Statistics from the Data Covariance Matrix
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.
KeywordsUnit Root Test Trend Test Positive Eigenvalue Stochastic Trend Deterministic Trend
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