Package 'fRegression'

Regression Modelling Package

Description

The Rmetrics "fRegression" package is a collection of functions for linear and non-linear regression modelling.

Details

Package:	fRegression
Type:	Package
Version:	R 3.0.1
Date:	2014
License:	GPL Version 2 or later
Copyright:	(c) 1999-2014 Rmetrics Association
Repository:	R-FORGE
URL:	https://www.rmetrics.org

1 Introduction

Regression modelling, especially linear modelling, LM, is a widely used application in financial engineering. In finance it mostly appears in form that a variable is modelled as a linear or more complex relationship as a function of other variables. For example the decision of buying or selling in a trading model may be triggered by the outcome of a regression model, e.g. neural networks are a well known tool in this field.

2 Fitting Regression Models

Rmetrics has build a unique interface to several regression models available in the base and contributed packages of R. The following regression models are interfaced and available through a common function regFit. The argument use allows to select the desired model:

    regFit          fits regression models
    - lm            fits a linear model [stats]
    - rlm           fits a LM by robust regression [MASS]
    - glm           fits a generliazed linear model [stats]
    - gam           fits a generlized additive model [mgcv]
    - ppr           fits a projection pursuit regression model [stats]
    - nnet          fits a single hidden-layer neural network model [nnet]
    - polymars      fits an adaptive polynomial spline regression [polspline]
    

An advantage of the regFit function is, that all the underlying functions of its family can be called with the same list of arguments, and the value returned is always an unique object, an object of class "fREG" with the following slots: @call, @formula, @method, @data, @fit, @residuals, @fitted, @title, and @description.

Furthermore, independent of the selected regression model applied we can use the same S4 methods for all types of regressions. This includes, print,plot, summary, predict, fitted, residuals, coef, vcov, and formula methods.

It is possible to add further regression models to this framework either his own implementations or implementations available through other contributed R packages. Suggestions include biglm, earth amongst others.

2 Simulation of Regression Models

contains a function to simulate artificial regression models, mostly used for testing.

    regSim          simulates artificial regression model data sets
    

3 Extractor Functions

These generic functions are:

    fitted          extracts fitted values from a fitted 'fREG' object
    residuals       extracts residuals from a fitted 'fREG' object
    coef            extracts coefficients from a fitted 'fREG' object
    formula         extracts formula expression from a fitted 'fREG' object
    vcov            extracts variance-covariance matrix of fitted parameters
    

4 Forecasting

The function predict returns predicted values based on the fitted model object.

    predict         forecasts from an object of class 'fREG'
    

4 Reporting Functions

For printing and plotting use the functions:

    print           prints the results from a regression fit
    plot            plots the results from a gression fit
    summary         returns a summary report       
    

About Rmetrics:

The fRegression Rmetrics package is written for educational support in teaching "Computational Finance and Financial Engineering" and licensed under the GPL.

---

REG coefficients Methods

Description

Extracts coefficients from a fitted regression model.

Methods

object = "ANY"

Generic function.

object = "fREG"

Extractor function for coefficients.

Note

coef is a generic function which extracts the coefficients from objects returned by modeling functions, here the regFit and gregFit parameter estimation functions.

Author(s)

Diethelm Wuertz for the Rmetrics R-port.

Examples

## regSim -
   x = regSim(model = "LM3", n = 50)
  
## regFit -
   fit = regFit(Y ~ X1 + X2 + X3, data = x, use = "lm")
   
## coef - 
   coef(fit)

---

Extract Regression Model Fitted Values

Description

Extracts fitted values from a fitted regression model.

Methods

object = "ANY"

Generic function

object = "fREG"

Extractor function for fitted values.

Note

fitted is a generic function which extracts fitted values from objects returned by modeling functions, here the regFit and gregFit parameter estimation functions.

The class of the fitted values is the same as the class of the data input to the function regFit or gregFit. In contrast the slot fitted returns a numeric vector.

Author(s)

Diethelm Wuertz for the Rmetrics R-port.

Examples

## regSim -
   x.df = regSim(model = "LM3", n = 50)
  
## regFit -
   # Use data.frame input:
   fit = regFit(Y ~ X1 + X2 + X3, data = x.df, use = "lm")
   
## fitted - 
   val = slot(fit, "fitted")
   head(val)
   class(val)
   val = fitted(fit)
   head(val)
   class(val)
   
## regFit -
   # Convert to dummy timeSeries Object:
   library(timeSeries)
   x.tS = as.timeSeries(x.df)
   fit = regFit(Y ~ X1 + X2 + X3, data = x.tS, use = "lm")
   
## fitted - 
   val = slot(fit, "fitted")
   head(val)
   class(val)
   val = fitted(fit)
   head(val)
   class(val)

---

Extract Regression Model formula

Description

Extracts formula from a fitted regression model.

Methods

object = "ANY"

Generic function

object = "fGARCH"

Formula

Note

formula is a generic function which extracts the formula expression from objects returned by modeling functions, here the regFit and gregFit parameter estimation function.

Author(s)

Diethelm Wuertz for the Rmetrics R-port.

Examples

## regSim -
   x = regSim(model = "LM3", n = 50)
  
## regFit -
   fit = regFit(Y ~ X1 + X2 + X3, data = x, use = "lm")
   
## formula - 
   formula(fit)

---

Class "fREG"

Description

The class 'fREG' represents a fitted model of an heteroskedastic time series process.

Objects from the Class

Objects can be created by calls of the function regFit. The returned object represents parameter estimates of linear and generalized linear models.

Slots

call:

Object of class "call": the call of the garch function.

formula:

Object of class "formula": the formula used in parameter estimation.

family:

Object of class "character": the family objects provide a convenient way to specify the details of the models used by function grefFit For details we refer to the documentation for the function glm in R's base package on how such model fitting takes place.

method:

Object of class "character": a string denoting the regression model in use, i.e. one of those listed in the use argument of the function regFit or gregFit.

data:

Object of class "list": a list with at least two entries named x containing the data frame used for the estimation, and data with the object of the rectangular input data.

fit:

Object of class "list": a list with the results from the parameter estimation. The entries of the list depend on the selected algorithm, see below.

residuals:

Object of class "numeric": a numeric vector with the residual values.

fitted:

Object of class "numeric": a numeric vector with the fitted values.

title:

Object of class "character": a title string.

description:

Object of class "character": a string with a brief description.

Methods

show

signature(object = "fREG"): prints an object of class 'fREG'.

plot

signature(x = "fREG", y = "missing"): plots an object of class 'fREG'.

summary

signature(object = "fREG"): summarizes results and diagnostic analysis of an object of class 'fREG'.

predict

signature(object = "fREG"): forecasts mean and volatility from an object of class 'fREG'.

fitted

signature(object = "fREG"): extracts fitted values from an object of class 'fREG'.

residuals

signature(object = "fREG"): extracts fresiduals from an object of class 'fREG'.

coef

signature(object = "fREG"): extracts fitted coefficients from an object of class 'fREG'.

formula

signature(x = "fREG"): extracts formula expression from an object of class 'fREG'.

Author(s)

Diethelm Wuertz and Rmetrics Core Team.

---

Regression Model Plot Methods

Description

Plots results obtained from a fitted regression model.

Usage

## S4 method for signature 'fREG,missing'
plot(x, which = "ask", ...)

Arguments

x	an object of class 'fREG'.
which	a character string selecting which plot should be displayed. By default which="ask" which allows to generate plots interactively.
...	additional arguments to be passed to the underlying plot functions.

Details

The plots are a set of graphs which are common to the regression models implemented in the function regFit. This includes linear regression models use="lm", robust linear regression models use="rlm", generalized linear regression models use = "glm", generalized additive regression models use = "gam", projection pursuit regression models use = "ppr", neural network regression models use = "nnet", and polychotomous MARS models use = "polymars".

In addition one can also use the original plot functions of the original models, .e.g. plot(slot(object, "fit").

Methods

x = "ANY", y = "ANY"

Generic function.

x = "fREG", y = "missing"

Plot function to display results obtained from a fitted regression model.

Author(s)

Diethelm Wuertz for the Rmetrics R-port.

Examples

## regSim -
   x = regSim(model = "LM3", n = 50)
  
## regFit -
   fit = regFit(Y ~ X1 + X2 + X3, data = x, use = "lm")
   
## plot -

---

Regression Models Prediction Function

Description

Predicts a time series from a fitted regression model.

Usage

## S4 method for signature 'fREG'
predict(object, newdata, se.fit = FALSE, type = "response", ...)

Arguments

newdata	new data.
object	an object of class fREG as returned from the function regFit().
se.fit	a logical flag. Should standard errors be included? By default FALSE.
type	a character string by default "response".
...	arguments to be passed.

Value

returns ...

Methods

object = "ANY"

Generic function

object = "fREG"

Predict method for regression models.

Author(s)

Diethelm Wuertz for the Rmetrics R-port.

Examples

## regSim -
   x <- regSim(model = "LM3", n = 50)
  
## regFit -
   fit <- regFit(Y ~ X1 + X2 + X3, data = x, use = "lm")

---

Regression Modelling

Description

Estimates the parameters of a regression model.

Usage

regFit(formula, data, family = gaussian, 
    use = c("lm", "rlm", "glm", "gam", "ppr", "nnet", "polymars"), 
    title = NULL, description = NULL, ...)

Arguments

data	data is the data frame containing the variables in the model. By default the variables are taken from environment(formula), typically the environment from which lm is called.
description	a brief description of the project of type character.
family	a description of the error distribution and link function to be used in glm and gam models. See glm and family for more details.
formula	a symbolic description of the model to be fit. A typical glm predictor has the form response ~ terms where response is the (numeric) response vector and terms is a series of terms which specifies a (linear) predictor for response. For binomial models the response can also be specified as a factor. A gam formula, see also gam.models, allows that smooth terms can be added to the right hand side of the formula. See gam.side.conditions for details and examples.
use	denotes the regression method by a character string used to fit the model. method must be one of the strings in the default argument. "lm", for linear regression models, "rlm", for robust linear regression models, "glm" for generalized linear modelling, "gam" for generalized additive modelling, "ppr" for projection pursuit regression, "nnet" for feedforward neural network modelling, and "polymars" for polychotomous MARS.
title	a character string which allows for a project title.
...	additional optional arguments to be passed to the underlying functions. For details we refer to inspect the following help pages: lm, glm, gam, ppr, polymars, or nnet.

Details

The function regFit was created to provide a selection of regression models working together with Rmetrics' "timeSeries" objects and providing a common S4 object as the returned value. These models include linear modeling, robust linear modeling, generalized linear modeling, generalized additive modelling, projection pursuit regression, neural networks, and polychotomous MARS models.

LM – Linear Modelling:

Univariate linear regression analysis is a statistical methodology that assumes a linear relationship between some predictor variables and a response variable. The goal is to estimate the coefficients and to predict new data from the estimated linear relationship.

R's base function

lm(formula, data, subset, weights, na.action, method = "qr",
model = TRUE, x = FALSE, y = FALSE, qr = TRUE, singular.ok = TRUE,
contrasts = NULL, offset, ...)

is used to fit linear models. It can be used to carry out regression, single stratum analysis of variance and analysis of covariance, although aov may provide a more convenient interface for these.

Rmetrics' function

regFit(formula, data, use = "lm", ...)

calls R's base function lm but with the difference that the data argument, may be any rectangular object which can be transferred by the function as.data.frame into a data frame with named columns, e.g. an object of class "timeSeries". The function regFit returns an S4 object of class "fREG" whose slot @fit is the object as returned by the function "lm". In addition we have S4 methods fitted and residuals which allow to retrieve the fitted values and the residuals as objects of same class as defined by the argument data.

The function plot.lm provides four plots: a plot of residuals against fitted values, a Scale-Location plot of sqrt(| residuals |) against fitted values, a normal QQ plot, and a plot of Cook's distances versus row labels.
[stats:lm]

LM – Robust Linear Modelling:

To fit a linear model by robust regression using an M estimator R offers the function

rlm(formula, data, weights, ..., subset, na.action,
method = c("M", "MM", "model.frame"),
wt.method = c("inv.var", "case"),
model = TRUE, x.ret = TRUE, y.ret = FALSE, contrasts = NULL)

from package MASS. Again we can use the Rmetrics' wrapper

regFit(formula, data, use = "rlm", ...)

which allows us to use for example S4 timeSeries objects as input and to get the output as an S4 object with the known slots.
[MASS::rlm]

GLM – Generalized Linear Models:

Generalized linear modelling extends the linear model in two directions. (i) with a monotonic differentiable link function describing how the expected values are related to the linear predictor, and (ii) with response variables having a probability distribution from an exponential family.

R's base function from package stats comes with the function

glm(formula, family = gaussian, data, weights, subset,
na.action, start = NULL, etastart, mustart, offset,
control = glm.control(...), model = TRUE, method = "glm.fit",
x = FALSE, y = TRUE, contrasts = NULL, ...)

Again we can use the Rmetrics' wrapper

regFit(formula, data, use = "gam", ...)

[stats::glm]

GAM – Generalized Additive Models:

An additive model generalizes a linear model by smoothing individually each predictor term. A generalized additive model extends the additive model in the same spirit as the generalized linear model extends the linear model, namely for allowing a link function and for allowing non-normal distributions from the exponential family.
[mgcv:gam]

PPR – Projection Pursuit Regression:

The basic method is given by Friedman (1984), and is essentially the same code used by S-PLUS's ppreg. It is observed that this code is extremely sensitive to the compiler used. The algorithm first adds up to max.terms, by default ppr.nterms, ridge terms one at a time; it will use less if it is unable to find a term to add that makes sufficient difference. The levels of optimization, argument optlevel, by default 2, differ in how thoroughly the models are refitted during this process. At level 0 the existing ridge terms are not refitted. At level 1 the projection directions are not refitted, but the ridge functions and the regression coefficients are. Levels 2 and 3 refit all the terms; level 3 is more careful to re-balance the contributions from each regressor at each step and so is a little less likely to converge to a saddle point of the sum of squares criterion. The plot method plots Ridge functions for the projection pursuit regression fit.
[stats:ppr]

POLYMARS – Polychotomous MARS:

The algorithm employed by polymars is different from the MARS(tm) algorithm of Friedman (1991), though it has many similarities. Also the name polymars has been used for this algorithm well before MARS was trademarked.
[polyclass:polymars]

NNET – Feedforward Neural Network Regression:

If the response in formula is a factor, an appropriate classification network is constructed; this has one output and entropy fit if the number of levels is two, and a number of outputs equal to the number of classes and a softmax output stage for more levels. If the response is not a factor, it is passed on unchanged to nnet.default. A quasi-Newton optimizer is used, written in C.
[nnet:nnet]

Value

returns an S4 object of class "fREG".

Author(s)

The R core team for the lm functions from R's base package,
B.R. Ripley for the glm functions from R's base package,
S.N. Wood for the gam functions from R's mgcv package,
N.N. for the ppr functions from R's modreg package,
M. O' Connors for the polymars functions from R's ? package,
The R core team for the nnet functions from R's nnet package,
Diethelm Wuertz for the Rmetrics R-port.

References

Belsley D.A., Kuh E., Welsch R.E. (1980); Regression Diagnostics; Wiley, New York.

Dobson, A.J. (1990); An Introduction to Generalized Linear Models; Chapman and Hall, London.

Draper N.R., Smith H. (1981); Applied Regression Analysis; Wiley, New York.

Friedman, J.H. (1991); Multivariate Adaptive Regression Splines (with discussion), The Annals of Statistics 19, 1–141.

Friedman J.H., and Stuetzle W. (1981); Projection Pursuit Regression; Journal of the American Statistical Association 76, 817-823.

Friedman J.H. (1984); SMART User's Guide; Laboratory for Computational Statistics, Stanford University Technical Report No. 1.

Green, Silverman (1994); Nonparametric Regression and Generalized Linear Models; Chapman and Hall.

Gu, Wahba (1991); Minimizing GCV/GML Scores with Multiple Smoothing Parameters via the Newton Method; SIAM J. Sci. Statist. Comput. 12, 383-398.

Hastie T., Tibshirani R. (1990); Generalized Additive Models; Chapman and Hall, London.

Kooperberg Ch., Bose S., and Stone C.J. (1997); Polychotomous Regression, Journal of the American Statistical Association 92, 117–127.

McCullagh P., Nelder, J.A. (1989); Generalized Linear Models; Chapman and Hall, London.

Myers R.H. (1986); Classical and Modern Regression with Applications; Duxbury, Boston.

Rousseeuw P.J., Leroy, A. (1987); Robust Regression and Outlier Detection; Wiley, New York.

Seber G.A.F. (1977); Linear Regression Analysis; Wiley, New York.

Stone C.J., Hansen M., Kooperberg Ch., and Truong Y.K. (1997); The use of polynomial splines and their tensor products in extended linear modeling (with discussion).

Venables, W.N., Ripley, B.D. (1999); Modern Applied Statistics with S-PLUS; Springer, New York.

Wahba (1990); Spline Models of Observational Data; SIAM.

Weisberg S. (1985); Applied Linear Regression; Wiley, New York.

Wood (2000); Modelling and Smoothing Parameter Estimation with Multiple Quadratic Penalties; JRSSB 62, 413-428.

Wood (2001); mgcv: GAMs and Generalized Ridge Regression for R. R News 1, 20-25.

Wood (2001); Thin Plate Regression Splines.

There exists a vast literature on regression. The references listed above are just a small sample of what is available. The book by Myers' is an introductory text book that covers discussions of much of the recent advances in regression technology. Seber's book is at a higher mathematical level and covers much of the classical theory of least squares.

Examples

## regSim -
   x <- regSim(model = "LM3", n = 100)
  
   # LM       
   regFit(Y ~ X1 + X2 + X3, data = x, use = "lm") 
   # RLM      
   regFit(Y ~ X1 + X2 + X3, data = x, use = "rlm") 
   # AM       
   regFit(Y ~ X1 + X2 + X3, data = x, use = "gam")                
   # PPR      
   regFit(Y ~ X1 + X2 + X3, data = x, use = "ppr") 
   # NNET     
   regFit(Y ~ X1 + X2 + X3, data = x, use = "nnet") 
   # POLYMARS
   regFit(Y ~ X1 + X2 + X3, data = x, use = "polymars")

---

Regression Tests

Description

A collection and description of functions to test linear regression models, including tests for higher serial correlations, for heteroskedasticity, for autocorrelations of disturbances, for linearity, and functional relations.

The methods are:

"bg"	Breusch--Godfrey test for higher order serial correlation,
"bp"	Breusch--Pagan test for heteroskedasticity,
"dw"	Durbin--Watson test for autocorrelation of disturbances,
"gq"	Goldfeld--Quandt test for heteroskedasticity,
"harv"	Harvey--Collier test for linearity,
"hmc"	Harrison--McCabe test for heteroskedasticity,
"rain"	Rainbow test for linearity, and
"reset"	Ramsey's RESET test for functional relation.

There is nothing new, it's just a wrapper to the underlying test functions from R's contributed package lmtest. The functions are available as "Builtin" functions. Nevertheless, the user can still install and use the original functions from R's lmtest package.

Usage

lmTest(formula, method = c("bg", "bp", "dw", "gq", "harv", "hmc", 
    "rain", "reset"), data = list(), ...)
    
bgTest(formula, order = 1, type = c("Chisq", "F"), data = list())
bpTest(formula, varformula = NULL, studentize = TRUE, data = list())
dwTest(formula, alternative = c("greater", "two.sided", "less"),
    iterations = 15, exact = NULL, tol = 1e-10, data = list())
gqTest(formula, point=0.5, order.by = NULL, data = list())
harvTest(formula, order.by = NULL, data = list())
hmcTest(formula, point = 0.5, order.by = NULL, simulate.p = TRUE, 
    nsim = 1000, plot = FALSE, data = list()) 
rainTest(formula, fraction = 0.5, order.by = NULL, center = NULL, 
    data = list())
resetTest(formula, power = 2:3, type = c("fitted", "regressor", "princomp"), 
    data = list())

Arguments

alternative	[dwTest] - a character string specifying the alternative hypothesis, either "greater", "two.sided", or "less".
center	[rainTest] - a numeric value. If center is smaller than 1 it is interpreted as percentages of data, i.e. the subset is chosen that n*fraction observations are around observation number n*center. If center is greater than 1 it is interpreted to be the index of the center of the subset. By default center is 0.5. If the Mahalanobis distance is chosen center is taken to be the mean regressor, but can be specified to be a k-dimensional vector if k is the number of regressors and should be in the range of the respective regressors.
data	an optional data frame containing the variables in the model. By default the variables are taken from the environment which lmTest and the other tests are called from.
exact	[dwTest] - a logical flag. If set to FALSE a normal approximation will be used to compute the p value, if TRUE the "pan" algorithm is used. The default is to use "pan" if the sample size is < 100.
formula	a symbolic description for the linear model to be tested.
fraction	[rainTest] - a numeric value, by default 0.5. The percentage of observations in the subset is determined by fraction*n if n is the number of observations in the model.
iterations	[dwTest] - an integer specifying the number of iterations when calculating the p-value with the "pan" algorithm. By default 15.
method	the test method which should be applied.
nsim	[hmcTest] - an integer value. Determines how many runs are used to simulate the p value, by default 1000.
order	[bgTest] - an integer. The maximal order of serial correlation to be tested. By default 1.
order.by	[gqTest][harvTest] - a formula. A formula with a single explanatory variable like ~ x. Then the observations in the model are ordered by the size of x. If set to NULL, the default, the observations are assumed to be ordered (e.g. a time series). [rainTest] - either a formula or a string. A formula with a single explanatory variable like ~ x. The observations in the model are ordered by the size of x. If set to NULL, the default, the observations are assumed to be ordered (e.g. a time series). If set to "mahalanobis" then the observations are ordered by their Mahalanobis distance of the data.
plot	[hmcTest] - a logical flag. If TRUE the test statistic for all possible breakpoints is plotted, the default is FALSE.
point	[gqTest][hmcTest] - a numeric value. If point is smaller than 1 it is interpreted as percentages of data, i.e. n*point is taken to be the (potential) breakpoint in the variances, if n is the number of observations in the model. If point is greater than 1 it is interpreted to be the index of the breakpoint. By default 0.5.
power	[resetTest] - integers, by default 2:3. A vector of positive integers indicating the powers of the variables that should be included. By default it is tested for a quadratic or cubic influence of the fitted response.
simulate.p	[hmcTest] - a logical. If TRUE, the default, a p-value will be assessed by simulation, otherwise the p-value is NA.
studentize	[bpTest] - a logical value. If set to TRUE Koenker's studentized version of the test statistic will be used. By default set to TRUE.
tol	[dwTest] - the tolerance value. Eigenvalues computed have to be greater than tol=1e-10 to be treated as non-zero.
type	[bgTest] - the type of test statistic to be returned. Either "Chisq" for the Chi-squared test statistic or "F" for the F test statistic. [resetTest] - a string indicating whether powers of the "fitted" response, the "regressor" variables (factors are left out) or the first principal component, "princomp", of the regressor matrix should be included in the extended model.
varformula	[bpTest] - a formula describing only the potential explanatory variables for the variance, no dependent variable needed. By default the same explanatory variables are taken as in the main regression model.
...	[regTest] - additional arguments passed to the underlying lm test. Some of the tests can specify additional optional arguments like for alternative hypothesis, the type of test statistic to be returned, or others. All the optional arguments have default settings.

Details

bg – Breusch Godfrey Test:

Under $H_0$ the test statistic is asymptotically Chi-squared with degrees of freedom as given in parameter. If type is set to "F" the function returns the exact F statistic which, under $H_0$, follows an $F$ distribution with degrees of freedom as given in parameter. The starting values for the lagged residuals in the supplementary regression are chosen to be 0.
[lmtest:bgtest]

bp – Breusch Pagan Test:

The Breusch–Pagan test fits a linear regression model to the residuals of a linear regression model (by default the same explanatory variables are taken as in the main regression model) and rejects if too much of the variance is explained by the additional explanatory variables. Under $H_0$ the test statistic of the Breusch-Pagan test follows a chi-squared distribution with parameter (the number of regressors without the constant in the model) degrees of freedom.
[lmtest:bptest]

dw – Durbin Watson Test:

The Durbin–Watson test has the null hypothesis that the autocorrelation of the disturbances is 0; it can be tested against the alternative that it is greater than, not equal to, or less than 0 respectively. This can be specified by the alternative argument. The null distribution of the Durbin-Watson test statistic is a linear combination of chi-squared distributions. The p value is computed using a Fortran version of the Applied Statistics Algorithm AS 153 by Farebrother (1980, 1984). This algorithm is called "pan" or "gradsol". For large sample sizes the algorithm might fail to compute the p value; in that case a warning is printed and an approximate p value will be given; this p value is computed using a normal approximation with mean and variance of the Durbin-Watson test statistic.
[lmtest:dwtest]

gq – Goldfeld Quandt Test:

The Goldfeld–Quandt test compares the variances of two submodels divided by a specified breakpoint and rejects if the variances differ. Under $H_0$ the test statistic of the Goldfeld-Quandt test follows an F distribution with the degrees of freedom as given in parameter.
[lmtest:gqtest]

harv - Harvey Collier Test:

The Harvey-Collier test performs a t-test (with parameter degrees of freedom) on the recursive residuals. If the true relationship is not linear but convex or concave the mean of the recursive residuals should differ from 0 significantly.
[lmtest:harvtest]

hmc – Harrison McCabe Test:

The Harrison–McCabe test statistic is the fraction of the residual sum of squares that relates to the fraction of the data before the breakpoint. Under $H_0$ the test statistic should be close to the size of this fraction, e.g. in the default case close to 0.5. The null hypothesis is reject if the statistic is too small.
[lmtest:hmctest]

rain – Rainbow Test:

The basic idea of the Rainbow test is that even if the true relationship is non-linear, a good linear fit can be achieved on a subsample in the "middle" of the data. The null hypothesis is rejected whenever the overall fit is significantly inferior to the fit of the subsample. The test statistic under $H_0$ follows an F distribution with parameter degrees of freedom.
[lmtest:raintest]

reset – Ramsey's RESET Test

RESET test is popular means of diagnostic for correctness of functional form. The basic assumption is that under the alternative, the model can be written by the regression $y = X\beta + Z\gamma + u$. Z is generated by taking powers either of the fitted response, the regressor variables or the first principal component of X. A standard F-Test is then applied to determine whether these additional variables have significant influence. The test statistic under $H_0$ follows an F distribution with parameter degrees of freedom.
[lmtest:reset]

Value

A list with class "htest" containing the following components:

statistic	the value of the test statistic.
parameter	the lag order.
p.value	the p-value of the test.
method	a character string indicating what type of test was performed.
data.name	a character string giving the name of the data.
alternative	a character string describing the alternative hypothesis.

Note

The underlying lmtest package comes wit a lot of helpful examples. We highly recommend to install the lmtest package and to study the examples given therein.

Author(s)

Achim Zeileis and Torsten Hothorn for the lmtest package,
Diethelm Wuertz for the Rmetrics R-port.

References

Breusch, T.S. (1979); Testing for Autocorrelation in Dynamic Linear Models, Australian Economic Papers 17, 334–355.

Breusch T.S. and Pagan A.R. (1979); A Simple Test for Heteroscedasticity and Random Coefficient Variation, Econometrica 47, 1287–1294

Durbin J. and Watson G.S. (1950); Testing for Serial Correlation in Least Squares Regression I, Biometrika 37, 409–428.

Durbin J. and Watson G.S. (1951); Testing for Serial Correlation in Least Squares Regression II, Biometrika 38, 159–178.

Durbin J. and Watson G.S. (1971); Testing for Serial Correlation in Least Squares Regression III, Biometrika 58, 1–19.

Farebrother R.W. (1980); Pan's Procedure for the Tail Probabilities of the Durbin-Watson Statistic, Applied Statistics 29, 224–227.

Farebrother R.W. (1984); The Distribution of a Linear Combination of $\chi^2$ Random Variables, Applied Statistics 33, 366–369.

Godfrey, L.G. (1978); Testing Against General Autoregressive and Moving Average Error Models when the Regressors Include Lagged Dependent Variables, Econometrica 46, 1293–1302.

Goldfeld S.M. and Quandt R.E. (1965); Some Tests for Homoskedasticity Journal of the American Statistical Association 60, 539–547.

Harrison M.J. and McCabe B.P.M. (1979); A Test for Heteroscedasticity based on Ordinary Least Squares Residuals Journal of the American Statistical Association 74, 494–499.

Harvey A. and Collier P. (1977); Testing for Functional Misspecification in Regression Analysis, Journal of Econometrics 6, 103–119.

Johnston, J. (1984); Econometric Methods, Third Edition, McGraw Hill Inc.

Kraemer W. and Sonnberger H. (1986); The Linear Regression Model under Test, Heidelberg: Physica.

Racine J. and Hyndman R. (2002); Using R To Teach Econometrics, Journal of Applied Econometrics 17, 175–189.

Ramsey J.B. (1969); Tests for Specification Error in Classical Linear Least Squares Regression Analysis, Journal of the Royal Statistical Society, Series B 31, 350–371.

Utts J.M. (1982); The Rainbow Test for Lack of Fit in Regression, Communications in Statistics - Theory and Methods 11, 1801–1815.

Examples

## bg | dw -
   # Generate a Stationary and an AR(1) Series:
   x = rep(c(1, -1), 50)
   y1 = 1 + x + rnorm(100)
   # Perform Breusch-Godfrey Test for 1st order serial correlation:
   lmTest(y1 ~ x, "bg")
   # ... or for fourth order serial correlation:
   lmTest(y1 ~ x, "bg", order = 4)    
   # Compare with Durbin-Watson Test Results:
   lmTest(y1 ~ x, "dw")
   y2 = filter(y1, 0.5, method = "recursive")
   lmTest(y2 ~ x, "bg") 
   
## bp -
   # Generate a Regressor:
   x = rep(c(-1, 1), 50)
   # Generate heteroskedastic and homoskedastic Disturbances
   err1 = rnorm(100, sd = rep(c(1, 2), 50))
   err2 = rnorm(100)
   # Generate a Linear Relationship:
   y1 = 1 + x + err1
   y2 = 1 + x + err2
   # Perform Breusch-Pagan Test
   bp = lmTest(y1 ~ x, "bp")
   bp
   # Calculate Critical Value for 0.05 Level
   qchisq(0.95, bp$parameter)
   lmTest(y2 ~ x, "bp")
   
## dw -
   # Generate two AR(1) Error Terms 
   # with parameter rho = 0 (white noise) 
   # and rho = 0.9 respectively
   err1 = rnorm(100)
   # Generate Regressor and Dependent Variable
   x = rep(c(-1,1), 50)
   y1 = 1 + x + err1
   # Perform Durbin-Watson Test:
   lmTest(y1 ~ x, "dw")
   err2 = filter(err1, 0.9, method = "recursive")
   y2 = 1 + x + err2
   lmTest(y2 ~ x, "dw")
   
## gq -
   # Generate a Regressor:
   x = rep(c(-1, 1), 50)
   # Generate Heteroskedastic and Homoskedastic Disturbances:
   err1 = c(rnorm(50, sd = 1), rnorm(50, sd = 2))
   err2 = rnorm(100)
   # Generate a Linear Relationship:
   y1 = 1 + x + err1
   y2 = 1 + x + err2
   # Perform Goldfeld-Quandt Test:
   lmTest(y1 ~ x, "gq")
   lmTest(y2 ~ x, "gq")
   
## harv -
   # Generate a Regressor and Dependent Variable:
   x = 1:50
   y1 = 1 + x + rnorm(50)
   y2 = y1 + 0.3*x^2
   # Perform Harvey-Collier Test:
   harv = lmTest(y1 ~ x, "harv")
   harv
   # Calculate Critical Value vor 0.05 level:
   qt(0.95, harv$parameter)
   lmTest(y2 ~ x, "harv")
   
## hmc -
   # Generate a Regressor:
   x = rep(c(-1, 1), 50)
   # Generate Heteroskedastic and Homoskedastic Disturbances:
   err1 = c(rnorm(50, sd = 1), rnorm(50, sd = 2))
   err2 = rnorm(100)
   # Generate a Linear Relationship:
   y1 = 1 + x + err1
   y2 = 1 + x + err2
   # Perform Harrison-McCabe Test:
   lmTest(y1 ~ x, "hmc")
   lmTest(y2 ~ x, "hmc")
   
## rain -
   # Generate Series:
   x = c(1:30)
   y = x^2 + rnorm(30, 0, 2)
   # Perform rainbow Test
   rain = lmTest(y ~ x, "rain")
   rain
   # Compute Critical Value:
   qf(0.95, rain$parameter[1], rain$parameter[2]) 
   
## reset -
   # Generate Series:
   x = c(1:30)
   y1 = 1 + x + x^2 + rnorm(30)
   y2 = 1 + x + rnorm(30)
   # Perform RESET Test:
   lmTest(y1 ~ x , "reset", power = 2, type = "regressor")
   lmTest(y2 ~ x , "reset", power = 2, type = "regressor")

---

Regression Model Simulation

Description

Simulates regression models.

Usage

regSim(model = "LM3", n = 100, ...)

LM3(n = 100, seed = 4711)
LOGIT3(n = 100, seed = 4711)
GAM3(n = 100, seed = 4711)

Arguments

model	a character string defining the function name from which the regression model will be simulated.
n	an integer value setting the length, i.e. the number of records of the output series, an integer value. By default n=100.
seed	an integer value, the recommended way to specify seeds for random number generation.
...	arguments to be passed to the underlying function specified by the model argument.

Details

The function regSim allows to simulate from various regression models defined by one of the three example functions LM3, LOGIT3, GAM3 or by a user specified function.

The examples are defined in the following way:

# LM3:
> y = 0.75 * x1 + 0.25 * x2 - 0.5 * x3 + 0.1 * eps

# LOGIT3:
> y = 1 / (1 + exp(- 0.75 * x1 + 0.25 * x2 - 0.5 * x3 + eps))

# GAM3:
> y = scale(scale(sin(2 * pi * x1)) + scale(exp(x2)) + scale(x3))
> y = y + 0.1 * rnorm(n, sd = sd(y))

"LM3" models a liner regression model, "LOGIT3" a generalized linear regression model expressed by a logit model, and "GAM" an additive model. x1, x2, x3, and eps are random normal deviates of length n.

The model function should return an rectangular series defined as an object of class data.frame, timeSeries or mts which can be accepted from the parameter estimation functions regFit and gregFit.

Value

The function garchSim returns an object of the same class as returned by the underlying function match.fun(model). These may be objects of class data.frame, timeSeries or mts.

Note

This function is still under development. For the future we plan, that the function regSim will be able to generate general regression models.

Author(s)

Diethelm Wuertz for the Rmetrics R-port.

Examples

## LM2 -
   # Data for a user defined linear regression model:
   LM2 = function(n){
     x = rnorm(n)
     y = rnorm(n)
     eps = 0.1 * rnorm(n)
     z = 0.5 + 0.75 * x + 0.25 * y + eps
     data.frame(Z = z, X = x, Y = y)
   }
   for (FUN in c("LM2", "LM3")) {
     cat(FUN, ":\n", sep = "")
     print(regSim(model = FUN, n = 10))
   }

---

Extract Regression Model Residuals

Description

Extracts residuals from a fitted regression object.

Usage

## S4 method for signature 'fREG'
residuals(object)

Arguments

object

an object of class fREG as returned from the function regFit() or gregFit().

Methods

object = "ANY"

Generic function

object = "fREG"

Residuals

Note

residuals is a generic function which extracts residual values from objects returned by modeling functions.

Author(s)

Diethelm Wuertz for the Rmetrics R-port.

Examples

## regSim -
   x = regSim(model = "LM3", n = 50)
  
## regFit -
   fit = regFit(Y ~ X1 + X2 + X3, data = x, use = "lm")
   
## residuals - 
   residuals(fit)

---

Regression Modelling Show Methods

Description

Show methods for regression modelling.

Details

The show or print method returns the same information for all supported regression models through the use argument in the function regFit.

These are the 'title', the 'formula', the 'family' and the 'model parameters'.

Methods

object = "ANY"

Generic function.

object = "fREG"

Print method for objects of class 'fREG'.

Author(s)

Diethelm Wuertz for the Rmetrics R-port.

Examples

## regSim -
   x <- regSim(model = "LM3", n = 50)
  
## regFit -
   fit <- regFit(Y ~ X1 + X2 + X3, data = x, use = "lm")
   
## print -
   print(fit)

---

Regression Summary Methods

Description

Summary methods for regressing modelling.

Methods

object = "ANY"

Generic function

object = "fREG"

Summary method for objects of class 'fREG'.

Author(s)

Diethelm Wuertz for the Rmetrics R-port.

Examples

## regSim -
   x <- regSim(model = "LM3", n = 50)
  
## regFit -
   fit <- regFit(Y ~ X1 + X2 + X3, data = x, use = "lm")
   
## summary
   summary(fit)

---

Regression Model Plot Methods

Description

Plots results obtained from a fitted regression model.

Usage

## S3 method for class 'fREG'
termPlot(model, ...)

Arguments

model	an object of class 'fREG'.
...	additional arguments to be passed to the underlying functions.

Methods

x = "ANY"

Generic function.

x = "fREG"

Term plot function.

Author(s)

Diethelm Wuertz for the Rmetrics R-port.

Examples

## regSim -
   x <- regSim(model = "LM3", n = 50)
  
## regFit -
   fit <- regFit(Y ~ X1 + X2 + X3, data = x, use = "lm")

---

Regression Model Plot Methods

Description

Plots results obtained from a fitted regression model.

Usage

## S4 method for signature 'fREG'
terms(x, ...)

Arguments

x	an object of class 'fREG'.
...	additional arguments to be passed to the underlying functions.

Methods

x = "ANY"

Generic function.

x = "fREG"

Terms extractor function.

Author(s)

Diethelm Wuertz for the Rmetrics R-port.

Examples

## regSim -
   x <- regSim(model = "LM3", n = 50)
  
## regFit -
   fit <- regFit(Y ~ X1 + X2 + X3, data = x, use = "lm")

---

Extract Regression Model vcov

Description

Extracts vcov from a fitted regression model.

Methods

object = "ANY"

Generic function

object = "fREG"

Extractor function for vcov.

Note

vcov is a generic function which extracts fitted values from objects returned by modeling functions, here the regFit and gregFit parameter estimation functions.

Author(s)

Diethelm Wuertz for the Rmetrics R-port.

Examples

## regSim -
   x <- regSim(model = "LM3", n = 50)
  
## regFit -
   fit <- regFit(Y ~ X1 + X2 + X3, data = x, use = "lm")
   
## vcov - 
   vcov(fit)

---