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We propose a flexible link forecast methodology for weighted temporal networks. Our probabilistic model estimates the evolving link dynamics among a set of nodes through Wasserstein barycentric coordinates arising within the optimal transport theory. Optimal transport theory is employed to interpolate among network evolution sequences and to compute the probability distribution of forthcoming links. Besides generating point link forecasts for weighted networks, the methodology provides the probability that a link attains weights in a certain interval, namely a quantile of the weights distribution. We test our approach to forecast the link dynamics of the worldwide Foreign Direct Investments network and of the World Trade Network, comparing the performance of the proposed methodology against several alternative models. The performance is evaluated by applying non-parametric diagnostics derived from binary classifications and error measures for regression models. We find that the optimal transport framework outperforms all the competing models when considering quantile forecast. On the other hand, for point forecast, our methodology produces accurate results that are comparable with the best performing alternative model. Results also highlight the role played by model constraints in the determination of future links emphasising that weights are better predicted when accounting for geographical rather than economic distance.

The way agents form their expectations, and which equilibria are consequently selected, are issues that come over macroeconomic growth models that study the transition between states characterized by different levels of capital. Existing theoretical works in this area have been mostly dominated by perfect foresight models although alternative expectations schemes have been proposed in the last decades. In the present paper we propose an OLG model with capital accumulation assuming that agents employ simple yet heterogeneous rules to forecast the future course of the interest rate and the real wage. The forecasting rules are selected according to an evolutionary mechanism that, on the basis of a fitness measure, evaluates the performance of the heuristics agents are currently adopting. We analytically show that, in the heterogeneous setting we consider, multiple equilibria are possible, but the evolutionary heuristic switching allows for a decrease in the number of possible steady states, drives their selection, and reduces the actual possibility that the economy might be locked into a poverty trap.

The SARS-CoV-2 epidemics outbreak has shocked global financial markets, inducing policymakers to put in place unprecedented interventions to inject liquidity and to counterbalance the negative impact on worldwide financial systems. Through the lens of statistical physics, we examine the financial volatility of the reference stock and bond markets of the United States, United Kingdom, Spain, France, Germany and Italy to quantify the effects of country-specific socio-economic and political announcements related to the epidemics. Main results show that financial markets exhibit heterogeneous behaviours towards news on the epidemics, with the Italian and German bond markets responding with major delays to shocks. Additionally, credit markets tend to be slower than equity markets in adjusting prices after shocks, hence being slower at incorporating the effects of such news.

We study a financial market populated by heterogeneous agents, whose decisions are driven by “animal spirits”. Each agent may have either correct, optimistic or pessimistic beliefs about the fundamental value, which can change from time to time based on an evolutionary mechanism. The evolutionary selection of beliefs depends on a weighted evaluation of the general market sentiment perceived by the agents and on a profitability measure of the existent strategies. As the relevance given to the sentiment index increases, a herding phenomenon in agent behavior may occur and animal spirits can drive the market toward polarized economic regimes, which coexist and are characterized by persistent high or low levels of optimism and pessimism. This conduct is detectable from agents polarized shares and beliefs, which in turn influence the price level. Such polarized regimes can consist in stable steady states or can be characterized by endogenous dynamics, generating persistent alternating waves of optimism and pessimism, as well as return distributions displaying the typical features of financial time series, such as fat tails, excess volatility and multifractality. Moreover, we show that if the sentiment has no or low relevance on belief selection, those stylized facts are abated or are missing from the simulated time series.

This work investigates financial volatility cascades generated by SARS-CoV-2 related news using concepts developed in the field of seismology. We analyze the impact of socio-economic and political announcements, as well as of financial stimulus disclosures, on the reference stock markets of the United States, United Kingdom, Spain, France, Germany and Italy. We quantify market efficiency in processing SARS-CoV-2 related news by means of the observed Omori power-law exponents and we relate these empirical regularities to investors’ behavior through the lens of a stylized Agent-Based financial market model. The analysis reveals that financial markets may underreact to the announcements by taking a finite time to re-adjust prices, thus moving against the efficient market hypothesis. We observe that this empirical regularity can be related to the speculative behavior of market participants, whose willingness to switch toward better performing investment strategies, as well as their degree of reactivity to price trend or mispricing, can induce long-lasting volatility cascades.

The paper presents a new methodology aimed at detecting the modularity structure of an evolving weighted directed network, identifying communities and central nodes inside each of them, and tracking their common activity over time. The method is based on tensor factorization and it is applied to the Consolidated Banking Statistic, provided by the Bank of International Settlements. Findings show that data are well represented by three communities. The temporal pattern of each community varies according to the events involving the member nodes, showing a decrease of activities during crisis periods, such as the 2008 financial crisis and the European sovereign debt crisis.

This paper considers a standard New Keynesian model with heterogeneous expectations on the future level of inflation and output. A biased perception of the target pursued by the Central Bank may arise due to idiosyncrasies in information processing, leading to heterogeneous beliefs about the target. We consider an arbitrarily large number of agents’ beliefs and apply the concept of Large Type Limit. We find that an increase in the sensitivity of agents in selecting the optimal prediction strategy or in the spread of beliefs is crucial for the extent of the Central Bank to stabilize the economy. When the predictors are largely dispersed around the target, the Taylor principle is a requisite for stability since it prevents the self-fulfilling reinforcement mechanism between the realizations of the relevant macroeconomic variables and the forecasts of the agents. When the set of beliefs is somehow anchored to the target, stability can be achieved with weaker monetary policy.

Financial markets are complex systems where investors interact using competing strategies that generate behaviours in which herding and positive feedbacks may lead to endogenous instabilities. This paper develops a novel methodology to detect the emergence of such phases by quantifying the intensity of self-organizing processes arising from stock returns’ co-movements and self-similarities. Our methodology identifies a group of stocks, the Leading Temporal Module, whose statistical properties reflect the transition of the market into a crisis state. We define a topological indicator of the emergence of market discontinuity based on the autocovariance of the stocks in the Leading Temporal Module and on the ratio between the stocks’ correlations within this group and the correlations between these stocks and those outside the leading module. This indicator provides early-warning market signals useful for policy-makers and investors by mapping the evolution of the topological properties of the leading module in different points in time.

In this paper we consider a nonlinear model for the real economy described by a multiplier–accelerator setup. The model comprises the government sector, which influences the output dynamics by means of the fiscal policy, and the money market, where the money supply depends upon the fluctuations in the economic activity. Through rigorous analytical tools combined with numerical simulations, we investigate the stability conditions of the unique steady state and the emergence of different kinds of endogenous dynamics, which are the results of the action of the fiscal and the monetary policy through their reactivity degrees. Such policies, if properly tuned, can lead the economy toward the desired full employment target but, on the other hand, can also generate endogenous fluctuations in the pace of the economic activity, associated with the occurrence of closed invariant curves and multistability phenomena.

This paper considers a pure exchange overlapping generations model in which the money-growth rate is endogenous and follows a feedback rule. Different specifications for the monetary policy rule are analyzed, namely a so-called current, forward, or backward-looking feedback rule, depending on whether the monetary authority uses the actual, expected, or last observed values of the inflation rate to set the monetary policy. We study how the responsiveness of the policy rule with respect to inflation affects the determinacy of the monetary equilibrium. A policy rule is called aggressive (moderate) if it responds strongly (moderately) to inflation deviations from the target. We show how aggressive feedback rules, depending on the considered timing, can reinforce mechanisms that lead to indeterminacy or may lead the inflation rate to fluctuate around the monetary equilibrium at which monetary policy is aggressive. A leaning against the wind policy seems to be more desirable from an equilibrium determinacy point of view. On the contrary, a leaning with the wind policy could not be the recommended policy for the Central Bank.

In the learning-to-forecast laboratory experiments in Hommes et al. (2005), three different types of aggregate asset price behavior have been observed: monotonic convergence to the stable fundamental steady state, dampened price oscillations and permanent price oscillations. We present a simple behavioral 2-type heuristics switching model explaining individual as well as aggregate behavior in the experiment. Based on relative performance, agents switch between a simple trend following and an anchor and adjustment heuristic that differ in how much weight is given to the long run average price level. The nonlinear switching model exhibits path dependence through co-existence of a locally stable fundamental steady state and a stable (quasi-)periodic orbit, created via a so-called Chenciner bifurcation. Depending on initial states, agents coordinate individual expectations either on a stable fundamental steady state path or on almost self-fulfilling persistent price fluctuations around the fundamental steady state.

In this paper, we show how a rich variety of dynamics may arise in a simple multiplier–accelerator model when a nonlinearity is introduced in the investment function. A specific sigmoidal functional form is used to model investments with respect to the variation in national income, in order to bound the level of investments. In fact, due to obvious material constraints, business strategies cannot sustain infinite investments. With the aid of analytical and numerical tools, we investigate the stability conditions, bifurcations, as well as periodic and chaotic dynamics for different specifications of the model that may include or not an endogenous government expenditure. We obtain some comparative statics results that shed light on the stabilizing or destabilizing role of the various parameters in the model. Globally, we study multistability phenomena, i.e., the coexistence of different kinds of attractors with a rich and complex dynamic structure.

This paper proposes a new methodology based on non-negative matrix factorization to detect communities and to identify central nodes in a network as well as within communities. The method is specifically designed for directed weighted networks and, consequently, it has been applied to the interbank network derived from the e-MID interbank market. In an interbank network indeed links are directed, representing flows of funds between lenders and borrowers. Besides distinguishing between Systemically Important Borrowers and Lenders, the technique complements the detection of systemically important banks, revealing the community structure of the network, that proxies the most plausible areas of contagion of institutions’ distress.

We survey the theoretical findings and a set of applications in several different fields concerning two-dimensional maps characterized by a vanishing denominator. This family of maps, and the related dynamic properties, were originally brought to the attention of the researchers through their appearance in an economic application. Nowadays such maps have also been used in other research areas such as ecology or biology confirming a broader application of the theoretical results about plane maps with vanishing denominator to various research areas.

We investigate the dynamics of a cobweb type model with nonlinear demand and supply curves in which producers make forecasts on future prices with a backward looking expectation formation mechanism: the expected price for the next period is obtained by a weighted average of the prices observed in the last two periods. The study herewith presents aims at confirming the existence of a locally stabilising effect due to the presence of memory, but an increase of memory in price expectations can be globally qualitatively destabilising, in the sense that it leads to coexistence of different attractors with their respective basins of attraction.

In this paper we investigate the dynamics of a Cournot duopoly game with differentiated goods in which boundedly rational firms apply a gradient adjustment mechanism to update the quantity produced in each period. As in Ahmed et al. (2015), the demand functions are derived from an underlying CES utility function. The present analysis reveals that a higher degree of product differentiation may destabilize the Nash equilibrium. Through local analysis we provide conditions for the stability of the market equilibrium and through global analysis we investigate some bifurcations which cause qualitative changes in the structure of the attractors and of their basins as some parameters are allowed to vary. Since a higher degree of product differentiation tends to reduce competition and may generate undesirable fluctuations, an implication of our findings is that a stronger competition could help in stabilizing the unique Nash equilibrium.

This chapter describes some properties of the nonlinear dynamics emerging from two oligopoly models in discrete time. The target of this chapter is the investigation of some local and global bifurcations which are responsible for the changes in the qualitative behaviors of the trajectories of discrete dynamical systems.Two different kinds of oligopoly models are considered: the first one deals with the presence of differentiated goods and gradient adjustment mechanism, while the second considers the demand function of the producers to be dependent on advertising expenditures and adaptive adjustment of themoves. In bothmodels the standard local stability analysis of the Cournot-Nash equilibrium points is performed, as well as the global bifurcations of both attractors and (their) basins of attraction are investigated.

Contrary to the general belief, systemic risk does not only regard the risk posed by balance sheet relationships and interdependencies among institutions. It also features a temporal dimension related to the inappropriate responses of financial market participants to changes in risk over time. This paper proposes a method to simultaneously address the cross-sectional and the time dimension in which systemic risk materializes. The method is based on the TOPHITS algorithm. It provides three scores, namely borrowing, lending and time scores: the first two represent the systemic importance of the borrowing and the lending activity associated with each financial institution,while the third represents an empirical Early Warning Signal of the financial crisis. Our findings reveal that the identification of the time score as an indicator for an incoming market distress could be relevant to design macro prudential policies.

We present a cobweb model to explain price adjustment in a competitive market with homogeneous firms based on assumptions from Prospect Theory. Price changes are evaluated with respect to a psychological reference price which enters directly into the demand function. Accordingly, firms face a downward-sloping demand curve that is kinked at the consumers' reference price. Differently from the traditional cobweb model, the economy is described by a discontinuous map. Without assuming specic non-linearities and keeping the essential underlying mechanics of the model intact, we nd that the implementation of several features from Prospect Theory into our simple cobweb model may signicantly influence the market dynamics. Behavioral parameters play an important role for the market stability by reducing fluctuations and by directly affecting consumers' demand as well as production decisions.

In this paper, we analyze the dynamics of a monopoly model with constant elasticity in which the monopolist faces a form of bounded rationality due to limited accessibility to information. We assume the firm adopts a gradient mechanism to adjust the output level, and we investigate how the introduction of fixed and continuously distributed delays within the resulting continuous-time system may affect the long-run dynamics. We find that the stability of the equilibrium depends on the weighting function adopted to model continuously distributed delays, and the convergence of the realized output toward the steady state is crucially affected by the choice of the delay type which, in turn, reflects the availability and the weight assigned to information. Indeed, depending on the assumptions on modeling delays, the equilibrium point may undergo a Hopf bifurcation after which a limit cycle arises.

We develop a dynamic partial equilibrium model of the housing market, in which the dynamics of the house price is determined by the interaction between chartists and fundamentalists. The model endogenously generates episodes of boom and bust in the house price and can replicate the recent US house price dynamics, and points to endogenous and exogenous behavioral factors as the main determinants of such dynamics.

This article considers a Cournot duopoly model in a continuous-time framework and analyze its dynamic behavior when the competitors are heterogeneous in determining their output decision. Specifically the model is expressed in the form of differential equations with discrete delays. The stability conditions of the unique Nash equilibrium of the system are determined and the emergence of Hopf bifurcations is shown. Applying some recent mathematical techniques (stability switching curves) and performing numerical simulations, the paper confirms how different time delays affect the stability of the economy.

Resonances generate complicated bifurcation sequences. To design a picture of the bifurcation sequence occurring in the presence of a particular strong resonance, the two-parameters bifurcation of the equilibrium of a monopoly model with gradient adjustment mechanism and log-concave demand function is analyzed. Locally, the equilibrium may be destabilized through a period doubling or a supercritical Neimark–Sacker bifurcation. From a global perspective, it is also shown that the model undergoes strong resonances, such as the resonance 1:4, by using a continuation procedure. One of the interesting issue to tackle for this resonance is the investigation of the heteroclinic bifurcations which occur when pairs of saddles form connections near the 1:4 resonance. Therefore, the global analysis, through the description of the phase portraits and the basins of attractions, illustrates the theoretical features associated with the resonance and display interesting and complex dynamical behaviors, like the emergence of square and clover orbits.

The recent macroeconomic literature has been stressing the importance of considering heterogeneous expectations while addressing monetary policy. In this paper we consider a standard New Keynesian model, described by a two-dimensional nonlinear map, to analyze the bifurcation structure when agents have heterogeneous expectations and update their beliefs based on past performance. Depending on the degree of reactivity of the monetary policy to inflation and output deviations from the target equilibrium, different kind of dynamics may occur. We find that multiple equilibria and complicated dynamics, associated to codimension-2 bifurcations, may arise even if the monetary policy is set to respond more than point for point to inflation, as the Taylor principle prescribes. We show that if the monetary policy accommodates for a sufficient degree of output stabilization, complicated dynamics can be avoided and the number of coexisting equilibria reduces.

In this paper we analyze the topological properties of the network of the Euro Area banking market network, with the primary aim of assessing the importance of a bank in the financial system with respect to ownership and control of other credit institutions. The network displays power law distributions in both binary and weighted degree metrics indicating a robust yet fragile structure and a direct link between an increase of control diversification and a rise in the market power. Therefore while in good time the network is seemingly robust, in bad times many banks can simultaneously go into distress. This behavior paves the way for Central bank’s actions. In particular we investigate whether the Single Supervisory Mechanism introduced by the European Central Banks and based on banks’ total asset is a good proxy to quantify their systemic importance. Results indicate that not all the financial institutions with high value total assets are systemically important but only few of them. Moreover the network structure reveals that control is highly concentrated, with few important shareholders approximately controlling a separate subset of banks.

One of the main lessons of the recent financial crisis is that the network structure of the banking system has to be taken into account to assess systemic risk. In this chapter, we analyze the topological properties of the network of the Euro Area banking sector with the primary aim of assessing the importance of a bank in the financial system with respect to ownership and control of other credit institutions. The network displays power law distributions in both binary and weighted degree metrics indicating a robust yet fragile structure and a direct nexus between an increase of control diversification and a rise in the market power. Therefore, while in good time the network is seemingly robust, in bad times many banks can go into distress simultaneously. This behavior opens a narrow for Central bank's actions. In particular, we investigate whether the Single Supervisory Mechanism introduced by the European Central Banks and based on banks' total asset is a good proxy to quantify their systemic importance. Results indicate that not all the financial institutions with high value of total asset are systemically important but only few of them.

Il testo, nato dalle lezioni del corso di matematica per le applicazioni economiche e finanziarie, è destinato a studenti di corsi di laurea triennale e magistrale in economia interessati ad approfondire alcuni strumenti matematici utili nella trattazione di problemi economici e finanziari. Completano il manuale una raccolta di esercizi, alcune appendici e utili approfondimenti.

This article tackles the issue of local and global dynamics in a nonlinear duopoly with quantity setting (managerial) firms and horizontal product differentiation. It studies how the dynamics of a two-dimensional discrete time map evolves by focusing on changes either in the degree of product differentiation or the managerial power in the market share bonus. By combining mathematical techniques and numerical experiments, it shows that the Nash equilibrium of the game may not describe the long-term outcomes of the market. This holds because the fixed point of the map may be unstable or because different attractors (simple or chaotic) may capture the long-term dynamics of the model. The article also analyzes market dynamics when a new potential entrant tries to enter or, alternatively, a firm that was already in the market closes down and then tries to re-enter in a context where there is already an incumbent with a strictly positive quantity. The potential entrant may be subject to entry barriers or enters the market depending on the structure of consumers’ preferences and the demands of products of both varieties. In particular, the article analyzes some economic consequences of the non-invertibility of the map in the entry process.

The dynamics of a pure exchange overlapping generations model with endogenous money growth rule is investigated. We consider a nonlinear monetary policy rule which, in each period, bounds the money growth rate so that money is determined by the deviation of the inflation rate from its target. More precisely, we introduce such a mechanism through a sigmoidal money adjustment mechanism characterized by the presence of two asymptotes that bound the money variation, and thus the dynamics. It is shown that, depending on the timing of the monetary policy and the degree of reaction of the Central Bank, the target equilibrium may be destabilized via different types of bifurcations. Multistability and coexistence of attractors may also occur and the study of the basins of attraction allows us to analyze the global dynamic properties of the economy under scrutiny. We find that active monetary policy rules may be relevant for their stabilizing properties, but they also may open the door to equilibrium cycles of any periodicity and even chaos.

In the present paper, we investigate the dynamics of a model in which the real part of the economy, described within a multiplier-accelerator framework, interacts with a financial market with heterogeneous speculators, in order to study the channels through which the two sectors influence each other. Employing analytical and numerical tools, we investigate stability conditions as well as bifurcations and possible periodic, quasi-periodic, and chaotic dynamics, enlightening how the degree of market interaction, together with the accelerator parameter and the intervention of the fiscal authority, may affect the business cycle and the course of the financial market. In particular, we show that even if the steady state is locally stable, multistability phenomena can occur, with several and complex dynamic structures coexisting with the steady state. Finally, simulations reveal that the proposed model is able to explain several statistical properties and stylized facts observed in real financial markets, including persistent high volatility, fat-tailed return distributions, volatility clustering, and positive autocorrelation of absolute returns. Published by AIP Publishing.

The recent macroeconomic literature has been stressing the role of heterogeneous expectations in the formulation of monetary policy and recent laboratory experiments provided more evidence about this phenomenon. We use a simple model made up by the standard aggregate demand function, the New Keynesian Phillips curve and a Taylor rule to deal with di fferent issues, such as the stabilizing eff ect of diff erent monetary policies in a system populated by heterogeneous agents. The dynamic properties of the system depends crucially on the set of forecasting rules, on agent sensitivity in choosing the best predictor and on Central Bank's reaction to inflation. In particular we investigate whether the policy makers can sharpen macroeconomic stability in the presence of heterogeneous expectations about future in inflation and output gap and how this framework is able to reduce volatility and distortion in the whole system.