PARTIAL AUTOCORRELATION PROPERTIES FOR NON-STATIONARY AUTOREGRESSIVE MOVING-AVERAGE MODELS

Abstract. We are primarily interested in relating the partial autocorrelation behaviour of an autoregressive integrated moving-average process of order ( p, d, q ), { Z _i } say, with those of its D -differenced processes {(1 - B ) ^D Z _i } ( D = 1, …, d ). To this end, we evaluate the early partial correlations corresponding to serial correlations which initially follow a slow linear decline from unity. These partials, to a first approximation, take a small constant negative value from lag 2 onwards. We also demonstrate a relationship between the theoretical partials π _k and π _k (Δ) for a once-integrated process { Z _i } and its first-differenced process {(1 - B ) Z _i } respectively. These results carry over to cases where the non-stationary zeros are at -1, and differencing is replaced by the corresponding 'simplifying' transformation implicit in the operator 1 + B.     

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.     

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.     

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 .     

ON STATIONARITY OF THE SOLUTION OF A DOUBLY STOCHASTIC MODEL

Abstract. Consider the discrete parameter process {X_I} satisfying the doubly stochastic model X_t =ø_tX_t-1+ε_t where {ø} and {ε_t} are also stochastic processes. Necessary and sufficient conditions on {ø} are given for { X₁ } to be a second order process. When {ø_t} is a strictly stationary process, some sufficient conditions in terms of {ø} are given which guarantee the wide sense stationarity of {X_t} . It turns out that for these problems the distribution and dependence structure of the process {log |ø|} play an important role.     

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.     

Low Temperature Coefficient of Resonance Frequency Composition in the System Pb(Zr,Ti)O3—Pb(Mn1/3 Nb2/3)O3

Pb(Zr,Ti)O₃–Pb(Mn_1/3 Nb_2/3)O₃ (PZT–PMnN) system has been studied for high-power piezoelectric applications. This study investigates this system to find out the composition with high-power density piezoelectric characteristics and low tem-perature coefficient of resonance frequency (TCF). It was found that the composition 0.9PZT–0.1PMnN (Zr/Ti = 0.51/0.49) modified with 6 mol% Sr exhibits a TCF of −8 ppm/°C (−20 to +80°C). Further, the dielectric and piezoelectric properties of this composition are as follows: k _p= 0.53; Q _m= 800; d ₃₃= 274; ε₃₃/ε₀= 1290 and tan δ=1.1%, which shows the suitability of this composition for ultrasonic devices used under fluctuating thermal environment.     

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.     

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.     

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         

FREQUENCY DOMAIN TESTS OF MULTIVARIATE GAUSSIANITY AND LINEARITY

A stationary multivariate time series { X _t } is defined as linear if it can be written in the form X _t = ∑^∞ _{j =−∞} A _j e _{t − j} where A _j are square matrices and e _t are independent and identically distributed random vectors. If the e _t } are normally distributed, then { X _t is a multivariate Gaussian linear process. This paper is concerned with the testing of departures of a vector stationary process from multivariate Gaussianity and linearity using the bispectral approach. First the definition and properties of cumulants of random matrices are used to obtain the expressions for the higher-order cumulant and spectral vectors of a linear vector process as defined above. Then it is shown that linearity of a vector process implies constancy of the modulus square of its normalized higher-order spectra whereas the component of such a vector process does not necessarily have a linear representation. Finally, statistics for the testing of multivariate Gaussianity and linearity are proposed.     

ON THE INVERTIBILITY OF MULTIVARIATE LINEAR PROCESSES

Abstract. It is shown that a multivariate linear stationary process whose coefficients are absolutely summable is invertible if and only if its spectral density is regular everywhere. This general characterization of invertibility is applied later to the case of a linear process having an autoregressive moving-average (ARMA) representation. Under the usual assumptions, it is deduced that a process Y described by an ARMA(φ, TH) model is invertible if and only if the polynomial detTH( z ) has no roots on the unit circle. Given an invertible process Y which has an ARMA representation, it is finally shown that the process YT , where YT , =ε _i =0^l S _i Y _t-i , is invertible if and only if the matrix S ( z ) =ε _{i =0}^l S _i z ⁱ is of full rank for all z of modulus 1. It follows, in particular, that any subprocess of an invertible ARMA process is also invertible.     

Optical and X-Ray Single Crystal Studies of the Monoclinic ⇌ Tetragonal Transition in ZrO2

The monoclinic ⇌ tetragonal phase transition in ZrO₂ single crystals was studied at temperature by transmission optical microscopy and X-ray diffraction techniques. A series of timelapse photographs illustrated the relations between the events that occur during the transition. The events themselves were recognized by direct observation using a high-temperature microscope stage and by scrutiny of several high-temperature Laue photographs. During heating the monoclinic phase transforms to the tetragonal by the motion of an interface parallel to the (100) _m plane; simultaneous twinning also occurs behind the advancing interface. The tetragonal phase is usually twinned on the (1 2) _bct or ( 12) _bct plane, and the extent of twinning is influenced by the heating rate. Cooling transforms the untwinned tetragonal form into a twinned monoclinic form with the orientation of the monoclinic twins parallel to the trace of the (001) _m plane when observations are made in the (100) _m plane. Transformation of a twinned tetragonal crystal results in twins on the {110} _m and {001} _m planes. Orientation relations in the ZrO₂ transformation are: (100) _m ‖(110) _bct , [010] _m ‖[001] _bct , and by the virtue of twinning, (100) _m ‖(110) _bct , [001] _m ‖[001] _bct . During cooling the same topotaxial relations are maintained.     

ON THE CONSISTENCY OF LEAST SQUARES ESTIMATORS FOR A THRESHOLD AR(1) MODEL

Abstract. For the SETAR (2; 1,1) model

where {a_t(i)} are i.i.d. random variables with mean 0 and variance σ²(i), i = 1,2, and {a_t(l)} is independent of {a_t(2)}, we consider estimators of φ₁, φ ₂ and r which minimize weighted sums of the sum of squares functions for σ²(1) and σ²(2). These include as a special case the usual least squares estimators. It is shown that the usual least squares estimators of φ₁, φ₂ and r are consistent. If σ²(1) ≠σ²(2) conditions on the weights are found under which the estimators of r and φ₁ or φ₂ are not consistent.     

STATIONARITY OF THE SOLUTION OF Xt= AtXt-1+εt AND ANALYSIS OF NON-GAUSSIAN DEPENDENT RANDOM VARIABLES

Abstract. We give general and concrete conditions in terms of the coefficient (stochastic) process {A_t} so that the (doubly) stochastic difference equation X_t= A_tX_t-1+ε_t has a second-order strictly stationary solution. It turns out that by choosing {A_t} and the "innovation" process {ε_t} properly, a host of stationary processes with non-Gaussian marginals and long-range dependence can be generated using this difference equation. Examples of such nowGaussian marginals include exponential, mixed exponential, gamma, geometric, etc. When {A_t} is a binary time series, the conditional least-squares estimator of the parameters of this model is the same as those of the parameters of a Galton-Watson branching process with immigration.     

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.     

COMPARISON OF SOME NON-LINEAR AUTOREGRESSIVE PROCESSES

Abstract. We investigate autoregressive processes {X_n} ({Y_"}) satisfying X_n+1=Λ(X_n) + Z_n+1( Y_n+1=μ(Y_n) + Z_n+1) , n = 0, 1, 2, …, where Λ and μ are odd increasing functions and the Z_n's are i.i.d. with unimodal symmetric distribution. The process {|X_n|} is then stochastically monotone on [0, ∞). If |Λ| ≤ |μ| then |X_n is stochastically dominated by |Y_n| for n = 1, 2, …. Some related asymptotic properties, e.g., stationary distributions, are also discussed.     

VECTOR AUTOREGRESSIVE MODELS WITH UNIT ROOTS AND REDUCED RANK STRUCTURE:ESTIMATION. LIKELIHOOD RATIO TEST, AND FORECASTING

Abstract. The nonstationary multivariate autoregressive (AR) model Φ ( L ) Y _t =ε _t is considered for an m -dimensional process { Y _t }, where it is assumed that det {Φ( L )}= 0 has d < m unit roots and all other roots are outside the unit circle, and also that rank {Φ(1)}= r ( r = m – d ). Limiting distribution results obtained by Ahn and Reinsel for the least-squares and the Gaussian reduced rank (unit roots imposed) estimators for this AR model are extended to a model where the AR parameters possess additional structure such as nested reduced rank, and based on these results the asymptotic distribution of the likelihood ratio test statistic for testing the number d of unit roots is obtained. An analysis of three US monthly interest rate series is presented to illustrate the testing and estimation procedures. A small simulation study is also performed to examine the finite-sample properties of the likelihood ratio test and the prediction performance of models which impose different numbers of unit roots.     

Controlled Oxygen Partial Pressure Sintering of (Pb,La)(Zr,Ti)O3 Ceramics

Transparent PLZT(7/60/40) ceramics with large piezoelectric coefficients were obtained using a two-step sintering process with controlled oxygen partial pressure. Specifically, low-oxygen-pressure and low-temperature sintering were used in the first step, followed by a high-oxygen-pressure, high-temperature sintering cycle. High-density ceramics with small grain sizes of about 3 µm were prepared. As a result, k _p= 0.71, k ₃₃= 0.78, d ₃₃= 850 × 10^-12 C/N, and a transparency of 15% (λ= 610 nm, thickness of 1 mm) have been achieved; 20% improvement of d ₃₃ was gained compared to conventional processed PLZT ceramics ( d ₃₃= 710 × 10^-12 C/N).     

Effects of Sintering Atmosphere on Densification Behavior and Piezoelectric Properties of Pb(Ni1/3Nb2/3)O3–PbZrO3 Ceramics

Densification of polycrystalline Pb(Ni_1/3Nb_2/3)O₃–PbTiO₃–PbZrO₃ (PNN–PT–PZ) specimens was enhanced as the partial pressure of O₂ in the sintering atmosphere was increased. This observation was attributed to the increase in the internal pressure of a closed pore due to the thermal decomposition of PbO at a low partial pressure of O₂. The relative dielectric permittivity (ε_r), d ₃₃, k _p, and grain size of sintered specimens were also increased as the partial pressure of O₂ in the sintering atmosphere was increased. The observed dependence of piezoelectric properties on the partial pressure of O₂ was discussed in terms of the enhanced formation of the A-site vacancy ( V "_Pb) or the suppression of the B-site defect ( V ¨_O) as the oxygen potential increased.