Page 162 - Special Topic Session (STS) - Volume 4

STS577 Md. Sabiruzzaman et al.
not only the time interval which makes a difference in risk estimation, but also
the sampling rule employed to construct a particular time series. As for
example, in constructing a monthly time series from daily data, the last
business day of each month might be accepted as a representative of that
month. However, there is no reason why the day before the last business day
of each month should be a representative day of each month should be a
representative day or two business days before the last business day and so
on. Wavelet analysis works as a magic tool to analyze the data at different time
horizon. The wavelet analysis decomposes the time series at the highest
possible frequency into different time scale. Hence it provides a natural
platform to investigate the risk at different time scale without losing any time
point. Therefor it gives benefit to the researcher to analyze the time series at
different time horizon at a time without giving much effort.
The maximum overlap discrete wavelet transformation (MODWT), a
modified version of discrete wavelet transformation (DWT), is applicable to
both dyadic and non-dyadic time series, capable of preserving information of
the original signal, shift invariant and more efficient than DWT for variance
analysis (Gencay et al., 2002). Applications of MODWT for time scale
decomposition of time series are found in number of recent literature (Al.Wadi
et al., 2013; Alves et al., 2014; Gallegati et al., 2014; Reboredo et al., 2014). For
more on MODWT, we refer Percival and Walden (2000).
This study proposes a new algorithm for volatility prediction containing
multi-scale information and illustrates with weekly index of Dhaka Stock
Exchange (DSEX). Multiresolution analysis with MODWT is performed to
obtain variability at different time scale corresponding to different level of
investor and estimate risk at different scale of time using GARCH model. Time
scale variations are then incorporated in volatility prediction of the original
series through wavelet reconstruction.

2. Volatility prediction with Wavelet-GARCH Approach
In this study, we proposed wavelet-GARCH approach to predict volatility
and compare it with econometric approach. The prediction scheme capture
time scale variation at different levels from the wavelet domain instead of just
applying a forecasting algorithm directly on the raw data as many econometric
models do. Thus, having information of multiple scales and using an adequate
model for financial time series, the prediction accuracy is improved. In our
proposed approach, we first decompose the return series using MODWT
decomposed to obtain approximations and detail coefficients. While
approximations represent the location, detail coefficients represent variability
of the series at different scales. For example, detail at level j, dj represents the
variation at scale 2j. The detail coefficients are, therefore, modelled with an
appropriate GARCH equation to estimate volatility at different scales. Detail
coefficients at each level are then replaced with estimated GARCH volatility

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