This function estimates various ACD models with various assumed error term distributions, using Maximum Likelihood Estimation.
The currently available models (conditional mean specifications) are:
Standard ACD, Log-ACD (two alternative specifications), AMACD, ABACD, SNIACD and LSNIACD.
The currently available distributions are:
Exponential (also used for QML), Weibull, Burr, generalized Gamma, and generalized F.
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,
exogenousVariables = NULL, control = list())
a list of class "acdFit" with the following slots:
the durations object used to fit the model.
a vector of the estimated conditional mean durations
the residuals from the fitted model, calculated as durations/mu
the model for the conditional mean durations
the order of the model
the assumed error term distribution
the internal code used to represent the distribution
a vector of the estimated conditional mean duration parameters
a vector of the estimated error distribution parameters
total number of parameters
a data.frame with the log likelihood, AIC, BIC, and MSE calculated as the mean squared deviation of the durations and the estimated conditional durations.
a data.frame with the estimated coefficients and their standard errors and p-values
the value of the unfree distribution parameter. If 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.
the numerical hessian of the log likelihood evaluated at the estimate
number of observations
number of log-likelihood evaluations needed for the maximization algorithm
if the maximization algorithm converged, this value is zero. (see the help file optim, nlminb or solnp)
time required for estimation
who fitted the model and when
only available for QML estimation (choosing the exponential distribution) for the standard ACD(p, q) model. The robust correlation matrix of the parameter estimates.
either (1) a data frame including, at least, a column named 'durations' or 'adjDur' (for adjusted durations), or (2) a vector of durations
the conditional mean model specification. Must be one of "ACD", "LACD1", "LACD2", "AMACD", "BACD",
"ABACD", "SNIACD" or "LSNIACD". See 'Details' for detailed model specification.
the assumed error term distribution. Must be one of "exponential", "weibull", "burr", "gengamma", "genf", "qweibull", "mixqwe", "mixqww", or "mixinvgauss". See 'Details' for detailed model specification.
a vector detailing the order of the particular ACD model. For example an ACD(p, q) specification should have order = c(p, q).
a vector with parameter values to start the maximization algorithm from. Must be in the correct order according to the model specification (see Details).
if 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.
Specifies which optimization function to use for the estimation. "optim", "nlminb", "solnp", and "optimx" are available.
Argument passed to the optimization function if optimFnc = "optim" or optimFnc = "optimx" were chosen. Specifies the optimization algorithm. See the help files for optim, nlminb or solnp.
if FALSE the estimation results won't be printed.
if TRUE the standard errors will be computed by using bootstrap simulations. Currently only works with the standard ACD model.
if TRUE the expectation of the error terms' distribution will be forced to be 1, otherwise the distribution parameter specifying the mean will be set to 1 to ensure identification.
a logical vector of 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 parameter will be held fixed when estimating the other parameters.
used only for the SNIACD or LSNIACD model. A vector of break points.
specifies the columns in the durations data.frame that should be used as exogenous variables when fitting the model. Must be a vector, either with the column positions or the names of the columns. It is highly recommended to standardize the exogenous variables before running the estimation.
a list of control values,
maximum number of iterations performed by the numerical maximization algorithm.
an integer. If this is set to diffrent to 0, the values of the parameters each time the optimization function calls the log likelihood function. This search path of the MLE will then be plotted. Also passed on to the optimization function, see the help files for optim, nlminb or solnp.
number of bootstrap samples
Markus Belfrage
The 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.
When exogenous variables are used, they are added in the form of $$\sum_{j=1}^{k} \xi_j z_j$$ to the right hand side of the equations, where \(z_j\) are the exogenous variables.
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 \((model parameters, a, q)\).
Bauwens, L., and P. Giot (2000) The logarithmic ACD model: an application to the bid-ask quote process of three NYSE stocks. Annales d'Economie et de Statistique, 60, 117-149.
Engle R.F, Russell J.R. (1998) Autoregressive Conditional Duration: A New Model for Irregularly Spaced Transaction Data, Econometrica, 66(5): 1127-1162.
Grammig, J., and Maurer, K.-O. (2000) Non-monotonic hazard functions and the autoregressive conditional duration model. Econometrics Journal 3: 16-38.
Hautsch, N. (2003) Assessing the Risk of Liquidity Suppliers on the Basis of Excess Demand Intensities. Journal of Financial Econometrics (2003) 1 (2): 189-215
Hautsch, N. (2012) Econometrics of Financial High-Frequency Data. Berlin, Heidelberg: Springer.
Lunde, A. (1999): A generalized gamma autoregressive conditional duration model, Working paper, Aalborg University.
fitModel <- acdFit(durations = adjDurData, model = "ACD",
dist = "exponential", order = c(1,1), dailyRestart = 1)
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