F.type.test.statistic {acrt} R Documentation

## Computation of F-type test statistics

### Description

This function computes test statistics of the form T_w and T_{E, \mathsf{W}} as defined in Pötscher and Preinerstorfer (2016). The weights for T_w and for T_{E, \mathsf{W}} are obtained from a kernel function (Bartlett, Parzen, or Quadratic Spectral kernel, which provide nonnegative definite covariance estimators) and a bandwidth parameter. See also the description of the argument `ker` below for further details concerning the weights. The class of test statistics of the form T_w or T_{E, \mathsf{W}} includes F-type tests based on covariance estimators with data-independent bandwidth parameters and without prewhitening as considered in, e.g., Newey and West (1987), Andrews (1991), Kiefer and Vogelsang (2002, 2005), cf. also Preinerstorfer and Pötscher (2016).

### Usage

```F.type.test.statistic(y, R, r, X, bandwidth, ker, Eicker = FALSE, cores = 1)
```

### Arguments

 `y` Either an observation vector, or a matrix the columns of which are observation vectors. The number of rows of an observation vector must coincide with the number of rows of the design matrix `X`. `R` The restriction matrix. `F.type.test.statistic` computes a test statistic 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`. `r` The restriction vector. `F.type.test.statistic` computes a test statistic for the hypothesis R β = r. `r` needs to be a vector with the same number of coordinates as the number of rows of `R`. `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`. `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}}. `Eicker` Determines the test statistic computed. If `Eicker = TRUE`, then T_{E, \mathsf{W}} (with \mathsf{W}_{ij}=ker((i-j)/bandwidth)) is computed on the input observation vector(s) `y`. If `Eicker = FALSE`, then T_{w} (with w(j, n) = ker(j/bandwidth)) is computed on the input observation vector(s) `y` (cf. Pötscher and Preinerstorfer (2016) for a precise definition of these test statistics). Default is `Eicker = FALSE`. `cores` The number of CPU cores used in the (parallelized) computation of the test statistics. Default is 1. This can be used to speed up the computation in case `y` is a matrix with many columns. Parallelized computation is enabled only if the compiler used to build acrt supports OpenMP.

### Details

For details concerning the test statistics please see the relevant sections in Pötscher and Preinerstorfer (2016) .

### Value

The function returns a list consisting of:

 `test.val` Either a vector the entries of which correspond to the values of the test statistic evaluated at each column of the input matrix `y`, or, if `y` is a vector, the test statistic evaluated at `y`.

### References

Andrews, D. W. K. (1991). Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica, 59 817-858.

Kiefer, N. M. and Vogelsang, T. J. (2002). Heteroskedasticity - autocorrelation robust standard errors using the Bartlett kernel without truncation. Econometrica, 70 2093-2095.

Kiefer, N. M. and Vogelsang, T. J. (2005). A new asymptotic theory for heteroskedasticity - autocorrelation robust tests. Econometric Theory, 21 1130-1164.

Newey, W. K. and West, K. D. (1987). A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica, 55 703-708.

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

Preinerstorfer, D. and Pötscher, B. M. (2016). On size and power of heteroskedasticity and autocorrelation robust tests. Econometric Theory, 32 261-358.

`kweights`.

### Examples

```n <- 100
y <- rnorm(n)
X <- cbind(rep(1, length = n), rnorm(n))
R <- matrix(c(0, 1), nrow = 1, ncol = 2)
r <- 0
bandwidth <- n/10
ker <- "Bartlett"
F.type.test.statistic(y, R, r, X, bandwidth, ker)
```

[Package acrt version 1.0.1 Index]