Taro Takimoto

A numerical method for factorizing the rational spectral density matrix

Improving Rozanov (1967, Stationary Random Processes. San Francisco: Holden-day.)s algebraic-analytic solution to the canonical factorization problem of the rational spectral density matrix, this article presents a feasible computational procedure for the spectral factorization. We provide numerical comparisons of our procedure with the Bhansalis (1974, Journal of the Statistical Society, B36 , 61.) and Wilsons (1972 SIAM Journal on Applied Mathematics, 23 , 420) methods and illustrate its application in estimation of invertible MA representation. The proposed procedure is usefully applied to linear predictor construction, causality analysis and other problems where a canonical transfer function specification of a stationary process in question is required. Copyright Copyright 2010 Blackwell Publishing Ltd

The Measures of One-Way Effect, Reciprocity, and Association

To characterize the interdependent structure of a pair of two jointly second-order stationary processes , this chapter introduces the (overall as well as frequency-wise) measures of one-way effect, reciprocity, and association. Section 2.2 defines the Granger and Sims non-causality and establishes their equivalence for a general class of (not necessarily stationary) second-order processes. Sections 2.3 and 2.4 define the overall and frequency-wise one-way effect measures and provide three ways of deriving the frequency-wise measure. One is based on direct canonical factorization of the spectral density matrix. The other two are based on distributed-lag representation and innovation orthogonalization, respectively. Each approach provides a different representation of the same quantity. Section 2.5 introduces the overall and the frequency-wise measures of reciprocity and association.

Representation of the Partial Measures

This chapter extends the measures introduced in the previous chapter to partial measures in the presence of third-series involvement. Third-series intervention is known to sometimes incur phenomena such as spurious or indirect causality attributable to possible feedback from the series. To address the problem, this chapter introduces an operational way to define the partial causality and allied concepts between a pair of processes. The third-effect elimination is of the one-way effect component of the third series from a pair of subject-matter series to preserve the inherent feedback structure of the pair of interest.

Inference on Changes in Interdependence Measures

The causal relationship between the time series can be characterized with the moments of distributions for the series and the parameters of models such as the vector ARMA model from previous chapters. Thus, the changes in the moments of the time series and the model parameters suggest the possibility of a change in causal relationships as we expected. However, the changes in the moments and the model parameters do not tell us much about the magnitude of the change in causal relationships. In this chapter, we provide a measure of the change in causal relationships between a time series and the test statistic to determine whether such a change is associated with a structural change and is statistically significant. The properties of the measure and the test statistic are examined through a Monte Carlo simulation, and empirical examples are provided.

Inference Based on the Vector Autoregressive and Moving Average Model

Based on the stationary vector ARMA process, this chapter shows how the partial measures of interdependence introduced in Sect. 3.3 are numerically evaluated and applied to practical situations. Section 4.1 discusses the statistical inference on those measures using the standard asymptotic theory of the Whittle likelihood inference for stationary multivariate ARMA processes. The point is the use of simulation-based estimations of the covariance matrix of each measure-related statistic. In Sect. 4.2, we investigate the small sample performance of partial one-way effect measure estimates using Monte Carlo data generated by a pair of trivariate data generating processes, the VAR(2) and VARMA(1,1) models. All model parameter estimates are produced using an improved version of the Takimoto and Hosoya (2004, 2006) procedure. The partial frequency-wise measures of the one-way effect are evaluated using spectral factorization, and the parameters are substituted with a modified Whittle estimate. To illustrate the analysis of interdependence in the frequency domain, Sect. 4.3 provides an empirical analysis of US interest rates and economic growth data.