In macro-econometrics more attention needs to be paid to the relationships among deterministic trends of different variables, or co-trending, especially when economic growth is of concern. The number of relationships, i.e., the co-trending rank, plays an important role in evaluating the veracity of propositions, particularly relating to the Japanese economic growth in view of the structural changes involved within it. This book demonstrates how to determine the co-trending rank from a given set of time series data for different variables. At the same time, the method determines how many of the…mehr
In macro-econometrics more attention needs to be paid to the relationships among deterministic trends of different variables, or co-trending, especially when economic growth is of concern. The number of relationships, i.e., the co-trending rank, plays an important role in evaluating the veracity of propositions, particularly relating to the Japanese economic growth in view of the structural changes involved within it. This book demonstrates how to determine the co-trending rank from a given set of time series data for different variables. At the same time, the method determines how many of the co-trending relations also represent cointegrations. This enables us to perform statistical inference on the parameters of relations among the deterministic trends. Co-trending is an important contribution to the fields of econometric methods, macroeconomics, and time series analyses.
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.
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.
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