Product sum matrix

In statistics, the product sum matrix or moment matrix is a symmetrical matrix that results from the product of the data matrix with its transpose . The inverse of the product sum matrix plays a major role in the calculation of the least squares estimator and in the calculation of projection matrices. The product sum matrix measures the information contained in the regressors.

definition

The product sum matrix is ​​defined as follows:

${\displaystyle \mathbf {X} ^{\top }\mathbf {X} ={\begin{pmatrix}\sum x_{t1}^{2}&\sum x_{t1}x_{t2}&\sum x_{t1}x_{t3}&\cdots &\sum x_{t1}x_{tK}\\\sum x_{t2}x_{t1}&\sum x_{t2}^{2}&\sum x_{t2}x_{t3}&\cdots &\sum x_{t2}x_{tK}\\\sum x_{t3}x_{t1}&\sum x_{t3}x_{t2}&\sum x_{t3}^{2}&\cdots &\sum x_{t3}x_{tK}\\\vdots &\vdots &\vdots &\ddots &\vdots \\\sum x_{tK}x_{t1}&\sum x_{tK}x_{t2}&\sum x_{tK}x_{t3}&\cdots &\sum x_{tK}^{2}\end{pmatrix}}=\sum \mathbf {x} _{i}\mathbf {x} _{i}^{\top }}$,

where the data matrix ${\displaystyle \mathbf {X} }$

${\displaystyle \mathbf {X} ={\begin{pmatrix}x_{11}&x_{12}&\cdots &x_{1k}&\cdots &x_{1K}\\x_{21}&x_{22}&\cdots &x_{2k}&\cdots &x_{2K}\\\vdots &\vdots &\ddots &\vdots &\ddots &\vdots \\x_{t1}&x_{t2}&\cdots &x_{tk}&\cdots &x_{tK}\\\vdots &\vdots &\ddots &\vdots &\ddots &\vdots \\x_{T1}&x_{T2}&\cdots &x_{Tk}&\cdots &x_{TK}\end{pmatrix}}}$

represents.

Used in the least squares estimator

The least squares estimator is the product of the inverse sum of products matrix with the product of with the vector of the endogenous variables: ${\displaystyle \mathbf {X} ^{\top }}$

${\displaystyle \mathbf {b} ={\begin{pmatrix}b_{1}\\b_{2}\\b_{3}\\\vdots \\b_{K}\end{pmatrix}}={\begin{pmatrix}\sum x_{t1}^{2}&\sum x_{t1}x_{t2}&\sum x_{t1}x_{t3}&\cdots &\sum x_{x1}x_{tK}\\\sum x_{t2}x_{t1}&\sum x_{t2}^{2}&\sum x_{t2}x_{t3}&\cdots &\sum x_{t2}x_{tK}\\\sum x_{t3}x_{t1}&\sum x_{t3}x_{t2}&\sum x_{t3}^{2}&\cdots &\sum x_{t3}x_{tK}\\\vdots &\vdots &\vdots &\ddots &\vdots \\\sum x_{tK}x_{t1}&\sum x_{tK}x_{t2}&\sum x_{tK}x_{t3}&\cdots &\sum x_{tK}^{2}\end{pmatrix}}^{-1}\cdot {\begin{pmatrix}\sum x_{t1}y_{t}\\\sum x_{t2}y_{t}\\\sum x_{t3}y_{t}\\\vdots \\\sum x_{tK}y_{t}\end{pmatrix}}=(\mathbf {X} ^{\top }\mathbf {X} )^{-1}\mathbf {X} ^{\top }\mathbf {y} }$.

The vector corresponds to the endogenous variable

${\displaystyle \mathbf {y} ={\begin{pmatrix}y_{1}\\y_{2}\\\vdots \\y_{t}\\\vdots \\y_{T}\end{pmatrix}}}$.

Asymptotic results

The product sum matrix averaged over n summands converges to a positively definite matrix , ${\displaystyle \mathbf {V} }$

${\displaystyle \lim _{n\to \infty }{\frac {\mathbf {X} _{n}^{\top }\mathbf {X} _{n}}{n}}=\mathbf {V} }$,

which plays an important role in determining the asymptotic properties of the KQ estimator.

Individual evidence

1. ↑ Winfried Schröder: Data Mining: Theoretical Aspects and Applications , p. 136
2. ↑ Gholamreza Nakhaeizadeh: Newer statistical methods and modeling in geoecology , p. 113