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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

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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.

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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 (1.7)

1 θˆ n = (ω ˆ n  αˆ n  βˆ n ) = arg min θ∈Θ n

n  t=1

t (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 ηˆ t = ηt (θˆ n ), 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 γˆ n = log(αˆ n ηˆ 2t + βˆ n ) 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.

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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 (αˆ n  βˆ n ) 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(αˆ n − α0  βˆ n − 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 (αˆ n  βˆ n ) 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 αˆ n and βˆ n 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.

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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 αˆ n → α0 

βˆ n → β0

and ω ˆ n → ω0

a.s.

as n → ∞

(ii) When γ0 > 0, if P(η1 = 0) = 0, then and βˆ n → β0

αˆ n → α0

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 and βˆ n → β0

αˆ n → α0

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(θˆ n − 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(αˆ n − α0  βˆ n − 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 (αˆ n  βˆ n ) 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.

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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 (αˆ n  βˆ n ) is asymptotically normal and its asymptotic distribution is given by (2.3). 2.2. Estimating the Asymptotic Variance of (αˆ n  βˆ n ) Without Assuming Stationarity In view of (2.1) and (2.2), when γ0 < 0, the asymptotic distribution of the QMLE (αˆ n  βˆ n ) of (α0  β0 ) is given by (2.5)

√ d n(αˆ n − α0  βˆ n − 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 (θˆ n ) Jαβαβ = n t=1 σt4 (θˆ n ) ∂(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 (αˆ n  βˆ n ) 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 κˆ η = n−1 t=1 ηˆ 4t , where ηˆ t = t /σt (θˆ n ). I∗ → I∗ a.s. as n → ∞. (i) When γ0 < 0, we have κˆ η → κη and  (ii) When γ0 > 0, we have κˆ η → κη and  I∗ → I a.s. (iii) When γ0 = 0, we have κˆ η → κη and if A is satisfied,  I∗ → I in probability. In any case, (κˆ η − 1) I∗−1 is a consistent estimator of the asymptotic variance of the QMLE of (α0  β0 ). 2.3. Inconsistency of ω ˆ n 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.

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The inconsistency of ω ˆ n 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 (αˆ n  βˆ n ) 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 (2.7)

1 t (ω α β) (αβ)∈Θαβ n t=1

(αˆ (ω) βˆ cn (ω)) = arg min c n

n

was studied by Jensen and Rahbek (2004b). They proved that when γ0 > 0, (2.3) continues to hold when the global QMLE (αˆ n  βˆ n ) is replaced by the local and constrained QMLE (αˆ cn (ω) βˆ cn (ω)) In Appendix A.3, we prove that, under the assumptions of Theorem 2.2, in particular, γ0 = 0 (2.8)

√ d n(αˆ cn (ω) − α0  βˆ cn (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)

(αˆ cn (ω) βˆ cn (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.

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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(aαˆ n + bβˆ n − c)



:=  > −1 (1 − α) (κˆ η − 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)

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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(γˆ n − 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 γˆ n 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(αˆ n ηˆ t + βˆ n )} − log(αˆ n ηˆ t + βˆ n )  σˆ u = n t=1 n t=1 σˆ γ2 = σˆ u2 + (κˆ η − 1){aˆ  J−1 aˆ − (1 − νˆ 1 )2 }   n βˆ n 1 1 − νˆ 1 νˆ 1   νˆ 1 =  aˆ = 0 αˆ n n t=1 αˆ n ηˆ 2t + βˆ n βˆ n COROLLARY 3.2: Under the assumptions of Theorem 3.1, (3.6)

σˆ γ2 → 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   σˆ γ −1 σˆ γ −1 γˆ n − √  (1 − α/2) γˆ n + √  (1 − α/2)  n n The following result provides asymptotic critical regions for the strict stationarity testing problems.

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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   √ γˆ n ST −1 C = Tn := n >  (1 − α) (3.7) σˆ u 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 γˆ n defined by (1.8), where θˆ n 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).

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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→∞ (αˆ n  βˆ n ) a.s., γˆ n → Γ

and

σˆ u2 → σ∗2

a.s.

for some constant

σ∗2 > 0

If Γ < 0, then, under regularity conditions that imply the strong consistency of θˆ n 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 ∗ σˆ → Var log α 2 ∗ + β > 0 a.s. σt (θ ) 

∗

γˆ n → 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 (αˆ n  βˆ n ) is always CAN, whereas ω ˆ n 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 =

√ γˆ n − γ0 √ γ0 n + n → −∞ σˆ u σˆ u

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 αˆ n . In any case, the value of αˆ n + βˆ n 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 αˆ n βˆ n 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 (αˆ n  βˆ n ) 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 (ω ˆ n  αˆ n  βˆ n ) = 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 )

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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)).

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It remains to prove the asymptotic normality of (αˆ n  βˆ n ) when γ0 > 0 Notice that we cannot use the fact that the derivative of the criterion cancels at θˆ n = (ω ˆ n  αˆ n  βˆ n ), since we have no consistency result for ω ˆ n . 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 (αˆ n  βˆ n ) → (α0  β0 ) and θ0 belongs to the interior of Θ. Hence, an expansion of the criterion derivative gives ⎞ ⎛ n 1  ∂ n ˆ t (θn ) ⎟ √ 1  ∂ ⎜ √n ∂ω t (θ0 ) + Jn n(θˆ n − 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 θˆ n 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  ∂ √  ˆ n − ω0 ) → 0 a.s. t (θ) (A.23) Jn (i 1) n(ω ˆ n − ω0 ) ≤ sup  √n (ω ∂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.

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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(ω ˆ n − 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(ω ˆ n − 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(ω ˆ n − 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 κˆ η = κη (θˆ n ) and 1  1 ∂σt2 ∂σt2 ˆ (θn ) n t=1 σt2 (θˆ n ) ∂θi ∂θj n

around θ0 , and the ergodic theorem together with the consistency of θˆ n 

TESTING AND ESTIMATION OF GARCH MODELS

851

Now consider the case (ii). For some θ∗ = (ω∗  α∗  β∗ ) between θˆ n and θ0 , we have n n 1 ∂σt2 (θ∗ ) ˆ 1  4 2  4t (A.27) κˆ η = ηt − (θn − 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 |ω ˆ n − ω0 | + dtα (α∗  β∗ )|αˆ n − α0 | + dtβ (α∗  β∗ )|βˆ n − β0 |) = o(1) a.s. where the last equality follows from the strong consistency of αˆ n and βˆ n , the fact that |ω ˆ n − 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 ≤ ˆn) α ˆ n σt2 (θ t=1

βˆ nj−1

j=1

ht

≤K

n  t=1

ht ρt σ (θˆ n ) 2 t

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 (θˆ n ) > 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 = βˆ j−1 2 n t=1 σt2 (θˆ n ) 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

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C. FRANCQ AND J.-M. ZAKOÏAN

where, for θ∗ such that θ0 − θ∗ ≤ θ0 − θˆ n ) , |Sn | is bounded by t  (β∗ )j−1 2t−j 2  n K  j=1

n

t=1

σt2 (θ∗ )

× {ρt |ω ˆ n − ω0 | + dtα (α∗  β∗ )|αˆ n − α0 | + dtβ (α∗  β∗ )|βˆ n − 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∗ )

|βˆ n − 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∗ )|αˆ n − α0 | + dtβ (α∗  β∗ )|βˆ n − β0 |) Therefore, |Rn | = oP (1) by Proposition A.1(iii), the weak consistency of αˆ n and βˆ n , and the existence of moments for dtα (α∗  β∗ ), dtβ (α∗  β∗ ), and ht /σt2 (θ2∗ ). Hence κˆ η → κ in probability. Now, in view of the first inequality of (A.28), n we have nJωα ≤ K t=1 σ 2h(θtˆ ) 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 θˆ n existed, then the test of critical region C = { θˆ n − θ1 > θˆ n − θ1∗ } would have null asymptotic errors of the first and second kind, in contradiction with Proposition 3.1. Q.E.D.

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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)

(αˆ cn (ω) βˆ cn (ω)) → (α0  β0 ) in probability as n → ∞

A Taylor expansion of the criterion derivative gives ∂ 1  t (ω αˆ cn (ω) βˆ cn (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(αˆ cn (ω) − α0  βˆ cn (ω) − β0 )  n

where (α∗  β∗ ) is between (αˆ cn (ω) βˆ cn (ω)) and (α0  β0 ). Another Taylor expansion yields, for (α∗∗  β∗∗ ) between (αˆ cn (ω) βˆ cn (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)

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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 {αˆ cn (ω) αˆ cn (ω)} = Ln (α0  β0 ) + +

∂Ln (α˜ n  β˜ n ){αˆ cn (ω) − α0 } ∂α

∂Ln (α˜ n  β˜ n ){βˆ cn (ω) − β0 } ∂β

TESTING AND ESTIMATION OF GARCH MODELS

855

where (α˜ n  β˜ n ) is between (αˆ cn (ω) βˆ cn (ω)) and (α0  β0 ). Note that since E4t < ∞, almost surely, for β < 1,     ∂Ln  (α β0 ) lim sup sup sup n→∞ α β 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 .

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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.

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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 (αˆ n  βˆ n ) 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 (αˆ n  βˆ n ) o(1) 1 ˆ log αˆ n log + βn = γˆ n = n t=1 n t=1 wt (αˆ n  βˆ n ) wt (αˆ n  βˆ n ) →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 σˆ u2 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 σˆ u2 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 )

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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.