21.2 FORECASTING DURATIONS

21.2 FORECASTING DURATIONS

Interest in predicting the length of time it takes before some event unfolds arises in numerous situations. For example, we might be interested in predicting the length of an unemployment spell, the length of a bull market run for stock prices, or the time it takes before the Federal Reserve changes the Federal funds rate. This section studies the most common prediction models that have been used to model and predict duration data.

Specifically, let $t_i$ denote an arrival time such as the time of execution of a financial transaction, the start of a recession, or the data for a change in the Federal Reserve’s funds rate, while $y_i=t_i-t_{i-1}$ is the interval between two arrival times ${(}^{\alpha }{\mathrm{events}}^{\prime \prime })$ the duration of which we are interested in modeling. Furthermore, following Engle and Russell (1998) let ${\psi }_i$ be the conditional expectation of the ith duration given past durations,

$${\psi }_i\left(y_{i-1},\dots ,y_1;\theta \right)\equiv {\psi }_i=E\left[y_i|y_{i-1},\dots ,y_1\right].\text{\hspace{1em}(21.5)}$$

Engle and Russell (1998) discuss various parameterizations of (21.5). For the case with a multiplicative error structure, they use the decomposition

$$y_i={\psi }_i{\epsilon }_i,\text{\hspace{1em}(21.6)}$$

where ${\epsilon }_i\sim i.i.d$ . has unit mean and density $p_{\epsilon }(\varphi )$ defined over a positive support since durations cannot be negative. Note that the duration, ${\gamma }_i,$ is modeled here as a latent variable $\left({\psi }_i\right)$ times a positive random variable $\left({\epsilon }_i\right)$ . The parameters $\theta$ and $\varphi$ in (21.5) and (21.6) are assumed to be constant.

The autoregressive conditional duration (ACD) class of models specifies different densities $p_{\epsilon }(\varphi )$ in (21.6) and different functional forms for the expected duration in (21.5). The resulting duration models are very similar to the GARCH specifications considered in section 13.2. If the expected duration depends on only the most recent m durations (lags of $y_i)$ , we have

$${\psi }_i=\omega +\sum _{j=1}^m{\alpha }_j{y}_{i-j}.\text{\hspace{1em}(21.7)}$$

Engle and Russell (1998) also propose a dynamic $\mathrm{ACD}(m,q)$ specification without the limited memory feature of (21.7):

$${\psi }_i=\omega +\sum _{j=1}^m{\alpha }_j{y}_{i-j}+\sum _{j=1}^q{\beta }_j{\psi }_{i-j}.\text{\hspace{1em}(21.8)}$$

Under (21.8) and appropriate conditions ensuring stationarity, the unconditional expectation of the duration for this model becomes

$$E\left[y_i\right]=\frac{\omega }{1-{\sum }_{j=1}^m{\alpha }_j-{\sum }_{j=1}^q{\beta }_j}.$$

As for GARCH models, (21.8) is similar to an $\mathrm{ARMA}(m,q)$ process for durations, and so we can use iterative methods such as the chain rule to form predictions of future durations.

One specification of particular interest is the ACD(1,1) model:

$${\psi }_i=\omega +{\alpha }_1{y}_{i-1}+{\beta }_1{\psi }_{i-1}.\text{\hspace{1em}(21.9)}$$

Assuming that ${\alpha }_1,{\beta }_1\ge 0$ and $\omega >0$ , this model can capture excess dispersion in durations, consistent with much empirical evidence. Estimation of the parameters of the ACD model can be based on maximum likelihood methods provided that a distribution $p_{\epsilon }(\varphi )-\mathrm{e}.\mathrm{g}.$ ., the exponential distribution—has been specified. Engle and Russell (1998) show empirically that ACD models can successfully be applied to model durations between transactions of IBM shares.

Bauwens et al. (2004) consider various extensions to these models. Within the context of the ACD(1,1) model (21.9) they consider densities $P_{\epsilon }$ such as the exponential, Weibull, and generalized gamma. The exponential distribution uses only a single parameter and yields a flat hazard function. The Weibull distribution nests the exponential distribution as a special case and implies a monotonic hazard function. The Weibull distribution ${\epsilon }_i\sim \mathrm{W}(\gamma ,1)$ has (noncentral) moments $E\left[{\epsilon }_i^n\right]=$ $\Gamma (1+n/\gamma )$ and is easy to work with. The generalized gamma has two shape parameters and offers more flexibility.

Using any one of these densities, a parametric (conditional) density forecast of $\mathbf{y}$ can be computed as $p_{\epsilon }(y_i{\psi }_i^{-1}){\psi }_i^{-1}$

Bauwens et al. (2004) discuss logarithmic ACD models of the form

$$\begin{array}{r}\log ({\psi }_i)=\omega +{\alpha }_1\log (y_{i-1})+{\beta }_1\log ({\psi }_{i-1}).\end{array}$$

This specification has the advantage that it need not impose sign restrictions on the right-hand side; while durations must be nonnegative, log-durations need not satisfy this requirement. Given a density, $p_{\epsilon }.$ for the shocks ${\epsilon }_i$ in $y_i=\exp ({\psi }_i){\epsilon }_i$ , density forecasts from log-ACD models can be generated from $p_{\epsilon }(y_i(\exp ({\psi }_i){)}^{-1})(\exp ({\psi }_i){)}^{-1}$

A third class of duration models that allows for a single dynamic stochastic factor has been proposed by Bauwens and Veredas (2004). The factor, ${\tilde{\psi }}_i$ , is assumed to follow the process

$$\begin{array}{r}\log ({\tilde{\psi }}_i)=\omega +{\beta }_1\log ({\tilde{\psi }}_{i-1})+u_i,u_i\sim \mathrm{ind}\mathrm{N}(0,{\sigma }^2),\end{array}$$

where, again, $y_i=\exp ({\tilde{\psi }}_i){\epsilon }_i$ , and ${\epsilon }_i$ and $u_i$ are mutually independent. This is labeled the stochastic conditional duration model. For example, Bauwens and Veredas show that the mean and variance of the durations implied by this model with a Weibull distribution, ${\epsilon }_i\sim \mathrm{W}(\gamma ,1)$ , are

$$E\left[y_i\right]=\Gamma (1+1/\gamma )\exp \left(\frac{\omega }{1-{\beta }_1}+\frac{{\sigma }^2}{2(1-{\beta }_1^2)}\right),$$

$$\mathrm{Var}(y_i)=E{\left[y_i\right]}^2\left(\frac{\Gamma (1+2/\gamma )}{\Gamma (1+1/\gamma {)}^2}\exp \left(\frac{{\sigma }^2}{1-{\beta }_1^2}\right)-1\right).$$

These expressions can be used to assess the properties of the model and to generate forecasts of the mean and variance of the durations.

Bauwens et al. (2004) evaluate their duration forecasts using the probability integral transform described in chapter 18. Probability integral transform methods are well suited for the suite of models described above which are fully parametric and mostly available in closed form with the exception of the stochastic conditional duration model in (21.11), whose density forecasts have to be computed by simulation. As a benchmark a simple constant-parameter exponential distribution with i.i.d. increments (corresponding to a Poisson model) can be used, although more sophisticated dynamic benchmarks could be based on any one of the models described here.