# The ACD model

This model was introduced by Engle and Russell (1998) to represent the dynamics of durations between trades on stock or exchange rate markets. Typically, intertrade durations are generated by a computerized order matching system which automatically selects trading partners who satisfy elementary matching criteria. Therefore, the timing of such automatically triggered transactions is a priori unknown and adds a significant element of randomness to the trading process. From the economic point of view, research on intertrade durations is motivated by the relevance of the time varying speed of trading for purposes

such as strategic market interventions. Typically the market displays episodes of accelerated activity and slowdowns which reflect the varying liquidity of the asset. This concept is similar to the notion of velocity used in monetary macroeconomics to describe the rate of money circulation (Gourieroux, Jasiak, and Le Fol, 1999). In the context of stock markets, periods of intense trading are usually associated with short intertrade durations accompanied by high price volatilities and large traded volumes.

The intertrade durations plotted against time display dynamic patterns similar to stock price volatilities, termed in the literature the clustering effect. This means that long durations have a tendency to be followed by long durations, while short durations tend to be followed by short durations. To capture this behavior Engle and Russell (1998) proposed the ACD model. It accommodates the duration clustering in terms of temporal dependence of conditional means of durations. From the time series perspectives it is an analog of the GARCH model representing serial correlation in conditional variances of stock prices (see Engle, 1982) and from the duration analysis point of view it is an accelerated hazard model.

Let N be the number of events observed at random times. The N events are indexed by i = 1,…, N from the first observed event to the last. The ith duration is the time between the ith event and the (i – 1)th event. The distribution of the sequence of durations is characterized by the form of the conditional distribution of the duration Y; given the lagged durations Yj_1 = {Y;-1, Y;-2,…}. The ACD(p, q) model is an accelerated hazard model, where the effect of the past is summarized by the conditional expectation y; = E(Y; | Yti):

where f0 is a baseline distribution with unitary mean, and the conditional mean у, satisfies:

y; = w + a(L)Y; + P(L)y;, (21.27)

where L denotes the lag operator, a(L) = a1L + a 2L2 + … + aqLq and P(L) = p1L +

P2L2 + … + epLp are lag polynomials of degrees p and q respectively. The coefficients a, j = 1,…, q, Р/, j = 1,…, p are assumed to be nonnegative to ensure the positivity of у;. They are unknown and have to be estimated jointly with the baseline distribution. This specification implies that the effect of past durations on the current conditional expected value decays exponentially with the lag length. Indeed, the ACD( p, q) process may be rewritten as an ARMA(m, p) process in Y:

[1 – a(L) – P(L)]Y; = w + [1 – p(L)]v, (21.28)

where m = max(p, q), and v; = Y; – y; = Y; – E(Y; | Yi-1) is the innovation of the duration process. The stationarity condition requires that the roots of [1 – a(L) – P(L)] and [1 – P(L)] lie outside the unit circle, or equivalently since aj7 Pj are nonnegative (LjO-j + XjPj) < 1.

The baseline distribution can be left unspecified or constrained to belong to a parametric family, such as the Weibull family.

The ACD model was a pioneering specification in the domain of duration dynamics. Further research has focused on providing refinements of this model in order to improve the fit. Empirical results show for example that many duration data display autocorrelation functions decaying at a slow, hyperbolic rate. Indeed, the range of temporal dependence in durations may be extremely large suggesting that duration processes possess long memory. This empirical finding is at odds with the exponential decay rate assumed by construction in the ACD model. To accommodate the long persistence, a straightforward improvement consists in accounting for fractional integration in the ACD model (Jasiak, 1999). The corresponding fractionally integrated process is obtained by introducing a fractional differencing operator:

Ф (W – L)dYi = w + [1 – P(L)K (21.29)

where the fractional differencing operator (1 – L)d is defined by its expansion with respect to the lag operator:

(1 – L)d = £ Г(к – d)r(k + 1)-1r(-d)-1Lk = £ nkLk, say, (21.30)

k = 0 k=0

where Г denotes the gamma function and 0 < d < 1.

Although this extension successfully captures serial correlation at long lags, it fails to solve the major drawback of the basic ACD model, which consists of tying together the movements of conditional mean and conditional variance by supposing that E(Yt | YM, Yt_2…) = y and V(Yt | YM, У-2…) = ?, where

the value k0 depends on the baseline distribution. We see that even though overdispersion arises whenever k0 > 1, its magnitude is supposed to be path independent. This is a stringent assumption which in practice is often violated by the data. Empirical results based on intertrade durations on stock markets suggest on the contrary, the presence of path dependent (under)overdispersion as well as the existence of distinct dynamic patterns of the conditional mean and dispersion. As an illustration we display in Figure 21.5 a scatterplot of squared means and variances of intertrade durations of the Alcatel stock traded on the Paris bourse. We observe that the cluster is relatively dispersed with a significant number of observations featuring (conditional) underdispersion. Therefore, the data provide evidence supporting the co-existence of (conditional) under – and overdispersion, generating marginal overdispersion.

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