Estimation in nonstationary random coefficient autoregressive models

Abstract.  We investigate the estimation of parameters in the random coefficient autoregressive (RCA) model X _k  = ( φ  +  b _k ) X _{k −1} +  e _k , where ( φ ,  ω ²,  σ ²) is the parameter of the process,     ,     . We consider a nonstationary RCA process satisfying E  log | φ  +  b ₀| ≥ 0 and show that σ ² cannot be estimated by the quasi-maximum likelihood method. The asymptotic normality of the quasi-maximum likelihood estimator for ( φ ,  ω ²) is proven so that the unit root problem does not exist in the RCA model.     

Kernel deconvolution of stochastic volatility models

Abstract.  In this paper, we study the problem of the nonparametric estimation of the function m in a stochastic volatility model h _t  =  exp( X _t /2λ) ξ _t , X _t  =  m ( X _{t −1}) +  η _t , where ξ _t is a Gaussian white noise. We show that the model can be written as an autoregression with errors-in-variables. Then an adaptation of the deconvolution kernel estimator proposed by Fan and Truong [ Annals of Statistics , 21, (1993) 1900] estimates the function m with the optimal rate, which depends on the distribution of the measurement error. The rates vary from powers of n to powers of  ln( n ) depending on the rate of decay near infinity of the characteristic function of this noise. The performance of the method are studied by some simulation experiments and some real data are also examined.     

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         

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.     

Estimation and testing for the parameters of ARCH(q) under ordered restriction

Abstract.  In this paper, we study a stationary ARCH( q ) model with parameters α ₀, α ₁, α ₂,…, α _q . It is known that the model requires all parameters α _i to be non-negative, but sometimes the usual algorithm based on Newton–Raphson's method leads us to obtain some negative solutions. So this study proposes a method of computing the maximum likelihood estimator (MLE) of parameters under the non-negative restriction. A similar method is also proposed for the case where the parameters are restricted by a simple order: α ₁≥ α ₂≥⋯≥ α _p . The strong consistency of the above two estimators is discussed. Furthermore, we consider the problem of testing homogeneity of parameters against the simple order restriction. We give the likelihood ratio (LR) test statistic for the testing problem and derive its asymptotic null distribution.     

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.     

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.     

The Limiting Distribution of the Residual Processes in Nonstationary Autoregressive Processes

The weak limit of the partial sums of the normalized residuals in an AR(1) process y _t = ρ y _{t −1} + e _t is shown to be a standard Brownian motion W ( x ) when |ρ| ≠ 1. However, when |ρ| = 1, the weak limit is W ( x ) plus an extra term due to estimation of ρ. Asymptotic behaviour of the partial sums is investigated with ρ = exp( c )/ n ) in the vicinity of unity, yielding a c -dependent weak limit as n ←∞, whose limit is again W ( x ) as | c | ←∞. An extension is made to nonstationary AR( p ) processes with multiple characteristic roots on the unit circle. The weak limit of the partial sums has close resemblance to that for the polynomial regression.     

THIRD-ORDER ASYMPTOTIC PROPERTIES OF ESTIMATORS IN GAUSSIAN ARMA PROCESSES WITH UNKNOWN MEAN

Abstract. This paper deals with the third-order asymptotic theory for Gaussian autoregressive moving-average (ARMA) processes with unknown mean μ. We are interested in the estimation of ρ = ( α₁…, α_p, β₁…, β_q ), where α ₁…, α_ρ and β ₁…, β_q are the coefficients of the autoregressive part and the moving-average part, respectively. First, we investigate the third-order asymptotic optimality of the bias adjusted maximum likelihood estimator (MLE) of ρ in the presence of the nuisance parameters μ and ² (innovation variance). Next, for a Gaussian AR(1μ μ, ²), we propose a mean corrected estimator α_c1c2 of the autoregressive coefficient. We make a comparison between the bias adjusted estimator α_c1c2* and the bias adjusted MLE, in terms of their probabilities of concentration around the true value, or equivalently, in terms of their mean squared errors. Finally some numerical studies are provided in order to verify the third-order asymptotic theory.     

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.     

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.     

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.     

NON-NEGATIVE AUTOREGRESSIVE MODELS

Abstract. Consider a stationary non-negative autoregressive (AR) model given x _t = b ₁ x _t -1, +…+ b _p x _t-p + e _t , where the e _t are independent identically distributed non-negative variables and b ₁, …, b _p are non-negative parameters, and all the roots of the equation 1 – b ₁ u –…– b _p u ^p = 0 are outside the unit circle. The stationary solution of the above AR model is called a stationary non-negative AR process. Let x ₁, x ₂, … x _n be an example of a stationary non-negative AR process. Under very general conditions strongly consistent estimators of the AR parameters b ₁, b ₂, …, b _p have been studied. In this paper a new procedure is proposed to estimate not only b ₁, b ₂, …, b _p but also b _o which is the essential lower bound of the variable e _t . We shall show that the new estimators obtained using the new procedure are consistent estimators of b _o, b ₁, …, b _p under the weakest condition which guarantees that the stationary non-negative AR model has a stationary non-degenerative solution.     

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.     

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.     

Likelihood analysis of a first-order autoregressive model with exponential innovations

Abstract. This paper derives the exact distribution of the maximum likelihood estimator of a first-order linear autoregression with an exponential disturbance term. We also show that, even if the process is stationary, the estimator is T -consistent, where T is the sample size. In the unit root case, the estimator is T ²-consistent, while, in the explosive case, the estimator is ρ ^T -consistent. Further, the likelihood ratio test statistic for a simple hypothesis on the autoregressive parameter is asymptotically uniform for all values of the parameter.     

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.     

A Class of Non-Embeddable ARMA Processes

We show that a stationary ARMA( p , q ) process { X _n = 0, 1, 2, ...} whose moving-average polynomial has a root on the unit circle cannot be embedded in any continuous-time autoregressive moving-average (ARMA) process { Y }( t ), t ≥ 0}, i.e. we show that it is impossible to find a continuous-time ARMA process { Y }( t )} whose autocovariance function at integer lags coincides with that of { X _n }. This provides an answer to the previously unresolved question raised in the papers of Chan and Tong ( J. Time Ser. Anal. 8 (1987), 277–81), He and Wang ( J. Time Ser. Anal. 10 (1989), 315–23) and Brockwell ( J. Time Ser. Anal. 16 (1995), 451–60).     

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.     

Synthesis of Y-Doped BaTiO3–(Bi1/2K1/2)TiO3 Lead-Free Positive Temperature Coefficient of Resistivity Ceramics and Their PTC Effects

As a lead-free positive temperature coefficient of resistivity (PTCR) material, (1– x mol%) BaTiO₃– x mol% (Bi_1/2K_1/2) TiO₃– y mol% Y₂O₃–0.5 mol% TiO₂ (BT– x BKT–2 y Y–0.5TiO₂) systems were prepared by the conventional solid-state reaction method. All samples containing <2 mol% BKT sintered in air possessed relatively low room-temperature resistivity (ρ₂₅) and high positive temperature coefficient (PTC) effect. However, when the BKT content exceeded 2 mol%, the sample was not semiconductive after sintering in air. The effects of sintering schedule on the properties of PTCR ceramics were discussed. The results showed that the optimum composition of BT–1BKT–0.2Y–0.5TiO₂, sintered at 1330°C for not-soaking and then fast quenched in air, achieved rather low ρ₂₅ of 28 Ω·cm and a high jump of resistivity (maximum resistivity [ρ_max]/minimum resistivity [ρ_min]) of 4.0 orders of magnitude with T _c about 155°C. The ρ₂₅ of the as-sintered sample could be further reduced to about 10 Ω·cm by annealing in N₂ at 450°C for 30 min, accompanied decrease on the PTC effect.