ON TESTING FOR SERIAL CORRELATION WITH A WAVELET-BASED SPECTRAL DENSITY ESTIMATOR IN MULTIVARIATE TIME SERIES

A wavelet-based consistent test for serial correlation of unknown form is proposed. As a spatially adaptive estimation method, wavelets can effectively detect local features such as peaks and spikes in a spectral density, which can arise as a result of strong autocorrelation or seasonal or business cycle periodicities in economic and financial time series. The proposed test statistic is constructed by comparing a wavelet-based spectral density estimator and the null spectral density. It is asymptotically one-sided N(0,1) under the null hypothesis of no serial correlation and is consistent against serial correlation of unknown form. The test is expected to have better power than a kernel-based test (e.g., Hong, 1996, Econometrica 64, 837–864) when the true spectral density has significant spatial inhomogeneity. This is confirmed in a simulation study. Because the spectral densities of time series arising in practice usually have unknown smoothness, the wavelet-based test is a useful complement to the kernel-based test in practice.

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Quantile cross-spectral density: A novel and effective tool for clustering multivariate time series

Expert Systems with Applications ◽

10.1016/j.eswa.2021.115677 ◽

2021 ◽

pp. 115677

Author(s):

Ángel López-Oriona ◽

José A. Vilar

Keyword(s):

Time Series ◽

Spectral Density ◽

Multivariate Time Series ◽

Cross Spectral Density

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ASYMPTOTIC THEORY FOR SPECTRAL DENSITY ESTIMATES OF GENERAL MULTIVARIATE TIME SERIES

Econometric Theory ◽

10.1017/s0266466617000068 ◽

2017 ◽

Vol 34 (1) ◽

pp. 1-22 ◽

Cited By ~ 8

Author(s):

Wei Biao Wu ◽

Paolo Zaffaroni

Keyword(s):

Time Series ◽

Spectral Density ◽

Multivariate Time Series ◽

Rates Of Convergence ◽

Stationary Time Series ◽

Regularity Conditions ◽

Dynamic Factor ◽

Asymptotic Results ◽

Density Estimates ◽

Convergence Results

We derive uniform convergence results of lag-window spectral density estimates for a general class of multivariate stationary processes represented by an arbitrary measurable function of iid innovations. Optimal rates of convergence, that hold as both the time series and the cross section dimensions diverge, are obtained under mild and easily verifiable conditions. Our theory complements earlier results, most of which are univariate, which primarily concern in-probability, weak or distributional convergence, yet under a much stronger set of regularity conditions, such as linearity in iid innovations. Based on cross spectral density functions, we then propose a new test for independence between two stationary time series. We also explain the extent to which our results provide the foundation to derive the double asymptotic results for estimation of generalized dynamic factor models.

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AN AUTOREGRESSIVE SPECTRAL DENSITY ESTIMATOR AT FREQUENCY ZERO FOR NONSTATIONARITY TESTS

Econometric Theory ◽

10.1017/s0266466698145024 ◽

1998 ◽

Vol 14 (5) ◽

pp. 560-603 ◽

Cited By ~ 37

Author(s):

Pierre Perron ◽

Serena Ng

Keyword(s):

Spectral Density ◽

Serial Correlation ◽

Mean Squared Error ◽

Spectral Density Function ◽

Weighted Sums ◽

Density Estimator ◽

Correlated Errors ◽

Squared Error ◽

Cointegration Tests ◽

Asymptotic Analyses

Many unit root and cointegration tests require an estimate of the spectral density function at frequency zero of some process. Commonly used are kernel estimators based on weighted sums of autocovariances constructed using estimated residuals from an AR(1) regression. However, it is known that with substantially correlated errors, the OLS estimate of the AR(1) parameter is severely biased. In this paper, we first show that this least-squares bias induces a significant increase in the bias and mean-squared error (MSE) of kernel-based estimators. We then consider a variant of the autoregressive spectral density estimator that does not share these shortcomings because it bypasses the use of the estimate from the AR(1) regression. Simulations and local asymptotic analyses show its bias and MSE to be much smaller than those of a kernel-based estimator when there is strong negative serial correlation. We also include a discussion about the appropriate choice of the truncation lag.

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Asymptotic Expansions of the Distributions of Statistics Related to the Spectral Density Matrix in Multivariate Time Series and Their Applications

Econometric Theory ◽

10.1017/s0266466600004928 ◽

1990 ◽

Vol 6 (1) ◽

pp. 75-96 ◽

Cited By ~ 2

Author(s):

Masanobu Taniguchi ◽

Koichi Maekawa

Keyword(s):

Time Series ◽

Density Matrix ◽

Spectral Density ◽

Asymptotic Expansions ◽

Multivariate Time Series ◽

Principal Component ◽

Autoregressive Process ◽

Reduced Form ◽

Gaussian Stationary Process ◽

Spectral Density Matrix

Let {X(t)} be a multivariate Gaussian stationary process with the spectral density matrix f0(ω), where θ is an unknown parameter vector. Using a quasi-maximum likelihood estimator θ̂ of θ, we estimate the spectral density matrix f0(ω) by fθ̂(ω). Then we derive asymptotic expansions of the distributions of functions of fθ̂(ω). Also asymptotic expansions for the distributions of functions of the eigenvalues of fθ̂(ω) are given. These results can be applied to many fundamental statistics in multivariate time series analysis. As an example, we take the reduced form of the cobweb model which is expressed as a two-dimensional vector autoregressive process of order 1 (AR(1) process) and show the asymptotic distribution of θ̂, the estimated coherency, and contribution ratio in the principal component analysis based on θ̂ in the model, up to the second-order terms. Although our general formulas seem very involved, we can show that they are tractable by using REDUCE 3.

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