Y.M. Kaniovski

Modeling Industrial Dynamics with Innovative Entrants

The paper analyzes some generic features of industrial dynamics whereby innovative change is carried, stochastically, by new entrants. Relying on the formal representation suggested in Winter et al. (1997), it studies both the asymptotic properties of such processes and their appropriability to account for a few empirical stylized facts, including persistent entry and exit, skewed size distributions and turbulence in market shares.

On "badly behaved" dynamics (Some applications of generalized urn schemes to technological and economic change)

Adaptive (path dependent) processes of growth modeled by urn schemes are important for several fields of applications: biology, physics, chemistry, economics. In this paper we present a general introduction to urn schemes, together with some new results. We review the studies that have been done in the technological dynamics by means of such schemes. Also several other domains of economic dynamics are analysed by the same machinery and its new modifications allowing to tackle non-homogeneity of the phase space. We demonstrate the phenomena of multiple equilibria, different vonvergence rates for different limit patterns, locally positive and locally negative feedbacks, limit behavior associated with non-homogeneity of economic environment where producers (firms) are operating. It is also shown that the above urn processes represent a natural and convenient stochastic replicator dynamics which can be used in evolutionary games.

Generalized urn schemes and technological dynamics

Abstract Adaptive (path dependent) processes of growth modeled through urn schemes find important applications to economic dynamics (and also to other disciplines, such as biology, physics, chemistry). The paper presents some further properties of generalized urn schemes and studies dynamic stochastic processes characterized by both positive and, possibly, negative feedbacks of a functional form as ‘badly behaved’ as possible. Two applicantions to technological diffusion are considered. One of the models tackles the case when there is a separation within the pool of adopters which can be interpreted as the outcome of adaptive learning on the features of the new technologies by imperfectly informed agents. Other examples deal with dependence of final market shares of two technologies on the pricing policies of the firms which produce them. The stochasticity of the processes is caused by some mixed strategies used by the adopters or/and imperfectness of the information which they possess.

Price Expectations and Cobwebs Under Uncertainty

There is given a market for several perishable goods, supplied under technological randomness and price uncertainty. We study whether and how producers eventually may learn rational price expectations. The model is of cobweb type. Its dynamics fit standard forms of stochastic approximation with either variable or constant stepsizes. Relying upon quite weak and natural assumptions we prove new convergence results.

Limit theorems for stationary distributions of birth-and-death processes

Birth-and-death processes or, equivalently, finite Markov chains with three-diagonal transition matrices proved to be adequate models for processes in physics [12], biology [4,5], sociology [13] and economics [1,3,10]. The analysis in this case quite often relies on the stationary distribution of the chain. Representing it as a Gibbs distribution, we study its limit behavior as the number of states increases. We show that the limit nests on the set of global minima of the limit Gibbs potential. If the set consists of a finite number k of singletons α i where the second derivatives α i of the potential are positive, the limit distribution assigns probability 1 / √α i Σ k j ) =1 1/√α j to α i , When at some points the second derivative is zero, the limit distribution nests only on them, we describe it explicitly. If the set of minima consists of a finite number of singletons and intervals, the limit distribution concentrates onlv on intervals. We obtain a formula for it.

Non-standard limit theorems for urn models and stochastic approximation procedures

The adaptive processes of growth modeled by a generalized urn scheme have proved to be an efficient tool for the analysis of complex phenomena in economics, biology, and physical chemistry. They demonstrate non-ergodic limit behavior with multiple limit states. There are two major sources of complex feedbacks governing these processes: non-linearity (even local which is caused by non-differentiability of the functions driving them) and multiplicity of limit states stipulated by the non-linearity. The authors suggest an analytical approach for studying some of the patterns of complex limit behavior. The approach is based on conditional limit theorems. The corresponding limits are, in general, not infinitely divisible. They show that convergence rates could vary for different limit states. The rates depend upon the smoothness (in neighborhoods of the limit states) of the functions governing the processes. Since the mathematical machinery allows us to treat a quite general class of recursive stochastic discrete-time processes, we also derive corresponding limit theorems for stochastic approximation procedures. The theorems yield new insight into the limit behavior of stochastic approximation procedures in the case of non-differentiable regression functions with multiple roots.

Traces of business cycles in credit-rating migrations

Using migration data of a rating agency, this paper attempts to quantify the impact of macroeconomic conditions on credit-rating migrations. The migrations are modeled as a coupled Markov chain, where the macroeconomic factors are represented by unobserved tendency variables. In the simplest case, these binary random variables are static and credit-class-specific. A generalization treats tendency variables evolving as a time-homogeneous Markov chain. A more detailed analysis assumes a tendency variable for every combination of a credit class and an industry. The models are tested on a Standard and Poor’s (S&P’s) dataset. Parameters are estimated by the maximum likelihood method. According to the estimates, the investment-grade financial institutions evolve independently of the rest of the economy represented by the data. This might be an evidence of implicit too-big-to-fail bail-out guarantee policies of the regulatory authorities.

Identification of hidden Markov chains governing dependent credit-rating migrations

ABSTRACT Three models of dependent credit-rating migrations are considered. Each of them entails a coupling scheme and a discrete-time Markovian macroeconomic dynamics. Every credit-rating migration is modeled as a mixture of an idiosyncratic and a common component. The larger is the pool of debtors affected by the same common component, the stronger is the dependence among migrations. The distribution of the common component depends on macroeconomic conditions. At every time instant, the resulting allocation of debtors to credit classes and industries follows a mixture of multinomial distributions. Dealing with M non default credit classes, there are 2M theoretically possible macroeconomic outcomes. Only few of them occur with a positive probability. Restricting the macroeconomic dynamics to such outcomes simplifies estimation. A heuristics for identifying them is suggested. Using the maximum likelihood method, it was tested on a Standard and Poors (S&Ps) data set.

Modeling of Dependent Credit Rating Transitions Governed by Industry-Specific Markovian Matrices

Two coupling schemes where probabilities of credit rating migrations vary across industry sectors are introduced. Favorable and adverse macroeconomic factors, encoded as values 1 and 0, of credit class- and industry-specific unobserved tendency variables, modify the transition probabilities rendering individual evolutions dependent. Unlike in the known coupling schemes, expansion in some industry sectors and credit classes coexists with shrinkage in the rest. The schemes are tested on Standard and Poor’s data. Maximum likelihood estimators and MATLAB optimization software were used.