acdFit(durations = NULL, model = "ACD", dist = "exponential",
order = NULL, startPara = NULL, dailyRestart = 0, optimFnc = "optim",
method = "Nelder-Mead", output = TRUE, bootstrapErrors = FALSE,
forceErrExpec = TRUE, fixedParamPos = NULL, bp = NULL, control = list())"ACD", "LACD1", "LACD2", "AMACD", "BACD", "ABACD", "SNIACD" or "LSNIACD". See 'Details' for de"exponential", "weibull", "burr", "gengamma", "genf", "qweibull", "mixqwe", "mixqww", or "mixorder = c(p, q).TRUE the conditional duration will start fresh every new trading day. Can only be used if the durations arguments included the clock time of the durations, or if the time argument was provided."optim", "nlminb", "solnp", and "optimx" are available.optimFnc = "optim" or optimFnc = "optimx" were chosen. Specifies the optimization algorithm.FALSE the estimation results won't be printed.TRUE the standard errors will be computed by using bootstrap simulations. Currently only works with the standard ACD model.TRUE error terms distribution will be forced to be 1, otherwise the distribution parameter specifying the mean will be set to 1 to ensure identification.TRUE and FALSE. Can only be used if the argument startPara were provided, and should be of the same length. Each element represents the respective start parameter and if TRUE, this pa"acdFit" with the following slots:forceErrExpec = TRUE were used, this parameter is a function of the other distribution parameters, to force the mean of the distribution to be one. Otherwise the parameter was fixed at 1 to ensure identification.startPara argument is a vector of the parameter values to start from. The length of the vector naturally depends on the model and distribution. The first elements represent the model parameters, and the last elements the distribution parameters. For example for an ACD(1,1) with Weibull errors the first 3 elements are $\omega, \alpha_1, \beta_1$ for the model, and the last is $\gamma$ for the Weibull distribution.
The family of ACD models are
$$x_i = \mu_i \epsilon_i,$$
where different specifications of the conditional mean duration $\mu_i$ and the error term $\epsilon_i$ give rise to different models as shown below.
Conditional mean duration $\mu_i$ specifications according to the model argument:
ACD(p, q) specification: (Engle and Russell, 1998)
$$\mu_i = \omega + \sum_{j=1}^{p} \alpha_j x_{i-j} + \sum_{j=1}^{q} \beta_j \mu_{i-j}$$
The element order of the startPara vector is $(\omega, \alpha_j...,\beta_j...)$.
LACD1(p, q): (Bauwens and Giot, 2000)
$$\ln\mu_i = \omega + \sum_{j=1}^{p} \alpha_j \ln \epsilon_{i-j} + \sum_{j=1}^{q} \beta_j \ln \mu_{i-j}$$
The element order of the startPara vector is $(\omega, \alpha_j...,\beta_j...)$.
LACD2(p, q): (Lunde, 1999)
$$\ln\mu_i = \omega + \sum_{j=1}^{p} \alpha_j \epsilon_{i-j} + \sum_{j=1}^{q} \beta_j \ln \mu_{i-j}$$
The element order of the startPara vector is $(\omega, \alpha_j...,\beta_j...)$.
AMACD(p, r, q) (Additive and Multiplicative ACD): (Hautsch , 2012)
$$\mu_i = \omega + \sum_{j=1}^{p} \alpha_j x_{i-j} + \sum_{j=1}^{r} \nu_j \epsilon_{i-j} + \sum_{j=1}^{q} \beta_j \mu_{i-j}$$
The element order of the startPara vector is $(\omega, \alpha_j...,\nu_j...,\beta_j...)$.
ABACD(p, q) (Augmented Box-Cox ACD): (Hautsch, 2012)
$$\mu_i^{\delta_1} = \omega + \sum_{j=1}^{p} \alpha_j \left( |\epsilon_{i-j}-\nu|+c_j|\epsilon_{i-j}-b| \right)^{\delta_2} + \sum_{j=1}^{q} \beta_j \mu_{i-j}^{\delta_1}$$
The element order of the startPara vector is $(\omega, \alpha_j..., c_j..., \beta_j..., \nu, \delta_1, \delta_2)$.
BACD(p, q) (Box-Cox ACD): (Hautsch, 2003)
$$\mu_i^{\delta_1} = \omega + \sum_{j=1}^{p} \alpha_j \epsilon_{i-j}^{\delta_2} + \sum_{j=1}^{q} \beta_j \mu_{i-j}^{\delta_1}$$
The element order of the startPara vector is $(\omega, \alpha_j..., \beta_j...)$.
SNIACD(p, q, M) (Spline News Impact ACD): (Hautsch, 2012, with a slight difference)
$$\mu_i = \omega + \sum_{j=1}^{p} (\alpha_{j-1}+c_0) \epsilon_{i-j} + \sum_{j=1}^{p} \sum_{k=M}^{r} (\alpha_{j-1}+c_k)1_{(\epsilon_{i-j} \le \bar{\epsilon_k})}+\sum_{j=1}^{q} \beta_j \mu_{i-j},$$
where $1_{()}$ is an indicator function and $\alpha_0=0$.
The element order of the startPara vector is $(\omega, c_k..., \alpha_j..., \beta_j...)$ (The number of $\alpha$-parameters are $p-1]$).
The distribution of the error term $\epsilon_i$ specifications according to the dist argument:
Exponential distribution, dist = "exponential":
$$f(\epsilon)=\exp(-\epsilon)$$
Weibull distribution, dist = "weibull":
$$f(\epsilon)=\theta \gamma \epsilon^{\gamma-1}e^{-\theta \epsilon^{\gamma}} ,$$
where $\theta=[\Gamma(\gamma^{-1}+1)]^{\gamma}$ if forceErrExpec = TRUE.
Burr distribution, dist = "burr":
$$f(\epsilon)= \frac{\theta \kappa \epsilon^{\kappa-1}}{(1+\sigma^2 \theta \epsilon^{\kappa})^{\frac{1}{\sigma^2}+1}},$$
where,
$$\theta= \sigma^{2 \left(1+\frac{1}{\kappa}\right)} \frac{\Gamma \left(\frac{1}{\sigma^2}+1\right)}{\Gamma \left(\frac{1}{\kappa}+1\right) \Gamma \left(\frac{1}{\sigma^2}-\frac{1}{\kappa}\right)},$$
if forceErrExpec = TRUE.
The element order of the startPara vector is $(model parameters, \kappa, \sigma^2)$.
Generalized Gamma distribution, dist = "gengamma":
$$f(\epsilon)=\frac{\gamma \epsilon^{\kappa \gamma - 1}}{\lambda^{\kappa \gamma}\Gamma (\kappa)}\exp \left{{-\left(\frac{\epsilon}{\lambda}\right)^{\gamma}}\right}$$
where $\lambda=\frac{\Gamma(\kappa)}{\Gamma(\kappa+\frac{1}{\gamma})}$ if forceErrExpec = TRUE.
The element order of the startPara vector is $(model parameters, \kappa, \gamma)$.
Generalized F distribution, dist = "genf":
$$f(\epsilon)= \frac{\gamma \epsilon^{\kappa \gamma -1}[\eta+(\epsilon/\lambda)^{\gamma}]^{-\eta-\kappa}\eta^{\eta}}{\lambda^{\kappa \gamma}B(\kappa,\eta)},$$
where $B(\kappa,\eta)=\frac{\Gamma(\kappa)\Gamma(\eta)}{\Gamma(\kappa+\eta)}$, and if forceErrExpec = TRUE,
$$\lambda=\frac{\Gamma(\kappa)\Gamma(\eta)}{\eta^{1/\gamma}\Gamma(\kappa+1/\gamma)\Gamma(\eta-1/\gamma)}.$$
The element order of the startPara vector is $(model parameters, \kappa, \eta, \gamma)$.
q-Weibull distribution, dist = "qweibull":
$$f(\epsilon) = (2-q)\frac{a}{b^a} \epsilon^{a-1} \left[1-(1-q)\left(\frac{\epsilon}{b}\right)^a\right]^{\frac{1}{1-q}}$$
where if forceErrExpec = TRUE,
$$b = \frac{(q-1)^{\frac{1+a}{a}}}{2-q}\frac{a\Gamma(\frac{1}{q-1})}{\Gamma(\frac{1}{a}) \Gamma(\frac{1}{q-1}-\frac{1}{a}-1)}.$$
The element order of the startPara vector is $(model parameters, a, q)$.fitModel <- acdFit(durations = adjDurData, model = "ACD",
dist = "exponential", order = c(1,1), dailyRestart = 1)Run the code above in your browser using DataLab