Co-trending: A Statistical System Analysis of Economic Trends

1 Introduction.- 1.1 Motivation and Results of the Research.- 1.2 Review of the Literature.- 1.3 Exploration of Different Approaches.- 1.4 Notations.- 2 Co-trending.- 2.1 Co-trending in the Non-parametric Approach.- 2.2 Co-trending in the Parametric Approach.- 2.3 Trend Functions.- 2.4 Co-trending and Co-breaking.- 2.5 A Survey on the Direct and the Indirect Modelling of Trend.- 3 Statistics from the Data Covariance Matrix.- 3.1 Three-way Decomposition of the n-Dimensional Space.- 3.2 Group ?.- 3.3 Group 2.- 3.4 Group 1.- 3.5 An Approximation of P(X).- 3.6 Plan for the Rest of the Book.- 4 Principal Components.- 4.1 Group ?, i = 1, ...., n - r.- 4.2 Group 2, i = n - r + 1, ...., n - r1, r2 > 0.- 4.3 Group 1, i = n - r1 + 1, ...., n, r1 > 0.- 5 Unit Root Tests.- 5.1 Univariate Unit Root Test.- 5.2 Group ?: The Case Where B???,i is not a Cointegration Vector.- 5.3 Group ?: The Case Where B???,i is a Cointegration Vector.- 5.4 Group 2: The Case Where r2 = 1, and Group 1.- 5.5 Multivariate Unit Root Test.- 5.6 Group 2: The Case Where r2 > 2.- 5.7 Mistaken Constituents of Groups.- 5.8 Supplement on the Martingale Covariance.- 6 Trend Tests.- 6.1 Discrimination Between Group 1 and Group ? with B???,i as a Cointegration Vector.- 6.2 Failure of the Standard Trend Tests in Discriminating Groups ? and 2.- 6.3 Test Statistic and Limit Distribution for the Discrimination Between Group 2 and Group ? with B???,i not in the Cointegrating Space.- 6.4 Reduced Rank Regression of X on S H.- 6.5 A Finite Sample Analysis of the Reduced Rank Regression.- 6.6 Asymptotic Analyses of the Reduced Rank Regression.- 6.7 Data-Dependent Critical Value.- 6.8 Mistaken Constituents of the Groups ? and 2.- 7 Sequential Decision Rule.- 7.1 Grouping Method.- 7.2Sequential Decision Rule.- 7.3 Tests of a Grouping Method.- 7.4 Justification of the Sequential Decision Rule.- 7.5 Supplement.- 8 Simulation Studies.- 8.1 DGP.- 8.2 Details of the Trend Test for I(0).- 8.3 A Table of Asymptotic Probabilities.- 8.4 Results of Simulation Studies.- 8.5 Empirical Example.- A Mathematical Proofs.- A.l Mathematical Proofs of Propositions and Lemmas in Chapter 2.- A.2 Mathematical Proofs of Propositions and Lemmas in Chapter 3.- A.3 Mathematical Proofs of Propositions and Lemmas in Chapter 4.- A.4 Mathematical Proofs of Propositions and Lemmas in Chapter 5.- A.5 Mathematical Proofs of Propositions and Lemmas in Chapter 6.- B.3 Estimation of the VAR, (B.3).- C Determination of the Co-breaking Rank.- C.1 The Number of Common Breaks.- C.2 Detrended Covariance Matrix.- C.3 UURT.- C.4 MURT.- C.5 TRT.- Notes.- References.