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## ‘FRACTIONAL INTEGRATION’ IN FINANCE IS MITTAG-LEFFLER HYPOTHESIS

Take a bird’s eye view of financial econometrics. You can see a development of technical tools that begins with Box-Jenkins in 1976. They establish methods for time series analysis for univariate time series models and establish ARIMA models. The basic question is stationarity. An autoregressive model for an observed time series modeled by an autoregressive process $x_t = a_1 x_{t-1} + \cdots a_{12} x_{t-12}$ is stationary if the estimated coefficients all lie in the unit circle. If the observed series is stationary, it’s $I(0)$. If not, take differences and when the resulting differenced series is stationary, the original series is $I(1)$. This method is then generalized with some new issues in 1980. Christopher Sims et. al. established the multiple time series version of this analysis. The new issues are co-integration. So consider the vector of two variables $z_t = [x_t,y_t]$ so the analogue of the univariate autoregressive model is $z_t = A_1 z_{t-1} + \cdots + A_{12} z_{t-12}$ now with 2×2 matrices $A_1,\dots, A_{12}$ replacing scalar coefficients. when the residuals of a regression of $x_t$ on $y_t$ is $I(0)$ then $x_t,y_t$ are co-integrated. In the univariate cases, between $I(0)$ and $I(1)$ are fractional integration. The long memory revolve around these intermediate types of series. The analogous concept is fractional cointegration: two $I(d)$ series have regressionresiduals with smaller fractional differencing parameter than $d$.

For the central example in financial economics, which is equity volatility we have shown that the ‘long memory’ aspect is more concretely approached via a Mittag-Leffler function in the AR(12) parameters.  A natural hypothesis is that in fact this is the standard situation.  Thus it is natural to conjecture that these time series tools are being developed in vast generality where the underlying data have a concrete structure of Mittag-Leffler functions in the autoregressive parameters.  This will be my working hypothesis.

For a nice detailed presentation of vector autoregressive models, this looks very good:  Watson_VARs_handbook_94

In order to check universality, we choose a monthly macro series, the US inflation as well as daily volatility from two commodities gold and crude oil and show that the Mittag-Leffler structure for autoregressive structure of volatility holds in all cases.  Please excuse the labelling as these are AR(20) fits rather than AR(12) but the main discovery here is the Mittag-Leffler structure in the coefficients universal across instruments and frequency.  This is a fundamental feature of volatility-like objects in finance and economics that I expect to hold for all series where the ‘long memory’ or ‘fractional integration’ and so on occur.