If 𝐙 is the n ×k matrix of standardized regressors and 𝐓 is...

If $\mathbf{Z}$ is the $n \times k$ matrix of standardized regressors and $\mathbf{T}$ is the $k \times k$ upper, triangular matrix in Eq. $(10.3),$ show that the transformed regressors $\mathbf{W}=\mathbf{Z} \mathbf{T}^{-1}$ are orthogonal and have unit variance.

Key Concepts

Matrix Transformation and Inversion

Matrix transformation involves applying a linear mapping to a set of variables, which may change their scale, orientation, or correlation. When using an invertible matrix like T inverse, the transformation W = ZT?¹ redefines the regressors in a way that can simplify their inter-relationships, often aiming to achieve properties like orthogonality and ease of interpretation.

Orthogonalization and Unit Variance

Orthogonalization is the process of transforming a set of vectors into a new set in which each pair of different vectors is uncorrelated (or orthogonal), meaning their inner product is zero. When combined with standardization, the resulting transformed regressors possess unit variance as well as mutual orthogonality. This property is especially valuable in regression analysis because it leads to more stable and interpretable parameter estimates.

Standardized Regressors

Standardized regressors are variables that have been adjusted to have a mean of zero and a variance of one. This is important in statistical regression because it puts all variables on a comparable scale, which can simplify analysis and interpretation while reducing issues related to differences in units or scales.

Upper Triangular Matrix

An upper triangular matrix, typically created during decomposition methods in linear algebra such as the Gram-Schmidt process or Cholesky decomposition, is a matrix in which all the entries below the main diagonal are zero. In this context, the upper triangular matrix T contains the coefficients needed to map the correlation structure of the original standardized regressors, facilitating further transformations.

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