Co-trending: A Statistical System Analysis of Economic Trends

To invite the readers to the present book we begin Section 1.1 with what has motivated the authors to study the relations among deterministic trends. The rest of this section consists of an expository explanation of co-trending and presentation of highlights of the authors’ results. In Section 1.2 the literature on the relations among trends will be reviewed. We have explored a number of different approaches before settling on the one in this book, and they are discussed briefly in Section 1.3. The mathematical notations are assembled in Section 1.4.

Our basic research strategy is to model the relations among deterministic trends directly without using the parameters of stochastic parts of models. Direct modelling is divided into the parametric approach and the non-parametric approach. The parametric approach is based on a list of parametrically specified functions of time, which will be called trend functions, while the non-parametric approach does not use such functions. We begin with the non-parametric approach in Section 2.1, which will then be followed by the parametric approach in Section 2.2. Section 2.3 will present an example of trend functions that is hoped will be useful for empirical studies of the Japanese economy. Section 2.4 will compare our co-trending with the co-breaking in Hendry and Mizon (1998) and Clements and Hendry (1999). Section 2.5 will briefly survey the history of the direct modelling and the indirect modelling in the time series analyses. We had long seen the indirect modelling only, but Schmidt and Phillips (1992) have advocated the direct modelling.

We shall utilise the information contained in the data covariance matrix to investigate the co-trending relations. Given an eigenvalue of the matrix, (i) the eigenvalue itself, (ii) the eigenvector associated with it, and (iii) the principal component associated with it form one set of statistics. Altogether n sets of statistics are available, and they are ordered in descending order in terms of the eigenvalues. In Section 3.1 below, the relations among the deterministic trends are classified into those that hold among the stochastic trends as well and those that do not hold among the stochastic trends. Let r₁ and r₂, respectively, represent the numbers of the former and the latter. By definition we have that r = r₁ + r₂. Section 3.2 will show that the first n - r sets of statistics provide the information on the common deterministic trends. It will be shown in Section 3.3 that the next r₂ sets of statistics represent the relations that hold among the deterministic trends but do not hold among the stochastic trends. Section 3.4 will show that the last r₁ sets of statistics are related to the relations that hold among both the deterministic trends and the stochastic trends. Sections 3.5 and 3.6 will indicate how these results provide the bases for the testing procedures which will be given after Chapter 4. Section 3.6 contains an overview of the rest of this book.

The results in the previous chapter have been derived by the non-parametric approach. Hereafter, the co-trending will be modelled by the parametric approach using Assumption 2.2 with some elementary trend functions. The purpose is to make the DGP of principal components sufficiently specific so that statistical tests can be applied to the principal components. Propositions 3.1 through 3.3 of the previous chapter have been derived from Assumptions 2.1, 3.1, 3.2, 3.3, and 3.4 based on the non-parametric approach. In translating these results onto the parametric approach, Assumptions 2.1, 3.2, 3.3, and 3.4 are derived from Assumption 2.2 and Definition 2.2. Definition 2.2 specifies the conditions to be met by the elementary trend functions, and it will be considered as a part of Assumption 2.2. Thus our basic assumptions in the parametric approach are Assumptions 2.2 and 3.1.

We shall prove that the minor, nonstandard terms in principal components do not affect the applicability of the unit root tests. Both the univariate and the multivariate unit root tests are considered. The univariate unit root test, which is the well known method in Perron (1989), is expected to be applied to each individual principal component in Group ⊥, one principal component in Group 2 when r₂ = 1, and each individual component in Group 1. The test will be described in Section 5.1. It will be followed in Sections 5.2 and 5.3 by our proofs of its applicability to the principal components in Group ⊥, and in Section 5.4 by its applicability to Group 2 and Group 1. The multivariate unit root test is a test for zero cointegration rank in Johansen et al. (2000). It is expected to be applied to the entire set of principal components in Group 2 when r₂ ≥ 2. The method will be described in Section 5.5, and our proof of its applicability to Group 2 will be given in Section 5.6. So far we have assumed that the correct division among Groups ⊥, 2, and 1 is known, and that the division is used to assign the correct unit root tests to the principal components. In Section 5.7, we shall analyse the effects that incorrect divisions among the three Groups would bring about.

The following two major points, (a) and (b), have emerged in the previous chapter.

We are now ready to consider how to determine n - r, r₂, and r₁. In Chapter 3 i = 1, ..., n is the running index to denote the descending order of eigenvalues of the data covariance matrix. i is also associated with the principal component that corresponds to the i-th eigenvalue. Given n - r, r₂, and r₁, the principal components with i = 1, ..., n - r, with i = n - r + 1, ..., n - r + r₂ ≡ n - r₁, and with i = n - r₁ + 1, ..., n each reveal asymptotically distinctive features. Thus i = 1, ..., n has been divided in three groups, Group ⊥ that consists of i = 1,..., n - r, Group 2 that consists of i = n - r + 1, ..., n - r + r₂ ≡ n - r₁, and Group 1 that consists of i = n - r₁+1,...,n.

In the previous chapters all the reasoning has proceeded on the asymptotic theories, while the methods derived there anticipate their applications to macroeco-nomic studies involving about 100 to 200 quarterly observations. The present chapter presents our simulation studies for the case where n = 2, p = 2, T = 100 or 200, and S H = [t, dt (b)] with b = 0.5. Section 8.1 will describe seven DGPs that incorporate a variety of structures of deterministic trends as well as different cointegration structures. Section 8.2 will give a detailed description of the trend test for I(0) for the special case of n, p, and S H, because the test was only outlined in Section 6.1 for the general case. Section 8.3 will begin with the reiteration of the asymptotic reasoning on the special case of n, p, and S H, which leads to a table of probabilities, Table 8.4, which we should expect if we had an infinite sample size. The table is arranged so that it can be easily compared with the tables of probabilities produced by simulations on finite sample sizes. In Section 8.3 we shall also give instruction on the last grouping method, which was left unclear in Chapter 7. Section 8.4 will present our simulations studies. The design of simulations incorporates data-dependence of the critical values of the trend test for I(1). Some discrepancies are discovered between the asymptotic theory and the simulation results with T = 100 and 200, and the discrepancies are analysed. Section 8.5 will present an example of applications to Japanese macroeconomic time series.