theta {wbsd}

theta

Description

This function computes theta as discussed in the article Pötscher and Preinerstorfer (2020).

Usage

theta(
X, R, r, hcmethod, restr.cov, wilddist, wildmult, 
wildmult.restr, boot.res.restr,  boot.center.restr,    
tol = 1e-07, as.tol = 1e-07, in.tol = 1e-05, comp.meth = "exact", 
Boot.supp = NULL, checks = TRUE, cores = 1)

Arguments

X

Matrix (n times k, rank k).

R

Matrix (q times k, rank q).

r

Vector (q dimensional).

hcmethod

Integer in [-1, 4]. Determines the method used in the construction of the covariance estimator used in the test statistic. The value -1 corresponds to uncorrected F statistic without df adjustment; the value 0 corresponds to the HC0 estimator; ...; the value 4 corresponds to the HC4 estimator. Cf. Pötscher and Preinerstorfer (2020) and the references there for details.

restr.cov

TRUE or FALSE. Covariance matrix estimator based on null-restricted (TRUE) or unrestricted (FALSE) residuals.

wilddist

Distribution used to generate the bootstrap sample. Either “rademacher” or “mammen”. This is only used in case comp.meth = exact.

wildmult

Integer in [0, 4]. Determines the wild-bootstrap multiplier weights used. The value 0 corresponds to constant multiplieres of 1; the value 1 corresponds to multipliers that are obtained by taking the square of the HC1 weights; ...; the value 4 corresonds to multipliers that are obtained by taking the square of the HC4 weights.

wildmult.restr

TRUE or FALSE. Bootstrap multiplier weights computed on null-restricted (TRUE) or unrestricted (FALSE) projection matrix.

boot.res.restr

TRUE or FALSE. Bootstrap residuals based on null-restricted sample residuals (TRUE) or unrestricted sample residuals (FALSE)

boot.center.restr

TRUE or FALSE. Bootstrap sample centered at null-restricted predictor (TRUE; or FALSE)

tol

Real number. Tolerance parameter used in checking invertibility of the covariance matrix in the test statistic. This tolerance parameter is also used in checking R\hat{\beta}(e_i) \neq 0 in the computation of \vartheta_{2, Het} and \vartheta_{2, uc} (it is checked whether \|R\hat{\beta}(e_i)\|_{\infty} > tol. Default is 1e-07. A negativ input value will be converted to the machine epsilon. To check the stability of the numerical computations, it is recommended to try different values of tol and compare the obtained value of theta. Larger values of tol will lead to larger theta.

as.tol

Real number. Tolerance parameter used in checking Assumptions 1 or 2. Default is 1e-07. A negativ input value will be converted to the machine epsilon. To check the stability of the numerical computations, it is recommended to try different values of as.tol and compare the obtained value of theta. Larger values of as.tol will lead to larger theta.

in.tol

Real number. Tolerance parameter used in checking a strict inequality in the theta computation. Default is 1e-05. A negativ input value will be converted to the machine epsilon. To check the stability of the numerical computations, it is recommended to try different values of in.tol and compare the obtained value of theta. Larger values of in.tol will lead to larger theta.

comp.meth

“exact” or “approximation”. Determines whether probabilities over the bootstrap distribution are computed exactly or approximately. Exact computations are only feasible for small sample sizes.

Boot.supp

Bootstrap approximation distribution (e.g., obtained by drawing vectors each with n independent entries distributed according to a Rademacher distribution, or a Mammen distribution); (matrix, n rows, columns = number of bootstrap samples). This is only used if comp.meth = “approximation”; note that these vectors are further multiplied by wildmult weights.

checks

TRUE (default) or FALSE. Determines whether input checks are conducted (TRUE) or not (FALSE).

cores

Positive integer. Maximal number of cores used in the computations.

Details

For details see the relevant sections in Pötscher and Preinerstorfer (2020).

Value

theta

The value of theta; in case Assumptions are not satisfied NA is returned.

Aspt.sat

TRUE if checks passed, FALSE if checks did not pass.

Max.ind

The index that led to the minimum in the computation of theta. If Astp.sat = FALSE, then Max.ind = NA. Furthermore, Max.ind = NA in case theta was determined by the sufficient condition (k = q, boot.res.restr = boot.center.restr = TRUE) for theta = 1 discussed in the paper

References

Pötscher, B. M. and Preinerstorfer, D. (2020). How Reliable are Bootstrap-based Heteroskedastiity Robust Tests? <arXiv:2005.04089>

Examples

v1 <- rnorm(10)
X <- cbind(rep(1, 10), rnorm(10), rnorm(10), sign(v1)*exp(v1))                         
R <- rbind(c(0, 0, 1, 0), c(0, 0, 0, 1))
r <- c(0, 0)

theta(X, R, r, 3, FALSE, "rademacher", 3, FALSE, TRUE, TRUE)