The R package
bootUR implements several bootstrap tests for unit
roots, both for single time series and for (potentially) large systems
of time series.
The package can be installed from CRAN using
The development version of the
bootUR package can be installed from
# install.packages("devtools") devtools::install_github("smeekes/bootUR")
When installing from GitHub, in order to build the package from source, you need to have the appropriate R development tools installed (such as Rtools on Windows.)
If you want the vignette to appear in your package when installing from GitHub, use
# install.packages("devtools") devtools::install_github("smeekes/bootUR", build_vignettes = TRUE, dependencies = TRUE)
instead. As building the vignette may take a bit of time (all bootstrap code below is run), package installation will be slower this way.
After installation, the package can be loaded in the standard way:
bootUR provides a few simple tools to check if your data are suitable
to be bootstrapped.
The bootstrap tests in
bootUR do not work with missing data, although
multivariate time series with different start and end dates (unbalanced
panels) are allowed.
bootUR provides a simple function to check if
your data contain missing values. We will illustrate this on the
MacroTS dataset of macroeconomic time series that comes with the
data("MacroTS") check_missing_insample_values(MacroTS) #> GDP_BE GDP_DE GDP_FR GDP_NL GDP_UK CONS_BE CONS_DE CONS_FR CONS_NL CONS_UK #> FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE #> HICP_BE HICP_DE HICP_FR HICP_NL HICP_UK UR_BE UR_DE UR_FR UR_NL UR_UK #> FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
If your time series have different starting and end points (and thus
some series contain
NAs at the beginning and/or end of your sample,
the resampling-based moving block bootstrap (MBB) and sieve bootstrap
(SB) cannot be used.
bootUR lets you check the start and end points as
sample_check <- find_nonmissing_subsample(MacroTS) # Provides the number of the first and last non-missing observation for each series: sample_check$range #> GDP_BE GDP_DE GDP_FR GDP_NL GDP_UK CONS_BE CONS_DE CONS_FR CONS_NL #> first 1 1 1 5 1 1 1 1 5 #> last 100 100 100 100 100 100 100 100 100 #> CONS_UK HICP_BE HICP_DE HICP_FR HICP_NL HICP_UK UR_BE UR_DE UR_FR UR_NL #> first 1 9 9 9 9 9 1 1 1 1 #> last 100 100 100 100 100 100 100 100 100 100 #> UR_UK #> first 1 #> last 100 # Gives TRUE if the time series all start and end at the same observation: sample_check$all_equal #>  FALSE
If you have
ggplot2 installed, you can also plot the missing data
patterns in your series to get a quick overview. You may need to
manipulate some arguments to get the plot properly sized (therefore it
is not run here automatically).
plot_missing_values(MacroTS, show_names = TRUE, axis_text_size = 5, legend_size = 6)
As the standard test for unit roots,
bootUR also has an implementation
of the standard, non-bootstrap, augmented Dickey-Fuller (ADF) test
(though its use is not recommended if sample sizes are small). For this
adf() function can be used. The function allows to set
many options. First, one can choose between the classical single-step
two_step = FALSE), in which deterministic components are
directly included in the test regression, and the more flexible and
modern two-step procedure (
two_step = TRUE) where deterministic
components are first removed before applying the unit root test to
detrended data. For the standard ADF test, the two specifications
generally yield nearly identical results.
Lag length selection is done automatically in the ADF regression; the
default is by the modified Akaike information criterion (MAIC) proposed
by Ng and Perron (2001) with the correction of Perron and Qu (2008).
Other options include the regular Akaike information criterion (AIC), as
well as the Bayesian information criterion and its modified variant. In
addition, the rescaling suggested by Cavaliere et al. (2015) is
implemented to improve the power of the test under heteroskedasticity;
this can be turned off by setting
criterion_scale = FALSE. To
overwrite data-driven lag length selection with a pre-specified lag
length, simply set both the minimum
min_lag and maximum lag length
max_lag for the selection algorithm equal to the desired lag length.
We illustrate the ADF test here on Dutch GDP for the two-step specification, including a linear trend in the specification.
GDP_NL <- MacroTS[, 4] adf(GDP_NL, deterministics = "trend") #> #> Two-step ADF test ( with intercept and trend ) on a single time series #> #> data: GDP_NL #> tstat = -2.5153, p-value = 0.3202 #> alternative hypothesis: true gamma is less than 0 #> sample estimates: #> gamma #> -0.05286343
To perform a bootstrap version of the ADF unit root test on a single
time series, use the
boot_adf() function. The function allows to set
many options, including the bootstrap method used (option
the deterministic components included (option
deterministics) and the
type of detrending used (option
detrend = "OLS"
gives the standard ADF test,
detrend = "QD" provides the powerful
DF-GLS test of Elliott, Rothenberg and Stock (1996). Here we use the
terminology Quasi-Differencing (QD) rather than GLS as this conveys the
meaning less ambiguously and is the same terminology used by Smeekes and
Taylor (2012) and Smeekes (2013). In all cases, two-step detrending is
We illustrate the bootstrap ADF test here on Dutch GDP, with the sieve
bootstrap = SB) as in the specification used by Palm,
Smeekes and Urbain (2008) and Smeekes (2013). To get the well-known test
proposed by Paparoditis and Politis (2003), set
bootstrap = "MBB". We
set only 399 bootstrap replications (
B = 399) to prevent the code from
running too long. We add an intercept and a trend
deterministics = "trend") and OLS detrending. The console gives you
live updates on the bootstrap progress. To turn these off, set
show_progress = FALSE. The bootstrap loop can be run in parallel by
do_parallel = TRUE (the default).
As random number generation is required to draw bootstrap samples, we first set the seed of the random number generator to obtain replicable results.
set.seed(155776) boot_adf(GDP_NL, B = 399, bootstrap = "SB", deterministics = "trend", detrend = "OLS", do_parallel = FALSE) #> Progress: |------------------| #> ******************** #> #> SB Bootstrap OLS test ( with intercept and trend ) on a single time #> series #> #> data: GDP_NL #> tstat = -2.5153, p-value = 0.1454 #> alternative hypothesis: true gamma is less than 0 #> sample estimates: #> gamma #> -0.05286343
boot_union() for a test based on the union of rejections of 4
tests with different number of deterministic components and different
type of detrending (Smeekes and Taylor, 2012). The advantage of the
union test is that you don’t have to specify these (rather influential)
specification tests. This makes the union test a safe option for quick
or automatic unit root testing where careful manual specification setup
is not viable. Here we illustrate it with the sieve wild bootstrap as
proposed by Smeekes and Taylor (2012).
boot_union(GDP_NL, B = 399, bootstrap = "SWB", do_parallel = FALSE) #> Progress: |------------------| #> ******************** #> #> SWB Bootstrap Union test on a single time series #> #> data: GDP_NL #> tstat = -0.71148, p-value = 0.614 #> alternative hypothesis: true gamma is less than 0 #> sample estimates: #> gamma #> NA
boot_panel performs a test on a multivariate (panel) time
series by testing the null hypothesis that all series have a unit root.
A rejection is typically interpreted as evidence that a ‘significant
proportion’ of the series is stationary, although how large that
proportion is - or which series are stationary - is not given by the
test. The test is based on averaging the individual test statistics,
also called the Group-Mean (GM) test in Palm, Smeekes and Urbain (2011).
Palm, Smeekes and Urbain (2011) introduced this test with the moving
block bootstrap (
bootstrap = "MBB"). However, this resampling-based
method cannot handle unbalancedness, and will therefore give an error
when applied to
boot_panel(MacroTS, bootstrap = "MBB", B = 399, do_parallel = FALSE) #> Error in check_inputs(data = data, boot_sqt_test = boot_sqt_test, boot_ur_test = boot_ur_test, : Resampling-based bootstraps MBB and SB cannot handle unbalanced series.
Therefore, you should switch to one of the wild bootstrap methods. Here we illustrate it with the dependent wild bootstrap (DWB) of Shao (2010) and Rho and Shao (2019).
By default the union test is used for each series (
union = TRUE), if
this is set to
FALSE the deterministic components and detrending
methods can be specified as in the univariate Dickey-Fuller test.
Although the sieve bootstrap method
"SWB" can be used
(historically they have been popular among practitioners), Smeekes and
Urbain (2014b) show that these are not suited to capture general forms
of dependence across units. The code will give a warning to recommend
using a different bootstrap method.
boot_panel(MacroTS, bootstrap = "DWB", B = 399, do_parallel = FALSE) #> Progress: |------------------| #> ******************** #> #> Panel DWB Bootstrap Group-Mean Union test #> #> data: MacroTS #> tstat = -0.86213, p-value = 0.1103 #> alternative hypothesis: true gamma is less than 0 #> sample estimates: #> gamma #> NA
To perform individual ADF tests on multiple time series simultaneously,
boot_ur() can still be used. As the bootstrap is
performed for all series simultaneously, resampling-based bootstrap
"SB" cannot be used directly in case of unbalanced
panels. If they are used anyway, the function will revert to splitting
the bootstrap up and performing it individually per time series. In this
case a warning is given to alert the user.
ADFtests_out <- boot_ur(MacroTS[, 1:5], bootstrap = "MBB", B = 399, union = FALSE, deterministics = "trend", detrend = "OLS", do_parallel = FALSE) #> Warning in check_inputs(data = data, boot_sqt_test = boot_sqt_test, boot_ur_test #> = boot_ur_test, : Missing values cause resampling bootstrap to be executed for #> each time series individually. #> Progress: |------------------| #> ******************** print(ADFtests_out) #> #> MBB Bootstrap ADF test ( with intercept and trend ) on each individual #> series (no multiple testing correction) #> #> data: MacroTS[, 1:5] #> alternative hypothesis: true gamma is less than 0 #> #> Tests performed on each series: #> estimate statistic p-value #> GDP_BE -0.06958 -2.792 0.1955 #> GDP_DE -0.10895 -2.774 0.1003 #> GDP_FR -0.03451 -2.049 0.5113 #> GDP_NL -0.05286 -2.515 0.1930 #> GDP_UK -0.03996 -2.449 0.2882
boot_ur (intentionally) does not provide a correction for
multiple testing; of course, if we perform each test with a significance
level of 5%, the probability of making a mistake in all these tests
becomes (much, if
N is large) more than 5%. To explicitly account for
multiple testing, use the functions
boot_sqt() performs the Bootstrap Sequential Quantile
Test (BSQT) proposed by Smeekes (2015). Here we split the series in
groups which are consecutively tested for unit roots, starting with the
group most likely to be stationary (having the smallest ADF statistics).
If the unit root hypothesis cannot be rejected for the first group, the
algorithm stops; if there is a rejection, the second group is tested,
and so on.
Most options are the same as for
boot_panel. The parameter
controls the significance level of the individual tests performed in the
sequence, with a default value of 0.05. The other important new
parameter to set here is the group sizes. These can either be set in
units, or in fractions of the total number of series (i.e. quantiles,
hence the name) via the parameter
steps. If we have
N time series,
steps = 0:N means each unit should be tested sequentially. To
split the series in four equally sized groups (regardless of many series
there are), use
steps = 0:4 / 4. By convention and in accordance with
notation in Smeekes (2015), the first entry of the vector should be
equal to zero, while the second entry indicates the end of the first
group, and so on. However, if the initial zero is accidentally omitted,
it is automatically added by the function. Similarly, if the final value
is not equal to
1 (in case of quantiles) or
N to end the last group,
this is added by the function.
Testing individual series consecutively is easiest for interpretation,
but is only meaningful if
N is small. In this case the method is
equivalent to the bootstrap StepM method of Romano and Wolf (2005),
which controls the familywise error rate, that is the probability of
making at least one false rejection. This can get very conservative if
N is large, and you would typically end up not rejecting any null
hypothesis. The method is illustrated with the autoregressive wild
bootstrap of Smeekes and Urbain (2014a) and Friedrich, Smeekes and
N <- ncol(MacroTS) # Test each unit sequentially boot_sqt(MacroTS, steps = 0:N, bootstrap = "AWB", B = 399, do_parallel = FALSE) #> Progress: |------------------| #> ******************** #> #> AWB Bootstrap Sequential Quantile Union test #> #> data: MacroTS #> alternative hypothesis: true gamma is less than 0 #> #> Sequence of tests: #> H0: # I(0) H1: # I(0) tstat p-value #> Step 1 0 1 -1.661 0.02256 #> Step 2 1 2 -1.413 0.12281 # Split in four equally sized groups (motivated by the 4 series per country) boot_sqt(MacroTS, steps = 0:4 / 4, bootstrap = "AWB", B = 399, do_parallel = FALSE) #> Progress: |------------------| #> ******************** #> #> AWB Bootstrap Sequential Quantile Union test #> #> data: MacroTS #> alternative hypothesis: true gamma is less than 0 #> #> Sequence of tests: #> H0: # I(0) H1: # I(0) tstat p-value #> Step 1 0 5 -1.052 0.05013
boot_fdr() controls for multiple testing by controlling
the false discovery rate (FDR), which is defined as the expected
proportion of false rejections relative to the total number of
rejections. This scales with the total number of tests, making it more
suitable for large
N than the familywise error rate.
The bootstrap method for controlling FDR was introduced by Romano,
Shaikh and Wolf (2008), who showed that, unlike the classical way to
control FDR, the bootstrap is appropriate under general forms of
dependence between series. Moon and Perron (2012) applied this method to
unit root testing; it is essentially their method which is implemented
boot_fdr() though again with the option to change the bootstrap
used (their suggestion was MBB). The arguments to be set are the same as
for the other multivariate unit root tests, with the exception of
FDR_level wihch controls the FDR level. As BSQT, the method only
report those tests until no rejection occurs.
We illustrate it here with the final available bootstrap method, the block wild bootstrap of Shao (2011) and Smeekes and Urbain (2014a).
N <- ncol(MacroTS) boot_fdr(MacroTS[, 1:10], FDR_level = 0.1, bootstrap = "BWB", B = 399, do_parallel = FALSE) #> Progress: |------------------| #> ******************** #> #> BWB Bootstrap Union test with False Discovery Rate control #> #> data: MacroTS[, 1:10] #> alternative hypothesis: true gamma is less than 0 #> #> Sequence of tests: #> tstat critical value #> GDP_DE -1.077 -1.581
Generally the unit root tests above would only be used as a single step
in a larger algorithm to determine the orders of integration of the time
series in the dataset. In particular, many economic datasets contain
variables that have order of integration 2, and would so need to be
differenced twice to eliminate all trends. A standard unit root test
cannot determine this however. For this purpose, we add the function
order_integration() which performs a sequence of unit root tests to
determine the orders of each time series.
Starting from a maximum order (by default equal to 2), it differences the data time until there can be at most one unit root. If the test is not rejected for a particular series, we know this series if of order . The series for which we do reject are integrated once (such that they are differenced times from their original level), and the test is repeated. By doing so until we have classified all series, we obtain a full specification of the orders of all time series.
The function allows us to choose which unit root test we want to use.
Here we take
boot_fdr. We don’t only get the orders out, but also the
appropriately differenced data.
out_orders <- order_integration(MacroTS[, 11:15], method = "boot_fdr", B = 399, do_parallel = FALSE) #> Progress: |------------------| #> ******************** #> Progress: |------------------| #> ******************** # Orders out_orders$order_int #> HICP_BE HICP_DE HICP_FR HICP_NL HICP_UK #> 0 0 1 0 1 # Differenced data stationary_data <- out_orders$diff_data
To achieve the differencing,
order_integration() uses the function
diff_mult() which is also available as stand-alone function in the
package. Finally, a function is provided to plot the found orders (not
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