size: Computing the Size of Heteroskedasticity Robust Tests
In hrt: Heteroskedasticity Robust Testing

Description Usage Arguments Details Value References See Also Examples

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.

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)

`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 β = 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 Σ_j in Stage 0 of the algorithm; the diagonal entries of each Σ_j sum up to 1) are chosen is as follows: If q = 1 and lower = 0, one of the initial values Σ_j is a matrix which maximizes the expectation of the quadratic form y \mapsto y'Σ^{1/2} A_C Σ^{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). One starting value Σ_j is a diagonal matrix with constant diagonal entries. If `lower` is zero, then (i) \lceil Mp/4 \rceil - 1 covariance matrices Σ_j are drawn by sampling their diagonals τ_1^2, ..., τ_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 Σ_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 τ_1^2, ..., τ_n^2 of Σ_j via (t_1^2, ..., t_n^2)/∑_{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`. 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. 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 Σ is normalized to sum up to 1; if lower > 0, then lower corresponds to what is denoted τ_* 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.

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

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`.

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

davies, constrOptim.

#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)