Multiscale GARCH Modeling and Inference

Abstract

The motivation behind this thesis is the shortage of formal statistical inference methods in the literature for testing whether a time series model is consistent with a sample at multiple sampling frequencies simultaneously. Most existing statistical methods for time series data focuses on a particular frequency of sample. However, in the statistical modeling of financial time series and applications, having a modeling being consistent with data at multiple frequencies can provide better interpretation of the underlying phenomenon and provide convenience in practical applications.

Mantegna and Stanley (1995, 1996) and Ghashghaie, et.al.(1996)are among the pioneers in pointing out the distinctive scaling behavior in financial asset return distributions. Mandelbrot, et.al. (1997) explicitly pointed out the need to look at financial time series at multiple frequencies and use the scaling property of the data to help identify a model. Engle and Patton (2001) raised the question of whether a GARCH(1,1) model, acceptable for modeling return volatility at each single time scale from 1-day to 1-week, is consistent across scales.

It is the purpose of this thesis to propose formal statistical inference methods for testing whether a given time series of ARMA and GARCH type is consistent with a sample at multiple frequencies simultaneously. To do so, we first examine the problem of model temporal aggregation. Then, based on temporal aggregation relations, we propose a novel statistical inference methods based on empirical likelihood with estimating equations. The proposed method can be used to formally test hypotheses of the following types: (i) whether a model with a fixed set of parameter values is consistent with sample at multiple frequencies; (ii) whether the model itself is capable of being consistent with the sample at multiple frequencies. Some related problems on GARCH model parameterization and parameter estimation with temporally aggregated data are also addressed.