An unbiased autoregressive conditional intraday seasonal variance filtering process

Abstract

We develop a new autoregressive conditional seasonal variance (ARCSV) process that captures both the changes in and the persistency of the intraday seasonal (U-shape) pattern of volatility. Unlike other procedures for seasonality, this approach allows for the intraday volatility pattern to change over time, resulting in an increase in the filtering performance over the extant deterministic filtering models. We quantify the gains in the filtering performance by comparing our model with the flexible Fourier form (FFF) model of Andersen and Bollerslev [J. Empir. Finance, 1997a, 4, 115–158]. Moreover, the ARCSV model does not create any statistical distortion in the filtered series, as occurs with other de-seasoning processes. We prove that the ARCSV model satisfies the spectral criteria required to be judged as a good filtering process. Monte Carlo simulation results show that the performance of the ARCSV model is superior to the FFF model. In particular, the seasonal adjustment performance of the ARCSV model is robust under the condition that the innovation of the underlying seasonal variance process is large and the daily non-seasonal variance process is misspecified.

Keywords:

• Seasonality
• Seasonal variance
• Filtering
• U-shape
• Seasonal adjustment

Acknowledgements

We would like to thank Paul Pfleiderer for his encouragement and Tom Barkley, Zhiyao Chen, and Zhiguang Wang for suggestions on earlier versions. An earlier draft of this paper was presented at the 2008 Financial Management Meetings in Dallas.

Notes

†Bollerslev and Ghysels (1996) developed a new ARCH model, namely Periodic GARCH or PGARCH, to adjust the conditional volatility for the periodic volatility component. For simplicity, PGARCH(1,1) is as follows: and , where is the cumulative intraday index, n is the intraday index (or stage), and n = 1, … , N, with N being the number of intraday intervals in a day. Dummy variables and sinusoids can be used for the seasonal period specific coefficients (Martens et al. 2002). The PGARCH model is efficient in describing conditional heteroskedasticity when the seasonal volatility component is present together with the non-seasonal volatility component. However, unlike the formulation for the conditional volatility in Andersen and Bollerslev (1997a), the volatility formulation in PGARCH is unable to split the periodic (seasonal) volatility component from the non-periodic (non-seasonal) volatility component. Hence, in this study, the PGARCH model is not considered as a process that filters out the seasonality in volatility. Martens et al. (2002) provide empirical results that the PGARCH model is more efficient in forecasting intraday volatility than is the FFF model when intraday seasonal volatility is present. Note that Martens et al. (2002) do not compare the PGARCH with the FFF as filtering models.

‡Hughes et al. (2008) employ the Component ARCH(1,1) model (developed by Engle and Lee 1993) to examine the U-shaped volatility of T-bills. In contrast to the ARCSV model, the component ARCH model cannot be used as a ‘filtering’ procedure. The component ARCH model works to examine the periodicity in volatility as follows. If the coefficient of a weekday dummy is large, then there is a periodic high volatility on that weekday. Similarly, if the coefficient of a weekday dummy is small, then there is a periodic low volatility on that weekday. However, the component ARCH model cannot be used to filter the periodic volatility component out of the volatility series because the permanent component and the periodic component are not separately estimated in the permanent volatility equation. Put differently, the combined volatility of the periodic and the permanent volatility components is estimated in the permanent volatility equation. Consequently, a filtering process cannot be applied in this situation.

§For a discussion of the stochastic component of the seasonal mean and seasonal variance, see Granger (1976), Hillmer and Tiao (1982), Bell and Hillmer (1984), Hylleberg (1986), Andersen and Bollerslev (1997a), Beltratti and Morana (2001), and Omrane and Bodt (2007).

¶This FFF model in Andersen and Bollerslev (1997a) induces statistical noise into the S&P 500 futures after the intraday volatility is deseasonalized. These results occur because the FFF model used to filter the seasonal volatility includes the interaction terms between the interday volatility and the sinusoid terms. Since the interday volatility is not a part of the intraday seasonal component, the seasonal volatility determined by the FFF procedure includes a part of the non-seasonal component. Consequently, the seasonally filtered returns will contain noise from the FFF filtering procedure (see Andersen and Bollerselv 1997a, p. 148 for the filtering results of the S&P 500 futures contract).

∥Beltratti and Morana (2001, p. 208) note “We model c(i,t,n) [the stochastic variance component] the fundamental daily frequency, as stochastic while its harmonics are modeled as deterministic as for Andersen and Bollerslev (1997a).”

⊥See figure 3 of Omrane and Bodt (2007). The autocorrelation functions of the volatility deseasonalized by the neural network method still exhibit periodic fluctuations.

†The multiplicative model of the variance series is . Taking the logarithm of both sides will result in the equivalent additive form.

‡If a time series X _τ has a seasonal component on the basis of period K, for , then the mean of the series can be approximated by a trigonometric series:

where , and (see equation (2.1) of Nerlove (1964)). Note that and . In the preceding trigonometric series the set of the index for the seasonal frequencies is . Let the distinct seasonal frequencies ω _S be with as defined in definition 1. The approximated trigonometric series can also be represented by a complex form, on the distinct frequencies where . In the complex representation of the approximated trigonometric series the set of the index for the distinct seasonal frequencies () is also because the complex form can be expressed by (Fuller 1995, p. 131).

§Note that a one day lag is identical to K intraday lags, for example , , etc. An example of interpreting and is to view the seasonal variance at the intraday interval (n) = 9 a.m. today (time t) and 9 a.m. yesterday (time t−1).

¶This definition is also given by Nerlove (1964) and Granger (1976).

†See also Jones and Brelsford (1967), Pagano (1978), and Vecchia (1985a, b) regarding PARMA models.

‡Since the focus of this study concerns the development of an intraday seasonal adjustment procedure, this study does not explicitly specify the daily variance process.

§Providing that the mean model in a two-step estimation is correctly specified, as Engle and Sheppard (2001) point out, the standard errors of the variance filtering model are not affected by the parameters of the mean model because the expected cross partial derivatives of the log-likelihood function with respect to the mean and the seasonal variance parameters are zero when using the normal likelihood procedure (see also Greene 2000, pp. 108 and 131). A maximum likelihood estimation procedure is employed to determine the parameters in (4). See also Engle and Sheppard (2001), Engle (2002), and Engle et al. (2006) for the application of the two-step estimation and the zero-mean specification for their variance models.

¶It is interesting that, in the FFF model of Anderson and Bollerslev (1997a), the adjustment for the seasonal variance component is not the same seasonal adjustment procedure as defined in definition 4. See the proof in appendix C.

†It is worth mentioning that the seasonal variances () in (7) are not known values, and therefore the coefficients (, and ) are not filters that will determine a signal. Rather, the seasonal variances and the coefficients are unknown values and are determined simultaneously. Therefore, there is not an issue of a non-zero phase with regard to the determination of the seasonal variance by the ARCSV process in (7). The phase is measured between realized values of a variable and its signal determined by filters using the realized values. Furthermore, in this study the phase is measured between the original variances and the seasonally adjusted variances. For the same reason, there is no issue in applying the coherence to the original variances and the seasonally adjusted variances determined by the ARCSV model. As mentioned previously, theorem 1 shows that the phase between the original variances and the seasonally adjusted variances determined by the ARCSV process is zero at all frequencies, and the coherence equals one at the non-seasonal frequencies and is less than one at the seasonal frequencies. See Grether and Nerlove (1970), Granger (1976), Bell and Hillmer (1984) and Hylleberg (1986), among others, for a discussion of issues on applying the spectral evaluation criteria to seasonal adjustment procedures.

‡Also see appendix E for covariance stationarity of the residual process ().

†The subscript tau (τ) is the cumulative intraday index, defined as for and . We use the smoothed periodograms as estimators of the spectral densities of and , where the unsmoothed periodogram is defined as the variance of the time series at a specific frequency. The unsmoothed periodogram is not a consistent estimator because its variance does not approach zero as the sample size grows. To overcome the aforementioned inconsistency problem with the unsmoothed periodogram, smoothing is applied to the periodogram by employing a weighted average of the unsmoothed periodograms at neighboring frequencies. The number of neighboring frequencies to be included is determined by the width of the window, which is called the bandwidth. The bandwidth needs to be large enough to ensure the consistency of the spectral density and other cross-spectral quantities, such as the coherence and the phase spectrum. However, widening the bandwidth too much causes a distortion of the spectral density at neighboring frequencies within the bandwidth, creating a ‘leakage’ problem. If there are very large variances at some frequencies, then the large variances can be leaked into neighboring frequencies through the weighted average. Therefore, widening the bandwidth creates a trade-off between consistency gain and leakage. The determination of the bandwidth depends on the researcher's judgment. In this study, spectral density refers to the smoothed periodogram. Here a rectangular weighting scheme with a bandwidth of 11 is applied to the seasonally unfiltered and filtered series. There was no difference in the results from using other bandwidths.

†The daily variance is not estimated properly in the Mote Carlo simulation in this study when the underlying daily variance series is generated by the NARCH(1,1) with a gamma of 0.5, while the daily variance is estimated with the GARCH(1,1) process.

‡The estimated spectral density in figure 2 and the squared coherence in figure 3 below period 10 are removed to enhance the visibility of the results.