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The Cointegrated Matrix Autoregressive Model

Emanuele Lopetuso, Massimiliano Caporin

Abstract

Traditional econometric analyzes represent observations as vectors despite the inherent complexity of empirical data structures. When data are organized along dual classification dimensions, a matrix representation provides a more natural and interpretable framework. Building on recent advances in matrix autoregressive (MAR) modeling, this study introduces a novel error correction representation tailored for matrix-structured data. Through comparative analysis with existing methodologies, we demonstrate two critical advancements. First, the proposed model preserves the interpretative foundations of conventional cointegration analysis, with coefficients that explicitly capture dynamics rooted in adjustment toward steady-state positions. Second, in contrast to previous formulations, our error correction framework allows for an equivalent matrix autoregressive representation, preserving the fundamental structure of the data in both specifications. This ensures that the matrix representation reflects an intrinsic characteristic of the data.

The Cointegrated Matrix Autoregressive Model

Abstract

Traditional econometric analyzes represent observations as vectors despite the inherent complexity of empirical data structures. When data are organized along dual classification dimensions, a matrix representation provides a more natural and interpretable framework. Building on recent advances in matrix autoregressive (MAR) modeling, this study introduces a novel error correction representation tailored for matrix-structured data. Through comparative analysis with existing methodologies, we demonstrate two critical advancements. First, the proposed model preserves the interpretative foundations of conventional cointegration analysis, with coefficients that explicitly capture dynamics rooted in adjustment toward steady-state positions. Second, in contrast to previous formulations, our error correction framework allows for an equivalent matrix autoregressive representation, preserving the fundamental structure of the data in both specifications. This ensures that the matrix representation reflects an intrinsic characteristic of the data.

Paper Structure

This paper contains 43 sections, 10 theorems, 155 equations, 3 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Cointegrated-MAR model
  3. Cointegrated-MAR of order p
  4. Alternative Matrix Representation for Cointegrated Systems.
  5. Estimation
  6. Determination of the Cointegration Ranks
  7. Testing Hypotheses on Cointegration and Adjustment Matrices
  8. Simulations
  9. Finite-Sample Performance of the ECC-MAR Estimator
  10. Identification of the Cointegration Ranks
  11. Finite-Sample Performance of Hypothesis Testing
  12. Empirical Illustration
  13. Comparison with Standard VECM
  14. Conclusions
  15. Proofs
  16. ...and 28 more sections

Key Result

Theorem 1

Let $x_t:=\operatorname{vec}(X_t)$. If Assumption Ass_1 holds, then $x_t$ is a $I(1)$ cointegrated process, with $(m-r_1)(n-r_2)$ common stochastic trends and cointegration rank $\blacktriangleleft$$\blacktriangleleft$

Figures (3)

  • Figure 1: Boxplots of the distance between the estimated and true cointegration spaces over 500 Monte Carlo replications. For each configuration of matrix dimensions and cointegration ranks, results are reported for three sample sizes: $T=100$ (first area), $T=500$ (second area), and $T=1000$ (third area). Within each area, two boxplots are shown, corresponding to the ECC-MAR estimator (left) and the CVAR estimator (right).
  • Figure 2: Index: Production in industry for Belgium, Germany, and Austria. Monthly data from Jan-2000 to Dec-2019. Seasonally Adjusted.
  • Figure 3: Consumer Opinion Surveys: Consumer Prices: Future Tendency for Belgium, Germany, and Austria. Monthly data from Jan-2000 to Dec-2019. Seasonally Adjusted.

Theorems & Definitions (19)

  • Theorem 1
  • Lemma 1
  • Theorem 2
  • Theorem 3
  • Lemma 2
  • Lemma 3
  • Theorem 4
  • Lemma 4
  • Theorem 5
  • Proposition 1
  • ...and 9 more