Performs wild bootstrap and calculates bootstrap-t confidence intervals and
p-values for the mean, or the regression coefficients from a linear model.
-- Function File: bootwild (y)
-- Function File: bootwild (y, X)
-- Function File: bootwild (y, X, CLUSTID)
-- Function File: bootwild (y, X, BLOCKSZ)
-- Function File: bootwild (y, X, ..., NBOOT)
-- Function File: bootwild (y, X, ..., NBOOT, ALPHA)
-- Function File: bootwild (y, X, ..., NBOOT, ALPHA, SEED)
-- Function File: bootwild (y, X, ..., NBOOT, ALPHA, SEED, L)
-- Function File: STATS = bootwild (y, ...)
-- Function File: [STATS, BOOTSTAT] = bootwild (y, ...)
-- Function File: [STATS, BOOTSTAT, BOOTSSE] = bootwild (y, ...)
-- Function File: [STATS, BOOTSTAT, BOOTSSE, BOOTFIT] = bootwild (y, ...)
'bootwild (y)' performs a null hypothesis significance test for the
mean of y being equal to 0. This function implements wild bootstrap-t
resampling of Webb's 6-point distribution of the residuals, and computes
p-values after imposing the null hypothesis (H0) [1-4]. The following
statistics are printed to the standard output:
- original: the mean of the data vector y
- std_err: heteroscedasticity-consistent standard error(s)
- CI_lower: lower bound(s) of the 95% bootstrap-t confidence interval
- CI_upper: upper bound(s) of the 95% bootstrap-t confidence interval
- tstat: Student's t-statistic
- pval: two-tailed p-value(s) for the parameter(s) being equal to 0
- fpr: minimum false positive risk for the corresponding p-value
By default, the confidence intervals are symmetric, two-sided
bootstrap-t confidence intervals. The p-values are computed
following both of the guidelines by Hall and Wilson [5]. The minimum
false positive risk (FPR) is computed according to the Sellke-Berger
approach as described in [6,7].
'bootwild (y, X)' also specifies the design matrix (X) for least squares
regression of y on X. X should be a column vector or matrix the same
number of rows as y. If the X input argument is empty, the default for X
is a column of ones (i.e. intercept only) and thus the statistic computed
reduces to the mean (as above). The statistics calculated and returned in
the output then relate to the coefficients from the regression of y on X.
'bootwild (y, X, CLUSTID)' specifies a vector or cell array of numbers
or strings respectively to be used as cluster labels or identifiers.
Rows in y (and X) with the same CLUSTID value are treated as clusters
with dependent errors. Rows of y (and X) assigned to a particular
cluster will have identical resampling during wild bootstrap. If empty
(default), no clustered resampling is performed and all errors are
treated as independent. The standard errors computed are cluster robust.
'bootwild (y, X, BLOCKSZ)' specifies a scalar, which sets the block size
for bootstrapping when the residuals have serial dependence. Identical
resampling occurs within each (consecutive) block of length BLOCKSZ
during wild bootstrap. Rows of y (and X) within the same block are
treated as having dependent errors. If empty (default), no block
resampling is performed and all errors are treated as independent.
The standard errors computed are cluster robust.
'bootwild (y, X, ..., NBOOT)' specifies the number of bootstrap resamples,
where NBOOT must be a positive integer. If empty, the default value of
NBOOT is 1999.
'bootwild (y, X, ..., NBOOT, ALPHA)' is numeric and sets the lower and
upper bounds of the confidence interval(s). The value(s) of ALPHA must
be between 0 and 1. ALPHA can either be:
o scalar: To set the (nominal) central coverage of SYMMETRIC
bootstrap-t confidence interval(s) to 100*(1-ALPHA)%.
For example, 0.05 for a 95% confidence interval.
o vector: A pair of probabilities defining the (nominal) lower and
upper bounds of ASYMMETRIC bootstrap-t confidence interval(s)
as 100*(ALPHA(1))% and 100*(ALPHA(2))% respectively. For
example, [.025, .975] for a 95% confidence interval.
The default value of ALPHA is the scalar: 0.05, for a symmetric 95%
bootstrap-t confidence interval.
'bootwild (y, X, ..., NBOOT, ALPHA, SEED)' initialises the Mersenne
Twister random number generator using an integer SEED value so that
'bootwild' results are reproducible.
'bootwild (y, X, ..., NBOOT, ALPHA, SEED, L)' multiplies the regression
coefficients by the hypothesis matrix L. If L is not provided or is empty,
it will assume the default value of 1 (i.e. no change to the design).
'STATS = bootwild (...) returns a structure with the following fields:
original, std_err, CI_lower, CI_upper, tstat, pval, fpr and the sum-of-
squared error (sse).
'[STATS, BOOTSTAT] = bootwild (...) also returns a vector (or matrix) of
bootstrap statistics (BOOTSTAT) calculated over the bootstrap resamples
(before studentization).
'[STATS, BOOTSTAT, BOOTSSE] = bootwild (...) also returns a vector
containing the sum-of-squared error for the fit on each bootstrap
resample.
'[STATS, BOOTSTAT, BOOTSSE, BOOTFIT] = bootwild (...) also returns an
N-by-NBOOT matrix containing the N fitted values for each of the NBOOT
bootstrap resamples.
Bibliography:
[1] Wu (1986). Jackknife, bootstrap and other resampling methods in
regression analysis (with discussions). Ann Stat.. 14: 1261–1350.
[2] Cameron, Gelbach and Miller (2008) Bootstrap-based Improvements for
Inference with Clustered Errors. Rev Econ Stat. 90(3), 414-427
[3] Webb (2023) Reworking wild bootstrap-based inference for clustered
errors. Can J Econ. https://doi.org/10.1111/caje.12661
[4] Cameron and Miller (2015) A Practitioner’s Guide to Cluster-Robust
Inference. J Hum Resour. 50(2):317-372
[5] Hall and Wilson (1991) Two Guidelines for Bootstrap Hypothesis Testing.
Biometrics, 47(2), 757-762
[6] Colquhoun (2019) The False Positive Risk: A Proposal Concerning What
to Do About p-Values, Am Stat. 73:sup1, 192-201
[7] Sellke, Bayarri and Berger (2001) Calibration of p-values for Testing
Precise Null Hypotheses. Am Stat. 55(1), 62-71
bootwild (version 2023.07.05)
Author: Andrew Charles Penn
https://www.researchgate.net/profile/Andrew_Penn/
Copyright 2019 Andrew Charles Penn
This program is free software: you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation, either version 3 of the License, or
(at your option) any later version.
This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
You should have received a copy of the GNU General Public License
along with this program. If not, see http://www.gnu.org/licenses/
The following code
% Input univariate dataset heights = [183, 192, 182, 183, 177, 185, 188, 188, 182, 185].'; % Test statistics and p-values (H0 = 0) bootwild (heights); % Please be patient, the calculations will be completed soon...
Produces the following output
Summary of wild bootstrap null hypothesis significance tests for linear models ******************************************************************************* Bootstrap settings: Function: X \ y Resampling method: Wild bootstrap-t Number of resamples: 1999 Standard error calculations: Heteroscedasticity-Consistent (HC0) Confidence interval (CI) type: Symmetric bootstrap-t interval Nominal central coverage: 95% Null value (H0) used for hypothesis testing (p-values): 0 Test Statistics: original std_err CI_lower CI_upper t-stat p-val FPR +184.5 +1.243 +181.6 +187.4 +148. <.001 .010
The following code
% Input bivariate dataset
X = [ones(43,1),...
[01,02,03,04,05,06,07,08,09,10,11,...
12,13,14,15,16,17,18,19,20,21,22,...
23,25,26,27,28,29,30,31,32,33,34,...
35,36,37,38,39,40,41,42,43,44]'];
y = [188.0,170.0,189.0,163.0,183.0,171.0,185.0,168.0,173.0,183.0,173.0,...
173.0,175.0,178.0,183.0,192.4,178.0,173.0,174.0,183.0,188.0,180.0,...
168.0,170.0,178.0,182.0,180.0,183.0,178.0,182.0,188.0,175.0,179.0,...
183.0,192.0,182.0,183.0,177.0,185.0,188.0,188.0,182.0,185.0]';
% Compute test statistics and p-values
bootwild (y, X);
% Please be patient, the calculations will be completed soon...
Produces the following output
Summary of wild bootstrap null hypothesis significance tests for linear models ******************************************************************************* Bootstrap settings: Function: X \ y Resampling method: Wild bootstrap-t Number of resamples: 1999 Standard error calculations: Heteroscedasticity-Consistent (HC0) Confidence interval (CI) type: Symmetric bootstrap-t interval Nominal central coverage: 95% Null value (H0) used for hypothesis testing (p-values): 0 Test Statistics: original std_err CI_lower CI_upper t-stat p-val FPR +175.5 +2.502 +170.0 +181.0 +70.1 <.001 .010 +0.1904 +0.08261 +0.009462 +0.3714 +2.31 .040 .259
Package: statistics-resampling