Causal Vector Autoregression Enhanced with Covariance and Order Selection
Abstract
:1. Introduction
2. Materials and Methods
 Reduced form VAR$\left(p\right)$ model: for given integer $p\ge 1$, it is$${\mathbf{X}}_{t}+{\mathit{M}}_{1}{\mathbf{X}}_{t1}+\cdots +{\mathit{M}}_{p}{\mathbf{X}}_{tp}={\mathbf{V}}_{t},\phantom{\rule{1.em}{0ex}}t=p+1,p+2,\dots ,$$Here, the ordering of the components of ${\mathbf{X}}_{t}$ does not matter: if it is changed (with some permutation of $\{1,\dots ,d\}$), then clearly the rows of the matrices ${\mathit{M}}_{j}$s and, furthermore, the rows and columns of $\mathsf{\Sigma}$ are permuted accordingly.
 Structural form SVAR$\left(p\right)$ model: for given integer $p\ge 1$, it is$$\mathit{A}{\mathbf{X}}_{t}+{\mathit{B}}_{1}{\mathbf{X}}_{t1}+\cdots +{\mathit{B}}_{p}{\mathbf{X}}_{tp}={\mathbf{U}}_{t},\phantom{\rule{1.em}{0ex}}t=p+1,p+2,\dots ,$$Here, the ordering of the components of ${\mathbf{X}}_{t}$ does matter: if it is changed (with some permutation of $\{1,\dots ,d\}$), then the matrices $\mathit{A}$, ${\mathit{B}}_{j}$ and $\Delta $ cannot be obtained in a simple way; they profoundly change under the given permutation.However, there is a onetoone correspondence between the reduced and structural model; since $\mathit{A}$ is invertible, from Equation (2), Equation (1) can be obtained (and vice versa):$${\mathbf{X}}_{t}+{\mathit{A}}^{1}{\mathit{B}}_{1}{\mathbf{X}}_{t1}+\cdots +{\mathit{A}}^{1}{\mathit{B}}_{p}{\mathbf{X}}_{tp}={\mathit{A}}^{1}{\mathbf{U}}_{t},\phantom{\rule{1.em}{0ex}}t=p+1,p+2,\dots ,$$
 Causal CVAR$\left(p\right)$ unrestricted model: it also obeys Equation (2), but here the ordering of the components follows a causal ordering, given e.g., by an expert’s knowledge. This is a recursive ordering along a “complete” DAG, where the permutation (labeling) of the graph nodes (assigned to the components of ${\mathbf{X}}_{t}$) is such that ${X}_{t,i}$ can be caused by ${X}_{t,j}$ whenever $i<j$, which means a $j\to i$ directed edge. Here, the causal effects are meant contemporaneously, and reflected by the upper triangular structure of the matrix $\mathit{A}$.It is important that, in any ordering of the jointly Gaussian variables, a Bayesian network (in other words, a Gaussian directed graphical model) can be constructed, in which every node (variable) is regressed linearly with the variables corresponding to higher label nodes. The partial regression coefficients behave like path coefficients, also used in SEM. If the DAG is complete, then there are no zero constraints imposed on the partial regression coefficients. Here, building the DAG just aims at finding a sensible ordering of the variables.
 Causal CVAR$\left(p\right)$ restricted model: here, an incomplete DAG is built, based on partial correlations.First, we build an undirected graph: do not connect i and j if the partial correlation coefficients of ${X}_{i}$ and ${X}_{j}$, eliminating the effect of the other variables is 0 (theoretically), or less than a threshold (practically). Such an undirected graphical model is called Markov random field (MRF). It is known (see Rao (1973) and Lauritzen (2004)) that partial correlations can be calculated from the concentration matrix (inverse of the covariance matrix). However, here the upper left block of the inverse of the large block matrix, containing the first p autocovariance matrices, is used. If this undirected graph is triangulated, then in a convenient (socalled perfect) ordering of the nodes, the zeros of the adjacency matrix form an RZP. We can find such a (not necessarily unique) ordering of the nodes with the maximal cardinality search (MCS) algorithm, together with cliques and separators of a socalled junction tree (JT); see Lauritzen (2004), Koller and Friedman (2009), and Bolla et al. (2019). In this ordering (labeling) of the nodes, a DAG can also be constructed, which is Markov equivalent to the undirected one (it has no socalled sink V configuration); for further details, see Section 3.2.Having an RZP in the CVAR restricted model, we use the incomplete DAG for estimation. With the covariance selection method of Dempster (1972), the starting concentration matrix is reestimated by imposing zero constraints for its entries in the RZP positions (symmetrically). By the theory (see, e.g., Bolla et al. (2019)), this will result in zero entries of $\mathit{A}$ in the no directed edge positions.
3. Results
3.1. The Unrestricted Causal VAR(p) Model
 Upper left block: ${\mathit{C}}^{1}\left(p\right0,\dots ,p1)$;
 Upper right block: ${\mathit{C}}^{1}\left(p\right0,\dots ,p1){\mathit{C}}^{T}(1,\dots ,p){\mathfrak{C}}_{p}^{1}$;
 Lower left block: ${\mathfrak{C}}_{p}^{1}\mathit{C}(1,\dots ,p){\mathit{C}}^{1}\left(p\right0,\dots ,p1)$;
 Lower right block: ${\mathfrak{C}}_{p}^{1}+{\mathfrak{C}}_{p}^{1}\mathit{C}(1,\dots ,p){\mathit{C}}^{1}\left(p\right0,\dots ,p1){\mathit{C}}^{T}(1,\dots ,p){\mathfrak{C}}_{p}^{1}$,
3.2. The Restricted Causal VAR(p) Model
 G is triangulated (with other words, chordal), i.e., every cycle in G of a length of at least four has a chord.
 G has a perfect numbering of its nodes such that, in this labeling, $\mathrm{ne}\left(i\right)\cap \{i+1,\dots ,d\}$ is a complete subgraph, where $\mathrm{ne}\left(i\right)$ is the set of neighbors of i, for $i=1,\dots ,d$. It is also called single node elimination ordering (see Wainwright (2015)), and obtainable with the maximal cardinality search (MCS) algorithm of Tarjan and Yannakakis (1984); see also Koller and Friedman (2009).
 G has the following running intersection property: we can number the cliques of it to form a socalled perfect sequence ${C}_{1},\dots ,{C}_{k}$ where each combination of the subgraphs induced by ${H}_{j1}={C}_{1}\cup \dots \cup {C}_{j1}$ and ${C}_{j}$ is a decomposition $(j=2,\dots ,k)$, i.e., the necessarily complete subgraph ${S}_{j}={H}_{j1}\cap {C}_{j}$ is a separator. More precisely, ${S}_{j}$ is a node cutset between the disjoint node subsets ${H}_{j1}\backslash {S}_{j}$ and ${R}_{j}={C}_{j}\backslash {S}_{j}={H}_{j}\backslash {H}_{j1}$. This sequence of cliques is also called a junction tree (JT).Here, any clique ${C}_{j}$ is the disjoint union of ${R}_{j}$ (called residual), the nodes of which are not contained in any ${C}_{i}$, $i<j$ and of ${S}_{j}$ (called separator) with the following property: there is an ${i}^{*}\in \{1,\dots ,j1\}$ such that$${S}_{j}={C}_{j}\cap \left({\cup}_{i=1}^{j1}{C}_{i}\right)={C}_{j}\cap {C}_{{i}^{*}}.$$This (not necessarily unique) ${C}_{{i}^{*}}$ is called parent clique of ${C}_{j}$. Here, ${S}_{1}=\varnothing $ and ${R}_{1}={C}_{1}$. Furthermore, if such an ordering is possible, a version may be found in which any prescribed set is the first one. Note that the junction tree is indeed a tree with nodes ${C}_{1},\dots ,{C}_{k}$ and one less edge that are the separators ${S}_{2},\dots ,{S}_{k}$.
 There is a labeling of the nodes such that the adjacency matrix contains a reducible zero pattern (RZP). It means that there is an index set $I\subset \left\{\right(i,j):\phantom{\rule{0.166667em}{0ex}}1\le i<j\le d\}$ which is reducible in the sense that, for each $(i,j)\in I$ and $h=1,\dots ,i1$, we have $(h,i)\in I$ or $(h,j)\in I$ or both.Indeed, this convenient labeling is a perfect numbering of the nodes.
 The following Markov chain property also holds: $f\left({\mathbf{x}}_{{R}_{j}}\phantom{\rule{0.166667em}{0ex}}\right\phantom{\rule{0.166667em}{0ex}}{\mathbf{x}}_{{C}_{1}\cup \dots \cup {C}_{j1}})=f\left({\mathbf{x}}_{{R}_{j}}\phantom{\rule{0.166667em}{0ex}}\right\phantom{\rule{0.166667em}{0ex}}{\mathbf{x}}_{{S}_{j}})$.Therefore, if we have a perfect sequence ${C}_{1},\dots ,{C}_{k}$ of the cliques with separators ${S}_{1}=\varnothing ,{S}_{2},\dots ,{S}_{k}$, then, for any state configuration $\mathbf{x}$, we have the following factorized form of the density:$$f\left(\mathbf{x}\right)=\frac{{\prod}_{j=1}^{k}f\left({\mathbf{x}}_{{C}_{j}}\right)}{{\prod}_{j=2}^{k}f\left({\mathbf{x}}_{{S}_{j}}\right)}=\prod _{i=1}^{k}f\left({\mathbf{x}}_{{R}_{j}}\right{\mathbf{x}}_{{S}_{j}}).\phantom{\rule{1.em}{0ex}}$$
4. Applications with Order Selection
4.1. Financial Data
4.2. IMR (Infant Mortality Rate) Longitudinal Data
 Case 1: $\left\{\mathrm{IMR},\mathrm{MMR},\mathrm{HepB},\mathrm{OPExp},\mathrm{HExp},\mathrm{GDP}\right\}$.
 Case 2: $\left\{\mathrm{IMR},\mathrm{MMR},\mathrm{HepB},\mathrm{GDP},\mathrm{OPExp},\mathrm{HExp}\right\}$.
5. Discussion
6. Conclusions and Further Perspectives
Supplementary Materials
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
VAR  Vector AutoRegression 
SVAR  Structural Vector AutoRegression 
CVAR  Causal Vector AutoRegression 
SEM  Structural Equation Modeling 
DAG  Directed Acyclic Graph 
JT  Junction Tree 
MCS  Maximal Cardinality Search 
IPS  Iterative Proportional Scaling 
RZP  Reducible Zero Pattern 
MRF  Markov Random Field 
AIC  Akaike Information Criterion 
AICC  Akaike Information Criterion Corrected 
BIC  Bayesian Information Criterion 
HQ  Hannan and Quinn’s criterion 
MLE  Maximum Likelihood Estimate 
PLS  Partial Least Squares regression 
RMSE  Root Mean Square Error 
IMR  Infant Mortality Rate 
LDL  variant of the Cholesky decomposition for a symmetric, positive semidefinite matrixas $\mathit{L}$ (lower triangular)$\times \mathit{D}$ (diagonal)$\times {\mathit{L}}^{T}$ 
Appendix A. Proofs of the Main Theorems
Appendix A.1. Proof of Theorem 1
Appendix A.2. Algorithm for the Block LDL Decomposition of Appendix A.1
 Outer cycle (columnwise). For $j=1,\dots ,d$: ${\delta}_{j}^{1}={k}_{jj}{\sum}_{h=1}^{j1}{\ell}_{jh}{\delta}_{h}^{1}{\ell}_{jh}$ (with the reservation that ${\delta}_{1}^{1}={k}_{11}$);
 Inner cycle (rowwise). For $i=j+1,\dots ,d$:$${\ell}_{ij}=\left({k}_{ij}\sum _{h=1}^{j1}{\ell}_{ih}{\delta}_{h}^{1}{\ell}_{jh}\right){\delta}_{j}$$$${\ell}_{d+1,j}=\left({\mathit{k}}_{d+1,j}\sum _{h=1}^{j1}{\ell}_{d+1,h}{\delta}_{h}^{1}{\ell}_{jh}\right){\delta}_{j}$$(with the reservation that, in the $j=1$ case, the summand is zero), where ${\mathit{k}}_{d+1,j}$ for $j=1,\dots ,d$ is $d\times 1$ vector in the bottom left block of $\mathit{K}$.
Appendix A.3. Proof of Theorem 2
Appendix A.4. Algorithm for the Block LDL Decomposition of Appendix A.3
 Outer cycle (columnwise). For $j=1,\dots ,d$: ${\delta}_{j}^{1}={k}_{jj}{\sum}_{h=1}^{j1}{\ell}_{jh}{\delta}_{h}^{1}{\ell}_{jh}$ (with the reservation that ${\delta}_{1}^{1}={k}_{11}$);
 Inner cycle (rowwise). For $i=j+1,\dots ,d$:$${\ell}_{ij}=\left({k}_{ij}\sum _{h=1}^{j1}{\ell}_{ih}{\delta}_{h}^{1}{\ell}_{jh}\right){\delta}_{j}$$$${\ell}_{d+1,j}=\left({\mathit{k}}_{d+1,j}\sum _{h=1}^{j1}{\ell}_{d+1,h}{\delta}_{h}^{1}{\ell}_{jh}\right){\delta}_{j}$$(with the reservation that, in the $j=1$ case, the summand is zero), where ${\mathit{k}}_{d+1,j}$ for $j=1,\dots ,d$ are $pd\times 1$ vectors in the bottom left block of $\mathit{K}$.
Appendix B. Pseudocodes
Algorithm A1: Constructing an undirected graph and a causal ordering of variables 
Input :$\mathit{D}$, $n\times d$ data matrix p, order of the CVAR model ${r}^{*}$, threshold for the partial correlation statistical test Output: undirected graph G and its perfect ordering

Algorithm A2: Constructing an unrestricted CVAR model 
Input :$\mathit{D}$: $n\times d$ data matrix or the existing $\mathit{K}$ from Algorithm A1 p: order of the CVAR model $({i}_{1},\dots ,{i}_{d})$, causal ordering of the d observed variables. Output: parameter matrices $\mathit{A},{\mathit{B}}_{1},\dots ,{\mathit{B}}_{p}$

Algorithm A3: Constructing a restricted CVAR model 
Input :$\mathit{D}$: $n\times d$ data matrix p: order of the CVAR model G, (undirected) chordal graph for observed variables $({i}_{1},\dots ,{i}_{d})$, causal ordering of observed variables. Output: parameter matrices $\widehat{\mathit{A}},{\widehat{\mathit{B}}}_{1},\dots ,{\widehat{\mathit{B}}}_{p}$

Note
1  Please see the main text for suggestions on graphs that are not triangulated when moralization and running the IPS algorithm is needed. The default ${r}^{*}$ threshold is usually set according to a significance level (e.g., $\alpha =0.05$) for the partial correlation test. This can be changed based on the sample size and the effect size. 
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NIK  EU  ISE  EM  BVSP  DAX  FTSE  SP  

NIK  0.016 *  0.035 *  0.522  −0.260  −0.019 *  −0.076  0.024 *  
EU  0.016 *  0.217  0.034 *  0.067  0.687  0.747  0.018 *  
ISE  0.035 *  0.217  0.358  −0.157  −0.077  −0.059  0.034 *  
EM  0.522  0.034 *  0.358  0.546  0.048  0.086  −0.184  
BVSP  −0.260  0.067  −0.157  0.546  −0.093  −0.045  0.533  
DAX  −0.019 *  0.687  −0.077  0.048  −0.093  −0.203  0.191  
FTSE  −0.076  0.747  −0.059  0.086  −0.045  −0.203  0.057  
SP  0.024 *  0.018 *  0.034 *  −0.184  0.533  0.191  0.057 
NIK  EU  ISE  EM  BVSP  DAX  FTSE  SP  

NIK  1  0.0264  0.0042  −0.8902  0.2030  0.0170  0.0781  −0.0336 
EU  0  1  −0.0418  −0.0146  −0.0239  −0.3746  −0.5255  −0.0033 
ISE  0  0  1  −0.9518  0.1613  −0.1658  −0.3129  −0.1413 
EM  0  0  0  1  −0.3507  −0.1182  −0.2464  0.1077 
BVSP  0  0  0  0  1  −0.0129  −0.2782  −0.6375 
DAX  0  0  0  0  0  1  −0.8102  −0.2336 
FTSE  0  0  0  0  0  0  1  −0.6100 
SP  0  0  0  0  0  0  0  1 
NIK${}_{1}$  EU${}_{1}$  ISE${}_{1}$  EM${}_{1}$  BVSP${}_{1}$  DAX${}_{1}$  FTSE${}_{1}$  SP${}_{1}$  

NIK  0.1845  −0.1685  −0.0874  0.0852  0.0635  0.0205  −0.1236  −0.2798 
EU  −0.0131  0.1219  −0.0044  0.0291  −0.0124  −0.0393  −0.0979  0.0011 
ISE  0.0677  0.2811  −0.0657  0.2473  −0.2940  −0.0543  0.0098  −0.1442 
EM  −0.0016  −0.0569  −0.0159  0.1076  −0.0917  −0.0945  0.0875  −0.1071 
BVSP  −0.0140  0.0704  0.0142  −0.1046  0.1397  −0.1497  0.1188  −0.0812 
DAX  −0.0034  0.2021  −0.0342  −0.0044  −0.0352  −0.0476  −0.0670  −0.0673 
FTSE  0.0293  −0.0168  −0.0109  0.0420  −0.1129  0.2141  0.0805  −0.2641 
SP  0.0417  0.2603  −0.0261  0.0112  −0.0026  −0.0709  −0.2850  0.1240 
NIK  EU  ISE  EM  BVSP  DAX  FTSE  SP  

NIK  1  −0.0114  0.0103  −0.8822  0.1995  0.0233  0.0856  −0.0214 
EU  0  1  −0.0426  −0.0110  −0.0240  −0.3745  −0.5137  −0.0128 
ISE  0  0  1  −0.9788  0.1701  −0.1669  −0.3139  −0.1361 
EM  0  0  0  1  −0.3450  −0.1154  −0.2375  0.0922 
BVSP  0  0  0  0  1  −0.0047  −0.2655  −0.6601 
DAX  0  0  0  0  0  1  −0.8120  −0.2339 
FTSE  0  0  0  0  0  0  1  −0.6320 
SP  0  0  0  0  0  0  0  1 
NIK${}_{1}$  EU${}_{1}$  ISE${}_{1}$  EM${}_{1}$  BVSP${}_{1}$  DAX${}_{1}$  FTSE${}_{1}$  SP${}_{1}$  

NIK  0.2063  −0.1826  −0.1106  0.1063  0.0731  0.0187  −0.1502  −0.2580 
EU  −0.0037  0.1364  −0.0010  0.0232  −0.0150  −0.0371  −0.0996  −0.0107 
ISE  0.0409  0.2476  −0.0771  0.2274  −0.2772  −0.0447  0.0331  −0.1284 
EM  0.0489  −0.0200  −0.0030  0.1360  −0.1150  −0.0996  0.0468  −0.1162 
BVSP  −0.0066  0.0931  0.0261  −0.1091  0.1312  −0.1573  0.1161  −0.0935 
DAX  −0.0123  0.2146  −0.0319  0.0073  −0.0406  −0.0536  −0.0727  −0.0694 
FTSE  0.0852  0.0019  0.0275  0.0145  −0.1117  0.2377  0.1035  −0.3427 
SP  0.0530  0.2759  −0.0565  −0.0033  0.0024  −0.0945  −0.3106  0.1789 
NIK${}_{2}$  EU${}_{2}$  ISE${}_{2}$  EM${}_{2}$  BVSP${}_{2}$  DAX${}_{2}$  FTSE${}_{2}$  SP${}_{2}$  

NIK  −0.0402  −0.1695  −0.0410  0.0156  0.0998  −0.0406  0.1367  −0.0091 
EU  0.0017  0.0771  −0.0065  0.0054  0.0037  0.0192  −0.0762  −0.0394 
ISE  −0.0142  −0.1725  −0.0276  −0.0088  0.0389  0.1167  0.0826  0.0357 
EM  −0.0054  0.0650  −0.0322  0.1155  −0.0695  −0.0959  −0.0162  −0.0270 
BVSP  −0.0423  0.0332  −0.0449  0.2878  −0.0717  −0.0221  −0.0381  −0.0120 
DAX  −0.0372  0.0177  0.0130  0.0658  −0.0360  −0.0108  −0.0202  0.0059 
FTSE  0.0491  0.3107  −0.0820  0.0693  0.0299  0.0153  −0.0840  −0.3038 
SP  0.0447  −0.0628  0.0804  −0.1824  0.0785  0.0133  −0.1775  0.1284 
NIK  EU  ISE  EM  BVSP  DAX  FTSE  SP  

NIK  1  0  0  −0.8193  0.2080  0  0  0 
EU  0  1  −0.0421  0  −0.0269  −0.3782  −0.5297  0 
ISE  0  0  1  −0.9386  0.1653  −0.1675  −0.3161  −0.1477 
EM  0  0  0  1  −0.3419  −0.1184  −0.2464  0.0997 
BVSP  0  0  0  0  1  −0.0130  −0.2729  −0.6423 
DAX  0  0  0  0  0  1  −0.8102  −0.2336 
FTSE  0  0  0  0  0  0  1  −0.6104 
SP  0  0  0  0  0  0  0  1 
NIK${}_{1}$  EU${}_{1}$  ISE${}_{1}$  EM${}_{1}$  BVSP${}_{1}$  DAX${}_{1}$  FTSE${}_{1}$  SP${}_{1}$  

NIK  0.1811  −0.1797  −0.0856  0.0842  0.0739  −0.0058  −0.1146  −0.2662 
EU  −0.0131  0.1213  −0.0046  0.0304  −0.0130  −0.0415  −0.0969  0.0002 
ISE  0.0676  0.2814  −0.0658  0.2483  −0.2941  −0.0567  0.0120  −0.1472 
EM  −0.0016  −0.0567  −0.0158  0.1067  −0.0908  −0.0951  0.0890  −0.1085 
BVSP  −0.0139  0.0704  0.0142  −0.1041  0.1391  −0.1488  0.1195  −0.0828 
DAX  −0.0034  0.2019  −0.0342  −0.0046  −0.0353  −0.0474  −0.0669  −0.0672 
FTSE  0.0292  −0.0171  −0.0109  0.0419  −0.1130  0.2142  0.0807  −0.2642 
SP  0.0417  0.2608  −0.0261  0.0115  −0.0026  −0.0713  −0.2853  0.1239 
NIK  EU  ISE  EM  BVSP  DAX  FTSE  SP  

NIK  1  0  0  −0.8191  0.2076  0  0  0 
EU  0  1  −0.0423  0  −0.0293  −0.3811  −0.5192  0 
ISE  0  0  1  −0.9662  0.1790  −0.1713  −0.3112  −0.1470 
EM  0  0  0  1  −0.3361  −0.1153  −0.2372  0.0835 
BVSP  0  0  0  0  1  −0.0069  −0.2544  −0.6664 
DAX  0  0  0  0  0  1  −0.8128  −0.2336 
FTSE  0  0  0  0  0  0  1  −0.6319 
SP  0  0  0  0  0  0  0  1 
NIK${}_{1}$  EU${}_{1}$  ISE${}_{1}$  EM${}_{1}$  BVSP${}_{1}$  DAX${}_{1}$  FTSE${}_{1}$  SP${}_{1}$  

NIK  0.2009  −0.1869  −0.1098  0.1089  0.0824  −0.0079  −0.1493  −0.2428 
EU  −0.0038  0.1387  −0.0013  0.0260  −0.0153  −0.0410  −0.1027  −0.0086 
ISE  0.0353  0.2865  −0.0750  0.2479  −0.2741  −0.0639  0.0101  −0.1418 
EM  0.0494  −0.0218  −0.0027  0.1338  −0.1144  −0.0990  0.0500  −0.1177 
BVSP  −0.0107  0.1202  0.0276  −0.0947  0.1327  −0.1674  0.0987  −0.1030 
DAX  −0.0110  0.2072  −0.0322  0.0034  −0.0412  −0.0503  −0.0677  −0.0675 
FTSE  0.0824  0.0176  0.0281  0.0224  −0.1104  0.2309  0.0928  −0.3463 
SP  0.0506  0.2898  −0.0560  0.0040  0.0037  −0.1010  −0.3199  0.1760 
NIK${}_{2}$  EU${}_{2}$  ISE${}_{2}$  EM${}_{2}$  BVSP${}_{2}$  DAX${}_{2}$  FTSE${}_{2}$  SP${}_{2}$  

NIK  −0.0455  −0.1847  −0.0391  0.0264  0.0906  −0.0486  0.1427  0.0089 
EU  0.0017  0.0755  −0.0058  0.0047  0.0033  0.0179  −0.0765  −0.0370 
ISE  −0.0161  −0.1634  −0.0290  −0.0021  0.0352  0.1113  0.0821  0.0313 
EM  −0.0056  0.0659  −0.0330  0.1189  −0.0701  −0.0959  −0.0167  −0.0283 
BVSP  −0.0430  0.0415  −0.0456  0.2906  −0.0729  −0.0258  −0.0389  −0.0168 
DAX  −0.0369  0.0163  0.0130  0.0656  −0.0356  −0.0100  −0.0203  0.0064 
FTSE  0.0485  0.3142  −0.0820  0.0716  0.0290  0.0128  −0.0845  −0.3054 
SP  0.0442  −0.0606  0.0805  −0.1825  0.0778  0.0117  −0.1773  0.1281 
p  AIC  AICC  BIC  HQ 

1  −76.81  −33,222.68  −76.07  −76.52 
2  −76.85  −33,173.98  −75.60  −76.36 
3  −76.84  −33,095.75  −75.08  −76.15 
4  −76.83  −33,011.98  −74.55  −75.94 
5  −76.77  −32,893.23  −73.97  −75.67 
6  −76.69  −32,766.33  −73.37  −75.39 
7  −76.58  −32,612.38  −72.74  −75.08 
8  −76.48  −32,457.38  −72.11  −74.77 
9  −76.41  −32,316.33  −71.52  −74.49 
p  AIC  AICC  BIC  HQ 

1  −76.87  −33,239.11  −76.19  −76.60 
2  −76.91  −33,190.27  −75.71  −76.44 
3  −76.93  −33,129.67  −75.22  −76.26 
4  −77.00  −33084.90  −74.77  −76.13 
5  −76.94  −32,969.37  −74.19  −75.86 
6  −76.92  −32,869.31  −73.65  −75.64 
7  −76.81  −32,718.36  −73.02  −75.33 
8  −76.80  −32,612.63  −72.49  −75.11 
9  −76.78  −32,495.56  −71.94  −74.88 
IMR  MMR  HepB  OPExp  HExp  GDP  

IMR  1.0  −1.1259  −0.0161  0.0003  0.0176  −0.1348 
MMR  0.0  1.0000  0.3594  0.0492  −0.0684  0.7135 
HepB  0.0  0.0000  1.0000  −0.1626  0.2510  −0.8196 
OPExp  0.0  0.0000  0.0000  1.0000  −0.6876  −0.4229 
HExp  0.0  0.0000  0.0000  0.0000  1.0000  0.6749 
GDP  0.0  0.0000  0.0000  0.0000  0.0000  1.0000 
IMR1  MMR1  HepB1  OPExp1  HExp1  GDP1  

IMR  0.2986  −0.3589  −0.0076  0.0042  −0.0115  −0.0639 
MMR  −0.0149  −0.7469  −0.2358  0.0540  0.0193  −0.5577 
HepB  13.0541  −15.7658  −1.1915  −0.3506  0.2170  −1.7902 
OPExp  7.1616  −8.0906  −0.2994  −0.1038  −0.0215  −0.7720 
HExp  1.6861  −2.9922  −0.4650  −0.1566  −0.0681  −1.0913 
GDP  −11.0674  13.2182  0.3254  0.3204  −0.3099  1.2129 
IMR  MMR  HepB  GDP  OPExp  HExp  

IMR  1.0  0.0736  0.0052  0.0111  0.0  0.0040 
MMR  0.0  1.0000  0.0237  0.1180  0.0  −0.0116 
HepB  0.0  0.0000  1.0000  −0.3418  0.0  0.0236 
GDP  0.0  0.0000  0.0000  1.0000  0.0  −0.0035 
OPExp  0.0  0.0000  0.0000  0.0000  1.0  −0.7854 
HExp  0.0  0.0000  0.0000  0.0000  0.0  1.0000 
IMR1  MMR1  HepB1  GDP1  OPExp1  HExp1  

IMR  −0.9198  −0.0592  −0.0021  0.0053  −0.0013  −0.0049 
MMR  −1.0175  0.2511  −0.0002  0.0607  −0.0047  0.0039 
HepB  3.6850  4.4606  −0.9058  −0.4230  −0.0811  −0.0950 
GDP  0.5849  −0.4453  0.0640  −0.8573  0.0896  0.1086 
OPExp  3.6432  −3.7486  −0.1413  −0.4380  0.0273  −0.0926 
HExp  1.9561  −3.4687  −0.5221  −0.6302  −0.2298  −0.1171 
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Bolla, M.; Ye, D.; Wang, H.; Ma, R.; Frappier, V.; Thompson, W.; Donner, C.; Baranyi, M.; Abdelkhalek, F. Causal Vector Autoregression Enhanced with Covariance and Order Selection. Econometrics 2023, 11, 7. https://doi.org/10.3390/econometrics11010007
Bolla M, Ye D, Wang H, Ma R, Frappier V, Thompson W, Donner C, Baranyi M, Abdelkhalek F. Causal Vector Autoregression Enhanced with Covariance and Order Selection. Econometrics. 2023; 11(1):7. https://doi.org/10.3390/econometrics11010007
Chicago/Turabian StyleBolla, Marianna, Dongze Ye, Haoyu Wang, Renyuan Ma, Valentin Frappier, William Thompson, Catherine Donner, Máté Baranyi, and Fatma Abdelkhalek. 2023. "Causal Vector Autoregression Enhanced with Covariance and Order Selection" Econometrics 11, no. 1: 7. https://doi.org/10.3390/econometrics11010007