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 <markus.belfrage@gmail.com>

References

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

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

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.

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.

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

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

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.

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

Details

The finite mixture of inverse Gaussian distributions was used by Gomes-Deniz and Perez-Rodrigues (2013) 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, E. and Perez-Rodriguez, J.V. (2016) Mixture Inverse Gaussian for Unobserved Heterogeneity in the Autoregressive Conditional Duration Model. Communications in Statistics - Theory and Methods, http://dx.doi.org/10.1080/03610926.2016.1200094

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

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

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, BACD, BCACD, AACD, EXACD, SNIACD, LSNIACD, TACD, and TAMACD.

The currently available distributions are: Exponential (also used for QML), Weibull, Burr, generalized Gamma, generalized F, Q-Weibull, and Mixture inverse Gaussian.

Usage

acdFit(durations = NULL, model = "ACD", dist = "exponential", 
    order = NULL, startPara = NULL,  dailyRestart = FALSE, optimFnc = "optim",
    method = "Nelder-Mead", output = TRUE, bootstrapErrors = FALSE, 
    forceErrExpec = TRUE, fixedParamPos = NULL, bp = NULL, 
    exogenousVariables = NULL, optimFncArgs = list(),
    control = list())

Arguments

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|\epsilon_{i-j}-\nu| \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..., \beta_j..., c, \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...).

EXACD(p, q) (Exponential ACD): (Dufour and Engle, 2000)

\ln\mu_i = \omega + \sum_{j=1}^{p} \left( \alpha_j \epsilon_{i-j} + \delta_j |\epsilon_{i-j} - 1| + \sum_{j=1}^{q} \beta_j \mu_{i-j} \right)

The element order of the startPara vector is (\omega, \alpha_j..., \beta_j..., \delta_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=1}^{M} (\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.

Note that this is a diffrent parameterization then the one in stats::dweibull. While the shape parameter in stats::dweibull is the same as \gamma, the scale parameter is equal to \theta^{-\gamma}.

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:

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.

Dufour, A., & Engle, R. F. (2000) The ACD Model: Predictability of the Time Between Consecutive Trades (Discussion Papers in Finance: 2000-05). ISMA Centre, University of Reading.

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

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

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

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,
        plotDur = TRUE, plotAdjDur = TRUE, plotResi = TRUE)

Arguments

Value

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

Author(s)

Markus Belfrage

Examples

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

data(durData)
f <- acf_acd(durData)
f

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

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:

Author(s)

Markus Belfrage

Examples

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

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

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.

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

Value

Author(s)

Markus Belfrage

Examples

data(durData)
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)

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

Examples

## Not run: 
data(transData)
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

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: 

data(adjDurData)
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

Author(s)

Markus Belfrage

Examples

data(durData)
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 = NULL, param2sequence = NULL, startpoint = NULL, 
       length.out = NULL, returnOutput = FALSE)

Arguments

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:

Author(s)

Markus Belfrage

Examples

## Not run: 


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

Examples

data(durData)
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

Author(s)

Markus Belfrage

Examples

## Not run: 

data(adjDurData)
# 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)

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

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.

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

Examples

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

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

Author(s)

Markus Belfrage

Examples

## Not run: 
data(adjDurData)
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.

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, bp = NULL)

Arguments

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

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 (currently only works for model = "ACD").

Usage

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

Arguments

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 of:

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

data(adjDurData)
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 (currently only works for model = "ACD").

Usage

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

Arguments

Value

a list of:

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, testTVACD.

Examples

data(adjDurData)
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 (currently only works for model = "ACD").

Usage

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

Arguments

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:

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

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

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