VECM for multivariate/vector-valued time series

In the presence of cointegration, the stacked VECM is

\[ \boldsymbol{\Delta} \mathbf{y}_t = \mathbf{d} + \boldsymbol{\alpha} \boldsymbol{\beta}'\mathbf{y}_{t-1} + \sum_{j=1}^{p-1} \boldsymbol{\phi}_j \boldsymbol{\Delta} \mathbf{y}_{t-j} + \mathbf{e}_t \]

• \(\mathbf{d} \in \mathbb{R}^{N}\) is the deterministic term
• \(\boldsymbol{\beta} \in \mathbb{R}^{N \times r}\) is the cointegrating matrix for the economic indicators and countries
• \(\boldsymbol{\alpha} \in \mathbb{R}^{N \times r}\) is the corresponding adjustment coefficient matrix
• \(\boldsymbol{\phi}_j \in \mathbb{R}^{N \times N}\) is the autoregressive coefficient matrix

Matrix Error Correction Model

The Error Correction Model with matrix-valued observations is \[ \boldsymbol{\Delta} \mathbf{Y}_t = \mathbf{D} + \mathbf{U}_1 \mathbf{U}_3'\mathbf{Y}_{t-1} \mathbf{U}_4 \mathbf{U}_2' + \sum_{j=1}^{p-1} \boldsymbol{\phi}_{1,j} \boldsymbol{\Delta} \mathbf{y}_{t-j} \boldsymbol{\phi}_{2,j}' + \mathbf{E}_t \]

\[ \mathbf{E}_t \sim MVN(\mathbf{0}, \boldsymbol{\Sigma}_1, \boldsymbol{\Sigma}_2) \]

• \(\mathbf{U}_3 \in \mathbb{R}^{N_1 \times r_1}\) and \(\mathbf{U}_4 \in \mathbb{R}^{N_2 \times r_2}\) are the cointegrating matrices for the rows (economic indicators) and columns (countries)
• \(\mathbf{U}_1 \in \mathbb{R}^{N_1 \times r_1}\) and \(\mathbf{U}_2 \in \mathbb{R}^{N_2 \times r_2}\) are the adjustment coefficients
• \(\boldsymbol{\phi}_1 \in \mathbb{R}^{N_1 \times N_1}\) and \(\boldsymbol{\phi}_2 \in \mathbb{R}^{N_2 \times N_2}\) are the matrix AR coefficients.

Matrix Error Correction Model restrictions

Vectorizing the MECM¹ yields \[ \boldsymbol{\Delta} \mathbf{y}_t = \mathbf{d} + \underbrace{(\mathbf{U}_2 \otimes \mathbf{U}_1)}_{\boldsymbol{\alpha}} \underbrace{(\mathbf{U}_4 \otimes \mathbf{U}_3)'}_{\boldsymbol{\beta}'} \mathbf{y}_{t-1} + \sum_{j=1}^{p-1} \underbrace{(\boldsymbol{\phi}_{2,j} \otimes \boldsymbol{\phi}_{1,j})}_{\boldsymbol{\phi}_j} \boldsymbol{\Delta} \mathbf{y}_{t-j} + \mathbf{e}_t \] \[ \operatorname{vec}(\mathbf{E}_t) \sim N(\operatorname{vec}(\mathbf{0}), \boldsymbol{\Sigma}_2 \otimes \boldsymbol{\Sigma}_1) \]

• This form has been proposed by Li and Xiao (2024)
• We complement their study by allowing for different numbers of cointegrating relations and different lag orders

Log-Likelihood

The log-likelihood (up to a constant) of the MECM(p) for fixed rank \(r_1\) and \(r_2\) is \[ \mathcal{L} (\boldsymbol{\Theta}) = -\frac{T N_2}{2} \log | \boldsymbol{\Sigma}_1 | - \frac{T N_1}{2} \log | \boldsymbol{\Sigma}_2 | - \frac{1}{2} \sum_{t=1}^T \operatorname{tr} \left(\boldsymbol{\Sigma}_1^{-1} \mathbf{E}_t \boldsymbol{\Sigma}_2^{-1} \mathbf{E}_t'\right) \] where \(\mathbf{E}_t = \boldsymbol{\Delta} \mathbf{Y}_t - \mathbf{U}_1 \mathbf{U}_3' \mathbf{Y}_{t-1} \mathbf{U}_4 \mathbf{U}_2' - \sum_{j=1}^{p-1} \boldsymbol{\phi}_{1,j} \boldsymbol{\Delta} \mathbf{Y}_{t-j} \boldsymbol{\phi}_{2,j}' - \mathbf{D}\) and \(\boldsymbol{\Theta}\) collects all parameters. This can be solved using gradient descent.

Algorithm Details

Selection of Cointegration ranks

In practice, the ranks \(r_1\) and \(r_2\) are unknown and must be selected. We use Akaike Information Criteria (AIC) and Bayesian Information Criteria (BIC). \[ \text{AIC}(r_1, r_2, p) = -2 \mathcal{L}(\widehat{\boldsymbol{\Theta}}) + 2 \psi(r_1, r_2, p) \\ \text{BIC}(r_1, r_2, p) = -2 \mathcal{L}(\widehat{\boldsymbol{\Theta}}) + \ln(T) \psi(r_1, r_2, p) \]

where \[ \psi(r_1, r_2, p) = r_1 (2 N_1 - r_1) + r_2 (2 N_2 - r_2) + (p-1) (N_1^2 + N_2^2) \] is the number of parameters to estimate.

First Simulation Study

• Selecting ranks with AIC and BIC
• DGP is the MECM with 0 lags

Fully reduced

True rank of \(\mathbf{r} = (1,1)\)

Reduced Rank in First Dimension

True rank of \(\mathbf{r} = (1,4)\)

Reduced Rank in Second Dimension

True rank of \(\mathbf{r} = (3,1)\)

No rank reduction

True rank of \(\mathbf{r} = (3,4)\)

Second Simulation Study

• DGP is the MECM with 1 lag

Fully reduced

True rank of \(\mathbf{r} = (1,1)\)

Reduced Rank in First Dimension

True rank of \(\mathbf{r} = (1,4)\)

Reduced Rank in Second Dimension

True rank of \(\mathbf{r} = (3,1)\)

No rank reduction

True rank of \(\mathbf{r} = (3,4)\)

Macroeconomic Indicators for various countries

Estimated Coefficient Values

• Each \(\widehat{\mathbf{U}}_1\) and \(\widehat{\mathbf{U}}_2\) entry shows how strongly a row (indicator) or column (country) responds to deviations from the long-run equilibrium
• \(\widehat{\mathbf{U}}_3\) (indicators) and \(\widehat{\mathbf{U}}_4\) (countries) capture the linear combination of variables that form the long-run equilibrium

Cointegrating relation

• The cointegrated series \((\widehat{\mathbf{U}}_4 \otimes \widehat{\mathbf{U}}_3)' \operatorname{vec}(\mathbf{Y}_{t-1})\) shows two spikes at the beginning of the 90s but stays stable around the mean value of -0.99

Conclusion

• Investigated the matrix error-correction model for cointegration separately across row and column dimensions of matrix time series
• Proposed AIC/BIC for joint rank and lag selection and showed finite sample performance across simulation settings
• Applied the approach to international macroeconomic indicators, finding one stable long-run relation across both indicators and countries

Appendix

Pseudo-Code for the Gradient Descent Algorithm

• Initialize parameters using Nearest Kronecker Product (Van Loan 2000)
• Cycle through parameter blocks (block coordinate updates): \(\mathbf{D}, \mathbf{U}_1, \mathbf{U}_2, \mathbf{U}_3, \mathbf{U}_4, \boldsymbol{\Sigma}_1, \boldsymbol{\Sigma}_2, \boldsymbol{\phi}_1, \boldsymbol{\phi}_2\)
• Each block is updated by taking a step in the negative gradient direction with its own step size (calculated using the Lipschitz constant)
• Certain matrices are renormalized after each update
• Continue until loss improvement falls below the tolerance threshold

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