Estimation and Moment Recursion Relations for Multimodal Distributions of the Exponential Family

Catastrophe theory and the financial crisis

“…This equation can be estimated using Cobb's (1981) method of moments estimator for multimodal density function. We use the maximum likelihood approach developed by Cobb and Watson (1980) and Cobb et al (1983) and later extended by Oliva et al (1987) using subspace fitting methods which is freely available in the R package "cusp".…”

A MODEL OF THE DISTRIBUTION OF PRICES1

“…Integrating both sides with respect to p gives a first-order linear differential equation: k=f+ (f*da2/dp+ a2df/dp) where k is a constant of integration. Solving for d/dp gives: d2k 24u+da2/dp dpa2 a2 Finally, the stationary density is given as the solution of this differential equation: (11) where i is the normalising constant which is determined by the boundary conditions of (6) and is given by: 1 logJ1 r" 2M(s)+da2(s)/ds\ 1 L Jo a(s) ) ds] dp (12) = exp This density represents a generalized exponential distribution which has been studied by Cobb, Koppstein and Chen (1983); Cobb and Zacks (1985); 71 (10) Martin (1990); Lye and Martin (1993a), and applied by Fischer and Jammernegg (1986) to the analysis of the Phillips curve.…”

“…Secondly, the generalized gamma distribution is non-standard, so the usual econometric software is no longer appropriate. However, Cobb, Koppstein and Chen (1983)…”

“…This paper shows how the distribution of prices can be derived explicitly from an economic model. The approach generates a generalized gamma distribution which belongs to the generalized exponential family (see Cobb, Koppstein and Chen, 1983;Cobb and Zacks, 1985;and Lye and Martin, 1993a). This distribution is extremely flexible as it contains the gamma, Rayleigh, and exponential distribution as special cases, and is related to the generalized gamma distribution studied by McDonald (1984).…”

Threshold Time Series Models as Multimodal Distribution Jump Processes

“…The relationship between the SETAR and MATS models is further highlighted when the distributional properties of the two models are investigated. First, the MATS model is related to stochastic catastrophe theory (see Cobb et al, 1983). Tong (1983) has also shown how the SETAR model is related to catastrophe theory via the theory of discontinuous decision processes.…”