DISCRIMINANT ANALYSIS FOR STATIONARY VECTOR TIME SERIES

Abstract. In this paper, we shall consider the case where a stationary vector process { X_t } belongs to one of two categories described by two hypotheses π ₁ and π ₂. These hypotheses specify that { X_t } has spectral density matrices f (Λ) and g (Λ) under π ₁ and π ₂, respectively. Although Gaussianity of { X_t } is not assumed, we can formally make the Gaussian likelihood ratio (GLR) based on X (1),… X ( T ). Then an approximation I ( f : g ) of the GLR is given in terms of f (Λ) and g (Λ). If f (Λ) and g (Λ) are known, we can use I ( f : g ) as a classification statistic. It is shown that I ( f : g ) is a consistent classification criterion in the sense that the misclassification probabilities converge to zero as T →∝. When g is contiguous to f , we discuss non-Gaussian robustness of I ( f : g ). A sufficient condition for the non-Gaussian robustness will be given. Also a numerical example will be given.     

Some Results Concerning the Asymptotic Distribution of Sample Fourier Transforms and Periodograms for a Discrete-time Stationary Process with a Continuous Spectrum

We consider a stationary process ( X_t , t = 0, ±1, ...) with a continuous spectrum. Denote by D_n (λ) a tapered Fourier transform of ( X ₀, X ₁, ..., X _{n −1}) at (angular) frequency λ. We obtain the asymptotic distribution of D_n (λ) and the joint asymptotic distribution of { D_n (λ _j ), 1 ≤ j ≤ k } with continuity of the spectral density f (.) at the relevant frequencies as the only assumption concerning the second-order structure of ( X_t ); all other assumptions required are easily stated. The results are extended to processes for which f (.) is continuous except at λ = 0, with lim_λ←0 f (λ)λ^{2 d} = K , a constant, where 0 < d < ½, as is typical of certain types of processes with long-range dependence. Results for the sample periodogram, proportional to | D_n (λ)|², follow immediately.     

Functional Coefficient Autoregressive Models: Estimation and Tests of Hypotheses

In this paper, we study nonparametric estimation and hypothesis testing procedures for the functional coefficient AR (FAR) models of the form X_t = f ₁( X _{t − d} ) X _{t − 1}+ ... + f_p ( X _{t − d} ) X _{t − p} +ε _t , first proposed by Chen and Tsay (1993). As a direct generalization of the linear AR model, the FAR model is a rich class of models that includes many useful parametric nonlinear time series models such as the threshold AR models of Tong (1983) and exponential AR models of Haggan and Ozaki (1981). We propose a local linear estimation procedure for estimating the coefficient functions and study its asymptotic properties. In addition, we propose two testing procedures. The first one tests whether all the coefficient functions are constant, i.e. whether the process is linear. The second one tests if all the coefficient functions are continuous, i.e. if any threshold type of nonlinearity presents in the process. The results of some simulation studies as well as a real example are presented.     

Robustness of the autoregressive spectral estimate for linear processes with infinite variance

Consider a discrete-time linear process { x _t }, a one-sided moving average of independent identically distributed random variables {ε _t }, with the common distribution in the domain of attraction of a symmetric stable law of index δ∈ (0, 2) and the moving-average coefficients b ( j ) such that ε _t is invertible in terms of the present and possibly infinite past values of { x _t }. By treating { x _t } as if it is second-order stationary, a normalized spectral density function f (μ) is defined in terms of the b ( j ) and, having observed x ₁, ..., x _T , an autoregression of order k is fitted by the well-known Yule–Walker and least squares methods and the normalized autoregressive spectral estimators are constructed. On letting k ←∞ as T ←∞, but sufficiently slowly, these estimators are shown to be uniformly consistent for f (μ), the convergence rate being T ^−1/φ, φ > δ. The finite sample behaviour is investigated by a simulation study which also examines possible effects of considering 'non-invertible' models.     

ON LINEAR PROCESSES WITH GIVEN MOMENTS

Abstract. Let X _t = c ₀ Y _t + c ₁ Y _{t -1}+… be a linear process with known coefficients c _k , where Y _t is a strict white noise. Let m ₁, …, m _2r be given numbers. A method is presented to determine whether there exists a distribution of Y _t such that EX ^k _t = m _k for k = 1, …, 2 r . In the positive case, such a distribution of Y _t is described. Some explicit formulas for AR(1) and AR(2) models are derived. The results can be used for simulating a process with given moments of its stationary distribution. The procedure also enables proof that some stationary distributions cannot belong to the given linear process.     

AUTOREGRESSIVE PROCESSES WITH NORMAL STATIONARY DISTRIBUTIONS

Abstract. For the strictly stationary AR( k ) process Z _t = Λ ( Z _{t -1}) + α _t , with Λ : R ^k → R , Z _{t -1}= [ Z _{t -1}, Z _{t -2},…, Z _t-k ] and { α _t } an independent identically distributed white noise process, we partially characterize the Λ for which the stationary distribution of Z _t is normal.     

NON-NEGATIVE AUTOREGRESSIVE PROCESSES

Abstract. Consider a stationary autoregressive process given by X _t = b ₁ X _{t -1}+…+ b _p X _t-p + Y _t , where the Y _t are independent identically distributed positive variables and b ₁,…, b _p are non-negative parameters. Let the variables X ₁,…, X _n be given. If p = 1 then it is known that b ₁*= min( X _t / X _{t -1}) is a strongly consistent estimator for b ₁ under very general conditions. In this paper the case p = 2 is analysed in detail. It is proved that min( X _t / X _{t -1})→ b ₁ almost surely (a.s.) and min( X _t / X _{t -2})→ b ₂+ b ₁² a.s. as n → 8. The convergence is very slow. Denote by b ₁* and b ₂* values of b ₁ and b ₂ respectively which maximize b ₂+ b ₂ under the conditions X _t - b ₁ X _{t -1}- b ₂ X _{t -2}≥ 0 for t = 3,…, n . We prove that b ₁* b ₁ and b ₂* b ₂ a.s. Simulations show that b ₁* and b ₂* are better than the least-squares estimators of the autoregressive coefficients when the distribution of Y _t is exponential.     

Regression Models with Time Series Errors

In models of the form Y_t = r ( X_t ) + Z_t , where r is an unknown function and { X_t } is a covariate process independent of the stationary error { Z_t }, we give conditions under which estimators based on residuals Z ₁, ..., Z _n obtained from linear smoothers are asymptotically equivalent to those based on the actual errors Z ₁, ..., Z_n .     

A Test of Linearity for Functional Autoregressive Models

We propose a new test for linearity in time series. We consider an asymptotically stationary functional AR( p ) model on ℜ ^d of the form
X _n = f ( X _{n −1}, ..., X _{n − p} ) + ξ _n ( n ∈ N).
The testing procedure is based on a suitably normalized sum of quadratic deviations between two different estimates of the function f evaluated at q distinct points of ℜ ^dp . The estimators are f^ _n , a recursive version of the non-parametric kernel estimator of f , and  _n , a least squares estimator well suited to the linear case. The main result states that the test statistic has a χ² limit distribution under the null hypothesis. A similar result is derived under the alternative hypothesis for the test statistic corrupted by a non-linear term. Our simulations indicate that our asymptotic results hold for moderate sample sizes when the testing procedure is used carefully     

Choice of thresholds for wavelet shrinkage estimate of the spectrum

We study the problem of estimating the log-spectrum of a stationary Gaussian time series by thresholding the empirical wavelet coefficients. We propose the use of thresholds t _{j , n} depending on sample size n , wavelet basis ψ and resolution level j . At fine resolution levels ( j = 1, 2, ...) we propose
t _{j , n} = α _j log n
where {α _j } are level-dependent constants and at coarse levels ( j ≫ 1)
t _{j , n} = (π/√3)(log n )^1/2.
The purpose of this thresholding level is to make the reconstructed log-spectrum as nearly noise-free as possible. In addition to being pleasant from a visual point of view, the noise-free character leads to attractive theoretical properties over a wide range of smoothness assumptions. Previous proposals set much smaller thresholds and did not enjoy these properties.     

TESTING FOR CYCLICAL NON-STATIONARITY IN AUTOREGRESSIVE PROCESSES

This paper deals with the distributions evolving from the likelihood-ratio test for the factor 1 − B ⁿ in the lag polynomial Φ( B ) under the basic assumption that the data series is generated by the autoregressive model Φ( B ) X _t = ε _t where {ε _t } denotes Gaussian white noise. A characterization of the statistic and its asymptotic properties is given. Asymptotic and finite-sample significance points are tabulated. The test procedure is illustrated by an economics example.     

A note on L1 density estimation for linear processes

In this paper, we consider the L ₁ performance of a kernel estimator, f^_n of the density of a linear process X_t ∑^∞ _{k =0} a _k Z _t−k , a ₀ = 1, where { Z _t } is a sequence of independent and identically distributed (i.i.d.) random variables with E | Z ₁|^ε< ∞, for some ε > 1, and { a_k } is a sequence of reals converging to zero at a certain rate. Asymptotic minimizations of the integrated L ₁ risk of f_n and its upper bounds are considered. This paper extends the earlier results for the i.i.d. case by Devroye and Gyorfi ( Nonparametric Density Estimation: The L₁ View. New York: Wiley, 1985) and by Hall and Wand (Minimizing the L ₁ distance in nonparametric density estimation, J. Multivariate Anal. 26 (1988), 59–88) to the linear process case. Numerical examples to illustrate the performance of f_n are also presented.     

Sign Invariance in Goodness-of-Fit Tests for Time Series

A goodness-of-fit test for a stationary stochastic process may be based on a functional of the difference between the sample standardized spectral distribution and a hypothesized standardized spectral distribution. Theorems are given to show that under certain conditions the distribution of such a functional based on observations from a process { y_t } indexed by a parameter θ is the same for θ=θ₀ and for θ=−θ₀. The results are illustrated by three examples of time series processes.     

On the Spectral Density of the Wavelet Transform of Fractional Brownian Motion

We consider the wavelet transform { W _a ( t ), −∞ < t } < ∞}, at scale a > 0, of a fractional Brownian motion. A simple and mathematically rigorous proof is given to establish the existence of the spectral density f _Wa (λ) of the wavelet transform and provide an expression for it.     

State-space Models with Finite Dimensional Dependence

We consider nonlinear state-space models, where the state variable (ζ _t ) is Markov, stationary and features finite dimensional dependence (FDD), i.e. admits a transition function of the type: π(ζ _t |ζ _{t −1}) =π(ζ _t ) a '(ζ _t ) b (ζ _{t −1}), where π(ζ _t ) denotes the marginal distribution of ζ _t , with a finite number of cross-effects between the present and past values. We discuss various characterizations of the FDD condition in terms of the predictor space and nonlinear canonical decomposition. The FDD models are shown to admit explicit recursive formulas for filtering and smoothing of the observable process, that arise as an extension of the Kitagawa approach. The filtering and smoothing algorithms are given in the paper.
JEL. C4.     

ESTIMATION OF THE COEFFICIENTS OF A MULTIVARIATE LINEAR FILTER USING THE INNOVATIONS ALGORITHM

It is shown under mild conditions that the estimators of the coefficient matrices obtained by applying the innovations algorithm to the sample covariances of observations of the multivariate linear time series X _t = ∑^∞ _{j =0}ψ _i Z _t , t = 0, ±1, ±2, . . ., are consistent. The asymptotic distribution of the estimators is found to have a very simple form which generalizes the corresponding univariate result of Brockwell and Davis (Simple consistent estimation of the coefficients of a linear filter. In Stochastic Processes and Their Applications . Amsterdam: North- Holland, pp. 47--59). The asymptotic distribution of the corresponding estimator of the spectral density matrix is also derived. Some simulation results are presented to illustrate the small-sample behaviour of the estimators.     

A note on estimation by least squares for harmonic component models

Abstract. Let observations ( X ₁,…, X _n ) be generated by a harmonic model such that X _t = A ₀ cos  ω ₀ t + B ₀ sin  ω ₀ t + ε _t , where A ₀, B ₀, ω ₀ are constants and ( ε _t ) is a stationary process with zero mean and finite variance. The estimation of A ₀, B ₀, ω ₀ by the method of least squares is considered. It is shown that, without any restriction on ω in the minimization procedure, the estimate     is an n -consistent estimate of ω ₀, and hence (     ) has the usual asymptotic distribution.
The extension to a harmonic model with k >1 components is discussed. The case k =2 is considered in detail, but it was only found possible to establish the result under the restriction that both angular frequencies lie in the interval         

TIME SERIES RESIDUALS WITH APPLICATION TO PROBABILITY DENSITY ESTIMATION

Abstract. A linear stationary and invertible process y _t models the second-order properties of T observations on a discrete time series, up to finitely many unknown parameters θ. Two estimators of the residuals or innovations ɛ _t of y _t are presented, based on a θ estimator which is root- T consistent with respect to a wide class of ɛ _t distributions, such as a Gaussian estimator. One sets unobserved y _t equal to their mean, the other treats y _t as a circulant and may be best computed via two passes of the fast Fourier transform. The convergence of both estimators to ɛ _t is investigated. We apply the estimated ɛ _t to estimate the probability density function of ɛ _t . Kernel density estimators are shown to converge uniformly in probability to the true density. A new sub-class of linear time series models is motivated.     

A Median-Unbiased Estimator of the AR(1) Coefficient

A proof is given that the median of the ratios of consecutive observations of a stationary first-order autoregressive process X_t = α X _{t −1} + Y_t with P ( Y_t ≥ 0) = P ( Y_t ≤ 0) = 1/2 and P ( X_t = 0) = 0 is a median-unbiased estimator of α.     

MULTIVARIATE LOCAL POLYNOMIAL REGRESSION FOR TIME SERIES:UNIFORM STRONG CONSISTENCY AND RATES

Abstract. Local high-order polynomial fitting is employed for the estimation of the multivariate regression function m ( x₁ ,… x_d ) = E {φ( Y_d )φ X ₁= x ₁,…, X_d = x_d }, and of its partial derivatives, for stationary random processes { Y_i , X_i }. The function φ may be selected to yield estimates of the conditional mean, conditional moments and conditional distributions. Uniform strong consistency over compact subsets of R_d , along with rates, are established for the regression function and its partial derivatives for strongly mixing processes.