American Institute of Mathematical Sciences

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July  2020, 7(3): 239-252. doi: 10.3934/jdg.2020017

Network ANOVA random effects models for node attributes

Gabriel Montes-Rojas 1,, and  Pedro Elosegui 2,

1. 

Instituto Interdisciplinario de Economía Política de Buenos Aires, (IIEP-BAIRES-UBA)-CONICET, Facultad de Ciencias Económicas, Universidad de Buenos Aires, Av. Córdoba 2122 2do piso, C1120AAQ, Ciudad Autónoma de Buenos Aires, Argentina
2. 

Banco Central de la República Argentina and Universidad Nacional de La Plata

* Corresponding author: Gabriel Montes-Rojas

We are grateful to the Guest Editors, Viktoriya Semeshenko, Gabriel Brida and Andrea Roventini, and to two anonymous referees for helpful comments and suggestions. Matías Pardini and Emilio Sáenz Guillén provided invaluable research assistance. We also thank the Gerencia Principal de Estadísticas Económicas of Banco Central de la República Argentina for the data used in the empirical application.

Received  February 2020 Revised  April 2020 Published  July 2020

This paper develops a subgraph network random effects error components structure for network data to perform analysis of variance. In particular, it proposes a model for evaluating the network interdependence of nodes attributes allowing for edge and triangle specific components. The latter serve as a basal model for modeling more general network effects. Consistent estimators of the variance components and Lagrange Multiplier specification tests for evaluating the appropriate model of random components in networks structures is proposed. Monte Carlo simulations show that the tests have good performance in finite samples. The proposed tests is applied to the unsecured (Call) interbank market network in Argentina.

Keywords: Networks, Nodes, Links, Triangles, Clusters.
Mathematics Subject Classification: Primary:05Cxx; Secondary:68R10.
Citation: Gabriel Montes-Rojas, Pedro Elosegui. Network ANOVA random effects models for node attributes. Journal of Dynamics and Games, 2020, 7 (3) : 239-252. doi: 10.3934/jdg.2020017
References:
[1]

D. Acemoglu, A. Ozdaglar and A. Tahbaz-Salehi, Systemic risk and stability in financial networks, American Economic Review, 105 (2015), 564-608.  doi: 10.3386/w18727.

[2]

P. Angelini, A. Nobili and C. Picillo, The interbank market after August 2007: What has changed and why?, Journal of Money, Credit and Banking, 43 (2011), 923-958. 

[3]

L. Anselin, A. Bera, R. Florax and M. Yoon, Simple diagnostic tests for spatial dependence, Regional Science and Urban Economics, 26 (1996), 77-104.  doi: 10.1016/0166-0462(95)02111-6.

[4]

L. Bargigli, G. di Iasio, L. Infante, F. Lillo and F. Pierobon, The multiplex structure of interbank networks, Quantitative Finance, 15 (2015), 673-691.  doi: 10.1080/14697688.2014.968356.

[5]

S. Battiston, M. Puliga, R. Kaushik, P. Tasca and G. Caldarelli, Debtrank: Too central to fail? Financial networks, the fed and systemic risk, Scientific Reports, 2. doi: 10.1038/srep00541.

[6]

M. L. Bech, J. T. E. Chapman and R. J. Garratt, Which bank is the central bank?, Journal of Monetary Economics, 57. doi: 10.1016/j.jmoneco.2010.01.002.

[7]

A. Bera and Y. Bilias, Rao's score, Neyman's $c(\alpha)$ and Silvey's lm tests: An essay on historical developments and some new results, Journal of Statistical Planning and Inference, 97 (2001), 9-44.  doi: 10.1016/S0378-3758(00)00343-8.

[8]

A. Bera, G. Montes-Rojas and W. Sosa-Escudero, General specification testing with locally misspecified models, Econometric Theory, 26 (2010), 1838-1845.  doi: 10.1017/S0266466609990818.

[9]

A. Bera, G. Montes-Rojas and W. Sosa-Escudero, A new robust and most powerful test in the presence of local misspecification, Communications in Statistics - Theory and Methods, 46 (2017), 8187-8198.  doi: 10.1080/03610926.2016.1177077.

[10]

A. Bera and M. Yoon, Specification testing with locally misspecified alternatives, Econometric Theory, 9 (1993), 649-658.  doi: 10.1017/S0266466600008021.

[11]

G. G. Booth, U. G. Gurun and H. Zhang, Financial networks and trading in bond markets, Journal of Financial Markets, 18 (2014), 126-157.  doi: 10.1016/j.finmar.2013.08.001.

[12] A. G. Chandrasekhar, Econometrics of network formation, in The Oxford Handbook of the Economics of Networks, Oxford University Press, Oxford, 2016. 
[13]

A. de Paula, Econometrics of network models, CEMMAP Working Paper CWP52/15, (2017).

[14]

S. de Raco and V. Semeshenko, Labor mobility and industrial space in Argentina, Journal of Dynamics & Games, 6 (2019), 107-118.  doi: 10.3934/jdg.2019008.

[15]

B. S. Graham, Network data, NBER Working Paper 26577, (2019).

[16]

H. H. Kelejian and I. R. Prucha, A generalized moments estimator for the autoregressive parameter in a spatial model, International Economic Review, 40 (1999), 509-533.  doi: 10.1111/1468-2354.00027.

[17]

H. H. Kelejian and I. R. Prucha, Specification and estimation of spatial autoregressive models with autoregressive and heteroskedastic disturbances, Journal of Econometrics, 157 (2010), 53-67.  doi: 10.1016/j.jeconom.2009.10.025.

[18]

Y. Kumagai, Social networks and global transactions, Journal of Dynamics & Games, 6 (2019), 211-219.  doi: 10.3934/jdg.

[19]

S. Langfield, Z. Liu and T. Ota, Mapping in the UK interbank system, Journal of Banking & Finance, 45 (2014), 288-303. 

[20]

J.-L. Molina-Borboa, S. Martinez-Jaramillo, F. Lopez-Gallo and M. van der Leij, A multiplex network analysis of the mexican banking system: Link persistence, overlap and waiting times, Journal of Network Theory in Finance, 1 (2015), 99-138.  doi: 10.21314/JNTF.2015.006.

[21]

S. Poledna, J. L. Molina-Borboa, S. Martínez-Jaramillo, M. van der Leij and S. Thurner, The multi-layer network nature of systemic risk and its implications for the costs of financial crises, Journal of Financial Stability, 20 (2015), 70-81.  doi: 10.1016/j.jfs.2015.08.001.

[22]

A. Temizsoy, G. Iori and G. Montes-Rojas, The role of bank relationship in the interbank market, Journal of Economic Dynamics & Control, 59 (2015), 118-141.  doi: 10.1016/j.jedc.2015.07.008.

[23]

A. Temizsoy, G. Iori and G. Montes-Rojas, Network centrality and funding rates in the e-mid interbank market, Journal of Financial Stability, 33 (2017), 346-365.  doi: 10.1016/j.jfs.2016.11.003.

[24]

C. Upper, Simulation methods to assess the danger of contagion in interbank markets, Journal of Financial Stability, 7 (2011), 111-125.  doi: 10.1016/j.jfs.2010.12.001.

show all references

References:
[1]

D. Acemoglu, A. Ozdaglar and A. Tahbaz-Salehi, Systemic risk and stability in financial networks, American Economic Review, 105 (2015), 564-608.  doi: 10.3386/w18727.

[2]

P. Angelini, A. Nobili and C. Picillo, The interbank market after August 2007: What has changed and why?, Journal of Money, Credit and Banking, 43 (2011), 923-958. 

[3]

L. Anselin, A. Bera, R. Florax and M. Yoon, Simple diagnostic tests for spatial dependence, Regional Science and Urban Economics, 26 (1996), 77-104.  doi: 10.1016/0166-0462(95)02111-6.

[4]

L. Bargigli, G. di Iasio, L. Infante, F. Lillo and F. Pierobon, The multiplex structure of interbank networks, Quantitative Finance, 15 (2015), 673-691.  doi: 10.1080/14697688.2014.968356.

[5]

S. Battiston, M. Puliga, R. Kaushik, P. Tasca and G. Caldarelli, Debtrank: Too central to fail? Financial networks, the fed and systemic risk, Scientific Reports, 2. doi: 10.1038/srep00541.

[6]

M. L. Bech, J. T. E. Chapman and R. J. Garratt, Which bank is the central bank?, Journal of Monetary Economics, 57. doi: 10.1016/j.jmoneco.2010.01.002.

[7]

A. Bera and Y. Bilias, Rao's score, Neyman's $c(\alpha)$ and Silvey's lm tests: An essay on historical developments and some new results, Journal of Statistical Planning and Inference, 97 (2001), 9-44.  doi: 10.1016/S0378-3758(00)00343-8.

[8]

A. Bera, G. Montes-Rojas and W. Sosa-Escudero, General specification testing with locally misspecified models, Econometric Theory, 26 (2010), 1838-1845.  doi: 10.1017/S0266466609990818.

[9]

A. Bera, G. Montes-Rojas and W. Sosa-Escudero, A new robust and most powerful test in the presence of local misspecification, Communications in Statistics - Theory and Methods, 46 (2017), 8187-8198.  doi: 10.1080/03610926.2016.1177077.

[10]

A. Bera and M. Yoon, Specification testing with locally misspecified alternatives, Econometric Theory, 9 (1993), 649-658.  doi: 10.1017/S0266466600008021.

[11]

G. G. Booth, U. G. Gurun and H. Zhang, Financial networks and trading in bond markets, Journal of Financial Markets, 18 (2014), 126-157.  doi: 10.1016/j.finmar.2013.08.001.

[12] A. G. Chandrasekhar, Econometrics of network formation, in The Oxford Handbook of the Economics of Networks, Oxford University Press, Oxford, 2016. 
[13]

A. de Paula, Econometrics of network models, CEMMAP Working Paper CWP52/15, (2017).

[14]

S. de Raco and V. Semeshenko, Labor mobility and industrial space in Argentina, Journal of Dynamics & Games, 6 (2019), 107-118.  doi: 10.3934/jdg.2019008.

[15]

B. S. Graham, Network data, NBER Working Paper 26577, (2019).

[16]

H. H. Kelejian and I. R. Prucha, A generalized moments estimator for the autoregressive parameter in a spatial model, International Economic Review, 40 (1999), 509-533.  doi: 10.1111/1468-2354.00027.

[17]

H. H. Kelejian and I. R. Prucha, Specification and estimation of spatial autoregressive models with autoregressive and heteroskedastic disturbances, Journal of Econometrics, 157 (2010), 53-67.  doi: 10.1016/j.jeconom.2009.10.025.

[18]

Y. Kumagai, Social networks and global transactions, Journal of Dynamics & Games, 6 (2019), 211-219.  doi: 10.3934/jdg.

[19]

S. Langfield, Z. Liu and T. Ota, Mapping in the UK interbank system, Journal of Banking & Finance, 45 (2014), 288-303. 

[20]

J.-L. Molina-Borboa, S. Martinez-Jaramillo, F. Lopez-Gallo and M. van der Leij, A multiplex network analysis of the mexican banking system: Link persistence, overlap and waiting times, Journal of Network Theory in Finance, 1 (2015), 99-138.  doi: 10.21314/JNTF.2015.006.

[21]

S. Poledna, J. L. Molina-Borboa, S. Martínez-Jaramillo, M. van der Leij and S. Thurner, The multi-layer network nature of systemic risk and its implications for the costs of financial crises, Journal of Financial Stability, 20 (2015), 70-81.  doi: 10.1016/j.jfs.2015.08.001.

[22]

A. Temizsoy, G. Iori and G. Montes-Rojas, The role of bank relationship in the interbank market, Journal of Economic Dynamics & Control, 59 (2015), 118-141.  doi: 10.1016/j.jedc.2015.07.008.

[23]

A. Temizsoy, G. Iori and G. Montes-Rojas, Network centrality and funding rates in the e-mid interbank market, Journal of Financial Stability, 33 (2017), 346-365.  doi: 10.1016/j.jfs.2016.11.003.

[24]

C. Upper, Simulation methods to assess the danger of contagion in interbank markets, Journal of Financial Stability, 7 (2011), 111-125.  doi: 10.1016/j.jfs.2010.12.001.

Figure 1.  LM tests for edge effects, $ \sigma_\mu^2=0 $, Erdös-Rényi random graph

Notes: Monte carlo experiments based on 2000 replications. Solid line: $ LM_{\mu} $. Dashed line: $ LM_{\mu\delta} $. Dotted line: $ LM_{\mu(\delta)} $

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Figure 2.  LM tests for edge effects, $ \sigma_\mu^2=0 $, Queen spatial structure

Notes: Monte carlo experiments based on 2000 replications. Solid line: $ LM_{\mu} $. Dashed line: $ LM_{\mu\delta} $. Dotted line: $ LM_{\mu(\delta)} $.

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Figure 3.  LM tests for triangle effects, $ \sigma_\delta^2=0 $, Erdös-Rényi random graph

Notes: Monte carlo experiments based on 2000 replications. Solid line: $ LM_{\delta} $. Dashed line: $ LM_{\mu\delta} $. Dotted line: $ LM_{\delta(\mu)} $. Dash-dot line: $ LM_{\delta-\mu} $

Figure Options

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Figure 4.  LM tests for triangle effcts, $ \sigma_\delta^2=0 $, Queen spatial structure

Notes: Monte carlo experiments based on 2000 replications. Solid line: $ LM_{\delta} $. Dashed line: $ LM_{\mu\delta} $. Dotted line: $ LM_{\delta(\mu)} $. Dash-dot line: $ LM_{\delta-\mu} $

Figure Options

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Figure 5.  Subgraph joint tests for edge and triangle effects and spatial Moran's Ⅰ LM test

Note: P-values in log-scale. Dashed line is the 10% critical value and dotted line to the 5% critical values. Horizontal axis corresponds to joint test for edge and triangle effects (LMµ, δ). Vertical axis corresponds to Moran's Ⅰ LM tests for spatial error.

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Figure 6.  Subgraph tests for edge and triangle effects

Note: P-values in log-scale. Dashed line is the 10% critical value and dotted line to the 5% critical values. Horizontal axis corresponds to tests for edge effects (LMµ). Vertical axis corresponds to tests for triangle effects (LMδ).

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Figure 7.  Robust subgraph tests for edge and triangle effects

Note: P-values in log-scale. Dashed line is the 10% critical value and dotted line to the 5% critical values. Horizontal axis corresponds to tests for edge effects robust to triangle effects (LMµ(δ)). Vertical axis corresponds to tests for triangle effects robust to edge effects (LMδ(µ)).

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Table 1.  Empirical size
N LMµ LMδ LMµ,δ LMµ(δ) LMδ(µ) LMδ−µ
Erdös-Rényi random graph
Size 1%
100 0.009 0.016 0.0145 0.009 0.0165 0.0115
225 0.012 0.0115 0.015 0.013 0.012 0.009
400 0.013 0.012 0.0085 0.0095 0.0075 0.007
Size 5%
100 0.043 0.05 0.0465 0.042 0.052 0.041
225 0.052 0.0485 0.0495 0.052 0.0495 0.041
400 0.047 0.0475 0.049 0.046 0.046 0.0435
Size 10%
100 0.082 0.0885 0.0855 0.089 0.092 0.0765
225 0.1045 0.092 0.102 0.098 0.0995 0.0875
400 0.089 0.087 0.093 0.0965 0.099 0.0915
Spatial queen structure
Size 1%
100 0.0115 0.0105 0.0105 0.01 0.011 0.0115
225 0.0075 0.0065 0.012 0.0145 0.0135 0.014
400 0.0085 0.0085 0.0095 0.012 0.011 0.011
Size 5%
100 0.0475 0.0515 0.047 0.048 0.044 0.046
225 0.045 0.039 0.0565 0.0595 0.052 0.0525
400 0.046 0.0465 0.049 0.0535 0.049 0.0505
Size 10%
100 0.0965 0.0975 0.0955 0.094 0.09 0.097
225 0.0965 0.09 0.1 0.1085 0.1115 0.1125
400 0.0935 0.0965 0.098 0.096 0.0995 0.1015
Notes: Monte carlo experiments based on 2000 replications.
Table Options

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N LMµ LMδ LMµ,δ LMµ(δ) LMδ(µ) LMδ−µ
Erdös-Rényi random graph
Size 1%
100 0.009 0.016 0.0145 0.009 0.0165 0.0115
225 0.012 0.0115 0.015 0.013 0.012 0.009
400 0.013 0.012 0.0085 0.0095 0.0075 0.007
Size 5%
100 0.043 0.05 0.0465 0.042 0.052 0.041
225 0.052 0.0485 0.0495 0.052 0.0495 0.041
400 0.047 0.0475 0.049 0.046 0.046 0.0435
Size 10%
100 0.082 0.0885 0.0855 0.089 0.092 0.0765
225 0.1045 0.092 0.102 0.098 0.0995 0.0875
400 0.089 0.087 0.093 0.0965 0.099 0.0915
Spatial queen structure
Size 1%
100 0.0115 0.0105 0.0105 0.01 0.011 0.0115
225 0.0075 0.0065 0.012 0.0145 0.0135 0.014
400 0.0085 0.0085 0.0095 0.012 0.011 0.011
Size 5%
100 0.0475 0.0515 0.047 0.048 0.044 0.046
225 0.045 0.039 0.0565 0.0595 0.052 0.0525
400 0.046 0.0465 0.049 0.0535 0.049 0.0505
Size 10%
100 0.0965 0.0975 0.0955 0.094 0.09 0.097
225 0.0965 0.09 0.1 0.1085 0.1115 0.1125
400 0.0935 0.0965 0.098 0.096 0.0995 0.1015
Notes: Monte carlo experiments based on 2000 replications.
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