critical.value {hrt}

Critical Values for Heteroskedasticity Robust Testing

Description

This function provides an implementation of Algorithm 3 in Pötscher and Preinerstorfer (2021), based on Algorithm 1 (if q = 1) or Algorithm 2 (if q > 1) in the same reference as the auxiliary algorithm \mathsf{A}. Which of the two algorithms is used is automatically determined as a function of q, the number of rows of R.

The user is referred to Pötscher and Preinerstorfer (2021) for definitions, a detailed description of the problems solved by the algorithms, and for a detailed description of the algorithms themselves.

Most of the input parameters to critical.value are actually used in the auxiliary Algorithm 1 or 2, respectively. Algorithm 1 is based on the function davies from the package CompQuadForm. The parameters lim and acc for davies can be supplemented by the user. Algorithms 1 and 2 are implemented using the function constrOptim from stats in Stages 1 and 2; this function is used with default parameters, but control parameters can be supplied by the user.

After determining a critical value for a given testing problem via the function critical.value, it is recommended that: (i) the user applies the function size to compute the size of the test corresponding to the critical value obtained; and (ii) to check whether the size obtained does coincide with (or is close to) the targeted level of significance (that is alpha). If (ii) is not the case, this is an indication of numerical issues, which potentially can be avoided by changing the input parameters responsible for the accuracy of the computations.

Usage

critical.value(alpha, R, X, hcmethod, restr.cov, Mp, M1, M2, 
N0 = NULL, N1 = NULL, N2 = NULL, tol = 1e-08, 
control.1 = list("reltol" = 1e-02, "maxit" = dim(X)[1]*20),
control.2 = list("reltol" = 1e-03, "maxit" = dim(X)[1]*30),
cores = 1, lower = 0, eps.close = .0001, lim = 30000, acc = 0.001, 
size.tol = .001, maxit = 25, as.tol = 1e-08)

Arguments

alpha

Significance level. A real number in the interval (0, 1).

R

The restriction matrix. critical.value computes the (smallest) size-controlling critical value for a test of the hypothesis R \beta = r. R needs to be of full row rank, and needs to have the same number of columns as X.

X

The design matrix X needs to be of full column rank. The number of columns of X must be smaller than the number of rows of X.

hcmethod

Integer in [-1, 4]. Determines the method applied in the construction of the covariance estimator used in the test statistic. The value -1 corresponds to unadjusted (i.e., classical) F statistic without df adjustment; the value 0 corresponds to the HC0 estimator; ...; the value 4 corresponds to the HC4 estimator. Note that in case restr.cov is TRUE the null-restricted versions of the covariance estimators are computed. Cf. Pötscher and Preinerstorfer (2021) and the references there for details.

restr.cov

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

Mp

This input is used in Algorithm 1 or 2, respectively. Mp is a positive integer (should be chosen large, e.g., 50000; but the feasibility depends on the dimension of X, etc). Mp determines M_0 in Algorithm 1 or 2 (i.e., \mathsf{A}), respectively, that is, the number of initial values chosen in Stage 0 of that algorithm. The way initial values (i.e., the sets of variance covariance matrices \Sigma_j in Stage 0 of the algorithm; the diagonal entries of each \Sigma_j sum up to 1) are chosen is as follows:

1. If q = 1 and lower = 0, one of the initial values \Sigma_j is a matrix which maximizes the expectation of the quadratic form y \mapsto y'\Sigma^{1/2} A_C \Sigma^{1/2}y under an n-variate standard normal distribution. Here, A_C is a matrix that is defined Pötscher and Preinerstorfer (2021). If diagonal entries of this maximizer are 0, then they are replaced by the value of eps.close (and the other values are adjusted so that the diagonal sums up to 1).

2. One starting value \Sigma_j is a diagonal matrix with constant diagonal entries.

3. If lower is zero, then (i) \lceil Mp/4 \rceil - 1 covariance matrices \Sigma_j are drawn by sampling their diagonals \tau_1^2, ..., \tau_n^2 from a uniform distribution on the unit simplex in R^n; and (ii) the remaining M_p - (\lceil Mp/4 \rceil - 1) covariance matrices \Sigma_j are each drawn by first sampling a vector (t_1, ..., t_n)' from a uniform distribution on the unit simplex in R^n, and by then obtaining the diagonal \tau_1^2, ..., \tau_n^2 of \Sigma_j via (t_1^2, ..., t_n^2)/\sum_{i = 1}^n t_i^2. If lower is nonzero, then the initial values are drawn analogously, but from a uniform distribution on the subset of the unit simplex in R^n corresponding to the restriction imposed by the lower bound lower.

4. n starting values equal to covariance matrices with a single dominant diagonal entry and all other diagonal entries constant. The size of the dominant diagonal entry is regulated via the input parameters eps.close and lower. In case lower is nonzero, the size of the dominant diagonal entry equals 1-(n-1)*(lower + eps.close). In case lower is zero, the size of the dominant diagonal entry equals 1-eps.close.

M1

This input is used in Algorithm 1 or 2, respectively. A positive integer (should be chosen large, e.g., 500; but the feasibility depends on the dimension of X, etc). Corresponds to M_1 in the description of Algorithm 1 and 2 in Pötscher and Preinerstorfer (2021). M1 must not exceed Mp.

M2

This input is used in Algorithm 1 or 2, respectively. A positive integer. Corresponds to M_2 in the description of Algorithm 1 and 2 in Pötscher and Preinerstorfer (2021). M2 must not exceed M1.

N0

This input is needed in Algorithm 2. Only used in case q > 1 (i.e., when Algorithm 2 is used). A positive integer. Corresponds to N_0 in the description of Algorithm 2 in Pötscher and Preinerstorfer (2021).

N1

This input is needed in Algorithm 2. Only used in case q > 1 (i.e., when Algorithm 2 is used). A positive integer. Corresponds to N_1 in the description of Algorithm 2 in Pötscher and Preinerstorfer (2021). N1 should be greater than N0.

N2

This input is needed in Algorithm 2. Only used in case q > 1 (i.e., when Algorithm 2 is used). A positive integer. Corresponds to N_2 in the description of Algorithm 2 in Pötscher and Preinerstorfer (2021). N2 should be greater than N1.

tol

This input is used in Algorithm 1 or 2, respectively. (Small) positive real number. Tolerance parameter used in checking invertibility of the covariance matrix in the test statistic. Default is 1e-08.

control.1

This input is used in Algorithm 1 or 2, respectively. Control parameters passed to the constrOptim function in Stage 1 of Algorithm 1 or 2, respectively. Default is control.1 = list("reltol" = 1e-02, "maxit" = dim(X)[1]*20).

control.2

This input is used in Algorithm 1 or 2, respectively. Control parameters passed to the constrOptim function in Stage 2 of Algorithm 1 or 2, respectively. Default is control.2 = list("reltol" = 1e-03, "maxit" = dim(X)[1]*30).

cores

The number of CPU cores used in the (parallelized) computations. Default is 1. Parallelized computation is enabled only if the compiler used to build hrt supports OpenMP.

lower

Number in [0, n^{-1}) (note that the diagonal of \Sigma is normalized to sum up to 1; if lower > 0, then lower corresponds to what is denoted \tau_* in Pötscher and Preinerstorfer (2021)). lower specifies a lower bound on each diagonal entry of the (normalized) covariance matrix in the covariance model for which the user wants to obtain a critical value that achieves size control. If this lower bound is nonzero (which is the non-standard choice), then the size is only computed over all covariance matrices, which are restricted such that their minimal diagonal entry is not smaller than lower. The relevant optimization problems in Algorithm 1 and 2 are then carried out only over this restricted set of covariance matrices. The size will then in general depend on lower. See the relevant discussions concerning restricted heteroskedastic covariance models in Pötscher and Preinerstorfer (2021). Default is 0, which is the recommended choice, unless there are strong reasons implying a specific lower bound on the variance in a given application.

eps.close

(Small) positive real number. This determines the size of the dominant entry in the choice of the initial values as discussed in the description of the input Mp above. Default is 1e-4.

lim

This input is needed in Algorithm 1. Only used in case q = 1 (i.e., when Algorithm 1 is used). Input parameter for the function davies. Default is 30000.

acc

This input is needed in Algorithm 1. Only used in case q = 1 (i.e., when Algorithm 1 is used). Input parameter for the function davies. Default is 1e-3.

size.tol

(Small) positive real number. \epsilon in Algorithm 3. Default is 1e-3.

maxit

Maximum number of iterations in the while loop of Algorithm 3. Default is 25.

as.tol

(Small) positive real number. Tolerance parameter used in checking rank conditions for verifying Assumptions 1, 2, and for checking a non-constancy condition on the test statistic in case hcmethod is not -1 and restr.cov is TRUE. as.tol is also used in the rank computations required for computing lower bounds for size-controlling critical values. Furthermore, as.tol is used in checking the sufficient conditions for existence of a size-controlling critical value provided in Pötscher and Preinerstorfer(2021). Default is 1e-08.

Details

For details see the relevant sections in Pötscher and Preinerstorfer (2021), in particular the description of Algorithms 1 and 2 in the Appendix.

Value

The output of critical.value is the following:

critical.value

The critical value obtained by Algorithm 3.

approximate.size

The approximate size of the test based on the returned critical value.

iter

The number of iterations performed. If iter is smaller than maxit, then the algorithm determined because the required level of accuracy was achieved.

References

Pötscher, B. M. and Preinerstorfer, D. (2021). Valid Heteroskedasticity Robust Testing. <arXiv:2104.12597>

See Also

davies, constrOptim.

Examples

#critical value for the classical (uncorrected) F-test in a location model
#with unrestricted heteroskedasticity

#it is known that (in this very special case) the conventional critical value 
#C <- qt(.975, df = 9)^2
#is size-controlling (thus the resulting size should be 5% (approximately))

R <- matrix(1, nrow = 1)
X <- matrix(rep(1, length = 10), nrow = 10, ncol = 1)
hcmethod <- -1
restr.cov <- FALSE
Mp <- 1000
M1 <- 5
M2 <- 1

#here, the parameters are chosen such that the run-time is low
#to guarantee a high accuracy level in the computation, 
#Mp, M1 and M2 should be chosen much higher

critical.value(alpha = .05, R, X, hcmethod, restr.cov, Mp, M1, M2)