Journal of Financial Market Infrastructures

SPEI’s diary: econometric analysis of a dynamic network

Miguel Angel Gavilan-Rubio and Biliana Alexandrova-Kabadjova

  • This paper identifies the forces driving SPEI, the Mexican payment system, between 2005 and 2015 using a TVC-FAVAR model and network metrics within FMI daily data.
  • System trend and stability of the system are the two main forces driving the system.
  • There is feedback between the forces driving SPEI and the network metrics.
  • Banks adjust their strategies of liquidity management in response to changes in the network structure.

This paper identifies the determinants behind the dynamics of the real-time settlement payment system in Mexico, SPEI, during the period January 2005–December 2015. To that end, we use a two-step econometric strategy. First, we estimate a time-varying coefficients factor-augmented vector autoregressive (TVC-FAVAR) model to identify the underlying dynamic factors. Second, we regress the fitted dynamic factors on SPEI’s network metrics, such as centrality, distance and bilateral relationships. These metrics capture different market aspects and help us to understand banks’ strategy patterns. We find that there are five factors driving the dynamics of the Mexican payment system, and linear combinations of the network metrics explain changes in these dynamics. The two main forces are (i) the trend, affected by most of the metrics; and (ii) the stability of the system, affected by centrality and the bilateral relationships. We also extend our analysis to understand how these forces affect funding patterns and liquidity management among banks. We find that changes in participants’ network position force banks to adjust their strategies of liquidity management.

1 Introduction

Financial market infrastructures (FMIs) constantly evolve into more complex dynamic systems. Globalization, financial innovation and new means-of-payment adoption push the day-to-day performance of payment systems to their limits, with the result that all are continually seeking new ways to assist consumers and businesses in how to make financial transactions. For instance, our once cash-based society is crossing a digital frontier of mobile platforms (eg, Apple Pay, Android Pay), and new players are similarly performing traditional functions (eg, PayPal, Square). In the latter case, systems such as bitcoin are attempting to find alternatives to the existing payment arrangement. Beyond these examples, the market of payment services is more dynamic than ever.

The dynamics present a constant challenge to central banks in their exercise of liquidity provision. As these systems have grown and incorporated new technologies, the rules and standards governing them must evolve to accommodate such advances. In this way, understanding the determinants behind these dynamics helps policy makers to design better policies that ensure payment-system integrity as well as identify any effect among participants. This ensures that all participants have the same incentives and prevents anomalies as a result of government policies that apply to some players but not others.

The real-time settlement payment system in Mexico, known by its Spanish acronym SPEI, has experienced important changes since its birth in August 2004. It grew at an average of 9.8% per year in nominal volume and at an average of 47.4% per year in number of payments between 2005 and 2015. The number of participants has increased from 24 to 130, and it now spans not only credit institutions but also brokerages, nonbank financial institutions and other FMIs. All of these changes affect the dynamics of the system as well as participants’ intraday liquidity management. The dynamics prompt the correct operation of the payment system, while the intraday liquidity management determines liquidity pressure.

These elements are just some of the challenges that FMI authorities have to deal with in order to ensure the stability of the system. A retrospective assessment of the dynamic network and its effects on intraday liquidity management must be carried out to help policy makers understand the implications of their decisions. A question arises: what forces drive the dynamics in SPEI?

This paper seeks to pin down the determinants of SPEI’s dynamics between 2005 and 2015. Our hypothesis is that network metrics are behind it. As we have said, payment systems are affected by globalization, financial innovation and means-of-payment incorporation, and changes to those three elements embody changes to the value and number of payments traded in those infrastructures. In this way, values and quantities are a result of market forces, and network metrics can be read as indicators of these forces. Let us give an example using the degree centrality in a three-participant payment system. This metric is defined as the number of links incident upon a node. In the case that one participant, say A, implements a new successful means of payment, we expect that it increases the number of payments toward the other two participants, B and C. A’s degree centrality would increase along with the system’s centrality. However, we also expect a weaker competitive system as a result of A’s new comparative advantage, and effects on system growth are uncertain given the effects on other metrics. Thus, network metrics mirror market structure in a payment system and affect its dynamics.

We use a time-varying coefficients factor-augmented vector autoregressive model (TVC-FAVAR) to test our hypothesis and answer our question. A FAVAR model and its variants describe dynamic systems based on the underlying forces driving a system, which are known as the factors that explain the behavior of a system. The time-varying specification also allows testing for structural changes, which are the conditions that occur when a market changes how it functions.

We follow a two-step strategy to test our hypothesis. First, we fit the TVC-FAVAR model to the empirical data, obtaining the underlying dynamic factors and the loadings (participants’ reactions to the factors). The data set contains the daily network of payments in SPEI between August 14, 2004 and June 30, 2016, but we present the results for the period January 2005–December 2015. The second step is factor identification, in which we regress the dynamic factors on the network metrics. Significant coefficients in this step imply that network metrics explain the performance of the forces driving the dynamics, which supports our hypothesis.

The main contribution of this paper is to apply, for the first time, econometric methods in order to identify the network metrics as forces driving a payment system. This approach expands the concept of controllability introduced by Galbiati et al (2013) and implemented in Galbiati and Stanciu-Vizeteuz (2015). While both papers identify the relevant nodes as a set with the ability to influence all other nodes, leading the network to a particular configuration or state, we alternatively propose that the network metrics of the whole system are the forces driving such dynamics.

The second innovation of this paper is to show the behavioral implications of our approach for liquidity management. This novelty seeks changes in the use of two funding resources as a response to changes in the forces driving the system. These funds are identified following the algorithm introduced by Alexandrova-Kabadjova et al (2015) and are defined as (i) reused payments, incoming payments used for covering new outgoing payments; and (ii) external funds, participants’ funds received from liquidity-provision channels outside the large-value payment system (LVPS). Thus, we show the changes in the composition of both resources as a result of changes in the network metrics.

We find that five factors drive the (Mexican) payment system, two of them most significantly, and that their performance is explained by network metrics. Factor 1 measures the system trend and is affected by market structure conditions. Factor 2 is an indicator of system stability (the inverse of liquidity pressure). Moreover, we verify changes in the performance of the system and the response of the participants to such changes.

The remainder of this paper is structured as follows. Section 2 presents a brief literature review. Section 3 describes the evolution of the overall dynamics in SPEI, taking into account the external funds and reused payments. In Section 4, we present the methodology to answer our research question and identify the determinants driving the dynamics in SPEI. Section 5 shows the factor identification results and the behavioral implications for liquidity management. Finally, in Section 6, we present our conclusions and some potential future extensions of this line of research.

2 Literature review

The aim of this paper is to identify the forces that are driving SPEI. This question is intrinsically related to the concept of controllability. This section provides an overview of the main studies dedicated to analyzing the concept. We also extend this section with the main studies dedicated to analyzing bank-funding patterns in liquidity management, since we show the implications of the driving forces in reused payments and external funds. Finally, given our transactional analysis, we also discuss topics related to FMI (data) analysis.

According to control theory, a dynamical system is controllable if, with a suitable choice of inputs, it can be driven from any initial state to any desired final state with infinite time (see Kalman 1963; Luenberger 1979; Slotine and Li 1991).

Liu et al (2011) develop analytical tools, based on nodes’ weight identification, to study the controllability of an arbitrary complex directed network, identifying the set of driver nodes with time-dependent control that can guide the system’s entire dynamics.

The authors find that the number of driver nodes is determined mainly by the network’s degree distribution and show that sparse inhomogeneous networks, which emerge in many real complex systems, are the most difficult to control. However, they can be controlled using a few driver nodes; counterintuitively, Liu et al (2011) discover that in both model and real systems the driver nodes tend to avoid the high-degree nodes.

While these authors set the analytical framework using the characteristics of real networks, such as trust in college students and prison inmates, Galbiati et al (2013) suggest the same analytical framework in payment systems and interbank lending markets. Galbiati and Stanciu-Vizeteuz (2015) describe TARGET2 (T2) by means of classic network measures and apply these methods to uncover two additional features of T2: driver nodes and communities.

By looking at T2 drivers, Galbiati and Stanciu-Vizeteuz (2015) were able to single out banks that might have a dominant influence on systemic liquidity flow. These authors focus on the largest eighty-nine banks in the system, and they are able to single out thirteen banks, which would not have all been detected by looking at bank size or other centrality measures.

While previous papers identify the controllability of the system through driver nodes, we propose a different approach. We use network metrics as external regressors of the underlying forces, which are first estimated using econometric methods. Thus, the definition of driver nodes is not excluded in our approach, since system network metrics are driven by participant weights.

Our study also examines the implications of network metrics changes in liquidity management and funding patterns. The concept of reused payments, and external funds as its complement, is based on the liquidity game among banks. Bech and Garratt (2003) examined banks’ intraday behaviors in a game theoretic framework for the case of two players. Their paper models intraday settlements as two-stage games in which each player basically determines whether to make a payment early or late. Under a different regime, these authors find that both early and delayed payments are possible equilibriums. However, for certain levels of delay and liquidity costs, participants might be in a prisoner’s dilemma, in which the dominant strategy for both is to delay payments.

Becher et al (2008) investigate the factors influencing the timing and funding of payments in the CHAPS sterling system. Their results show that participants in CHAPS sterling often use incoming funds to make payments, a process known as liquidity recycling.

The authors explain that liquidity recycling can be problematic if participants delay their outgoing payments in anticipation of incoming funds. Their analysis of CHAPS payment activity shows that the level of liquidity recycling, though high, is stable throughout the day. This is a consequence of (i) the settlement of time-critical payments in CHAPS supplies liquidity early in the day, (ii) the sterling system throughput guidelines providing a centralized coordination mechanism that essentially limits any tendency toward payment delay, and (iii) the relatively small direct membership of the system facilitating coordination, enabling members to maintain a constant flux of payments during the day.

Diehl (2013) addresses the question of how free-riding in LVPSs should be measured properly. He developed several measures for identifying free-riding. His study shows that a combination of at least two measures is recommended for capturing the effects of free-riding. Based on nine important banks in the German part of T2, free-riding was measured at the single bank level. The evaluated measures show stable payment behavior of most of the participants over time. However, some remarkable regime shifts were observed.

Massarenti et al (2012) studied the intraday patterns and timing of all T2 interbank payments. They provide insight into the intraday patterns of T2 payments and into the evolution of settlement delay. Their analysis provides a system-wide view and can be used by operators and overseers of central banks. One of their results indicates that in terms of the numbers of payments processed, the first hour and a half is the most critical time during the system’s operating hours, whereas the last hour is crucial in terms of value of payments.

Heijmans and Heuver (2014) study the T2 transaction data in order to identify different liquidity elements. They develop a method to identify liquidity stress in the market as a whole and at the individual bank level. The stress indicators look at liquidity provided by the central bank (monetary loans), unsecured interbank loans (value and interest rate), the use of collateral and payments on behalf of their own business and that of their clients. In the case of Mexico, two simulation studies have been conducted by Alexandrova-Kabadjova and Solís-Robleda (2012, 2013) to analyze the settlement rules of SPEI and the liquidity management of commercial banks that are direct participants.

These authors study the behavior of the selected financial institutions related to the reuse of incoming payments. They find that despite the growing volume of low-value payments processed in real time through SPEI, settlement is performed efficiently. Further, the authors argue that, according to the observed patterns in commercial banks’ intraday behavior, there are particular hours in which low-value payments are preferably sent.

What these papers have in common is that they use transaction-level data from a payment system, FMI or real-time gross settlement (RTGS) system. However, they do not look at operational level per type of direct participant.

In the next section, we present a description of SPEI performance, in aggregate and at the level of funding patterns.

3 SPEI’s diary: dynamic evolution

As mentioned, SPEI has experienced significant growth since its birth in August 2004. It grew at an average of 9.8% per year in nominal volume between 2005 and 2015, 5.8% in real terms, assuming the consumer price index as a proxy to the deflator of payment services.11We exclude 2004 in the calculation of the rate because SPEI was not operating the whole year. The value of total payments in 2005 was 247.5% of the Mexican gross domestic product (GDP), increasing to 335% in 2015.

The change in the size of the infrastructure is the result of different elements, such as more direct participants operating in the system and the inclusion of new types of participants. These elements, among some others, have been affected by changes in the regulatory framework, changing the SPEI ecosystem and liquidity management. In this section, we present the changes in the organizational scheme of SPEI and the main dynamics in the funding patterns of liquidity management.

3.1 SPEI organizational scheme

The present study focuses on identifying the dynamics of SPEI. This system, alongside the Swiss interbank clearing (SIC) and Turkish RTGS (TIC-RTGS) systems, is among the few examples worldwide of RTGS systems being used to simultaneously settle both large- and low-value payments. The SPEI organizational scheme refers to three aspects of the system’s connectivity: direct participants, FMIs and the liquidity provision mechanism.

SPEI started operations on August 13, 2004 and was meant to substitute SPEUA, a Mexican RTGS system that had been processing large-value payments since 1994.22“SPEUA” is the Spanish acronym for the extended-use electronic payments system. Since August 19, 2005, SPEI has been the only system settling all-size direct credit electronic payment transactions in real time. This condition has led to the development of new intraday-liquidity-management skills required to deal with the challenge of settling low-value payments in real time. Parts (a) and (b) of Figure 1 show the organizational scheme of SPEI in 2005 and 2015, respectively.

Banco de México (BdM) uses two channels to provide liquidity to commercial banks, development banks and brokerage houses. The first channel is via collateralized overdrafts and is available to participants with accounts in SIAC.33SIAC is the system in charge of managing the accounts of commercial and development banks, brokerages and government entities that by law should have an account at BdM. This system is operated by the central bank and provides liquidity to the FMIs in Mexico. The second is through repurchase agreement (repo) transactions using a computational module known as RSP, which is specifically designed for settling these operations.44Repos and overdrafts should be fully collateralized. The rules applying to commercial banks are established in numeral M.73 of circular 2019/95 (repealed by circular 3/2011); the rules followed by development banks are given in numeral BD.51 of circular 1/2006, whereas numeral CB.2 of circular 115/2002 establishes the rules on brokerage houses.

Several changes in the regulatory framework of the Mexican FMIs took place in 2008. The Mexican peso (MXN) was incorporated into the set of currencies that the continuous linked settlement (CLS) system operates in on foreign exchange markets.55CLS operates under the payment versus payment (PvP) scheme. A direct connection was established between SPEI and CLS. Later, the authorities introduced a new security settlement system, known as the DALÍ, which operates under the delivery versus payment (DvP) scheme.66This FMI replaced the interactive system for securities depository (SIDV) in November 2008. Since then, participants that have access to DALÍ and SPEI have been allowed to transfer funds to their own accounts from one system to the other. Finally, nonbank financial institutions were allowed direct access to SPEI but with no access to the intraday liquidity provided by the central bank.

Regarding the participants, they are allowed to make fund transfers and payment obligations on their own behalf or on behalf of a third party (clients). In 2005, direct participants in SPEI were only private multiple-purpose banks (CBs) and public development banks (DBs). Both groups are credit institutions and have current accounts in the SIAC. However, since 2008, four groups of direct participants in SPEI have been identified:

  1. (i)

    private multiple-purpose banks (CBs),

  2. (ii)

    public development banks (DBs),

  3. (iii)

    brokerage houses (Bs), and

  4. (iv)

    other nonbank financial institutions (ONBFI).

Further, DALÍ and the CLS system are FMIs connected to SPEI.

Organizational scheme of SPEI.
Figure 1: Organizational scheme of SPEI. (a) Initial scheme (2005). (b) Current scheme (2015).

3.2 Funding patterns and liquidity management in SPEI

The changes in the organizational scheme of SPEI affect the funding patterns and the liquidity management of the participants in the system. Over 90% of the overall volume processed is considered to be low-value payments, and given that the system operates as an RTGS scheme, low-value payments push up the liquidity pressure.77If we compare that scheme with a deferred net settlement scheme (DNS). Participants with direct access to BdM liquidity assess the funds needed to settle their obligations throughout the day in advance. Then, based on their pre-assessment, they transfer funds from their accounts in SIAC to their accounts in SPEI, and they use their accounts in SPEI to settle payments during the day.

Following the algorithm proposed in Alexandrova-Kabadjova et al (2015) and using transactional data from SPEI, we can distinguish between two funding resources: (i) reused payments, or incoming payments used for covering new outgoing payments; and (ii) external funds, or participants’ funds received from liquidity-provision channels outside the LVPS. The settlement process in SPEI is organized through cycles, and the authors develop an algorithm based on the date of these settlement cycles.

The algorithm works as follows. Let be the set of participants in the system, and let ? be the set of cycles in one day. We denote the sum of the incoming payments as Pi,trec and the sum of the outgoing payments as Pi,tsent by each i and in each cycle t?, respectively. We define the difference between these two amounts by participant in each cycle t? as

  Ai,t=Pi,trec-Pi,tsent.   (3.1)

Let Si be the positive balance resulting from aggregating the value of Ai,t for each t?, given that Si=0 for all it1. We denote by Fi the sum of funds each i has to spend in those cycles, in which Si0, given that Fi=0 for all it1. In this way, thanks to the register of the cycles, both resources can be observed.

SPEI 2005–15: funding patterns.
Figure 2: SPEI 2005–15: funding patterns. (a) Total payments in the banking system (Mexican new pesos). (b) Proportion of funds (with respect to 1).

Figure 2 illustrates the use of external funds and reused payments for commercial and development banks in SPEI. Figure 2(a) shows the levels of external funds, reused payments and a twenty-one-day moving average of the external funds. Figure 2(b) shows the performance of the same funds as a ratio.

In aggregate terms, the external funds ratio has followed a relatively stable path. It has been fluctuating around 20%. We observe that in the first period between 2005 and 2007, external funds are stable. This fact is related to the stable growth of the number of payments. We also observe that the external-funds-to-total-payments ratio is around 15%. After this period, we observe a reduction in the number of payment transactions, without significant changes on the level of external funds. During this period (2008 to mid-2010), the external funds ratio was around 19%. Between July 2010 and August 2011, we observe strong growth of the payments and an increment in the level of external funds used to cover participants’ obligations. Consequently, the proportion of external funds on average during this period is 21%. In the period between September 2011 and June 2013, the volume of payments remained at the same level, and the external funds ratio was 19%. Finally, for the period between July 2013 and December 2015, a new rise in the volume of payments is observed. Nevertheless, the average proportion of extra funds used has fallen to 16%.

4 Methodology: econometric analysis

As stated, we attempt to identify the underlying forces driving SPEI, and we point to network metrics as the driver of the dynamics in this system. We use a two-step strategy to test this hypothesis.

In the first step, we estimate the underlying dynamic factors driving the system. To this end, we estimate a TVC-FAVAR model with the time series of participant payments. As a result, we obtain an unbiased estimation of the dynamic factors and the participant responses to the factors (these are known as loadings).

In the second step, we regress the estimated dynamic factors on eight different network metrics. We use a variant of a time series regression, a seasonal autoregressive integrated moving average with exogenous variables (SARIMAX) model. If the coefficients in these regressions were significant, network metrics would explain the behavior of the estimated dynamic factors, supporting our initial hypothesis.

In this section, we present a detailed description of the data set and methodology implemented to test our hypothesis.

4.1 The data

The payment system data is constructed in the following way. The initial database contains the daily transactional payments from August 13, 2004 to June 30, 2016. First, we create an aggregate daily network of payments. This network compounds the bilateral transactions among sixty-four participants over 2983 days. With this network, we calculate the network metrics of the system as a time series.

Second, we aggregate the daily network of payments by participants, ie, we create a panel with the total sent payments by participant over 2983 days (190 912 observations). We use this panel to estimate the TVC-FAVAR model and obtain the dynamic underlying factors.

As mentioned, we are interested in showing the implications of this factor analysis in liquidity management. To that end, following Alexandrova-Kabadjova et al (2015), the external funds and reused payments are distinguished from the initial database (the transactional data). Therefore, and as a result of the algorithm, we finally have three panels: the total payments, the external funds and the reused payments, for sixty-four participants over 2983 days (in Mexican pesos).

The three panels as well as the daily networks have the same participants. There are sixty-two direct participants, of which fifty-seven are commercial banks, four are development banks and one is BdM. There are also two FMIs with accounts in SPEI: DALÍ and the CLS system.

4.2 Step 1: econometric framework of the dynamic factors

SPEI can be assumed to be a dynamic system. This is based on the idea that there exists a relationship that describes a time dependence in the set of participants, or, alternatively, a network whose status changes over time. These changes refer to the evolution of the payments among participants, the ability to create subgroups and the structure of connections among them. SPEI as a dynamic system accepts the following state space representation:

  Xt =ΛFt+ϵt,   (4.1)
  Ft =Φ(L)Ft+νt.   (4.2)

Equation (4.1) states that Xt, a vector with N total payments88One for every participant. at time t, is a linear combination of Ft, the underlying factors at the same time t. The factors represent the state of the world, and they fully describe the system and its response to any given change. Thus, they behave like an engine driving SPEI. Λ is the loading factor and reflects the response of the payments to the unknown factors. Equation (4.2) represents the dynamics of the underlying factors as differential equations.99In our case, it follows a vector-autoregressive process. Φ(L) denotes a polynomial of a lag operator of order L, ie, it produces the Lth previous elements of Ft.1010We can expand this polynomial as Φ(L)Ft=Φ1Ft-1+Φ2Ft-2++ΦLFt-L. Note that this specification assumes linearity.1111Nonlinear dynamic systems are known as chaotic systems. They are ruled by chaos theory and are highly sensitive to initial conditions.

FAVAR models can be used to estimate dynamic systems. The main advantage is based on the fact that it exploits a large set of economic information, and particularly for this case, a large number of participants. This data-rich environment results in a more extensive analysis, as it allows us to isolate the effects of the factors on a vast number of different variables at the same time. The model results in a consistent estimation of the dynamic factors and loadings when N grows faster than T0.5 (see Bai and Ng 2006).1212In our case, N=64 and T0.5=54.62. As argued by Stock and Watson (2008) and Banerjee et al (2008), the factors are still estimated consistently even if there is some time variation in the loading parameters. Thus, TVC-FAVAR represents a suitable framework for estimating the dynamic system and analyzing empirical micro-effects.

The reader could argue against this methodology due to the difficulties in understanding the meaningless factors or the possible bias as a result of unbalanced panels. The meanings are not crucial from the perspective of a dynamic system that is mainly focused on predicting the performance of the system. Despite this advantage, we carry out a factor-identification in the second step of our strategy. The effects of unbalanced panels are mitigated under the assumption of large data sets: as stated, we work with panels with 2983 periods of observations.

Moreover, we introduce the time-varying coefficients, which implies that the effects of the factors may change across time. In other words, the dynamic system can evolve over time, which allows testing for structural changes to the system. The TVC-FAVAR model accepts an equation-by-equation (for every participant) representation and is given by

  xi,t =λi,tFt+ϵi,t,ϵtN(0,Vt),   (4.3)
  Ft =ϕg,t(L)Ft-L+νg,t,νtN(0,Qt),   (4.4)
  λi,t =λi,t-1+ωi,t,ωtN(0,Wt),   (4.5)
  ϕg,t =ϕg,t-1+ηg,t,ηtN(0,Rt),   (4.6)

where the subindex i denotes each participant in SPEI; xi,t is the observable time series of participant i – total payments, external funds or reused payments – at period t; λi,t=[λ1,i,t,λ2,i,t,,λG,i,t] denotes the factor loading; and Ft=[f1,t,f2,t,,fG,t] denotes the underlying G factors driving the dynamic into the system at period t. Equations (4.5) and (4.6) show that the dynamics of the coefficients follow random walks, and they model the evolution of the responses to factors or changes in the behavior of the participants.1313Thanks to this specification, one can analyze the causes of behavioral changes across time, for example, by regressing the dynamic slopes to the economic and financial characteristics of each institution.

Regarding the error terms, they are assumed to be Gaussian and are estimated recursively using simulation-free variance matrix discounting methods (see, for example, Quintana and West 1988). We use exponentially weighted moving average (EWMA) estimators. These depend on two decay factors. Such recursive estimators are trivial computationally. In addition, the EWMA is an accurate approximation to an integrated generalized autoregressive conditional heteroscedasticity (GARCH) model. All variance–covariance matrixes are assumed to be diagonal.

The model is estimated by a two-step Kalman restricted estimation, following the procedure in Koop and Korobilis (2013). The algorithm is as follows.

  1. (1)

    Initial values:

    1. (a)

      obtain the principal component estimates of the factors Ft^;

    2. (b)

      initialize all parameters from the ordinary least squares static estimation {Λ0,Φ0,V0,Q0,R0,W0}.

  2. (2)

    Estimate the time-varying coefficients {Λt,Φt,Vt,Qt,Rt,Wt} given Ft^:

    1. (a)

      estimate {Vt,Qt,Rt,Wt} using a variance discounting procedure;

    2. (b)

      estimate {Λt,Φt}, given {Vt,Qt,Rt,Wt}, using the Kalman filter and smoother.

  3. (3)

    Estimate the dynamic factors F~t given {Λt,Φt,Vt,Qt,Rt,Wt} using the Kalman filter and smoother.

Note that Λt=[λ1,t,λ2,t,,λN,t] is an N-by-G matrix.

This empirical model is fitted for three SPEI unbalanced dynamic panels. First, we estimate the model for the total payments, keeping the factor loadings Λt and the dynamic factors F~t. Given that the total payments can be decomposed into the sum of external funds and reused payments, we estimate the model for both resources to analyze the implications for liquidity management using the dynamic factors from the total-payment model.

4.3 Step 2: network measurements and factor identification

Our hypothesis is that network metrics are driving the dynamics of the system. The underlying factors represent the engines driving the system, and as engines they need fuel to accelerate. However, they can slow down as a consequence of erosion and frictions. In order to identify the factors, we estimate SARIMAX models for every underlying factor:

  fg,t=αg+ΓgFt-1+ΥgNMt+HgSVt+μg,t,g=1,,G,   (4.7)

where denotes the first-difference operator, NMt denotes the network metrics, SVt denotes seasonal lags, and μg,t is an error term that follows a seasonal moving-average process.1414See Appendix A (available online) for a detailed description of the calculus of the measurements. We use three different centrality measurements – out-eigenvector centrality, out-degree centrality and harmonic distance – that capture the involvement in the cohesiveness of the network. We also use two different measurements of the structure of the network – community structure and clustering coefficient – that measure the degree to which participants in a network tend to congregate together. The rest of the measurements – reciprocity, relative net-trade value and reactivity – provide a measure of the quality of the bilateral relationships and its relative importance in the network.

In particular, we develop a new measurement, reactivity, which attempts to capture the responsiveness of sent payments to incoming payments. More specifically, we define apparent reactivity as the incoming payment elasticity of sent payments.

Thus, the significance and sign of the coefficients show the effects in the dynamic of the system. Positive signs reflect earnings or acceleration, while negative signs represent losses or deceleration.

In the next section, we explain the results of the factor identification and their implications for liquidity management.

5 Empirical results

We fit a TVC-FAVAR model with five factors and six lags.1515We determine the number of factors with the eigenvalue ratio test, as in Ahn and Horenstein (2013). We estimate the factors using principal component (PC) analysis. We calculate the PC on the covariance matrix of the standardized and seasonal-adjusted series. We determine the number of lags equal to 6 using the Bayesian information criterion (BIC); see Schwarz (1978). That specification explains more than 95% of the system variance, ie, any linear combination of the five dynamic factors explains above 95% of the variability of the participants in the system.

5.1 Network measurements as engines of the dynamic

The dynamic factors represent the common trends in the system. The main factor reflects the aggregate dynamic in the system, as we can see in Figure 4.1616See Appendix A (available online) for a detailed description of the measures and their evolutions. However, the rest of the factors identify complex interrelationships among participants that are part of different concepts. The interpretation of these concepts can be abstract because of the topology behind the factor method. For example, they can mean simple concepts, such as weight or length, as well as more complex measurements, such as the moment of inertia. In our case, we can state that network metrics explain a large fraction of factor behavior.

We select eight out of eighteen different network measurements to identify the underlying factors of the system. On the one hand, we drop ten measures because they are strongly correlated with the selected ones. They would create multicollinearity problems in the identification of the contributions. Thus, the other ten network measurements do not provide substantial new information. On the other hand, the eight selected measurements capture different properties of the network. In particular, they measure centrality, distance, clustering and bilateral relationship. Table 1 shows the results of the factor identification made with an equation-by-equation SARIMAX model.

Factor 1 identifies the common stochastic trend driving the nonstationary series of payments. This implies that SPEI has an evolutionary direction. It could be understood as the aggregate trend of the system. The empirical evidence suggests that changes in centrality, network structure and bilateral relative net-trades drive the changes in the dynamics. Thus, changes in the trend are the result of changes in the market structure’s conditions.

SPEI 2005–15: dynamic factors.
Figure 3: SPEI 2005–15: dynamic factors. (a) Evolution of dynamic factors (F~t). (b) Evolution of dynamic factors volatility (Qt~).

According to Table 1, centrality negatively affects the aggregate dynamics. Let us think in terms of two extreme cases: a fully connected payment system versus a star-structured one. In the first case, every participant can send and receive payments from every participant. The star structure forces the participants to use the central participant as intermediary. The velocity of the payments would increase in the fully connected system, and it would drop in the star structure. Thus, SPEI would maximize the velocity of transactions under a cohesive system, ie, with no intermediaries.

Both the community-structure probability and clustering coefficient also support this result. They have a negative sign in their contributions to the inertia. The existence of microsystems slows down the aggregate dynamic of the system. In the context of payment systems, clusters prompt the members to face higher risk. If they are close to the rest of the system, they become more vulnerable, since codependent relationships can lead to liquidity constraints. Cluster banks trade less with noncluster banks, and vice versa. Therefore, liquidity pressure increases, since access to reused payments is limited.

Moreover, the relative net-trade value (RNTV) shows a negative effect. This fact suggests that asymmetric bilateral relationships create resistance in the dynamics. A higher RNTV implies that participants do not send proportional payments with respect to those received. This creates asymmetries in the trade of payments. Funding problems arise when there are unbalanced relationships. For example, a small participant would require large use of external funds to meet its obligations. This slows down the liquidity flows, and therefore reduces the inertia of the entire system.

To sum up, changes in factor 1 are a result of adjustments to the market structure’s conditions. Centrality, structure and RNTV capture the market positions on the network. As a result, the dynamics of the system can shift. Thus, the maximal growth rate of total payments would be achievable under a perfect competition scheme.

Table 1: Factor identification in SPEI. [Standard deviations between brackets. * represents significance at 10%, ** at 5% and *** at 1%.]
Variable Factor 1 Factor 2 Factor 3 Factor 4 Factor 5
Constant -0.2273*** -0.0101 -0.0002 0.1195*** -0.0156***
  (0.04) (0.01) (0.00) (0.01) (0.00)
Eigenvector centrality -0.1720*** 0.0392*** 0.0474*** 0.0798*** -0.0184*
  (0.05) (0.01) (0.01) (0.03) (0.01)
Out-degree centrality -3.5579*** 0.2764*** 0.2019*** 1.2091*** -0.0808
  (0.24) (0.07) (0.06) (0.13) (0.05)
Community structure -1.4321*** 0.0591 0.1310** 0.2680** -0.0325
  (0.24) (0.06) (0.06) (0.13) (0.06)
Clustering coefficient -0.0350*** 0.0014 -0.0008 0.0161** 0.0004
  (0.01) (0.00) (0.00) (0.01) (0.00)
Harmonic distance 0.4586 0.0040 -0.0077* -0.2067*** 0.0254***
  (0.01) (0.00) (0.00) (0.01) (0.00)
Reciprocity 0.0036 0.0007 0.0036*** -0.0085*** -0.0005
  (0.00) (0.00) (0.00) (0.00) (0.00)
RNTV -0.0885*** -0.0130*** 0.0017 0.0423*** -0.0054***
  (0.01) (0.00) (0.00) (0.00) (0.00)
Reactivity -0.0016 -0.0012** -0.0016*** 0.0003 -0.0002
  (0.00) (0.00) (0.00) (0.00) (0.00)
Factor 1 (t-1) 0.9876*** -0.0057 -0.0008 0.0182** -0.0074
  (0.00) (0.01) (0.01) (0.01) (0.01)
Factor 2 (t-1) -0.0970 0.9858*** -0.0024 0.0712** 0.0462***
  (0.07) (0.00) (0.02) (0.03) (0.01)
Factor 3 (t-1) -0.3106*** 0.0545** 0.9679*** -0.0055 0.0657***
  (0.07) (0.02) (0.00) (0.03) (0.02)
Factor 4 (t-1) -0.0504 0.0013 0.0474*** 0.9613*** -0.0118
  (0.03) (0.02) (0.01) (0.01) (0.01)
Factor 5 (t-1) -0.0794 0.0470** 0.0651*** 0.0077 0.9617***
  (0.07) (0.02) (0.02) (0.04) (0.01)
θ1 (MA) -0.5691*** -0.3295*** -0.2278*** -0.5663*** -0.1964***
  (0.02) (0.02) (0.02) (0.03) (0.02)
Φ1 (SAR) 1.0746*** 1.0869*** 1.0664*** 1.0568*** 1.1006***
  (0.02) (0.02) (0.02) (0.02) (0.02)
Φ2 (SAR) -0.0757*** -0.0947*** -0.0887*** -0.0611*** -0.1163***
  (0.02) (0.01) (0.02) (0.02) (0.02)
Θ1 (SMA) -0.9719*** -0.9574*** -0.9182*** -0.9587*** -0.9336***
  (0.01) (0.01) (0.01) (0.01) (0.01)

According to Table 1, factor 2 accelerates because of increments in centrality. In addition, bilateral relationships make it decelerate. Thus, factor 2 seems to be a measure of stability in the network structure. Well-connected and big participants give persistence and cohesiveness to the network. Removing any of these participants would increase demand for liquidity from the whole system.

By contrast, RNTV and reactivity erode the centrality effects. An asymmetric and very reactive payment system becomes less stable. This fact is due to the funding capacity and dependence on payments. On the one hand, asymmetric trade defines a structure of classes among participants. A participant with a large deficient trade faces self-liquidity pressure and needs external funds; but this is the mirror image of a surplus participant. Thus, there would be classes in terms of funding capacity. On the other hand, a reactive participant responds by changing outgoing payments in a way that is more than proportional to changes in incoming payments. However, this also implies a higher dependency, since incoming payments can represent an important fraction of the funds. More than proportional responses lead the system in explosive growth that is not adjusted for liquidity. This results in liquidity constraints and higher risk. The opposite is also true, however: less than proportional responses slow the system, resulting in excess liquidity.

Let us illustrate the performance of network stability measurement (NSM) with an example. Say participant A gets an advantage with respect to the rest, and it becomes more important than the rest in its centrality. It would start to exhibit a larger surplus trade, and the total system could grow. However, some other participants would react to this asymmetry. If the response is more than proportional, the system could explode or collapse. So, the only way to compensate for the centrality advantage and to control the system would be by reducing the reactivity. Thus, the stability of the payment system is driven by a fragile equilibrium between the centrality, asymmetry and reactivity of its participants.

The remaining factors have a less straightforward identification. Factor 3 seems to be capturing interactions among participants. It is affected by centrality, community structure, harmonic distance, reciprocity and reactivity. This cocktail of measurements could be capturing the system strength of participants’ relationships. This measure would be an alternative approach and bring new meaning to the weighted network. Factor 3 would refer to a more abstract concept, the interconnectivity strength of the network. There can be inactive participants in the day-to-day trading of a payment system. Factor 3 could measure whether participants were actively using the system with most of the rest of the participants.

Factor 4 seems to be the effect of most measurements in the payment system business cycle. It could measure the business cycle of the system. Note that it is not affected by reactivity, but it is affected by the trend of the system as well as the stability of the system (factors 1 and 2). Centrality, distance and RNTV affect factor 5. This suggests that it could be reflecting the evolution in the rate of change of payment velocity.

Thus, factor identification is crucial and moves network analysis to a new step. We find that the trend of the system is driven by network positions, that factor 2 measures the stability of the network and that factor 3 captures a sort of interconnectivity strength. Factors 4 and 5 have a more complex meaning. Once we know the meaning of the main factors, we can also analyze the structural changes in SPEI and the implications for liquidity management.

5.2 Behavioral implications and liquidity management

The TVC-FAVAR model allows us to check structural changes in the system. We differentiate between two kinds of changes. On the one hand, we have changes to the conditions of the system, measured through changes in the performance of the factors. We consider big changes in levels and volatilities. On the other hand, we have changes in the loadings or reactions to the factors, which are reflected by changes in λr,i,t. Given the large number of participants, we analyze the average participant’s response and the funding breakdown, ie, the adjustment via external funds and reused payments.

5.2.1 Changes on the conditions of SPEI

We focus on factors 1 and 2, since they have the clearest meaning. As seen in Figure 3(a), we do not observe big changes in the levels of factor 1. However, its volatility reveals two clear structural breaks, as shown in Figure 3(b).

The first change captures the effect of new participants in the system. In particular, DALÍ, the securities settlement system in Mexico, experienced a change in performance in 2008. This affected the volatility of the trend. We could read this finding in two different ways. First, the increased volatility may be related to higher risk: in this case, the risk would be associated with the uncertainty of a new “big” participant in the system and its impact on liquidity pressure. Following Barvell (2002), new participants should encourage competition among old participants in order to promote efficient and low-cost payment services. However, there could be a cost if these participants are other FMIs (DALÍ’s case). The introduction of a big FMI could result in excessive legal, financial or operational risks. In addition, as we extract from the volatility of factor 1, old participants perceived higher risk with the introduction of DALÍ.

Second, this break may be interpreted as an increase in the system’s heterogeneity. Originally, there were commercial and development banks participating in the system. However, after some reforms in the regulatory framework in 2008, the heterogeneity of the participants increased: brokerages, other nonbank financial institutions, two infrastructures and BdM started to operate in SPEI. This change in the typology of participants resulted in a rise in system heterogeneity.

The second structural change reflects the effects of the 2007 financial crisis, which affected the Mexican economy one year later. It is represented by the highest peak in factor 1’s volatility. Analogously to DALÍ’s introduction, the financial crisis increased uncertainty in the financial market. Participants were unable to know for certain the quality of other participants’ balance sheets. Thus, there were some tensions, which resulted in a convulsed period.

A third change in performance is shown by the constant growth in volatility that factor 1 exhibits from mid-2010 onward. The management of governmental payments through SPEI explains this trend. Payments to suppliers, transfers and taxes, for example, started to be executed in SPEI. Given their nature of nonregularity and volume, volatility has been increasing since then.

Thus, we find that the conditions of the system, measured by the volatility of the trend, change across time. The system is sensitive to diverse changes, and these affect the total system.

5.2.2 Responses to the factors

As we explained in Section 4, the empirical model estimates loadings or factor responses assuming time-varying coefficients: the evolution of λr,i,t. Under this specification, the response to a change in the factors is different across time. This means, for example, that an increase in the total volume of payments has different effects in the system today than it will tomorrow. In addition, as we segregate these effects by funding origin, we can analyze the adjustment process between external funds and reused payments.

We define “system effects” as the changes in the responses to factor 1, which reflect the main inertia of the system. We can observe these changes in parts (a) and (b) of Figure 4. The first change we identify is the learning process during the early years of SPEI. We define this process as a transition period with a high heterogeneity in the response among participants to changes in the dynamics of the system. We establish this fact as the increment in the standard deviation of the responses in Figure 4(b); this reflects the diversity of the responses among the participants to factor 1. Thus, there is no common performance among the participants.

SPEI 2005–15: responses to the factors.
Figure 4: SPEI 2005–15: responses to the factors. (a) Average response to factor 1 (λ~¯1,i,t). (b) Standard deviation among participants in response to factor 1 (sd(λ~1,i,t)). (c) Average response to factor 2 (λ~¯2,i,t). (d) Standard deviation among participants in response to factor 2 (sd(λ~2,i,t)).

This pattern is also reflected in the evolution of external and reused funds. It implies that (i) the learning process did not only affect the decision to use one kind of fund and (ii) the participants were adjusting to each other. We can observe that external funds behave in a more unstable fashion than reused payments. This suggests that the use of external funds broadly responds to participants’ decisions and not to adjustments to the system. Thus, participants are surer about the evolution of incoming payments and make excessive use of external funds.

However, reused payments behave in a more stable manner and are more synchronized with the system. Participants improve the expectation mechanism of incoming payments during the learning process. Thus, there is a trade-off between both funds: in other words, participants learn how to replace external funds with reused payments. This trade-off not only defines their individual strategies, but also affects liquidity pressure. An intensive use of reused payments can increase liquidity pressure, but it reduces the underlying cost of the external funds. Pricing policy in the payment system must ensure self-sufficiency via reused payments, financing payments without having to resort excessively to external funding sources. Participants learn this mechanism during the learning process.

After the learning process, the system remains stable until mid-2008. From mid-2008 until 2012, the variability in the responses to the trend grows. This reflects the adjustment of the banking system to the entrance of new participants. After this period of new participant proliferation, participants homogenize their responses, and the system remains stable. However, we observe that the responses of the external funds stabilize. This fact implies that participants become more efficient in their intraday liquidity management. They have a prior strategy in the use of incoming payments. The use of external funds responds less to changes in the dynamics. Thus, the empirical evidence suggests an improvement in the efficiency of SPEI.

System effects show that participants have learned to cover their intraday liquidity needs. As a result, participants have made their responses uniform. In addition, they have improved the use of external funds versus reused payments. Therefore, the system has become more efficient regarding liquidity pressure.

Stability effects are defined as the changes in the responses to factor 2, which reflect the stability of the network. The heterogeneity in responses has been decreasing since SPEI began, as we can see in parts (c) and (d) of Figure 4. External funds and reused payments follow the same pattern: responses are becoming homogenous. This indicates that responses to changes in the stability of the system are relatively similar among participants, and their asymmetries, in terms of their responses, are lessening.

However, if we look at the funding components, we observe different behaviors. External funds and reused payments respond differently to changes in stability. Before mid-2009, both resources follow a nondeterministic trend, and there is no clear adjustment between them. After mid-2009, there is a long-run trade-off between them that goes on until mid-2014. As we saw in Figure 3(a), factor 2 indicates high stability of the network during this period. High stability in the network makes reused payments decline and external funds increase. This reduces liquidity pressure. If liquidity pressure decreases, the system becomes more stable. However, given the opportunity cost of the external funds, the dynamics could slow down. There is feedback, though, between both elements: the funding adjustment reflects cause and effect. For this reason, we observe that the responses of total payments to system stability follow a random walk. But its evolution depends on which effect dominates: velocity of payments or system stability.

Note that this effect is strongly related to the reactivity and the RNTV. As we saw in the factor identification, the reactivity of the system helps to stabilize the network. There is a negative relationship between reactivity and network stability. This indicates that a reduction in the dependency on incoming payments leads to an improvement in stability. The response of funding resources to factor 2 shows this mechanism of adjustment.

Since mid-2014, the dynamics of the system have changed, and the funding patterns have experienced a period of less stability. Now, reused payments react positively, but the opposite is true for the external funds. This would accelerate the system, but it would become less stable. Again, the evolution depends on which effect dominates. Reused payments are a larger fraction of the payments, so the final effect is based on them.

To sum up, the responses to factor 2 show the mechanism of adjustment between the funding resources and the stability of the system. At the same time, its interactions with the dynamics and the velocity of payments are also visible. For this reason, we can consider factor 2 as an inverse measurement of the liquidity pressure.

6 Final remarks and conclusions

The research presented in this paper seeks to identify the determinants of the dynamics in SPEI and their effects on liquidity management. We also give some implications for participants’ funding performance. Given this methodology, we identify the changes in the structure of the system. Thus, we analyze the controllability of a payment system.

We find that five forces are driving SPEI. The two main forces determining its performance are the trend and the network stability. The trend measures the growth of the payment system, and it has inertia, but this inertia changes as a consequence of the market structure conditions. The more competitive the payment system, the faster it grows. Thus, market structure conditions in a payment system are a function of centrality, grouping and trading. Changes in the dynamics reflect adjustments in the competitiveness of the system. Network stability refers to the notion of a system that exhibits controllable or uncontrollable changes. These changes may be detrimental to the performance of the system. The stability of the network results from a fragile equilibrium between centrality and bilateral interactions. It determines the liquidity pressure. This is the result of two forces with opposite directions: earnings from big participants’ persistence and the cost of asymmetry. Therefore, the interaction between both forces explains the two-faced problem in a payment system: velocity of payments versus liquidity pressure.

Thus, the main conclusion from these results is that network metrics are the representation of the underlying factors in a different reference system (both are expressed using different bases for the same vector space). Knowing the dynamic performance of the networks metrics will help us to understand the policy effects in a payment system.

Moreover, we also find changes in the responses to the forces, and we are able to state different processes. Responses to the trend are different among participants. We identify a learning process, which explains the ability of participants to mix external funds and reused payments. We assess the effect of a new FMI in the market. It increases the risk and entropy among participants’ behavior. The financial crisis also contributed to this effect.

The responses to the stability are diverse as well. However, they have been converging since the creation of SPEI. External funds increase during stable periods, providing more stability at the same time. Reused payments increase during unstable periods, but they reduce stability. Thus, there is feedback between stability and funding composition. Both results suggest that the system tends to converge. This implies that there exists an underlying optimal equilibrium.

Finally, we confirm an improvement in efficiency of the participants operating in SPEI. This result suggests that for more than a decade the liquidity settlement mechanism of SPEI has been able to deal with a continuous increase in the volume of payments, without creating extra pressure regarding the level of external funds needed to cover the new payment obligations.

This paper opens the door to new lines of research. On the one hand, optimal pricing policy must ensure the least liquidity cost. This paper could motivate a theoretical analysis in this regard. On the other hand, the idea of underlying optimal equilibrium suggests optimality in payments. The residuals of the model could be meant as error expectations. Thus, optimality in payment systems is the next step.

Declaration of interest

The authors report no conflicts of interest. The authors alone are responsible for the content and writing of the paper.


We would like to thank Miguel Angel Díaz Díaz and Othón Moreno González for their support of this research. We appreciate the comments from participants in the Financial Market Infrastructure Conference II: New Thinking in a New Era and the Bank of Canada and Payments Canada Workshop on the Modelling and Simulation of Payments and Other Financial System Infrastructures. This paper has benefited as well from the remarks of several anonymous referees – Matti Hellqvist, Ronald Heijmans, Luca Arciero, Stefan Niemann and Sheri Markose – and the audience of the Seminar in Market Infrastructures and Liquidity Management at the University of Essex. We also want to thank Angélica Paola González Lozada for her participation as research assistant. The views and conclusions presented in this paper are exclusively the authors’ and do not necessarily reflect those of BdM.


  • Ahn, S. C., and Horenstein, A. R. (2013). Eigenvalue ratio test for the number of factors. Econometrica 81(3), 1203–1227 (
  • Alexandrova-Kabadjova, B., and Solís-Robleda, F. (2012). The Mexican Experience in How the Settlement of Large Payments Is Performed in the Presence of a High Volume of Small Payments, Chapter 16, pp. 431–460. Bank of Finland, Helsinki.
  • Alexandrova-Kabadjova, B., and Solís-Robleda, F. (2013). Managing Intraday Liquidity: The Mexican Experience, Chapter 8, pp. 143–160. IGI Global, Hershey, PA (
  • Alexandrova-Kabadjova, B., Serguieva, A., Heijmans, R., and Garcia-Ochoa, L. (2015). Direct Participants’ Behavior through the Lens of Transactional Analysis: The Case of SPEI, Chapter 19, pp. 205–215. Springer.
  • Bai, J., and Ng, S. (2006). Confidence intervals for diffusion index forecasts with a large number of predictors and inference for factor-augmented regressions. Econometrica 74(4), 1133–1150 (
  • Banerjee, A., Marcellino, M., and Masten, I. (2008). Forecasting Macroeconomic Variables Using Diffusion Indexes in Short Samples with Structural Change, Chapter 4. Elsevier.
  • Barvell, K. (2002). Risks and developments in payment systems. In Seminar on Current Developments in Monetary and Financial Law. IMF Legal Department/IMF Institute.
  • Bech, M., and Garratt, R. (2003). The intraday liquidity management game. Journal of Economic Theory 109(2), 198–219 (
  • Becher, C., Galbiati, M., and Tudela, M. (2008). The timing and funding of CHAPS sterling payments. Economic Policy Review 14(2), 113–133 (
  • Dekker, A. (2005). Conceptual distance in social network analysis. Journal of Social Structure 6, 3. URL:
  • Diehl, M. (2013). Measuring free riding in large-value payment systems: the case of TARGET2. The Journal of Financial Market Infrastructures 1(3), 31–53.
  • Freeman, L. C. (1978). Centrality in social networks conceptual clarification. Social Networks 1(3), 215–239 (
  • Galbiati, M., and Stanciu-Vizeteuz, L. (2015). Communities and driver nodes in the TARGET2 payment system. The Journal of Financial Market Infrastructures 3(4), 31–61 (
  • Galbiati, M., Delpini, D., and Battiston, S. (2013). The power to control. Nature Physics 9, 126–128.
  • Garlaschelli, D., and Loffredo, M. I. (2004). Patterns of link reciprocity in directed networks. Physical Review Letters 93, 268701 (
  • Heijmans, R., and Heuver, R. (2014). Is this bank ill? The diagnosis of doctor TARGET2. The Journal of Financial Market Infrastructures 2(3), 3–36 (
  • Kalman, R. E. (1960). A new approach to linear filtering and prediction problems. Journal of Basic Engineering 82(1), 35–45 (
  • Kalman, R. E. (1963). Mathematical description of linear dynamical systems. Journal of the Society for Industrial and Applied Mathematics A: Control 1(2), 152–192 (
  • Koop, G., and Korobilis, D. (2013). A new index of financial conditions. MPRA Paper 45463, University Library of Munich (
  • Liu, Y.-Y., Slotine, J.-J., and Barabasi, A.-L. (2011). Controllability of complex networks. Nature 473(7346), 167–173 (
  • Luenberger, D. G. (1979). Introduction to Dynamic Systems: Theory, Models, & Applications. Wiley.
  • Massarenti, M., Petriconi, S., and Lindner, J. (2012). Intraday patterns and timing of TARGET2 interbank payments. The Journal of Financial Market Infrastructures 1(2), 3–24 (
  • Nadakuditi, R. R., and Newman, M. E. J. (2012). Graph spectra and the detectability of community structure in networks. Physical Review Letters 108, 188701 (
  • Newman, M. E. J. (2006). The mathematics of networks. In The New Palgrave Encyclopedia of Economics, Blume, L. E., and Durlauf, S. N. (eds). Palgrave.
  • Newman, M. E. J., Forrest, S., and Balthrop, J. (2002). Email networks and the spread of computer viruses. Physical Review E 66(3), 35101–35105 (
  • Quintana, J., and West, M. (1988). Time Series Analysis of Compositional Data, 3rd edn, pp. 747–756. Oxford University Press.
  • Schwarz, G. E. (1978). Estimating the dimension of a model. Annals of Statistics 6(2), 461–464 (
  • Slotine, J.-J. E., and Li, W. (1991). Applied Nonlinear Control. Prentice-Hall, Englewood Cliffs, NJ.
  • Stock, J., and Watson, M. (2008). Forecasting in Dynamic Factor Models Subject to Structural Instability. Oxford University Press.
  • Watts, D. J., and Strogatz, S. H. (1998). Collective dynamics of “small-world” networks. Nature 393(6684), 440–442 (

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