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bootUR: Bootstrap Unit Root Tests

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The R package bootUR implements several bootstrap tests for unit roots, both for single time series and for (potentially) large systems of time series.

Installation and Loading

Installation

The package can be installed from CRAN using

install.packages("bootUR")

The development version of the bootUR package can be installed from GitHub using

# 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 (Rtools on Windows, or these tools on Mac).

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.

Load Package

After installation, the package can be loaded in the standard way:

library(bootUR)

Preliminary Analysis: Missing Values

bootUR provides a few simple tools to check if your data are suitable to be bootstrapped.

Inspect Data for Missing Values

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 package.

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

Checking Start and End Points of Time Series

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 follows:

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
#> [1] FALSE

Visualizing Missing Data

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)

Augmented Dickey-Fuller Test

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 purpose the adf() function can be used. The function allows to set many options. First, one can choose between the classical single-step procedure (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 selection

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.

Implementation

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 trend) on a single time series
#> 
#> data: GDP_NL
#> null hypothesis: Series has a unit root
#> alternative hypothesis: Series is stationary
#> 
#>        estimate largest root statistic p-value
#> GDP_NL                0.9471    -2.515  0.3202

Univariate Bootstrap Unit Root Tests

Augmented Dickey-Fuller Test

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 bootstrap), the deterministic components included (option deterministics) and the type of detrending used (option detrend). While 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 used.

Implementation

We illustrate the bootstrap ADF test here on Dutch GDP, with the sieve bootstrap (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 setting 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
#> null hypothesis: Series has a unit root
#> alternative hypothesis: Series is stationary
#> 
#>        estimate largest root statistic p-value
#> GDP_NL                0.9471    -2.515  0.1454

Union of Rejections Test

Use 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
#> null hypothesis: Series has a unit root
#> alternative hypothesis: Series is stationary
#> 
#>        estimate largest root statistic p-value
#> GDP_NL                    NA   -0.7115   0.614

Panel Unit Root Test

The function 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 averagi