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Register a new userSymbolic Dynamics in a Matching Labour Market Model
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In this paper we apply the techniques of symbolic dynamics to the analysis of a labor market which shows large volatility in employment flows. In a recent paper, Bhattacharya and Bunzel cite{BB} have found that the discrete time version of the Pissarides-Mortensen matching model can easily lead to chaotic dynamics under standard sets of parameter values. To conclude about the existence of chaotic dynamics in the numerical examples presented in the paper, the Li-Yorke theorem or the Mitra sufficient condition were applied which seems questionable because they may lead to misleading conclusions. Moreover, in a more recent version of the paper, Bhattacharya and Bunzel cite{BB1} present new results in which chaos is completely removed from the dynamics of the model. Our paper explores the matching model so interestingly developed by the authors with the following objectives in mind: (i) to show that chaotic dynamics may still be present in the model for standard parameter values; (ii) to clarify some open questions raised by the authors in cite{BB}, by providing a rigorous proof of the existence of chaotic dynamics in the model through the computation of topological entropy in a symbolic dynamics setting.
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We propose a general methodology to measure labour market dynamics, inspired by the search and matching framework, based on the estimate of the transition rates between labour market states. We show how to estimate instantaneous transition rates starting from discrete time observations provided in longitudinal datasets, allowing for any number of states. We illustrate the potential of such methodology using Italian labour market data. First, we decompose the unemployment rate fluctuations into inflow and outflow driven components; then, we evaluate the impact of the implementation of a labour market reform, which substantially changed the regulations of temporary contracts.
Visibility algorithms are a family of geometric and ordering criteria by which a real-valued time series of N data is mapped into a graph of N nodes. This graph has been shown to often inherit in its topology non-trivial properties of the series structure, and can thus be seen as a combinatorial representation of a dynamical system. Here we explore in some detail the relation between visibility graphs and symbolic dynamics. To do that, we consider the degree sequence of horizontal visibility graphs generated by the one-parameter logistic map, for a range of values of the parameter for which the map shows chaotic behaviour. Numerically, we observe that in the chaotic region the block entropies of these sequences systematically converge to the Lyapunov exponent of the system. Via Pesin identity, this in turn suggests that these block entropies are converging to the Kolmogorov- Sinai entropy of the map, which ultimately suggests that the algorithm is implicitly and adaptively constructing phase space partitions which might have the generating property. To give analytical insight, we explore the relation k(x), x in[0,1] that, for a given datum with value x, assigns in graph space a node with degree k. In the case of the out-degree sequence, such relation is indeed a piece-wise constant function. By making use of explicit methods and tools from symbolic dynamics we are able to analytically show that the algorithm indeed performs an effective partition of the phase space and that such partition is naturally expressed as a countable union of subintervals, where the endpoints of each subinterval are related to the fixed point structure of the iterates of the map and the subinterval enumeration is associated with particular ordering structures that we called motifs.
The symbolic dynamics technique is well-known for low-dimensional dynamical systems and chaotic maps, and lies at the roots of the thermodynamic formalism of dynamical systems. Here we show that this technique can also be successfully applied to time series generated by complex systems of much higher dimensionality. Our main example is the investigation of share price returns in a coarse-grained way. A nontrivial spectrum of Renyi entropies is found. We study how the spectrum depends on the time scale of returns, the sector of stocks considered, as well as the number of symbols used for the symbolic description. Overall our analysis confirms that in the symbol space transition probabilities of observed share price returns depend on the entire history of previous symbols, thus emphasizing the need for a modelling based on non-Markovian stochastic processes. Our method allows for quantitative comparisons of entirely different complex systems, for example the statistics of symbol sequences generated by share price returns using 4 symbols can be compared with that of genomic sequences.
We use symbolic dynamics to study discrete-time dynamical systems with multiple time delays. We exploit the concept of avoiding sets, which arise from specific non-generating partitions of the phase space and restrict the occurrence of certain symbol sequences related to the characteristics of the dynamics. In particular, we show that the resulting forbidden sequences are closely related to the time delays in the system. We present two applications to coupled map lattices, namely (1) detecting synchronization and (2) determining unknown values of the transmission delays in networks with possibly directed and weighted connections and measurement noise. The method is applicable to multi-dimensional as well as set-valued maps, and to networks with time-varying delays and connection structure.
The extent to which a matching engine can cloud the modelling of underlying order submission and management processes in a financial market remains an unanswered concern with regards to market models. Here we consider a 10-variate Hawkes process with simple rules to simulate common order types which are submitted to a matching engine. Hawkes processes can be used to model the time and order of events, and how these events relate to each other. However, they provide a freedom with regards to implementation mechanics relating to the prices and volumes of injected orders. This allows us to consider a reference Hawkes model and two additional models which have rules that change the behaviour of limit orders. The resulting trade and quote data from the simulations are then calibrated and compared with the original order generating process to determine the extent with which implementation rules can distort model parameters. Evidence from validation and hypothesis tests suggest that the true model specification can be significantly distorted by market mechanics, and that practical considerations not directly due to model specification can be important with regards to model identification within an inherently asynchronous trading environment.
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