Diagnostic tests for non-causal time series with infinite variance

Journal of Statistical Planning and Inference 147 (2014) 117–131

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Journal of Statistical Planning and Inference journal homepage: www.elsevier.com/locate/jspi

Diagnostic tests for non-causal time series with infinite variance Yunwei Cui a, Thomas J. Fisher b,n, Rongning Wu c a b c

Department of Mathematics and Statistics, University of Houston-Downtown, Houston, TX 77002, USA Department of Statistics, Miami University, Oxford, OH 45056, USA Department of Statistics and Computer Information Systems, Baruch College, The City University of New York, New York, NY 10010, USA

a r t i c l e i n f o

abstract

Article history: Received 2 February 2013 Received in revised form 5 August 2013 Accepted 30 October 2013 Available online 8 November 2013

Goodness-of-fit testing for non-causal autoregressive time series with non-Gaussian stable noise is studied. To model time series exhibiting sharp spikes or occasional bursts of outlying observations, the exponent of the stable errors is assumed to be less than two. Under such a condition, the innovation variables have no finite second moment. We prove that the sample autocorrelation functions of the trimmed residuals are asymptotically normal. Nonparametric tests are also investigated. An assortment of test statistics is suggested for model assessment. & 2013 Elsevier B.V. All rights reserved.

Keywords: AR process Portmanteau Test α-stable distributions

1. Introduction Infinite variance autoregressive (AR) time series models have various practical applications. For example, Resnick (1997) fitted such a model to interarrival times between packet transmissions on a computer network, Gallagher (2001) studied differenced sea surface temperatures and fitted a symmetric α-stable AR model, and Ling (2005) examined the daily log-returns of the Hang Seng Index in the Hong Kong stock market. When modeling infinite variance autoregressive processes, non-Gaussian α-stable distributions (i.e. the exponent parameter α o2) are often adopted to specify the innovation process due to their ability to model sharp spikes observed in the data. This rich class of probability distributions allows heavy tails and skewness, the features exhibited in many observed time series including signal processing in electrical engineering, see Stuck and Kleiner (1974) and Sheng and Chen (2011); portfolio selection, see Rachev et al. (2004); and asset allocation, see Tokat and Schwartz (2002). So, the assumption of α-stable innovations in AR models appears well justified both theoretically and empirically. When studying AR processes, causality (all roots of the AR polynomial are outside the unit circle) is conventionally assumed. However, such an assumption is only needed when the study is carried out within the classical Gaussian framework, in order to ensure the identifiability of model parameters. Indeed, for every non-causal Gaussian AR process there exists an equivalent causal representation in the sense that the two processes have the same mean and autocorrelation functions, see Brockwell and Davis (1991). Since a Gaussian distribution is uniquely determined by its first two moments, the two processes necessarily possess the identical probability structure and are indistinguishable. In contrast, as demonstrated in Breidt et al. (1990, 1991), in the non-Gaussian setting, a non-causal AR process will have a different probability structure than its causal representation. In other words, for a non-Gaussian AR process the model parameters are identifiable and the model can be

n

Corresponding author. Tel.: þ 513 529 2176; fax: þ513 529 0989. E-mail address: [email protected] (T.J. Fisher).

0378-3758/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jspi.2013.10.010

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configured uniquely without being confined to the causal case, see Breidt and Davis (1992) and Rosenblatt (2000, Section 1.4) for additional information on identifiability. In this work we consider diagnostic procedures for non-causal non-Gaussian α-stable AR processes. Andrews et al. (2009) derived maximum likelihood estimators (MLEs) for such processes. They showed that, when fitting trading volumes of the Wal-Mart stock, a non-causal model yielded a better description of the observed data in the sense that the residuals are more compatible with the assumption of independent innovations than the residuals produced by its causal representation. Lanne et al. (2012) considered forecasting of the AR time series and demonstrated the improvements in the change-ofdirection forecasts when relaxing causality in the AR model fitted to the US inflation series. Recently Andrews and Davis (2013) developed a procedure of model identification for infinite variance AR processes and showed that minimizing Gaussian-based AIC yields a consistent estimator of the AR order. Like some work in the literature, they propose all-pass models to estimate the order of non-causality. The technique is also widely used in the fitting of non-causal models with innovations having finite variance (Rosenblatt, 2000 and the references therein and Breidt et al., 2001). Also a two-step procedure using all-pass models to fit noninvertible ARMA models was developed in Andrews et al. (2006). Compared to the efforts devoted to parameter estimation and model identification for non-causal non-Gaussian α-stable AR processes, model diagnostics have not been fully addressed so far. Motivated by the recent results of Lee and Ng (2010) and Bouhaddioui and Ghoudi (2012) we develop portmanteau test procedures for checking the goodness-of-fit of the noncausal α-stable AR model, where the model parameters are fitted using maximum likelihood estimation from Andrews et al. (2009). As second moments do not exist for infinite variance models, the asymptotic behavior of the sample autocorrelation of the residuals from the fitted model is difficult to harness for model diagnostics. To circumvent the difficulty, we use the trimmed residuals or a nonparametric procedure based on the ranks of the residuals and the squared residuals. We show that the sample autocorrelation of trimmed residuals at a given lag for an adequately fitted non-causal AR process is asymptotically normal; hence the commonly used portmanteau tests in the classical Gaussian framework that are based on the sample autocorrelation function, those of Box and Pierce (1970) and Ljung and Box (1978), can be easily extended to an infinite variance setting. We also show that the rank correlations of the residuals and the squared residuals are asymptotically normal and nonparametric tests of the empirical process could be developed for model diagnostic purpose. Extending the results in Bouhaddioui and Ghoudi (2012) to include squared residuals allows us to construct diagnostic procedures to check the order of non-causality. The rest of the paper is organized as follows. In Section 2, we introduce the necessary background material to derive the asymptotic distribution of trimmed residuals and also propose nonparametric methods. Several diagnostic statistics are introduced along with a discussion on a suggested two-step model fitting and diagnostic procedure. In Section 3, we examine the finite sample performance of the proposed procedures through simulation studies by checking and comparing the empirical sizes and powers of the tests and make suggestions for model diagnostic procedures in general. In Section 4, the test procedures are applied to empirical data. All technical proofs are relegated to the Appendix. 2. Theoretical results 2.1. Preliminaries Let fY t g be the autoregressive process satisfying the stochastic difference equation ϕðBÞY t ¼ Z t ;

ð1Þ

where the AR characteristic polynomial has no zeros on the unit circle, ϕðzÞ≔1  ϕ1 z  ⋯ ϕp z a0 for jzj ¼ 1, and the independent identically distributed (i.i.d.) innovation variables fZ t g come from a stable distribution with exponent αA ð0; 2Þ. We also assume that the AR characteristic polynomial could be written as the product of causal and purely non-causal polynomials p

ϕðzÞ ¼ ð1 θ1 z ⋯ θr zr Þð1  θr þ 1 z ⋯  θr þ s zs Þ;

ð2Þ

where 1  θ1 z  ⋯  θr z a 0 for jzjr 1, 1  θr þ 1 z ⋯ θr þ s z a0 for jzj Z1, p ¼ r þ s is the order of AR process and s 4 0 is the order of non-causality. Then the unique strictly stationary solution to (1) is given by Y t ¼ ∑1 j ¼  1 ψ j Z t  j , where ψj's are j determined by the Laurent series expansion for 1=ϕðzÞ ¼ ∑1 j ¼  1 ψ j z . It is well known that the coefficients fψ j g are geometrically decaying; namely there exist C 1 4 0 and 0 o D1 o 1 such that jψ j j o C 1 Djjj 1 for all j. Now let ψ j ¼ ψ  j , for j 40. From results in Chapters 3 and 13 of Brockwell and Davis (1991) we rewrite the solution as r

s

1

1

j¼0

j¼1

Y t ¼ ∑ ψ j Zt  j þ ∑ ψ jZt þ j :

ð3Þ

D b be the MLE by Andrews et al. (2009). Then n1=α ðϕ b  ϕÞFor the AR model defined in (1), let ϕ S, where S is a vector of random variables from a delta-method transformation of ξ, the maximizer to the random function WðÞ in Andrews et al. (2009, Section 1.3). It can be shown following some algebra that n  1=α ¼ oðn  1=δ þ 1=2 Þ for any δ on the interval ð2α=ðα þ 2Þ; minðα; 1ÞÞ. Suppose the observed time series is represented as fY  p þ 1 ; …; Y 0 ; Y 1 ; …; Y n g from (1) and an AR(p) is fit to the series satisfying the assumptions:

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119

1. Zt are i.i.d. non-Gaussian stable. 2. The AR(p) model is estimated by the MLE method of Andrews et al. (2009). n Let fZ^ t gt ¼ 1 be the residuals of the fitted model

Z^t ¼ Y t  ϕ^ 1 Y t  1 ⋯  ϕ^ p Y t  p :

ð4Þ ^L M n

and For some predetermined lower percentile λL and upper percentile λU , let n statistics of fZ^ t gt ¼ 1 , respectively. We define the following trimmed residuals υ^ t ¼ Z^ t 1

L

U

^ o Z^ t o M ^ Þ ðM n n

^U M n

be the ðnλL Þ-th and ðnλU Þ -th order

:

The goal is to test the hypotheses where the null (H0) is that the non-causal α-stable AR model defined by Eq. (1) is adequately identified. For the trimmed residuals, the sample autocorrelation (ACF) at lag k, ρ^ k , is computed by the formula ρ^ k ¼

ð∑nt ¼ k þ 1 υ^ t υ^ t  k Þ  ð∑nt ¼ k þ 1 υ^ t Þð∑nt ¼ k þ 1 υ^ t  k Þ=ðn  kÞ ð∑nt ¼ 1 υ^ 2t Þ  ð∑nt ¼ 1 υ^ t Þ2 =n

:

ð5Þ

The sample partial autocorrelation (PACF) at lag k, π^ k , can be derived by Durbin–Levinson algorithm: 1

π^ k ¼

ρ^ k  ρ^ Tðk  1Þ R^ k  2 ρ^ nðk  1Þ 1 1  ρ^ Tðk  1Þ R^ k  2 ρ^ ðk  1Þ

;

ð6Þ

where ρ^ ðk  1Þ ¼ ðρ^ 1 ; …; ρ^ k  1 ÞT , R^ k  2 ¼ ðρ^ ji  jj Þki;j¼11 (i.e. the symmetric Toeplitz matrix generated by ð1; ρ^ 1 ; …; ρ^ k  2 Þ), and ρ^ nðk  1Þ ¼ ðρ^ k  1 ; …; ρ^ 1 ÞT . We now give the asymptotic distributions of ρ^ k and π^ k . Theorem 1. If the model (1) is correctly identified by the MLE method, then, for any positive integer m, we have pffiffiffi D nρ^ ðmÞ -Nð0; Im Þ; pffiffiffi D nπ^ ðmÞ -Nð0; Im Þ; D

where - is convergence in distribution, ρ^ ðmÞ ≔ðρ^ 1 ; …; ρ^ m ÞT , π^ ðmÞ ≔ðπ^ 1 ; …; π^ m ÞT , and Im is the m  m identity matrix. The results of Theorem 1 allow for the construction of the so-called portmanteau test statistics for time series goodnessof-fit. A Box–Pierce or Ljung–Box type statistic can be constructed, for example ρ^ k : n k ¼ 1 k m

ð7Þ

Q ℓb ðmÞ ¼ nðn þ2Þ ∑

Under the null hypothesis, the Ljung–Box type statistic will behave as a chi-square random variable with m degrees of freedom as the sample size n increases. A Monti type statistic, see Monti (1994), can be constructed utilizing the partial autocorrelation function of trimmed residuals and Theorem 1 π^ 2k : k ¼ 1 nk m

ð8Þ

Q mt ðmÞ ¼ nðn þ 2Þ ∑

It also will be asymptotically distributed as a chi-square random variable with m degrees of freedom for a given positive integer m. We note that unlike in the classical AR fitting scenario, we do not lose any degrees of freedom in the chi-square approximation when utilizing trimmed residuals. Nonparametric tests can also be developed. The following result provides the foundation for nonparametric tests based on the empirical process of the residuals or the squared residuals. First define the rank and the normalized rank of the residual Z^ j by r j ¼ ∑ni¼ 1 1ðZ^ r Z^ Þ and r~ j ¼ ∑ni¼ 1 1ðZ^ r Z^ Þ =n, respectively. Then the rank correlation is computed via i

j

i

j

i ~ ~ γ^ i ¼ ∑nt ¼ 1 ðr t  1=2Þðr t þ i  1=2Þ and the Cramèr–von Mises function is given by " #     2 n m ri  1 þ j 3 rk  1 þ j ri þ j  1 ri þ j  1 1 1 n n m 1 nðn 1Þð2n  1Þ m  m1 ∑ ∏ 1þ τ€ ðmÞ ¼ ∑ ∑ ∏ 1    ; þn 2 2 3 ni¼1k¼1j¼1 n n n n 6n 2 i¼1j¼1

where a 3 b is the maximum of a and b. In the same fashion, for the squared residuals, one could also define the rank correlation and the Cramèr–von Mises function, denoted by γ^ ni and τ€ n ðmÞ, respectively. Theorem 2. If the model (1) is correctly identified by the MLE method, then, for any positive integer m, we have pffiffiffi D 12 nγ^ ðmÞ -Nð0; Im Þ; pffiffiffi D 12 nγ^ nðmÞ -Nð0; Im Þ;

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where γ^ ðmÞ ≔ð^γ 1 ; …; γ^ m ÞT , γ^ nðmÞ ≔ð^γ n1 ; …; γ^ nm ÞT and Im is the m  m identity matrix and D

τ€ ðmÞ-τm ; D

τ€ n ðmÞ-τm ; R1 R1 where τm ¼ 0 … 0 C~ du1 …dum , and C~ is the sequential empirical copula process of a sequence of i.i.d. random variables. Theorem 2 provides the basis for nonparametric tests. In addition to the two statistics, τ€ ðmÞ and τ€ n ðmÞ, whose asymptotics could be obtained through the simulation of i.i.d. uniformly distributed random variables, we propose two additional statistics m

Q rk ðmÞ ¼ 144n ∑ γ^ 2k ; k¼1 m

Q srk ðmÞ ¼ 144n ∑ ð^γ nk Þ2 : k¼1

Both are asymptotically chi-square distributed with m degrees of freedom. 2.2. Goodness-of-fit testing procedure Here we discuss the diagnostic procedure in a broader context. We remove the assumption of causality or non-causality and refer to such processes as general α-stable AR processes. Our theoretical results above are motivated by the work of Lee and Ng (2010) and Bouhaddioui and Ghoudi (2012) which are strictly based on residuals and are for the purely causal cases. Theorems 1 and 2 hold true for both the causal and non-causal cases; i.e. the order of non-causality s ¼0 or 0 o s rp, respectively. For a general α-stable AR model, even if the overall AR order is correctly identified, an inappropriate model could be constructed if the non-causality order s is incorrectly identified. From our simulation results we found that this particular problem can be serious under certain situations. Therefore, in the general α-stable AR processes case, model fitting and diagnostics should be performed as part of a two step process as described in Andrews and Davis (2013): first, identify the AR order p by minimizing the AIC statistic, computed using Gaussian likelihood, and fit a causal model with Yule–Walker method; second, explore if a non-causal model of the same order p has a better fit than the causal model. The Yule–Walker method can be used for fitting the causal α-stable AR model (Davis and Resnick, 1986). Furthermore, according to Andrews and Davis (2013), even though the Yule–Walker method cannot be used to estimate the true AR parameters when the model is non-causal, it consistently estimates a causal AR model with the same number of parameters and all-pass innovations. The all-pass stable process with α o 2 can behave as an uncorrelated process despite the fact that non-Gaussian all-pass AR processes of order greater than zero are known to be dependent. In practice, this is similar to the phenomenon of GARCH processes that appear uncorrelated although dependent. As seen in our simulations, the proposed tests based on the residuals can have difficulty detecting the serial dependence among the all-pass stable processes. As a consequence, an inappropriate causal AR model, instead of a more suitable non-causal model, may be fitted to the data if one is satisfied with the results that the residuals of the fitted causal model are uncorrelated without checking the second order correlations of the residuals. A preliminary solution to the problem is to develop tests for identifying the second order correlations among the residuals. For a quick and easy method, we can resort to the type of test proposed in McLeod and Li (1983) to detect the serial correlation of the squares of the residuals. Let fat g denote the squares of an i.i.d. sequence of α-stable distributed random variables fZ t g with α o2. For some predetermined upper percentile λ define Mn as the ðnλÞ-th order statistic of fat gnt ¼ 1 and obtain the trimmed squared residuals by ξt ¼ at 1ðat o Mn Þ : We can calculate the sample autocorrelation function value for fξt gnt ¼ 1 at lag k using (5) and denote it by ω^ k . It is easy to show that pffiffiffi D ^ ðmÞ -Nð0; Im Þ; nω ^ ðmÞ ≔ðω^ 1 ; …; ω^ m ÞT and Im is the m  m identity matrix. Unfortunately an analogous result is much more difficult to where ω show for the residuals of a correctly fitted general α-stable AR process with infinite variance. In the simulation study, we apply the Ljung–Box type test (7) and the Monti type test (8) to the trimmed squared residuals (denoted by Q sℓb and Qsmt, respectively) and use a chi-square with m degrees of freedom as the cutoff. Although the theoretical asymptotic distribution is currently unknown, this cutoff has intuitive appeal and appears to produce reasonable Type I error rates in general. The statistics show striking improvements in terms of power compared to those based on the trimmed residuals in detecting the order of non-causality when the overall order p is correctly identified. The results in Theorem 2 provide a theoretical justification for using the squared residuals in the nonparametric test and they can be utilized freely. However, in practice, when a causal infinite variance AR model is identified and passes all the proposed tests, practitioners should use caution when utilizing Q sℓb and Qsmt to study the serial correlations among the squared residuals to further investigate if a non-causal model would be preferred. For this purpose, before theoretical

Y. Cui et al. / Journal of Statistical Planning and Inference 147 (2014) 117–131

121

rationale is available, one may borrow the mass simulation procedure originally developed by Andrews and Davis (2013) for model identification to detect all-pass processes. The procedure generates confidence bands for the sample correlations of the absolute values and squares of the mean-corrected residuals from the fitted causal AR model. Alternatively, the distributional properties of Q sℓb and Qsmt could be obtained through Monte Carlo methods like those described in Lin and McLeod (2006). 2.3. Other portmanteau statistics Besides the Ljung–Box and Monti type statistics for the trimmed residuals proposed previously, other portmanteau tests are also widely used for Gaussian or causal AR model diagnostics. We briefly introduce these tests and give the asymptotic results when they are applied to the trimmed residuals. Recent work in the literature has suggested that asymmetric statistics may be more powerful in some situations than the symmetric (i.e. equally weighted) Ljung–Box and Monti type statistics. Define R^ m as the Toeplitz matrix of autocorrelations 2 3 … ρ^ m 1 ρ^ 1 6 7 1 … ρ^ m  1 7 6 ρ^ 1 7: R^ m ¼ 6 6 ⋮ ⋮ ⋱ ⋮ 7 4 5 1 ρ^ m ρ^ m  1 … Peña and Rodríguez (2002) suggested a statistic based on the likelihood ratio test from multivariate analysis. Their statistic is D^ ¼ nð1 jR^ m j1=m Þ. Utilizing the asymptotic normality from Theorem 1 and an application of the delta-method, the asymptotic distribution under the null hypothesis can be shown to satisfy m ^ D ∑ m  k þ 1 χ2; Dk m k¼1

ð9Þ

2

where each χk is a chi-square random variable with one degree of freedom. This distribution is difficult to write explicitly but can be well approximated by a Gamma distribution; see Peña and Rodríguez (2002) for details. In Peña and Rodríguez (2006) they suggested the sum of the log of one minus the squared partial autocorrelation function. Utilizing Theorem 1, that statistic can also be shown to satisfy (9). Mahdi and McLeod (2012) generalized the result of Peña and Rodríguez (2002, 2006) to the multivariate time series setting. In the univariate case their statistic is      3n log R^ m ; Q mm ðmÞ ¼ ð10Þ 2m þ 1 and the asymptotic distribution follows a result similar to (9) and can be approximated with a chi-square with ð3=2Þmðm þ 1Þ=ð2m þ 1Þ degrees of freedom. Recently, Fisher and Gallagher (2012) suggested an alternative asymmetric test compared to those based on the determinant of the matrix R^ m . They proposed a weighted Ljung–Box: m  k þ1 ρ^ 2k ; m n k k¼1 m

Q fg ðmÞ ¼ nðn þ 2Þ ∑

ð11Þ

which is shown to satisfy the distribution in (9) and can be well approximated by a Gamma distribution with shape 3mðm þ 1Þ=ð8m þ 4Þ and scale 2ð2m þ1Þ=3m. Likewise, a weighted Monti statistic is also introduced that follows the same asymptotic distribution under the null hypothesis. We note that asymmetric versions of the trimmed squared residual tests can also be constructed. Namely Qsmm(m) and Qsfg(m) where ρ^ k are replaced by ω^ k , the autocorrelation of the trimmed squared residuals ξt. Furthermore, an asymmetric nonparametric test could be constructed as Qrk is of the same form as the Ljung–Box type statistic, but we exclude that in this paper for brevity. 3. Simulation studies The goal of the simulation study is to demonstrate the validity of the theoretical results and to show the general effectiveness of the various test statistics introduced. It has been established in the literature that different test types work well for different innovations and alternative models. The primary objective here is to show the overall effectiveness of the diagnostic procedures as a whole. We leave it to the practitioner to make any specific recommendations for specific alternatives. An assortment of models, fits and α-stable innovation parameters are explored in these studies. Computation on α-stable distributions has been well studied and is known to be computationally intensive; see Nolan (1997, 1999), and Belov (2005). Our studies were performed in the GNU-licensed R-Project utilizing the stable distribution in the stabledist package with parameterization method zero. Due to the computational intensity in optimizing the likelihood function, our studies were run in a parallel framework. Similar to Andrews et al. (2009), when optimizing the likelihood function we found that the initial condition for the optimization was as important as the nonlinear optimizer. We generated 1200 random initial conditions; the likelihood function was found for each, and then the Nelder–Mead

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optimization routine was run on the best eight. The parameters for the maximum likelihood function of those eight were chosen as the MLE for the general AR process with α-stable innovations. In our studies we compare the Ljung–Box type statistic, Q ℓb ðmÞ in (7), the Monti type, Qmt(m) in (8), the Mahdi–McLeod type, Qmm(m) in (10), the weighted Ljung–Box, Qfg(m) in (11), the nonparametric test Qrk(m) for the residuals, Qsrk(m) for the squared residuals, and the empirical process statistics τ€ ðmÞ and τ€ n ðmÞ; note we drop the function notation ðmÞ below. For comparison we also include Q sℓb , Qsmt, Qsmm and Qsfg, the McLeod and Li (1983) type results utilizing the trimmed squared residuals. The Mahdi–McLeod is chosen over the suggestions in Peña and Rodríguez (2002, 2006) since it is numerically stable (see Lin and McLeod, 2006), generally has conservative Type I error performance and is implemented in the portes package. The statistics from Fisher and Gallagher (2012) are available in the WeightedPortTest package and include unweighted versions as well; i.e. the traditional Ljung–Box and Monti types. Mathematically, it can be shown that the test in Mahdi and McLeod (2012) is essentially an asymmetric Monti type test and that in Fisher and Gallagher (2012) is an asymmetric Ljung–Box type test. For all tests except the nonparametric tests, the residuals are trimmed at the 1st and 99th percentiles. The squared residuals are trimmed at the 99th percentile.

3.1. Empirical size We check the finite sample empirical size of the proposed tests for different AR(1) and AR(2) models at the nominal level of 5%. For each selected model, 1000 realizations of the model are generated and properly fit using the algorithm from Andrews et al. (2009). Table 1 reports the number of rejections (empirical size) when n ¼100 sample generated as a noncausal AR(1) with parameter ϕ ¼ 2 is properly fit for a variety of lags and α values. Table 2 provides the empirical size when a non-causal AR(2) with parameters, ϕ1 ¼ 1:2, ϕ2 ¼ 1:6, are correctly fitted for sample size n ¼100. Several α (tail-index) and β (skewness) parameters for the innovation distribution were explored and a few are provided here. Overall we see the tests all appear to have satisfactory Type I error performance and many tend to be slightly conservative. Only in a few cases (boldfaced) do we see significantly (based on one-sided hypothesis test) inflated size and most are with the non-parametric test τ€ . Furthermore we note from the results that the β parameter appears to have limited influence on the empirical size. Table 3 looks at a similar study for a properly fit non-causal AR(2) with a larger sample size of n ¼500; we exclude the statistics based on squared residuals and only display the results for two lags for brevity. We note that with a larger sample size, the parameters for the innovation sequence begin to influence performance. As the α term decreases below 1.2, the empirical size becomes liberal. Furthermore the β term appears to inflate the size when α is small. Fortunately, as pointed out in Mittnik et al. (2000), in practical problems the α values are generally greater than 1.5. We also note in these size studies that the lag m at which the statistics are calculated appears to have little effect on the empirical size. Overall, the proposed asymptotic distributions approximate these test statistics satisfactorily when α Z1:5. All tests appear to perform equally well in these cases, though some tests may seem a little bit conservative and the nonparametric test τ€ appears liberal in a few cases, particularly for larger values of m.

Table 1 Empirical size (number of rejections based on 1000 realizations) at nominal 5% level for AR(1) model, s ¼1, with ϕ ¼ 2, n¼ 100, β ¼ 0:5, γ ¼ 1, δ ¼ 0 and α varies. α

m

Q ℓb

Qmt

Qmm

Q fg

Qrk

τ€

Qslb

Qsmt

Qsmm

Qsfg

Qsrk

τ€ n

1.8

5 10 15 20

40 48 56 59

41 44 46 42

32 36 33 32

34 40 46 52

25 29 32 24

59 90 74 72

35 39 38 38

38 50 38 35

30 41 37 37

26 37 38 38

37 32 37 33

53 45 48 46

1.5

5 10 15 20

35 39 46 52

31 47 47 47

32 31 28 26

33 34 35 45

22 28 36 26

42 66 71 64

41 38 22 17

51 43 32 29

44 42 36 29

36 33 33 28

41 40 40 32

50 53 41 42

1.2

5 10 15 20

42 43 48 44

40 53 47 48

40 38 35 31

42 36 42 49

30 27 38 32

47 71 72 66

54 38 26 24

58 45 43 31

45 42 44 36

44 39 35 29

44 37 37 28

38 55 49 45

0.9

5 10 15 20

51 43 43 34

53 50 49 38

52 44 45 41

49 43 45 43

34 38 37 27

57 67 68 53

40 45 31 15

44 50 56 40

39 44 42 40

36 39 36 36

32 29 33 28

40 54 51 46

Y. Cui et al. / Journal of Statistical Planning and Inference 147 (2014) 117–131

123

Table 2 Empirical size (number of rejections based on 1000 realizations) at nominal 5% level for AR(2) model, s¼ 1, with ϕ1 ¼  1:2, ϕ2 ¼ 1:6, n¼ 100, β ¼ 0, γ ¼ 1, δ ¼ 0 and α varies. α

m

Q ℓb

Qmt

Qmm

Q fg

Qrk

τ€

Qslb

Qsmt

Qsmm

Qsfg

Qsrk

1.8

5 10 15 20

41 54 62 67

39 49 46 40

30 41 31 30

36 48 53 64

25 33 27 26

31 61 67 60

40 39 38 41

40 47 40 35

34 30 29 24

30 29 33 37

36 31 29 22

46 32 44 45

1.5

5 10 15 20

43 46 59 66

46 46 35 36

40 32 27 26

44 40 48 51

33 34 30 28

30 35 36 36

54 38 25 24

58 48 43 33

49 45 37 33

45 45 30 29

37 33 36 30

34 34 39 39

1.2

5 10 15 20

55 43 51 48

52 42 39 30

50 44 37 29

47 45 47 47

41 41 43 37

43 41 44 40

49 51 26 22

53 63 49 38

48 50 51 47

45 45 46 30

37 42 35 27

43 43 49 40

0.9

5 10 15 20

54 47 47 44

54 55 46 38

54 51 48 43

54 54 49 44

83 61 54 51

85 75 69 55

41 44 24 28

45 54 49 39

40 46 40 40

38 40 34 29

71 60 55 50

102 90 88 87

τ€ n

Table 3 Empirical size (number of rejections based on 1000 realizations) at nominal 5% level for AR(2) model, s ¼1, with ϕ1 ¼  1:2, ϕ2 ¼ 1:6, n¼ 500, γ ¼ 1, δ ¼ 0 and (α, β) varies. m

Q ℓb

Qmt

α ¼ 1:5 β¼0

10 20

28 48

30 49

α ¼ 0:8 β¼0

10 20

81 80

α ¼ 1:5 β ¼ 0:5

10 20

α ¼ 0:8 β ¼ 0:5

10 20

Qmm

τ€

Q fg

Qrk

32 37

29 37

30 35

20 69

78 86

87 83

88 79

85 74

107 104

43 44

38 39

37 34

34 37

31 33

38 41

112 101

113 104

112 105

112 101

91 94

122 109

3.2. Empirical power The previous subsection showed that the suggested statistics have good type I error performance in general. In this section we explore the performance of the statistics in detecting inadequately fitted models. The simulation studies for diagnostics of wrongly fitted causal models have been examined extensively in the literature, so here we focus on the noncausal cases. We explore non-causal AR(2) models of size n ¼100, with innovations with varying α, that are improperly fit. Fig. 1 displays the empirical power for significance level 5% when a non-causal AR(2) process with one root inside the unit circle is underfit as a non-causal AR(1). To help with the readability of the image, we exclude the Monti type and Mahdi–McLeod type statistics here as they perform similar to the Ljung–Box and Fisher–Gallagher types, respectively. The power of the statistics based on the residuals is displayed in black while those based on the squared residuals are in gray. It appears that the proposed tests all perform reasonably well and have satisfactory powers for detecting underfitted models. The statistics based on the residuals and trimmed residuals appear to be more powerful compared to the squared residuals. Furthermore, there appears to be a slight improvement when utilizing the asymmetric statistic. As aforementioned, an interesting quirk can arise when modeling a non-causal process. A practitioner could correctly identify the overall autoregressive order p and still underfit the non-causal order s. Fig. 2 displays the powers when a noncausal AR(2) model is fit with a purely causal AR(2) model. It is worth noting that the tests based on the residuals perform poorly when the α-parameter is close to 2. The tests based on the squared residuals do a much better job in detecting the underfit non-causal component. We note that as the α-parameter decreases the tests based on the residuals slightly improve but never beat those based on squared residuals. Another phenomenon in fitting a non-causal AR process is the possibility to overfit the non-causal component. Fig. 3 explores the performance of the methods when an AR(2) with s¼1 is fit with a purely non-causal AR(2) model with s¼2; that is, the non-causal order is overfit. We see from this study that most of the statistics perform reasonably well in determining the inadequacy of the fit, with no clear best performer. It can also be seen that the relationship between α and

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Empirical Power as a function of α 1.0

0.8 Test Stats Q lb Q fg Q rk τ

Power

0.6

Q sl b Q sf g Q srk τ*

0.4

0.2

0.0 1.0

1.2

1.4

1.6

1.8

2.0

α values n = 100, m = 10 based on 1000 iterations

Fig. 1. AR(2) with ϕ1 ¼  1:2, ϕ2 ¼ 1:6, n¼ 100, (β ¼ 0, γ ¼ 1, δ ¼ 0) underfit as AR(1), s ¼ 1.

Empirical Power as a function of α 1.0 Test Stats Q lb Q fg Q rk τ

0.8

Q sl b Q sf g Q srk τ*

Power

0.6

0.4

0.2

0.0 1.0

1.2

1.4

1.6

1.8

2.0

α values n = 100, m = 5 based on 1000 iterations

Fig. 2. AR(2) with ϕ1 ¼  1:2, ϕ2 ¼ 1:6, n ¼100, (β ¼ 0, γ ¼ 1, δ ¼ 0) improperly fit as AR(2), s ¼0.

power is particularly complicated in this case whereas in the previous studies (generally) as α decreased away from Gaussian power increased. Lastly, we explore the power as a function of the sample size n. Fig. 4 displays the power when the same non-causal AR(2) process is underfit as a non-causal AR(1). Here, we set α ¼ 1:7 and β ¼ 0, and explore power as the sample size increases from 50 to 500. We exclude the Ljung–Box type statistic here as its power functions are somewhere in between the Fisher–Gallagher and rank tests (same with the trimmed squared residuals). We see that all of the statistics appear to be consistent for this particular alternative. We note that the convergence of the power function for the Cramèr–von Mises function statistic is quite slow under this particular alternative and lag m. Overall we see the proposed methods can be effective as a diagnostic tool in the fitting of non-causal AR models. The statistics based on residuals appear to be the most powerful when the AR order, p, has been underfit or the non-causal order, s, has been overfit. The tests based on the squared residuals appear to be the most powerful when s is underfit. The asymmetric variants from Mahdi and McLeod (2012) and Fisher and Gallagher (2012) can improve power. Additionally, as m increases, we observed the decrease in power that is well known in the diagnostics literature; this was particular prevalent in the Cramèr–von Mises function statistic.

Y. Cui et al. / Journal of Statistical Planning and Inference 147 (2014) 117–131

125

Empirical Power as a function of α 1.0

Test Stats Q lb Q fg Q rk τ

0.8

Q sl b Q sf g Q srk τ*

Power

0.6

0.4

0.2

0.0 1.0

1.2

1.4

1.6

1.8

2.0

α values n = 100, m = 20 based on 1000 iterations

Fig. 3. AR(2) with ϕ1 ¼  1:2, ϕ2 ¼ 1:6, n¼ 100, (β ¼ 0, γ ¼ 1, δ ¼ 0) improperly fit as AR(2), s¼ 2.

Empirical Power as a function of n 1.0

0.8

Power

0.6

0.4 Test Stats Q sf g Q fg Q srk Q rk τ* τ

0.2

0.0 0

100

200

300

400

500

n values m = 20 based on 1000 iterations

Fig. 4. AR(2) with ϕ1 ¼  1:2, ϕ2 ¼ 1:6, (α ¼ 1:7, β ¼ 0, γ ¼ 1, δ ¼ 0) improperly fit as AR(1), s ¼1.

4. Example As an example, we apply our tests to the natural logarithms of the volumes of Wal-Mart (WMT) stock traded on the New York Stock Exchange from December 1, 2003 to December 31, 2004. We display the time series of the Wal-Mart data, together with its normal QQ-plot and sample ACF and PACF plots in Fig. 5. From the normal QQ-plot, one can see that the data appear to have heavier tails than Gaussian. Moreover, both Jarque–Bera and Shapiro–Wilk tests of normality reject normality with strong evidence of p-values less than 0.0001. The ACF and PACF plots indicate that an AR model with p ¼1 or p ¼2 is adequate for the data. In addition, AIC achieves its smallest value at p ¼2. Since AIC is a consistent order selection tool for general AR processes (Andrews and Davis, 2013), it is reasonable to fit an AR(2) model with non-Gaussian stable noise. Three different AR(2) models are fit to the Wal-Mart data using the MLE method from Andrews et al. (2009), namely, a causal model, a non-causal model with one root inside the unit circle, and a purely non-causal model. Table 4 shows the fitted model and the p-values of different test procedures (the τ€ ðmÞ and τ€ n ðmÞ tests are excluded as they suggest that all models are adequate). Only the non-causal model with one root inside the unit circle passes all tests. In addition, in all of our fits, α was estimated to be about 1.6, which provides confidence in our results based on the simulation studies. The same data has been investigated by Andrews et al. (2009) and Wu and Davis (2010). They all suggested that

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Sample Quantiles

17.0

16.0

17.0

16.0

15.0

15.0 0

50

100

150

200

250

−3

−2

−1

0.8

0.8

0.4

0.4

0.0

0.0 0

10

0

1

2

3

Theoretical Quantiles

t

20

30

40

0

10

time lag

20

30

40

time lag

Fig. 5. (a) The natural logarithms of the volumes of Wal-Mart stock, (b) the normal Q–Q plot, (c) the sample ACF, and (d) the sample PACF.

Table 4 p-values of the tests of fit for various AR(2) models fitted to the natural logarithms of the volumes of Wal-Mart stock. Test

Q ℓb

s ¼0 m¼ 5 m¼ 10 m¼ 15 m¼ 20

Qmt

Qmm

Qfg

Qrk

Qslb

Qsmt

Qsmm

Qsfg

Qsrk

ϕðzÞ ¼ 1  0:4326z  0:2122z2 10  11 10  10 10  9 10  8 10  8 10  7 10  6 10  5

10  16 10  15 10  14 10  12

10  15 10  15 10  13 10  12

0.22 0.13 0.32 0.36

0.01 0.02 0.12 0.18

0.01 0.03 0.16 0.18

10  4 10  3 0.01 0.03

10  3 10  3 10  3 0.03

0.22 0.13 0.32 0.36

s ¼1 m¼ 5 m¼ 10 m¼ 15 m¼ 20

ϕðzÞ ¼ 1 þ 2:0766z  2:0772z2 0.97 0.97 0.70 0.73 0.89 0.85 0.97 0.95

0.95 0.86 0.88 0.92

0.95 0.84 0.87 0.92

0.24 0.17 0.44 0.53

0.65 0.34 0.60 0.78

0.64 0.32 0.57 0.75

0.46 0.52 0.53 0.62

0.47 0.52 0.53 0.60

0.23 0.17 0.44 0.53

s ¼2 m¼ 5 m¼ 10 m¼ 15 m¼ 20

ϕðzÞ ¼ 1 þ 1:0240z  2:4993z2 0.24 0.25 0.42 0.42 0.37 0.29 0.63 0.43

0.21 0.23 0.30 0.33

0.21 0.23 0.32 0.40

0.01 0.02 0.09 0.13

0.14 0.06 0.05 0.15

0.10 0.03 0.03 0.07

0.10 0.03 0.02 0.02

0.11 0.04 0.03 0.05

0.01 0.02 0.09 0.13

a non-causal AR(2) model with one root inside the unit circle is more suitable for the data, for which we have the same conclusion according to the proposed tests.

5. Conclusions In this work we studied how to diagnose non-causal α-stable AR processes. Extending previous results, an assortment of portmanteau tests is proposed. The effectiveness of the tests is evaluated under the broader context of general α-stable AR processes and some under-fitting and over-fitting problems were studied. Overall the proposed methods appear effective but we found they could have a reduced power when determining the order of non-causality. As a remedy, McLeod–Li type tests are proposed. Like Andrews and Davis (2013), we suggested an overall two step procedure for fitting a general α-stable AR model. First, fit a causal model using AIC. Second, either use the simulation method for the all-pass processes of Andrews and Davis (2013) or use the suggested tests to validate the adequacy of fitted non-causal models and consequently choose the best fit of all possible causal and non-causal models.

Y. Cui et al. / Journal of Statistical Planning and Inference 147 (2014) 117–131

127

Acknowledgments The authors would like to thank the referees and associate editor for their work in reviewing this paper; we feel the changes are reflected in a more complete and thorough manuscript. The first author would like to thank the University of Houston-Downtown ORCA fund for partial support in this research. The second author would like to express his appreciation toward his previous institution, the University of Missouri-Kansas City, for its support of this research. Appendix A. Appendix To prove Theorem 1, we follow the method used in Lee and Ng (2010). Since Proposition 5.2 of their work is true for the innovation process in general, we can use it freely. The key is to establish the technical lemmas in their paper for the non-causal model. In the following, Propositions 1 and 2 correspond to Propositions 5.1 and 5.3 of Lee and Ng (2010), respectively. Let φt ¼ Z t  Z^ t , for t ¼ 1; …; n. From (3) we get p

p

1

p

1

j¼1

j¼1

k¼0

j¼1

k¼1

φt ¼ ∑ ðϕj  ϕ^ j ÞY t  j ¼ ∑ ðϕj  ϕ^ j Þ ∑ ψ k Z t  j  k þ ∑ ðϕj  ϕ^ j Þ ∑ ψ k Z t  j þ k : By changing the order of summation      p   1 minðj;pÞ  1 1 minðj;pÞ     b ^ ^  ∑ ðϕj  ϕ j Þ ∑ ψ k Z t  j  k  r  ∑ ∑ ðϕk  ϕ k Þψ j  k Z t  j  r ∑ ∑ ‖ϕ ϕ‖jψ j  k jjZ t  j j; j ¼ 1    j¼1 k¼1 j¼1 k¼1 k¼0 and

ð12Þ

ð13Þ

       p   1 p  p  1 p  1        ∑ ðϕj  ϕ^ j Þ ∑ ψ k Z t  j þ k  r  ∑ ∑ ðϕk  ϕ^ k Þψ j þ k Z t þ j  þ  ∑ ∑ ðϕk  ϕ^ k Þψ k  j Z t  j  j ¼ 1      j¼0k¼1 j ¼ 1 k ¼ jþ1 k¼1 1

p1

p

b J jψ j þ k jjZ t þ j jþ ∑ r ∑ ∑ Jϕϕ

p

∑

j ¼ 1 k ¼ jþ1

j¼0k¼1

b J jψ k  j jjZ t  j j; Jϕϕ

ð14Þ

^ By Andrews et al. (2009) the MLE of the AR polynomial coefficients, ϕ, b where J ϕ ϕ^ J is the Euclidean distance of ϕ and ϕ. D b  ϕÞS. By our assumption that 0 oα o 2, we can find a δ converges to a vector of random variables in distribution, n1=α ðϕ with 2α=ðαþ 2Þ o δo minfα; 1g such that n  1=α ¼ oðn  1=δ þ 1=2 Þ. Note that for any given ε 40 there always exists a ς1 4 0 such b J oς1 n  1=α g then there exists N1 4 0 that PðjSj 4ς1 Þ o ε=2, where jSj denotes the maximum norm. If we define An ¼ f J ϕ ϕ such that PðAn Þ 4 1 ε whenever n 4 N1 . Under the condition of An, we can obtain an upper bound for (13)    p  1 1 minðj;pÞ   ð15Þ  ∑ ðϕj  ϕ^ j Þ ∑ ψ k Z t  k  r ς1 n  1=α ∑ ∑ jψ j  k jjZ t  j j; j ¼ 1  j¼0 j¼1 k¼1 and an upper bound for (14)    p  p p1 p 1 1    ∑ ðϕj  ϕ^ j Þ ∑ ψ k Z t þ k  rς1 n  1=α ∑ ∑ jψ j þ k jjZ t þ j j þ ς1 n  1=α ∑ ∑ jψ k  j jjZ t  j j: j ¼ 1  j ¼ 1 k ¼ jþ1 j¼0k¼1 k¼1

ð16Þ

jψ j  k j, ψ nj ¼ ∑pk ¼ 1 jψ j þ k j, and ψ 1;…;p ¼ ∑pk ¼ 1 jψ k j. Assuming An is true, an upper bound for jφt j is given by Let ψ nj ¼ ∑minðj;pÞ k¼1 1

1

p1

j¼1

j¼0

j¼1

jφt jr ς1 n  1=α ∑ ψ nj jZ t  j j þ ς1 n  1=α ∑ ψ nj jZ t þ j j þς1 n  1=α ∑ ψ 1;…;p jZ t  j j:

ð17Þ

Proposition 1. For (17), the following are true:  δ  1    n E ∑ ψ j jZ t  j j o1; j ¼ 1   δ  1    n E ∑ ψ j jZ t þ j j o1; j ¼ 0  8  δ  δ 9   1 = n <  1     δ  δ=α n n ¼ oðnÞ: ς1 n ∑ E ∑ ψ j jZ t  j j þE ∑ ψ j jZ t þ j j    ; : t¼1 j¼1 j¼0 δ 1 n δ Proof. The coefficients fψ j g and fψ j g are geometrically decaying as j-1. As a result, ∑1 j ¼ 1 jψ j j o1 and ∑j ¼ 1 jψ j j o 1. Change the order of summation and apply the triangle inequality, then we have 1

1 minðj;pÞ

∑ jψ nj jδ r ∑

j¼1

1

∑ jψ j  k jδ ¼ p ∑ jψ j jδ o 1;

j¼1 k¼1

j¼0

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Y. Cui et al. / Journal of Statistical Planning and Inference 147 (2014) 117–131

and 1

1

1

p

∑ jψ j n jδ r ∑ ∑ jψ j þ k jδ r p ∑ jψ j jδ o 1:

j¼0

j¼0k¼1

j¼1

Also by the triangle inequality (see for example, Brockwell and Davis, 1991, page 537) and EjZ t  j jδ o 1  δ ( )  1  1   n n δ δ E ∑ ψ j jZ t  j j r E ∑ jψ j j jjZ t  j j o 1; j ¼ 1  j¼1  δ ( )  1  1   □ E ∑ ψ nj jZ t þ j j rE ∑ jψ nj jδ jjZ t þ j jδ o1: j ¼ 0  j¼0 Given a fixed number 0 o λ o1 and fβn g, a predetermined sequence of real numbers, let χ t ¼ Z t  Z ð½nλÞ  βn . Proposition 5.3 of Lee and Ng (2010) is also true for the non-causal AR sequences. Proposition 2. For any ς2 4 0,   n P n  1=2 ∑ 1ðjφt j 4 jχ t jÞ 1An 4 ς2 -0: t¼1

Proof. As in Lee and Ng (2010), we can pick a constant ς3 4 0 such that PðjZ sð½nλÞ j 4 ς3 Þ is arbitrarily small in which sðkÞ ¼ j if Zj is the kth largest number among fZ 1 ; …; Z n g. To show the result it is sufficient to get n  ∑ P ðjφt j 4jχ t jÞ \ An \ ðjZ sð½nλÞ j o ς3 Þ ¼ oðn1=2 Þ:

t¼1

By Lee and Ng (2010), for any t A f1; …; ng, Pfðjφt j4 jχ t jÞ \ An \ ðjZ sð½nλÞ j oς3 Þg r

 1 n 1 þ Efjφt jδ jχ t j  δ 1ðjZ sð½nλÞ j o ς3 Þ 1An t asð½nλÞg: n n

Using the triangle inequality and (17) n

δ

∑ Efjφt j jχ t j

t¼1

δ

8 < n   1ðjZ sð½nλÞ j o ς3 Þ 1An t asð½nλÞg r ∑ E ςδ1 n  δ=α t¼1 :

8 < þ ∑ E ςδ1 n  δ=α t¼1 :

1

n

∑ ψ j jZ t þ j j

j¼0

8 < þ ∑ E ςδ1 n  δ=α t¼1 :

jχ t j

∑ ψ j jZ t  j j

1ðjZ sð½nλÞ j o ς3 Þ jsð½nλÞ at

9 =

∑ ψ 1;…;p jZ t  j j

j¼1

jχ t j

δ

jχ t j

δ

1ðjZ sð½nλÞ j o ς3 Þ jsð½nλÞ at

9 = ;

ð18Þ

ð19Þ

; 9 =

!δ

p1

n

δ

n

j¼1

!δ n

!δ

1

1ðjZ sð½nλÞ j o ς3 Þ jsð½nλÞ at : ;

ð20Þ

To finish the proof, we next show (18)–(20) are oðn1=2 Þ. Conditional on sð½nλÞ ¼ t  j and sð½nλÞ at  j, (18) is bounded above by n

1

t¼1

j¼1

∑ ςδ1 ςδ3 n  δ=α ∑ jψ nj jδ Efjχ t j  δ jsð½nλÞ atg (

n

ð21Þ )

1

þ ∑ E ςδ1 n  δ=α ∑ jψ nj jδ jZ t  j jδ jχ t j  δ 1ðsð½nλÞ a t  jÞ jsð½nλÞ a t : t¼1

ð22Þ

j¼1

We apply Proposition 1, (5.23) in Proposition 5.3 of Lee and Ng (2010), and the fact that n  1=α ¼ oðn  1=δ þ 1=2 Þ to (21) and get n

1

t¼1

j¼1

∑ ςδ1 ςδ3 n  δ=α ∑ jψ nj jδ Efjχ t j  δ jsð½nλÞ atg ¼ oðn1=2 Þ:

For (22), two cases, 1 rj r t 1 and jZ t, are considered. When 1 rj rt  1, ( ) t 1

n

∑ E ςδ1 n  δ=α ∑ jψ nj jδ jZ t  j jδ jχ t j  δ 1ðsð½nλÞ a t  jÞ jsð½nλÞ a t

t¼1

j¼1

n

r ∑

t¼1

ςδ1 n  δ=α

n2 E n1

(

t 1

) n δ

δ

∑ jψ j j jZ t  j j jχ t j

j¼1

δ

jsð½nλÞ a t; t j

 ¼ o n1=2 ;

Y. Cui et al. / Journal of Statistical Planning and Inference 147 (2014) 117–131

129

n o δ δ 1 n δ since by (5.22) of Lee and Ng (2010), E ∑tj  jsð½nλÞ a t; t  j ¼ Oð1Þ. When j Zt, Z t  j is in the set ¼ 1 jψ j j jZ t  j j jχ t j fZ 0 ; Z  1 ; …g. Hence, Z t  j is independent of sð½nλÞ, which implies that Efjψ nj jδ jZ t  j jδ jχ t j  δ 1ðsð½nλÞ a t  jÞ jsð½nλÞ a tg ¼ jψ nj jδ EfjZ t  j jδ gEfjχ t j  δ 1ðsð½nλÞ a t  jÞ jsð½nλÞ atg: Now use Proposition 1 and (5.23) of Lee and Ng (2010) again (

n

)

1

n

∑ E ςδ1 n  δ=α ∑ jψ nj jδ jZ t  j jδ jχ t j  δ 1ðsð½nλÞ a t  jÞ jsð½nλÞ a t r ∑ ςδ1 n  δ=α

t¼1

t¼1

j¼t

 n1 1 ∑ jψ n jδ EfjZ t  j jδ gEfjχ t j  δ 1ðsð½nλÞ a t  jÞ jsð½nλÞ a tg ¼ o n1=2 : n j¼t j

Therefore, (18) is oðn1=2 Þ . In the same way, the result also holds for (19) and (20).

□

Proof of Theorem 1. Proof. Let qL and qU be the ðλL Þth and ðλU Þ-th quantiles of Zt. Denote the mean and standard deviation of the trimmed random variable Z t 1ðqL o Z t o qU Þ by μ and s, μ ¼ E½Z t 1ðqL o Z t o qU Þ 

and

s2 ¼ Var½Z t 1ðqL o Z t o qU Þ :

Let Z μt ¼ Z t 1ðqL o Z t o qU Þ μ, then directly from Lemma 4.1 in Lee and Ng (2010) ( ) n

n  1=2

∑

t ¼ kþ1 n

D

Z μt Z μt  k

-Nð0; s4 I m Þ; k ¼ 1;2;…;m

D

n  1=2 ∑ Z μt -Nð0; κ2 Þ; t¼1

n

n

P

∑ ðZ μt Þ2 -s2 ;

1

ð23Þ

t¼1

with κ being a certain constant associated with the distribution of Zt and ðqL ; qU Þ. L U ^ L and M ^ U be the ðnλL Þ-th and ðnλU Þ-th Now let Mn and Mn be the ðnλL Þ-th and ðnλU Þ-th order statistics of fZ t gnt ¼ 1 and M n n n μ ^ ^ ^ order statistics of fZ t gt ¼ 1 and define Z t ¼ Z t 1 ^ L ^ ^ U Þ  μ. It follows from Propositions 1 and 2 and the proof of Lemma ðM n o Z t o M n 4.2 in Lee and Ng (2010) that   n  P μ μ n  1=2 ∑ Z μt Z μt  k  Z^ t Z^ t  k -0; for k ¼ 1; 2; …; m; t ¼ kþ1

  n  μ P n  1=2 ∑ Z μt  Z^ t -0; t¼1

   μ  P n  1 ∑ ðZ μt Þ2  ðZ^ t Þ2 -0: n

ð24Þ

t¼1

Now note that μ μ μ μ 1 ffi 1 ffi 1 pffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi ∑nt ¼ k þ 1 Z^ t Z^ t  k  pffiffiffiffiffiffiffi ∑nt ¼ k þ 1 Z^ t n  ∑nt ¼ k þ 1 Z^ t  k k nk : n  kρ^ k ¼ n  k μ μ 1 n 1 ∑t ¼ 1 ðZ^ t Þ2  2 ð∑nt ¼ 1 Z^ t Þ2 n n

Combining (23) and (24) yields the result.

□

Proof of Theorem 2. Let the empirical copula of the residuals be defined as nmþ1 m m

1 C m;n ðu1 ; …; um Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∑ ∏ I r~ i þ j  1 r uj  ∏ uj ; nmþ1 i ¼ 1 j ¼ 1 j¼1

where ðu1 ; …; um Þ A ½0; 1m . Cline and Brockwell (1985) showed lim

t-1

P½jY 1 j4 t ¼ P½jZ 1 j 4 t

1  ∑ ψ j jα :

ð25Þ

j ¼ 1

As a result lim nP½jY 1 j4 an t ¼

t-1

1

∑

j ¼ 1

jψ j jα t  α ;

ð26Þ

for all t 40, where an ¼ infft : nP½jZ 1 j 4t r1g. Then we can follow the same lines of Theorem 3.4 and related technical Lemmas in Bouhaddioui and Ghoudi (2012) to prove that the empirical copula of the residuals C m;n converges to a ~ the sequential empirical process of a sequence of i.i.d. random variables identified in Genest and continuous process C, Rémillard (2004), for which there is no simple expression. However, as in Proposition 2.1 of Genest and Rémillard (2004), the Möbius transformation of C m;n , M, leads to some simple results. Let A be a subset of f1; …; mg with jAj 41, the Möbius

130

Y. Cui et al. / Journal of Statistical Planning and Inference 147 (2014) 117–131

transformation of C m;n indexed by A is nmþ1



 1 MA C m;n ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∑ ∏ I r~ i þ j  1 ruj uj : nmþ1 i ¼ 1 jAA

~ and furthermore MA ðCÞ ~ and MA′ ðCÞ ~ are Then MA ðC m;n Þ converges to continuous centered Gaussian processes MA ðCÞ asymptotically independent whenever two sets A aA′. Letting A ¼ f1; k þ 1g, then the serial rank correlation γ^ i could be derived, as in Bouhaddioui and Ghoudi (2012), from the Möbius transformation of C m;n through Z

1 γ^ i ¼ pffiffiffi MA C m;n du n Z 1Z 1

1 C m;n u1 ; 1; …; 1; uk þ 1 ; 1; …; 1 du1 duk þ 1 ¼ pffiffiffi n 0 0   1 ni ∑ ðr~ t  1=2Þðr~ t þ k  1=2Þ ; ¼ ð27Þ n t¼1 pffiffiffi and nγ^ i is asymptotically normal with mean zero and variance 1=122 . The same result carries over to the case of the squared residuals as discussed in Bouhaddioui and Ghoudi (2012). Also the asymptotics of τ€ and τ€ n follow as a direct consequence of the convergence of C m;n . □ References Andrews, B., Calder, M., Davis, R.A., 2009. Maximum likelihood estimation for α-stable autoregressive processes. Ann. Stat. 37 (4), 1946–1982, http://dx.doi. org/10.1214/08-AOS632. Andrews, B., Davis, R.A., 2013. Model identification for infinite variance autoregressive processes. J. Econom. 172 (2), 222–234, http://dx.doi.org/10.1016/j. jeconom.2012.08.009. Andrews, B., Davis, R.A., Breidt, F.J., 2006. Maximum likelihood estimation for all-pass time series models. J. Multivar. Anal. 97 (7), 1638–1659, http://dx.doi. org/10.1016/j.jmva.2006.01.005. Belov, I.A., 2005. 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