Working Paper Series
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National Centre of Competence in Research Financial Valuation and Risk Management
Working Paper No. 66
Robust GMM Tests for Structural Breaks
Patrick Gagliardini Fabio Trojani Giovanni Urga
First version: May 2003 Current version: June 2003
This research has been carried out within the NCCR FINRISK project on
“Interest Rate and Volatility Risk”.
___________________________________________________________________________________________________________
First Version: May 2003. This Version: June 2003
Robust GMM Tests for Structural Breaks
∗Abstract
We propose a class of new robust GMM tests for endogenous structural breaks. The tests are based on supremum and average statistics derived from robust GMM estimators with a bounded influence function.
They imply a bounded linearized asymptotic bias of size and power under local model misspecifications.
We illustrate the trade-off between robustness and efficiency of the new robust tests in some Monte Carlo simulation where we find that the performance of classical GMM tests is highly unstable under local model misspecifications. In particular, the loss in power can be very substantial in some models. Similarly, the size of the classical tests can be very unstable, especially in models based on nonlinear orthogonality functions. Robust testing procedures yield important power improvements under a misspecified model while under an exact reference model they yield a performance comparable to the one of classical tests.
Moreover, robust procedures stabilize the size under a model misspecification for those models where important size instabilities of the classical tests have been observed. Finally, classical testing procedures based on the supremum functional seem to be less stable than those based on the average, implying larger power improvements of robust procedures in this case when compared with the case of an average statistic.
Keywords: Robust Tests, Structural Breaks, Monte Carlo J.E.L. Classification Numbers: C10, C12, C13, C15.
Patrick Gagliardini, Institute of Finance, University of Southern Switzerland, Via Buffi 13,
CH-6900 Lugano, e-mail: [email protected].
Fabio Trojani, Institute of Finance, University of Southern Switzerland, Via Buffi 13,
CH-6900 Lugano, e-mail: [email protected].
Giovanni Urga, Faculty of Finance, Cass Business School, 106 Bunhill Row, London EC1Y 8TZ (U.K.).
Tel.: +/44/20/70408696, Fax.: ++/44/20/70408885, e-mail: [email protected]
http://www.staff.city.ac.uk/~giourga.
∗We wish to thank Loriano Mancini, Claudio Ortelli, Paolo Porchia and participants to the International Con- ference on MODELLING STRUCTURAL BREAKS, LONG MEMORY AND STOCK MARKET VOLATILITY (held in London at Cass Business School, 6-7 December 2002) for discussions and helpful comments. G. Urga wishes to thank ESRC under Grant number R000238145 (”New methods for detecting multiple structural breaks in time series ”) for funding this research. Patrick Gagliardini and Fabio Trojani gratefully acknowledge the financial support of the Swiss National Science Foundation (grant 12-65196.01 and NCCR FINRISK).
1 Introduction
We propose a class of new Generalized Method of Moments (GMM, Hansen (1982)) tests for en- dogenous structural breaks (Andrews (1993)) that are robust to local model misspecifications of a given reference model. GMM based test statistics defining tests for structural breaks are typi- cally obtained as the supremum, the average or some related functional of sequences of quadratic GMM statistics (see also Andrews and Ploberger (1994)), each being asymptotically chi-square distributed under the null of no break1 . Such GMM functionals are typically evaluated at some GMM model parameter estimate, conditionally on a given break date. A general GMM statistical functional (as for instance a GMM estimator or the size/power of a GMM test) is robust to local model misspecifications if and only if it is based on a GMM model with a bounded orthogonal- ity function. Moreover, non robust quadratic GMM statistics can be robustified by applying a weighted orthogonality function that bounds optimally the influence of a general model misspeci- fication (see Ronchetti and Trojani (2001)). This defines robust GMM (RGMM) estimation and testing procedures of simple parametric hypotheses in a fairly general GMM setting. In this paper, we propose a class of robust supremum and average tests for structural breaks, which are defined over sequences of quadratic RGMM statistics based on a bounded orthogonality function.
The need for robust statistical procedures for estimation and testing has been stressed by many authors and is now widely recognized (see for instance, Hampel (1974), Koenker and Bassett (1978), Huber (1981), Peracchi (1990, 1991), Heritier and Ronchetti (1994), Krishnakumar and Ronchetti (1997), Ronchetti and Trojani (2001), Ortelli and Trojani (2002) and Mancini, Ronchetti and Trojani (2003)). In particular, it is now well-known that a necessary robustness requirement for a Maximum-Likelihood (ML)-type GMM test based on an asymptotically chi-square distributed statistic is a bounded influence function (Hampel (1974)) of the GMM estimator defining the statistic. Thus, GMM estimators with unbounded influence function do not imply robust inference
1 Asymptotic critical values for GMM tests of structural breaks are derived by the Functional Central Limit Theorem and have typically to be computed by simulation; see for instance Andrews (1993) and Andrews (2003).
Analytical approximations have been proposed in Hansen (1997).
procedures for tests based on ML-type, asymptotically chi-square distributed, GMM statistics.
This implies that tests for structural breaks defined via functionals of sequences of such statistics cannot ensure the local robustness of inferences on structural breaks. Therefore, we construct supremum and average tests for breaks using sequences of RGMM statistics with bounded influence function. In the GMM setting, a bounded influence function of a test statistic is equivalent to the boundedness of the given orthogonality function. Therefore, RGMM statistics are obtained by truncating appropriately the unbounded orthogonality function of a nonrobust GMM model setting; see again Ronchetti and Trojani (2001).
We quantify the performance of our robust testing procedures for structural breaks in some Monte Carlo experiments of a few GMM model settings. This issue has been so far largely unexplored in the literature. For the estimation problem, Fiteni (2002) derived the asymptotic properties of a robust break date estimator defined through the supremum functional over a sequence of robustified loss functions. These results apply to a standard linear regression model setting. To our knowledge, the robust testing problem has been not addressed so far in the literature. We propose a class of general RGMM tests, applying to linear and nonlinear model settings, and quantify the contribution of robustness when testing for breaks in a few Monte Carlo simulations.
Our findings are as follows. First, the performance of classical GMM tests is highly unsta- ble under local model misspecifications. In particular, the loss in power in the presence of local departures from conditional normality can be substantial in some models. Similarly, the size of the classical GMM tests can be very unstable, especially in GMM models based on nonlinear orthogonality functions. Second, robust testing procedures for structural breaks yield important power improvements under a locally misspecified model. At the same time, under an exact refer- ence model, they have power properties comparable to those of the classical tests. Third, robust procedures stabilize the size under a model misspecification for those models where important size instabilities of the classical tests have been observed. Finally, classical testing procedures
based on the supremum functional seem to be less stable than those based on the average, im- plying larger improvements of robust procedures upon classical methodologies, when applied to supremum functionals.
The rest of the paper is organized as follows. Section 2 reviews Andrews (1993) GMM testing setting for structural breaks. Section 3 introduces robust GMM tests for structural breaks and studies formally their local robustness properties in neighborhoods of a reference model. Section 4 analyzes by Monte Carlo simulations the empirical properties of the new robust tests in linear and nonlinear GMM testing settings, while Section 5 concludes and gives suggestions for further developments.
2 GMM tests for structural breaks
We briefly review Andrews (1993) GMM testing setting for structural breaks and we write the relevant statistics as functionals on a suitable set of probability distributions. This formalism will allow us to analyze in a second step the robustness properties of GMM tests for structural breaks. In the following we adopt the symbol =⇒ to denote weak convergence in the sense of Pollard (1984, pp. 64-66) for sequences of random elements of a space of bounded Euclidean valued cadlag functions on Π ⊂ [0, 1], equipped by the supremum norm topology and by the corresponding Borel sigma algebra. The symbol →d denotes convergence in distribution, ∇ denotes the gradient operator, B¡
Rk¢
is the Borel sigma algebra on Rk, k·k is the Euclidean norm.
2.1 Hypotheses of structural changes
We consider a parametric model indexed by parameters (βt, δ0) ∈ Θ = B × ∆ ⊂ Rp× Rq, for t = 1, 2, .., and test the null hypothesis of parameter stability:
H0: βt= β0 for all t ≥ 1 and some β0∈ B ⊂ Rp . (1)
Several alternative hypotheses may be of interest in the present setting. The simple one time change alternative with known change point2 π ∈ Π ⊂ (0, 1) is given by:
H1T(π) : βt=
β1(π) for t = 1, .., T π β2(π) for t = T π + 1, .., T
, (2)
for some constant vectors β1(π), β2(π) ∈ B. A natural alternative where the change point π ∈ Π ⊂ (0, 1) is unknown is:
HA(Π) = [
π∈Π
H1T(π) .
In this case one tests for the presence of a break in the known interval Π. Finally, when applying tests for structural breaks as general diagnostics tools, a natural alternative may be
H1: βs6= βt for some s, t ≥ 1 .
Although this hypothesis is more general than ∪Π⊂(0,1)HA(Π) the robust GMM tests for structural breaks considered in this paper have power3 also against H1.
2.2 Full sample GMM estimators
Let W = {Wt: t ≥ 1} be a stochastic process with values in W ⊂ Rk, defined on a measur- able space (Ω, F), and let m : Rk × Θ → Rυ be an orthogonality function4 . GMM tests for structural breaks can be defined based on full sample or partial sample GMM estimators (see An- drews (1993)). Since in Section 2.3 below we will focus on testing procedures based on Lagrange Multiplier (LM) statistics, we can consider here for brevity only the first class of GMM estimators.
A full sample GMM (FS-GMM) estimator eθ = µeβ
0
, eδ
0¶0
is the asymptotic functional solution of the orthogonality condition
0 = lim
T→∞
1 T
XT t=1
EP[m (Wt, β, δ)] , (3)
2For technical reasons Π is assumed to be a closed set (see also Andrews (1993)).
3See Andrews (1993) and Andrews and Ploberger (1994).
4 For brevity and easy notation we focus on the exactly identified case (ν = p + q), even if the overidentified case can be treated along the same lines.
where P is a probability measure on (Ω, F) such that the solution of (3) is well-defined and unique under the null (1)5 . In the following it will be convenient to work with the finite dimensional distributions Pt of Wt, defined by
Pt(A) = P (Wt∈ A) , for any A ∈ B¡
Rk¢
and t ≥ 1. Defining PT = T1 XT t=1
Pt, we impose for the rest of the paper the following asymptotic stationarity assumption, ensuring existence of a weak limit P∞ for the sequence©
PT : T ≥ 1ª .
Assumption 1 There exists a probability measure P∞ on¡ Rk, B¡
Rk¢¢
such that P∞is the weak limit of©
PT : T ≥ 1ª :
PT → P∞ , weakly as T → ∞.
If the functional Q 7→ EQ[m (W, β, δ)] is weakly continuous for any (β, δ) ∈ Θ, equation (3) can be written in the equivalent form
0 = EP
∞[m (W, β, δ)] , (4)
where P∞= limT→∞PT. A sufficient condition for the weak continuity of Q 7→ EQ[m (W, β, δ)]
is the boundedness of the orthogonality function m. In fact, boundedness of m is precisely the condition that is required in order to ensure the local robustness of a general GMM estimator; see also Section 3 below. The solution eθ¡
P∞¢ :=³
eβ¡ P∞¢0
, eδ¡ P∞¢0´0
of (4) for any suitable P∞ defines the asymptotic functional structure of the FS-GMM estimator eθ =
µeβ
0
, eδ
0¶0 .
To define a FS-GMM estimator associated with a sample WT := {Wt: 1 ≤ t ≤ T } let PWT :=
1 T
PT
t=1δWt be the empirical distribution of WT, where δWt is the measure with point mass at Wt. Under standard regularity conditions one has PWT → P∞ weakly, P -almost surely, as T → ∞. The finite sample FS-GMM estimator is bθ :=
µβb
0
, bδ
0¶0
:=³ βe¡
PWT¢0 , eδ¡
PWT¢0´0
, i.e.
the solution of (4) for PWT. Specifically, this gives the finite sample estimating equations 0 = EP
WT
h m³
W, bβ, bδ´i
= 1 T
XT t=1
m³
Wt, bβ, bδ´ .
5This defines implicitly a domain for the functional eθ.
Under the null hypothesis (1) of no break, weak convergence under P of a finite sample FS-GMM estimator bθ to eθ¡
P∞¢
can be ensured with Assumption 1 in Andrews (1993). We report this result in the next theorem using a functional notation.
Theorem 2 Under regularity conditions on (Θ, m, P ) and under the null hypothesis (1) (see An- drews (1993), Assumption 1 p. 830), every sequencen
bθ : T ≥ 1o
of FS-GMM estimators satisfies
√T³ bθ − eθ¡
P∞¢´
→dN0¡ P∞¢−1
S (P )12B (1) , where the matrix
N¡ P∞¢
=£ M¡
P∞¢ Mδ¡
P∞¢ ¤ , is given by
M¡ P∞¢
= EP
∞
h∇β0m³ W, eθ¡
P∞¢´i , Mδ¡
P∞¢
= EP
∞
h∇δ0m³ W, eθ¡
P∞¢´i ,
and where {B (π) : π ∈ [0, 1]} is a standard Brownian motion in Rυ. The matrix S (P ) is given by:
S (P ) = lim
T→∞V arP Ã 1
√T XT t=1
m³ Wt, eθ¡
P∞¢´! .
For simplicity of exposition we assume in the sequel that the processn m³
Wt, eθ¡ P∞¢´
: t ≥ 1o is a martingale difference6 under P , implying
S(P ) = S(P∞) := EP
∞
· m³
W, eθ¡ P∞¢´
m³ W, eθ¡
P∞¢´0¸ ,
if m is bounded and under the null (1) of no break. Under standard conditions, consistent estimators of S(P∞) and M¡
P∞¢ are S¡
PWT¢
and M¡ PWT¢
, respectively.
The next section introduces Andrews (1993) LM-GMM tests for structural breaks. They are obtained from the FS-GMM estimators defined above.
2.3 Test statistics
Consistent asymptotically equivalent GMM test statistics for testing H0against H1T(π) are Wald- type, Lagrange Multiplier-type (LM) or Likelihood ratio-type statistics. Without loss of generality
6This assumption is satisfied in many applications, such as for instance those of Section 4.
we focus in the sequel on LM test functionals. In our setting a LM test can be defined by means of the statistic (see also Andrews (1993), p. 837)
LMdT(π) = T
π (1 − π)· LM¡
π, PWT¢
= T
π (1 − π)U0¡
π, PWT¢ U¡
π, PWT¢ ,
where
U¡
π, PWT¢
= H¡ P∞¢12
EP
WT ,π
hm³
W, bβ, bδ´i
= H¡
P∞¢12 1 T
XT π t=1
m³
Wt, bβ, bδ´ ,
with
H¡ P∞¢
= S¡ P∞¢−1
M¡ P∞¢
Σ¡ P∞¢
M0¡ P∞¢
S¡ P∞¢−1
, Σ¡
P∞¢
= h
M0¡ P∞¢
S¡ P∞¢−1
M¡
P∞¢i−1 ,
and with the empirical finite measure PWT,π= T1 PT π t=1δWt.
The LM statistic dLMT(π) is particularly simple to compute, since it requires only the com- putation of a single FS-GMM estimator. This is a clear advantage when working with RGMM statistics, because RGMM estimators for time series are typically more computationally intensive than classical ones (see Section 3 below).
As PWT,π→ πP∞weakly when T → ∞, P almost surely, the asymptotic functional structure of the functional U is determined as
U¡ π, P∞¢
= H¡ P∞¢12
EπP
∞
hm³ W, eθ¡
P∞¢´i
. (5)
Therefore, the asymptotic functional structure of LM¡
·, P∞¢
is given by
LM¡ π, P∞¢
= U0¡ π, P∞¢
U¡ π, P∞¢
. (6)
In this paper we focus on a class of supremum and average statistics bξsupT and bξaveT , respectively, to test H0 against alternatives of the form HA(Π) or H1. The test statistic bξsupT is defined by
bξsupT := sup
π∈Π
d
LMT(π) = T · eξsup¡ PWT¢
, (7)
where
eξsup¡ PWT¢
= sup
π∈Π
µ 1
π (1 − π)LM¡
π, PWT¢¶ . The test statistic bξaveT is defined by
bξaveT :=
Z
Π
LMdT(π) dλ (π) = T · eξave¡ PWT¢
, (8)
where
eξave¡ PWT¢
= Z
Π
µ 1
π (1 − π)LM¡
π, PWT¢¶
dλ (π) ,
and λ is the Lebesgue measure on Π. The asymptotic functional structure of eξsup and eξave is obtained from (7), (8) and (6) as
eξsup¡ P∞¢
= sup
π∈Π
µ 1
π (1 − π)LM¡
π, P∞¢¶
, eξave¡ P∞¢
= Z
Π
1
π (1 − π)LM¡ π, P∞¢
dλ (π) . (9) From (6) and (9) we deduce that the robustness properties of the functionals eξsup(·) and eξave(·) are completely determined by the ones of the functional U . Therefore, one can expect to obtain a class of tests for structural breaks with better robustness properties when working with quadratic functionals based on a robust functional U . This in turn will require working with GMM test statistics and estimators based on a GMM setting with a bounded orthogonality function m; see also Section 3 below.
In a likelihood setting, statistics of the form (8) define an optimal test in terms of a weighted average power criterion based on an uniform prior for the break date π ∈ Π (see Andrews and Ploberger (1994)). Specifically, average type tests can be interpreted as the optimal test for structural breaks in the case of alternative hypotheses very near to the null7 . When constructing robust tests for structural breaks in a likelihood setting we can therefore expect robust versions of the bξaveT statistic to produce the highest weighted power at the model, when compared for instance with robust versions of bξsupT .
7 To test for more global structural break alternatives, optimal tests of the average exponential form can be applied (Andrews and Ploberger (1994)). Robust versions of such tests can be constructed readily with the same procedure used in Section 3, where robust versions of bξsupT and bξaveT are obtained.
The null asymptotic distribution of dLMT(·) as a process indexed by π implies, by the Func- tional Central Limit theorem, the one of (7) and (8). This is summarized by the next result due to Andrews (1993).
Theorem 3 Under regularity conditions on (Θ, m, P ) and under the null hypothesis (1) (see Andrews (1993), Assumption 1 p. 830 and Assumption 3 p. 835) it follows:
1. dLMT(·) =⇒ Qp(·) as a process indexed by π ∈ Π, where Qp(π) := 1
π (1 − π)Jp(π)0Jp(π) :=(Bp(π) − πBp(1))0(Bp(π) − πBp(1))
π (1 − π) ,
and Bp(·) is a p−dimensional standard Brownian motion on [0, 1].
2. bξsupT →dsupπ∈ΠQp(π) under P . 3. bξaveT →d
R
ΠQp(π) dλ (π) under P .
Based on this result, critical values for bξsupT , bξaveT can be computed by simulation of the process Qp(·). Results analogous to the one in Theorem 3 can be derived for the asymptotic distribution of dLMT(·) under appropriate sequences of parametric local alternatives to the null hypothesis (1); see Theorem 3 in Andrews (1993) for details.
3 Robust GMM tests for structural breaks
Statistical robustness deals with inference procedures that are based on smooth statistical func- tionals. In this paper we focus on inference procedures for structural breaks which are locally robust in nonparametric neighborhoods of a parametric reference model, that is procedures that are not excessively sensible to small deviations from a reference model.
Our goal is to develop smooth test functionals for breaks around some fix parametric reference model belonging to a (nonparametric) neighborhood of relevant model distributions. Therefore, a minimal robustness requirement is continuity of such functionals. A second stronger requirement is their Fréchet differentiability (see for instance Bednarsky (1993)). Based on (9), we can expect the power and level functionals of robust tests for breaks based on the supremum or the average to satisfy the first or the second requirement, respectively, if and only if the statistical functional
U in (5) satisfies the first or the second requirement, respectively. Therefore, a first focus is on working in GMM settings where such statistical functionals are Fréchet differentiable.
3.1 Fréchet differentiability
Let P be a probability measure on (Ω, F) satisfying the conditions for Theorem 2 and 3 to hold.
This will be the reference model in our robust setting8 . The boundedness of the orthogonality function m is a necessary condition for a general GMM statistic like the FS-GMM estimator eθ or the functional U in (5) to be Fréchet differentiable. More specifically, an orthogonality function m is unbounded if and only if the influence function of a GMM statistic is unbounded.
Unboundedness of the influence function implies an unbounded asymptotic bias of a GMM statistic in a neighborhood of P∞(see for instance Ronchetti and Trojani (2001)). For the rest of the paper, we therefore assume a GMM setting based on a bounded orthogonality function m.
Assumption 4 The orthogonality function m is such that kmk∞:= sup
(w,θ)∈W×Θkm (w, θ)k < ∞ .
Under Assumption 4 and further regularity conditions which are typically a subset of those suffi- cient for Theorem 2 and 3 to hold, Fréchet differentiability of the GMM functionals eθ and U at P∞can be implied (see for instance Clarke (1986) and Heritier and Ronchetti (1994)). Therefore, we assume in the sequel their Fréchet differentiability.
Assumption 5 The functionals eθ and U are Fréchet differentiable.
The important property of Fréchet differentiable testing functionals for robust inference purposes is their uniform convergence in distribution over sequences of distributional asymptotic neighbor- hoods of the given reference model. This feature provides a way to compute uniform asymptotic expansions where the linearized asymptotic bias under contamination in the level and the power of the test can be formally bounded over neighborhoods of the reference model. We address this issue in the next section for our robust supremum and average tests for structural breaks.
8Typically, this will be a parametric distribution; see also the examples in Section 4 below.
3.2 Uniform convergence
To study the robustness properties of GMM testing procedures for structural breaks we consider sequences Uε := {Uε,T : T ≥ 1} of local neighborhoods of P . Denote by M the vector space of finite signed measures on (Ω, F) and by |·|V the variation norm on this space. A local neighborhood Uε,T of P is defined by
Uε,T =n
Qε,T ∈ M : Qε,T is probability measure :¯¯Qε,T − P¯¯
V < ε/√ To
.
Let further Qε,T :=n
Qε,Tt : t = 1, .., To
be the corresponding sequence of finite dimensional dis- tributions of {Wt: t = 1, ..T } under Qε,T, i.e.
Qε,Tt (A) = Qε,T(Wt∈ A) ,
for any A ∈ B¡ Rk¢
. Uniform weak convergence of U¡
·, PWT
¢ as a process indexed by π over
sequences Uεof asymptotic neighborhoods of the reference model is defined next.
Definition 6 The sequence © U¡
·, PWT
¢: T ≥ 1ª
converges weakly as a process indexed by π to Jp(·), uniformly over the sequence Uε, if
Lε,T³√
T³ U¡
·, PWT
¢− U³
·, Qε,TT
´´´
=⇒ Jp(·) (T → ∞)
uniformly in Qε,T, where Lε,T is the process distribution under Qε,T whereas Jp(·) is the Brownian Bridge process in Theorem 3.
Notice that, by construction,¯¯¯Qε,Tt − Pt
¯¯
¯V < ε/√
T for any t, implying¯¯¯Qε,TT − PT
¯¯
¯V < ε/√ T , where for brevity we denote now by |·|V the variation norm on the space of finite signed measures on¡
Rk, B¡ Rk¢¢
. Therefore, Uε,T neighborhoods of probability measures on (Ω, F) translate in a natural way into neighborhoods of probability measures on¡
Rk, B¡ Rk¢¢
.
Uniform convergence in distribution over a sequence Uεgives us a way to study the robustness properties of the asymptotic size and power functionals of bξsupT and bξaveT under a sequence of local contaminations Qε,TT , T ≥ 1. For brevity we focus on the robustness of the asymptotic size functionals limT→∞αsup(·) and limT→∞αave(·) of supremum and average tests for structural
breaks, respectively. They are defined under the null (1) by
Tlim→∞αsup³ Qε,TT ´
= lim
T→∞Qε,TT ³
bξsupT > ξsup0 ´ ,
Tlim→∞αave³ Qε,TT ´
= lim
T→∞Qε,TT ³
bξaveT > ξave0 ´ ,
where ξsup0 and ξave0 are the critical values of the tests for a given nominal size α0. Given the Fréchet differentiability Assumption 5 we can assume uniform convergence in distribution of the sequence©
U¡
·, PWT
¢: T ≥ 1ª
; see also Clarke (1986) and Heritier and Ronchetti (1994) for more details on the relation between Fréchet differentiability and uniform convergence in distribution.
Assumption 7 The sequence© U¡
·, PWT
¢: T ≥ 1ª
converges weakly as a process indexed by π to Jp(·), uniformly over the sequence Uε.
The Fréchet differentiability Assumption 5 and Assumption 7 imply a uniformly bounded bias of power and level of tests for structural breaks, over asymptotic distributional neighborhoods of the reference model. This gives formally the robustness of a testing procedure for structural breaks under local misspecifications of a reference model.
3.3 Asymptotic level expansions of tests for structural breaks
We next provide an expansion of the asymptotic size functional of robust supremum and average tests for structural breaks over sequences of asymptotic neighborhoods of the reference model P . For simplicity of exposition we work in the sequel with asymptotic ε−contaminated neighborhoods.
Let M∞ be a set of asymptotically stationary measures defined by
M∞=©
Q ∈ M : Q is a probability measure and QT → Q∞weakly as T → ∞ª .
An ε−contaminated neighborhood Uε,T of P is defined by:
Uε,T =
½ Qη,T =
µ 1 − η
√T
¶ P + η
√TQ : Q ∈ M∞and η < ε
¾ .
The asymptotic behaviour of the size of tests for structural breaks under local model misspecifi- cations is characterized by the next theorem9 .
9The proof is in the Appendix.
Theorem 8 Under Assumption 4, 7 and under the null hypothesis (1) it follows, for any η < ε:
Tlim→∞αsup³ Qη,TT ´
= α0+ η2µsup· d2¡
P∞, Q∞¢ + o¡
η2¢ , and
Tlim→∞αave³ Qη,TT ´
= α0+ η2µave· d2¡
P∞, Q∞¢ + o¡
η2¢ , where d2¡
P∞, Q∞¢
is given by d2¡
P∞, Q∞¢
= EQ
∞
³ m³
W, eθ¡
P∞¢´´0 H¡
P∞¢ EQ
∞
³ m³
W, eθ¡ P∞¢´´
,
µsup= − ∂Lsup(ξsup0 , x)
∂x
¯¯
¯¯
x=0
, µave= − ∂Lave(ξave0 , x)
∂x
¯¯
¯¯
x=0
, and Lsup(·, x), Lave(·, x), are the cumulative distribution function of
sup
π∈Π
Jp∗(π, x)0Jp∗(π, x) π (1 − π) ,
Z
Π
Jp∗(π, x)0Jp∗(π, x)
π (1 − π) dλ (π) , respectively, where the process Jp∗(·, x) is defined by
Jp∗(π, x) = Jp(π) − π(1 − π) · x .
Theorem 8 shows that the asymptotic linearized bias in the level of tests for structural breaks is proportional to d2¡
P∞, Q∞¢
. In particular, if m is bounded the bias in the level of supremum and average tests for breaks is bounded over sequences of asymptotic neighborhoods of the ref- erence model. This implies formally the robustness of GMM tests for breaks based on bounded orthogonality functions.
Theorem 8 can be also used to give asymptotic bounds on the maximal bias in the level of tests based on a bounded orthogonality function. In particular, an orthogonality function such that
sup
(w,θ)∈W×Θ
¯¯m (w, θ)0H¡ P∞¢
m (w, θ)¯¯ < c2 , (10) for some constant c >√
υ, implies up to terms of order o¡ η2¢
:
¯¯
¯ limT
→∞αsup³ Qη,TT ´
− α0
¯¯
¯ ≤ η2µsup· c2 , ¯¯¯ limT
→∞αave³ Qη,TT ´
− α0
¯¯
¯ ≤ η2µave· c2 . (11)
For testing purposes the bounds (11) can be used to choose the constant c in dependence of the maximal amount of contamination expected (ε) and the maximal bias in the asymptotic level
which a researcher is willing to accept. In this case, the derivatives µsup, µave, will have to be computed numerically, by simulating the distribution of
sup
π∈Π
Jp∗(π, x)0Jp∗(π, x) π (1 − π) ,
Z
Π
Jp∗(π, x)0Jp∗(π, x) π (1 − π) ,
for several values of x in a neighborhood of 0. For instance, the local robustness of tests for struc- tural breaks in dependence of model parameters like α0 or Π could be studied, thereby producing information about the degree of robustness required in a particular model setting. Notice that in the case where Π = {π0} and λ = δπ0 (the Dirac distribution at π0) the above tests collapse to a test for a break at a known break date. In this case (but only in this case) the results of Theorem 8 coincide with those of Theorem 1 in Ronchetti and Trojani (2001), where the local robustness of the size functional for standard Maximum Likelihood-type GMM tests has been characterized.
Since in this case the distribution of the random variable ¡
Jp∗(π0, x)0Jp∗(π0, x)¢
/ (π0(1 − π0)) is noncentral chis-quare, full analytical expressions for µsup= µave become available.
The bound (10) is satisfied by RGMM estimators with bounded self-standardized sensitivity, i.e. such that10
sup
(w,θ)∈W×Θ
¯¯
¯m (w, θ)0S¡ P∞¢−1
m (w, θ)¯¯¯ < c2 .
Therefore, we consider supremum and average tests for breaks based on such RGMM estimators and their orthogonality functions. Details on the definition and the computation of such GMM estimators and their orthogonality functions in a general GMM setting are provided in Ronchetti and Trojani (2001), p. 45-48.
1 0Indeed:
¯¯m (w, θ)0H¡ P∞¢
m (w, θ)¯
¯ ≤¯¯¯m (w, θ)0S¡
P∞¢−1m (w, θ)¯¯¯ , since the matrix
S¡ P∞¢12
H¡ P∞¢
S¡ P∞¢12 is an orthogonal projection matrix.
4 Monte Carlo simulations
This section studies by means of Monte Carlo simulation the size and power properties of the above robust GMM tests for breaks in different model settings. We compare the performance of the RGMM tests with the one of the classical GMM tests by focusing on the efficiency under ideal conditions and the robustness under local departures from a conditionally normal reference model.
4.1 Testing for structural breaks in a linear regression model
We first consider tests for a break in the slope coefficient of a linear regression model with an autoregressive regressor. The model is given by:
yt= γ + βtxt+ σut
xt= α + ρxt−1+ σεεt
, (12)
where:
βt=
β1, for t = 1, ..., T π0 β2, for t = T π0+ 1, ...T
,
for some π0 ∈ Π. The error term εt in the process xt is i.i.d.N (0, 1) distributed. For the error term ut in the linear regression model (12) we simulate a set of different distributions according to Model 1a-1e below. Specifically, we set:
• Model 1a: ut∼ i.i.d. N (0, 1) ,
• Model 1b: ut∼ i.i.d. t5/p 5/3,
• Model 1c: ut∼ i.i.d. t3/√ 3,
• Model 1d: ut∼ i.i.d. CN (0.05, 0, 3) /√ 1.4,
• Model 1e: ut∼ i.i.d. CN (0.1, 0, 3) /√ 1.8,
where tnis a Student distribution with n degrees of freedom and CN (x, 0, 3) is a standard normal distribution contaminated with probability x by a further zero mean normal distribution having
standard deviation 3. All error distributions have been standardized. The standard orthogonality conditions for a least squares estimation of the linear regression model (12) are based on an orthogonality function given by
ψ(Wt, β, δ) =
ψ1(Wt, β, δ) ψ2(Wt, β, δ)
=
1 xt
(yt− γ − βxt)
(yt− γ − βxt)2/σ2− 1
,
where Wt = (yt, xt)0, δ = ¡ γ, σ2¢0
. Classical FS-GMM estimators are obtained by using such orthogonality conditions. Since ψ is unbounded these estimators are not robust. Hence, also tests based on LM-type GMM statistics derived from such estimators are not robust. Robust FS-RGMM estimators of (12) can be constructed by applying orthogonality conditions based on a truncated orthogonality function given by
m(Wt, β, δ) =
m1(Wt, β, δ) m2(Wt, β, δ)
=
A1ψ1(Wt, β, δ)wc1(A1[ψ1(Wt, β, δ)]) A2[ψ2(Wt, β, δ) − τ2] wc2(A2[ψ2(Wt, β, δ) − τ2])
, (13) where wc(z) := min(1, c/ kzk), defines a set of Huber’s weights that downweight observations which are influential (in terms of asymptotic bias) for a classical least squares estimation of the model; see also Hampel et al. (1986), Section 4.4., for more details. The constants c1 > √
2, c2> 1 are tuning constants that control the amount of robustness in the estimation of (γ, β)0 and σ2, respectively. The matrix A1∈ R2×2 and the scalars A2, τ2 are determined as the solution of the implicit equations:
0 = EP0[m(Wt, β0, δ0)] , I = EP0
h
m(Wt, β0, δ0)m(Wt, β0, δ0)0i ,
where P0 is the reference model distribution of a linear regression model (12) with normally distributed error terms ut. Specifically, since under P0 the random variable (yt− γ0− β0xt)2/σ20 is X12 distributed, irrespectively of (β0, δ0), the correction constanta A2, τ2 have to be computed
only once, before starting the robust estimation algorithm.
Table 1a-1e present the results of our Monte Carlo simulations for Models 1a-1e, respectively, in the given linear regression setting. The break date is fixed as t0= 0.5 · T , where T = 200,300, respectively. In the simulations for T = 200 we have set α = γ = 0, β1 = 1, β2 = 1, 1.1, 1.2, 1.3, σu = σe = 1, ρ = 0.2, 0.5, 0.8. In the simulations for T = 300 the values of β2− β1 applied for T = 200 have been multiplied by a factor p
2/3 in order to obtain comparable local alternatives across the different sample sizes. The tuning constants for the RGMM test have been set at c1 = 3, c2 = 3. We also fixed Π = [p0, 1 − p0], where p0 = 0.25. The nominal level of the tests is α0 = 0.05. Finally, for the classical tests we provide the results of two GMM statistics. The first (OLS) uses a restricted estimate of S¡
P∞¢
which uses explicitly the independence of (xt) and (ut). The second statistic (GMM) is the standard one which uses an unrerstricted estimate of S¡
P∞¢
. RGMM is the robust GMM statistic based on an unrestricted estimate of S¡ P∞¢
using the bounded orthogonality function (13).
Table 1a shows that the power loss under normality of supremum and average RGMM tests for breaks is moderate, with losses relatively to the classical GMM tests that are typically below 10%. As expected, the power of classical and robust average tests is above the one of tests based on the supremum functional. Table 1b shows that already under a t5 error distribution the power curves of classical and robust GMM tests are virtually undistinguishable. Notice, however, that classical GMM average tests are in this setting systematically oversized, especially for the cases T = 200 and ρ = 0.2, 0.5. This effect seems to derive from the estimation of the asymptotic covariance matrix S¡
P∞¢
of m. Indeed, the GMM average tests using an OLS based estimation of S¡
P∞¢
present virtually no such oversize. However, their power curves are systematically below the ones of the robust average tests. In Table 1c, under a t3error distribution, these issues are more pronounced with a clearly higher power of robust GMM tests relatively to standard procedures. Notice that in this model setting the systematic oversize of classical average tests is particularly pronounced. For example, the empirical sizes of the classical GMM tests are for all
parameter choices ρ = 0.2, 0.5, 0.8 and T = 200 above 0.075. Table 1d shows similar patterns as for a t5 distribution when simulating under a CN (0.05, 0, 3) error distribution: the power curves of classical and robust GMM tests under such a setting are virtually undistinguishable and the classical average test shows a tendence to a slight oversize when ρ = 0.2, 0.5, T = 200 and when T = 300. Under the CN (0.1, 0, 3) setting in Table 1e these issues are more pronounced with the power curves of supremum and average tests that are above the ones of classical procedures and with again an oversize of classical average tests for breaks.
4.2 Testing for structural breaks in an ARCH model
To investigate the properties of classical and robust GMM tests for structural breaks in nonlinear models, we now consider an ARCH model setting (Engle (1982)). We analyze two types of tests for breaks in this setting by testing for a break in the intercept and the slope, respectively, of the conditional variance equation in an ARCH(1) model.
4.2.1 Breaks in the autoregressive parameter of the conditional variance equation
The model specification is given by:
yt = σtut,
σ2t = α0+ α1,ty2t−1,
where
α1,t=
α1,1, for t = 1, ..., T π0 α2,1, for t = T π0+ 1, ...T
,
for some π0 ∈ Π. For utwe simulate again a set of distributions near to a standard normal one, according to Models 2a-2e below:
• Model 2a: ut∼ i.i.d. N (0, 1) ,
• Model 2b: ut∼ i.i.d. t7/p 7/5
• Model 2c: ut∼ i.i.d. t5/p 5/3
• Model 2d: ut∼ i.i.d. CN (0.05, 0, 3) /√ 1.4
• Model 2e: ut∼ i.i.d. CN (0.1, 0, 3) /√ 1.8
All error distributions have been standardized. The orthogonality conditions for a classical FS- GMM estimation of the model are defined by an orthogonality function given by:
ψ(Wt, β, δ) =
1 y2t−1
1 σ2t
µy2t σ2t − 1
¶ ,
where σ2t = α0 + α1yt2−1, Wt = (yt, yt−1), β = α1, δ = α0. This orthogonality function is unbounded, so that the implied GMM estimators are not robust. The orthogonality function for a robust GMM estimation of the model is given by:
m(Wt, β, δ) = A [ψ(Wt, β, δ) − τ (yt−1)] wc(A [ψ(Wt, β, δ) − τ (yt−1)]) , (14)
for some tuning constant c > √
2. The matrix A ∈ R2 and the vector function τ are defined by the implicit equations:
0 = EP0
h
m(Wt, β0, δ0) | yt−1
i , I = EP0
h
m(Wt, β0, δ0)m(Wt, β0, δ0)0i ,
(15)
where P0 is the reference model distribution of an ARCH(1) model with conditional normally distributed error terms ut. The shift factors τ (yt−1), t = 1, ..., T , can be computed by using an analytical Laplace approximation of the integrals involved in the solution of (15), as proposed in Mancini, Ronchetti and Trojani (2003). This avoids the numerical computation of such integrals and largely reduces the computation time of robust GMM estimators in the present and related settings. Details on the computation of τ (yt−1) and the corresponding robust FS-GMM estimator for the moment conditions (15) are given in the Appendix.
Table 2a-2e present the results of our Monte Carlo simulations for Models 2a-2e, respectively, in the given ARCH(1) model setting. The break date is fixed as t0= 0.5 · T , where T = 500, 1000.
In the simulations for T = 1000 we have set α0 = 0.01, α1 = 0.2, α2 = 0.2, 0.3, 0.4, 0.5. In the
simulations for T = 500 the values of β2− β1 applied for T = 1000 have been multiplied by a factor√
2 in order to obtain comparable local alternatives across the different sample sizes. The tuning constants for the RGMM test have been set at c = 6.18. We also fixed Π = [p0, 1 − p0], where p0 = 0.45. The nominal level of the tests is α0 = 0.05. Finally, for the classical tests we provide the results of two GMM statistics. The first (OLS) is based on a restricted estimate of S¡
P∞¢
which uses explicitly the independence of utand ut−1. The second statistic (GMM) is the standard one which uses an unrestricted estimate of S¡
P∞¢
. RGMM is the robust GMM statistic based on an unrestricted estimate of S¡
P∞¢
using the bounded orthogonality function (14).
Table 2a shows that the power loss under normality of average RGMM test for breaks is moderate and always below 10%. Moreover, even under normality, the power of RGMM supremum tests is above the one of classical GMM procedures. Finally, classical GMM procedures based on the supremum imply a too conservative size behavior while RGMM tests based on the supremum do correct such a size distorsion in the right direction. As expected, the power of classical and robust average tests is above the one of tests based on the supremum functional. Table 2b shows that under a t7 error distribution the gain in power of RGMM tests based on the supremum is very large, with increases that are sometimes around 80%-100%. Also in the case of the average statistic, RGMM procedures do produce clear power increases in this setting. Both supremum and average classical statistics yield a too conservative size behavior that is corrected in the right direction by RGMM tests for breaks. Such issues are even more apparent in Table 2c, under a t5distribution, where also in the case of the average statistic RGMM procedures yield very large power improvements. The results in Tables 2d and 2e (for a CN (0.05, 0, 3) and a CN (0.1, 0, 3) error distribution, respectively) are qualitatively similar to the ones of Table 2b and 2c, with effects that are however quantitatively even larger than in the case of a Student t error distribution.
4.2.2 Breaks in the intercept parameter of the conditional variance equation
In this case the model is defined by:
yt = σtut,
σ2t = α0,t+ α1yt2−1 ,
where:
α0,t=
α0,1, for t = 1, ..., T π0 α0,2, for t = T π0+ 1, ...T
,
for some π0 ∈ Π. The set of distributions simulated for ut is the same as the one of the last section, and will be listed according to Models 3a-3e, respectively. The functions ψ, m defining the orthogonality function for a classical and a robust FS-GMM estimation of the model are as above with the relevant arguments that are now β = α0and δ = α1. In particular, the matrix M used for the computation of the LM tests is now different from the one used in Section 4.2.1.
Table 3a-3e present the results of our Monte Carlo simulations for Models 3a-3e, respectively, in the given ARCH(1) model setting. In the simulations for T = 1000 we have set α0,1= 0.01, α0,2= 0.01, 0.0125, 0.015 and α2= 0.2. In the simulations for T = 500 the values of α0,2− α0,1 applied for T = 1000 have been multiplied by a factor√
2 in order to obtain comparable local alternatives across the different sample sizes. All further parameters are the same as the ones of Section 4.2.1.
Table 3a shows that the power loss under normality of average RGMM test for breaks is in this case virtually negligible. Moreover, the classical average test presents a slight tendence to an oversize which is corrected in the right direction by the robust average test. As for the tests of Section 4.2.1, the power under normality of RGMM supremum tests is above the one of classical GMM procedures. Similarly, classical GMM procedures based on the supremum imply a too conservative size behavior while RGMM tests based on the supremum do correct such a size distorsion in the right direction. As expected, the power of classical and robust average tests is again above the one of tests based on the supremum functional. Table 2b shows that under a t7
error distribution the gain in power of RGMM tests based on the supremum is large, with increases that are sometimes around 30%-40%. In the case of the average statistic, RGMM procedures do produce power increments of about 10% in this setting. Finally, the too conservative behavior of classical supremum tests is corrected in the right direction by RGMM tests for breaks. In Table 3c, under a t5 distribution, both supremum and average RGMM procedures yield large power improvements relatively to their classical counterparts. Again, the size distorsions of classical tests based on the supremum are corrected in the right direction by RGMM tests for breaks.
The results in Tables 3d and 3e (for a CN (0.05, 0, 3) and a CN (0.1, 0, 3) error distribution, respectively) are similar to the findings in Table 3b and 3c, with effects that are however again quantitatively larger than in the case of a Student t error distribution.
5 Conclusions
We proposed a class of new supremum and average RGMM tests for structural breaks, which imply a bounded asymptotic bias of size and power under local model misspecifications. Robustness of the new tests is obtained by computing the supremum or the average over a sequence of GMM Lagrange Multiplier statistics in a setting based on a bounded orthogonality function. Monte Carlo simulation showed the new robust GMM tests to perform very well already under slight deviations from the reference model, both in terms of the efficiency and the robustness of the inference procedure, when compared with standard (nonrobust) GMM tests for structural breaks.
Due to the intrinsic difficulty in developing perfectly specified parametric models for econometric applications, it is expected that RGMM tests for breaks can help in providing some more robust and consistent evidence on the presence of breaks in the statistical analysis of economic data series.
6 Appendix 1: Proof of Theorem 8
By Assumption 7:
√T³ U¡
·, PWT
¢− U³
·, Qη,TT
´´
=⇒ Jp(·) ,
uniformly over the sequence Uε as T → ∞. Moreover, a von Mises (1947) expansion under the null hypothesis (1) gives up to terms of order O³
η/√ T´
:
√T · H¡ P∞¢−12
U³
·, Qη,TT
´
= √
T EπQη,T
T
³ m³
W, eθ³
Qη,TT ´´´
= ηEπP
T
³∇θ0m³ W, eθ¡
PT¢´´
D³ eθ´ ¡
PT, QT¢ +ηEπQ
T
³ m³
W, eθ¡ PT
¢´´· π
where
QT 7−→ D³ eθ´ ¡
PT, QT¢
is the Fréchet derivative of eθ at PT in direction QT−PT, under the null hypothesis (1). Consistency of eθ at the model implies as T → ∞ under further mild regularity conditions
√T · H¡
P∞¢−1/2
U³ Qη,TT ´
→ η · b¡
π, P∞, Q∞¢ ,
where
b¡
π, P∞, Q∞¢
= πh EP
∞
³
∇θ0m³ W, eθ¡
P∞¢´´
D³ eθ´ ¡
P∞, Q∞¢ + πEQ
∞
³ m³
W, eθ¡
P∞¢´´i
= πh N0¡
P∞¢ D³
eθ´ ¡
P∞, Q∞¢ + πEQ
∞
³m³ W, eθ¡
P∞¢´´i
. (16)
To compute b¡
π, P∞, Q∞¢
note that under the null hypothesis (1) one has that the full sample moment conditions
0 = EP
∞
³m³ W, eθ¡
P∞¢´´
, (17)
hold for the FS-GMM estimator eθ. The boundedness of m then gives the Fréchet derivative D³
eθ´ ¡
P∞, Q∞¢
by implicitly differentiating (17) in the direction Q∞− P∞ to get
0 = EP
∞
³
∇θ0m³ W, eθ¡
P∞¢´´
D³ eθ´ ¡
P∞, Q∞¢ + EQ
∞
³ m³
W, eθ¡ P∞¢´´
,
