Package 'ACDm'

ACD Modelling

Description

Package for Autoregressive Conditional Duration (ACD, Engle and Russell, 1998) models. Creates trade, price or volume durations from transactions (tic) data, performs diurnal adjustments, fits various ACD models and tests them.

Credit

The author would like to thank the department of statistics at Hanken School of Economics, as the bulk of this work was done there while working as a research assistant.

Author(s)

Markus Belfrage

Maintainer: Markus Belfrage <[email protected]>

References

Engle R.F, Russell J.R. (1998) Autoregressive Conditional Duration: A New Model for Irregularly Spaced Transaction Data, Econometrica, 66(5): 1127-1162.

---

ACD (Autoregressive Conditional Duration) Model Fitting

Description

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.

Usage

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())

Arguments

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 "ACD", "LACD1", "LACD2", "AMACD", "BACD", "ABACD", "SNIACD" or "LSNIACD". See 'Details' for detailed model specification.
dist	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.
order	a vector detailing the order of the particular ACD model. For example an ACD(p, q) specification should have order = c(p, q).
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 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.
optimFnc	Specifies which optimization function to use for the estimation. "optim", "nlminb", "solnp", and "optimx" are available.
method	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.
output	if FALSE the estimation results won't be printed.
bootstrapErrors	if TRUE the standard errors will be computed by using bootstrap simulations. Currently only works with the standard ACD model.
forceErrExpec	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.
fixedParamPos	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.
bp	used only for the SNIACD or LSNIACD model. A vector of break points.
exogenousVariables	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.
control	a list of control values, maxit maximum number of iterations performed by the numerical maximization algorithm. trace 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. B number of bootstrap samples

Details

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 $(model parameters, a, q)$.

Value

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 p-values
forcedDistPara	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.
comments
hessian	the numerical hessian of the log likelihood evaluated at the estimate
N	number of observations
evals	number of log-likelihood 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.

Author(s)

Markus Belfrage

References

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.

Examples

fitModel <- acdFit(durations = adjDurData, model = "ACD", 
            dist = "exponential", order = c(1,1), dailyRestart = 1)

---

Methods for class acdFit

Description

residuals.acdFit() returns the residuals and coef.acdFit() returns the coefficients of a fitted ACD model of class 'acdFit', while print.acdFit() prints the essential information. predict.acdFit() predicts the next N durations by thier expected value.

Usage

## S3 method for class 'acdFit'
residuals(object, ...)
## S3 method for class 'acdFit'
coef(object, returnCoef = "all", ...)
## S3 method for class 'acdFit'
print(x, ...)
## S3 method for class 'acdFit'
predict(object, N = 10, ...)

Arguments

object	the fitted ACD model of class 'acdFit' (as returned by the function acdFit).
x	same as object, ie. an object of class 'acdFit'.
returnCoef	on of "all", "distribution", or "model". Specifies whether all estimated parameters should be returned or only the distribution parameters or the model (for the conditional mean duration) parameters.
N	the number of the predictions in predict.
...	additional arguments to print.

---

Autocorrelation function plots for ACD models

Description

plots the ACF (Auto Correlation Function) for the durations, diurnally adjusted durations, and residuals.

Usage

acf_acd(fitModel = NULL, conf_level = 0.95, max = 50, min = 1)

Arguments

fitModel	a fitted model of class "acdFit", or a data.frame containing at least one the columns "durations", "adjDur", or "residuals". Can also be a vector of durations or residuals.
conf_level	the confidence level of the confidence bands
max	the largest lag to plot
min	the smallest lag to plot

Value

returns a data.frame with the values of the sample autocorrelations for each lag and variable.

Author(s)

Markus Belfrage

Examples

fitModel <- acdFit(adjDurData)
acf_acd(fitModel, conf_level = 0.95, max = 50, min = 1)

f <- acf_acd(durData)
f

---

The Burr Distribution

Description

Density, distribution function, quantile function, random generation and calculation of the expected value for the Burr distribution with parameters theta, kappa and sig2.

Usage

dburr(x, theta = 1, kappa = 1.2, sig2 = 0.3, forceExpectation = F)
pburr(x, theta = 1, kappa = 1.2, sig2 = .3, forceExpectation = F)
qburr(p, theta = 1, kappa = 1.2, sig2 = .3, forceExpectation = F)
rburr(n = 1, theta = 1, kappa = 1.2, sig2 = .3, forceExpectation = F)
burrExpectation(theta = 1, kappa = 1.2, sig2 = .3)

Arguments

x	vector of quantiles.
p	vector of probabilities.
n	number of observations..
theta, kappa, sig2	parameters, see 'Details'.
forceExpectation	logical; if TRUE, the expectation of the distribution is forced to be 1 by letting theta be a function of the other parameters.

Details

The PDF for the Burr distribution is (as in e.g. Grammig and Maurer, 2000):

$f(x)=\frac{\theta \kappa x^{\kappa - 1}}{(1 + \sigma^2 x^{\kappa)^{\frac{1}{\sigma^2}+1}}}$

Value

dburr gives the density (PDF), qburr the quantile function (inverted CDF), rburr generates random deviates, and burrExpectation returns the expected value of the distribution, given the parameters.

Author(s)

Markus Belfrage

References

Grammig, J., and Maurer, K.-O. (2000) Non-monotonic hazard functions and the autoregressive conditional duration model. Econometrics Journal 3: 16-38.

---

Durations computation

Description

Computes durations from a data.frame containing the time stamps of transactions. Trade durations, price durations and volume durations can be computed (if the appropriate data columns are given).

Usage

computeDurations(transactions, open = "10:00:00", close = "18:25:00", 
rm0dur = TRUE, type = "trade", priceDiff = .1, cumVol = 10000)

Arguments

transactions	a data.frame with, at least, transaction time in a column named 'time' (see Details)
open	the opening time of the exchange. Transactions done outside the trading hours will be ignored.
close	the closing time of the exchange.
rm0dur	if TRUE zero-durations will be removed and transactions done on the same second will be aggregated, e.g. price will then be the volume weighted avrage price of the aggregated transactions.
type	the type of durations to be computed. Either "trade", "price", or "volume".
priceDiff	only if type = "price". Price durtions are (here) defind as the duration until the price has changed by at least 'priceDiff' in absolute value.
cumVol	only if type = "cumVol". Volume durtions are (here) defind as the duration until the cumulative traded volume since the last duration has surpassed 'cumVol'.

Details

The data.frame must include a column named 'time' with the time of each transaction, in a time format recognizable by POSIXlt or strings in format "yyyy-mm-dd hh:mm:ss". If the column 'price' or 'volume' is included its also possible to compute price- and volume durations (see arguments priceDiff and cumVol)

Value

a data.frame with columns:

time	the calander time of the start of each duration spell.
price	the volume weighted avrage price of the shares traded during the spell of the duration.
volume	the volume (total shares traded) during the duration spell.
Ntrans	number of transactions done during the spell.
durations	the computed duration.

Author(s)

Markus Belfrage

Examples

## Not run: 
#only the first 3 days of data:
durDataShort <- computeDurations(transData[1:56700, ]) 
str(durDataShort)
head(durDataShort)
## End(Not run)

---

Time Series Data Sets

Description

The data file transData is the base data used in all of the examples. It is a data.frame with rows representing a single transaction and has the columns 'time', 'price', giving the trade price, and 'volume', giving the number of shares traded for the transaction. The data set is based on real transactions but has been obfuscated by transforming the dates, price and volume, for proprietary reasons. It covers two weeks of nearly 100 000 transactions, recorded with 1 second precision.

The durData data.frame is simply the trade durations formed from transData using the function durData <- computeDurations(transData)

The adjDurData data object is in turn created by adjDurData <- diurnalAdj(durData, aggregation = "all") to add diurnally adjusted durations.

defaultSplineObj is an estimated cubic spline of the diurnal component using the sample data. It is used when simulating from sim_ACD() with the argument diurnalFactor set to TRUE, when no user splineObj is provided.

---

The generalized F distribution

Description

Density and distribution function for the generalized F distribution. Warning: the distribution function pgenf and genfHazard are computed numerically, and may not be precise!

Usage

dgenf(x, kappa = 5, eta = 1.5, gamma = .8, lambda = 1, forceExpectation = F)
pgenf(q, kappa = 5, eta = 1.5, gamma = .8, lambda = 1, forceExpectation = F)
genfHazard(x, kappa = 5, eta = 1.5, gamma = .8, lambda = 1, forceExpectation = F)

Arguments

x, q	vector of quantiles.
kappa, eta, gamma, lambda	parameters, see 'Details'.
forceExpectation	logical; if TRUE, the expectation of the distribution is forced to be 1 by letting theta be a function of the other parameters.

Details

The PDF for the generelized F distribution is:

$f(\epsilon)= \frac{\gamma \epsilon^{\kappa \gamma -1}[\eta+(\epsilon/\lambda)^{\gamma}]^{-\eta-\kappa}\eta^{\eta}}{\lambda^{\kappa \gamma}B(\kappa,\eta)},$

where $B(\kappa,\eta)=\frac{\Gamma(\kappa)\Gamma(\eta)}{\Gamma(\kappa+\eta)}$ is the beta function.

---

Discreet mix of the q-Weibull and the exponential distributions

Description

Density (PDF), distribution function (CDF), and hazard function for a discreetly mixed distribution of the q-Weibull and the exponential distributions.

Usage

dmixqwe(x, pdist = .5, a = .8, qdist = 1.5, lambda = .8, b = 1, forceExpectation = F)
pmixqwe(q, pdist = .5, a = .8, qdist = 1.5, lambda = .8, b = 1, forceExpectation = F)
mixqweHazard(x, pdist = .5, a = .8, qdist = 1.5, lambda = .8, b = 1, forceExpectation = F)

Arguments

x, q	vector of quantiles.
pdist, a, qdist, lambda, b	parameters, see 'Details'.
forceExpectation	logical; if TRUE, the expectation of the distribution is forced to be 1 by letting b be a function of the other parameters.

Details

The PDF for the mixed distribution is:

$f(x) = p(2-q)\frac{a}{b^a} x^{a-1} \left[1-(1-q)\left(\frac{x}{b}\right)^a\right]^{\frac{1}{1-q}} + (1-p)\frac{1}{\lambda}exp(-\frac{x}{\lambda})$

if forceExpectation = TRUE the b parameter is a function of the other parameters to force the expectation to be 1.

See Also

qWeibullDist for the Q-Weibull distribution and pmixqww for Q-Weibull mixed with the ordinary Weibull.

---

Discreet mix of the q-Weibull and the ordinary Weibull distributions

Description

Density (PDF), distribution function (CDF), and hazard function for a discreetly mixed distribution of the q-Weibull and the ordinary Weibull distributions.

Usage

dmixqww(x, pdist = .5, a = 1.2, qdist = 1.5, theta = .8, gamma = 1, b = 1,
  forceExpectation = F)
        
pmixqww(q, pdist = .5, a = 1.2, qdist = 1.5, theta = .8, gamma = 1, b = 1,
  forceExpectation = F)
        
mixqwwHazard(x, pdist = .5, a = 1.2, qdist = 1.5, theta = .8, gamma = 1, b = 1,
  forceExpectation = F)

Arguments

x, q	vector of quantiles.
pdist, a, qdist, theta, gamma, b	parameters, see 'Details'.
forceExpectation	logical; if TRUE, the expectation of the distribution is forced to be 1 by letting b be a function of the other parameters.

Details

The PDF for the mixed distribution is:

$f(x) = p(2-q)\frac{a}{b^a} x^{a-1} \left[1-(1-q)\left(\frac{x}{b}\right)^a\right]^{\frac{1}{1-q}} + (1-p)\theta \gamma x^{-\theta x^{\gamma}}$

if forceExpectation = TRUE the b parameter is a function of the other parameters to force the expectation to be 1.

See Also

qWeibullDist for the Q-Weibull distribution and pmixqwe for Q-Weibull mixed with the exponential distribution.

---

Dirunal adjustment for durations

Description

Performs a diurnal adjustment of the durations, i.e. removes a daily seasonal component. Four different methods of diurnal adjustment are available, namely "cubicSpline", "supsmu" (Friedman's SuperSmoother), "smoothSpline" (smoothed version of the cubic spline), or "FFF" (Flexible Fourier Form).

Usage

diurnalAdj(dur, method = "cubicSpline", nodes = c(seq(600, 1105, 60), 1105),
aggregation = "all", span = "cv", spar = 0, Q = 4, returnSplineFnc = FALSE)

Arguments

dur	a data.frame containing the columns durations, containing durations, and time, containing the time stamps.
method	the method used. One of "cubicSpline", "supsmu", "smoothSpline", or "FFF".
nodes	only for method = "cubicSpline" or method = "smoothSpline". A vector of nodes to use for the spline function, in the unit minutes after midnight. The first and last element of the vector must be the start and end of the trading day. The nodes given are actually the limits of intervalls, of wich the midpoints will be set as the nodes using the means of the intervals.
aggregation	what type of aggregation to use. Either "weekdays", "all", or "none". If for example "weekdays" is chosen, all Mondays will have the same daily seasonal component, and so on.
span	argument passed to supsmu if method = "supsmu" were chosen. Affects the smoothness of the curve, see supsmu.
spar	argument passed to smooth.spline if method = "smooth.spline" were chosen. Affects the smoothness of the curve, see smooth.spline.
Q	number of trigonometric function pairs for method = "FFF".
returnSplineFnc	if TRUE instead or returning the adjusted durations a list of spline objects will be returned, containing the coefficents of the spline function. Only available for method = "cubicSpline".

Value

If returnSplineFnc is FALSE (default): the input data.frame dur with an added column of the diurnally adjusted durations called 'adjDur'.

Otherwise, a list of spline objects containing the coefficents of the spline function.

Author(s)

Markus Belfrage

Examples

diurnalAdj(durData, aggregation = "none", method = "supsmu")

## Not run: 

head(durData)
f <- diurnalAdj(durData, aggregation = "weekdays", method = "FFF", Q = 3)
head(f)

f <- diurnalAdj(durData, aggregation = "all", returnSplineFnc = TRUE)
f

## End(Not run)

---

Finite mixture of inverse Gaussian Distribution

Description

Density (PDF), distribution function (CDF), and hazard function for Finite mixture of inverse Gaussian Distributions.

Usage

dmixinvgauss(x, theta = .2, lambda = .1, gamma = .05, forceExpectation = F)
pmixinvgauss(q, theta = .2, lambda = .1, gamma = .05, forceExpectation = F)
mixinvgaussHazard(x, theta = .2, lambda = .1, gamma = .05, forceExpectation = F)

Arguments

x, q	vector of quantiles.
theta, lambda, gamma	parameters, see 'Details'.
forceExpectation	logical; if TRUE, the expectation of the distribution is forced to be 1..

Details

The finite mixture of inverse Gaussian distributions was used by Gomes-Deniz and Perez-Rodrigues (201X) for ACD-models. Its PDF is:

$f(x) = \frac{\gamma + x}{\gamma + \theta} \sqrt{\frac{\lambda}{2 \pi x^3}} \exp \left[ - \frac{\lambda(x-\theta)^2}{2 x \theta^2}\right].$

If forceExpectation = TRUE the distribution is transformed by dividing the random variable with its expectation and using the change of variable function.

References

Gomez-Deniz Perez-Rodriguez (201X) Non-exponential mixtures, non-monotonic financial hazard functions and the autoregressive conditional duration model. Working paper. Retrieved June 16, 2015, from http://dea.uib.es/digitalAssets/254/254084_perez.pdf.

---

The generelized Gamma distribution

Description

Density (PDF), distribution function (CDF), quantile function (inverted CDF), random generation and hazard function for the generelized Gamma distribution with parameters gamma, kappa and lambda.

Usage

dgengamma(x, gamma = 0.3, kappa = 1.2, lambda = 0.3, forceExpectation = F)
pgengamma(x, gamma = .3, kappa = 3, lambda = .3, forceExpectation = F)
qgengamma(p, gamma = .3, kappa = 3, lambda = .3, forceExpectation = F)
rgengamma(n = 1, gamma = .3, kappa = 3, lambda = .3, forceExpectation = F)
gengammaHazard(x, gamma = .3, kappa = 3, lambda = .3, forceExpectation = F)

Arguments

x	vector of quantiles.
p	vector of probabilities.
n	number of observations..
gamma, kappa, lambda	parameters, see 'Details'.
forceExpectation	logical; if TRUE, the expectation of the distribution is forced to be 1 by letting theta be a function of the other parameters.

Details

The PDF for the generelized Gamma distribution is:

$f(x)=\frac{\gamma x^{\kappa \gamma - 1}}{\lambda^{\kappa \gamma}\Gamma (\kappa)}\exp \left\{{-\left(\frac{x}{\lambda}\right)^{\gamma}}\right\}$

Value

dgengamma gives the density (PDF), pgengamma gives the distribution function (CDF), qgengamma gives the quantile function (inverted CDF), rgenGamma generates random deviates, and genGammaHazard gives the hazard function.

Author(s)

Markus Belfrage

---

Transactions plots

Description

Plots (1) the price over time, (2) volume traded over time for a given interval, and (3) number of transactions over time for a given interval.

Usage

plotDescTrans(trans, windowunit = "hours", window = 1)

Arguments

trans	a data.frame with the column 'time', 'price', and 'volume'. Currently only works if all of those are available.
windowunit	the unit of the time interval. One of "secs", "mins", "hours", or "days".
window	a positive integer giving the length of the interval.

Examples

## Not run: 
plotDescTrans(transData, windowunit = "hours", window = 1)
## End(Not run)

---

Hazard function plot

Description

Estimates and plots the hazard function from an estimatated ACD model.

Usage

plotHazard(fitModel, breaks = 20, implied = TRUE, xstop)

Arguments

fitModel	an estimated model of class acdFit. Can also be a numerical vector.
breaks	the number of quantiles used to estimate the hazard.
implied	a logical flag. If TRUE then the implied hazard function using the distribution parameter estimates will be plotted together with the nonparametric estimate of the error term hazard function.
xstop	where to stop plotting the implied hazard.

Details

This estimator of the hazard function is based on the one used by Engle and Russell (1998). It is modified sligthly to decrease its bias and inconsistency. However, the estimator is still not fully consistent when using a fixed number of breaks (quantiles).

Author(s)

Markus Belfrage

References

Engle, R.F and Russell, J.R. (1998) Autoregressive Conditional Duration: A New Model for Irregularly Spaced Transaction Data. Econometrica, 66(5): 1127-1162.

Examples

## Not run: 

fitModelWei <- acdFit(adjDurData, dist = "wei")
plotHazard(fitModelWei)

## End(Not run)

---

Mean duration plot

Description

Plots the mean duration over time at chosen interval length

Usage

plotHistAcd(durations, windowunit = "mins", window = 1)

Arguments

durations	a data.frame containing the durations and their time stamps.
windowunit	the unit of the time interval. One of "secs", "mins", "hours", or "days".
window	a positive integer giving the length of the interval.

Author(s)

Markus Belfrage

Examples

plotHistAcd(durData, windowunit = "days", window = 1)

## Not run: 

plotHistAcd(durData, windowunit = "mins", window = 30)

## End(Not run)

---

Plots the response surface of the log likelihood of a fitted model.

Description

Plots the log likelihood for a fitted model against either one or two of the parameters at a time. This can help to find issues with for example poor identification of a model.

Usage

plotLL(fitModel, parameter1 = 1, parameter2 = NULL, 
param1sequence, param2sequence, startpoint = NULL, returnOutput = FALSE)

Arguments

fitModel	a fitted model of class acdFit.
parameter1	the first parameter for the log likelihood to be plotted against. Either the index of the parameter as an integer, or the name of the parameter.
parameter2	the second parameter for the log likelihood to be plotted against. Either the index of the parameter as an integer, or the name of the parameter. If left empty, a plot with only the parameter1 will be drawn.
param1sequence, param2sequence	the sequence of points from with the log likelihood is computed. If left empty, the log likelihood will be computed at 21 points spanning between MLE-3*SD and MLE+3*SD in the one dimensional case, and the 11x11 points for the same range in the two dimensional case.
startpoint	a vector of size equal to the number of parameters in the model. If this is supplied, the log likelihood will be evaluated at this point instead of the point of the MLE (for the parameters not in parameter1 and parameter2).
returnOutput	a logical flag. If set to TRUE, the values of the response surface will be returned. See 'value' below.

Value

Only if returnOutput = TRUE

1. For the one dimensional case: a data.frame with the columns 'logLikelihood', and 'param1sequence' for all the values of the parameter1 witch the log likelihood was evaluated at

2. For the two dimensional case: a list with the following items:

para1	a vector with the sequence of the parameter1 values.
para2	a vector with the sequence of the parameter2 values.
z	a matrix with the log likelihood values. The element at the ith row and jth column is evaluated at the ith para1 value and jth para2 value.

Author(s)

Markus Belfrage

Examples

## Not run: 

#Indicates identification issues with the generelized gamma distibution:
#(Try a diffrent 'startPara' in acdFit() to get slightly a better fit)
fitModel2 <- acdFit(durations = adjDurData[1:3000, ], dist = "gengamma")
seq1 <- seq(500, 1000, 50)
seq2 <- seq(.02, 0.045, 0.001)
plotLL(fitModel = fitModel2, parameter1 = "kappa", parameter2 = "gamma", 
       param1sequence = seq1, param2sequence = seq2)

## End(Not run)

---

Plots rolling means of durations

Description

Plots rolling means of durations

Usage

plotRollMeanAcd(durations, window = 500)

Arguments

durations	a data.frame containing the column 'time' and 'durations'.
window	the length of the rolling window.

Examples

plotRollMeanAcd(durData, window = 500)

---

Scatter plot for ACD models

Description

Function to help scatter plot different variables of a fitted ACD model and superimposes a smoothed conditional mean using ggplot2. Can be used to investigate the possible need for non-linear models and issues with the diurnal adjustment.

Usage

plotScatterAcd(fitModel, x = "muHats", y = "residuals", xlag = 0, ylag = 0,
                           colour = NULL, xlim = NULL, ylim = NULL, alpha = 1/10,
                           smoothMethod = "auto")

Arguments

fitModel	a fitted model of class "acdFit"
x	the variable used on the x-axis. One of "muHats", "residuals", "durations", "adjDur", "dayTime", "time", or "index".
y	the variable used on the y-axis. One of "muHats", "residuals", "durations", "adjDur", "dayTime", "time", or "index".
xlag	number of lags used for the variable shown on the x-axis.
ylag	number of lags used for the variable shown on the y-axis.
colour	a possible third variable to be represented with a colour scale. One of "muHats", "residuals", "durations", "adjDur", "dayTime", or "time".
xlim	a vector of the limits of the x-axis to possibly zoom in on a certain region.
ylim	a vector of the limits of the y-axis to possibly zoom in on a certain region.
alpha	alpha parameter passed to ggplot2. For large data sets many data points will overlap. The alpha parameter can make the points transparent, making it easier to distinguish the density of different region. Takes the value between 1 (opaque) and 0 (completely transparent).
smoothMethod	value passed as smooth argument to ggplot2. See stat_smooth.

Author(s)

Markus Belfrage

Examples

## Not run: 

# The mean residuals are too small for small values of the estimated conditional 
# mean, suggesting a need for a different conditional mean model specification:
fitModel <- acdFit(adjDurData)
plotScatterAcd(fitModel, x = "muHats", y = "residuals")

## End(Not run)

---

Quantile-Quantile plot of the residuals

Description

Plots a QQ-plot of the residuals and the theoretical quantiles implied by the model estimates.

Usage

qqplotAcd(fitModel, xlim = NULL, ylim = NULL)

Arguments

fitModel	a fitted ACD model, i.e. an object of class "acdFit"
xlim	an optional vector of limits for the x-axis
ylim	an optional vector of limits for the y-axis

Examples

fitModelExp <- acdFit(adjDurData, dist = "exp")
qqplotAcd(fitModelExp)

---

The q-Weibull distribution

Description

Density (PDF), distribution function (CDF), quantile function (inverted CDF), random generation, exepcted value, and hazard function for the q-Weibull distribution.

Usage

dqweibull(x, a = .8, qdist = 1.2, b = 1, forceExpectation = F)
pqweibull(q, a = .8, qdist = 1.2, b = 1, forceExpectation = F)
qqweibull(p, a = .8, qdist = 1.2, b = 1, forceExpectation = F)
rqweibull(n = 1, a = .8, qdist = 1.2, b = 1, forceExpectation = F)
qweibullExpectation(a = .8, qdist = 1.2, b = 1)
qweibullHazard(x, a = .8, qdist = 1.2, b = 1, forceExpectation = F)

Arguments

x, q	vector of quantiles.
p	vector of probabilities.
n	number of observations.
a, qdist, b	parameters, see 'Details'.
forceExpectation	logical; if TRUE, the expectation of the distribution is forced to be 1 by letting b be a function of the other parameters.

Details

The PDF for the q-Weibull distribution is:

$f(\epsilon) = (2-q)\frac{a}{b^a} \epsilon^{a-1} \left[1-(1-q)\left(\frac{\epsilon}{b}\right)^a\right]^{\frac{1}{1-q}}$

The distribution was used for ACD models by Vuorenmaa (2009).

References

Vuorenmaa, T. (2009) A q-Weibull Autoregressive Conditional Duration Model with an Application to NYSE and HSE data. Available at SSRN: http://ssrn.com/abstract=1952550.

---

Residual Density Histogram

Description

Plots a density histogram of the residuals and superimposes the density implied by the model estimates.

Usage

resiDensityAcd(fitModel, xlim = NULL, binwidth = .1, density = FALSE)

Arguments

fitModel	a fitted ACD model, i.e. an object of class "acdFit"
xlim	an optional vector of limits for the x-axis
binwidth	the width of the bins of the density histogram.
density	if TRUE a kernel density estimate will be added

Author(s)

Markus Belfrage

Examples

## Not run: 
fitModelBurr <- acdFit(adjDurData, dist = "burr")
resiDensityAcd(fitModelBurr)
## End(Not run)

---

ACD simulation

Description

Simulates a sample from a specified ACD model and error term distribution dist. The error terms can also be sampled from residuals. The possibility of including a diurnal seasonal component in the simulated sample is included.

Usage

sim_ACD(N = 1000, model = "ACD", dist = "exponential", param = NULL, order = NULL,
    Nburn = 50, startX = c(1), startMu = c(1), errors = NULL, sampleErrors = TRUE, 
    roundToSec = FALSE, rm0 = FALSE, diurnalFactor = FALSE, splineObj = NULL,
    open = NULL, close = NULL)

Arguments

N	sample size
model	the class of conditional mean duration specification. One of "ACD", "LACD1", "LACD2", "AMACD","ABACD", "SNIACD" or "LSNIACD".
dist	the distribution of the error terms (only if errors are left out). Must be one of "exponential", "weibull", "burr", "gengamma" or "genf".
param	a vector of the parameters of the DGP (data generating process).
order	a vector describing the order of the conditional mean duration specification, e.g. order = c(1,1) for an ACD(1,1) model.
Nburn	the number of burned observations. Used to lower the effect of the start values of the simulated series.
startX	a vector of values to start the simulation from.
startMu	a vector of conditional mean values to start the simulation from.
errors	a vector of error terms. If provided and sampleErrors = TRUE the errors will be sampled from this vector (with replacement). If instead sampleErrors = FALSE the error terms will be matched by the errors vector non stochastic (must then be of the same length as N + Nburn)
sampleErrors	logical flag, see errors above. Default is TRUE.
roundToSec	if TRUE the simulated sample will be discretized with 1 second(unit) precision.
rm0	if TRUE zero durations will be removed. Will the result in a smaller sample than N.
diurnalFactor	if TRUE the simulated data will include a diurnal factor. The diurnal factor is from a fitted cubic spline given as argument to splineObj. If the argument splineObj is empty, a default fitted cubic spline from transData using aggregation over weekdays will be used.
splineObj	a cubic spline return by diurnalAdj(). Currently only works with cubic splines fitted with weekday aggregation. Also see diurnalFactor above.
open	only used if diurnalFactor = TRUE and a splineObj were provided. The time the exchange opens trading (as used in the fitted splineObj), for example open = "10:00:00".
close	only used if diurnalFactor = TRUE and a splineObj were provided. The time the exchange close trading (as used in the fitted splineObj), for example close = "18:25:00".

Value

a numerical vector of simulated ACD durations

Author(s)

Markus Belfrage

Examples

x <- sim_ACD() #simulates 1000 observations from an ACD(1,1) with exp. errors as default
acdFit(x)

---

Residual standardization

Description

Standardizes residuals from a fitted ACD model of class 'acdFit' by a probability integral transformation (taking the CDF, using the estimated distribution parameters, of the residuals) or by returning the Cox-Snell residuals.

Usage

standardizeResi(fitModel, transformation = "probIntegral")

Arguments

fitModel	a fitted ACD model of class 'acdFit'.
transformation	type of transformation done, either "probIntegral", or "cox-snell".

Details

The probability integral transformation is done by taking the CDF of the residuals from the model estimation, using the estimated distribution parameters. Under correct specification the probability integral transformed residuals should be iid. uniform(0, 1).

The Cox-Snell residuals is the computed by taking the integrated hazard of the residuals from the model estimation, using the estimated distribution parameters. Under correct specification the probability integral transformed residuals should be iid. unit exponentially distributed.

---

LM test of no Remaining ACD (Meitz and Terasvirta, 2006)

Description

Tests if there is any remaining ACD structure in the residuals

Usage

testRmACD(fitModel, pStar = 2, robust = TRUE)

Arguments

fitModel	a fitted ACD model, i.e. an object of class "acdFit".
pStar	the number of alpha parameters in the alternative hypothesis. See $p*$ under 'Details'.
robust	if TRUE the LM statistic will be calculated using the "robust" version, making its asymptotic behavior unaffected by possible misspecification of the error term distribution (Meitz and Terasvirta, 2006).

Details

For the model

$x_i = \mu_i \phi_i \epsilon_i,$

$\mu_i = \omega + \sum_{j=1}^{p} \alpha_j x_{i-j} + \sum_{j=1}^{q} \beta_j \mu_{i-j},$

$\phi_i = 1 + \sum_{j=1}^{p*} \frac{x_{i-j}}{\mu_{i-j}},$

the function tests the null hypothesis

$H_0: \phi_i = 1.$

Value

a list containing:

chi2	the value of the LM statistic.
pv	the pvalue of the test statistic.

Author(s)

Markus Belfrage

References

Meitz, M. and Terasvirta, T. (2006). Evaluating models of autoregressive conditional duration. Journal of Business and Economic Statistics 24: 104-124.

See Also

testTVACD, testSTACD.

Examples

fitModel3000obs <- acdFit(adjDurData[1:3000,])
testRmACD(fitModel3000obs, pStar = 2, robust = TRUE)

---

LM test against Smooth Transition ACD models (Meitz and Terasvirta, 2006)

Description

Tests if the alpha parameters and the constant should be varying with the value of the lagged durations, according to a logistic transition function.

Usage

testSTACD(fitModel, K = 2, robust = TRUE)

Arguments

fitModel	a fitted ACD model, i.e. an object of class "acdFit".
K	the order of the logistic transition function used for the alternative hypothesis.
robust	if TRUE the LM statistic will be calculated using the "robust" version, making its asymptotic behavior unaffected by possible misspecification of the error term distribution (Meitz and Terasvirta, 2006).

Value

a list of:

chi2	the value of the LM statistic.
pv	the pvalue of the test statistic.

See Also

testRmACD, testTVACD.

Examples

fitModel3000obs <- acdFit(adjDurData[1:3000,])
testSTACD(fitModel3000obs, K = 2, robust = TRUE)

---

LM test against Time-Varying ACD models (Meitz and Terasvirta, 2006)

Description

Tests if the parameters are time-varying.

Usage

testTVACD(fitModel, K = 2, type = "total", robust = TRUE)

Arguments

fitModel	a fitted ACD model, i.e. an object of class "acdFit".
K	the order of the logistic transition function used for the alternative hypothesis.
type	either "total" or "intraday". If "total", the possible time varying parameters under the alternative varies over the total time of the sample, whereas for "intraday", the time variable is time of the day. See 'Details'
robust	if TRUE the LM statistic will be calculated using the "robust" version, making its asymptotic behavior unaffected by possible misspecification of the error term distribution (Meitz and Terasvirta, 2006).

Details

This function tests the fitted standard ACD model against the TVACD model of Meitz and Terasvirta (2006). The TVACD model lets the ACD parameters vary over time by a logistic transition function.

In one specification, the time variable is total time, and a test rejecting the null in favor of this alternative specification would indicate that the ACD parameters are changing over time over the total sample.

The other specification lets the parameters be intraday varying, by letting the transition variable be the time of the day. Failing this test could indicate that the diurnal adjustment was inadequate at removing any diurnal component.

Value

a list of:

chi2	the value of the LM statistic.
pv	the pvalue of the test statistic.

Author(s)

Markus Belfrage

References

Meitz, M. and Terasvirta, T. (2006). Evaluating models of autoregressive conditional duration. Journal of Business and Economic Statistics 24: 104-124.

See Also

testRmACD, testSTACD.

Examples

fitModel5000obs <- acdFit(adjDurData[1:5000,])
testTVACD(fitModel5000obs, K = 2, type = "total", robust = TRUE)

testTVACD(fitModel5000obs, K = 2, type = "intraday", robust = TRUE)

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