Accurate and Robust Indirect Inference for
diffusion Models
Veronika Czellar and Elvezio Ronchetti
No 2008.01
Cahiers du département d’économétrie
Faculté des sciences économiques et sociales
Université de Genève
Août 2008
Département d’économétrie
Université de Genève, 40 Boulevard du Pont d’Arve, CH -1211 Genève 4
http://www.unige.ch/ses/metri/
Accurate and Robust Indirect Inference for Diffusion Models
Veronika Czellar∗ HEC Paris Elvezio Ronchetti† University of Geneva August 4, 2008 AbstractIndirect inference (Smith, 1993; Gouri´eroux, Monfort and Renault, 1993) is a simulation-based estimation method dealing with econometric models whose likelihood function is intractable. Typical examples are diffusion models described by stochastic differential equations. A potential problem that arises when estimating a diffusion model is the possible model misspecification which can lead to biased estimators and misleading test results. To correct the bias due to model misspecification, Genton and Ronchetti (2003) proposed robust indirect inference. The standard asymptotic approx-imation to the finite sample distribution of the robust indirect estimators and tests, however, can be very poor and can lead to misleading inference. To improve the finite sample accuracy, we propose in this paper an optimal choice of the auxiliary discretized model and a new test based on asymptotically equivalent M-estimators of the robust indirect estimators. We apply the robust indirect saddlepoint tests using an optimal choice of discretization to various contaminated diffusion models and we illustrate the gain in finite sample accuracy when using the new technique.
Keywords: indirect inference,M-estimators, influence function, robust statistics, sad-dlepoint approximations.
∗
HEC Paris, 1 rue de la Lib´eration, 78351 Jouy en Josas, France. email: [email protected]. The authors are grateful to Eric Zivot, Cristophe Croux and Fallaw Sowell for helpful comments. V. Czellar would like to thank the University of Washington Statistics Department for use of its computer cluster.
†Department of Econometrics, University of Geneva, Blv. Pont d’Arve 40, 1211 Geneva, Switzerland.
1
Introduction
Indirect inference (Smith, 1993; Gouri´eroux, Monfort and Renault, 1993), hereafter II, is a simulation-based estimation method dealing with econometric models whose likelihood function is intractable. For example, in diffusion models described by stochastic differ-ential equations, it is often difficult or impossible to carry out likelihood based inference. Such diffusion models are typically used in finance when modeling and forecasting interest rate behaviour; see e.g. Merton (1980), Gallant and Tauchen (1996), Ait-Sahalia (1996, 1999). Other applications are discussed in the overviews by Jiang and Turnbull (2004) and Heggland and Frigessi (2004).
II requires chosing an auxiliary model, easier to estimate than the structural model, simulating pseudo-data from the structural model and minimizing the distance between the estimators obtained with real and simulated data. For example, in the case of a dif-fusion model, the auxiliary model can be the discretized version of the difdif-fusion model. Gouri´eroux, Monfort and Renault (1993), hereafter GMR, proved that under certain reg-ularity conditions, II provides consistent estimators of the parameters of the structural model which are asymptotically normal. Moreover, they proposed Wald-type, score-type and likelihood ratio-type tests for testing hypotheses on the parameters and they showed that the three tests are asymptotically equivalent andχ2-distributed.
A potential problem that arises when estimating a diffusion model is the possible model misspecification. This is an important issue when estimating and forecasting interest rates since diffusion models are at best approximate descriptions of the underlying structure of the data. Model misspecification can lead to biased estimators and misleading test results. Genton and Ronchetti (2003) introduced robust IIand showed that to ensure that the II estimator can cope with small deviations from the diffusion model, the influence function (Hampel, 1974; Hampel, Ronchetti, Rousseeuw and Stahel, 1986) of the auxiliary estimator has to be bounded. Ortelli and Trojani (2005) extended these results to the setup of the efficient methods of moments. Finally, Czellar, Karolyi and Ronchetti (2007) proposed the indirect robust generalized method of moments (IRGMM) as an II estimator with bounded auxiliary influence function, applied the IRGMM technique to estimate and forecast short-term interest rate processes, and illustrated its predictive performance on various monthly Eurocurrency rate series.
To test hypotheses on the parameters using robust II estimators, Genton and Ronchetti (2003) suggested to use an asymptoticallyχ2-distributed LR-type test based on auxiliary
estimators with bounded influence functions (hereafter robust LR-type test). However, not much is known about the finite-sample accuracy of classical or robust LR-type tests.
The contributions of our paper can be summarized as follows. First we quantify the effects of discretization and model misspecification on II estimators. This allows to evalu-ate the impact of each deviation on the bias of II estimators. In particular, this leads to a criterion for the choice of the discretization. Secondly, we investigate the asymptoticχ2 ap-proximation to the finite sample distribution of the II LR tests. We show that the accuracy of this approximation can be very poor and can lead to misleading inference. Therefore, we provide a new robust II test with good accuracy in moderate to small sample sizes. It is based on the fact that we can associate a robust II estimator with an asymptotically equivalent M-estimator defined by the influence function of the II estimator. Using the saddlepoint test for multivariateM-estimates of Robinson, Ronchetti and Young (2003) we develop a robust II saddlepoint test which is asymptoticallyχ2-distributed with arelative error of orderO(n−1). We investigate the robust and small sample properties of this new
test on contaminated diffusion models and illustrate its good performance both in finite samples and in the presence of deviations from the assumed model distribution.
The paper is organized as follows. In Section 2, we review II estimators and classical LR-type tests in the case of diffusion models. In Section 3 we study the bias of classical and robust II estimators due to discretization and model misspecification and we propose a choice of an auxiliary discretized model. In Section 4, we show the poor finite sample accuracy of classical and robust LR-type tests. In Section 5, we introduce the robust II saddlepoint test, apply it to contaminated diffusion models, and illustrate its good accuracy and robustness properties. Section 6 concludes the paper with some open problems and discussion. The proofs of the propositions presented in the paper can be found in the Appendix.
2
Robust and finite sample properties of classical II
estima-tors and tests
2.1 Estimating diffusion models via II
Given a sample of observations {yt}t=0,1,...,n assumed to be generated from a diffusion model
withWt a standard Wiener process, consider an auxiliary discretized model:
yt =yt−1+η(yt−1, µ) +σ(yt−1, µ)t, (2)
wheret are identically distributed standard normal variables. Suppose that the auxiliary estimator is defined by:
˜
µ= arg max µ
˜
Q({yt}t=0,1,...,n;µ), (3)
where ˜Q(·;µ) is the objective function of an estimation technique associated with the auxiliary model (2). For example, in the case when the auxiliary estimator is the maximum likelihood estimator (MLE) of the model (2),
˜ Q({yt}t=0,1,...,n;µ) = 1 n n X t=1 l(yt;yt−1, µ),
wherel(yt;yt−1, µ) =log f(yt;yt−1, µ) andf(yt;yt−1, µ) is the conditional density ofyt|yt−1.
We consider in this paper the II estimator ˆθdefined according to the following steps: 1) Compute the auxiliary estimator ˜µdefined by (3) using the original data{yt}t=0,1,...,n. 2) Simulatepseudo-observationsfrom the diffusion model (1). If an exact discretization is not available for (1), generate data from a fine discretization of (1) by dividing the time interval ∆t= 1 intom= 1/δ subintervals of lengthδ. The Euler approximation corresponding to the time intervalδ is the process {ykδ}k=0,1,...,mn defined by
y(k+1)δ=ykδ+δη(ykδ, θ) +
√
δσ(ykδ, θ)k, (4)
wherekis a standard normal variable. Selecting data at timeskδ∈N, gives pseudo-data{yt(s)(θ)}t=0,1,...,n. SimulateS pseudo-data sets from this model:
{yt(s)(θ)}t=0,1,...,n, s= 1, . . . , S, S ≥1. (5)
3) For eachs, calculate the function ˜Q {yt(s)(θ)};µand the pseudo-auxiliary estimator for the simulated data:
˜ µS(θ) = arg max µ X s ˜ Q {yt(s)(θ)};µ. (6)
4) The II estimator ˆθis the minimizer of the distance between the auxiliary estimators ˜ µand ˜µS(θ): ˆ θ= arg min θ (˜µ−µ˜S(θ)) TΩ(˜µ−µ˜ S(θ)), (7) where Ω is a positive definite symmetric matrix.
Other simulation-based II estimators can be defined by choosing in 3) either ˜µS(θ) =
1 S PS s=1µ˜({y (s) t (θ)}t=0,1,...,n) or ˜µS(θ) = ˜µ({yt(θ)}t=0,1,...,Sn), where {yt(θ)}t=0,1,...,Sn is a long pseudo-data series of sizeSn.
2.2 LR-type tests
To test a hypothesis defined byH0 : θ(1) =θ10, whereθ= (θ(1), θ(2))T, θ(1) ∈Rp1 , θ(2)∈
Rp−p1, GMR define the LR-type test statistic in the following way. Consider the optimal
weight matrix ˜ Ω∗= ˜MµΣ ˜˜Mµ, (8) where ˜ Mµ=− ∂2Q˜ ∂µ∂µT({yt}t=0,1,...,n; ˜µ), Σ =˜ 1 n n X t=1 ∂l ∂µ(yt;yt−1,µ˜) ∂l ∂µT(yt;yt−1,µ˜) −1 , (9)
and let ˆθ be the II estimator obtained using the optimal weight matrix ˜Ω∗. The LR-type
test statistic is defined by
LRS(θ10) = Sn S+ 1 h ˜ µ−µ˜S(ˆθ(0))TΩ∗ µ˜−µ˜S(ˆθ(0))− µ˜−µ˜S(ˆθ)TΩ∗ µ˜−µ˜S(ˆθ)i, (10)
where ˆθ(0) is the constrained II estimator obtained by minimizing the quadratic form in (7) underH0 : θ(1) =θ10∈Rp1. GMR show that under regularity conditions, LRS(θ10) is
asymptotically χ2
p1 distributed underH0.
In the following we will show that theχ2approximation of the LR-type tests can provide inaccurate p-values and hence misleading test results in small to moderate sample sizes. This is due to two main reasons. First of all, for most diffusion models, the associated stochastic differential equation cannot be solved and we cannot generate pseudo-data from the exact model (1) in step 2). In this case, the peudo-data is generated from the fine Euler discretization (4) which converges to the structural model only whenδ → 0. Secondly, in empirical applications the structural model (1) is at best an approximate description of
the true model that generated the original data{yt}t=0,1,...,n.
In the next section we first investigate the bias of the II estimator for various choices of the Euler discretization and various contaminations of the true data generating model. The effects of these deviations on the tests will be discussed in Section 4.
3
Bias of II estimators
Denote byFθ the structural model (1) and byFδ, θ, 0≤δ ≤1, the model used to generate pseudo-data {y(ts)(θ)}ts=0=1,,...,S1,...,n. In particular, for δ = 0, F0, θ = Fθ. Moreover, denote by Gε, θ = (1−ε)Fθ +εG, with 0 ≤ ε ≤ 1 and G an arbitrary and unknown distribution, the model that generated the true data {yt}t=0,1,...,n. If ε = 0 then G0, θ = Fθ and the structural model is the true data generating process. In order to investigate the asymptotic bias of II estimators, we define the functional associated with II estimators.
Define anauxiliary functionalTethat for a given distributionF associates the parameter
e T(F)∈Rr, r≥p, satisfying e T(F(n)) = ˜µ , (11) where F(n)(x) = 1 n n X t=1 ∆yt(x),
where ∆yt denotes the pointmass distribution which puts probability 1 on yt, and ˜µis the
auxiliary estimate defined in (3). An II-functional T associates a distribution F with the parameter T(F) = arg min υ ne T(F)−µ(υ)TΩ eT(F)−µ(υ)o= arg min υ QF(υ), (12) whereµ(υ) is a function fromRp to Rr and Ω∈Rr×Rr is a symmetric, positive definite
matrix. For example,µ(θ) can be
(a) Te(Fθ) for nonsimulation-based II estimators when the structural modelFθis tractable; (b) Te(Fδ, θ) for nonsimulation-based II estimators in the case when the structural model
is intractable, but the approximated modelFδ, θ is tractable;
(c) ˜µS(θ) defined in (6) for simulation-based II estimators where both the structural and approximated models,Fθ and Fδ, θ, are intractable.
The II-functionalT defined by (c) and evaluated atF(n) is the II estimator ˆθdescribed in Section 2.1, i.e.
T(F(n)) = ˆθ . (13) and the asymptotic bias of the II estimator ˆθ underGε, θ is given by
asbias(ˆθ) =T(Gε, θ)−θ . (14)
Proposition 1 shows that the consistency of the auxiliary estimator for the auxiliary parameterµis not a necessary condition to ensure the consistency of the II estimator for the structural parameter.
Proposition 1 Given a structural model Fθ, denote by T the functional associated with
the II estimator defined in (12) with µ(θ) given in (a). Suppose that (i) the quadratic form QFθ(υ) has a unique minimum.
Then, the II estimator T is Fisher consistent, i.e.
T(Fθ) =θ . (15)
The proof of Proposition 1 can be found in Appendix A.1.
Notice that Proposition 1 shows that the II estimator defined with µ(θ) = Te(Fθ) has no asymptotic bias. In most of the applications this type of II estimator is unfeasable since
e
T(Fθ) does not have a closed form. We turn now to study the bias of II estimators with other choices ofµ(θ).
Proposition 2 allows to calculate the asymptotic bias of the II estimator in the case when the auxiliary parameter µand the II parameter θhave the same dimension.
Proposition 2 Given a structural modelFθ and a contaminated model Gε, θ= (1−ε)Fθ+ ε G, 0≤ ε <1 denote by T the functional associated with an II estimator given by (12). Suppose that,
(i) the quadratic form QGε,θ(υ) has a unique minimum;
(iii) ∂µ∂υ T(Gε,θ)
exists and is nonsingular. Then,
T(Gε, θ) =µ−1(Te(Gε,θ)). (16) The proof of Proposition 2 can be found in Appendix A.2.
3.1 Illustration on the Geometric Brownian motion with drift
Suppose that the structural modelFθ is a geometric Brownian motion with drift (GBM):
dyt =βytdt+σytdWt, (17)
whereWt is a standard Wiener process andθ= (β, σ2)T. Consider the auxiliary model ˜Fµ which is a crude discretization of (17):
yt = (1 +β)yt−1+σyt−1t, (18)
where t is a standard normal variable and µ = (β, σ2)T. In step 1) of the II estimation procedure, consider the maximum likelihood estimator ˜µ associated with the auxiliary model ˜Fµ defined by the objective function:
˜ Q({yt};β, σ2) =− n 2log(σ 2)− 1 2σ2 n X t=1 (rt−(1 +β))2,
wherert =yt/yt−1. The resulting maximum likelihood estimators ofβ and σ2 are:
˜ β= 1 n n X t=1 (rt−1), (19) ˜ σ2 = 1 n n X t=1 (rt−r)2 , (20)
where r = n1Prt. Denote by ˜µ = ( ˜β,σ˜2)T the auxiliary estimator. The pseudo-data generating processFδ, θneeded in step 2) is the Euler discretization (4) withδ = m1, m∈N∗.
An exact discretization of (17) can be obtained using Itˆo’s lemma:
logrt=β− σ2
where εt is a standard normal variable. In the following we suppose that the structural modelFθ is the model (21).
Suppose that the original data is generated from a contaminated model Gε, θ = (1− ε)Fθ+εG. Typically G is an uknown distribution but for the sake of illustration, in this section we consider the perturbationεt ∼(1−ε)N(0,1) +εN(0, τ2),τ ≥1. Hence by (21) underGε, θ rt ∼ (1−ε) logN(β− σ2 2 , σ 2) +εlogN(β−σ2 2 , σ 2τ2). (22)
The caseτ = 1 corresponds to a model without contamination, i.e. Gε, θ =Fθ.
When the II estimator is nonsimulation-based, using Proposition 2, we can calculate the asymptotic bias of the II estimator analytically. Since for diffusion models, in step 2) of the II procedure, the pseudo-data generating process is the Euler discretizationFδ, θ, in the following we calculate the asymptotic bias of the II estimator ˆθδ,ε where the II-functional T in (12) is defined withµ given in (b).
From (21) and using (16), the asymptotic bias of the II estimator with underlying model
Gε, θ andµ defined in (b) is:
asbias(ˆθδ,ε) = (1−ε)eβ+εeβ+σ2(τ2−1)/2 δ −1 δ −β (1−ε)e2β+σ2+εe2β−σ2+2σ2τ2 δ − (1−ε)eβ+εeβ+σ2(τ2−1)/2 2δ δ −σ2 . (23)
The proof of (23) is given in Appendix A.3.
Figures 1 and 2 show the asymptotic biases of II estimators of β and σ2 in (23) for various values ofεand δ, when the true parameter values are β =−0.05 and σ= 0.2, the discretization inFδ, θvaries between 1/δ = 1,2, . . . ,20 andτ in the contaminationGε, θis in the interval 1≤τ ≤3. On the left side of the plots, on the vertical axis atδ = 1 andτ = 1 (no contamination) we can read the maximal asymptotic bias due to discretization which are 1.23·10−3for ˆβ and−3.07·10−3 for ˆσ2. Whenδ = 1/20, the biases due to discretization
reduce to 6.24·10−5 and −1.60·10−4. Hence, on the vertical plane atδ= 1/20, the biases reflect the asymptotic bias due essentially only to contamination Gε, θ, and these biases areunbounded. These figures show that the bias of II estimators due to the contamination
3.2 Bias due to discretization Fδ, θ only: choice of δ
In this subsection we focus on the bias due to discretizationFδ, θ only and propose a choice ofδ. Consider a diffusion model (1) withθ∈Rp. We propose a choice of discretization such
that the relative asymptotic bias of the II estimator is below a certain relative threshold
α >0 that the user is willing to accept:
δ∗= maxnδ >0asbias(ˆθ (j) δ ) θ(j) < α, ∀j= 1, . . . , po, (24)
where ˆθ(j) is the jth component of the II estimator.
As an illustrative example, consider again the GBM model (17). Replacing ε by 0 in (23), we obtain the relative asymptotic biases of the II estimators ofβ and σ2:
asbias( ˆβδ) β = eδβ−1 δβ −1 asbias(ˆσδ2) σ2 = eδ(2β+σ2) −e2δβ δσ2 −1.
Figure 3 illustrates the relative asymptotic biases of the II estimators of the model GBM when the true parameter values are known and are β = −0.05 and σ = 0.2 for 1/δ= 1,2, . . . ,20. For example,δ = 1/8 ensures that the relative asymptotic biases of the II estimators ofβ andσ2 are less than 1%.
In general, the true parameter values are unknown and for simulation-based estimators,
µ(θ) is simulation-based and we cannot calculate analytically the asymptotic bias of the II estimator. In this case, we can replace the parameters θj by the auxiliary estimates and the asymptotic bias can be calculated via simulations by considering a large sample generated from a fine Euler discretization.
As an illustrative example, consider the model of Chan, Karolyi, Longstaff and Sanders (1992) (CKLS):
dyt = (α+β yt)dyt+σ ytγdWt, (25) withα= 0.05, β=−0.2,σ= 0.4 andγ = 0.75. We simulate a sample of sizen= 200,000 from an Euler discretization Fδ, θ with δ = 1/1000. We estimate θ = (α, β, σ, γ)T by II with µ(θ) = ˜µS(θ) with S = 1. Figure 4 illustrates the relative asymptotic biases of II estimators of α, β, σ and γ for 1/δ = 1,2, . . . ,50. This figure shows that a choice
of δ = 1/20 is sufficiently small to ensure that the relative asymptotic biases of the II estimators are all below 1%.
These results confirm Broze, Scaillet and Zako¨ıan’s (1998) findings that simulating data from an Euler scheme with a fine delta leads to II estimators with negligable discretization bias. We note that Broze et al. (1998) also found that simulating pseudo-data from a second-order discretization scheme of Mil’shtein (1976) did not lead to any improvements of the II estimators in terms of asymptotic bias.
3.3 Bias due to contamination Gε, θ only
In this subsection we focus on the bias due to contaminationGε, θ only and we suppose that we are able to simulate pseudo-data from the structural model Fθ. This case was investi-gated by Genton and Ronchetti (2003) for the GBM model with original data satisfying (22) and pseudo-data according to (21). Using our Proposition 2 with the II functionalT
based onµ(θ) =Te(Fθ) , we obtain the following asymptotic biases for the II estimators
asbias(ˆθε) = log (1−ε)eβ+εeβ+σ2 2 (τ2−1) −β log(1−ε)e2β+σ2 +εe2β−σ2+2σ2τ2 −2 log(1−ε)eβ+εeβ+σ 2 2 (τ2−1) −σ2 . (26) The first row of (26) corrects formula (4) in Genton and Ronchetti (2003) computed using an expansion of the bias of the II estimator by Gouri´eroux and Monfort (1996) which assumes that the original data-generating model is the structural model.
Genton and Ronchetti (2003) show that the influence function of the II estimator is bounded by the influence function of the auxiliary estimator. For the GBM model they use the following robust auxiliary estimators ( ˜βR,σ˜R2)T defined by:
n X t=1 ψc(rt−1− ˜ βR ˜ σR ) = 0, n X t=1 χc(rt−1− ˜ βR ˜ σR ) = 0, (27)
where ψc(z) = min{c,max(−c, z)}, χc(z) = ψ2c(z) −EΦψc2, Φ is the standard normal distribution, and c = 1.345. The bias of the robust auxiliary estimator and asymptotic bias of the robust II estimator using Proposition 2 can be computed numerically solving the implicit integral equations and using the functional Te(Fθ) = (Te1(Fθ),Te2(Fθ))T, associated
with the robust auxiliary estimators ( ˜βR,σ˜R2)T: Z ∞ −∞ ψc x−1−Te1(Fθ) q e T2(Fθ) dFθ(x) = 0, Z ∞ −∞ χc x−1−Te1(Fθ) q e T2(Fθ) dFθ(x) = 0, (28)
whereθ= (β, σ2)T and Fθ is the distribution function of a lognormal variableX such that logX ∼ N(β−σ22, σ2).
Figures 5 and 6 show these biases for the classical and robust auxiliary and II estimators. It appears that the auxiliary MLE and the associated II estimators are biased when the underlying distribution of the increments is contaminated withN(0, τ2),τ ≥1. The biases of ( ˜βM L,˜σ2M L) and ( ˆβI,σˆI2) increase exponentially withτ. On the other hand the biases of the robust auxiliary and robust II estimators are bounded for 1≤τ <3.5. The auxiliary estimators are biased even ifτ = 1 which corresponds to the case without contamination. This bias is mainly due to discretization. Under the structural model Fθ, II corrects the bias due to discretization of the auxiliary estimator. Forτ >1 the robust II estimators are those with the smallest bias in presence of contaminations. These figures further show that the robust II estimator is the only one which can correct both the bias due to contamination and discretization.
Robust II estimators for more general one factor diffusion models are considered in Czellar, Karolyi and Ronchetti (2007).
4
Classical tests
In this section we perform a small Monte Carlo simulation to show that classical LR-type tests can lead to inaccuratep-values and misleading inference.
We choose S = 20 to ensure that the asymptotic efficiency of the auxiliary estimator under the auxiliary model is greater than 95%; cf. Czellar and Zivot (2008). We generate 10,000 datasets of size n = 40 and n= 240 from (21) with β =−0.05 and σ = 0.2, and for each dataset we compute the II estimators and the corresponding value of the LR-type test statistic, using pseudo-datasets generated from the Euler discretization withδ = 1/20. According to Figure 3, for this choice of discretization, the relative asymptotic bias of the II estimator of both parametres due to discretization is less than 0.5%. We consider two cases: data generated from (21) without outliers and with 5% additive outliers from (22).
• the simple hypothesis: H0 : (β, σ)T = (−0.05,0.2)T, without outliers (on the left)
and with 5% additive outliers (on the right);
• the composite hypothesis onβ: H0 : β =−0.05, without outliers (on the left) and
with 5% additive outliers (on the right);
• the composite hypothesis onσ: H0 : σ= 0.2, without outliers (on the left) and with
5% additive outliers (on the right).
The solid line (“45◦ line”) represents the exact size. Figure 8 shows the samep-value plots
for a sample size ofn= 240.
Figure 7 shows that for n= 40 under the model Fθ both classical and robust LR-type simple and composite tests for σ tests are oversized. All the classical LR-type tests are highly oversized when adding 5% additive outliers while the robust LR-type tests are much less so. Figure 8 shows that for a sample size ofn= 240 with no outliers in the original data classical and robust LR-type tests are much less size-distorted than for n= 40. However, at significance levels α = 1% (α = 5%) composite classical and robust LR-type tests on
σ still exhibit actual sizes of 2.5% (respectively 7.5%). Adding 5% additive outliers to a sample of size 240 affects even more the size of the test than in the case ofn= 40.
These results suggest the development of a better test which will be presented in the next section.
5
Robust II saddlepoint tests
In the previous sections we have shown that inference based on the LR-type test can be misleading when there are deviations from the assumed underlying model. Moreover, the approximation of its distribution based on first-order asymptotic theory can be inaccurate in moderate sample sizes and in the tails. In this section we propose an alternative to the LR-type test which cope with both problems and lead to accurate inference in moderate to small samples and in the presence of deviations from the assumed model. This test is built on a test proposed by Robinson, Ronchetti and Young (2003) and based on multivariateM -estimates. Under the null hypothesis it is asymptotically distributed as aχ2-variate as the classical LR-type test but it exhibitsa relative error of orderO(n−1). This is an important
improvement from the classical statistic for which the asymptotic approximatimation has
accuracy of the new test especially in the tails; cf. Figure 9. To define the test we first need the following results which establish the relationship between II estimators and
M−estimators.
5.1 An asymptotically equivalent M-estimator
In order to associate II estimators with asymptotically equivalent M-estimators, we first recall the notion of influence function; cf. Hampel (1974), Hampel, Ronchetti, Rousseeuw and Stahel (1986). A functional U is called Gˆateau differentiable at F if there exists a functiona1 such that for all Gon a set of distribution functions
lim ε→0 U((1−ε)F +εG)−U(F) ε = Z a1(y)dG(y), (29)
wherea1is theGˆateau derivativeofU atF. ConsiderG= ∆x, the point mass distribution that puts probability 1 onx. The Gˆateau derivative ofU at F in the direction ofG= ∆x is called the influence functionand is denoted byIF(x;T, F):
IF(x;T, F) = lim ε→0
U((1−ε)F +ε∆x)−U(F)
ε . (30)
The next proposition shows that to a given estimator with known influence function, there exists an asymptotically equivalent M-estimator.
Proposition 3 Hampel et al. (1986), p. 231. Given a model Fθ and a Fisher consistent
estimator U for θ (i.e. U(Fθ) =θ) with influence function IF(x;U, Fθ), the M-estimator
defined by theM-functionalTM given by
EF
Ψ(X;TM)= 0,
where Ψ(x;θ) =IF(x;U, Fθ) has the same asymptotic distribution as U.
For the convenience of the reader, we give a proof of Proposition 3 in Appendix A.5.
We turn now to the case of II estimators with II functionalT. When the functionalT
is defined by (12) withµ(θ) given in (a) (nonsimulation-based II), the influence function of the II estimator is given by Genton and de Luna (2000)
IF(x;T, Fθ) = ∂µ(θ) ∂θ T Ω∂µ(θ) ∂θ −1∂µ(θ) ∂θ T ΩIF(x;T , Fe θ), (31)
whereIF(x;T , Fe θ) is the influence function of the auxiliary estimator Te computed at the underlying modelFθ.
From Proposition 3 we can associate an M-estimator TM defined by
EF[ζ(X;TM)] = 0,
ζ(x;θ) = ∂µ(θ)
∂θ T
ΩIF(x;T , Fe θ) (32) which is asymptotically equivalent to T. Notice that an M−estimator is defined by a Ψ function up to a multiplicative constant matrix and in (32) we can drop the first matrix appearing on the right hand side of (31).
The function ζ that definesTM takes a special form in the following cases:
• WhenTe is anM-estimator defined by EF[ψ(X;Te)] = 0 we have
ζ(x;θ) = ∂µ(θ) ∂θ T ΩE∂ψ(X;t) ∂t t=Te(Fθ) −1 ψ(x;Te(Fθ)). (33)
• In the case when the dimensions of the auxiliary and structural parameters are the same (r = p), the M-estimator asymptotically equivalent with the II estimator is defined by
ζ(x;θ) =IF(x;T , Fe θ). (34)
• Ifr =p and the auxiliary estimator is an M-estimator then from (33) and (34), we have:
ζ(x;θ) =ψ(x;Te(Fθ)). (35)
We are now ready to define a robust test which will cope with both the small sample and robustness issues. Starting with a robust auxiliary estimator Te we compute the function given by (32) and we use it to construct the saddlepoint test proposed by Robinsonet al.
5.2 A robust II saddlepoint test
Denote by T the functional associated with the robust II estimator of θ ∈ Rp for the
model Fθ. To test a hypothesis defined by H0 : θ(1) = θ10, where θ = (θ(1), θ(2))T,
θ(1) ∈Rp1 , θ
(2) ∈Rp−p1, we propose the robust II saddlepoint test statistic
h(T(1)M) (36)
whereh is defined in Robinson et al. (2003) by
h(y) = inf θ(2) sup λ {− Kζ(λ;θ)}, (37) Kζ(λ;θ) = logEF(0) θ h eλTζ(X;θ)i, (38)
ζ is the function given by (32), andθ(0) is such that θ(1)=θ10.
According to Robinson et al. (2003) p.1157, asn→ ∞,
2n·h(T(1)M)) ∼ χ2p1
underH0. The order of the relative error of the asymptotic distribution isO(n−1).
In our simulations we will replace the M-estimator TM in (36) by the II estimator which seems to be a more natural choice in our case.
This new test can be also applied to II estimators whose auxiliary estimators are non-robust. However, our simulation study in the next section shows that robust II saddlepoint tests provide more accurate inference even in cases when the original data does not contain any outliers.
Empirical version of the saddlepoint test for robust II estimators
Since the cumulant generating function (38) may not be available, Robinson et al.
(2003) define an empirical version of the saddlepoint test forM-estimates. This empirical version applied to II estimates is defined as follows.
Given a sample y1,· · · , yn of observations, we need to compute firstθ∗(2) defined by: θ∗ (2) = arg minθ (2) −κ(λ(θ1 0, θ(2)); (θ1 0, θ(2))) , where κ(λ;θ) = logh1 n n X i=1 exp λTζ(yi;θ) i , with λ(θ) = arg max λ [−κ(λ;θ)],
and the functionζ as defined in (32). Then, the weights wi are given by:
wi = exp λ(θ∗)Tζ(yi;θ∗) / n X j=1 exp λ(θ∗)Tζ(yj;θ∗) , i= 1, . . . , n , (39) whereθ∗ = (θ 1 0, θ∗(2))T.
• Computation of the function ˜h(y)
˜ h(y) = inf t(2) sup λ n −Kw λ; (y, t(2)) o , (40) where Kw(λ;θ) = log Xn t=1 wiexp λTζ(yi;θ) , (41)
and the weights {wi} are fixed and given by (39).
This construction guarantees that the discrete distribution defined by (39) is the closest distribution (with respect to the Kullback-Leibler distance) to the empirical distribution (n1,· · ·,n1) which satisfies the null hypothesis H0. Kw(λ;θ) is its cumulant generating function and it replaces (38) in the empirical version of the saddlepoint test.
5.3 Illustration on the GBM model
In the case of the GBM model the auxiliary ML estimator defined by (19) and (20) is an
for II estimators is defined by (36) with ζ(rt;θ) = rt−1−Te1(Fθ) (rt−1−Te1(Fθ))2−Te2(Fθ) ! , whereθ= (β, σ2)T and Te(F θ) = (eβ −1, e2β+σ 2
−e2β)T. The robust II is defined with an auxiliary M-estimator given by (27). Hence, the score function ζ is given by
ζ(rt;θ) = ψc rt−√1−Te1(Fθ) e T2(Fθ) χc rt−√1−Te1(Fθ) e T2(Fθ) . (42)
In this case, Te(Fθ) cannot be calculated analytically, however it can be approximated by ˜
µS(θ) given in (6).
Figures 8 and 9 present p-value plots comparing classical LR-type, robust LR-type, II saddlepoint and robust II saddlepoint tests for sample sizes ofn= 40 and 240. For fairness we calculated both saddlepoint tests by replacing Te(Fθ) by the simulated approximation ˜
µS(θ), although for the nonrobust version the analytical version is available. Figure 10 shows the relative error plots of the χ2 approximation. These plots confirm the excellent
performance of the robust II saddlepoint tests which show very small relative errors even in the tails.
6. Conclusion
In this paper we propose a new test which improves the finite sample properties of the classical and robust LR-type II tests. We show its good small sample and robust properties on contaminated diffusion models. The investigation of both robustness and finite sample properties of estimators and tests can be carried out for other models using the techniques presented in this paper. Moreover, new and more reliable statistical procedures can be constructed using these methods as e.g. in the case of Efficient Methods of Moments; cf. Czellar and Zivot (2008).
Appendix
A.1 Proof of Proposition 1
Since µ(θ) =Te(Fθ), from (12) we have:
QFθ(θ) = ne T(Fθ)−µ(θ) T Ω(Te(Fθ)−µ(θ) o = 0.
Moreover, QFθ(θ) ≥ 0 as Ω is positive definite. So, by uniqueness of the minimum of
QFθ(υ), θis the unique minimum of QFθ and
T(Fθ) = arg min
υ QFθ(υ) =θ .
A.2 Proof of Proposition 2
By definition we have T(Gε, θ) = arg min υ n˜ T(Gε, θ)−µ(υ) T ΩT˜(Gε, θ)−µ(υ) o .
DifferentiatingQGε, θ(υ) with respect toυand using (i) and (iii) T(Gε, θ) is defined by the
following implicit equation,
h e T(Gε, θ)−µ(T(Gε, θ)) iT Ω ∂µ ∂υT T(Gε, θ) = 0. (43) Since Ω and ∂υ∂µT T(Gε, θ)
are nonsingular, equation (43) is equivalent to:
e
T(Gε, θ) =µ(T(Gε, θ)),
and by the local inversion theorem, µis bijective in a neighborhood ofT(Gε, θ). Thus
T(Gε, θ) =µ−1 e T(Gε, θ) . A.3 Bias of the II estimators of the model GBM under Gε, θ and Fδ, θ
From Proposition 2, the asymptotic bias of the II estimator is given by
asbias(ˆθδ,ε) =µ−1(Te(Gε, θ))−θ , (44)
whereµ(θ) =Te(Fδ, θ). The auxiliary functional Teis defined by
e
T1(F) =EF[rt]−1, (45)
e
T2(F) =EF[r2t]−EF[rt]2. (46) Under the Euler discretization Fδ, θ with δ= 1/m we have
y(k+1)δ ykδ = 1 +δβ+√δσk ∼ N(1 +δβ, δσ2). Hence, rewriting rt = yt yt−1 = ytmδ y(tm−1)δ · y(tm−1)δ y(tm−2)δ · · · · · y((t−1)m+1)δ y(t−1)mδ , we have µ(θ) = EFδ, θ[rt]−1 EFδ, θ[rt2]−EFδ, θ[rt]2 ! = (1 +δβ) m−1 δσ2+ (1 +δβ)2m−(1 +δβ)2m ! . (47) Inverting (47) we get µ−1(ν) = (ν1+1)1/m−1 1/m ν2+(ν1+1)2 1/m −(ν1+1)2/m 1/m . (48)
Finally, using (22) we obtain
e T(Gε, θ) = EGε, θ[rt]−1 EGε, θ[r 2 t]−EGε, θ[rt] 2 ! = (1−ε)e β+εeβ+σ22(τ2−1) −1 (1−ε)e2β+σ2 +εe2β−σ2+2σ2τ2 −(1−ε)eβ+εeβ+σ2 2 (τ2−1) 2 . (49)
From (48) and (49), the result (23) follows.
A.4 Bias of the auxiliary and indirect estimators of the model GBM underGε, θ
and Fθ
Suppose the data is generated from a contaminated GBM (17) where the contamination is generated from (22). Since µ(θ) =Te(Fθ) we have
µ(θ) = EFθ[rt]−1 EFθ[r 2 t]−EFθ[rt] 2] ! = e β−1 e2β+σ2 −e2β ! and µ−1 ν1 ν2 ! = ln(ν1+ 1) ln( ν2 (ν1+1)2 + 1) ! .
The asymptotic biases of the auxiliary and II estimators of θ= (β, σ2) are
asbias(˜θ, Gε) =Te(Gε)−θ , asbias(ˆθ, Gε) =µ−1(Te(Gε))−θ , with e T(Gε) = EGε[rt]−1 VGε[rt] ! .
The mean and variance under the contaminated distributionGε are:
EGε[rt] = (1−ε)e β +ε eβ+σ22(τ2−1) , VGε[rt] = (1−ε)e 2β+σ2 +ε e2β−σ2+2σ2τ2−(EGε[rt]) 2,
and the result in (26) follows.
A.5 Proof of Proposition 3
From (29) and (30) we have
It follows thatTM is Fisher consistent since by construction EFθ[Ψ(X;T M(F θ))] =EFθ[IF(X;U, FTM(Fθ))] = 0, and hence TM(Fθ) =θ . (51)
TM is anM-functional and its influence function is
IF(x;TM, Fθ) = −EFθ h∂Ψ(X;t) ∂tT t=TM(F θ) i−1 Ψ(x;TM(Fθ)) =−EFθ h∂Ψ(x;θ) ∂θT i−1 Ψ(x;θ) =−EFθ h∂IF(x;U, F θ) ∂θT i−1 IF(x;U, Fθ).
Using the first term of the von Mises expansion (von Mises, 1947; Fernholz, 1983), we get
˜
θ−θ=U(Fθ˜)−U(Fθ) = Z
IF(x;U, Fθ)fθ˜(x)dx+o(kθ˜−θk)
and differentiating with respect to ˜θ at ˜θ=θgives with (50) and (51):
I = ∂U(Fθ˜) ∂θ˜T ˜ θ=θ= Z IF(x;U, Fθ) ∂fθ(x) ∂θT dx=− Z ∂IF(x;U, F θ) ∂θT fθ(x)dx=−EFθ h∂IF ∂θT i ,
where I stands for the p×p identity matrix. So IF(x;TM, Fθ) = IF(x;U, Fθ) and the M-estimator defined by Ψ andU have the same asymptotic properties.
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Figure 5 Asymptotic biases of estimators ofβ, when true parameters areβ=−0.05 andσ= 0.2.
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0.00 0.02 0.04 0.06 0.08 0.10 0.00 0.05 0.10 0.15 0.20 0.25 0.30
Simple tests, no outliers
Nominal size Actual size LR SP Robust LR Robust SP n=40 0.00 0.02 0.04 0.06 0.08 0.10 0.00 0.05 0.10 0.15 0.20 0.25 0.30
Simple tests, 5% outliers
Nominal size Actual size LR SP Robust LR Robust SP n=40 0.00 0.02 0.04 0.06 0.08 0.10 0.00 0.05 0.10 0.15 0.20 0.25 0.30
Composite tests on beta, no outliers
Nominal size Actual size LR SP Robust LR Robust SP n=40 0.00 0.02 0.04 0.06 0.08 0.10 0.00 0.05 0.10 0.15 0.20 0.25 0.30
Composite tests on beta, 5% outliers
Nominal size Actual size LR SP Robust LR Robust SP n=40 0.00 0.02 0.04 0.06 0.08 0.10 0.00 0.05 0.10 0.15 0.20 0.25 0.30
Composite tests on sigma, no outliers
Nominal size Actual size LR SP Robust LR Robust SP n=40 0.00 0.02 0.04 0.06 0.08 0.10 0.00 0.05 0.10 0.15 0.20 0.25 0.30
Composite tests on sigma, 5% outliers
Nominal size Actual size LR SP Robust LR Robust SP n=40
Figure 7p-value plots of simple and composite tests, when true parameters areβ=−0.05 andσ= 0.5
0.00 0.02 0.04 0.06 0.08 0.10 0.00 0.05 0.10 0.15 0.20 0.25 0.30
Simple tests, no outliers
Nominal size Actual size LR SP Robust LR Robust SP n=240 0.00 0.02 0.04 0.06 0.08 0.10 0.00 0.05 0.10 0.15 0.20 0.25 0.30
Simple tests, 5% outliers
Nominal size Actual size LR SP Robust LR Robust SP n=240 0.00 0.02 0.04 0.06 0.08 0.10 0.00 0.05 0.10 0.15 0.20 0.25 0.30
Composite tests on beta, no outliers
Nominal size Actual size LR SP Robust LR Robust SP n=240 0.00 0.02 0.04 0.06 0.08 0.10 0.00 0.05 0.10 0.15 0.20 0.25 0.30
Composite tests on beta, 5% outliers
Nominal size Actual size LR SP Robust LR Robust SP n=240 0.00 0.02 0.04 0.06 0.08 0.10 0.00 0.05 0.10 0.15 0.20 0.25 0.30
Composite tests on sigma, no outliers
Nominal size Actual size LR SP Robust LR Robust SP n=240 0.00 0.02 0.04 0.06 0.08 0.10 0.00 0.05 0.10 0.15 0.20 0.25 0.30
Composite tests on sigma, 5% outliers
Nominal size Actual size LR SP Robust LR Robust SP n=240
Figure 8p-value plots of simple and composite tests, when true parameters areβ=−0.05 andσ= 0.5
0.02 0.04 0.06 0.08 0.10 0.0 0.5 1.0 1.5 2.0
Simple tests, no outliers
Tail probabilities Relative error LR SP Robust LR Robust SP n=40 0.02 0.04 0.06 0.08 0.10 0.0 0.5 1.0 1.5 2.0
Simple tests, no outliers
Tail probabilities Relative error LR SP Robust LR Robust SP n=240 0.02 0.04 0.06 0.08 0.10 0.0 0.5 1.0 1.5 2.0
Composite tests on beta, no outliers
Tail probabilities Relative error LR SP Robust LR Robust SP n=40 0.02 0.04 0.06 0.08 0.10 0.0 0.5 1.0 1.5 2.0
Composite tests on beta, no outliers
Tail probabilities Relative error LR SP Robust LR Robust SP n=240 0.02 0.04 0.06 0.08 0.10 0.0 0.5 1.0 1.5 2.0
Composite tests on sigma, no outliers
Tail probabilities Relative error LR SP Robust LR Robust SP n=40 0.02 0.04 0.06 0.08 0.10 0.0 0.5 1.0 1.5 2.0
Composite tests on sigma, no outliers
Tail probabilities Relative error LR SP Robust LR Robust SP n=240
Figure 9 Relative error plots of simple and composite tests, when true parameters areβ=−0.05 and σ= 0.5 andn= 40,240.
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