Long-memory in high-frequency exchange rate volatility under temporal aggregation

Abstract

This paper applies log-periodogram estimators of the fractional difference parameter to the volatility of the US dollar exchange rate returns of 11 European currencies, and under temporal aggregation from an underlying half-hourly intra-day frequency. Particular attention is paid to the sequencing of the nonlinear transformation of returns and their temporal aggregation. The results reported confirm that long-memory in absolute returns constitutes an intrinsic and empirically significant characteristic of the exchange rates considered. At the practical level, our results lend support to the proposal that nonlinear transformation prior to temporal aggregation can return meaningful long-memory parameter estimates. Our findings also illustrate the advantages of long-memory parameter estimation based on the smoothed periodogram applied to absolute returns in controlling for noise induced by temporal aggregation in the processing of high-frequency data.

Keywords:

• Market efficiency
• Behavioural finance
• Empirical asset pricing
• Mutual funds

Notes

†Relatedly, the availability of high-frequency financial data has also afforded the opportunity of reappraising the forecast performance of volatility models. It has been demonstrated that the more frequent sampling of intra-day observations significantly reduces the inherent noise in traditional measures of true volatility based on squared returns, consequently reducing the attendant measurement error uncertainty in the evaluation of volatility forecasts. That is, through the approximation of realized volatility in continuous time using ‘integrated volatility’, constructed as the cumulative sum of squared intraday returns (Andersen and Bollerslev 1998, Andersen et al. 1999, 2003).

‡Further alternatives include the long memory ARCH model (Robinson 1991, Ding and Granger 1996), the heterogeneous arch (HARCH) model (Dacorogna et al. 1998), and the long memory nonlinear moving average model (Robinson and Zaffaroni 1996, Zaffaroni 2003). Also, novelly, a new family of long memory processes which accommodate long memory in the volatility correlation by measuring historical volatilities over a set of increasing time horizons and computing the resulting effective volatility using a sum with power law weights (Zumbach 2004).

§More formally, a parametric model is closed under temporal aggregation if a model from the same parametric class but with differing parameter values provides an appropriate characterization of the underlying data generating process at all observation frequencies. Models in the SV, FIGARCH and LMSV classes do not generally possess this property, and GARCH models are closed under temporal aggregation only under the restrictive conditions established by Drost and Nijman (1993) and Drost and Werker (1996).

¶For investigations of the high-frequency intra-day DM–USD rate (sample periods after colons), see Andersen and Bollerslev (1997a, b: 1/10/1992–29/9/1993), Andersen et al. (1999: 1/10/1987–30/9/1992) and Andersen et al . (2003: 1/12/1986–30/6/1999). For investigations of the high-frequency intra-day JPY–USD rate, see Bollerslev and Wright (2000: 1/12/1986–1/12/1996) and Andersen et al . (2003: 1/12/1986–30/6/1999).

†For more specific details of the data, and data availability, see http://www.olsendata.com. For previous analyses of the data set, see the special issue of the Journal of Empirical Finance (1999), Vol. 6, No. 5, and Proceedings of the HFDF-II Conference, Olsen & Associates, Zurich, Switzerland, 1998.

‡Bollerslev and Wright (2000), for example, report fractional difference parameter estimates, under various degrees of temporal aggregation, for the volatility of a 10-year sample of 5-min US dollar–Japanese yen spot exchange rate returns which comprises over half-a-million observations.

§To preserve commonality in the number of returns for each currency exchange rate, we make no corrections for worldwide or country-specific national holidays, or other calendar effects, during the sample period. Also note that there are a total of 12 575 observations instead of 12 576 (=262 × 48) due to the loss of the return for the first observation on 1 January 1996 in calculating returns from the underlying prices. This feature is accommodated in all subsequent data analysis.

¶See, for example, the references cited in the Introduction.

¶Whilst not reported here in full, and similar to results reported by Andersen and Bollerslev (1997a) for one-year of DEM–$ returns, we find that orders of J = 0 and P = (7,8) are sufficient to model intra-day periodicity in the range of the high-frequency exchange rates considered here (further details are available from the authors on request). It should also be noted that various alternative methods for modelling the systematic intra-day patterns in the volatility of high-frequency financial data have been proposed. Müller et al. (1990) propose a time scale transformation involving standardization by a measure of the degree of market activity, which they term ‘theta time’, based on polynomial approximations to activity in distinct geographical regions over the 24-h trading cycle. Baillie and Bollerslev (1991) suggest introducing a dummy variable for each intra-day time interval, while Andersen and Bollerslev (1997b) suggest standardizing by the mean absolute return value for each intra-day interval. Andersen and Bollerslev (1997b) utilize a low-pass filtering technique based on a two-sided weighted average of past and future absolute returns. Beltratti and Morana (1999) employ a stochastic volatility model with cyclical components, whilst Gençay et al. (2001) employ wavelet methodology. We do not pursue those alternatives here.

⊥Andersen and Bollerslev (1997b) have suggested that the long-run hyperbolic rate of decay implied by the aggregation of component processes described in the Introduction is robust to the presence of periodicity in intra-day volatility. Also, under the restrictive condition that the periodic component is independent, to generalization of the framework to allow ‘news’ arrivals to share a number of common factors relating to the state of the macro economy, provided that each of the common factors is covariance stationary. Here we control for such periodicity prior to estimation of fractional difference parameters in order not to compound the dynamics of volatility.

†The negative skewness in the data may be interpreted as indirect evidence of ‘volatility feedback’ (Campbell and Hentschel 1992) or ‘feedback (noise) trading’ (Sentana and Wadhwani 1992).

‡As emphasized by Goodhart and O’Hara (1997), in the standard sequential trade framework market makers set new trading prices equivalent to the conditional expected value of the asset, subsequent trading prices forming a martingale, such that ex ante prices and returns should be uncorrelated. However, where market makers have inventory concerns, price changes may exhibit negative serial correlation due to the efforts of the market maker to move their portfolio in a desired direction. More recently, Corsi et al. (2001) have argued that such market maker bias, and other microstructure effects which arise due to the multiple contributor structure of the FX market (e.g. informational asymmetries, ‘fighting-screen’ effects and delayed quotes), lead to an ‘incoherence’ in the price formation process. Consequently, at the tick frequency, the observed price in tick time is composed of the subordinate process of the true price and additive noise, resulting in a strong negative first-order autocorrelation in returns at relatively high frequencies. For an elaboration of this argument and a formal model, see Corsi et al. (2001).

§Values of the autocorrelation functions for absolute and squared returns at higher lag orders, and of Ljung–Box portmanteau test statistics for the joint significance of absolute (and squared) returns, are omitted here in the interests of brevity, but are available from the authors on request.

†Further details of these, now commonly applied, estimators are omitted here in the interests of brevity. For a particularly lucid account, see Reisen (1994), and for discussion of the formal properties of these estimators, see Robinson (1994a, b, 1995a, b) and Hurvich et al. (1998).

‡It is important to note that the effects of temporal aggregation differ for log-periodogram estimates of the fractional difference parameter in the mean and in volatility, respectively. Simulation results reported for the log-periodogram regression procedure for long-memory dependencies in the mean commonly suggest large finite sample biases in macroeconomic time series of realistic sample size (Agiakloglou et al. 1992, Cheung 1993, Hurvich and Beltrao 1994, Hurvich et al. 1998). Moreover, this small sample bias depends crucially on the time span of the data and simply increasing the frequency of the observations over a fixed time span does little to enhance the quality of the estimates. In sharp contrast, corresponding simulation results for estimating long memory in volatility by the same procedure demonstrate that the performance of the estimates may be greatly enhanced by increasing the observation frequency, over even relatively short calendar time spans (Bollerslev and Wright 2000).

†More explicitly, where [x] takes the sequence of values [x] _{t , n} then for n = 1, 2, …, K where K = 48/k and t = 1, 2, …, 260.

†Whilst not reported here, we have also considered the long-memory properties of absolute returns and squared returns temporally aggregated to the daily frequency after taking the nonlinear transformation. Those results, available from the authors on request, broadly confirm the simulation results reported by Bollerslev and Wright (2000). That is, for absolute returns at least, daily fractional difference parameter estimates are in fairly close accordance with those identified in the high-frequency data for a number of the exchange rates analysed, but are mostly statistically insignificant (especially under the d _GPH estimator). This finding confirms that a single year of daily data aggregated from the underlying high-frequency data cannot substitute for direct analysis of the high-frequency data itself and, as Bollerslev and Wright note, underscores the importance of high-frequency data availability for reliable inference concerning long-run volatility dependencies.

‡It is also of note that our results accord closely with the theoretical results concerning the properties of nonlinear transformations of fractionally differenced process reported by Dittmann and Granger (2002), that the square of a stationary Gaussian fractionally differenced process should exhibit less dependence than the initial process. Specifically, Dittmann and Granger establish that, for a Gaussian I(d) process, if d ≤ 1/4 the squared process is I(0), whilst if 1/4 < d < 1/2 the squared process has long memory with parameter 2d–1/2, and if 1/2 ≤ d < 1 the squared series has long memory with parameter d. In particular, whilst not reported here in full, assuming the initial process to be represented by absolute returns, tests of equality of the d _GPH estimates for squared returns with the parameter values implied by these relationships cannot be rejected on the basis of asymptotic t-tests for any of the currencies analysed.

†This latter finding therefore also offers direct empirical support for the observation that the analysis of instantaneous nonlinear transforms of long memory measures contained in Robinson (2001) does not formally apply to the squared temporally aggregated data (Bollerslev and Wright 2000).