Structural Breaks in Quantile Regression

Description

Methods for detecting structural breaks, determining the number of breaks, and estimating break locations in linear quantile regression, using a single or multiple quantiles, based on Qu (2008) and Oka and Qu (2011). Applicable to both time series and repeated cross-sectional data.

Main Functions

• rq.break: Main function for detecting structural breaks in quantile regression

• sq: Performs structural break testing using single quantile approach

• dq: Tests for breaks using multiple quantiles

• brdate: Estimates potential break dates

Author(s)

Maintainer: Zhongjun Qu qu@bu.edu

Authors:

• Tatsushi Oka oka.econ@gmail.com

Other contributors:

• Samuel Messer snmesser@bu.edu [contributor]

References

Qu, Z. (2008). Testing for Structural Change in Regression Quantiles. Journal of Econometrics, 146(1), 170-184 doi:10.1016/j.jeconom.2008.08.006

Oka, T., and Qu, Z. (2011). Estimating Structural Changes in Regression Quantiles. Journal of Econometrics, 162(2), 248-267 doi:10.1016/j.jeconom.2011.01.005

Estimating Break Dates in a Quantile Regression

Description

This function estimates break dates in a quantile regression model, allowing for up to m breaks. When m = 1, this function finds the optimal single-break partition within a segment by evaluating the objective function at each possible break point to determine the break date. When m > 1, a dynamic programming algorithm is used to efficiently determine the break dates.

Usage

brdate(y, x, n.size = 1, m, trim.size, vec.long)

Arguments

Details

The function first determines the optimal one-break partition. If multiple breaks are allowed (m > 1), it applies a dynamic programming algorithm as in Bai and Perron (2003) to minimize the objective function. The algorithm finds break dates by iterating over all possible partitions, returning optimal break locations and associated objective function values.

Value

An upper triangular matrix (⁠m × m⁠) containing estimated break dates. The row index represents break dates, and the column index corresponds to the permitted number of breaks before the ending date.

References

Bai, J. and P. Perron (2003). Computation and analysis of multiple structural change models. Journal of Applied Econometrics, 18(1), 1-22.

Oka, T. and Z. Qu (2011). Estimating structural changes in regression quantiles, Journal of Econometrics, 162(2), 248-267.

Examples

  # data
  data(gdp)
  y = gdp$gdp
  x = gdp[,c("lag1", "lag2")]
  n.size = 1
  T.size = length(y) # number of time periods

  # setting
  vec.tau   = seq(0.20, 0.80, by = 0.150)
  trim.e    = 0.2
  trim.size = round(T.size * trim.e)  #minimum length of a regime
  m.max     = 3

  # compute the objective function under all possible partitions
  out.long   = gen.long(y, x, vec.tau, n.size, trim.size)
  vec.long.m = out.long$vec.long  ## for break estimation using multiple quantiles

  # break date
  mat.date = brdate(y, x, n.size, m.max, trim.size, vec.long.m)
  print(mat.date)

Confidence Intervals for Break Dates

Description

This function constructs confidence intervals for break dates based on a single quantile or multiple quantiles (specified by the user).

Usage

ci.date.m(y, x, vec.tau, vec.date, n.size = 1, v.b = 2)

Arguments

Value

A numeric matrix where:

• The 1st column contains the break dates.

• The 2nd and 3rd columns contain the lower and upper bounds of the confidence intervals, respectively.

References

Oka, T. and Z. Qu (2011). Estimating Structural Changes in Regression Quantiles. Journal of Econometrics, 162(2), 248–267.

Examples

# data 
data(gdp)
y = gdp$gdp
x = gdp[,c("lag1", "lag2")] 

# quantiles 
vec.tau  = 0.8

# break dates (point estimates)
vec.date = c(146, 200)

# Calculate confidence intervals for break dates
res = ci.date.m(y, x, vec.tau, vec.date, n.size = 1, v.b = 2)
print(res)

Sequential Determination of the Number of Breaks Using the DQ Test

Description

This function determines the number of breaks in a model by sequentially applying the DQ(l | l+1) test. It tests for additional breaks by comparing the test statistic to critical values at various significance levels.

Usage

dq(
  y,
  x,
  vec.tau,
  q.L,
  q.R,
  n.size = 1,
  m.max,
  trim.size,
  mat.date,
  d.Sym,
  table.cv
)

Arguments

Value

A list containing:

test

A numeric vector of DQ test statistics.

cv

A numeric matrix of critical values for the DQ test at 10, 5, and 1 percent significance levels.

date

A numeric matrix of estimated break dates.

nbreak

A numeric vector indicating the number of detected breaks at different significance levels.

Examples

# This example may take substantial time for automated package 
# checks since it involves dynamic programming
data(gdp)
y = gdp$gdp
x = gdp[,c("lag1", "lag2")]
n.size = 1
T.size = length(y) # number of time periods

# setting
vec.tau   = seq(0.20, 0.80, by = 0.150)
trim.e    = 0.2
trim.size = round(T.size * trim.e)  #minimum length of a regime
m.max     = 3

# dynamic program algorithm to compute the objective function
out.long   = gen.long(y, x, vec.tau, n.size, trim.size)
mat.long.s = out.long$mat.long  ## for break estimation using individual quantile
vec.long.m = out.long$vec.long  ## for break estimation using multiple quantiles jointly

# break date
mat.date = brdate(y, x, n.size, m.max, trim.size, vec.long.m)

## qunatile ranges (left and right)
q.L   = 0.2
q.R   = 0.8
d.Sym = TRUE ## symmetric trimming of quantiles
table.cv = NULL ##covered by the response surface because d.Sym = TRUE

# determine the number of breaks
out.m = dq(y, x, vec.tau, q.L, q.R, n.size, m.max, trim.size, mat.date, d.Sym, table.cv)

# result
print(out.m)

Test for the Presence of a Break within a Range of Quantiles

Description

This procedure computes test statistics for detecting a single structural break within a range of quantiles. The null hypothesis is that there is no break in any quantile in the specified range; the alternative is that at least one quantile in the range is affected by a break.

Usage

dq.test.0vs1(y, x, q.L, q.R, n.size = 1)

Arguments

Value

A numeric scalar representing the DQ test statistic.

References

Koenker, R. and Bassett Jr, G. (1978). Regression quantiles. Econometrica, 46(1), 33–50.

Qu, Z. (2008). Testing for structural change in regression quantiles. Journal of Econometrics, 146(1), 170–184.

Examples

## data
data(gdp)
y = gdp$gdp
x = gdp[,c("lag1", "lag2")]

## qunatile range (left and right limits)
q.L = 0.2
q.R = 0.8

## N
n.size = 1

# dq-test
result = dq.test.0vs1(y, x, q.L, q.R, n.size)
print(result)

Sequential Test for Additional Breaks within a Range of Quantiles

Description

This function performs a sequential test to determine whether the number of breaks in a quantile regression model should be increased from L to L+1 using multiple quantiles.

Usage

dq.test.lvsl_1(y, x, q.L, q.R, n.size = 1, vec.date)

Arguments

Details

This procedure tests for the existence of L breaks against L+1 breaks based on multiple quantiles: H_0: L breaks vs. H_1: L+1 breaks.

Value

A numeric value representing the DQ test statistic.

References

Qu, Z. (2008). Testing for Structural Breaks in Regression Quantiles. Journal of Econometrics, 146(1), 170-184.

Examples

# Load data
data(gdp)
y = gdp$gdp
x = gdp[,c("lag1", "lag2")]

# Set quantile range (left and right limits)
q.L = 0.2
q.R = 0.8

# Set N parameter
n.size = 1

# Specify break date under H_0
vec.date = 146

# Run the test
result = dq.test.lvsl_1(y, x, q.L, q.R, n.size, vec.date)
print(result)

The Dataset for Young Drivers

Description

This is a repeated cross-sectional data set on young drivers (under 21 years old) involved in motor vehicle accidents in the state of California from 1983 to 2007 (quarterly data). The data are obtained from the National Highway Traffic Safety Administration (NHTSA), which include the blood alcohol concentration (BAC) of the driver, their age, gender, and whether the crash was fatal.

Usage

data(driver)

Format

A data frame with five variables:

• yq: Year and quarter ("Year Quarter" format, e.g., "1983 Q2").

• bac: The blood alcohol concentration (BAC) level of the driver.

• age: The driver's age.

• gender: A gender dummy, with 1 for male and 0 for female.

• winter: A dummy variable for the fourth quarter, with 1 for Q4 and 0 otherwise.

Details

Motor vehicle crashes are the leading cause of death among youth aged 15–20, with a high proportion involving drunk driving. The BAC level is an important measure of alcohol impairment. Oka and Qu (2011) used this data to examine whether and how young drivers' drinking behaviors have changed over time.

References

Oka, T. and Z. Qu (2011). Estimating Structural Changes in Regression Quantiles. Journal of Econometrics, 162(2), 248–267.

Examples

data(driver)
names(driver)
summary(driver)

US Real GDP Growth Data

Description

This dataset contains quarterly real GDP growth rates in the US from 1947 Q4 to 2009 Q2. It is used in Oka and Qu (2011) to examine whether and where distributional breaks occur in US GDP.

Usage

data(gdp)

Format

A data frame with four variables:

• yq A character vector representing the year and quarter (e.g., "1947 Q2").

• gdp A numeric vector of real GDP growth rates (annualized by multiplying by 4).

• lag1 A numeric vector representing the first-order lagged value of gdp.

• lag2 A numeric vector representing the second-order lagged value of gdp.

References

Oka, T. and Z. Qu (2011). Estimating Structural Changes in Regression Quantiles. Journal of Econometrics, 162(2), 248–267.

Examples

data(gdp)
names(gdp)
summary(gdp)

Dynamic Programming Algorithm

Description

This function computes the objective function values for all possible segments of the sample.

Usage

gen.long(y, x, vec.tau, n.size = 1, trim.size)

Arguments

Value

A list containing:

mat.long

A matrix of objective function values for separate quantiles.

vec.long

A matrix of objective function values for combined quantiles.

References

Bai, J and P. Perron (2003). Computation and Analysis of Multiple Structural Change Models. Journal of Applied Econometrics, 18(1), 1-22.

Examples

# This example may take substantial time for automated package checks
data(gdp)
y = gdp$gdp
x = gdp[,c("lag1", "lag2")]
n.size = 1
T.size = length(y) # number of time periods

# setting
vec.tau   = seq(0.20, 0.80, by = 0.150)
trim.e    = 0.2
trim.size = round(T.size * trim.e)  #minimum length of a regime

out.long   = gen.long(y, x, vec.tau, n.size, trim.size)

Compute Critical Values for the DQ test using a Response Surface

Description

This function returns critical values obtained from a response surface analysis. Note that this procedure only applies when the trimming is symmetric, i.e., q.R=1-q.L.

Usage

res.surface(p, l, q.L, q.R, d.Sym)

Arguments

Value

A numeric vector of length 3 containing critical values at the 10%, 5%, and 1% significance levels.

Examples

# The number of regerssors 
p = 5 
## The number of breaks under the null 
l = 2 

# qunatile range (left and right limits)
q.L = 0.2
q.R = 0.8 

# symmetric quantile trimming is true 
d.Sym = TRUE 

## critical values from response surface 
cvs = res.surface(p, l, q.L, q.R, d.Sym)

print(cvs)

Testing for Breaks and Estimating Break Dates and Sizes with Confidence Intervals

Description

This is the main function of this package for testing breaks in quantile regression models and estimating break dates and break sizes with corresponding confidence intervals.

Usage

rq.break(y, x, vec.tau, N, trim.e, vec.time, m.max, v.a, v.b, verbose)

Arguments

Value

A list containing:

• ⁠$s.out⁠: A list with break testing results, estimated break dates, confidence intervals, and coefficient estimates based on individual quantiles.

• ⁠$m.out⁠: A list with break testing results, estimated break dates, confidence intervals, and coefficient estimates obtained by testing and estimating breaks using multiple quantiles simultaneously.

Each list (s.out or m.out) contains:

• test_tau: A matrix of test statistics and critical values for break detection at quantile tau.

• nbreak_tau: The number of detected breaks at quantile tau.

• br_est_tau: A matrix of estimated break dates and their confidence intervals at quantile tau.

• br_est_time_tau: The same as br_est_tau, but with break dates reported in calendar format (if vec.time is provided and is not NULL).

• coef_tau: Estimated regression coefficients for each regime at quantile tau.

• bsize_tau: Break size estimates for each transition between regimes at quantile tau.

References

Koenker, R. and G. Bassett Jr. (1978). Regression quantiles. Econometrica, 46(1), 33-50.

Oka, T. and Z. Qu (2011). Estimating Structural Changes in Regression Quantiles. Journal of Econometrics, 162(2), 248-267.

Qu, Z. (2008). Testing for Structural Change in Regression Quantiles. Journal of Econometrics, 146(1), 170-184.

Examples

## Example 1
## Time series example, using GDP data
## data
data(gdp)
y        = gdp$gdp
x        = gdp[,c("lag1", "lag2")]
vec.time = gdp$yq

## the maximum number of breaks allowed
m.max = 3

## the signifance level for sequenatial testing
## 1, 2 or 3 for 10%, 5% or 1%, respectively
v.a = 2

## the significance level for the confidence intervals of estimated break dates.
## 1 or 2 for 90% and 95%, respectively.
v.b = 2

## the size of the cross-section
N = 1

## the trimming proportion for estimating the break dates
## (represents the minimum length of a regime; used to exclude
## the boundaries of the sample)
trim.e = 0.15

## quantiles
vec.tau = seq(0.20, 0.80, by = 0.150)

verbose = FALSE #do not print

## main estimation
res = rq.break(y, x, vec.tau, N, trim.e, vec.time, m.max, v.a, v.b, verbose)

Estimating Break Sizes and Confidence Intervals Given Break Dates

Description

This procedure estimates a linear quantile regression given a set of break dates. It is structured to compute break sizes between adjacent regimes and their confidence intervals.

Usage

rq.est.full(y, x, v.tau, vec.date, n.size = 1)

Arguments

Value

An object from the quantile regression estimates, rq(), with structural breaks.

References

Koenker, R. and G. Bassett Jr. (1978). Regression Quantiles. Econometrica, 46(1), 33–50.

Oka, T. and Z. Qu (2011). Estimating Structural Changes in Regression Quantiles. Journal of Econometrics, 162(2), 248–267.

Examples

## data
data(gdp)
y = gdp$gdp
x = gdp[,c("lag1", "lag2")]

## quantile
v.tau = 0.8

## break date
vec.date = 146

# cross-sectional size
n.size = 1

## estimation
rq.est.full(y, x, v.tau, vec.date, n.size)

Regime-Specific Coefficients and Confidence Intervals Given Break Dates

Description

This function estimates the coefficients for each regime, given the break dates.

Usage

rq.est.regime(y, x, v.tau, vec.date, n.size = 1)

Arguments

Value

A list containing the estimated coefficients for each regime.

Examples

## data
data(gdp)
y        = gdp$gdp
x        = gdp[,c("lag1", "lag2")]

## quantile
v.tau = 0.8

## break date
vec.date = 146

# cross-sectional size
n.size = 1

## estimation
result = rq.est.regime(y, x, v.tau, vec.date, n.size)
print(result)

Determine the Number of Breaks Using the SQ(l|l+1) Test

Description

This procedure sequentially applies the SQ test to determine the number of breaks, based on a single quantile.

Usage

sq(y, x, v.tau, n.size = 1, m.max, trim.size, mat.date)

Arguments

Value

A list with the following components:

test

A numeric vector of SQ test statistics.

cv

A numeric matrix of critical values for the SQ test, with the 1st, 2nd, and 3rd rows corresponding to the 10%, 5%, and 1% significance levels.

date

A numeric matrix of break dates, with the 1st, 2nd, and 3rd rows corresponding to the 10%, 5%, and 1% significance levels.

nbreak

A numeric vector indicating the number of breaks at the 10%, 5%, and 1% significance levels.

Examples

## data 
data(gdp)
y = gdp$gdp
x = gdp[,c("lag1", "lag2")] 

## quantile 
v.tau = 0.8 

# cross-sectional size 
n.size = 1 

# the maximum number of breaks 
m.max = 3

## trim 
T.size    = length(y)
trim.e    = 0.2
trim.size = round(T.size * trim.e)  #minimum length of a regime

# get.long 
out.long   = gen.long(y, x, v.tau, n.size, trim.size)
mat.long.s = out.long$mat.long  ## for individual quantile

# mat.date 
mat.date = brdate(y, x, n.size, m.max, trim.size, mat.long.s)

# sq 
result = sq(y, x, v.tau, n.size, m.max, trim.size, mat.date)
print(result)

Test for a Structural Break in a Conditional Quantile

Description

The function implements a break test to evaluate whether a single structural break exists at a given quantile.

Usage

sq.test.0vs1(y, x, v.tau, n.size = 1)

Arguments

Value

A numeric value representing the test statistic for the presence of a structural break.

References

Koenker, R. and G. Bassett Jr, (1978). Regression Quantiles. Econometrica, 46(1), 33–50.

Qu, Z. (2008). Testing for Structural Change in Regression Quantiles. Journal of Econometrics, 146(1), 170–184.

Examples

## data
data(gdp)
y = gdp$gdp
x = gdp[,c("lag1", "lag2")]

## quantile
v.tau = 0.8

# cross-sectional size
n.size = 1

# sq test: 0 vs 1
result = sq.test.0vs1(y, x, v.tau, n.size)
print(result)

Sequential Test for an Additional Break in a Conditional Quantile

Description

This function tests the null hypothesis of L breaks against the alternative hypothesis of L+1 breaks in a single conditional quantile.

Usage

sq.test.lvsl_1(y, x, v.tau, n.size = 1, vec.date)

Arguments

Details

The function sequentially tests for breaks by splitting the sample conditional on the break dates under the null hypothesis. At each step, it applies sq.test.0vs1() to compare the hypothesis of no additional break against one more break.

Value

A numeric value representing the test statistic.

References

Qu, Z. (2008). Testing for Structural Change in Regression Quantiles. Journal of Econometrics, 146(1), 170-184.

Oka, T. and Z. Qu (2011). Estimating Structural Changes in Regression Quantiles. Journal of Econometrics, 162(2), 248-267.

Examples

## data
data(gdp)
y = gdp$gdp
x = gdp[,c("lag1", "lag2")]

## quantile
v.tau = 0.8

# cross-sectional size
n.size = 1

## break date
vec.date = 146

## sq-test: 1 vs 2
result = sq.test.lvsl_1(y, x, v.tau, n.size, vec.date)
print(result)