The data sample used in this chapter is the standard equity futures contracts for the S&P 500 traded in the after-hours electronic market on the GLOBEX
7. ACD Modelling with Structural Breaks (Crisis Period) 129
Figure 7.2: Time Series Plot of Adjusted Durations for S&P 500 During Crisis Period
exchange from beginning of October 2006 to the end of December 20082. The
sample period is selected to include the global …nancial crisis of 2007 to end of 2008. The duration data in this chapter are …rst diurnally adjusted using a spline to remove the daily pattern as in Chapter 6. The adjusted duration time series data for the S&P 500 have 1,349,514 observations over the sample period. A time series plot of the adjusted durations is shown in Figure 7.2.
Apart from duration, volume is also considered in the model as an additional mark. Volume is also diurnally adjusted to remove the daily seasonality. All the transactions occurring at the same time are aggregated to avoid zero durations. The diurnal patterns for duration and volume are shown in panel (a) and (b) in Figure 7.3, where the horizontal axis represents the time of a day in the form of number of seconds from mid-night, and vertical axis is the adjusted diurnal series.
The duration and volume diurnal patterns during pre-crisis and crisis period
2The data sample in this chapter is a continuation of the data (July 2004-September 2006)
7. ACD Modelling with Structural Breaks (Crisis Period) 130
Figure 7.3: Diurnal (a) Duration and (b) Volume Pattern for S&P 500 During Crisis Period
are shown in Figure 7.3 and 7.43. The diurnal patterns for the crisis period
have clearly changed from the pre-crisis data. The di¤erences are clear for both adjusted duration and volume between mid-night and 8:30 am. Compared to the 2004-2006, the adjusted duration during crisis period drops and volume rises more quickly in the early morning before 8:30 am. This indicates that the trading activities were more intensive, and investors acted more quickly according to their available information during the morning period, which is consistent with Dungey, Fakhrutdinova, and Goodhart (2009). The quickly rising diurnal adjusted volume is consistent with this view.
The range of diurnal adjusted duration is between 10 and 80 seconds during the crisis period, compared to from 10 to 170 seconds before the crisis period,
7. ACD Modelling with Structural Breaks (Crisis Period) 131
indicating the average level of waiting time during the crisis is much shorter. However, the range of diurnal adjusted volume has actually decreased during the crisis. This might due to less clustering of trading during crisis period. In other words, compared with non-crisis period, the trading activities during the crisis period are more evenly distributed, but with shorter waiting time. Clustering behaviour is also discussed later in this chapter in association with changes in ACD model parameters.
From the comparison between Figure 7.3 and 7.4, it is clear that the adjusted volume pattern has shifted downwards overall during the crisis period. As the time period between mid-night and 8:30 am mostly corresponds trading activities from Europe, it is a good indication that worldwide investors trading behaviour has changed due to the global …nancial crisis.
Figure 7.4: Diurnal (a) Duration and (b) Volume Pattern for S&P 500 During 2004 to 2006 Period.
7. ACD Modelling with Structural Breaks (Crisis Period) 132
The LM-based tests of Andrews (1993) and Andrews and Ploberger (1994) are adopted for detecting structural breaks in the ACD model of this chapter. The LM-based tests treat the multiple change points detection as an extension of the single change point problem as described in chapter 6. The sample data are …rst …tted with a Weibull ACD (1, 1) model. The Weibull ACD (1, 1) model used in this chapter follows:
8 > > < > > : Xi = i"i i =!0+Pmj=0 jxi j +Pqj=0!j i j +vivol; "i i:i:d: (7.1)
where !; j and !k are parameters, and p and q represent the lag orders. The
adjusted volume is added as an additional mark in the model, with a parameterv1. Manganelli (2005) introduced a vector autoregression approach to model duration and volume simultaneously, producing greater feedback on volatility. However, this thesis does not pursue this approach as structural e¤ects during GFC period is our main focus. The Andrews and Ploberger (AP) tests are then applied to the derivative series at the maximizing ACD parameters. The date which generates the largest AP LM statistics is recognized as the most signi…cant break. Two separate Weibull ACD (1,1) models are again applied on the two sub-periods divided by the break as before. The new ACD model parameter derivative series are then used for succeeding round of the AP test. The same process continues until no further break points are located. Details of the Andrews and Ploberger LM-based tests have been covered in chapter 6, and the methodology of Weibull ACD model is also explained in chapter 4, and are therefore not repeated here.
7. ACD Modelling with Structural Breaks (Crisis Period) 133