Individual Unit Root Tests without multiple testing control

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.

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).

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.

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)
#> 
#> 	MBB bootstrap union test on each individual series (no multiple testing
#> 	correction)
#> 
#> data: MacroTS[, 1:2]
#> null hypothesis: Series has a unit root
#> alternative hypothesis: Series is stationary
#> 
#> Tests performed on each series: 
#>        estimate largest root statistic p-value
#> GDP_BE                    NA   -0.8123 0.34673
#> GDP_DE                    NA   -1.0958 0.08543
#>