HDTSA (version 1.0.6-1)

CP_Inference: Inference for the Double Projection Iterations (DPI) factor loading estimator in the tensor time series CP-factor model

Description

This function performs inference for the DPI estimator of a factor loading vector in the tensor CP-factor model (Chang et al., 2026+). Given a direction vector h, a factor index i, a mode index j, the observed tensor time series Y, and the output object returned by CP_TTS, the function returns the linear transformation of the debiased estimate, its estimated standard error, the original iterative loading estimator, and the estimated bias-correction term.

Usage

CP_Inference(
  h,
  i,
  j,
  Y,
  res.CP.DPI,
  var.est = c("plug-in", "long-run"),
  kernel = NULL,
  b_n = NULL
)

Value

A list with the following components:

aij.h.de

The debiased linear transformation \(\mathbf{h}^{\top}(\hat{\mathbf{a}}_{i,j} - \hat{\boldsymbol{\vartheta}}_{i,j})\).

se.h.ij

The estimated standard error of aij.h.de.

aij.iter

The original iterative estimator \(\hat{\mathbf{a}}_{i,j}\).

vartheta.ij

The estimated bias-correction term \(\hat{\boldsymbol{\vartheta}}_{i,j}\).

Arguments

h

A numeric vector of length \(d_j\). It specifies the linear transformation \(\mathbf{h}^{\top}(\hat{\mathbf{a}}_{i,j} - \hat{\boldsymbol{\vartheta}}_{i,j})\), where \(\hat{\boldsymbol{\vartheta}}_{i,j}\) is the bias-correction term.

i

A positive integer. The factor index.

j

A positive integer. The tensor mode index.

Y

An array containing the observed tensor time series with dimension \(n \times d_1 \times \cdots \times d_m\).

res.CP.DPI

An output object returned by CP_TTS.

var.est

A character string specifying the variance estimator. Available options are "plug-in" and "long-run". The default is "plug-in".

kernel

A kernel function used for the long-run variance estimator. If NULL, the quadratic spectral kernel is used. This argument is only used when var.est = "long-run".

b_n

A positive number specifying the bandwidth used for the long-run variance estimator. If NULL, a data-driven bandwidth rule is used. This argument is only used when var.est = "long-run".

Details

Let \(\hat{\mathbf{a}}_{i,j}\) be the DPI estimator of the loading vector for factor \(i\) and mode \(j\). The function computes the debiased estimator \(\hat{\mathbf{a}}_{i,j}-\hat{\boldsymbol{\vartheta}}_{i,j}\) and returns the linear transformation $$ \mathbf{h}^{\top} \left( \hat{\mathbf{a}}_{i,j} - \hat{\boldsymbol{\vartheta}}_{i,j} \right). $$ The reported standard error is $$ \sqrt{ \widehat{\mathrm{Var}} \left[ \mathbf{h}^{\top} \left( \hat{\mathbf{a}}_{i,j} - \hat{\boldsymbol{\vartheta}}_{i,j} \right) \right]/n } , $$ where \(n\) is the sample size. The variance can be estimated either by the plug-in estimator derived under the independence condition between the factors and idiosyncratic errors, or by a kernel-type long-run variance estimator. See Chang et al. (2026+) for more details on the construction of \(\hat{\boldsymbol{\vartheta}}_{i,j}\) and the estimated variance.

References

Chang, J., Huang, G., Yao, Q., & Yu, L. (2026+). CP-factorization for high dimensional tensor time series and double projection iterations. Preprint. tools:::Rd_expr_doi("doi:10.48550/arXiv.2606.08560").

Examples

if (FALSE) {
fit <- CP_TTS(Y)

out <- CP_Inference(
  h = h,
  i = 1,
  j = 2,
  Y = Y,
  res.CP.DPI = fit
)

out$aij.h.de
out$se.h.ij
out$aij.iter
out$vartheta.ij

## Use the long-run variance estimator
out.lr <- CP_Inference(
  h = h,
  i = 1,
  j = 2,
  Y = Y,
  res.CP.DPI = fit,
  var.est = "long-run"
)
}