SpecTest function

Global testing for spectral density matrix

SpecTest() implements a new global test proposed in Chang et al. (2022) for the following hypothesis testing problem: [REMOVE_ME]$H_0:f_{i,j}(\omega)=0 \mathrm{\ for\ any\ }(i,j)\in \mathcal{I}\mathrm{\ and\ }\omega \in \mathcal{J}\mathrm{\ \ versus\ \ }H_1:H_0\mathrm{\ is\ not\ true }\,, [REMOVE_ME_2]$

where $f_{i,j}(\omega)$ represents the cross-spectral density between $x_{t,i}$ and $x_{t,j}$ at frequency $\omega$ with $x_{t,i}$ being the $i$-th component of the $p \times 1$ times series ${\bf x}_t$. Here, $\mathcal{I}$ is the set of index pairs of interest, and $\mathcal{J}$ is the set of frequencies.

SpecTest(X, J.set, cross.indices, B = 1000, flag_c = 0.8)

Arguments

• X: An $n\times p$ data matrix ${\bf X} = ({\bf x}_1, \dots , {\bf x}_n)'$, where $n$ is the number of observations of the $p\times 1$ time series $\{{\bf x}_t\}_{t=1}^n$.

• J.set: A vector representing the set $\mathcal{J}$

  of frequencies.

• cross.indices: An $r \times 2$ matrix representing the set $\mathcal{I}$ of $r$ index pairs, where each row represents an index pair.

• B: The number of bootstrap replications for generating multivariate normally distributed random vectors when calculating the critical value. The default is 2000.

• flag_c: The bandwidth $c\in(0,1]$ of the flat-top kernel for estimating $f_{i,j}(\omega)$ [See (2) in Chang et al. (2022)]. The default is 0.8.

Returns

An object of class "hdtstest", which contains the following components:

• Stat: The test statistic of the test.

• pval: The p-value of the test.

• cri95: The critical value of the test at the significance level 0.05.

Description

SpecTest() implements a new global test proposed in Chang et al. (2022) for the following hypothesis testing problem:

where $f_{i,j}(\omega)$ represents the cross-spectral density between $x_{t,i}$ and $x_{t,j}$ at frequency $\omega$ with $x_{t,i}$ being the $i$-th component of the $p \times 1$ times series ${\bf x}_t$. Here, $\mathcal{I}$ is the set of index pairs of interest, and $\mathcal{J}$ is the set of frequencies.

Examples

# Example 1
## Generate xt
n <- 200
p <- 10
flag_c <- 0.8
B <- 1000
burn <- 1000
z.sim <- matrix(rnorm((n+burn)*p),p,n+burn)
phi.mat <- 0.4*diag(p)
x.sim <- phi.mat %*% z.sim[,(burn+1):(burn+n)]
x <- x.sim - rowMeans(x.sim)

## Generate the sets I and J
cross.indices <- matrix(c(1,2), ncol=2)
J.set <- 2*pi*seq(0,3)/4 - pi

res <- SpecTest(t(x), J.set, cross.indices, B, flag_c)
Stat <- res$statistic
Pvalue <- res$p.value
CriVal <- res$cri95

References

Chang, J., Jiang, Q., McElroy, T. S., & Shao, X. (2022). Statistical inference for high-dimensional spectral density matrix. arXiv preprint. tools:::Rd_expr_doi("doi:10.48550/arXiv.2212.13686") .

See Also

SpecMulTest