Package 'bootUR'

Augmented Dickey-Fuller Unit Root Test

Description

This function performs a standard augmented Dickey-Fuller unit root test on a single time series.

Usage

adf(data, data_name = NULL, deterministics = "intercept", min_lag = 0,
  max_lag = NULL, criterion = "MAIC", criterion_scale = TRUE,
  two_step = TRUE)

Arguments

data	A $T$-dimensional vector to be tested for unit roots. Data may also be in a time series format (e.g. ts, zoo or xts), or a data frame.
data_name	Optional name for the data, to be used in the output. The default uses the name of the 'data' argument.
deterministics	String indicating the deterministic specification. Only relevant if union = FALSE. Options are ⁠"none":⁠ no deterministics; ⁠"intercept":⁠ intercept only; ⁠"trend":⁠ intercept and trend. If union = FALSE, the default is adding an intercept (a warning is given).
min_lag	Minimum lag length in the augmented Dickey-Fuller regression. Default is 0.
max_lag	Maximum lag length in the augmented Dickey-Fuller regression. Default uses the sample size-based rule $12(T/100)^{1/4}$.
criterion	String for information criterion used to select the lag length in the augmented Dickey-Fuller regression. Options are: "AIC", "BIC", "MAIC", "MBIC". Default is "MAIC" (Ng and Perron, 2001).
criterion_scale	Logical indicator whether or not to use the rescaled information criteria of Cavaliere et al. (2015) (TRUE) or not (FALSE). Default is TRUE.
two_step	Logical indicator whether to use one-step (two_step = FALSE) or two-step (two_step = TRUE) detrending. The default is two-step detrending.

Details

The function encompasses the standard augmented Dickey-Fuller test. The reported p-values are MacKinnon's unit root p-values taken from the package urca.

Lag length selection is done automatically in the ADF regression with the specified information criterion. If one of the modified criteria of Ng and Perron (2001) is used, the correction of Perron and Qu (2008) is applied. For very short time series (fewer than 50 time points) the maximum lag length is adjusted downward to avoid potential multicollinearity issues in the bootstrap. 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.

Value

An object of class "bootUR", "htest" with the following components:

method	The name of the hypothesis test method;
data.name	The name of the variable on which the method is performed;
null.value	The value of the (gamma) parameter of the lagged dependent variable in the ADF regression under the null hypothesis. Under the null, the series has a unit root. Testing the null of a unit root then boils down to testing the significance of the gamma parameter;
alternative	A character string specifying the direction of the alternative hypothesis relative to the null value. The alternative postulates that the series is stationary;
estimate	The estimated value of the (gamma) parameter of the lagged dependent variable in the ADF regression;
statistic	The value of the test statistic of the ADF unit root test;
p.value	The p-value of the ADF unit root test.
specifications	The specifications used in the test.

Errors and warnings

Error: Multiple time series not allowed. Switch to a multivariate method such as boot_ur, or change argument data to a univariate time series.

The function provides a standard ADF test with asymptotic p-value. It does not support multiple time series

References

Smeekes, S. and Wilms, I. (2023). bootUR: An R Package for Bootstrap Unit Root Tests. Journal of Statistical Software, 106(12), 1-39.

Examples

# standard ADF test on GDP_BE
GDP_BE_adf <- adf(MacroTS[, 1], deterministics = "trend")

---

Bootstrap augmented Dickey-Fuller Unit Root Test

Description

This function performs a standard augmented Dickey-Fuller bootstrap unit root test on a single time series.

Usage

boot_adf(data, data_name = NULL, bootstrap = "AWB", B = 1999,
  block_length = NULL, ar_AWB = NULL, deterministics = "intercept",
  min_lag = 0, max_lag = NULL, criterion = "MAIC", detrend = "OLS",
  criterion_scale = TRUE, show_progress = TRUE, do_parallel = TRUE,
  cores = NULL)

Arguments

data	A $T$-dimensional vector to be tested for unit roots. Data may also be in a time series format (e.g. ts, zoo or xts), or a data frame.
data_name	Optional name for the data, to be used in the output. The default uses the name of the 'data' argument.
bootstrap	String for bootstrap method to be used. Options are "MBB" Moving blocks bootstrap (Paparoditis and Politis, 2003); "BWB" Block wild bootstrap (Shao, 2011); "DWB" Dependent wild bootstrap (Shao, 2010; Rho and Shao, 2019); "AWB" Autoregressive wild bootstrap (Smeekes and Urbain, 2014a; Friedrich, Smeekes and Urbain, 2020), this is the default; "SB" Sieve bootstrap (Chang and Park, 2003; Palm, Smeekes and Urbain, 2008; Smeekes, 2013); "SWB" Sieve wild bootstrap (Cavaliere and Taylor, 2009; Smeekes and Taylor, 2012).
B	Number of bootstrap replications. Default is 1999.
block_length	Desired 'block length' in the bootstrap. For the MBB, BWB and DWB bootstrap, this is a genuine block length. For the AWB bootstrap, the block length is transformed into an autoregressive parameter via the formula $0.01^(1/block_length)$ as in Smeekes and Urbain (2014a); this can be overwritten by setting ar_AWB directly. Default sets the block length as a function of the time series length T, via the rule $block_length = 1.75 T^(1/3)$ of Palm, Smeekes and Urbain (2011).
ar_AWB	Autoregressive parameter used in the AWB bootstrap method (bootstrap = "AWB"). Can be used to set the parameter directly rather than via the default link to the block length.
deterministics	String indicating the deterministic specification. Only relevant if union = FALSE. Options are ⁠"none":⁠ no deterministics; ⁠"intercept":⁠ intercept only; ⁠"trend":⁠ intercept and trend. If union = FALSE, the default is adding an intercept (a warning is given).
min_lag	Minimum lag length in the augmented Dickey-Fuller regression. Default is 0.
max_lag	Maximum lag length in the augmented Dickey-Fuller regression. Default uses the sample size-based rule $12(T/100)^{1/4}$.
criterion	String for information criterion used to select the lag length in the augmented Dickey-Fuller regression. Options are: "AIC", "BIC", "MAIC", "MBIC". Default is "MAIC" (Ng and Perron, 2001).
detrend	String indicating the type of detrending to be performed. Only relevant if union = FALSE. Options are: "OLS" or "QD" (typically also called GLS, see Elliott, Rothenberg and Stock, 1996). The default is "OLS".
criterion_scale	Logical indicator whether or not to use the rescaled information criteria of Cavaliere et al. (2015) (TRUE) or not (FALSE). Default is TRUE.
show_progress	Logical indicator whether a bootstrap progress update should be printed to the console. Default is FALSE.
do_parallel	Logical indicator whether bootstrap loop should be executed in parallel. Default is TRUE.
cores	The number of cores to be used in the parallel loops. Default is to use all but one.

Details

The options encompass many test proposed in the literature. detrend = "OLS" gives the standard augmented Dickey-Fuller test, while detrend = "QD" provides the DF-GLS test of Elliott, Rothenberg and Stock (1996). The bootstrap algorithm is always based on a residual bootstrap (under the alternative) to obtain residuals rather than a difference-based bootstrap (under the null), see e.g. Palm, Smeekes and Urbain (2008).

Lag length selection is done automatically in the ADF regression with the specified information criterion. If one of the modified criteria of Ng and Perron (2001) is used, the correction of Perron and Qu (2008) is applied. For very short time series (fewer than 50 time points) the maximum lag length is adjusted downward to avoid potential multicollinearity issues in the bootstrap. 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.

Value

An object of class "bootUR", "htest" with the following components:

method	The name of the hypothesis test method;
data.name	The name of the data on which the method is performed;
null.value	The value of the (gamma) parameter of the lagged dependent variable in the ADF regression under the null hypothesis. Under the null, the series has a unit root. Testing the null of a unit root then boils down to testing the significance of the gamma parameter;
alternative	A character string specifying the direction of the alternative hypothesis relative to the null value. The alternative postulates that the series is stationary;
estimate	The estimated value of the (gamma) parameter of the lagged dependent variable in the ADF regression.;
statistic	The value of the test statistic of the unit root test;
p.value	The p-value of the unit root test;
details	A list containing the detailed outcomes of the performed test, such as selected lags, individual estimates and p-values.
specifications	The specifications used in the test.

Errors and warnings

Error: Multiple time series not allowed. Switch to a multivariate method such as boot_ur, or change argument data to a univariate time series.

The function is a simple wrapper around boot_ur to facilitate use for single time series. It does not support multiple time series, as boot_ur is specifically suited for that.

References

Smeekes, S. and Wilms, I. (2023). bootUR: An R Package for Bootstrap Unit Root Tests. Journal of Statistical Software, 106(12), 1-39.

Chang, Y. and Park, J. (2003). A sieve bootstrap for the test of a unit root. Journal of Time Series Analysis, 24(4), 379-400.

Cavaliere, G. and Taylor, A.M.R (2009). Heteroskedastic time series with a unit root. Econometric Theory, 25, 1228–1276.

Cavaliere, G., Phillips, P.C.B., Smeekes, S., and Taylor, A.M.R. (2015). Lag length selection for unit root tests in the presence of nonstationary volatility. Econometric Reviews, 34(4), 512-536.

Elliott, G., Rothenberg, T.J., and Stock, J.H. (1996). Efficient tests for an autoregressive unit root. Econometrica, 64(4), 813-836.

Friedrich, M., Smeekes, S. and Urbain, J.-P. (2020). Autoregressive wild bootstrap inference for nonparametric trends. Journal of Econometrics, 214(1), 81-109.

Ng, S. and Perron, P. (2001). Lag Length Selection and the Construction of Unit Root Tests with Good Size and Power. Econometrica, 69(6), 1519-1554,

Palm, F.C., Smeekes, S. and Urbain, J.-P. (2008). Bootstrap unit root tests: Comparison and extensions. Journal of Time Series Analysis, 29(1), 371-401.

Paparoditis, E. and Politis, D.N. (2003). Residual-based block bootstrap for unit root testing. Econometrica, 71(3), 813-855.

Perron, P. and Qu, Z. (2008). A simple modification to improve the finite sample properties of Ng and Perron's unit root tests. Economic Letters, 94(1), 12-19.

Rho, Y. and Shao, X. (2019). Bootstrap-assisted unit root testing with piecewise locally stationary errors. Econometric Theory, 35(1), 142-166.

Smeekes, S. (2013). Detrending bootstrap unit root tests. Econometric Reviews, 32(8), 869-891.

Shao, X. (2010). The dependent wild bootstrap. Journal of the American Statistical Association, 105(489), 218-235.

Shao, X. (2011). A bootstrap-assisted spectral test of white noise under unknown dependence. Journal of Econometrics, 162, 213-224.

Smeekes, S. and Taylor, A.M.R. (2012). Bootstrap union tests for unit roots in the presence of nonstationary volatility. Econometric Theory, 28(2), 422-456.

Smeekes, S. and Urbain, J.-P. (2014a). A multivariate invariance principle for modified wild bootstrap methods with an application to unit root testing. GSBE Research Memorandum No. RM/14/008, Maastricht University

See Also

boot_ur

Examples

# boot_adf on GDP_BE
GDP_BE_adf <- boot_adf(MacroTS[, 1], B = 199, deterministics = "trend", detrend = "OLS",
                       do_parallel = FALSE, show_progress = FALSE)
print(GDP_BE_adf)

---

Bootstrap Unit Root Tests with False Discovery Rate control

Description

Controls for multiple testing by controlling the false discovery rate (FDR), see Moon and Perron (2012) and Romano, Shaikh and Wolf (2008).

Usage

boot_fdr(data, data_name = NULL, bootstrap = "AWB", B = 1999,
  block_length = NULL, ar_AWB = NULL, FDR_level = 0.05, union = TRUE,
  deterministics = NULL, detrend = NULL, min_lag = 0, max_lag = NULL,
  criterion = "MAIC", criterion_scale = TRUE, show_progress = TRUE,
  do_parallel = TRUE, cores = NULL)

Arguments

data	A $T$-dimensional vector or a ($T$ x $N$)-matrix of $N$ time series with $T$ observations to be tested for unit roots. Data may also be in a time series format (e.g. ts, zoo or xts), or a data frame, as long as each column represents a single time series.
data_name	Optional name for the data, to be used in the output. The default uses the name of the 'data' argument.
bootstrap	String for bootstrap method to be used. Options are "MBB" Moving block bootstrap (Paparoditis and Politis, 2003; Palm, Smeekes and Urbain, 2011); "BWB" Block wild bootstrap (Shao, 2011; Smeekes and Urbain, 2014a); "DWB" Dependent wild bootstrap (Shao, 2010; Smeekes and Urbain, 2014a; Rho and Shao, 2019); "AWB" Autoregressive wild bootstrap (Smeekes and Urbain, 2014a; Friedrich, Smeekes and Urbain, 2020), this is the default; "SB" Sieve bootstrap (Chang and Park, 2003; Palm, Smeekes and Urbain, 2008; Smeekes, 2013); "SWB" Sieve wild bootstrap (Cavaliere and Taylor, 2009; Smeekes and Taylor, 2012).
B	Number of bootstrap replications. Default is 1999.
block_length	Desired 'block length' in the bootstrap. For the MBB, BWB and DWB bootstrap, this is a genuine block length. For the AWB bootstrap, the block length is transformed into an autoregressive parameter via the formula $0.01^(1/block_length)$ as in Smeekes and Urbain (2014a); this can be overwritten by setting ar_AWB directly. Default sets the block length as a function of the time series length T, via the rule $block_length = 1.75 T^(1/3)$ of Palm, Smeekes and Urbain (2011).
ar_AWB	Autoregressive parameter used in the AWB bootstrap method (bootstrap = "AWB"). Can be used to set the parameter directly rather than via the default link to the block length.
FDR_level	Desired False Discovery Rate level of the unit root tests. Default is 0.05.
union	Logical indicator whether or not to use bootstrap union tests (TRUE) or not (FALSE), see Smeekes and Taylor (2012). Default is TRUE.
deterministics	String indicating the deterministic specification. Only relevant if union = FALSE. Options are ⁠"none":⁠ no deterministics; ⁠"intercept":⁠ intercept only; ⁠"trend":⁠ intercept and trend. If union = FALSE, the default is adding an intercept (a warning is given).
detrend	String indicating the type of detrending to be performed. Only relevant if union = FALSE. Options are: "OLS" or "QD" (typically also called GLS, see Elliott, Rothenberg and Stock, 1996). The default is "OLS".
min_lag	Minimum lag length in the augmented Dickey-Fuller regression. Default is 0.
max_lag	Maximum lag length in the augmented Dickey-Fuller regression. Default uses the sample size-based rule $12(T/100)^{1/4}$.
criterion	String for information criterion used to select the lag length in the augmented Dickey-Fuller regression. Options are: "AIC", "BIC", "MAIC", "MBIC". Default is "MAIC" (Ng and Perron, 2001).
criterion_scale	Logical indicator whether or not to use the rescaled information criteria of Cavaliere et al. (2015) (TRUE) or not (FALSE). Default is TRUE.
show_progress	Logical indicator whether a bootstrap progress update should be printed to the console. Default is FALSE.
do_parallel	Logical indicator whether bootstrap loop should be executed in parallel. Default is TRUE.
cores	The number of cores to be used in the parallel loops. Default is to use all but one.

Details

The false discovery rate FDR is defined as the expected proportion of false rejections relative to the total number of rejections.

See boot_ur for details on the bootstrap algorithm and lag selection.

Value

An object of class "bootUR", "mult_htest" with the following components:

method	The name of the hypothesis test method;
data.name	The name of the data on which the method is performed;
null.value	The value of the (gamma) parameter of the lagged dependent variable in the ADF regression under the null hypothesis. Under the null, the series has a unit root. Testing the null of a unit root then boils down to testing the significance of the gamma parameter;
alternative	A character string specifying the direction of the alternative hypothesis relative to the null value. The alternative postulates that the series is stationary;
estimate	The estimated values of the (gamma) parameter of the lagged dependent variable in the ADF regressions. Note that for the union test (union = TRUE), this estimate is not defined, hence NA is returned;
statistic	The value of the test statistic of the unit root tests;
p.value	A vector with NA values, as p-values are not available for the FDR method;
rejections	A vector with logical indicators for each time series whether the null hypothesis of a unit root is rejected (TRUE) or not (FALSE);
details	A list containing the detailed outcomes of the performed tests, such as selected lags, individual estimates and p-values. In addtion, the slot FDR contains a matrix with for each step the test statistics and critical value, up to non-rejection.
series.names	The names of the series that the tests are performed on;
specifications	The specifications used in the test(s).

Errors and warnings

Error: Resampling-based bootstraps MBB and SB cannot handle missing values.

If the time series in data have different starting and end points (and thus some series contain NA values at the beginning and/or end of the sample, the resampling-based moving block bootstrap (MBB) and sieve bootstrap (SB) cannot be used, as they create holes (internal missings) in the bootstrap samples. Switch to another bootstrap method or truncate your sample to eliminate NA values.

Warning: SB and SWB bootstrap only recommended for boot_ur; see help for details.

Although the sieve bootstrap methods "SB" and "SWB" can be used, Smeekes and Urbain (2014b) show that these are not suited to capture general forms of dependence across units, and using them for joint or multiple testing is not valid. This warning thereofre serves to recommend the user to consider a different bootstrap method.

Warning: Deterministic specification in argument deterministics is ignored, as union test is applied.

The union test calculates the union of all four combinations of deterministic components (intercept or intercept and trend) and detrending methods (OLS or QD). Setting deterministic components manually therefore has no effect.

Warning: Detrending method in argument detrend is ignored, as union test is applied.

The union test calculates the union of all four combinations of deterministic components (intercept or intercept and trend) and detrending methods (OLS or QD). Setting detrending methods manually therefore has no effect.

References

Smeekes, S. and Wilms, I. (2023). bootUR: An R Package for Bootstrap Unit Root Tests. Journal of Statistical Software, 106(12), 1-39.

Chang, Y. and Park, J. (2003). A sieve bootstrap for the test of a unit root. Journal of Time Series Analysis, 24(4), 379-400.

Cavaliere, G. and Taylor, A.M.R (2009). Heteroskedastic time series with a unit root. Econometric Theory, 25, 1228–1276.

Cavaliere, G., Phillips, P.C.B., Smeekes, S., and Taylor, A.M.R. (2015). Lag length selection for unit root tests in the presence of nonstationary volatility. Econometric Reviews, 34(4), 512-536.

Elliott, G., Rothenberg, T.J., and Stock, J.H. (1996). Efficient tests for an autoregressive unit root. Econometrica, 64(4), 813-836.

Friedrich, M., Smeekes, S. and Urbain, J.-P. (2020). Autoregressive wild bootstrap inference for nonparametric trends. Journal of Econometrics, 214(1), 81-109.

Moon, H.R. and Perron, B. (2012). Beyond panel unit root tests: Using multiple testing to determine the non stationarity properties of individual series in a panel. Journal of Econometrics, 169(1), 29-33.

Ng, S. and Perron, P. (2001). Lag Length Selection and the Construction of Unit Root Tests with Good Size and Power. Econometrica, 69(6), 1519-1554,

Palm, F.C., Smeekes, S. and Urbain, J.-P. (2008). Bootstrap unit root tests: Comparison and extensions. Journal of Time Series Analysis, 29(1), 371-401.

Palm, F. C., Smeekes, S., and Urbain, J.-.P. (2011). Cross-sectional dependence robust block bootstrap panel unit root tests. Journal of Econometrics, 163(1), 85-104.

Paparoditis, E. and Politis, D.N. (2003). Residual-based block bootstrap for unit root testing. Econometrica, 71(3), 813-855.

Perron, P. and Qu, Z. (2008). A simple modification to improve the finite sample properties of Ng and Perron's unit root tests. Economic Letters, 94(1), 12-19.

Rho, Y. and Shao, X. (2019). Bootstrap-assisted unit root testing with piecewise locally stationary errors. Econometric Theory, 35(1), 142-166.

Romano, J.P., Shaikh, A.M., and Wolf, M. (2008). Control of the false discovery rate under dependence using the bootstrap and subsampling. Test, 17(3), 417.

Shao, X. (2010). The dependent wild bootstrap. Journal of the American Statistical Association, 105(489), 218-235.

Shao, X. (2011). A bootstrap-assisted spectral test of white noise under unknown dependence. Journal of Econometrics, 162, 213-224.

Smeekes, S. (2013). Detrending bootstrap unit root tests. Econometric Reviews, 32(8), 869-891.

Smeekes, S. and Taylor, A.M.R. (2012). Bootstrap union tests for unit roots in the presence of nonstationary volatility. Econometric Theory, 28(2), 422-456.

Smeekes, S. and Urbain, J.-P. (2014a). A multivariate invariance principle for modified wild bootstrap methods with an application to unit root testing. GSBE Research Memorandum No. RM/14/008, Maastricht University

Smeekes, S. and Urbain, J.-P. (2014b). On the applicability of the sieve bootstrap in time series panels. Oxford Bulletin of Economics and Statistics, 76(1), 139-151.

See Also

boot_ur

Examples

# boot_fdr on GDP_BE and GDP_DE
two_series_boot_fdr <- boot_fdr(MacroTS[, 1:2], bootstrap = "MBB", B = 199,
                                do_parallel = FALSE, show_progress = FALSE)
print(two_series_boot_fdr)

---

Panel Unit Root Test

Description

Performs a test on a multivariate (panel) time series by testing the null hypothesis that all series have a unit root. The test is based on averaging the individual test statistics, also called the Group-Mean (GM) test in Palm, Smeekes and Urbain (2011).

Usage

boot_panel(data, data_name = NULL, bootstrap = "AWB", B = 1999,
  block_length = NULL, ar_AWB = NULL, union = TRUE,
  union_quantile = 0.05, deterministics = NULL, detrend = NULL,
  min_lag = 0, max_lag = NULL, criterion = "MAIC",
  criterion_scale = TRUE, show_progress = TRUE, do_parallel = TRUE,
  cores = NULL)

Arguments

data	A $T$-dimensional vector or a ($T$ x $N$)-matrix of $N$ time series with $T$ observations to be tested for unit roots. Data may also be in a time series format (e.g. ts, zoo or xts), or a data frame, as long as each column represents a single time series.
data_name	Optional name for the data, to be used in the output. The default uses the name of the 'data' argument.
bootstrap	String for bootstrap method to be used. Options are "MBB" Moving block bootstrap (Paparoditis and Politis, 2003; Palm, Smeekes and Urbain, 2011); "BWB" Block wild bootstrap (Shao, 2011; Smeekes and Urbain, 2014a); "DWB" Dependent wild bootstrap (Shao, 2010; Smeekes and Urbain, 2014a; Rho and Shao, 2019); "AWB" Autoregressive wild bootstrap (Smeekes and Urbain, 2014a; Friedrich, Smeekes and Urbain, 2020), this is the default; "SB" Sieve bootstrap (Chang and Park, 2003; Palm, Smeekes and Urbain, 2008; Smeekes, 2013); "SWB" Sieve wild bootstrap (Cavaliere and Taylor, 2009; Smeekes and Taylor, 2012).
B	Number of bootstrap replications. Default is 1999.
block_length	Desired 'block length' in the bootstrap. For the MBB, BWB and DWB bootstrap, this is a genuine block length. For the AWB bootstrap, the block length is transformed into an autoregressive parameter via the formula $0.01^(1/block_length)$ as in Smeekes and Urbain (2014a); this can be overwritten by setting ar_AWB directly. Default sets the block length as a function of the time series length T, via the rule $block_length = 1.75 T^(1/3)$ of Palm, Smeekes and Urbain (2011).
ar_AWB	Autoregressive parameter used in the AWB bootstrap method (bootstrap = "AWB"). Can be used to set the parameter directly rather than via the default link to the block length.
union	Logical indicator whether or not to use bootstrap union tests (TRUE) or not (FALSE), see Smeekes and Taylor (2012). Default is TRUE.
union_quantile	The quantile of the bootstrap distribution used for scaling the individual statistics in the union. Ideally this should equal the desired significance level of the test. Default is 0.05. This parameter is overwritten when a significance level is provided in the argument level.
deterministics	String indicating the deterministic specification. Only relevant if union = FALSE. Options are ⁠"none":⁠ no deterministics; ⁠"intercept":⁠ intercept only; ⁠"trend":⁠ intercept and trend. If union = FALSE, the default is adding an intercept (a warning is given).
detrend	String indicating the type of detrending to be performed. Only relevant if union = FALSE. Options are: "OLS" or "QD" (typically also called GLS, see Elliott, Rothenberg and Stock, 1996). The default is "OLS".
min_lag	Minimum lag length in the augmented Dickey-Fuller regression. Default is 0.
max_lag	Maximum lag length in the augmented Dickey-Fuller regression. Default uses the sample size-based rule $12(T/100)^{1/4}$.
criterion	String for information criterion used to select the lag length in the augmented Dickey-Fuller regression. Options are: "AIC", "BIC", "MAIC", "MBIC". Default is "MAIC" (Ng and Perron, 2001).
criterion_scale	Logical indicator whether or not to use the rescaled information criteria of Cavaliere et al. (2015) (TRUE) or not (FALSE). Default is TRUE.
show_progress	Logical indicator whether a bootstrap progress update should be printed to the console. Default is FALSE.
do_parallel	Logical indicator whether bootstrap loop should be executed in parallel. Default is TRUE.
cores	The number of cores to be used in the parallel loops. Default is to use all but one.

Details

See boot_ur for details on the bootstrap algorithm and lag selection.

Value

An object of class "bootUR", "htest" with the following components:

method	The name of the hypothesis test method;
data.name	The name of the variable on which the method is performed;
null.value	The value of the (gamma) parameter of the lagged dependent variable in the ADF regression under the null hypothesis. Under the null, the series has a unit root. Testing the null of a unit root then boils down to testing the significance of the gamma parameter;
alternative	A character string specifying the direction of the alternative hypothesis relative to the null value. The alternative postulates that the series is stationary;
estimate	For the union test, the estimated value of the (gamma) parameter of the lagged dependent variable in the ADF regression is not defined, hence NA is given;
statistic	The value of the test statistic of the unit root test;
p.value	The p-value of the unit root test;
details	A list containing the detailed outcomes of the performed tests, such as selected lags, individual estimates and p-values.
specifications	The specifications used in the test.

Errors and warnings

Error: Resampling-based bootstraps MBB and SB cannot handle missing values.

If the time series in data have different starting and end points (and thus some series contain NA values at the beginning and/or end of the sample, the resampling-based moving block bootstrap (MBB) and sieve bootstrap (SB) cannot be used, as they create holes (internal missings) in the bootstrap samples. Switch to another bootstrap method or truncate your sample to eliminate NA values.

Warning: SB and SWB bootstrap only recommended for boot_ur; see help for details.

Although the sieve bootstrap methods "SB" and "SWB" can be used, Smeekes and Urbain (2014b) show that these are not suited to capture general forms of dependence across units, and using them for joint or multiple testing is not valid. This warning thereofre serves to recommend the user to consider a different bootstrap method.

Warning: Deterministic specification in argument deterministics is ignored, as union test is applied.

The union test calculates the union of all four combinations of deterministic components (intercept or intercept and trend) and detrending methods (OLS or QD). Setting deterministic components manually therefore has no effect.

Warning: Detrending method in argument detrend is ignored, as union test is applied.

The union test calculates the union of all four combinations of deterministic components (intercept or intercept and trend) and detrending methods (OLS or QD). Setting detrending methods manually therefore has no effect.

References

Smeekes, S. and Wilms, I. (2023). bootUR: An R Package for Bootstrap Unit Root Tests. Journal of Statistical Software, 106(12), 1-39.

Chang, Y. and Park, J. (2003). A sieve bootstrap for the test of a unit root. Journal of Time Series Analysis, 24(4), 379-400.

Cavaliere, G. and Taylor, A.M.R (2009). Heteroskedastic time series with a unit root. Econometric Theory, 25, 1228–1276.

Cavaliere, G., Phillips, P.C.B., Smeekes, S., and Taylor, A.M.R. (2015). Lag length selection for unit root tests in the presence of nonstationary volatility. Econometric Reviews, 34(4), 512-536.

Elliott, G., Rothenberg, T.J., and Stock, J.H. (1996). Efficient tests for an autoregressive unit root. Econometrica, 64(4), 813-836.

Friedrich, M., Smeekes, S. and Urbain, J.-P. (2020). Autoregressive wild bootstrap inference for nonparametric trends. Journal of Econometrics, 214(1), 81-109.

Ng, S. and Perron, P. (2001). Lag Length Selection and the Construction of Unit Root Tests with Good Size and Power. Econometrica, 69(6), 1519-1554,

Palm, F.C., Smeekes, S. and Urbain, J.-P. (2008). Bootstrap unit root tests: Comparison and extensions. Journal of Time Series Analysis, 29(1), 371-401.

Palm, F. C., Smeekes, S., and Urbain, J.-.P. (2011). Cross-sectional dependence robust block bootstrap panel unit root tests. Journal of Econometrics, 163(1), 85-104.

Paparoditis, E. and Politis, D.N. (2003). Residual-based block bootstrap for unit root testing. Econometrica, 71(3), 813-855.

Perron, P. and Qu, Z. (2008). A simple modification to improve the finite sample properties of Ng and Perron's unit root tests. Economic Letters, 94(1), 12-19.

Rho, Y. and Shao, X. (2019). Bootstrap-assisted unit root testing with piecewise locally stationary errors. Econometric Theory, 35(1), 142-166.

Shao, X. (2010). The dependent wild bootstrap. Journal of the American Statistical Association, 105(489), 218-235.

Shao, X. (2011). A bootstrap-assisted spectral test of white noise under unknown dependence. Journal of Econometrics, 162, 213-224.

Smeekes, S. (2013). Detrending bootstrap unit root tests. Econometric Reviews, 32(8), 869-891.

Smeekes, S. and Taylor, A.M.R. (2012). Bootstrap union tests for unit roots in the presence of nonstationary volatility. Econometric Theory, 28(2), 422-456.

Smeekes, S. and Urbain, J.-P. (2014a). A multivariate invariance principle for modified wild bootstrap methods with an application to unit root testing. GSBE Research Memorandum No. RM/14/008, Maastricht University

Smeekes, S. and Urbain, J.-P. (2014b). On the applicability of the sieve bootstrap in time series panels. Oxford Bulletin of Economics and Statistics, 76(1), 139-151.

See Also

boot_ur

Examples

# boot_panel on GDP_BE and GDP_DE
two_series_boot_panel <- boot_panel(MacroTS[, 1:2], bootstrap = "AWB", B = 199,
                                    do_parallel = FALSE, show_progress = FALSE)
print(two_series_boot_panel)

---

Bootstrap Sequential Quantile Test

Description

Performs the Bootstrap Sequential Quantile Test (BSQT) proposed by Smeekes (2015).

Usage

boot_sqt(data, data_name = NULL, steps = 0:NCOL(data), bootstrap = "AWB",
  B = 1999, block_length = NULL, ar_AWB = NULL, SQT_level = 0.05,
  union = TRUE, deterministics = NULL, detrend = NULL, min_lag = 0,
  max_lag = NULL, criterion = "MAIC", criterion_scale = TRUE,
  show_progress = TRUE, do_parallel = TRUE, cores = NULL)

Arguments

data	A $T$-dimensional vector or a ($T$ x $N$)-matrix of $N$ time series with $T$ observations to be tested for unit roots. Data may also be in a time series format (e.g. ts, zoo or xts), or a data frame, as long as each column represents a single time series.
data_name	Optional name for the data, to be used in the output. The default uses the name of the 'data' argument.
steps	Numeric vector of quantiles or units to be tested. Default is to test each unit sequentially.
bootstrap	String for bootstrap method to be used. Options are "MBB" Moving block bootstrap (Paparoditis and Politis, 2003; Palm, Smeekes and Urbain, 2011); "BWB" Block wild bootstrap (Shao, 2011; Smeekes and Urbain, 2014a); "DWB" Dependent wild bootstrap (Shao, 2010; Smeekes and Urbain, 2014a; Rho and Shao, 2019); "AWB" Autoregressive wild bootstrap (Smeekes and Urbain, 2014a; Friedrich, Smeekes and Urbain, 2020), this is the default; "SB" Sieve bootstrap (Chang and Park, 2003; Palm, Smeekes and Urbain, 2008; Smeekes, 2013); "SWB" Sieve wild bootstrap (Cavaliere and Taylor, 2009; Smeekes and Taylor, 2012).
B	Number of bootstrap replications. Default is 1999.
block_length	Desired 'block length' in the bootstrap. For the MBB, BWB and DWB bootstrap, this is a genuine block length. For the AWB bootstrap, the block length is transformed into an autoregressive parameter via the formula $0.01^(1/block_length)$ as in Smeekes and Urbain (2014a); this can be overwritten by setting ar_AWB directly. Default sets the block length as a function of the time series length T, via the rule $block_length = 1.75 T^(1/3)$ of Palm, Smeekes and Urbain (2011).
ar_AWB	Autoregressive parameter used in the AWB bootstrap method (bootstrap = "AWB"). Can be used to set the parameter directly rather than via the default link to the block length.
SQT_level	Desired significance level of the sequential tests performed. Default is 0.05.
union	Logical indicator whether or not to use bootstrap union tests (TRUE) or not (FALSE), see Smeekes and Taylor (2012). Default is TRUE.
deterministics	String indicating the deterministic specification. Only relevant if union = FALSE. Options are ⁠"none":⁠ no deterministics; ⁠"intercept":⁠ intercept only; ⁠"trend":⁠ intercept and trend. If union = FALSE, the default is adding an intercept (a warning is given).
detrend	String indicating the type of detrending to be performed. Only relevant if union = FALSE. Options are: "OLS" or "QD" (typically also called GLS, see Elliott, Rothenberg and Stock, 1996). The default is "OLS".
min_lag	Minimum lag length in the augmented Dickey-Fuller regression. Default is 0.
max_lag	Maximum lag length in the augmented Dickey-Fuller regression. Default uses the sample size-based rule $12(T/100)^{1/4}$.
criterion	String for information criterion used to select the lag length in the augmented Dickey-Fuller regression. Options are: "AIC", "BIC", "MAIC", "MBIC". Default is "MAIC" (Ng and Perron, 2001).
criterion_scale	Logical indicator whether or not to use the rescaled information criteria of Cavaliere et al. (2015) (TRUE) or not (FALSE). Default is TRUE.
show_progress	Logical indicator whether a bootstrap progress update should be printed to the console. Default is FALSE.
do_parallel	Logical indicator whether bootstrap loop should be executed in parallel. Default is TRUE.
cores	The number of cores to be used in the parallel loops. Default is to use all but one.

Details

The parameter steps can either be set as an increasing sequence of integers smaller or equal to the number of series N, or fractions of the total number of series (quantiles). For N time series, setting steps = 0:N means each unit should be tested sequentially. In this case the method is equivalent to the StepM method of Romano and Wolf (2005), and therefore controls the familywise error rate. To split the series in K equally sized groups, use steps = 0:K / K.

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. If the initial 0 or final value (1 or N) are omitted, they are automatically added by the function.

See boot_ur for details on the bootstrap algorithm and lag selection.

Value

An object of class "bootUR", "mult_htest" with the following components:

method	The name of the hypothesis test method;
data.name	The name of the data on which the method is performed;
null.value	The value of the (gamma) parameter of the lagged dependent variable in the ADF regression under the null hypothesis. Under the null, the series has a unit root. Testing the null of a unit root then boils down to testing the significance of the gamma parameter;
alternative	A character string specifying the direction of the alternative hypothesis relative to the null value. The alternative postulates that the series is stationary;
estimate	The estimated values of the (gamma) parameter of the lagged dependent variable in the ADF regressions. Note that for the union test (union = TRUE), this estimate is not defined, hence NA is returned;
statistic	The value of the test statistic of the unit root tests;
p.value	A vector with NA values, as p-values per inidividual series are not available.The p-value for each test in the sequence can be found in details;
rejections	A vector with logical indicators for each time series whether the null hypothesis of a unit root is rejected (TRUE) or not (FALSE);
details	A list containing the detailed outcomes of the performed tests, such as selected lags, individual estimates and p-values. In addtion, the slot FDR contains a matrix with for each step the stationary units under the null and alternative hypothesis, the test statistic and the p-value;
series.names	The names of the series that the tests are performed on;
specifications	The specifications used in the tests.

Errors and warnings

Error: Resampling-based bootstraps MBB and SB cannot handle missing values.

If the time series in data have different starting and end points (and thus some series contain NA values at the beginning and/or end of the sample, the resampling-based moving block bootstrap (MBB) and sieve bootstrap (SB) cannot be used, as they create holes (internal missings) in the bootstrap samples. Switch to another bootstrap method or truncate your sample to eliminate NA values.

Error: Invalid input values for steps: must be quantiles or positive integers.

Construction of steps does not satisfy the criteria listed under 'Details'.

Warning: SB and SWB bootstrap only recommended for boot_ur; see help for details.

Although the sieve bootstrap methods "SB" and "SWB" can be used, Smeekes and Urbain (2014b) show that these are not suited to capture general forms of dependence across units, and using them for joint or multiple testing is not valid. This warning thereofre serves to recommend the user to consider a different bootstrap method.

Warning: Deterministic specification in argument deterministics is ignored, as union test is applied.

The union test calculates the union of all four combinations of deterministic components (intercept or intercept and trend) and detrending methods (OLS or QD). Setting deterministic components manually therefore has no effect.

Warning: Detrending method in argument detrend is ignored, as union test is applied.

The union test calculates the union of all four combinations of deterministic components (intercept or intercept and trend) and detrending methods (OLS or QD). Setting detrending methods manually therefore has no effect.

References

Smeekes, S. and Wilms, I. (2023). bootUR: An R Package for Bootstrap Unit Root Tests. Journal of Statistical Software, 106(12), 1-39.

Chang, Y. and Park, J. (2003). A sieve bootstrap for the test of a unit root. Journal of Time Series Analysis, 24(4), 379-400.

Cavaliere, G. and Taylor, A.M.R (2009). Heteroskedastic time series with a unit root. Econometric Theory, 25, 1228–1276.

Cavaliere, G., Phillips, P.C.B., Smeekes, S., and Taylor, A.M.R. (2015). Lag length selection for unit root tests in the presence of nonstationary volatility. Econometric Reviews, 34(4), 512-536.

Elliott, G., Rothenberg, T.J., and Stock, J.H. (1996). Efficient tests for an autoregressive unit root. Econometrica, 64(4), 813-836.

Friedrich, M., Smeekes, S. and Urbain, J.-P. (2020). Autoregressive wild bootstrap inference for nonparametric trends. Journal of Econometrics, 214(1), 81-109.

Ng, S. and Perron, P. (2001). Lag Length Selection and the Construction of Unit Root Tests with Good Size and Power. Econometrica, 69(6), 1519-1554,

Palm, F.C., Smeekes, S. and Urbain, J.-P. (2008). Bootstrap unit root tests: Comparison and extensions. Journal of Time Series Analysis, 29(1), 371-401.

Palm, F. C., Smeekes, S., and Urbain, J.-.P. (2011). Cross-sectional dependence robust block bootstrap panel unit root tests. Journal of Econometrics, 163(1), 85-104.

Paparoditis, E. and Politis, D.N. (2003). Residual-based block bootstrap for unit root testing. Econometrica, 71(3), 813-855.

Perron, P. and Qu, Z. (2008). A simple modification to improve the finite sample properties of Ng and Perron's unit root tests. Economic Letters, 94(1), 12-19.

Rho, Y. and Shao, X. (2019). Bootstrap-assisted unit root testing with piecewise locally stationary errors. Econometric Theory, 35(1), 142-166.

Romano, J. P. and Wolf, M. (2005). Stepwise multiple testing as formalized data snooping. Econometrica, 73(4), 1237-1282. #' @references Shao, X. (2010). The dependent wild bootstrap. Journal of the American Statistical Association, 105(489), 218-235.

Shao, X. (2011). A bootstrap-assisted spectral test of white noise under unknown dependence. Journal of Econometrics, 162, 213-224.

Smeekes, S. (2013). Detrending bootstrap unit root tests. Econometric Reviews, 32(8), 869-891.

Smeekes, S. (2015). Bootstrap sequential tests to determine the order of integration of individual units in a time series panel. Journal of Time Series Analysis, 36(3), 398-415.

Smeekes, S. and Taylor, A.M.R. (2012). Bootstrap union tests for unit roots in the presence of nonstationary volatility. Econometric Theory, 28(2), 422-456.

Smeekes, S. and Urbain, J.-P. (2014a). A multivariate invariance principle for modified wild bootstrap methods with an application to unit root testing. GSBE Research Memorandum No. RM/14/008, Maastricht University

Smeekes, S. and Urbain, J.-P. (2014b). On the applicability of the sieve bootstrap in time series panels. Oxford Bulletin of Economics and Statistics, 76(1), 139-151.

See Also

boot_ur

Examples

# boot_sqt on GDP_BE and GDP_DE
two_series_boot_sqt <- boot_sqt(MacroTS[, 1:2], bootstrap = "AWB", B = 199,
                                do_parallel = FALSE, show_progress = FALSE)
print(two_series_boot_sqt)

---

Bootstrap Union Test for Unit Roots

Description

Performs bootstrap unit root test based on the union of rejections of 4 tests with different number of deterministic components and different type of detrending (Harvey, Leybourne and Taylor, 2012; Smeekes and Taylor, 2012).

Usage

boot_union(data, data_name = NULL, bootstrap = "AWB", B = 1999,
  block_length = NULL, ar_AWB = NULL, min_lag = 0, max_lag = NULL,
  criterion = "MAIC", criterion_scale = TRUE, union_quantile = 0.05,
  show_progress = TRUE, do_parallel = TRUE, cores = NULL)

Arguments

data	A $T$-dimensional vector to be tested for unit roots. Data may also be in a time series format (e.g. ts, zoo or xts), or a data frame.
data_name	Optional name for the data, to be used in the output. The default uses the name of the 'data' argument.
bootstrap	String for bootstrap method to be used. Options are "MBB" Moving blocks bootstrap (Paparoditis and Politis, 2003); "BWB" Block wild bootstrap (Shao, 2011); "DWB" Dependent wild bootstrap (Shao, 2010; Rho and Shao, 2019); "AWB" Autoregressive wild bootstrap (Smeekes and Urbain, 2014a; Friedrich, Smeekes and Urbain, 2020), this is the default; "SB" Sieve bootstrap (Palm, Smeekes and Urbain, 2008); "SWB" Sieve wild bootstrap (Cavaliere and Taylor, 2009; Smeekes and Taylor, 2012).
B	Number of bootstrap replications. Default is 1999.
block_length	Desired 'block length' in the bootstrap. For the MBB, BWB and DWB bootstrap, this is a genuine block length. For the AWB bootstrap, the block length is transformed into an autoregressive parameter via the formula $0.01^(1/block_length)$ as in Smeekes and Urbain (2014a); this can be overwritten by setting ar_AWB directly. Default sets the block length as a function of the time series length T, via the rule $block_length = 1.75 T^(1/3)$ of Palm, Smeekes and Urbain (2011).
ar_AWB	Autoregressive parameter used in the AWB bootstrap method (bootstrap = "AWB"). Can be used to set the parameter directly rather than via the default link to the block length.
min_lag	Minimum lag length in the augmented Dickey-Fuller regression. Default is 0.
max_lag	Maximum lag length in the augmented Dickey-Fuller regression. Default uses the sample size-based rule $12(T/100)^{1/4}$.
criterion	String for information criterion used to select the lag length in the augmented Dickey-Fuller regression. Options are: "AIC", "BIC", "MAIC", "MBIC". Default is "MAIC" (Ng and Perron, 2001).
criterion_scale	Logical indicator whether or not to use the rescaled information criteria of Cavaliere et al. (2015) (TRUE) or not (FALSE). Default is TRUE.
union_quantile	The quantile of the bootstrap distribution used for scaling the individual statistics in the union. Ideally this should equal the desired significance level of the test. Default is 0.05. This parameter is overwritten when a significance level is provided in the argument level.
show_progress	Logical indicator whether a bootstrap progress update should be printed to the console. Default is FALSE.
do_parallel	Logical indicator whether bootstrap loop should be executed in parallel. Default is TRUE.
cores	The number of cores to be used in the parallel loops. Default is to use all but one.

Details

The union is taken over the combination of tests with intercept only and intercept plus trend, coupled with OLS detrending and QD detrending, as in Harvey, Leybourne and Taylor (2012) and Smeekes an Taylor (2012). The bootstrap algorithm is always based on a residual bootstrap (under the alternative) to obtain residuals rather than a difference-based bootstrap (under the null), see e.g. Palm, Smeekes and Urbain (2008).

Lag length selection is done automatically in the ADF regressions with the specified information criterion. If one of the modified criteria of Ng and Perron (2001) is used, the correction of Perron and Qu (2008) is applied. 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.

Value

An object of class "bootUR", "htest" with the following components:

method	The name of the hypothesis test method;
data.name	The name of the variable on which the method is performed;
null.value	The value of the (gamma) parameter of the lagged dependent variable in the ADF regression under the null hypothesis. Under the null, the series has a unit root. Testing the null of a unit root then boils down to testing the significance of the gamma parameter;
alternative	A character string specifying the direction of the alternative hypothesis relative to the null value. The alternative postulates that the series is stationary;
estimate	For the union test, the estimated value of the (gamma) parameter of the lagged dependent variable in the ADF regression is not defined, hence NA is given;
statistic	The value of the test statistic of the unit root test;
p.value	The p-value of the unit root test;
details	A list containing the detailed outcomes of the performed tests, such as selected lags, individual estimates and p-values.
specifications	The specifications used in the test.

Errors and warnings

Error: Multiple time series not allowed. Switch to a multivariate method such as boot_ur, or change argument data to a univariate time series.

The function is a simple wrapper around boot_ur to facilitate use for single time series. It does not support multiple time series, as boot_ur is specifically suited for that.

References

Smeekes, S. and Wilms, I. (2023). bootUR: An R Package for Bootstrap Unit Root Tests. Journal of Statistical Software, 106(12), 1-39.

Chang, Y. and Park, J. (2003). A sieve bootstrap for the test of a unit root. Journal of Time Series Analysis, 24(4), 379-400.

Cavaliere, G. and Taylor, A.M.R (2009). Heteroskedastic time series with a unit root. Econometric Theory, 25, 1228–1276.

Cavaliere, G., Phillips, P.C.B., Smeekes, S., and Taylor, A.M.R. (2015). Lag length selection for unit root tests in the presence of nonstationary volatility. Econometric Reviews, 34(4), 512-536.

Friedrich, M., Smeekes, S. and Urbain, J.-P. (2020). Autoregressive wild bootstrap inference for nonparametric trends. Journal of Econometrics, 214(1), 81-109.

Harvey, D.I., Leybourne, S.J., and Taylor, A.M.R. (2012). Testing for unit roots in the presence of uncertainty over both the trend and initial condition. Journal of Econometrics, 169(2), 188-195.

Ng, S. and Perron, P. (2001). Lag Length Selection and the Construction of Unit Root Tests with Good Size and Power. Econometrica, 69(6), 1519-1554,

Palm, F.C., Smeekes, S. and Urbain, J.-P. (2008). Bootstrap unit root tests: Comparison and extensions. Journal of Time Series Analysis, 29(1), 371-401.

Paparoditis, E. and Politis, D.N. (2003). Residual-based block bootstrap for unit root testing. Econometrica, 71(3), 813-855.

Perron, P. and Qu, Z. (2008). A simple modification to improve the finite sample properties of Ng and Perron's unit root tests. Economic Letters, 94(1), 12-19.

Rho, Y. and Shao, X. (2019). Bootstrap-assisted unit root testing with piecewise locally stationary errors. Econometric Theory, 35(1), 142-166.

Shao, X. (2010). The dependent wild bootstrap. Journal of the American Statistical Association, 105(489), 218-235.

Shao, X. (2011). A bootstrap-assisted spectral test of white noise under unknown dependence. Journal of Econometrics, 162, 213-224.

Smeekes, S. (2013). Detrending bootstrap unit root tests. Econometric Reviews, 32(8), 869-891.

Smeekes, S. and Taylor, A.M.R. (2012). Bootstrap union tests for unit roots in the presence of nonstationary volatility. Econometric Theory, 28(2), 422-456.

Smeekes, S. and Urbain, J.-P. (2014a). A multivariate invariance principle for modified wild bootstrap methods with an application to unit root testing. GSBE Research Memorandum No. RM/14/008, Maastricht University

See Also

boot_ur

Examples

# boot_union on GDP_BE
GDP_BE_df <- boot_union(MacroTS[, 1], B = 199, do_parallel = FALSE, show_progress = FALSE)
print(GDP_BE_df)

---

Individual Unit Root Tests without multiple testing control

Description

This function performs bootstrap unit root tests on each time series individually.

Usage

boot_ur(data, data_name = NULL, bootstrap = "AWB", B = 1999,
  block_length = NULL, ar_AWB = NULL, level = NULL, union = TRUE,
  union_quantile = 0.05, deterministics = NULL, detrend = NULL,
  min_lag = 0, max_lag = NULL, criterion = "MAIC",
  criterion_scale = TRUE, show_progress = TRUE, do_parallel = TRUE,
  cores = NULL)

Arguments

data	A $T$-dimensional vector or a ($T$ x $N$)-matrix of $N$ time series with $T$ observations to be tested for unit roots. Data may also be in a time series format (e.g. ts, zoo or xts), or a data frame, as long as each column represents a single time series.
data_name	Optional name for the data, to be used in the output. The default uses the name of the 'data' argument.
bootstrap	String for bootstrap method to be used. Options are "MBB" Moving block bootstrap (Paparoditis and Politis, 2003; Palm, Smeekes and Urbain, 2011); "BWB" Block wild bootstrap (Shao, 2011; Smeekes and Urbain, 2014a); "DWB" Dependent wild bootstrap (Shao, 2010; Smeekes and Urbain, 2014a; Rho and Shao, 2019); "AWB" Autoregressive wild bootstrap (Smeekes and Urbain, 2014a; Friedrich, Smeekes and Urbain, 2020), this is the default; "SB" Sieve bootstrap (Chang and Park, 2003; Palm, Smeekes and Urbain, 2008; Smeekes, 2013); "SWB" Sieve wild bootstrap (Cavaliere and Taylor, 2009; Smeekes and Taylor, 2012).
B	Number of bootstrap replications. Default is 1999.
block_length	Desired 'block length' in the bootstrap. For the MBB, BWB and DWB bootstrap, this is a genuine block length. For the AWB bootstrap, the block length is transformed into an autoregressive parameter via the formula $0.01^(1/block_length)$ as in Smeekes and Urbain (2014a); this can be overwritten by setting ar_AWB directly. Default sets the block length as a function of the time series length T, via the rule $block_length = 1.75 T^(1/3)$ of Palm, Smeekes and Urbain (2011).
ar_AWB	Autoregressive parameter used in the AWB bootstrap method (bootstrap = "AWB"). Can be used to set the parameter directly rather than via the default link to the block length.
level	The desired significance level of the test (optional). This is only used for multivariate series to be able to provide a boolean vector with rejections of the null hypothesis or not for easy post-processing. Default is NULL, in which case no such vector is given.
union	Logical indicator whether or not to use bootstrap union tests (TRUE) or not (FALSE), see Smeekes and Taylor (2012). Default is TRUE.
union_quantile	The quantile of the bootstrap distribution used for scaling the individual statistics in the union. Ideally this should equal the desired significance level of the test. Default is 0.05. This parameter is overwritten when a significance level is provided in the argument level.
deterministics	String indicating the deterministic specification. Only relevant if union = FALSE. Options are ⁠"none":⁠ no deterministics; ⁠"intercept":⁠ intercept only; ⁠"trend":⁠ intercept and trend. If union = FALSE, the default is adding an intercept (a warning is given).
detrend	String indicating the type of detrending to be performed. Only relevant if union = FALSE. Options are: "OLS" or "QD" (typically also called GLS, see Elliott, Rothenberg and Stock, 1996). The default is "OLS".
min_lag	Minimum lag length in the augmented Dickey-Fuller regression. Default is 0.
max_lag	Maximum lag length in the augmented Dickey-Fuller regression. Default uses the sample size-based rule $12(T/100)^{1/4}$.
criterion	String for information criterion used to select the lag length in the augmented Dickey-Fuller regression. Options are: "AIC", "BIC", "MAIC", "MBIC". Default is "MAIC" (Ng and Perron, 2001).
criterion_scale	Logical indicator whether or not to use the rescaled information criteria of Cavaliere et al. (2015) (TRUE) or not (FALSE). Default is TRUE.
show_progress	Logical indicator whether a bootstrap progress update should be printed to the console. Default is FALSE.
do_parallel	Logical indicator whether bootstrap loop should be executed in parallel. Default is TRUE.
cores	The number of cores to be used in the parallel loops. Default is to use all but one.

Details

The options encompass many test proposed in the literature. detrend = "OLS" gives the standard augmented Dickey-Fuller test, while detrend = "QD" provides the DF-GLS test of Elliott, Rothenberg and Stock (1996). The bootstrap algorithm is always based on a residual bootstrap (under the alternative) to obtain residuals rather than a difference-based bootstrap (under the null), see e.g. Palm, Smeekes and Urbain (2008).

Lag length selection is done automatically in the ADF regression with the specified information criterion. If one of the modified criteria of Ng and Perron (2001) is used, the correction of Perron and Qu (2008) is applied. For very short time series (fewer than 50 time points) the maximum lag length is adjusted downward to avoid potential multicollinearity issues in the bootstrap. 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.

Value

An object of class "bootUR", "\*", where "\*" is "mult_htest" for multiple time series or "htest" for single time series, with the following components:

method	The name of the hypothesis test method;
data.name	The name of the data on which the method is performed;
null.value	The value of the (gamma) parameter of the lagged dependent variable in the ADF regression under the null hypothesis. Under the null, the series has a unit root. Testing the null of a unit root then boils down to testing the significance of the gamma parameter;
alternative	A character string specifying the direction of the alternative hypothesis relative to the null value. The alternative postulates that the series is stationary;
estimate	The estimated value(s) of the (gamma) parameter of the lagged dependent variable in the ADF regressions. Note that for the union test (union = TRUE), this estimate is not defined, hence NA is returned;
statistic	The value(s) of the test statistic of the unit root test(s);
p.value	The p-value(s) of the unit root test(s);
rejections	For "mult_htest" only. A vector with logical indicators for each time series whether the null hypothesis of a unit root is rejected (TRUE) or not (FALSE). This is only supplied when an optional significance level is given, otherwise NULL is returned;
details	A list containing the detailed outcomes of the performed tests, such as selected lags, individual estimates and p-values.
series.names	For "mult_htest" only. The names of the series that the tests are performed on;
specifications	The specifications used in the test(s).

Warnings

The function may give the following warnings.

Warning: Missing values cause resampling bootstrap to be executed for each time series individually.

If the time series in data have different starting and end points (and thus some series contain NA values at the beginning and/or end of the sample, the resampling-based moving block bootstrap (MBB) and sieve bootstrap (SB) cannot be used directly, as they create holes (internal missings) in the bootstrap samples. These bootstrap methods are therefore not applied jointly as usual, but individually to each series.

Warning: Deterministic specification in argument deterministics is ignored, as union test is applied.

The union test calculates the union of all four combinations of deterministic components (intercept or intercept and trend) and detrending methods (OLS or QD). Setting deterministic components manually therefore has no effect.

Warning: Detrending method in argument detrend is ignored, as union test is applied.

The union test calculates the union of all four combinations of deterministic components (intercept or intercept and trend) and detrending methods (OLS or QD). Setting detrending methods manually therefore has no effect.

References

Smeekes, S. and Wilms, I. (2023). bootUR: An R Package for Bootstrap Unit Root Tests. Journal of Statistical Software, 106(12), 1-39.

Chang, Y. and Park, J. (2003). A sieve bootstrap for the test of a unit root. Journal of Time Series Analysis, 24(4), 379-400.

Cavaliere, G. and Taylor, A.M.R (2009). Heteroskedastic time series with a unit root. Econometric Theory, 25, 1228–1276.

Cavaliere, G., Phillips, P.C.B., Smeekes, S., and Taylor, A.M.R. (2015). Lag length selection for unit root tests in the presence of nonstationary volatility. Econometric Reviews, 34(4), 512-536.

Elliott, G., Rothenberg, T.J., and Stock, J.H. (1996). Efficient tests for an autoregressive unit root. Econometrica, 64(4), 813-836.

Friedrich, M., Smeekes, S. and Urbain, J.-P. (2020). Autoregressive wild bootstrap inference for nonparametric trends. Journal of Econometrics, 214(1), 81-109.

Ng, S. and Perron, P. (2001). Lag Length Selection and the Construction of Unit Root Tests with Good Size and Power. Econometrica, 69(6), 1519-1554,

Palm, F.C., Smeekes, S. and Urbain, J.-P. (2008). Bootstrap unit root tests: Comparison and extensions. Journal of Time Series Analysis, 29(1), 371-401.

Palm, F. C., Smeekes, S., and Urbain, J.-.P. (2011). Cross-sectional dependence robust block bootstrap panel unit root tests. Journal of Econometrics, 163(1), 85-104.

Paparoditis, E. and Politis, D.N. (2003). Residual-based block bootstrap for unit root testing. Econometrica, 71(3), 813-855.

Perron, P. and Qu, Z. (2008). A simple modification to improve the finite sample properties of Ng and Perron's unit root tests. Economic Letters, 94(1), 12-19.

Rho, Y. and Shao, X. (2019). Bootstrap-assisted unit root testing with piecewise locally stationary errors. Econometric Theory, 35(1), 142-166.

Shao, X. (2010). The dependent wild bootstrap. Journal of the American Statistical Association, 105(489), 218-235.

Shao, X. (2011). A bootstrap-assisted spectral test of white noise under unknown dependence. Journal of Econometrics, 162, 213-224.

Smeekes, S. (2013). Detrending bootstrap unit root tests. Econometric Reviews, 32(8), 869-891.

Smeekes, S. and Taylor, A.M.R. (2012). Bootstrap union tests for unit roots in the presence of nonstationary volatility. Econometric Theory, 28(2), 422-456.

Smeekes, S. and Urbain, J.-P. (2014a). A multivariate invariance principle for modified wild bootstrap methods with an application to unit root testing. GSBE Research Memorandum No. RM/14/008, Maastricht University

Examples

# boot_ur on GDP_BE and GDP_DE
two_series_boot_ur <- boot_ur(MacroTS[, 1:2], bootstrap = "MBB", B = 199,
                              do_parallel = FALSE, show_progress = FALSE)
print(two_series_boot_ur)

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Check Missing Values in Sample

Description

Check Missing Values in Sample

Usage

check_missing_insample_values(X)

Arguments

X	A ($T$x$N$)-matrix of $N$ time series with $T$ observations. Data may also be in a time series format (e.g. ts, zoo or xts) or data frame.

Value

$N$-dimensional vector, for each series whether missing values are present (TRUE) or not (FALSE)

References

Smeekes, S. and Wilms, I. (2023). bootUR: An R Package for Bootstrap Unit Root Tests. Journal of Statistical Software, 106(12), 1-39.

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Differences of Multiple Time Series

Description

Performs differencing of multiple time series, with possibly different orders for each time series.

Usage

diff_mult(data, d, keep_NAs = TRUE)

Arguments

data	A ($T$x$N$)-matrix of $N$ time series with $T$ observations. Data may also be in a time series format (e.g. ts, zoo or xts) or data frame.
d	An $N$-dimensional vector containing the orders
keep_NAs	Logical indicator whether or not to keep the NA values resulting from differencing at the beginning of the sample. Default is TRUE. If FALSE, the entire row containing the NA values is removed.

Value

The appropriately differenced data in the same format as the original data.

References

Smeekes, S. and Wilms, I. (2023). bootUR: An R Package for Bootstrap Unit Root Tests. Journal of Statistical Software, 106(12), 1-39.

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Find Non-Missing Subsamples

Description

Find Non-Missing Subsamples

Usage

find_nonmissing_subsample(X)

Arguments

X	A ($T$x$N$)-matrix of $N$ time series with $T$ observations. Data may also be in a time series format (e.g. ts, zoo or xts) or data frame. Assumes a prior check on missing values in-sample has been done.

Value

A list with the following components

range	(2x$N$)-dimensional matrix containing the first and last non-missing observation in each column of X.
all_equal	Logical value indicating whether all series have the same non-missing indices.

References

Smeekes, S. and Wilms, I. (2023). bootUR: An R Package for Bootstrap Unit Root Tests. Journal of Statistical Software, 106(12), 1-39.

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Macroeconomic Time Series

Description

Macroeconomic data from Eurostat on GDP, consumption, inflation and unemployment for Belgium, Germany, France, the Netherlands and the United Kingdom.

Usage

MacroTS

Format

A time series object containing 20 macroeconomic seasonally adjusted time series, quarterly observed from 1992-2019 for Belgium (BE), Germany (DE), France (FR), the Netherlands (NL) and the United Kingdom (UK).

⁠GDP_BE⁠

Gross domestic product at market prices (index, 2015=100) for Belgium.

⁠GDP_DE⁠

Gross domestic product at market prices (index, 2015=100) for Germany.

⁠GDP_FR⁠

Gross domestic product at market prices (index, 2015=100) for France.

⁠GDP_NL⁠

Gross domestic product at market prices (index, 2015=100) for the Netherlands.

⁠GDP_UK⁠

Gross domestic product at market prices (index, 2015=100) for the United Kingdom.

⁠CONS_BE⁠

Final consumption expenditure (index, 2015=100) for Belgium.

⁠CONS_DE⁠

Final consumption expenditure (index, 2015=100) for Germany.

⁠CONS_FR⁠

Final consumption expenditure (index, 2015=100) for France.

⁠CONS_NL⁠

Final consumption expenditure (index, 2015=100) for the Netherlands.

⁠CONS_UK⁠

Final consumption expenditure (index, 2015=100) for the United Kingdom.

⁠HICP_BE⁠

Harmonised Indices of Consumer Prices (annual rate of change, 2015=100) for Belgium.

⁠HICP_DE⁠

Harmonised Indices of Consumer Prices (annual rate of change, 2015=100) for Germany.

⁠HICP_FR⁠

Harmonised Indices of Consumer Prices (annual rate of change, 2015=100) for France.

⁠HICP_N⁠

Harmonised Indices of Consumer Prices (annual rate of change, 2015=100) for the Netherlands.

⁠HICP_UK⁠

Harmonised Indices of Consumer Prices (annual rate of change, 2015=100) for the United Kingdom.

⁠UR_BE⁠

Unemployment rate (percentage of the active population) for Belgium.

⁠UR_DE⁠

Unemployment rate (percentage of the active population) for Germany.

⁠UR_FR⁠

Unemployment rate (percentage of the active population) for France.

⁠UR_NL⁠

Unemployment rate (percentage of the active population) for the Netherlands.

⁠UR_UK⁠

Unemployment rate (percentage of the active population) for the United Kingdom.

Note

• Unemployment rates are seasonally but not calendar adjusted, all other series are both seasonally and calendar adjusted.

• Quarterly inflation rates are sampled from Eurostat's monthly series with annual rates of change as the final month of the respective quarter.

• The unemployment rate for France excludes overseas territories ('France continental' in the Eurostat database).

Source

https://ec.europa.eu/eurostat/data/database

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Determine Order of Integration

Description

Determines the order of integration for each time series in a dataset via a sequence of unit root tests, and differences the data accordingly to eliminate stochastic trends.

Usage

order_integration(data, max_order = 2, method = "boot_ur", level = 0.05,
  plot_orders = FALSE, data_name = NULL, ...)

Arguments

data	A ($T$x$N$)-matrix of $N$ time series with $T$ observations. Data may also be in a time series format (e.g. ts, zoo or xts) or data frame.
max_order	The maximum order of integration of the time series. Default is 2.
method	The unit root tests to be used in the procedure. For multiple time series the options are "boot_ur", "boot_sqt" and "boot_fdr", with "boot_ur" the default. For single time series the options are "adf", boot_adf", "boot_union" and "boot_ur", with the latter the default.
level	Desired significance level of the unit root test. Default is 0.05.
plot_orders	Logical indicator whether the resulting orders of integration should be plotted. Default is FALSE.
data_name	Optional name for the data, to be used in the output. The default uses the name of the 'data' argument.
...	Optional arguments passed to the chosen unit root test function.

Details

The function follows the approach laid out in Smeekes and Wijler (2020), where all series is differenced $d-1$ times, where $d$ is the specified maximum order, and these differenced series are tested for unit roots. The series for which the unit root null is not rejected, are classified as $I(d)$ and removed from consideration. The remaining series are integrated, and tested for unit roots again, leading to a classification of $I(d-1)$ series if the null is not rejected. This is continued until a non-rejection is observed for all time series, or the series are integrated back to their original level. The series for which the null hypothesis is rejected in the final stage are classified as $I(0)$.

Care must be taken when using boot_sqt when the argument steps is given as a sequence of integers. As at each step series are removed, one may end up with fewer series to test than indicated in steps. While integers larger than the number of series will automatically be removed - along with a warning - by the test, it is recommend to set steps in the form of quantiles.

Plotting the orders of integration requires the ggplot2 package to be installed; plot will be skipped and a warning is given if not. For plots the function plot_order_integration is called. The user may prefer to set plot_orders = FALSE and call this function directly using the returned value of order_int in order to have more control over plot settings and save the plot object.

Value

An object of class "bootUR", "order_integration" with the following components

order_int	A vector with the found orders of integration of each time series.
diff_data	The appropriately differenced data according to order_int in the same format as the original data.

References

Smeekes, S. and Wilms, I. (2023). bootUR: An R Package for Bootstrap Unit Root Tests. Journal of Statistical Software, 106(12), 1-39.

Smeekes, S. and Wijler, E. (2020). Unit roots and cointegration. In P. Fuleky (Ed.) Macroeconomic Forecasting in the Era of Big Data, Chapter 17, pp. 541-584. Advanced Studies in Theoretical and Applied Econometrics, vol. 52. Springer.

Examples

# Use "boot_ur" to determine the order of GDP_BE and GDP_DE
orders_tseries <- order_integration(MacroTS[, 1:2], method = "boot_ur", B = 199,
do_parallel = FALSE, show_progress = FALSE)

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Plot Missing Values

Description

Plots missing values of different types for a time series dataset.

Usage

plot_missing_values(y, show_names = FALSE, show_legend = TRUE,
  axis_text_size = NULL, legend_size = NULL, cols = NULL)

Arguments

y	A ($T$x$N$)-matrix of $N$ time series with $T$ observations. Data may also be in a time series format (e.g. ts, zoo or xts) or data frame.
show_names	Show the time series' names on the plot (TRUE) or not (FALSE). Default is TRUE.
show_legend	Logical indicator whether a legend should be displayed. Default is TRUE.
axis_text_size	Size of the text on the axis. Default takes ggplot2 defaults.
legend_size	Size of the text in the legend if show_legend = TRUE. Default takes ggplot2 defaults.
cols	Vector with colours for displaying the different types of data. If the default is overwritten, four colours must be supplied.

Details

The function distinguish four types of data: observed data (non-missing) and three missing types. Type "Balanced NA" indicates where entire rows are missing (NA). These do not cause unbalancedness as the missing rows can simply be deleted. Type "Unbalanced NA" are missing values on the beginning or end of the sample, which cause unbalancedness. These affect some (but not all) bootstrap methods, see e.g.~boot_fdr. Type "Internal NA" are missing values inside the sample, which need to be removed before the bootstrap unit root tests can be used.

This function requires the package ggplot2 to be installed. If the package is not found, plotting is aborted.

Value

A ggplot2 object containing the missing values plot.

References

Smeekes, S. and Wilms, I. (2023). bootUR: An R Package for Bootstrap Unit Root Tests. Journal of Statistical Software, 106(12), 1-39.

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Plot Orders of Integration

Description

Plots a vector with orders of integration of time series.

Usage

plot_order_integration(orders, show_names = TRUE, show_legend = TRUE,
  names_size = NULL, legend_size = NULL, cols = NULL)

Arguments

orders	A "bootUR", "order_integration" object obtained from the function order_integration, or a vector with found orders of integration.
show_names	Show the time series' names on the plot (TRUE) or not (FALSE). Default is TRUE.
show_legend	Logical indicator whether a legend should be displayed. Default is TRUE.
names_size	Size of the time series' names if show_names = TRUE. Default takes ggplot2 defaults.
legend_size	Size of the text in the legend if show_legend = TRUE. Default takes ggplot2 defaults.
cols	Vector with colours for displaying the orders of integration. At least as many colours as orders of integration need to be supplied. Default supplies 4 colours for displaying up to $I(3)$ series.

Details

This function requires the package ggplot2 to be installed. If the package is not found, plotting is aborted.

Value

A ggplot2 object containing the plot of the orders of integration.

References

Smeekes, S. and Wilms, I. (2023). bootUR: An R Package for Bootstrap Unit Root Tests. Journal of Statistical Software, 106(12), 1-39.

See Also

order_integration

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