1 |2 < ∞ (details are available from the
authors). This assumption is used to prove that
1 1
→0
√
n t=1 ht
n
(2.4)
as
n→∞
in L1 when γ0 = 0.6
THEOREM 2.2: Suppose that the assumptions of Theorem 2.1(iii) hold, in par◦
ticular, γ0 = 0. Then if θ0 ∈ Θ, κη ∈ (1 ∞), E| log η21 | < ∞, and Assumption A
is satisfied, the QMLE (α̂n β̂n ) is asymptotically normal and its asymptotic distribution is given by (2.3).
2.2. Estimating the Asymptotic Variance of (α̂n β̂n ) Without
Assuming Stationarity
In view of (2.1) and (2.2), when γ0 < 0, the asymptotic distribution of the
QMLE (α̂n β̂n ) of (α0 β0 ) is given by
(2.5)
√
d
n(α̂n − α0 β̂n − β0 ) → N {0 (κη − 1)I∗−1 }
as n → ∞
with
(2.6)
−1
I∗ = Jαβαβ − Jαβω Jωω
Jωαβ
and
1 ∂σt2 ∂σt2
(θ0 ) Jωω = E∞ 2
ht ∂ω ∂ω
∂σt2
1 ∂σt2
(θ0 ) Jαβαβ = E∞ 2
ht ∂(α β) ∂(α β)
6
In the case γ0 > 0, (2.4) holds a.s. from Proposition A.1(i) and Assumption A is not needed.
TESTING AND ESTIMATION OF GARCH MODELS
and
Jωαβ = Jαβω
= E∞
827
1 ∂σt2 ∂σt2
(θ
)
0
h2t ∂ω ∂(α β)
Letting
∂σt2
∂σt2
1 1
(θ̂n )
Jαβαβ =
n t=1 σt4 (θ̂n ) ∂(α β) ∂(α β)
n
and defining Jαβω Jωω , and Jωαβ accordingly, it can be shown that
−1 I∗ = Jαβαβ − Jαβω Jωω
Jωαβ
is a strongly consistent estimator of I∗ in the stationary case γ0 < 0. The following result shows that this estimator is also a consistent estimator of the
asymptotic variance of (α̂n β̂n ) in the nonstationary case γ0 ≥ 0.
THEOREM 2.3: Let the assumptions (i)–(iii) of Theorem 2.1 hold, assume κη ∈
n
(1 ∞), and let κ̂η = n−1 t=1 η̂4t , where η̂t = t /σt (θ̂n ).
I∗ → I∗ a.s. as n → ∞.
(i) When γ0 < 0, we have κ̂η → κη and (ii) When γ0 > 0, we have κ̂η → κη and I∗ → I a.s.
(iii) When γ0 = 0, we have κ̂η → κη and if A is satisfied, I∗ → I in probability.
In any case, (κ̂η − 1)
I∗−1 is a consistent estimator of the asymptotic variance of the
QMLE of (α0 β0 ).
2.3. Inconsistency of ω̂n When γ0 > 0
The previous results do not give any insight on the asymptotic behavior of
the QMLE of ω0 . From the proof of Theorem 2.1, it can be shown that the
log-likelihood is completely flat in the direction where (α0 β0 ) is fixed and ω0
varies. More precisely, we have
n
∂
d
0
0 0
λn
n =
t (θ0 ) → N 0 (κη − 1)
n
0 n−1/2 I2
0 I
∂θ
t=1
for any sequence λn tending to zero as n → ∞.
Thus, in general, as noted by Jensen and Rahbek (2004b), there is no consistent estimator of ω0 . Indeed, we have the following result.
PROPOSITION 2.1: Consider the GARCH(1 1) model (1.1) with ηt ∼
N (0 1). Assume that Θ contains two arbitrarily close points θ1 = (ω1 α1 β1 )
and θ1∗ = (ω∗1 α1 β1 ) such that E log(α1 η2t + β1 ) > 0 and ω1 = ω∗1 . There exists
no consistent estimator of θ0 ∈ Θ.
828
C. FRANCQ AND J.-M. ZAKOÏAN
The inconsistency of ω̂n is illustrated via simulations in Francq and Zakoïan
(2010, p. 150).
2.4. A Constrained QMLE of (α0 β0 )
Whereas the asymptotic behavior of the QMLE (α̂n β̂n ) is independent of
ω0 when γ0 > 0 and the QMLE of ω0 is generally inconsistent in view of Proposition 2.1, it seems natural to avoid estimating ω0 . To this aim, a constrained
QMLE of (α0 β0 ), in which the first component of θ is fixed to an arbitrary
value ω, can be introduced. The estimator
1
t (ω α β)
(α̂ (ω) β̂ (ω)) = arg min
(αβ)∈Θαβ n
t=1
n
(2.7)
c
n
c
n
was studied by Jensen and Rahbek (2004b). They proved that when γ0 > 0,
(2.3) continues to hold when the global QMLE (α̂n β̂n ) is replaced by the local and constrained QMLE (α̂cn (ω) β̂cn (ω)) In Appendix A.3, we prove that,
under the assumptions of Theorem 2.2, in particular, γ0 = 0
(2.8)
√
d
n(α̂cn (ω) − α0 β̂cn (ω) − β0 ) → N {0 (κη − 1)I −1 }
as n → ∞
However, the next result shows that the restricted QMLE of (α0 β0 ) is generally inconsistent in the stationary case.
PROPOSITION 2.2: Let (t ) be a stationary solution of the GARCH(1 1)
model with parameters ω0 , α0 , and β0 , such that E4t < ∞. If ω = ω0 , then
(2.9)
(α̂cn (ω) β̂cn (ω)) does not converge in probability to
(α0 β0 )
On the contrary, Theorem 2.1 and Theorem 2.2 show that
(2.10)
the QMLE of (α0 β0 ) is always CAN
(under Assumption A when γ0 = 0).
3. TESTING
The consequence of Theorem 2.3, from a practical point of view, is extremely
important. It means that we can get confidence intervals or tests for (α0 β0 )
without assuming stationarity/nonstationarity.
Before considering strict stationarity testing, we start with tests on the
GARCH parameters.
829
TESTING AND ESTIMATION OF GARCH MODELS
3.1. Testing the GARCH Coefficients
First consider a testing problem of the form
(3.1)
H0 : aα0 + bβ0 ≤ c
against H1 : aα0 + bβ0 > c
where a b, and c are given numbers. A case of particular interest is a = b =
c = 1, because E2t < ∞ if and only if α0 + β0 < 1. Note, however, that we
do not impose any constraint on a b, and c, so some values of θ0 satisfying
H0 may correspond to nonstationary GARCH models. A direct consequence
of Theorems 2.1–2.3 is the following result, in which denotes the N (0 1)
cumulative distribution function. Let α ∈ (0 1).
◦
COROLLARY 3.1: Assume that θ0 ∈ Θ and the assumptions of Theorem 2.3
hold. For the testing problem (3.1), the test defined by the critical region
(3.2)
α∗
α∗
n
C = T
√
n(aα̂n + bβ̂n − c)
:= > −1 (1 − α)
(κ̂η − 1)(a b)
I∗−1 (a b)
has the asymptotic significance level α and is consistent.
The following result, showing that no consistent test exists for ω0 , is used to
prove the inconsistency of any estimator of this parameter.
PROPOSITION 3.1: Consider the GARCH(1 1) model (1.1) with ηt ∼
N (0 1). Let θ1 = (ω1 α1 β1 ) and θ1∗ = (ω∗1 α1 β1 ) be two points of Θ such
that E log(α1 η2t + β1 ) > 0 and ω1 = ω∗1 . When ω1 and ω∗1 are sufficiently close,
there exists no consistent test for H0 : θ0 = θ1 against H1 : θ0 = θ1∗ at the asymptotic
level α ∈ (0 1/2).
Propositions 2.1 and 3.1 show that no asymptotically valid inference on ω0
can be done in the nonstationary case. It is thus of interest to test whether a
given series is stationary or not.
3.2. Strict Stationarity Testing
Consider the strict stationarity testing problems
(3.3)
H0 : γ0 < 0 against
H1 : γ0 ≥ 0
H0 : γ0 ≥ 0 against
H1 : γ0 < 0
and
(3.4)
830
C. FRANCQ AND J.-M. ZAKOÏAN
where γ0 = E log(α0 η21 + β0 ). These hypotheses are not of the form (3.1) because γ0 not only depends on α0 and β0 , but also on the unknown distribution
of η1 . The following result gives the asymptotic distribution of the empirical
estimator of γ0 defined by (1.8) under either the stationarity or the nonstationarity conditions.
THEOREM 3.1: Assume the following conditions are satisfied: Theorem 2.1(iv)
in the case γ0 < 0, Theorem 2.1(v) in the case γ0 > 0, and Theorem 2.2 (and, in
particular, Assumption A) when γ0 = 0. Moreover, assume that E|a0 (η1 )|2 < ∞.
Let ut = log a0 (ηt ) − γ0 and σu2 = Eu2t . Then
(3.5)
√
d
n(γ̂n − γ0 ) → N (0 σγ2 )
where
σγ2 =
as n → ∞
σu2 + (κη − 1){a J −1 a − (1 − ν1 )2 } when γ0 < 0,
σu2
when γ0 ≥ 0,
with a = (0 (1 − ν1 )/α0 ν1 /β0 ) and ν1 = E{β0 /a0 (η1 )}.
It can be seen from the proof that the term in brackets in the first expression
of σγ2 is positive, showing that the asymptotic variance of γ̂n is larger in the
stationary case than in the nonstationary case. The next result provides an estimator of σγ2 which is consistent in every situation (explosive and stationary). It
allows to construct a confidence interval for the top Lyapunov exponent. Let
n
2
n
1
1
2
2
2
2
{log(α̂n η̂t + β̂n )} −
log(α̂n η̂t + β̂n ) σ̂u =
n t=1
n t=1
σ̂γ2 = σ̂u2 + (κ̂η − 1){â J−1 â − (1 − ν̂1 )2 }
n
β̂n
1
1 − ν̂1 ν̂1
ν̂1 =
â = 0
α̂n
n t=1 α̂n η̂2t + β̂n
β̂n
COROLLARY 3.2: Under the assumptions of Theorem 3.1,
(3.6)
σ̂γ2 → σγ2
in probability (and a.s. when γ0 = 0)
as n → ∞
Therefore, at the asymptotic level α ∈ (0 1), a confidence interval for γ0 is
σ̂γ −1
σ̂γ −1
γ̂n − √ (1 − α/2) γ̂n + √ (1 − α/2) n
n
The following result provides asymptotic critical regions for the strict stationarity testing problems.
TESTING AND ESTIMATION OF GARCH MODELS
831
COROLLARY 3.3: Let the assumptions of Theorem 3.1 hold. For the testing
problem (3.3), the test defined by the (stationary (ST)) critical region
√ γ̂n
ST
−1
C = Tn := n > (1 − α)
(3.7)
σ̂u
has its asymptotic significance level bounded by α, has an asymptotic probability
of rejection α under γ0 = 0, and is consistent for all γ0 > 0.
For the testing problem (3.4), the test defined by the (nonstationary (NS)) critical
region
(3.8)
CNS = {Tn < −1 (α)}
has its asymptotic significance level bounded by α, has an asymptotic probability
of rejection α under γ0 = 0, and is consistent for all γ0 < 0.
4. TESTING NONSTATIONARITY IN NONLINEAR GARCH
In this section, we study the behavior of the stationarity tests in Section 3.2
when the data are generated by the GARCH-type model
t = ht ηt t = 1 2 (4.1)
ht = ω(ηt−1 ) + b0 (ηt−1 )ht−1 with an initial value h0 , under the same assumptions on (ηt ) as in model (1.1).
In this model, ω : R → [ω ω] and b0 : R → [b +∞) for some ω ω b > 0. It
is assumed that b0 is decreasing over (−∞ 0] and increasing over [0 +∞)
This model belongs to the so-called class of augmented GARCH models (see
Hörmann (2008)) and encompasses many classes of GARCH(1 1) models
introduced in the literature: for instance, with constant ω(·), the standard
GARCH(1 1) when b0 (x) = α0 x2 + β0 , and the GJR model (Glosten, Jaganathan, and Runkle (1993)) when b0 (x) = α1 (max{x 0})2 + α2 (min{x 0})2 +
β0 . It can be shown that if E max{0 log b0 (ηt )} < ∞, then
(4.2)
Γ := E log b0 (ηt ) < 0
is a necessary and sufficient condition for the strict stationarity of this model
(see, e.g., Francq and Zakoïan (2006)). Our aim is to test strict stationarity
without estimating the nonparametric model (4.1). We shall see that, surprisingly, the tests developed for the standard GARCH(1 1) model still work in
this framework.
We still consider the statistic γ̂n defined by (1.8), where θ̂n is the estimator
(1.7) of the standard GARCH(1 1) parameter, but the observations are generated by the augmented GARCH(1 1) model (4.1) instead of the standard
GARCH(1 1).
832
C. FRANCQ AND J.-M. ZAKOÏAN
PROPOSITION 4.1: Let 1 n denote observations from model (4.1). Assume 0 < E| log η21 |2 < ∞ and E| log b0 (η1 )|2 < ∞.
If Γ > 0, then, under regularity conditions that imply the existence and uniqueness of a pseudo-true value (α∗ β∗ ) = limn→∞ (α̂n β̂n ) a.s.,
γ̂n → Γ
and
σ̂u2 → σ∗2
a.s.
for some constant
σ∗2 > 0
If Γ < 0, then, under regularity conditions that imply the strong consistency of
θ̂n to the unique pseudo-true value,
2
(ω∗ α∗ β∗ ) = arg min E 2 t + log σt2 (θ)
θ∈Θ
σt (θ)
and if Var log 2t < ∞, we have, for some Γ ∗ ,
2t
∗
σ̂ → Var log α 2 ∗ + β > 0 a.s.
σt (θ )
∗
γ̂n → Γ < 0
and
2
u
∗
REMARK 4.1: In the ARCH(1) case, under the condition E{a(η1 )/η21 } < ∞,
the pseudo-true value is α∗ = E(a(η1 )/η21 ) when Γ > 0.
Thus, the (non)stationarity tests developed in the standard GARCH(1 1)
case lead, asymptotically, to the right decision, even if the GARCH(1 1) model
is misspecified, except in the limit case where Γ = 0. More precisely, we have
the following result.
COROLLARY 4.1: Let the assumptions of Proposition 4.1 hold.
If Γ > 0, then
P(CNS ) → 0
and P(CST ) → 1
where CST and CST are defined in Corollary 3.3.
If Γ < 0, then
P(CST ) → 0
and P(CNS ) → 1
5. NUMERICAL ILLUSTRATIONS
Before illustrating our asymptotic results for the tests, we study the behavior
of the QMLE in finite samples.
5.1. Finite Sample Properties of the QMLE
From Theorems 2.1 and 2.2 and Proposition 2.1, we know that (α̂n β̂n ) is
always CAN, whereas ω̂n is only consistent in the stationary case. In the Monte
833
TESTING AND ESTIMATION OF GARCH MODELS
TABLE I
BIAS (MEAN ERRORS) AND MSE (MEAN SQUARED ERRORS) FOR THE QMLE OF A
GARCH(1 1), WITH ηt ∼ N (0 1)a
2nd (γ0 = −0180)
n = 200
Bias
MSE
n = 4,000
Bias
MSE
ST (γ0 = −0038)
NS (γ0 = 0078)
ω
α
β
ω
α
β
ω
α
β
−021
058
0.01
0.01
0.01
0.01
−034
110
0.01
0.02
0.01
0.02
−051
377
0.02
0.03
0.02
0.03
000
001
0.00
0.00
0.00
0.00
−003
003
0.00
0.00
0.00
0.00
−051
495
0.00
0.00
0.00
0.00
a The parameters θ = (1 03 06), θ = (1 05 06), and θ = (1 07 06), correspond to second-order stationary
0
0
0
(2nd), strict stationary (ST), and nonstationary (NS) models. Bias and MSE are computed over 1,000 independent
simulations of length n = 200 or 4,000.
Carlo experiments that we conducted, the finite sample behavior of the QMLE
is in perfect agreement with these asymptotic results. Table I summarizes a few
of these simulation experiments. We simulated 1,000 independent trajectories
of size n = 200 and n = 4,000 of GARCH(1 1) models with ηt ∼ N (0 1) and
parameter θ0 corresponding to a second-order stationary process, a strictly stationary process without second-order moment, and a nonstationary process.
Since ηt ∼ N (0 1), the QMLE corresponds to the maximum likelihood estimator (MLE). Similar results were obtained with other distributions for ηt .
Concerning the estimation of (α0 β0 ), the results are very similar for the three
values of θ0 , confirming that stationarity is not necessary for the estimation of
these parameters. By contrast, the column ω in the nonstationary case confirms
the asymptotic results of Proposition 2.1 and illustrates the impossibility of estimating the parameter ω0 with a reasonable accuracy under the nonstationarity
condition (1.5). Note that the root mean squared error (RMSE) of estimation
of ω0 even worsens when the sample size increases (3.77 for n = 200 and 4.95
for n = 4,000).
5.2. Finite Sample Properties of the Tests
5.2.1. On Simulated Data
To assess the performance of the tests developed in Section 3, we simulated N = 1000 independent trajectories of size n = 500, 2,000, and 4,000 of
a GARCH(1 1) model of the form (1.1), with different values of θ0 and the
standardized Student distribution with 7 degrees of freedom for ηt . The standardized Student distribution is often employed as the distribution of GARCH
errors in applied works.
With this distribution, we have γ0 = 0 for, in particular, α0 = 02575 and
β0 = 08. Results concerning the test of the hypotheses (3.1) with a = 0 and
834
C. FRANCQ AND J.-M. ZAKOÏAN
TABLE II
RELATIVE FREQUENCY OF REJECTION (IN %) FOR THE TEST (3.2) OF THE NULL HYPOTHESIS
H0 : β0 ≤ 07 AGAINST H1 : β0 > 07a
β0
n
0.61
0.64
0.67
0.70
0.73
0.76
0.79
500
2,000
4,000
3.5
0.3
0.2
4.3
0.6
0.3
5.2
1.8
1.0
8.9
6.8
5.5
12.6
18.3
27.7
26.8
53.1
76.9
49.6
91.5
99.0
a Here a = 0, b = 1, and c = 07 in (3.1). The nominal level is α = 5% when α = 02. The value (α β ) = (02 07)
0
0 0
corresponds to a stationary process.
b = 1, that is, a test on the value of β0 , are presented in Tables II and III. Note
that the test (3.2) behaves similarly when the tested value corresponds to a
stationary solution (Table II) or to a nonstationary process (Table III).
We now illustrate the behavior of the strict stationarity tests (3.7) and (3.8),
through simulations of the GARCH(1 1) models with β0 = 08 and values of
α0 corresponding to γ0 < 0 (α0 ∈ {018 020 022}), γ0 = 0 (α0 = 02575), and
γ0 > 0 (α0 ∈ {028 030 031}). Tables IV and V show that, as expected, the
frequency of rejection of the CST test increases with γ0 , while, obviously, that
of the CNS test decreases. The rejection frequencies of the two tests approach
the nominal level when γ0 = 0 and n increases, although it remains far from the
theoretical value in Table V. Other simulation experiments (not reported here)
reveal that the error of the first kind is better controlled for tests of stationarity
of ARCH(1) models, which is not surprising, because the model is simpler.
Now consider testing strict stationarity in a generalized GARCH(1 1) model
using the tests developed for the standard GARCH(1 1). The generalized
GARCH that we considered is a GJR model of the form (4.1) with b0 (x) =
α1 (max{x 0})2 + α2 (min{x 0})2 + β0 . We keep the same standardized Student
distribution for ηt , and we take β0 = 08 and α2 = 02575, whereas α1 varies in
TABLE III
RELATIVE FREQUENCY OF REJECTION (IN %) FOR THE TEST (3.2) OF THE HYPOTHESIS
H0 : β0 ≤ 07 AGAINST H1 : β0 > 07 WHEN α0 = 05a
β0
n
0.61
0.64
0.67
0.70
0.73
0.76
0.79
500
2,000
4,000
0.3
0.0
0.0
0.5
0.0
0.0
2.8
0.1
0.1
9.9
6.2
6.1
25.5
41.6
61.0
47.7
81.8
96.2
67.2
97.0
99.7
a The value (α β ) = (05 07) corresponds to a nonstationary process.
0 0
835
TESTING AND ESTIMATION OF GARCH MODELS
TABLE IV
RELATIVE FREQUENCY OF REJECTION (IN %) OF THE TEST (3.7) OF THE STATIONARITY
HYPOTHESIS H0 : γ0 < 0 FOR THE GARCH(1 1) MODEL WITH β0 = 08a
α0
n
0.18
0.20
0.22
0.2575
0.28
0.30
0.31
500
2,000
4,000
0.0
0.0
0.0
0.0
0.0
0.0
0.1
0.0
0.0
7.5
6.3
5.3
27.8
67.8
92.4
61.4
98.6
100.0
75.2
99.9
100.0
a The nominal level is α = 5%. The parameter α = 02575 corresponds to γ = 0.
0
0
such a way that Γ < 0 when α1 ∈ {018 020 022}, Γ = 0 when α1 = 02575,
and Γ > 0 when α1 ∈ {028 030 031}.
Table VI confirms the theoretical result of Section 4. More precisely, for
n sufficiently large, the tests give the right conclusion when Γ < 0 and Γ > 0.
Note that when Γ = 0, the rejection frequency is close to the nominal 5% level,
which is not surprising because the model is a standard GARCH(1 1) in this
case. In general, when the test is applied to nonstandard GARCH models,
there is no guarantee that the asymptotic relative frequency of rejection will
be close to the nominal asymptotic level of the standard GARCH(1 1).
5.2.2. On Real Data
The strict stationarity tests were then applied to the daily returns of 11 major stock market indices. We considered the CAC, DAX, DJA, DJI, DJT, DJU,
FTSE, Nasdaq,7 Nikkei, SMI, and SP500, from January 2, 1990 to January 22,
2009, except for the indices for which such historical data do not exist. TaTABLE V
RELATIVE FREQUENCY OF REJECTION (IN %) FOR TESTING THE NONSTATIONARITY
HYPOTHESIS H0 : γ0 ≥ 0 WITH THE TEST (3.8) FOR THE GARCH(1 1) MODEL WITH β0 = 08a
α0
n
500
2,000
4,000
0.18
0.20
0.22
0.2575
0.28
0.30
0.31
98.3
100.0
100.0
91.7
100.0
100.0
69.3
98.3
100.0
19.8
11.1
9.1
4.1
0.1
0.0
0.7
0.0
0.0
0.4
0.0
0.0
a The nominal level is α = 5%.
7
Since the Nasdaq index level was halved on January 3, 1994, one outlier has been eliminated
for this series.
836
C. FRANCQ AND J.-M. ZAKOÏAN
TABLE VI
RELATIVE FREQUENCY OF REJECTION (IN %) FOR THE TEST (3.8) OF THE STATIONARITY
HYPOTHESIS H0 : α < 0 FOR A GJR MODELa
α1
n
0.18
0.20
0.22
0.2575
0.28
0.30
0.31
500
2,000
4,000
0.1
0.0
0.0
0.1
0.0
0.0
1.1
0.1
0.0
7.8
6.6
5.6
15.8
31.7
45.1
32.7
65.8
87.7
35.2
77.4
96.1
a The parameter α = 02575 corresponds to Γ = 0. The nominal level is α = 5%.
1
ble VII displays the test statistics Tn computed on each series. Note that as
n → ∞, a.s.
Tn =
√ γ̂n − γ0 √ γ0
n
+ n → −∞
σ̂u
σ̂u
when γ0 < 0 and Tn → +∞ when γ0 > 0. Because the values of Tn given in
Table VII are very small, a nonstationary augmented GARCH(1 1) model is
not plausible, for any of these series.
For individual stock returns, the opposite conclusion can occur as the following examples show. We estimated GARCH(1 1) models on the daily series of Icagen (NasdaqGM: ICGN), Monarch Community Bancorp (NasdaqCM: MCBF), KV Pharmaceutical (NYSE: KV-A), Community Bankers
Trust (AMEX: BTC), and China MediaExpress (NasdaqGS: CCME).8
Table VIII shows that for four of these five stocks, the nonstationarity assumption cannot be rejected at any reasonable significance level. Interestingly,
nonstationarity can occur with a small or a large ARCH coefficient α̂n . In any
case, the value of α̂n + β̂n does not give clear insight on the possible nonstationarity of the series. As an example, Figure 1 displays the sample path of the
TABLE VII
TEST STATISTIC Tn OF THE STRICT STATIONARITY TESTS (3.7) AND (3.8)a
CAC
−14.5
DAX
DJA
DJI
DJT
DJU
FTSE
Nasdaq
Nikkei
SMI
SP500
−15.8
−15.1
−13
−15.1
−14
−10.7
−8.5
−15.4
−23
−11.1
a The test statistic is the realization of a random variable which is asymptotically N (0 1) distributed when γ = 0,
0
tends to −∞ under the strict stationarity hypothesis γ0 < 0, and tends to +∞ when γ0 > 0.
8
The data range from May 31, 2007, August 28, 2007, March 31, 2006, June 29, 2007, and
March 31, 2009, respectively, to February 7, 2011.
TESTING AND ESTIMATION OF GARCH MODELS
837
TABLE VIII
TEST STATISTIC Tn AND p-VALUES OF THE NONSTATIONARITY TEST (3.8) FOR STOCK RETURNS
n
α̂n
β̂n
Tn
p-value
ICGN
MCBF
KV-A
BTC
CCME
928
0581
0696
−2297
0011
868
0.023
0.979
0.024
0.510
1,221
0.143
0.927
1.120
0.869
908
0.508
0.765
0.491
0.688
469
0.413
0.750
0.457
0.676
MCBF series. The positive estimated value of γ0 for this series is in accordance
with the seemingly increasing volatility along the sample path.
6. CONCLUSION
This paper develops a unified theory for the inference of both stationary
and nonstationary GARCH(1 1) processes. The practical implications of our
results are the following.
(i) If one is interested in inference on (α0 β0 ) in the GARCH(1 1) model,
then stationarity testing is unnecessary. The standard QMLE of (α0 β0 ) is
CAN in every (stationary or nonstationary) situation. A key result, allowing one to construct confidence intervals, is that the asymptotic variance of
(α̂n β̂n ) can be estimated without any stationarity restriction.
(ii) If one is interested in a GARCH(1 1) application in which ω0 is used
(e.g., estimating the variance of today’s or tomorrow’s conditional distribution
FIGURE 1.—Log returns (in %) of the MCBF stock series.
838
C. FRANCQ AND J.-M. ZAKOÏAN
for derivatives pricing or for computing forecast intervals), then stationarity
testing is necessary, because if γ0 > 0, the QMLE of ω0 is not consistent.
(iii) The constrained QMLE of (α0 β0 ), which was known to be CAN in the
nonstationary case, is inconsistent in the stationary case. As a consequence,
this estimator should only be used if nonstationarity is taken for granted.
(iv) Surprisingly, the tests developed for the standard GARCH(1 1) are
able to detect nonstationarity in more general GARCH(1 1) models.
To conclude, let us mention two possible extensions of this work. First, it
would be interesting to know whether the test in this present paper works for
detecting stationarity in other volatility models (not restricted to augmented
GARCH). Second, it may be worth developing specific stationarity tests for
particular augmented GARCH models.
APPENDIX: PROOFS AND COMPLEMENTARY RESULTS
A.1. Asymptotic Behaviors of (ht )
When γ0 = 0, the asymptotic behavior of the sequences (ht ) (defined by
(1.1)) and (ht∞ ) (defined by (1.4)) is the same and is easily obtained by the
Cauchy rule. When γ0 = 0, the asymptotic behavior of ht∞ can be obtained by
the Chung–Fuchs theorem. The behavior of ht is different in this case and is
described in the result below.
PROPOSITION A.1: For the GARCH(1 1) model (1.1), the following properties hold.
(i) When γ0 > 0, ht → ∞ a.s. at an exponential rate: for any ρ > e−γ0 ρt ht → ∞
and
ρt 2t → ∞
a.s. as t → ∞
if
E| log(η21 )| < ∞
(ii) Klüppelberg, Lindner, and Maller (2004). When γ0 = 0,
ht → ∞
and
if
E| log(η21 )| < ∞
2t → ∞
in probability.
(iii) Let ψ be a decreasing bijection from (0 ∞) to (0 ∞) such that Eψ(21 ) <
∞. When γ0 = 0 and α0 > 0,
(A.1)
ψ(2t ) → 0
and ψ(ht ) → 0 in L1 .
PROOF: To prove (i) we note that for t > 1,
t−1
(A.2)
a0 (ηt−1 ) · · · a0 (ηt−i ) + a0 (ηt−1 ) · · · a0 (η0 )h0
ht = ω0 1 +
i=1
≥ ω0
t−1
i=1
a0 (ηi )
TESTING AND ESTIMATION OF GARCH MODELS
839
Thus, for any constant ρ ∈ (e−γ0 1) we have
t−1
1
1
log ρω0 +
lim inf log ρt ht ≥ lim
log ρa0 (ηi )
t→∞ t
t→∞ t
i=1
= E log ρa0 (η1 ) = log ρ + γ0 > 0
It follows that log ρt ht , and hence ρt ht , tends to +∞ a.s. as n → ∞. Now
E| log η21 | < ∞ entails log η2t /t → 0 a.s. as t → ∞. Therefore, lim inft→∞ t −1 ×
log ρt η2t ht ≥ E log ρa0 (η1 ) > 0 and ρt 2t = ρt η2t ht → +∞ a.s. by already given
arguments.
The proof of (ii) follows from Klüppelberg, Lindner, and Maller (2004).
Their condition E| log(δ + λε12 )| < ∞ becomes in our notation E| log a0 (η1 )| <
∞, and this condition is satisfied because E log+ a0 (η1 ) − E log− a0 (η1 ) =
γ0 = 0 and E log+ a0 (η1 ) ≤ α0 + β0 , where for any real-valued function f ,
f + (x) = max{f (x) 0} and f − (x) = max{−f (x) 0}.
To prove (iii), note that since ht > α0 2t−1 with α0 > 0, we have ψ(ht ) <
∗
ψ (2t−1 ), where ψ∗ (x) = ψ(α0 x) satisfies the same assumptions as ψ(x)
Therefore, the second convergence in (A.1) follows from the first convergence.
It suffices to consider the case 0 = 0. Note that even if 2t does not increase
with probability 1, 2t+1 is stochastically greater than 2t because
2t+1 = {ω0 + ω0 a0 (ηt ) + · · · + ω0 a0 (ηt ) · · · a0 (η2 )
+ ω0 a0 (ηt ) · · · a0 (η1 )}η2t+1
≥ {ω0 + ω0 a0 (ηt ) + · · · + ω0 a0 (ηt ) · · · a0 (η2 )}η2t+1
d
= 2t d
where = stands for equality in distribution. The dominated convergence theorem and (i) and (ii) then entail
∞
Eψ(2t ) =
P{2t < ψ−1 (u)} du
0
∞
→
0
lim ↓ P{2t < ψ−1 (u)} du = 0
t→∞
which completes the proof.
Q.E.D.
A.2. Asymptotic Normality of the QMLE of (α0 β0 )
Let ω = inf{ω|θ ∈ Θ}, α = inf{α|θ ∈ Θ}, β = inf{β|θ ∈ Θ}, ω = sup{ω|θ ∈
Θ}, α = sup{α|θ ∈ Θ}, and β = sup{β|θ ∈ Θ}. Denote by K any constant whose
value is unimportant and can change throughout the proofs.
840
C. FRANCQ AND J.-M. ZAKOÏAN
Define the [0 ∞]-valued process
vt (α β) =
j−1
∞
αη2t−j j=1
a0 (ηt−j )
k=1
β
a0 (ηt−k )
j−1
with the convention k=1 = 1 when j ≤ 1. Let Θ0 = {θ ∈ Θ : β < eγ0 } and Θp =
{θ ∈ [0 ∞)3 : β < 1/a0 (η1 )
−1
p }.
LEMMA A.1: (i) When γ0 > 0, for any θ ∈ Θ0 , the process vt (α β) is stationary
and ergodic. Moreover, for any compact Θ∗0 ⊂ Θ0 2
σt (θ)
sup − vt (α β) → 0
ht
θ∈Θ∗0
a.s. as t → ∞
Finally, for any θ ∈
/ Θ0 , it holds that σt2 (θ)/ht → ∞ a.s.
(ii) When γ0 = 0, for any θ ∈ Θp with p ≥ 1, the process vt (α β) is stationary
and ergodic. Moreover, for any compact Θ∗p ⊂ Θp 2
σt (θ)
− vt (α β) → 0
sup ht
θ∈Θ∗p
in Lp PROOF
: Without loss of generality, assume that σ02 (θ) = 0. We then have
t
σ (θ) = j=1 βj−1 (ω + α2t−j ) and
2
t
(A.3)
j
t
σt2 (θ) j−1 ht−k ω + α2t−j
=
β
= at + bt ht
ht−k+1
ht−j
j=1
k=1
where
at =
t
j=1
j−1
β
j
t
ht−k
αη2t−j :=
atj
ht−k+1
j=1
k=1
and
bt =
t
j=1
βj−1
ω
ht
For θ ∈ Θ0 , by the Cauchy root test, the series vt (α β) in a.s. finite. As a
measurable function of {ηu u < t}, the process vt (α β) is thus stationary and
t
ergodic. We have, for β0 = sup{β|θ ∈ Θ∗0 }, bt ≤ Kω(t + β0 )/ ht → 0 a.s. by
Proposition A.1(i). It follows that supθ∈Θ∗ bt → 0 a.s. Now note that
0
(A.4)
ht−k
ht−k
1
=
≤
ht−k+1 ω0 + a0 (ηt−k )ht−k a0 (ηt−k )
TESTING AND ESTIMATION OF GARCH MODELS
841
As in the proof of Lemma 4 in Jensen and Rahbek (2004b), for any fixed t0 < t,
we thus have
0 ≤ vt (α β) − at ≤
t0
(vtj − atj ) +
j=1
∞
vtj j=t0 +1
∞
where vt (α β) = j=1 vtj . In the case γ0 > 0, supθ∈Θ∗ (vtj − atj ) → 0 a.s.
0
∞
as t → ∞ by Proposition A.1(i). Moreover, the series supθ∈Θ∗ j=t0 +1 vtj =
0
∞
j=t0 +1 vtj (α0 β0 ), with obvious notations, converges by the Cauchy root test.
The first convergence in (i) follows.
Now for θ ∈
/ Θ0 , for any t0 < t, we have, for ρ > e−γ0 ,
0
0
σt2 (θ) ≥
atj =
vtj + o(ρt t02 ) a.s.
ht
j=1
j=1
t
t
as t → ∞ by Proposition A.1(i). The proof of (i) is completed by noting that
t0
γ0
and by the
j=1 vtj → ∞ a.s. as t0 → ∞ by the Cauchy root test when β > e
γ0
Chung–Fuchs theorem when β = e .
Now we turn to (ii). Since vt (α β)
p < ∞, vt (α β) is a.s. finite, and the stationarity and ergodicity follow. By γ0 = 0 = E log(α0 η21 + β0 )−p and Jensen’s
inequality, we have (α0 η21 + β0 )−1 p > 1 Thus θ ∈ Θp entails β < 1. It follows that supθ∈Θ∗p bt → 0 in Lp using Proposition A.1(iii). Noting that vtj p ≤
(α/α0 )ρj−1 , where ρ = β/(α0 η1 + β0 )
p < 1, the rest of the proof follows from
arguments similar to those used in the proof of (i) with a.s. convergences reQ.E.D.
placed by Lp convergences.
LEMMA A.2: If θ ∈ Θ0 , we have
vt (α β) = 1
a.s. if and only if
(α β) = (α0 β0 )
PROOF: Straightforward algebra shows that
(A.5)
vt (α β)(α0 η2t−1 + β0 ) = βvt−1 (α β) + αη2t−1 Hence
{vt (α β) − 1}(α0 η2t−1 + β0 ) = βvt−1 (α β) − β0 + (α − α0 )η2t−1 It follows that vt (α β) = 1 a.s. if and only if (iff) βvt−1 (α β) − β0 + (α −
α0 )η2t−1 = 0. By strict stationarity, vt−1 (α β) = 1 a.s. and we have β − β0 + (α −
α0 )η2t−1 = 0. Because the distribution of η2t−1 is nondegenerate, the conclusion
follows.
Q.E.D.
842
C. FRANCQ AND J.-M. ZAKOÏAN
PROOF OF THEOREM 2.1(i)–(iii): The result stated in (i) is standard. Consider
case (ii). Note that (ω̂n α̂n β̂n ) = arg minθ∈Θ Qn (θ) where Qn (θ) =
the
n
n−1 t=1 {t (θ) − t (θ0 )} We have
n
σt2 (θ)
ht
1 2
−
1
+
log
ηt
= On (α β) + Rn (θ)
Qn (θ) =
n t=1
σt2 (θ)
ht
where
and
n
1
1 2
− 1 + log vt (α β)
On (α β) =
η
n t=1 t vt (α β)
n
1
σt2 (θ)
ht
1 2
−
+
log
Rn (θ) =
ηt
n t=1
σt2 (θ) vt (α β)
ht vt (α β)
Lemma A.1(i) entails that if θ ∈
/ Θ0 , then Qn (θ) → ∞ a.s. It thus suffices to
consider the case θ ∈ Θ∗0 , where Θ∗0 is an arbitrary compact subset of Θ0 . We
have by stationarity and ergodicity of vt (α β) a.s.
1
lim On (α β) = E
− 1 + log v1 (α β) ≥ 0
n→∞
v1 (α β)
because log x ≤ x − 1 for x > 0 The inequality is strict except when v1 (α β) =
1 a.s. By Lemma A.2, we thus have E{On (α β)} ≥ 0 with equality only if
(α β) = (α0 β0 )
To handle Rn (θ), we prove the following lemma. Let Θαβ be the compact set
of the (α β)’s such that (ω α β) ∈ Θ.
LEMMA A.3: Suppose that P(ηt = 0) = 0. Then, for any k > 0,
k
k
1
ht
E sup
< ∞ and E sup
< ∞
2
(αβ)∈Θαβ vt (α β)
θ∈Θ σt (θ)
PROOF: Let ε > 0 such that p(ε) := P(|ηt | ≤ ε) ∈ [0 1). If |ηt−1 | > ε, since
the sum vt (α β) is greater than its first term, we have
α0 η2t−1 + β0 α0
β0
1
α0
β0
≤
+
=
≤
+ 2 := K(ε)
vt (α β)
α
α
αε
αη2t−1
αη2t−1
Now if |ηt−1 | ≤ ε but |ηt−2 | > ε, minorizing the sum vt (α β) by its second term,
we have
1
a0 (ε)
α0
β0
a0 (ε)
≤
+
≤ K(ε)
vt (α β)
α
β
β
αη2t−2
843
TESTING AND ESTIMATION OF GARCH MODELS
Continuing in this manner, we can write
i−1
∞
1
a0 (ε)
≤ K(ε)
sup
1|ηt−1 |≤ε · · · 1|ηt−i+1 |≤ε 1|ηt−i |>ε
β
(αβ)∈Θαβ vt (α β)
i=1
Thus, for any integer k,
k
1
sup
(αβ)∈Θαβ vt (α β)
≤ {K(ε)}
k
∞
1|ηt−1 |≤ε · · · 1|ηt−i+1 |≤ε 1|ηt−i |>ε
i=1
It follows that
E
sup
(αβ)∈Θαβ
1
vt (α β)
a0 (ε)
β
k(i−1)
k
≤ {K(ε)}k {1 − p(ε)}
∞
p(ε)i−1
i=1
a0 (ε)
β
k(i−1)
Noting that limε→0 p(ε) = 0 and limε→0 a0 (ε) = β0 , we have p(ε)( a0β(ε) )k < 1
for ε sufficiently small. The first result of the lemma is thus proven.
Similarly, we have for |ηt−1 | > ε,
ω 0 α0
β0
ω0 + a0 (ηt−1 )ht−1
ht
≤
+
+ 2 := H(ε)
≤
2
σt2 (θ) ω + αht−1 η2t−1 + βσt−1
ω
α
αε
(θ)
and for |ηt−1 | ≤ ε and |ηt−2 | > ε,
ht
ω0 a0 (ε)
+
H(ε)
≤
σ (θ)
ω
β
2
t
More generally,
(A.6)
sup
θ∈Θ
ht
≤ Vt σt2 (θ)
where
Vt =
∞
i=1
1|ηt−1 |≤ε · · · 1|ηt−i+1 |≤ε 1|ηt−i |>ε
×
j i−1
i−2 ω0 a0 (ε)
a0 (ε)
+
H(ε) ω j=0
β
β
844
C. FRANCQ AND J.-M. ZAKOÏAN
The conclusion follows by the same arguments as before.
Q.E.D.
Now we show that
(A.7)
lim sup |Rn (θ)| = 0
n→∞ θ∈Θ∗
a.s.
0
where Θ∗0 is defined in Lemma A.1. Using Lemma A.1, the absolute value of
the first term in Rn (θ) satisfies, using (A.6),
n
1 2
1
h
(θ)
σ
t
(A.8)
sup vt (α β) − t
η2t 2
∗
σt (θ)
ht
vt (α β) θ∈Θ0 n t=1
Kε 2 Vt
η
n t=1 t vt (α β)
n
≤
for any ε > 0, when n is large enough. The right-hand side tends a.s. to Kε as
n → ∞ by the ergodic theorem and Lemma A.3. The second term in Rn (θ)
is handled similarly and (A.7) follows. The proof is completed by standard
arguments, invoking the compactness of Θ.
The proof of (iii) is identical, except that the a.s. convergence in (A.7) is
replaced by an L1 convergence. More precisely, by the Hölder inequality and
Lemma A.3, the expectation of the left-hand side of (A.8) is bounded by
n 2
K Vt sup vt (α β) − σt (θ) n t=1 vt (α β) q θ∈Θp
ht p
Q.E.D.
which tends to zero by Lemma A.1(ii).
We now need to introduce new [0 ∞]-valued processes. Let a(ηt ) = αη2t +β
and
d (α β) =
α
t
∞
η2t−j
j=1
dtβ (α β) =
a(ηt−j )
j−1
k=1
β
a(ηt−k )
j−1
∞
(j − 1)αη2t−j j=2
βa(ηt−j )
k=1
β
a(ηt−k )
LEMMA A.4: Assume γ0 ≥ 0 and Eη4t < ∞. We have
⎞
⎛ ∂
(θ
)
n
t
0
1 ⎜ ∂α
⎟ d
√
⎠ → N {0 (κη − 1)I } as n → ∞
⎝ ∂
n t=1
t (θ0 )
∂β
TESTING AND ESTIMATION OF GARCH MODELS
845
PROOF: Using the Wold–Cramér device, it suffices to show that for all λ =
(λ1 λ2 ) , the sequence
∇t =
⎛ ∂σ 2
t
⎞
(θ0 )
∂
∂
1 − η ⎜ ∂α
⎟
t (θ0 ) t (θ0 ) λ =
λ⎝ 2
⎠
∂σt
∂α
∂β
ht
(θ0 )
∂β
2
t
satisfies
1 d
∇t → N {0 (κη − 1)λ I λ}
√
n t=1
n
(A.9)
Since E log β/a(η1 ) < 0, by the Cauchy root test, the processes dtα (α β) and
dtβ (α β) are stationary and ergodic. Still assuming σ02 = 0, we have
∂σt2
(θ) =
βj−1 2t−j ∂α
j=1
t
∂σt2
(θ) =
(j − 1)βj−2 (ω + α2t−j )
∂β
j=2
t
Thus, using a direct extension of (A.4),
j
t
2
σt−k
2t−j
(θ)
1 ∂σt2
j−1
(θ)
=
≤ dtα (α β)
β
2
2
σt2 (θ) ∂α
σ
(θ)
σ
(θ)
t−j
j=1
k=1 t−k+1
j
t
2
σt−k
ω + α2t−j
(θ)
1 ∂σt2
j−2
(θ)
=
(j
−
1)β
2
σt2 (θ) ∂β
σ2
(θ) σt−j
(θ)
j=2
k=1 t−k+1
≤ dtβ (α β)
Moreover, we have
0 ≤ dtα (α β) −
1 ∂σt2
(θ) ≤ st0 + rt0 σt2 ∂α
where
st0 =
t0
η2t−j
j=1
rt0 =
2
2t−j βσt−k
(θ)
β
− 2
2
a(ηt−j ) k=1 a(ηt−k ) σt−j (θ) k=1 σt−k+1 (θ)
∞
j=t0 +1
j−1
η2t−j
a(ηt−j )
j−1
k=1
j
β
a(ηt−k )
846
C. FRANCQ AND J.-M. ZAKOÏAN
For all p ≥ 1, rt0 p → 0 as t0 → ∞ because β/a(η1 )
p < 1. Since η21 /
2
(θ)/σt2 (θ)
p < 1, and
a(η1 )
p < 1/α, βσt−1
2
β
(θ)
βσ
βω
t−1
a(η ) − σ 2 (θ) = a(η )σ 2 (θ) → 0
t
t
t
t
p
p
as t → ∞ by the dominated convergence theorem, st0 = st0 (t) converges to 0
in Lp as t → ∞ The same derivations hold true when dtα (α β) is replaced by
dtβ (α β). It follows that dtα (α β) and dtα (β) have moments of any order, and
1 ∂σt2
α
(A.10) 2
(θ) − dt (α β) → 0 and
σt (θ) ∂α
1 ∂σt2
β
σ 2 (θ) ∂β (θ) − dt (α β) → 0
t
in Lp for any p ≥ 1. Standard computations show that I = Ed1 d1 , where dt =
(dtα (α0 β0 ) dtβ (α0 β0 )) The same matrix was obtained by Jensen and Rahbek
(2004b).
Using (A.10) at θ = θ0 and the ergodic theorem, it then follows that, as n →
∞,
1 κη − 1 Var √
∇t =
E|λ dt |2 + o(1) → (κη − 1)λ I λ
n
n t=1
t=1
n
n
Moreover, for all ε > 0,
1
E∇t2 1{|∇t /√n|>ε} ≤ (κη − 1)E{|λ1 |d1α (β0 ) + |λ2 |d1β (α0 β0 )}2
n t=1
n
× 1{(|1−η2 |/√n)(|λ1 |dα (β0 )+|λ2 |dβ (α0 β0 ))>ε}
1
1
1
→ 0
We thus obtain (A.9) by the Lindeberg central limit theorem for martingale
differences (see Billingsley (1995, p. 476)).
Q.E.D.
LEMMA A.5: Let be an arbitrary compact subset of [0 ∞). Assume that
E log η21 < ∞. When γ0 > 0, we have
∞
∂
(A.11)
t (θ) < ∞ a.s.
sup
∂ω
t=1 θ∈Θ0
∞
∂2
(A.12)
t (θ)
sup
< ∞ a.s.
∂ω ∂θ
t=1 θ∈Θ0
(A.13)
(A.14)
TESTING AND ESTIMATION OF GARCH MODELS
847
n
1 ∂2 (ω α β )
t
0
0
− I (i j) = o(1) a.s. for all i j ∈ {1 2}
sup
∂θi+1 ∂θj+1
ω∈ n
t=1
n
1
∂3
sup
t (θ) = O(1) a.s. for all i j k ∈ {2 3}
n t=1 θ∈Θ ∂θi ∂θj ∂θk
When γ0 = 0,
n
1 ∂2 (ω α β )
t
0
0
(A.15) sup
− I (i j) = oP (1) for all i j ∈ {1 2}
∂θi+1 ∂θj+1
ω∈ n
t=1
n
3
∂
1
(A.16)
sup
t (θ) = OP (1) for all i j k ∈ {2 3}
n t=1 θ∈Θ4 ∂θi ∂θj ∂θk
PROOF: Let us first suppose γ0 > 0. Using Proposition A.1(i) and Lemma A.1(i), σt2 (θ) > α2t−1 , and using arguments similar to those used to show
bt → 0 in Lemma A.1, for θ ∈ Θ0 , there exist a random variable K and
ρ ∈ (βe−γ0 1) such that for t large enough,
t
−ht η2t
∂
η2t
1 j−1
t
(A.17) t (θ) = 4
+ 2
+1
a.s.
β ≤ Kρ
∂ω
σt (θ)
σt (θ) j=1
vt (α β)
∞
Since, in view of Lemma A.3, t=1 Kρt (η2t /vt (α β) + 1) has a finite expectation, it is a.s. finite, uniformly in θ ∈ Θ0 . Thus (A.11) is proved and (A.12) can
be obtained by the same arguments. For brevity, we only prove (A.13) in the
n
case i = 2 and j = 3. First note that I (1 2) = limn→∞ n−1 t=1 dtα (α0 β0 )dtβ (α0 β0 ) a.s. Moreover, we have
t
∂2 t (ω α0 β0 )
h
1
t
j−1
= 2η2t 2 − 1
β 2
∂α ∂β
σt
σt2 j=1 0 t−j
t
1 j−2
×
(j − 1)β0 (ω + α0 2t−j )
σt2 j=2
t
1
h
t
j−2
(j − 1)β0 2t−j + 1 − η2t 2
σt
σt2 j=2
where σt2 = σt2 (ω α0 β0 ) We obtain the result by showing that, a.s. as t → ∞,
t
(A.18)
j−1
β 2
j=1 0 t−j
α
− dt (α0 β0 ) → 0
ht
848
(A.19)
C. FRANCQ AND J.-M. ZAKOÏAN
t
j−2
(j − 1)β0 (ω + α0 2t−j )
j=2
β
sup
− dt (α0 β0 ) → 0
h
ω∈
t
t
j−2
(j − 1)β0 2t−j
j=2
1 β
(A.20) − dt (α0 β0 ) → 0
ht
α0
ht
(A.21) sup 2 − 1 → 0
ω∈ σt
t
j−1
In view of σt2 − ht = j=1 β0 (ω − ω0 ) and 1/ ht = o(ρt ) for some ρ ∈
(e−γ0 β−1
0 ), the convergence (A.21) holds. The convergences in (A.18)–(A.20)
are obtained by the arguments used to establish the first convergence in
Lemma A.1. Next we turn to (A.14). For instance consider the case i = j = 2
and k = 3. We have, now with σt2 = σt2 (θ),
2
t
∂3 t (θ)
1
h
t
= 2 − 6η2t 2
βj−1 2t−j
∂2 α ∂β
σt
σt2 j=1
t
1 ×
(j − 1)βj−2 (ω + α2t−j )
σt2 j=2
t
t
2
h
1
t
βj−1 2t−j
(j − 1)βj−2 2t−j
+ 2η2t 2 − 1
σt
σt2 j=1
σt2 j=2
1
(dtα (α β))2 dtβ (α β)
= 2 − 3η2t
vt (α β)
1
2
+ 2η2t
− 1 dtα (α β) dtβ (α β) + o(1) a.s.
vt (α β)
α
where the term o(1) is again obtained by arguments similar to those used to
show the first convergence in Lemma A.1. Noting that dtα (α β) and dtβ (α β)
admit moments of any order, (A.14) then follows from the ergodic theorem,
Lemma A.3, and the Cauchy–Schwarz inequality.
In the case γ = 0, the a.s. convergences of the proof of (A.13) and (A.14)
can be replaced by Lp convergences via the arguments used to show (A.10).
We then obtain (A.15) and (A.16) from Proposition A.1(iii).
Q.E.D.
PROOF OF THEOREM 2.1(iv) and (v): Part (iv) has already been proven (see
Berkes, Horváth, and Kokoszka (2003) and Francq and Zakoïan (2004)).
TESTING AND ESTIMATION OF GARCH MODELS
849
It remains to prove the asymptotic normality of (α̂n β̂n ) when γ0 > 0 Notice that we cannot use the fact that the derivative of the criterion cancels at
θ̂n = (ω̂n α̂n β̂n ), since we have no consistency result for ω̂n . Thus the minimum could lie on the boundary of Θ, even asymptotically. However, the partial
derivative with respect to (α β) is asymptotically equal to zero at the minimum, since (α̂n β̂n ) → (α0 β0 ) and θ0 belongs to the interior of Θ. Hence, an
expansion of the criterion derivative gives
⎞
⎛
n
1 ∂
n
t (θ̂n ) ⎟
√
1 ∂
⎜ √n
∂ω
t (θ0 ) + Jn n(θ̂n − θ0 )
(A.22) ⎝
⎠= √
t=1
n t=1 ∂θ
0
where Jn is a 3 × 3 matrix whose elements have the form
1 ∂2
Jn (i j) =
t (θi∗ )
n t=1 ∂θi ∂θj
n
where θi∗ = (ω∗i α∗i β∗i ) is between θ̂n and θ0 . Since θ0 ∈ Θ0 , we have θi∗ ∈ Θ0
for n large enough. By (A.12) in Lemma A.5 and the compactness of Θ, we
have, for i = 2 3,
2
∞
1
∂
√
t (θ)
(A.23) Jn (i 1) n(ω̂n − ω0 ) ≤
sup
√n (ω̂n − ω0 ) → 0 a.s.
∂ω ∂θ
t=1 θ∈Θ0
An expansion of the function
1 ∂2
t (ω∗2 α β)
n t=1 ∂α2
n
(α β) →
gives
1 ∂2
t (ω∗2 α0 β0 )
n t=1 ∂α2
n
Jn (2 2) =
1 ∂3 t (ω∗2 α∗ β∗ )
+
n t=1 ∂(α β) ∂α2
n
α∗2 − α0
β∗2 − β0
where (α∗ β∗ ) is between (α∗2 β∗2 ) and (α0 β0 ). Using (A.13), (A.14), and the
consistency of (α∗ β∗ ), we get Jn (2 2) → I (1 1) a.s., and similarly for i j =
1 2,
(A.24)
Jn (i + 1 j + 1) → I (i j) a.s.
The conclusion follows by considering the last two components in (A.22), and
from Lemma A.4, (A.23), and (A.24).
Q.E.D.
850
C. FRANCQ AND J.-M. ZAKOÏAN
PROOF OF THEOREM 2.2: The proof of the asymptotic normality still relies
on the Taylor expansion (A.22). The asymptotic distribution of the first term
in the right-hand side of (A.22) is still given by Lemma A.4. To deal with the
second term, we cannot use (A.23) because (A.12) requires γ0 > 0 Instead,
noting that
1 j−1 2
1
β t−j ≤
2
σt (θ) j=1
α
t
and β∗2 < 1 for n large enough, we obtain
√
(A.25) |Jn (2 1) n(ω̂n − ω0 )|
t
t
(β∗2 )j−1 2t−j
(β∗2 )j−1
n 2
K
2ht ηt
j=1
j=1
≤√
+1
σt4 (θ2∗ )
n t=1 σt2 (θ2∗ )
n ht 1
K 2ht η2t
+1
≤√
∗
2
2
σt (θ2∗ ) ht
n t=1 σt (θ2 )
Hence, by Lemma A.3,
n
√
K 1
E|Jn (2 1) n(ω̂n − ω0 )| ≤ √
E n t=1 ht
The same bound is obtained when Jn (2 1) is replaced by Jn (3 1). Moreover,
ht = ω0 (1 + Zt−1 + Zt−1 Zt−2 + · · · + Zt−1 · · · Z1 ) + Zt−1 · · · Z0 σ02 By Assumption A, it follows that, for i = 2 3,
√
(A.26) E|Jn (i 1) n(ω̂n − ω0 )| → 0 as n → ∞
Finally, using Theorem 2.1(iii), (A.15), and (A.16), the a.s. convergence (A.24)
can be replaced by the same convergence in probability. The conclusion follows
Q.E.D.
as in the case γ0 > 0
PROOF OF THEOREM 2.3: The convergence results in (i) can be shown in a
standard way, using Taylor expansions of the functions κ̂η = κη (θ̂n ) and
1 1 ∂σt2 ∂σt2
(θ̂n )
n t=1 σt2 (θ̂n ) ∂θi ∂θj
n
around θ0 , and the ergodic theorem together with the consistency of θ̂n TESTING AND ESTIMATION OF GARCH MODELS
851
Now consider the case (ii). For some θ∗ = (ω∗ α∗ β∗ ) between θ̂n and θ0 ,
we have
n
n
1 ∂σt2 (θ∗ )
1 4 2 4t
(A.27) κ̂η =
ηt −
(θ̂n − θ0 )
n t=1
n t=1 σt4 (θ∗ ) σt2 (θ∗ ) ∂θ
1 4
η + Rn n t=1 t
n
:=
By Proposition A.1 and already given arguments, for some ρ ∈ (0 1),
2
n
ht
K 4
|Rn | ≤
ηt
n t=1
σt2 (θ2∗ )
× (ρt |ω̂n − ω0 | + dtα (α∗ β∗ )|α̂n − α0 | + dtβ (α∗ β∗ )|β̂n − β0 |)
= o(1) a.s.
where the last equality follows from the strong consistency of α̂n and β̂n , the
fact that |ω̂n − ω0 | is bounded by compactness of Θ, and the existence moments
at any order for dtα (α∗ β∗ ), dtβ (α∗ β∗ ), and ht /σt2 (θ∗ ). Hence the first part of
(ii) is proven. Now, similarly to (A.25), we have
t
(A.28)
n
1 ht
nJωα ≤
α̂
n σt2 (θ̂n )
t=1
β̂nj−1
j=1
ht
≤K
n
t=1
ht
2
t
σ (θ̂n )
ρt
for ρ ∈ (0 1) when n is large enough, by Proposition A.1(i). It follows that
nJωα = O(1) a.s. Similarly nJωβ = O(1) a.s. Moreover, we have nJωω ≥
1/σ14 (θ̂n ) > 0. Thus we have shown that
−1 Jαβω Jωω
Jωαβ = o(1) a.s.
Now we turn to Jαβαβ . Considering the top left term, a Taylor expansion
around θ0 gives
2
n
t
1
1
(A.29) Jαα =
β̂j−1 2
n t=1 σt2 (θ̂n ) j=1 n t−j
1 α
=
{d (α0 β0 )}2
n t=1 t
n
2
n 1 ∂σt2
1
α
2
(θ0 ) − {dt (α0 β0 )} + Sn +
n t=1 σt2 (θ) ∂α
852
C. FRANCQ AND J.-M. ZAKOÏAN
where, for θ∗ such that θ0 − θ∗ ≤ θ0 − θ̂n )
, |Sn | is bounded by
t
(β∗ )j−1 2t−j 2
n
K j=1
n
t=1
σt2 (θ∗ )
× {ρt |ω̂n − ω0 | + dtα (α∗ β∗ )|α̂n − α0 | + dtβ (α∗ β∗ )|β̂n − β0 |}
K
n t=1
n
+
t
t
∗ j−1 2
(β ) t−j (j − 1)(β∗ )j−2 2t−j j=1
j=1
σt2 (θ∗ )
σt2 (θ∗ )
|β̂n − β0 |
= o(1) a.s.
by already used arguments. Moreover, the second term in the right-hand side
of (A.29) converges to 0 a.s. by (A.10), while the first term is equal to 1/α20 =
I (1 1) since dtα (α0 β0 ) = vt (α0 β0 )/α0 = 1/α0 by Lemma A.2. We thus have
shown that Jαα a.s. converges to I (1 1). The other two terms in Jαβαβ can be
handled similarly, which completes the proof of (ii).
Turning to (iii), we note that ∂σt2 (θ∗ )/∂ω ≤ K for n large enough, since
β0 < 1. Moreover, σt2 (θ∗ ) ≥ ω∗ + α∗ β∗ 2t−2 . Therefore, (A.27) continues to hold
with |Rn | bounded by
2
n
K 4
ht
1
ηt
∗
2
∗
n t=1
σt (θ2 ) ω + α∗ β∗ 2t−2
2
n
ht
K 4
+
η
n t=1 t σt2 (θ2∗ )
× (dtα (α∗ β∗ )|α̂n − α0 | + dtβ (α∗ β∗ )|β̂n − β0 |)
Therefore, |Rn | = oP (1) by Proposition A.1(iii), the weak consistency of α̂n and
β̂n , and the existence of moments for dtα (α∗ β∗ ), dtβ (α∗ β∗ ), and ht /σt2 (θ2∗ ).
Hence κ̂η → κ in probability. Now, in view of the first inequality of (A.28),
n
we have nJωα ≤ K t=1 σ 2h(θ̂t ) h1t and thus nJωα = OP (1) by Lemma A.3 and
n
t
the arguments used to show (A.26). Proceeding as in the proof of (i), we de−1 duce that Jαβω Jωω
Jωαβ = oP (1) By similar arguments, the right-hand side of
(A.29) converges to I (1 1) in probability.
Q.E.D.
PROOF OF PROPOSITION 2.1: If a consistent estimator θ̂n existed, then the
test of critical region C = {
θ̂n − θ1 > θ̂n − θ1∗ } would have null asymptotic errors of the first and second kind, in contradiction with Proposition 3.1.
Q.E.D.
TESTING AND ESTIMATION OF GARCH MODELS
853
A.3. Constrained QMLE of (α0 β0 )
PROOF OF (2.8): By the arguments used to prove Theorem 2.1(iii), we have
(A.30)
(α̂cn (ω) β̂cn (ω)) → (α0 β0 ) in probability as n → ∞
A Taylor expansion of the criterion derivative gives
∂
1 t (ω α̂cn (ω) β̂cn (ω))
0= √
n t=1 ∂(α β)
n
(A.31)
1 ∂
=√
t (ω α0 β0 )
n t=1 ∂(α β)
n
∂2
1
∗
∗
t (ω α β )
+
n t=1 ∂(α β) ∂(α β)
√
× n(α̂cn (ω) − α0 β̂cn (ω) − β0 ) n
where (α∗ β∗ ) is between (α̂cn (ω) β̂cn (ω)) and (α0 β0 ). Another Taylor expansion yields, for (α∗∗ β∗∗ ) between (α̂cn (ω) β̂cn (ω)) and (α0 β0 ),
n
1 ∂2
t (ω α0 β0 )
n
∂(α β) ∂(α β)
t=1
n
∂2
1
∗
∗ (ω
α
β
)
−
t
n t=1 ∂(α β) ∂(α β)
n ∂3
1 ∗
∗∗
∗∗ ≤ |α − α0 |
t (ω α β )
n t=1 ∂α ∂(α β) ∂(α β)
n ∂3
1 ∗∗
∗∗ + |β∗ − β0 |
(ω
α
β
)
t
n t=1 ∂β ∂(α β) ∂(α β)
n
∂3
1
∗
≤ |α − α0 |
t (ω α β)
sup
n t=1 θ∈Θ ∂α ∂(α β) ∂(α β)
n
3
∂
1
∗
+ |β − β0 |
t (ω α β)
sup
n t=1 θ∈Θ ∂β ∂(α β) ∂(α β)
= oP (1)
854
C. FRANCQ AND J.-M. ZAKOÏAN
using (A.30) and (A.16). Therefore, using (A.15), the term in parentheses in
(A.32) converges to I To conclude, it remains to prove that
1 ∂
d
t (ω α0 β0 ) → N (0 (κη − 1)I )
√
n t=1 ∂(α β)
n
(A.32)
We have
1 ∂
t (ω α0 β0 )
√
n t=1 ∂(α β)
n
1 ∂
=√
t (ω0 α0 β0 )
n t=1 ∂(α β)
n
ω − ω0 ∂2
t (ω∗ α0 β0 )
√
n t=1 ∂ω ∂(α β)
n
+
where ω∗ is between ω0 and ω. The last term tends to zero in probability, using
Assumption A, similarly to (A.26). The first term converges in distribution to
the normal law of (A.32) by Lemma A.4.
Q.E.D.
PROOF OF PROPOSITION 2.2: The ergodic theorem entails that, for β < 1,
almost surely
2t
1
+ log σt2 (ω α β)
Ln (α β) :=
n t=1 σt2 (ω α β)
ht
→ L(α β) = E∞ 2
+ log σt2 (ω α β)
σt (ω α β)
n
as n → ∞. The dominated convergence theorem implies that
1 ∂σt2
ht
∂L
(α0 β0 ) = E∞ 1 − 2
(ω α0 β0 )
∂α
σt (ω α0 β0 ) σt2 ∂α
1 ∂σt2
i
(ω α0 β0 ) = 0
= (ω − ω0 )
β0 E∞
σt4 ∂α
i≥0
It follows that the minimum of the function L(α β) is reached at (α∗ β∗ ) =
(α0 β0 ).
A Taylor expansion of Ln (·) yields
(A.33)
Ln {α̂cn (ω) α̂cn (ω)} = Ln (α0 β0 ) +
+
∂Ln
(α̃n β̃n ){α̂cn (ω) − α0 }
∂α
∂Ln
(α̃n β̃n ){β̂cn (ω) − β0 }
∂β
TESTING AND ESTIMATION OF GARCH MODELS
855
where (α̃n β̃n ) is between (α̂cn (ω) β̂cn (ω)) and (α0 β0 ). Note that since E4t <
∞, almost surely, for β < 1,
∂Ln
(α β0 )
lim sup sup sup
n→∞
α β<β ∂α
i
β 2t−1−i
n 1
2t i≥0
≤ lim
< ∞
1+
n→∞ n
ω
ω
t=1
∂Ln
lim sup sup sup
(α β0 )
n→∞
α β<β ∂β
i
2
β σt−1−i
n 2
i≥0
1
≤ lim
1+ t
< ∞
n→∞ n
ω
ω
t=1
Now suppose that
(A.34)
(α̂cn (ω) β̂cn (ω)) → (α0 β0 ) in probability as n → ∞
Then it follows from (A.33) that
Ln {α̂cn (ω) β̂cn (ω)} → L(α0 β0 )
in probability as n → ∞
Then taking the limit in probability in the inequality
Ln {α̂cn (ω) β̂cn (ω)} ≤ Ln (α∗ β∗ )
we find that L(α0 β0 ) ≤ L(α∗ β∗ ) which is in contradiction with the definition
Q.E.D.
of (α∗ β∗ ) = (α0 β0 ). Thus (A.34) cannot be true.
A.4. Stationarity Test
n
PROOF OF THEOREM 3.1: Let γn (θ) = n−1 t=1 log{αη2t (θ)+β} and ηt (θ) =
t /σt (θ). First consider the case γ0 < 0. A Taylor expansion gives
(A.35)
γ̂n = γn (θ0 ) +
∂γn (θ0 )
(θ̂n − θ0 ) + oP n−1/2
∂θ
with
(A.36)
n
−1 ∂γn (θ0 )
=
∂θ
n
t=1
⎧
⎛ ⎞⎫
0 ⎬
⎨
2
1
2 1 ∂σt (θ0 )
⎝
− η2t ⎠
α0 ηt
⎭
a0 (ηt ) ⎩
ht ∂θ
1
= −Ψ + oP (1)
856
C. FRANCQ AND J.-M. ZAKOÏAN
where, in view of α0 η21 /a0 (η1 ) = 1 − β0 /a0 (η1 ),
Ψ = (1 − ν1 )Ω − a
Ω = E∞
1 ∂σt2 (θ0 )
ht ∂θ
Moreover, the QMLE satisfies
(A.37)
n
√
1 ∂σt2 (θ0 )
−1 1
+ oP (1)
n(θ̂n − θ0 ) = −J √
(1 − η2t )
ht ∂θ
n t=1
In view of (A.35), (A.36), and (A.37), we have
√
(A.38)
n(γ̂n − γ0 )
1 1 ∂σt2 (θ0 )
1 + oP (1)
ut + Ψ J −1 √
(1 − η2t )
=√
ht ∂θ
n t=1
n t=1
n
n
Note that
1 1 ∂σt2 (θ0 )
1 ut Ψ J −1 √
(1 − η2t )
Cov √
ht ∂θ
n t=1
n t=1
n
n
= cΩ J −1 Ψ
where c = Cov(ut 1 − η2t ) The Slutsky lemma and the central limit theorem
for martingale differences thus entail
√
d
n(γ̂n − γ0 ) → N (0 σu2 + 2cΩ J −1 Ψ + (κη − 1)Ψ J −1 Ψ )
Now let θ0 = (ω0 α0 0) . Noting that θ0 ∂σt2 (θ0 )/∂θ = ht almost surely, we have
E
1 ∂σt2 (θ0 )
1 ∂σt2 (θ0 )
= 0
1−
θ
0
ht ∂θ
ht ∂θ
which entails J θ0 = Ω and Ω J −1 Ω = 1. It follows that Ω J −1 Ψ = 0. Noting
that Ψ J −1 Ψ = a J −1 a − (1 − ν1 )2 , the asymptotic distribution in (3.5) follows
in the case γ0 < 0.
Now consider the case γ0 ≥ 0. Let θn∗ be a sequence such that θn∗ − θ0 ≤
θ̂n − θ0 By Proposition A.1 (using Assumption A when γ0 = 0), we have
1 1 ∂σt2 (θn∗ )
√
n t=1 σt2 (θn∗ ) ∂ω
n
= o(1) a.s.
(resp., in probability) as
n→∞
TESTING AND ESTIMATION OF GARCH MODELS
857
when γ0 > 0 (resp., when γ0 = 0). It can be deduced that under the same con√ 2 γn (θn∗ )
= o(1) and
ditions, n ∂ ∂ω
∂θ
√
∂2 γn (θn∗ )
n(θ̂ − θ0 )
(θ̂ − θ0 ) = o(1)
∂θ ∂θ
which entails that (A.35) still holds. By the same arguments, (A.36) holds with
⎞
⎛ ⎞
⎞ ⎛
⎛
0
0
0
α
⎠
⎝
⎠
⎝
⎝
and Ψ = 0 ⎠ Ω = E dt (θ0 ) =
1/α0
β
ν1 /{β0 (1 − ν1 )}
0
dt (θ0 )
The conclusion follows.
Q.E.D.
The proof of Corollary 3.2 relies on arguments already used and is omitted.
PROOF OF COROLLARY 3.3: By arguments used in the proof of Theo2
surely to σu2 when γ < 0 or γ ≥ 0. Therefore,
rem 2.3,
√ σ̂u converges almost
√
Tn = n(γ̂n − γ0 )/σ̂u + nγ0 /σ̂u converges in probability to −∞ when γ < 0,
Q.E.D.
to +∞ when γ > 0, and in distribution to N (0 1) when γ0 = 0.
A.5. Inconsistency of Tests for ω
PROOF OF PROPOSITION 3.1: The most powerful test is the Neyman–
Pearson test of rejection region C = {Sn > cn }, where
Sn =
n
t=1
2t
2t
2
+
log
σ
− log σt2 (θ1∗ )
(θ
)
−
1
t
σt2 (θ1 )
σt2 (θ1∗ )
and cn is a positive constant corresponding to the α-quantile of the (continuous) distribution of Sn under H0 (see, e.g., Lehmann and Romano,
(2005, Theorem 3.2.1)). By Proposition A.1(i), noting that σt2 (θ1∗ ) − σt2 (θ1 ) =
t
j−1
∗
j=1 β1 (ω1 − ω1 ), there exists ρ ∈ (0 1) such that
2 ∗
2 ∗
σt (θ1 ) − σt2 (θ1 ) σt (θ1 ) − σt2 (θ1 ) t
≤ Kρt
(A.39) ≤ Kρ and σt2 (θ1∗ )
σt2 (θ1 )
under both H0 and H1 . Therefore, for some measurable function ϕ(·), as n →
∞,
Sn → S0 = ϕ(η21 η22 ; θ1 θ1∗ ) a.s. under H0
and
Sn → S1 = −ϕ(η21 η22 ; θ1∗ θ1 ) a.s. under H1 .
858
C. FRANCQ AND J.-M. ZAKOÏAN
More precisely, in the ARCH(1) case, we have
(A.40)
S0 =
∞
2
t t−1
ης
+ log(1 − ςt−1 )
S1 =
∞
t=1
∗
∗
η2t ςt−1
− log(1 + ςt−1
)
t=1
where
ςt−1 =
ω∗1 − ω1
t−1
ω∗1 + ω1
αk1 η2t−1 · · · η2t−k
k=1
∗
ςt−1
=
ω∗1 − ω1
t−1
ω1 + ω∗1
αk1 η2t−1 · · · η2t−k
k=1
In the GARCH(1 1) case, (A.40) still holds with
t
ςt−1 =
t
β1 (ω∗1 − ω1 )
j−1
j=1
2
t
∗
1
σ (θ )
∗
ςt−1
=
β1 (ω∗1 − ω1 )
j−1
j=1
σt2 (θ1 )
Using (A.39), it follows that
(A.41)
|S0 − S1 | ≤ K|ω1 − ω∗1 |
∞
ρt (η2t + 1)
t=t0
Since the laws of S0 and S1 are continuous when ω1 = ω∗1 , the power of the
Neyman–Pearson test tends to
lim PH1 (Sn > cn ) = P(S1 > c)
n→∞
where c is such that P(S0 > c) = α
For any ε > 0, we have
P(S1 > c) ≤ P(S0 + |S1 − S0 | > c) ≤ P(S0 > c − ε) + P(|S1 − S0 | > ε)
In the right-hand side of the last inequality, by continuity, the first probability is
close to α when ε is close to zero, and in view of (A.41), the second probability
is close to zero when |ω1 − ω∗1 | is small. It follows that when |ω1 − ω∗1 | is small,
P(S1 > c) < 1, which shows the inconsistency of the Neyman–Pearson test and
thus that of any test.
Q.E.D.
TESTING AND ESTIMATION OF GARCH MODELS
859
A.6. Stationarity Test in Nonlinear GARCH Models
PROOF OF PROPOSITION 4.1: We start by considering the case Γ > 0 By the
arguments given in the proof of Proposition A.1(i), ht → ∞ and 2t → ∞ a.s.
at an exponential rate as t → ∞. Moreover, it can be seen that the analog of
Lemma A.1(i) still holds. Indeed, for all θ such as β < eΓ , when t → ∞, then
|σt2 (θ)/ ht − wt (α β)| → 0 where, similar to (A.5), (wt (α β)) is defined as the
stationary solution of
(A.42)
b0 (ηt−1 )wt (α β) = βwt−1 (α β) + αη2t−1 Moreover, for all θ such as β ≥ eΓ , σt2 (θ)/ ht → ∞ a.s. as t → ∞.
The pseudo-true value is thus the solution of
(α∗ β∗ ) = arg min E{wt−1 (α β) − 1 + log wt (α β)}
(αβ)∈Θαβ
The consistency of (α̂n β̂n ) to (α∗ β∗ ) can then be shown by the arguments
used for the proof of Theorem 2.1(ii).
It follows that
n
n
η2t
1
a0 (ηt )wt+1 (α̂n β̂n )
o(1) 1
log α̂n
log
+ β̂n =
γ̂n =
n t=1
n t=1
wt (α̂n β̂n )
wt (α̂n β̂n )
→Γ
a.s. as n → ∞
For the previous convergence, we note that for some neighborhood V ∗
of (α∗ β∗ ), we have E sup(αβ)∈V ∗ | log w1 (α β)| < ∞, which entails
sup(αβ)∈V (θ∗ ) log wn+1 (α β)/n → 0 a.s. The latter moment condition comes
from the fact that the strict stationarity condition entails β∗ < eΓ , which entails
the existence of a moment of order s > 0 for wn+1 (α∗ β∗ ) (by the arguments
given in Lemma 2.3 of Berkes, Horváth, and Kokoszka (2003)).
The convergence of σ̂u2 to
a(η0 )v1 (α∗ β∗ )
σ∗2 = Var log
v0 (α∗ β∗ )
is obtained by similar arguments, noting that E| log w1 (α∗ β∗ )|2 < ∞ because
Ew1s (α∗ β∗ ) < ∞ for some s > 0 and because log w1 (α∗ β∗ ) ≥ α∗ log η20 −
log b0 (η0 ). By (A.42), the a.s. limit of σ̂u2 also can be written as Var log{α∗ η21 /
w1 (α∗ β∗ ) + β∗ }. This limit is positive, unless η21 is measurable with respect
to the sigma-field generated by {η − u u < 1}, which is impossible since the
assumption E| log η21 |2 > 0 entails that η21 has a nondegenerate distribution.
Now consider the case Γ < 0 Using standard arguments,
σ 2 (θ∗ ) − ω∗
2t
∗
∗
∗
Γ = E∞ log α 2 ∗ + β = E∞ log t+1 2 ∗
< 0
σt (θ )
σt (θ )
860
C. FRANCQ AND J.-M. ZAKOÏAN
The convergence of σ̂u2 to a positive limit is obtained by arguments already
used.
Q.E.D.
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VAART, A. W. (1998): Asymptotic Statistics. Cambridge, U.K.: Cambridge University
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University Lille 3 (EQUIPPE), BP 60149, 59653 Villeneuve d’Ascq cedex,
France and CREST; [email protected]
and
CREST, 15 Boulevard Gabriel Peri, 92245 Malakoff Cedex, France and University Lille 3 (EQUIPPE); [email protected].
Manuscript received July, 2010; final revision received August, 2011.