Time Series Analysis In Frequency Domain And It's Applications

In time series analysis, spectral analysis is of fundamental importance in thestudy of properties of stochastic processes. One hand, spectral analysis plays acentral role in the theory of linear predictions and filtering, on the other hand,the spectrum has an physical interpretation as a power-frequency distribution, itmay be estimated by fairly simple numerical techniques which don't require anyspecific assumptions on the structure of the process. In the dissertation, we studytwo kinds of model: autoregressive integrated moving average(ARIMA) model andautoregressive fractional integrated moving average(ARFIMA) model, which are alldeduced by the classical autoregressive moving average(ARMA) model, in frequencydomain, we discuss the periodogram properties of ARIMA process and the estima-tion of parameters for ARFIMA process.Let {X_t} be an ARIMA(p, 1,q), when p=q=0, {X_t} is said to be a ran-dom walk. The periodogram behavior of random walk process has been given inCrato(1996), in the thesis we deal with much more complicated case, and obtaintwo theorems about the periodogram of ARIMA(p, 1, q) process.Conclusion 1 Let {X_t} be an ARIMA(p,1,q), Y_t=(1-L)X_t, L repre-sents the backwards shift operator, Y_t is an autoregressive moving average(ARMA)model. Assuming that f_y(·) and r(·) are the spectral density and autocovariance ofY_t respectively, then |1-e^-iω|²I_n,X(ω)=I_n,Y(ω)+n^-1X_n²-R_n(ω)ω∈[0,Ï€], ER_n(ω)→0 as n→∞ω∈(0,Ï€).Consequently, |1-e^-iω|²EI_n,X(ω)→2Ï€(f_y(ω)+f_y(0))as n→∞.where I_n,X(·) denotes periodogram of {X_t}.Conclusion 2 Let X_t be a casual ARIMA(p, 1, q) process, Y_t=(1-L)X_t,Y_t=sum from j=0 to∞ψ_jε_t-j,sum from j=0 to∞|ψ_j|＜∞,ε_t～iid(0,σ²). Suppose f_y(·) and r(·) are the spectral density and autocovariance of Y_t respectively, then: if sum from j=0 to∞|ψ_jj^1/2|＜∞,Eε_t⁴=ησ⁴＜∞, we have ER_n²(ω_j)=2(4Ï€²f_y(ω_j)f_y(0))+O(n^-1/2)ω_j∈(0,Ï€).Moreover, for Fourier frequencyω_j,ω_j∈(0,Ï€), Cov(R_n(ω_j),R_n(ω_k))=O(n^-1／2)ω_j≠ω_k,and Cov(I_n,Y(ω_j),R_n(ω_k))={O(n^-1)ω_j≠ω_k O(n^-1／2)ω_j=ω_k.When d take fractional values, the process is known as autoregressive fractionalintegrated moving average(ARFIMA) process, when d∈(0,0.5), the process isstationary and shows long memory. In such case the correlations of process decayat hyperbolic rate, and the extent of memory is described by parameter d. In thethesis, we propose two methods in view of smoothing periodogram and bayesiananalysis, according to the estimation of memory parameter d.Conclusion 3 Let {x_t} be an ARFIMA process, under suitable conditions,the distribution of d is approximated by N(d,2(1/ν+1/ν²)[sum from j=1 to g(n)(x(j,n)-(?)(n,g(n))²]^-1)whereν=2n/(M integral from n=-∞to∞ω²(u)du, n is sample size and M denotes truncation point. Be-cause smoothing periodogram is a consistent estimator of the spectrum, the newestimator has smaller variance than that of GPH-estimator.Conclusion 4 In frequency domain, suppose the prior distribution of 2d isBeta(a, b), then we obtain the posterior density and bayesian estimator of d: f(d|y₁,y₂,…,y_m)∝π(d)δ/(ρ+λ)^m+1. (a-1)/(?)-2(b-1)/(1-2(?))+sum from j=1 to m x_j=(m+1)(sum from j=1 to m x_je^y_j+(?)x_j/(λ+sum from j=1 to m e^y_j+(?)x_jwhere y_j=lnI_n,x(ω_j),δ=multiply from j=1 to m e^y_j+dx_j,ρ=sum from j=1 to m e^y_j+dx_j,λ, a, b denote parametersof prior distributions. Moreover, we show the new estimator is more stable thanGPH-estimator according to different choices of frequencies used in the regression.For given data, we use AIC criterion and SIC criterion to select proper model, whichis used to fitting the data generate process.Conclusion 5 Furthermore, in long memory stochastic volatility model, weestimate memory parameter by bayesian method in frequency domain, and obtainthe posterior density and bayesian estimator of d: f(d|y₁,y₂,…,y_m)∝(β^αδ/Γ(α))(Γ(m+α)/(ρ+β)^m+α)Ï€(d). (a-1)/(?)-2(b-1)/(1-2(?))+sum from j=1 to m x_j=(m+α)(sum from j=1 to m x_je^y_j+(?)x_j)/(β+sum from j=1 to m e^y_j+(?)x_j)whereδ=multiply from j=1 to m e^y_j+dx_j,ρ=sum from j=1 to m e^y_j+dx_j,α,β, a, b denote parameters of priordistributions. According to the new estimator, we calculate it's mean and varianceby simulation study. Because of the introducing of prior information, the simula-tion mean and variance are smaller than those of GPH-estimator, moreover, thenew estimator is more stable than GPH-estimator according to different choices offrequencies. Finally, we deal with empirical data, and obtain results which accordwith simulation study.