size {hrt}

Computing the Size of Heteroskedasticity Robust Tests

Description

This function provides an implementation of Algorithm 1 (if q = 1) or 2 (if q > 1), respectively, in Pötscher and Preinerstorfer (2021). Which of the two algorithms is applied is automatically determined as a function of q.

The user is referred to the just-mentioned article for definitions, a detailed description of the problem solved the algorithms, and for a detailed description of the algorithms themselves.

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.

Usage

size(C, 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, 
levelCl = 0, LBcheck = FALSE, as.tol = 1e-08)

Arguments

C

Critical value. A positive real number (for negative critical values the size of the test equals 1).

R

The restriction matrix. size computes the size of a test for 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 the 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

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, 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.

5. If levelCl is nonzero (see the description of levelCl below for details concerning this input), then one further initial value may be obtained by: (i) checking whether C exceeds 5 times the critical value C_H, say, for which the rejection probability under homoskedasticity equals 1-levelCl; and (ii) if this is the case, running the function size (with the same input parameters, but with levelCl set to 0 and M2 set to 1) on the critical value C_H, and then using the output second.stage.parameter as a further initial value.

M1

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

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

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

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

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

(Small) positive real number. Tolerance parameter used in checking invertibility of the covariance matrix in the test statistic. Default is 1e-08.

control.1

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

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 compute the size. If this lower bound is nonzero, 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.

levelCl

Number in [0, 1). This enters via the choice of the initial values as discussed in the input Mp above. levelCl should be used in case C is unusually large. In this case, the additional set of starting values provided may help to increase the accuracy of the size computation. Default is 0.

LBcheck

Either FALSE (default) or TRUE. If TRUE, then C is compared to the theoretical lower bounds on size-controlling critical values in Pötscher and Preinerstorfer (2021). If the supplemented C is smaller than the respective lower bound, theoretical results imply that the size equals 1 and the function size is halted.

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. Furthermore, as.tol is used in the rank computations required for computing lower bounds for size-controlling critical values (in case LBcheck is TRUE or levelCl is nonzero). 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 size is the following:

starting.parameters

The rows of this matrix are the initial values (diagonals of covariance matrices) that were used in Stage 1 of the algorithm, and which were chosen from the pool of initial values in Stage 0.

starting.rejection.probs

The null-rejection probabilities corresponding to the initial values used in Stage 1.

first.stage.parameters

The rows of this matrix are the parameters (diagonals of covariance matrices) that were obtained in Stage 1 of the algorithm.

first.stage.rejection.probs

The null-rejection probabilities corresponding to the first.stage.parameters.

second.stage.parameters

The rows of this matrix are the parameters (diagonals of covariance matrices) that were obtained in Stage 2 of the algorithm.

second.stage.rejection.probs

The null-rejection probabilities corresponding to the second.stage.parameters.

convergence

Convergence codes returned from constrOptim in Stage 2 of the algorithm for each initial value.

size

The size computed by the algorithm, i.e., the maximum of the second.stage.rejection.probs.

References

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

See Also

davies, constrOptim.

Examples

#size of the classical (uncorrected) F-test in a location model
#with conventional t-critical value (5% level)

#it is known that (in this very special case) the conventional critical value 
#is size-controlling (i.e., the resulting size should be 5% (approximately))

C <- qt(.975, df = 9)^2
R <- matrix(1, nrow = 1)
X <- matrix(rep(1, length = 10), nrow = 10, ncol = 1)
hcmethod <- -1
restr.cov <- FALSE
Mp <- 100
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

size(C, R, X, hcmethod, restr.cov, Mp, M1, M2)