Description Usage Arguments Details Value Warnings References Examples
This function performs bootstrap unit root tests on each time series individually.
1 2 3 4 5 6  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)

data 
A Tdimensional 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. 
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

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 
Autoregressive parameter used in the AWB bootstrap method ( 
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 postprocessing. Default is 
union 
Logical indicator whether or not to use bootstrap union tests ( 
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 
deterministics 
String indicating the deterministic specification. Only relevant if
If 
detrend 
String indicating the type of detrending to be performed. Only relevant if 
min_lag 
Minimum lag length in the augmented DickeyFuller regression. Default is 0. 
max_lag 
Maximum lag length in the augmented DickeyFuller regression. Default uses the sample sizebased rule 12(T/100)^{1/4}. 
criterion 
String for information criterion used to select the lag length in the augmented DickeyFuller regression. Options are: 
criterion_scale 
Logical indicator whether or not to use the rescaled information criteria of Cavaliere et al. (2015) ( 
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. 
The options encompass many test proposed in the literature. detrend = "OLS"
gives the standard augmented DickeyFuller test, while detrend = "QD"
provides the DFGLS 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 differencebased 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 datadriven lag length selection with a prespecified 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.
An object of class "bootUR"
, "\*"
, where "\*"
is "mult_htest"
for multiple time series or "htest"
for single time series, with the following components:

The name of the hypothesis test method; 

The name of the data on which the method is performed; 

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; 

A character string specifying the direction of the alternative hypothesis relative to the null value. The alternative postulates that the series is stationary; 

The estimated value(s) of the (gamma) parameter of the lagged dependent variable in the ADF regressions. Note that for the union test ( 

The value(s) of the test statistic of the unit root test(s); 

The pvalue(s) of the unit root test(s); 

For 

For 

For 

The specifications used in the test(s). 
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 resamplingbased 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.
Chang, Y. and Park, J. (2003). A sieve bootstrap for the test of a unit root. Journal of Time Series Analysis, 24(4), 379400.
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), 512536.
Elliott, G., Rothenberg, T.J., and Stock, J.H. (1996). Efficient tests for an autoregressive unit root. Econometrica, 64(4), 813836.
Friedrich, M., Smeekes, S. and Urbain, J.P. (2020). Autoregressive wild bootstrap inference for nonparametric trends. Journal of Econometrics, 214(1), 81109.
Ng, S. and Perron, P. (2001). Lag Length Selection and the Construction of Unit Root Tests with Good Size and Power. Econometrica, 69(6), 15191554,
Palm, F.C., Smeekes, S. and Urbain, J.P. (2008). Bootstrap unit root tests: Comparison and extensions. Journal of Time Series Analysis, 29(1), 371401.
Palm, F. C., Smeekes, S., and Urbain, J..P. (2011). Crosssectional dependence robust block bootstrap panel unit root tests. Journal of Econometrics, 163(1), 85104.
Paparoditis, E. and Politis, D.N. (2003). Residualbased block bootstrap for unit root testing. Econometrica, 71(3), 813855.
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), 1219.
Rho, Y. and Shao, X. (2019). Bootstrapassisted unit root testing with piecewise locally stationary errors. Econometric Theory, 35(1), 142166.
Shao, X. (2010). The dependent wild bootstrap. Journal of the American Statistical Association, 105(489), 218235.
Shao, X. (2011). A bootstrapassisted spectral test of white noise under unknown dependence. Journal of Econometrics, 162, 213224.
Smeekes, S. (2013). Detrending bootstrap unit root tests. Econometric Reviews, 32(8), 869891.
Smeekes, S. and Taylor, A.M.R. (2012). Bootstrap union tests for unit roots in the presence of nonstationary volatility. Econometric Theory, 28(2), 422456.
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
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