Estimators of semi-parametric truncated regression models

Description

Functions for estimation of semi-parametric linear regression models with truncated response variables (fixed truncation point). Estimation using the Symmetrically Trimmed Least Squares (STLS) estimator (Powell 1986), Quadratic Mode (QME) estimator (Lee 1993) and Left Truncated (LT) estimator (Karlsson 2006).

Author(s)

Maintainer: Anita Lindmark anita.lindmark@umu.se [copyright holder]

Authors:

• Maria Karlsson maria.karlsson@umu.se

Air pollution data

Description

The data are a subsample of 500 observations from a data set that originates in a study where air pollution at a road is related to traffic volume and meteorological variables, collected by the Norwegian Public Roads Administration. The response variable consists of hourly values of the logarithm of the concentration of PM10 (particles), measured at Alnabru in Oslo, Norway, between October 2001 and August 2003. (Source: Statlib)

Usage

data(PM10)

Format

A data frame with 500 observations on the following 8 variables.

PM10

Hourly values of the logarithm of the concentration of PM10 (particles)

cars

The logarithm of the number of cars per hour

temp

Temperature 2 meters above ground (degree C)

wind.speed

Wind speed (meters/second)

temp.diff

The temperature difference between 25 and 2 meters above ground (degree C)

wind.dir

Wind direction (degrees between 0 and 360)

hour

Hour of day

day

Day number from October 1, 2001

Source

Aldrin, M. (2006) Improved predictions penalizing both slope and curvature in additive models, Computational Statistics & Data Analysis, 50, pp 267–284

References

Aldrin, M. (2006) Improved predictions penalizing both slope and curvature in additive models, Computational Statistics & Data Analysis, 50, pp 267–284

Examples

data(PM10)

Air pollution data (Truncated)

Description

Dataset PM10, truncated from the left at variable value PM10 = 2 (8 percent truncation).

Usage

data(PM10)

Format

A data frame with 460 observations on the following 8 variables.

PM10

Hourly values of the logarithm of the concentration of PM10 (particles)

cars

The logarithm of the number of cars per hour

temp

Temperature 2 meters above ground (degree C)

wind.speed

Wind speed (meters/second)

temp.diff

The temperature difference between 25 and 2 meters above ground (degree C)

wind.dir

Wind direction (degrees between 0 and 360)

hour

Hour of day

day

Day number from October 1, 2001

Examples

data(PM10trunc)

Function to obtain bootstrap confidence intervals (percentile)

Description

Function to obtain bootstrap confidence intervals (percentile)

Usage

bootconfint(bootrepl, level)

Arguments

Value

The function returns the bootstrap confidence intervals as a two-column matrix with one row per parameter

Function to extract model coefficients for objects of class "lt"

Description

Function to extract model coefficients for objects of class "lt"

Usage

## S3 method for class 'lt'
coef(object, ...)

Arguments

Function to extract model coefficients for objects of class "qme"

Description

Function to extract model coefficients for objects of class "qme"

Usage

## S3 method for class 'qme'
coef(object, ...)

Arguments

Function to extract model coefficients for objects of class "stls"

Description

Function to extract model coefficients for objects of class "stls"

Usage

## S3 method for class 'stls'
coef(object, ...)

Arguments

Function to obtain bootstrap covariance matrix

Description

Function to obtain bootstrap covariance matrix

Usage

covar.boot(bootrepl)

Arguments

Function to obtain fitted values from objects of class "lt"

Description

Function to obtain fitted values from objects of class "lt"

Usage

## S3 method for class 'lt'
fitted(object, ...)

Arguments

Function to obtain fitted values from objects of class "qme"

Description

Function to obtain fitted values from objects of class "qme"

Usage

## S3 method for class 'qme'
fitted(object, ...)

Arguments

Function to obtain fitted values from objects of class "stls"

Description

Function to obtain fitted values from objects of class "stls"

Usage

## S3 method for class 'stls'
fitted(object, ...)

Arguments

Function to be used by boot to obtain bootstrap replicates of the parameters estimated with QME

Description

Function to be used by boot to obtain bootstrap replicates of the parameters estimated with QME

Usage

funcLT(
  mf,
  mf_orig,
  d,
  formula,
  beta,
  bet,
  clower,
  const,
  cupper,
  point,
  direction,
  control
)

Arguments

Author(s)

Anita Lindmark and Maria Karlsson

Function to be used by boot to obtain bootstrap replicates of the parameters estimated with QME

Description

Function to be used by boot to obtain bootstrap replicates of the parameters estimated with QME

Usage

funcQME(
  mf,
  mf_orig,
  d,
  formula,
  beta,
  bet,
  cval,
  const,
  point,
  direction,
  control
)

Arguments

Author(s)

Anita Lindmark and Maria Karlsson

Function to be used by boot to obtain bootstrap replicates of the parameters estimated with STLS

Description

Function to be used by boot to obtain bootstrap replicates of the parameters estimated with STLS

Usage

funcSTLS(mf, mf_orig, d, formula, beta, bet, point, direction, control)

Arguments

Author(s)

Anita Lindmark and Maria Karlsson

LT objective function

Description

LT objective function

Usage

funcval.LT(bet, x, y, cl, cu)

Arguments

Value

Returns the value of the objective function for given parameter values

Author(s)

Anita Lindmark and Maria Karlsson

See Also

lt

QME objective function

Description

QME objective function

Usage

funcval.QME(bet, x, y, cv)

Arguments

Value

Returns the value of the objective function for given parameter values

Author(s)

Anita Lindmark and Maria Karlsson

See Also

qme

STLS objective function

Description

STLS objective function

Usage

funcval.STLS(bet, x, y)

Arguments

Value

Returns the value of the objective function for given parameter values

Author(s)

Anita Lindmark and Maria Karlsson

See Also

stls

Estimation of truncated regression models using the Left Truncated (LT) estimator

Description

Estimates linear regression models with truncated response variables (fixed truncation point), using the LT estimator (Karlsson 2006).

Usage

lt(
  formula,
  data,
  point = 0,
  direction = "left",
  clower = "ml",
  const = 1,
  cupper = 2,
  beta = "ml",
  covar = FALSE,
  na.action,
  ...
)

Arguments

Details

Minimizes the objective function described in Karlsson (2006) wrt the vector of regression coefficients, in order to find the LT estimates. The minimization is performed by optim using the "Nelder–Mead" method, and a maximum number of iterations of 2000. The maximum number of iterations can be adjusted by setting control = list(maxit = ...) (for more information see the documentation for optim).

It is recommended to use one of the methods for generating the starting values of the regression coefficients (see argument beta) rather than supplying these manually, unless one is confident that one has a good idea of what these should be. This because the starting values can have a great impact on the result of the minimization.

Note that setting cupper = 1 means that the LT estimates will coincide with the estimates from the Quadratic Mode Estimator (see function qme). For more detailed information see Karlsson and Lindmark (2014).

Value

lt returns an object of class "lt".

The function summary prints a summary of the results, including two types of confidence intervals (normal approximation and percentile method). The generic accessor functions coef, fitted, residuals and vcov extract various useful features of the value returned by lt

An object of class "lt", a list with elements:

Author(s)

Anita Lindmark and Maria Karlsson

References

Karlsson, M. (2006) Estimators of regression parameters for truncated and censored data, Metrika, 63, pp 329–341

Karlsson, M., Lindmark, A. (2014) truncSP: An R Package for Estimation of Semi-Parametric Truncated Linear Regression Models, Journal of Statistical Software, 57(14), pp 1–19, https://www.jstatsoft.org/article/view/v057i14

See Also

lt.fit, the function that does the actual fitting

qme, for estimation of models with truncated response variables using the QME estimator

stls, for estimation of models with truncated response variables using the STLS estimator

truncreg for estimating models with truncated response variables by maximum likelihood, assuming Gaussian errors

Examples

####################################
#          Simulated data          #
####################################
#Simulated data (asymmetrically distributed errors):
n <- 1000
set.seed(319993)
x1 <- runif(n, 0, 10)
x2 <- runif(n, 0, 10)
x3 <- runif(n, -5, 5)
eps <- rexp(n, 0.2) - 5
y <- 2 - 2*x1 + x2 + 2*x3 + eps
d <- data.frame(y = y, x1 = x1, x2 = x2, x3 = x3)


##Use a truncated subsample
dtrunc <- subset(d, y > 0)

##Use lt to consistently estimate the slope parameters
lt.sim <- lt(y ~ x1 + x2 + x3, dtrunc, point = 0, direction = "left", clower = "ml",
           const = 1, cupper = 2, beta = "ml", covar = FALSE)

summary(lt.sim)

####################################
#  Example using data "PM10trunc"  #
####################################

data(PM10trunc)

ltpm10 <- lt(PM10 ~ cars + temp + wind.speed + temp.diff + wind.dir + hour + day,
            data = PM10trunc, point = 2, control = list(maxit=2500))

summary(ltpm10)

An S4 class for objects from the lt function

Description

An S4 class for objects from the lt function

Slots

call

The matched call

coefficients

A matrix

startcoef

A matrix

cvalues

A data frame

value

Numeric

counts

Integer

convergence

Integer

message

A character vector

residuals

A matrix

fitted.values

A matrix

df.residual

Integer

covariance

A matrix

bootrepl

A matrix

Function for fitting LT

Description

Function to find LT estimates of the regression coefficients for regression models with truncated response variables. Uses optim. Intended to be called through lt, not on its own, since lt also transforms data into the correct form etc.

Usage

lt.fit(formula, mf, point, direction, bet, cl, cu, ...)

Arguments

Value

A list with components:

Author(s)

Anita Lindmark and Maria Karlsson

See Also

lt

Examples

require(utils)
##Model frame
n <- 1000
x <- rnorm(n,0,2)
y <- 2+x+4*rnorm(n)

d <- data.frame(y=y, x=x)
dl0 <- subset(d, y>0)
mf <- model.frame(y~x, data=dl0)

##Starting values and threshold values
lmmod <- lm(data=mf)
bet <- lmmod$coef
bet <- matrix(bet)
cl <- sqrt(deviance(lmmod)/df.residual(lmmod))
cu <- 2*cl

str(lt. <- lt.fit(y~x,mf,point=0,direction="left",bet,cl,cu))

Gives the starting coefficients from fitting a maximum likelihood model for truncated regression, assuming Gaussian errors

Description

Gives the starting coefficients from fitting a maximum likelihood model for truncated regression, assuming Gaussian errors. Uses the truncreg function from the truncreg package.

Usage

mlcoef(mf, mf_orig)

Arguments

Author(s)

Anita Lindmark and Maria Karlsson

Print function for objects of class "lt"

Description

Print function for objects of class "lt"

Usage

## S3 method for class 'lt'
print(x, digits = max(3, getOption("digits") - 3), ...)

Arguments

Print function for objects of class "qme"

Description

Print function for objects of class "qme"

Usage

## S3 method for class 'qme'
print(x, digits = max(3, getOption("digits") - 3), ...)

Arguments

Print function for objects of class "stls"

Description

Print function for objects of class "stls"

Usage

## S3 method for class 'stls'
print(x, digits = max(3, getOption("digits") - 3), ...)

Arguments

Print function for objects of class "summary.lt"

Description

Print function for objects of class "summary.lt"

Usage

## S3 method for class 'summary.lt'
print(x, digits = max(3, getOption("digits") - 3), ...)

Arguments

Print function for objects of class "summary.qme"

Description

Print function for objects of class "summary.qme"

Usage

## S3 method for class 'summary.qme'
print(x, digits = max(3, getOption("digits") - 3), ...)

Arguments

Print function for objects of class "summary.stls"

Description

Print function for objects of class "summary.stls"

Usage

## S3 method for class 'summary.stls'
print(x, digits = max(3, getOption("digits") - 3), ...)

Arguments

Estimation of truncated regression models using the Quadratic Mode Estimator (QME)

Description

Estimation of linear regression models with truncated response variables (fixed truncation point), using the Quadratic Mode Estimator (QME) (Lee 1993 and Laitila 2001)

Usage

qme(
  formula,
  data,
  point = 0,
  direction = "left",
  cval = "ml",
  const = 1,
  beta = "ml",
  covar = FALSE,
  na.action,
  ...
)

Arguments

Details

Finds the QME estimates of the regression coefficients by maximizing the objective function described in Lee (1993) wrt the vector of regression coefficients. The maximization is performed by optim using the "Nelder–Mead" method. The maximum number of iterations is set at 2000, but this can be adjusted by setting control=list(maxit=...) (for more information see the documentation for optim).

The starting values of the regression coefficients can have a great impact on the result of the maximization. For this reason it is recommended to use one of the methods for generating these rather than supplying the values manually, unless one is confident that one has a good idea of what the starting values should be. For more detailed information see Karlsson and Lindmark (2014).

Value

qme returns an object of class "qme".

The function summary prints a summary of the results, including two types of confidence intervals (normal approximation and percentile method). The generic accessor functions coef, fitted, residuals and vcov extract various useful features of the value returned by qme

An object of class "qme", a list with elements:

Author(s)

Anita Lindmark and Maria Karlsson

References

Karlsson, M. (2004) Finite sample properties of the QME, Communications in Statistics - Simulation and Computation, 5, pp 567–583

Karlsson, M., Lindmark, A. (2014) truncSP: An R Package for Estimation of Semi-Parametric Truncated Linear Regression Models, Journal of Statistical Software, 57(14), pp 1–19, https://www.jstatsoft.org/article/view/v057i14

Laitila, T. (2001) Properties of the QME under asymmetrically distributed disturbances, Statistics & Probability Letters, 52, pp 347–352

Lee, M. (1993) Quadratic mode regression, Journal of Econometrics, 57, pp 1-19

Lee, M. & Kim, H. (1998) Semiparametric econometric estimators for a truncated regression model: a review with an extension, Statistica Neerlandica, 52(2), pp 200–225

See Also

qme.fit, the function that does the actual fitting

stls, for estimation of models with truncated response variables using the STLS estimator

lt, for estimation of models with truncated response variables using the LT estimator

truncreg for estimating models with truncated response variables by maximum likelihood, assuming Gaussian errors

Examples

####################################
#          Simulated data          #
####################################
#Simulated data (asymmetrically distributed errors):
n <- 1000
set.seed(319993)
x1 <- runif(n, 0, 10)
x2 <- runif(n, 0, 10)
x3 <- runif(n, -5, 5)
eps <- rexp(n, 0.2) - 5
y <- 2 - 2*x1 + x2 + 2*x3 + eps
d <- data.frame(y = y, x1 = x1, x2 = x2, x3 = x3)

##Use a truncated subsample
dtrunc <- subset(d, y > 0)

##Use qme to consistently estimate the slope parameters
qme.sim <- qme(y ~ x1 + x2 + x3, dtrunc, point = 0, direction = "left",
              cval = "ml", const = 1, beta = "ml", covar = FALSE)


summary(qme.sim)

####################################
#  Example using data "PM10trunc"  #
####################################

data(PM10trunc)

qmepm10 <- qme(PM10 ~ cars + temp + wind.speed + temp.diff + wind.dir + hour + day,
              data = PM10trunc, point = 2, control = list(maxit=4500))

summary(qmepm10)

An S4 class for objects from the qme function

Description

An S4 class for objects from the qme function

Slots

call

The matched call

coefficients

A matrix

startcoef

A matrix

cval

A data frame

value

Numeric

counts

Integer

convergence

Integer

message

A character vector

residuals

A matrix

fitted.values

A matrix

df.residual

Integer

covariance

A matrix

bootrepl

A matrix

Function for fitting QME

Description

Function to find QME estimates of the regression coefficients for regression models with truncated response variables. Uses optim. Intended to be called through qme, not on its own, since qme also transforms data into the correct form etc.

Usage

qme.fit(formula, mf, point, direction, bet, cv, ...)

Arguments

Value

A list with components:

Author(s)

Anita Lindmark and Maria Karlsson

See Also

qme

Examples

require(utils)

##Model frame
n <- 1000
x <- rnorm(n,0,2)
y <- 2+x+4*rnorm(n)

d <- data.frame(y=y, x=x)
dl0 <- subset(d, y>0)
mf <- model.frame(y~x, data=dl0)

##Starting values and threshold value
lmmod <- lm(data=mf)
bet <- lmmod$coef
bet <- matrix(bet)
cv <- sqrt(deviance(lmmod)/df.residual(lmmod))

str(qme. <- qme.fit(y~x,mf,point=0,direction="left",bet,cv))

Function to extract model residuals from objects of class "lt"

Description

Function to extract model residuals from objects of class "lt"

Usage

## S3 method for class 'lt'
residuals(object, ...)

Arguments

Function to extract model residuals from objects of class "stls"

Description

Function to extract model residuals from objects of class "stls"

Usage

## S3 method for class 'qme'
residuals(object, ...)

Arguments

Function to extract model residuals from objects of class "stls"

Description

Function to extract model residuals from objects of class "stls"

Usage

## S3 method for class 'stls'
residuals(object, ...)

Arguments

Estimation of truncated regression models using the Symmetrically Trimmed Least Squares (STLS) estimator

Description

Function for estimation of linear regression models with truncated response variables (fixed truncation point), using the STLS estimator (Powell 1986)

Usage

stls(
  formula,
  data,
  point = 0,
  direction = "left",
  beta = "ml",
  covar = FALSE,
  na.action,
  ...
)

Arguments

Details

Uses optim ("Nelder–Mead" method) to minimize the objective function described in Powell (1986) wrt the vector of regression coefficients in order to find the STLS estimates (see Karlsson and Lindmark 2014 for more detailed information and background). The maximum number of iterations is set at 2000, but this can be adjusted by setting control=list(maxit=...) (for more information see the documentation for optim).

As the starting values of the regression coefficients can have a great impact on the result of the minimization it is recommended to use one of the methods for generating these rather than supplying the values manually (unless one is confident that one has a good idea of what the starting values should be).

Value

stls returns an object of class "stls".

The function summary prints a summary of the results, including two types of confidence intervals (normal approximation and percentile method). The generic accessor functions coef, fitted, residuals and vcov extract various useful features of the value returned by stls

An object of class "stls", a list with elements:

Author(s)

Anita Lindmark and Maria Karlsson

References

Karlsson, M., Lindmark, A. (2014) truncSP: An R Package for Estimation of Semi-Parametric Truncated Linear Regression Models, Journal of Statistical Software, 57(14), pp 1–19, https://www.jstatsoft.org/article/view/v057i14

Powell, J. (1986) Symmetrically Trimmed Least Squares Estimation for Tobit Models, Econometrika, 54(6), pp 1435–1460

See Also

stls.fit, the function that does the actual fitting

qme, for estimation of models with truncated response variables using the QME estimator

lt, for estimation of models with truncated response variables using the LT estimator

truncreg for estimating models with truncated response variables by maximum likelihood, assuming Gaussian errors

Examples

####################################
#          Simulated data          #
####################################
##Simulated data
n <- 1000
set.seed(319993)
x1 <- runif(n, 0, 10)
x2 <- runif(n, 0, 10)
x3 <- runif(n, -5, 5)
y <- 1 - 2*x1 + x2 + 2*x3 + rnorm(n,0,2)
d <- data.frame(y = y, x1 = x1, x2 = x2, x3 = x3)

##Use a truncated subsample
dtrunc <- subset(d, y > 0)

##Use stls to estimate the model
stls.sim <- stls(y ~ x1 + x2 + x3, dtrunc, point = 0,
                direction = "left", beta = "ml", covar = FALSE)

summary(stls.sim)

####################################
#  Example using data "PM10trunc"  #
####################################

data(PM10trunc)

stlspm10 <- stls(PM10 ~ cars + temp + wind.speed + temp.diff + wind.dir + hour + day,
               data = PM10trunc, point = 2)

summary(stlspm10)

An S4 class for objects from the stls function

Description

An S4 class for objects from the stls function

Slots

call

The matched call

coefficients

A matrix

startcoef

A matrix

value

Numeric

counts

Integer

convergence

Integer

message

A character vector

residuals

A matrix

fitted.values

A matrix

df.residual

Integer

covariance

A matrix

bootrepl

A matrix

Function for fitting STLS

Description

Function that utilizes optim to find STLS estimates of the regression coefficients for regression models with truncated response variables. Intended to be called through stls, not on its own, since stls also transforms data into the correct form etc.

Usage

stls.fit(formula, mf, point, direction, bet, ...)

Arguments

Value

A list with components:

Author(s)

Anita Lindmark and Maria Karlsson

See Also

stls

Examples

require(utils)

##Model frame
n <- 1000
x <- rnorm(n,0,2)
y <- 2+x+4*rnorm(n)

d <- data.frame(y=y, x=x)
dl0 <- subset(d, y>0)
mf <- model.frame(y~x, data=dl0)

##Starting values
lmmod <- lm(data=mf)
bet <- lmmod$coef
bet <- matrix(bet)

str(stls. <- stls.fit(y~x,mf,point=0,direction="left",bet))

Summary function for objects of class "lt"

Description

Summary function for objects of class "lt"

Usage

## S3 method for class 'lt'
summary(object, level = 0.95, ...)

Arguments

Value

A list with values:

An S4 class for objects from the summary.lt function

Description

An S4 class for objects from the summary.lt function

Slots

call

The matched call

coefficients

A matrix

startcoef

A matrix

cvalues

A data frame

value

Numeric

counts

Integer

convergence

Integer

message

A character vector

residuals

A matrix

fitted.values

A matrix

df.residual

Integer

covariance

A matrix

bootrepl

A matrix

level

Numeric

confint

A matrix

bootconfint

A matrix

Summary function for objects of class "qme"

Description

Summary function for objects of class "qme"

Usage

## S3 method for class 'qme'
summary(object, level = 0.95, ...)

Arguments

Value

A list with values:

An S4 class for objects from the summary.qme function

Description

An S4 class for objects from the summary.qme function

Slots

call

The matched call

coefficients

A matrix

startcoef

A matrix

cval

A data frame

value

Numeric

counts

Integer

convergence

Integer

message

A character vector

residuals

A matrix

fitted.values

A matrix

df.residual

Integer

covariance

A matrix

bootrepl

A matrix

level

Numeric

confint

A matrix

bootconfint

A matrix

Summary function for objects of class "stls"

Description

Summary function for objects of class "stls"

Usage

## S3 method for class 'stls'
summary(object, level = 0.95, ...)

Arguments

Value

A list with values:

An S4 class for objects from the summary.stls function

Description

An S4 class for objects from the summary.stls function

Slots

call

The matched call

coefficients

A matrix

startcoef

A matrix

value

Numeric

counts

Integer

convergence

Integer

message

A character vector

residuals

A matrix

fitted.values

A matrix

df.residual

Integer

covariance

A matrix

bootrepl

A matrix

level

Numeric

confint

A matrix

bootconfint

A matrix

Function to obtain the variance-covariance matrix for objects of class "lt"

Description

Function to obtain the variance-covariance matrix for objects of class "lt"

Usage

## S3 method for class 'lt'
vcov(object, ...)

Arguments

Function to obtain the variance-covariance matrix for objects of class "qme"

Description

Function to obtain the variance-covariance matrix for objects of class "qme"

Usage

## S3 method for class 'qme'
vcov(object, ...)

Arguments

Function to obtain the variance-covariance matrix for objects of class "stls"

Description

Function to obtain the variance-covariance matrix for objects of class "stls"

Usage

## S3 method for class 'stls'
vcov(object, ...)

Arguments