Bootstrap Sequential Quantile Test

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.

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.

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.

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.

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)
#> 
#> 	AWB bootstrap sequential quantile union test
#> 
#> data: MacroTS[, 1:2]
#> null hypothesis: Series has a unit root
#> alternative hypothesis: Series is stationary
#> 
#> Sequence of tests: 
#>        H0: # I(0) H1: # I(0)  tstat p-value
#> Step 1          0          1 -1.053  0.1407
#>