Stationary Martingale Differences Research Articles

On the adaptive Lasso estimator of AR(p) time series with applications to INAR(p) and Hawkes processes

We investigate the consistency and the rate of convergence of the adaptive Lasso estimator for the parameters of linear AR(p) time series with a white noise which is a strictly stationary and ergodic martingale difference. Roughly speaking, we prove that (i) If the white noise has a finite second moment, then the adaptive Lasso estimator is almost sure consistent (ii) If the white noise has a finite fourth moment, then the error estimate converges to zero with the same rate as the regularizing parameters of the adaptive Lasso estimator. Such theoretical findings are applied to estimate the parameters of INAR(p) time series and to estimate the fertility function of Hawkes processes. The results are validated by some numerical simulations, which show that the adaptive Lasso estimator allows for a better balancing between bias and variance with respect to the Conditional Least Square estimator and the classical Lasso estimator.

On the martingale central limit theorem for strictly stationary sequences

In this study, for strictly stationary martingale difference sequences, we completely characterize the domain of attraction for the standard normal law (a' la Lévy, Khinchine, and Feller). In particular, we obtain the Rosenthal type inequality (Theorem 4) and a new relative stability result (Theorem 2).

Estimation pitfalls when the noise is not i.i.d.

This paper extends Whittle estimation to linear processes with a general stationary ergodic martingale difference noise. We show that such estimation is valid for standard parametric time series models with smooth bounded spectral densities, e.g., ARMA models. Furthermore, we clarify the impact of the hidden dependence in the noise on such estimation. We show that although the asymptotic normality of the Whittle estimates may still hold, the presence of dependence in the noise impacts the limit variance. Hence, the standard errors and confidence intervals valid under i.i.d. noise may not be applicable and thus require correction. The goal of this paper is to raise awareness to the impact of a non-i.i.d. noise in applied work.

Weak invariance principle in some Besov spaces for stationary martingale differences

We extend the classical Donsker weak invariance principle to some Besov space framework. We consider polygonal line processes built from partial sums of stationary martingale differences and of independent and identically distributed random variables. The results obtained are shown to be optimal.

Holderian weak invariance principle under a Hannan type condition

We investigate the invariance principle in Hölder spaces for strictly stationary martingale difference sequences. In particular, we show that the sufficient condition on the tail in the i.i.d. case does not extend to stationary ergodic martingale differences. We provide a sufficient condition on the conditional variance which guarantee the invariance principle in Hölder spaces. We then deduce a condition in the spirit of Hannan one.

Sample quantile analysis for long-memory stochastic volatility models

This study investigates asymptotic properties of sample quantile estimates in the context of long-memory stochastic volatility models in which the latent volatility component is an exponential transformation of a linear long-memory time series. We focus on the least absolute deviation quantile estimator and show that while the underlying process is a sequence of stationary martingale differences, the estimation errors are asymptotically normal with the convergence rate which is slower than n and determined by the dependence parameter of the volatility sequence. A non-parametric resampling method is employed to estimate the normalizing constants by which the confidence intervals are constructed. To demonstrate the methodology, we conduct a simulation study as well as an empirical analysis of the Value-at-Risk estimate of the S&P 500 daily returns. Both are consistent with the theoretical findings and provide clear evidence that the coverage probabilities of confidence intervals for the quantile estimate are severely biased if the strong dependence of the unobserved volatility sequence is ignored.

CLT for linear random fields with stationary martingale-difference innovation

In [V. Paulauskas, On Beveridge–Nelson decomposition and limit theorems for linear random fields, J. Multivariate Anal., 101:621–639, 2010], limit theorems for linear random fields generated by independent identically distributed innovations were proved. In this paper, we present the central limit theorem for linear random fields with martingale-differences innovations satisfying the central limit theorem from [J. Dedecker, A central limit theorem for stationary random fields, Probab. Theory Relat. Fields, 110(3):397–426, 1998] and arranged in lexicographical order.

PARAMETER ESTIMATION IN NONLINEAR AR–GARCH MODELS

This paper develops an asymptotic estimation theory for nonlinear autoregressive models with conditionally heteroskedastic errors. We consider a general nonlinear autoregression of order p (AR(p)) with the conditional variance specified as a general nonlinear first-order generalized autoregressive conditional heteroskedasticity (GARCH(1,1)) model. We do not require the rescaled errors to be independent, but instead only to form a stationary and ergodic martingale difference sequence. Strong consistency and asymptotic normality of the global Gaussian quasi-maximum likelihood (QML) estimator are established under conditions comparable to those recently used in the corresponding linear case. To the best of our knowledge, this paper provides the first results on consistency and asymptotic normality of the QML estimator in nonlinear autoregressive models with GARCH errors.

A generalized skewness statistic for stationary ergodic martingale differences

We present a class of generalized skewness statistics depending on a parameter β > 0 and containing the usual skewness statistic when β = 3, but providing greater flexibility for modelling and testing skewness when β ≠ 3. The statistics’ suitability for financial applications is illustrated using a large data set from the Australian share market. Data is assumed to be observations on stationary ergodicmartingale differences with possibly leptokurtic marginals, rather than independent identically distributed samples. The statistics can be studentized for use in hypothesis testing. Proof is provided of their asymptotic distributions undermild assumptions. Rates of convergence and power of the tests against skewed alternatives are assessed using simulation.

COINTEGRATION RANK TESTING UNDER CONDITIONAL HETEROSKEDASTICITY

We analyze the properties of the conventional Gaussian-based cointegrating rank tests of Johansen (1996, Likelihood-Based Inference in Cointegrated Vector Autoregressive Models) in the case where the vector of series under test is driven by globally stationary, conditionally heteroskedastic (martingale difference) innovations. We first demonstrate that the limiting null distributions of the rank statistics coincide with those derived by previous authors who assume either independent and identically distributed (i.i.d.) or (strict and covariance) stationary martingale difference innovations. We then propose wild bootstrap implementations of the cointegrating rank tests and demonstrate that the associated bootstrap rank statistics replicate the first-order asymptotic null distributions of the rank statistics. We show that the same is also true of the corresponding rank tests based on the i.i.d. bootstrap of Swensen (2006, Econometrica 74, 1699–1714). The wild bootstrap, however, has the important property that, unlike the i.i.d. bootstrap, it preserves in the resampled data the pattern of heteroskedasticity present in the original shocks. Consistent with this, numerical evidence suggests that, relative to tests based on the asymptotic critical values or the i.i.d. bootstrap, the wild bootstrap rank tests perform very well in small samples under a variety of conditionally heteroskedastic innovation processes. An empirical application to the term structure of interest rates is given.

Co-Integration Rank Testing under Conditional Heteroskedasticity

We analyse the properties of the conventional Gaussian-based co-integrating rank tests of Johansen (1996) in the case where the vector of series under test is driven by globally stationary, conditionally heteroskedastic (martingale difference) innovations. We first demonstrate that the limiting null distributions of the rank statistics coincide with those derived by previous authors who assume either i.i.d. or (strict and covariance) stationary martingale difference innovations. We then propose wild bootstrap implementations of the co-integrating rank tests and demonstrate that the associated bootstrap rank statistics replicate the first-order asymptotic null distributions of the rank statistics. We show the same is also true of the corresponding rank tests based on the i.i.d. bootstrap of Swensen (2006). The wild bootstrap, however, has the important property that, unlike the i.i.d. bootstrap, it preserves in the re-sampled data the pattern of heteroskedasticity present in the original shocks. Consistent with this, numerical evidence suggests that, relative to tests based on the asymptotic critical values or the i.i.d. bootstrap, the wild bootstrap rank tests perform very well in small samples under a variety of conditionally heteroskedastic innovation processes. An empirical application to the term structure of interest rates is given.

Upper-Lower Class Tests for Weighted i.i.d. Sequences and Martingales

We study the upper-lower class behavior of weighted sums ∑ k=1 n a k X k , where X k are i.i.d. random variables with mean 0 and variance 1. In contrast to Feller’s classical results in the case of bounded X j , we show that the refined LIL behavior of such sums depends not on the growth properties of (a n ) but on its arithmetical distribution, permitting pathological behavior even for bounded (a n ). We prove analogous results for weighted sums of stationary martingale difference sequences. These are new even in the unweighted case and complement the sharp results of Einmahl and Mason obtained in the bounded case. Finally, we prove a general upper-lower class test for unbounded martingales, improving several earlier results in the literature.

Moderate Deviations for Linear Processes Generated by Martingale-Like Random Variables

In this paper we study the moderate deviation principle for linear statistics of the type Sn=∑i∈Zcn,iξi, where cn,i are real numbers, and the variables ξi are in turn stationary martingale differences or dependent sequences satisfying projective criteria. As an application, we obtain the moderate deviation principle and its functional form for sums of a class of linear processes with dependent innovations that might exhibit long memory. A new notion of equivalence of the coefficients allows us to study the difficult case where the variance of Sn behaves slower than n. The main tools are: a new type of martingale approximations and moment and maximal inequalities that are important in themselves.

On the exactness of the Wu–Woodroofe approximation

Let ( X i ) be a stationary process adapted to a filtration ( F i ) , E ( X i ) = 0 , E ( X i 2 ) < ∞ ; by S n = ∑ i = 0 n − 1 X i we denote the partial sums and σ n 2 = ‖ S n ‖ 2 2 . Wu and Woodroofe [Wei Biao Wu, M. Woodroofe, Martingale approximation for sums of stationary processes, Ann. Probab. 32 (2004) 1674–1690] have shown that if ‖ E ( S n ∣ F 0 ) ‖ 2 = o ( σ n ) then there exists an array of row-wise stationary martingale difference sequences approximating the partial sums S n . If ∑ n = 1 ∞ ‖ E ( S n ∣ F 0 ) ‖ 2 n 3 / 2 < ∞ then by [M. Maxwell, M. Woodroofe, Central limit theorems for additive functionals of Markov chains, Ann. Probab. 28 (2000) 713–724] there exists a stationary martingale difference sequence approximating the partial sums S n , and the central limit theorem holds. We will show that the process ( X i ) can be found so that ‖ E ( S n ∣ F 0 ) ‖ 2 = O ( n log 1 / 2 n ) , σ n 2 / n → constant but the central limit theorem does not hold. The linear growth of the variances σ n 2 is a substantial source of complexity of the construction.

Moment Inequalities for Sums of Dependent Random Variables under Projective Conditions

We obtain precise constants in the Marcinkiewicz-Zygmund inequality for martingales in $\mathbb{L}^{p}$ for p>2 and a new Rosenthal type inequality for stationary martingale differences for p in ]2,3]. The Rosenthal inequality is then extended to stationary and adapted sequences. As in Peligrad et al. (Proc. Am. Math. Soc. 135:541–550, [2007]), the bounds are expressed in terms of $\mathbb{L}^{p}$ -norms of conditional expectations with respect to an increasing field of sigma algebras. Some applications to a particular Markov chain are given.

Rates of convergence for minimal distances in the central limit theorem under projective criteria

In this paper, we give estimates of ideal or minimal distances between the distribution of the normalized partial sum and the limiting Gaussian distribution for stationary martingale difference sequences or stationary sequences satisfying projective criteria. Applications to functions of linear processes and to functions of expanding maps of the interval are given.

A Note on the Martingale Method of Proving the Central Limit Theorem for Stationary Sequences

Under appropriate assumptions, the martingale approximation method allows us to reduce the study of the asymptotic behavior of sums of random variables that form a stationary random sequence to a similar problem for sums of stationary martingale differences. In an early paper on the martingale method, the author have proposed certain sufficient conditions for the central limit theorem to hold. It is shown in the present note that these conditions, at least in one particular case, can be essentially relaxed. In the context of the central limit theorem for Markov chains, a similar observation was done in a recent Holzmann and author's work. Bibliography: 12 titles.

Stable limits of martingale transforms with application to the estimation of GARCH parameters

In this paper we study the asymptotic behavior of the Gaussian quasi maximum likelihood estimator of a stationary GARCH process with heavy-tailed innovations. This means that the innovations are regularly varying with index α∈(2,4). Then, in particular, the marginal distribution of the GARCH process has infinite fourth moment and standard asymptotic theory with normal limits and $\sqrt{n}$-rates breaks down. This was recently observed by Hall and Yao [Econometrica 71 (2003) 285–317]. It is the aim of this paper to indicate that the limit theory for the parameter estimators in the heavy-tailed case nevertheless very much parallels the normal asymptotic theory. In the light-tailed case, the limit theory is based on the CLT for stationary ergodic finite variance martingale difference sequences. In the heavy-tailed case such a general result does not exist, but an analogous result with infinite variance stable limits can be shown to hold under certain mixing conditions which are satisfied for GARCH processes. It is the aim of the paper to give a general structural result for infinite variance limits which can also be applied in situations more general than GARCH.

ON THE CENTRAL AND LOCAL LIMIT THEOREM FOR MARTINGALE DIFFERENCE SEQUENCES

Let [Formula: see text] be a Lebesgue space and T: Ω→Ω an ergodic measure-preserving automorphism with positive entropy. We show that there is a bounded and strictly stationary martingale difference sequence defined on Ω with a common nondegenerate lattice distribution satisfying the central limit theorem with an arbitrarily slow rate of convergence and not satisfying the local limit theorem. A similar result is established for martingale difference sequences with densities provided the entropy is infinite. In addition, the martingale difference sequence may be chosen to be strongly mixing.

A Random Functional Central Limit Theorem for Processes of Product Sums of Linear Processes Generated by Martingale Differences

A random functional central limit theorem is obtained for processes of partial sums and product sums of linear processes generated by non-stationary martingale differences. It develops and improves some corresponding results on processes of partial sums of linear processes generated by strictly stationary martingale differences, which can be found in [5].