acdFit  R Documentation 
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, LogACD (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 = "NelderMead", output = TRUE, bootstrapErrors = FALSE,
forceErrExpec = TRUE, fixedParamPos = NULL, bp = NULL,
exogenousVariables = NULL, control = list())
durations 
either (1) a data frame including, at least, a column named 'durations' or 'adjDur' (for adjusted durations), or (2) a vector of durations 
model 
the conditional mean model specification. Must be one of 
dist 
the assumed error term distribution. Must be one of 
order 
a vector detailing the order of the particular ACD model. For example an ACD(p, q) specification should have 
startPara 
a vector with parameter values to start the maximization algorithm from. Must be in the correct order according to the model specification (see Details). 
dailyRestart 
if 
optimFnc 
Specifies which optimization function to use for the estimation. 
method 
Argument passed to the optimization function if 
output 
if 
bootstrapErrors 
if 
forceErrExpec 
if 
fixedParamPos 
a logical vector of 
bp 
used only for the SNIACD or LSNIACD model. A vector of break points. 
exogenousVariables 
specifies the columns in the 
control 
a list of control values,

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_{ij} + \sum_{j=1}^{q} \beta_j \mu_{ij}
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_{ij} + \sum_{j=1}^{q} \beta_j \ln \mu_{ij}
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_{ij} + \sum_{j=1}^{q} \beta_j \ln \mu_{ij}
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_{ij} + \sum_{j=1}^{r} \nu_j \epsilon_{ij} + \sum_{j=1}^{q} \beta_j \mu_{ij}
The element order of the startPara
vector is (\omega, \alpha_j...,\nu_j...,\beta_j...)
.
ABACD(p, q) (Augmented BoxCox ACD): (Hautsch, 2012)
\mu_i^{\delta_1} = \omega + \sum_{j=1}^{p} \alpha_j \left( \epsilon_{ij}\nu+c_j\epsilon_{ij}b \right)^{\delta_2} + \sum_{j=1}^{q} \beta_j \mu_{ij}^{\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) (BoxCox ACD): (Hautsch, 2003)
\mu_i^{\delta_1} = \omega + \sum_{j=1}^{p} \alpha_j \epsilon_{ij}^{\delta_2} + \sum_{j=1}^{q} \beta_j \mu_{ij}^{\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_{j1}+c_0) \epsilon_{ij} + \sum_{j=1}^{p} \sum_{k=M}^{r} (\alpha_{j1}+c_k)1_{(\epsilon_{ij} \le \bar{\epsilon_k})}+\sum_{j=1}^{q} \beta_j \mu_{ij},
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 p1]
).
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^{\gamma1}e^{\theta \epsilon^{\gamma}} ,
where \theta=[\Gamma(\gamma^{1}+1)]^{\gamma}
if forceErrExpec = TRUE
.
Burr distribution, dist = "burr"
:
f(\epsilon)= \frac{\theta \kappa \epsilon^{\kappa1}}{(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(\eta1/\gamma)}.
The element order of the startPara
vector is (model parameters, \kappa, \eta, \gamma)
.
qWeibull distribution, dist = "qweibull"
:
f(\epsilon) = (2q)\frac{a}{b^a} \epsilon^{a1} \left[1(1q)\left(\frac{\epsilon}{b}\right)^a\right]^{\frac{1}{1q}}
where if forceErrExpec = TRUE
,
b = \frac{(q1)^{\frac{1+a}{a}}}{2q}\frac{a\Gamma(\frac{1}{q1})}{\Gamma(\frac{1}{a}) \Gamma(\frac{1}{q1}\frac{1}{a}1)}.
The element order of the startPara
vector is (model parameters, a, q)
.
a list of class "acdFit"
with the following slots:
durations 
the durations object used to fit the model. 
muHats 
a vector of the estimated conditional mean durations 
residuals 
the residuals from the fitted model, calculated as durations/mu 
model 
the model for the conditional mean durations 
order 
the order of the model 
distribution 
the assumed error term distribution 
distCode 
the internal code used to represent the distribution 
mPara 
a vector of the estimated conditional mean duration parameters 
dPara 
a vector of the estimated error distribution parameters 
Npar 
total number of parameters 
goodnessOfFit 
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. 
parameterInference 
a data.frame with the estimated coefficients and their standard errors and pvalues 
forcedDistPara 
the value of the unfree distribution parameter. If 
comments 

hessian 
the numerical hessian of the log likelihood evaluated at the estimate 
N 
number of observations 
evals 
number of loglikelihood evaluations needed for the maximization algorithm 
convergence 
if the maximization algorithm converged, this value is zero. (see the help file optim, nlminb or solnp) 
estimationTime 
time required for estimation 
description 
who fitted the model and when 
robustCorr 
only available for QML estimation (choosing the exponential distribution) for the standard ACD(p, q) model. The robust correlation matrix of the parameter estimates. 
Markus Belfrage
Bauwens, L., and P. Giot (2000) The logarithmic ACD model: an application to the bidask quote process of three NYSE stocks. Annales d'Economie et de Statistique, 60, 117149.
Engle R.F, Russell J.R. (1998) Autoregressive Conditional Duration: A New Model for Irregularly Spaced Transaction Data, Econometrica, 66(5): 11271162.
Grammig, J., and Maurer, K.O. (2000) Nonmonotonic hazard functions and the autoregressive conditional duration model. Econometrics Journal 3: 1638.
Hautsch, N. (2003) Assessing the Risk of Liquidity Suppliers on the Basis of Excess Demand Intensities. Journal of Financial Econometrics (2003) 1 (2): 189215
Hautsch, N. (2012) Econometrics of Financial HighFrequency 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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