critical.value: Critical Values for Autocorrelation Robust Testing
In acrt: Autocorrelation Robust Testing

Description Usage Arguments Details Value References See Also Examples

View source: R/critical.value.R

This function provides an implementation of Algorithm 1 in Pötscher and Preinerstorfer (2016), which we here abbreviate as (A1). The user is referred to this article for definitions, a detailed description of the problem solved by (A1), and for a detailed description of the algorithm itself.

critical.value(alpha, ar.order.max, bandwidth, ker, R, X, N0, N1, N2, Mp, M1, 
               M2, Eicker = FALSE, 
               opt.method.1 = "Nelder-Mead",
               opt.method.2 = "Nelder-Mead", 
               control.1 = list("reltol" = N1^(-.5), "maxit" = dim(X)[1]*20),
               control.2 = list("reltol" = N2^(-.5), "maxit" = dim(X)[1]*30),   
               cores = 1, margin = rep(1, length = ar.order.max))

`alpha`	Significance level. A real number in the interval (0, 1).
`ar.order.max`	Maximal order of the stationary autoregressive error process. A nonnegative integer. If `ar.order.max` is greater than sample size minus one, then `ar.order.max` is automatically replaced by sample size minus one. If `ar.order.max` equals sample size minus one, then the critical value approximated by (A1) controls size over the set of all stationary Gaussian error processes. If `ar.order.max` is set equal to 0, `critical.value` does not return the outcome of (A1), because no optimization is needed in that case. Instead, it draws a pseudorandom sample of size `N2` from the distribution of the test statistic under the null hypothesis and returns the corresponding 1-α quantile.
`bandwidth`	Bandwidth parameter used in the construction of the test statistic. A positive real number.
`ker`	Kernel function used in the construction of the test statistic. `ker` can take one of the values "Bartlett", "Parzen", or "Quadratic Spectral". The `kweights` function is used to generate the weights. For the test statistic T_{w} this implies the weights used via w(j, n) = ker(j/bandwidth). For the test statistic T_{E, \mathsf{W}} this implies the weights matrix \mathsf{W} via \mathsf{W}_{ij} = ker((i-j)/bandwidth), cf. Pötscher and Preinerstorfer (2016) for definitions of T_{w} and T_{E, \mathsf{W}}.
`R`	The restriction matrix. `critical.value` returns a critical value 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`.
`N0`	A positive integer. Corresponds to N_0 in the description of (A1) in Pötscher and Preinerstorfer (2016).
`N1`	A positive integer. Corresponds to N_1 in the description of (A1) in Pötscher and Preinerstorfer (2016). N1 should be greater than N0.
`N2`	A positive integer. Corresponds to N_2 in the description of (A1) in Pötscher and Preinerstorfer (2016). N2 should be greater than N1.
`Mp`	A positive integer. `Mp` determines M_0 in (A1), i.e., the number of starting values chosen in Stage 0 of (A1). The way initial values are generated depends on `ar.order.max`: If `ar.order.max` is 0 the choice of `Mp` is irrelevant, see the description of `ar.order.max` above. If `ar.order.max` is either 1 or 2, then the initial values are an i.i.d. pseudorandom sample of size `Mp` drawn from a distribution on the set of partial autocorrelation coefficients that induces a uniform distribution on the set of stationary autoregressive coefficients of order `ar.order.max` (cf. Jones (1987)). If `ar.order.max` is greater than 2, then starting values are generated as follows: Let A denote the set of integer multiples of 5 that are greater than 2 and smaller than `ar.order.max`. For every number l \in A \cup \{2, ar.order.max\} the algorithm in Jones (1987) is used to generate a pseudorandom sample of size `Mp` on the set of partial autocorrelation coefficients from a distribution that induces a uniform distribution on the set of stationary AR(l) coefficients. These samples are then used as initial values, cf. also the discussion in Pötscher and Preinerstorfer (2016).
`M1`	A positive integer. Corresponds to M_1 in the description of (A1) in Pötscher and Preinerstorfer (2016). M1 must not exceed Mp.
`M2`	A positive integer. Corresponds to M_2 in the description of (A1) in Pötscher and Preinerstorfer (2016). M2 must not exceed M1.
`Eicker`	Determines the test statistic used. If `Eicker = TRUE`, then critical values for T_{E, \mathsf{W}} (with \mathsf{W}_{ij}=ker((i-j)/bandwidth)) are computed. If `Eicker = FALSE`, then critical values for T_{w} (with w(j, n) = ker(j/bandwidth)) are computed (cf. Pötscher and Preinerstorfer (2016) for a precise definition of these test statistics). Default is `Eicker = FALSE`.
`opt.method.1`	The optimization method chosen in Stage 1 of (A1). Any optimization routine implemented in `optim` apart from "Brent" can be supplied. The default is "Nelder-Mead".
`opt.method.2`	The optimization method chosen in Stage 2 of (A1). Any optimization routine implemented in `optim` apart from "Brent" can be supplied. The default is "Nelder-Mead".
`control.1`	Control parameters passed to the `optim` function in Stage 1 of (A1). Default is `control.1 = list("reltol" = N1^(-.5), "maxit" = dim(X)[1]*20)`.
`control.2`	Control parameters passed to the `optim` function in Stage 2 of (A1). Default is `control.2 = list("reltol" = N2^(-.5), "maxit" = dim(X)[1]*30)`.
`cores`	The number of CPU cores used in the (parallelized) computation of the Monte-Carlo approximations in (A1). Default is 1. Parallelized computation is enabled only if the compiler used to build acrt supports OpenMP.
`margin`	The restrictions imposed on the partial autocorrelation coefficients. `margin` is an `ar.order.max`-dimensional vector of real numbers in (0, 1]. Default is `margin = rep(1, length = ar.order.max)`, which corresponds to no restriction.

For details see the relevant sections in Pötscher and Preinerstorfer (2016), in particular the description of Algorithm 1 in the Appendix.

The output of critical.value depends on ar.order.max. If ar.order.max is zero, the function critical.value returns a list consisting of:

critical.value

The critical value obtained by drawing a pseudorandom sample of size N2 from the distribution of the test statistic under the null hypothesis and by computing the corresponding 1-α quantile.

If ar.order.max is greater than zero, the function critical.value returns a list consisting of:

`starting.parameters`	The rows of this matrix are the initial values (partial autocorrelation coefficient vectors) that were used in Stage 1 of (A1), and which were chosen from the pool of randomly generated initial values in Stage 0. The rows correspond to ρ_{1:M_0}, ..., ρ_{M_1:M_0}, respectively, in the description of (A1).
`starting.quantiles`	Monte Carlo approximations of the 1-α quantiles corresponding to the initial values used in Stage 1 of (A1). The coordinates of this vector correspond to \tilde{F}^{-1}_{1:M_0}(1-α), ..., \tilde{F}^{-1}_{M_1:M_0}(1-α) in the description of (A1).
`first.stage.parameters`	The rows of this matrix are the parameters (partial autocorrelation coefficients) that were obtained in Stage 1 of (A1). The rows correspond to ρ^_{1}, ..., ρ^_{M_1}, respectively, in the description of (A1).
`first.stage.quantiles`	Monte Carlo approximations of the 1-α quantiles corresponding to the `first.stage.parameters`, i.e., \bar{F}_{j, ρ_j^}^{-1}(1-α)* for j = 1, ..., M_1 in the description of (A1).
`second.stage.parameters`	The rows of this matrix are the parameters (partial autocorrelation coefficients) that were obtained in Stage 2 of (A1). The rows correspond to ρ^{}_{1}, ..., ρ^{}_{M_2}, respectively, in the description of (A1).
`second.stage.quantiles`	Monte Carlo approximations of the 1-α quantiles corresponding to the `second.stage.parameters`, i.e., \bar{\bar{F}}_{j, ρ_j^{}}^{-1}(1-α) for j = 1, ..., M_2 in the description of (A1).
`convergence`	Convergence codes returned from `optim` in Stage 2 of (A1) for each initial value.
`critical.value`	The critical value obtained by (A1), i.e., the maximum of \bar{\bar{F}}_{j, ρ_j^{}}^{-1}(1-α) for j = 1, ..., M_2.

Jones, M. C. (1987). Randomly choosing parameters from the stationarity and invertibility region of autoregressive-moving average models. Applied Statistics, 36 134-138.

Pötscher, B.M. and Preinerstorfer, D. (2016). Controlling the size of autocorrelation robust tests. https://arxiv.org/abs/1612.06127/

optim, kweights.

## Not run: 
n <- 100
alpha <- .05
ar.order.max <- n-1
bandwidth <- 10
ker <- "Bartlett"
R <- matrix(c(0, 1), nrow = 1, ncol = 2)
X <- cbind(rep(1, length = n), rnorm(n))
N0 <- 1000
N1 <- 10000
N2 <- 50000
Mp <- 5000
M1 <- 10
M2 <- 2

critical.value(alpha, ar.order.max, bandwidth, ker, R, X, N0, N1, N2, Mp, M1, 
               M2)

## End(Not run)