Thanks to aiQUANT for his post on the Hurst Exponent. A copy of the paper he references is available here.

I tried to reproduce the results in excel but ran into a roadblock, mostly in understanding the log transformation. I wasn’t able to access the link referenced in the paper, but aiQUANT mentioned that there is a file on the Mathworks website, so I will check that out.

[Update1]
Bear Cave, I think a site[article] maintained[penned] by the author of the referenced Hurst paper, has a page about the Hurst exponent and its calculation. That information is available here.

Qian’s site also has the MATLAB code for the calculation. You can access the MATLAB file here.

I redid some of the calculations in the excel file, and now my results have narrowed the gap when compared to those presented in the original paper. I think I am messing up on the log transformation of R/S. I will look at the MATLAB code and see… right now I think I have the exponent estimated at around .59.

AUTOREGRESSIVE CONDITIONAL DURATION: A NEW MODEL FOR IRREGULARLY SPACED TRANSACTION DATA - Engle and Russell

This paper proposes a new statistical model for the analysis of data which arrive at irregular intervals. The model treats the time between events as a stochastic process and proposes a new class of point processes with dependent arrival rates. The conditional intensity is developed and compared with other self-exciting processes. Because the model focuses on the expected duration between events, it is called the autoregressive conditional duration (ACD) model. Asymptotic properties of the quasi maximum likelihood estimator are developed as a corollary to ARCH model results. Strong evidence is provided for duration clustering for the financial transaction data analyzed; both deterministic time-of-day effects and stochastic effects are important. The model is applied to the arrival times of trades and therefore is a model of transaction volume, and also to the arrival of other events such as price changes. Models for the volatility of prices are estimated with price-based durations, and examined from a market microstructure point of view.

KEYWORDS:Irregularly spaced time series data, dependent point process, high frequency data.

Co-integration and Error Correction, Representation, Estimation, and Testing - Granger and Engle

Abstract:

The relationship between co-integration and error correction models, first suggested in Granger (1981), is here extended and used to develop estimation procedures, tests, and empirical examples.If each element of a vector of time series x, first achieves stationarity after differencing, but a linear combination of a’x, is already stationary, the time series x, are said to be co-integrated with co-integrating vector a. There may be several such co-integrating vectors so that a becomes a matrix. Interpreting a’x, = 0 as a long run equilibrium, co-integration implies that deviations from equilibrium are stationary, with finite variance, even though the series themselves are nonstationary and have infinite variance.

The paper presents a representation theorem based on Granger (1983), which connects the moving average, autoregressive, and error correction representation for co-integrated systems. A vector autoregression in differenced variables is incompatible with these representations. Estimation of these models is discussed and a simple but asymptotically efficient two-step estimator is proposed. Testing for co-integration combines the problems of unit root tests and tests with parameters unidentified under the null. Seven statistics are formulated and analyzed. The critical values of these statistics are calculated based on a Monte Carlo simulation. Using these critical values, the power properties of the tests are examined and one test procedure is recommended for application.

In a series of examples it is found that consumption and income are co-integrated, wages and prices are not, short and long interest rates are, and nominal GNP is co-integrated with M2, but not M1, M3, or aggregate liquid assets.

KEYWORDS: Co-integration, vector autoregression, unit roots, error correction, multi-variate time series, Dickey-Fuller tests.