A Note on Tests for Nonlinearity in a Vector Time Series

1
1 Introduction
Nonlinear time series models have been used extensively in recent years to model complex dynamics that cannot be adequately represented using a linear model. Their use has been confined mainly to individual series, due to the additional complexity involved in modeling multivariate, correlated systems. There are examples, however, in which the relationship between series appears to be nonlinear. Hipel and McLeod (1994) hypothesize that, although a linear model may be adequate to describe average annual river flows, the relationship between daily river flows and precipitation may be nonlinear. Tong (1990) provides a nice review of nonlinear time series modeling, primarily in the univariate setting. Chen and Tsay (1993, 1996) and Lewis and Ray (1997) investigate techniques for obtaining semi-multivariate nonlinear models, in the sense that a single output series is modeled using several input series. Although briefly alluded to by Tong (1990, p.429) and Hipel and McLeod (1994, p.758), techniques for vector nonlinear time series modeling have only recently begun to be investigated. Granger and Tera¨svirta (1993) mention vector nonlinear autoregressive processes, vector nonlinear moving average processes, and multiple bilinear time series models in passing, but concentrate on statistical inference for univariate nonlinear models. A recent paper by Tsay (1997) discusses testing and modeling multivariate threshold autoregressive models; Wong and Li (1997) discuss vector processes having conditional heteroscedasticity.
2∞
2∞
fi( 1,t−1, 2,t−1ukl s,t−k u,t−l
(2)
s=1 k=1
s,u=1 k,l=1
2
∞
+
bi,suvklm s,t−k u,t−l v,t−m + · · · .
s,u,v=1 k,l,m=1
This representation is a generalization of the Volterra representation of a nonlinear stationary univariate time series. See, for example, Priestley (1988, p.26). In practice, a truncated representation involving a finite number of parameters is used to approximate the above structure. A bivariate linear process results if all the coefficients of the second and higher-order terms in (2) equal zero. Extensions of (1) and (2) to a general k-variate time series are clear.
jharvil@ and Bonnie K. Ray Dept. of Mathematical Sciences and Center for Applied Math and Statistics, New Jersey Institute of Technology, Newark, NJ 07102
borayx@
Summary
We investigate tests to detect general nonlinear behavior in a vector time series. The proposed tests are multivariate extensions of the univariate nonlinearity tests of Keenan (1985) and Tsay (1986). Simulation results show that, in general, the multivariate tests are more powerful than their univariate counterparts, especially for series having nonlinear structure that involves several components of the vector process and process error terms that are weakly or moderately cross-correlated. For illustration, we apply the tests to several standard multivariate data sets. We then analyze a set of seasonally adjusted quarterly capital expenditures and appropriations in U.S. manufacturing over the period 1953 to 1974 in more detail, to illustrate how one might build a vector nonlinear model and the gains that may achieved from allowing for nonlinear behavior in a vector series. Some key words: Likelihood Ratio; Multivariate time series; Nonlinearity
Before investigating specific nonlinear models for vector time series, however, it is desirable to have a test for general vector nonlinear structure. Univariate tests for examining particular types of nonlinear structure in individual series have been well studied. See, for example, De Gooijer and Kumar (1992) or Harvill (1998) for a review. But even if the components of a vector series exhibit nonlinear structure individually, a more powerful test of nonlinearity may be obtained if the component series are tested jointly. In this paper, we propose a multivariate test for general nonlinear structure in a vector series and show that, in most cases, it is more
(1)
X2,t = µ2 + f2( 1,t−1, 2,t−1, . . .) + 2,t,
where { 1,t} and { 2,t} are serially uncorrelated, but may be cross-correlated at lag zero, and f1(·), f2(·) are measurable real-valued functions. In general, fi(·), i = 1, 2 can be represented by a Volterra series of the form
2
powerful than the corresponding univariate test applied to each component series individually. Section 2 outlines the multivariate nonlinear framework considered in this paper, while Section 3 gives the new test statistic. Section 4 provides numerical comparisons of power for the univariate and multivariate test statistics based on simulations. Section 5 illustrates the tests on several real data sets and analyzes the multivariate nonlinear structure for one of the data sets in more detail. Some concluding remarks are given in Section 6.