https://www.nber.org/system/files/working_papers/t0264/t0264.pdf

Long memory and regime switching

The theoretical and empirical econometric literatures on long memory and regime switching have evolved largely independently, as the phenomena appear distinct. We argue, in contrast, that they are intimately related, and we substantiate our claim in several environments, including a simple mixture model, Engle and Lee's (1999) stochastic permanent break model, and Hamilton's (1989) Markov switching model. In particular, we show analytically that stochastic regime switching is easily confused with long memory, even asymptotically, so long as only a small' amount of regime switching occurs, in a sense that we make precise. A Monte Carlo analysis supports the relevance of the theory and produces additional insights.

Long Memory and Regime Switching

The paper gives some results and improves the derivation of the fractional Taylor's series of nondifferentiable functions obtained recently in the form f (@g + h) = E"@a (h^@aD"@g^@a)f(@g), 0 @a @? 1, where E"@a is the Mittag-Leffier function. Here, one defines fractional derivative as the limit of fractional difference, and by this way one can circumvent the problem which arises with the definition of the fractional derivative of constant using Riemann-Liouville definition. As a result, a modified Riemann-Liouville definition is proposed, which is fully consistent with the fractional difference definition and avoids any reference to the derivative of order greater than the considered one's. In order to support this F-Taylor series, one shows how its first term can be obtained directly in the form of a mean value formula. The fractional derivative of the Dirac delta function is obtained together with the fractional Taylor's series of multivariate functions. The relation with irreversibility of time and symmetry breaking is exhibited, and to some extent, this F-Taylor's series generalizes the fractional mean value formula obtained a few years ago by Kolwantar.

Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results

A simple construction that will be referred to as an error-duration model is shown to generate fractional integration and long memory. An error-duration representation also exists for many familiar ARMA models, making error duration an alternative to autoregression for explaining dynamic persistence in economic variables. The results lead to a straightforward procedure for simulating fractional integration and establish a connection between fractional integration and common notions of structural change. Two examples show how the error-duration model could account for fractional integration in aggregate employment and in asset price volatility.

/pdf/what-is-fractional-integration-4ijs7zf78t.pdf

What is fractional integration

By using the new fractional Taylor’s series of fractional order f ( x + h ) = E α ( h α D x α ) f ( x ) where E α ( . ) denotes the Mittag–Leffler function, and D x α is the so-called modified Riemann–Liouville fractional derivative which we introduced recently to remove the effects of the non-zero initial value of the function under consideration, one can meaningfully consider a modeling of fractional stochastic differential equations as a fractional dynamics driven by a (usual) Gaussian white noise. One can then derive two new families of fractional Black–Scholes equations, and one shows how one can obtain their solutions. Merton’s optimal portfolio is once more considered and some new results are contributed, with respect to the modeling on one hand, and to the solution on the other hand. Finally, one makes some proposals to introduce real data and virtual data in the basic equation of stock exchange dynamics.

Derivation and solutions of some fractional Black–Scholes equations in coarse-grained space and time. Application to Merton’s optimal portfolio

A class of micropulses and antipersistent fractional brownian motion

Alternative micropulses and fractional Brownian motion

Necessary conditions for the existence of conditional moments of stable random variables

How do conditional moments of stable vectors depend on the spectral measure

A class of a-stable, 0 < a < 2, processes is obtained as a sum of 'up-and-down' pulses determined by an appropriate Poisson random measure. Processes are H-self-affine (also frequently called 'self-similar') with H < 1/cr and have stationary increments. Their two-dimensional dependence structure resembles that of the fractional Brownian motion (for H < 1/2), but their sample paths are highly irregular (nowhere bounded with probability 1). Generalizations using different shapes of pulses are also discussed.

/pdf/stable-fractal-sums-of-pulses-the-cylindrical-case-3cmqkqvywe.pdf

Renata Cioczek-Georges

Papers

A class of micropulses and antipersistent fractional brownian motion

Alternative micropulses and fractional Brownian motion

Necessary conditions for the existence of conditional moments of stable random variables

How do conditional moments of stable vectors depend on the spectral measure

Stable fractal sums of pulses: the cylindrical case