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Inmathematics, amoment matrix is a special symmetric squarematrix whose rows and columns are indexed bymonomials. The entries of the matrix depend on the product of the indexing monomials only (cf.Hankel matrices.)

Moment matrices play an important role inpolynomial fitting, polynomial optimization (sincepositive semidefinite moment matrices correspond to polynomials which aresums of squares)^[1] andeconometrics.^[2]

A multiplelinear regression model can be written as

${\displaystyle y={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+\dots {\beta }_k{x}_k+u}$

where${\displaystyle y}$ is the dependent variable,${\displaystyle x_1,x_2\dots ,x_k}$ are the independent variables,${\displaystyle u}$ is the error, and${\displaystyle {\beta }_0,{\beta }_1\dots ,{\beta }_k}$ are unknown coefficients to be estimated. Given observations${\displaystyle {\left\{y_i,x_{i1},x_{i2},\dots ,x_{ik}\right\}}_{i=1}^n}$, we have a system of${\displaystyle n}$ linear equations that can be expressed in matrix notation.^[3]

or

${\displaystyle \mathbf{y}=\mathbf{X}\mathit{\beta }+\mathbf{u}}$

where${\displaystyle \mathbf{y}}$ and${\displaystyle \mathbf{u}}$ are each a vector of dimension${\displaystyle n\times 1}$,${\displaystyle \mathbf{X}}$ is thedesign matrix of order${\displaystyle N\times (k+1)}$, and${\displaystyle \mathit{\beta }}$ is a vector of dimension${\displaystyle (k+1)\times 1}$. Under theGauss–Markov assumptions, the best linear unbiased estimator of${\displaystyle \mathit{\beta }}$ is the linearleast squares estimator${\displaystyle \mathbf{b}={\left({\mathbf{X}}^{\mathsf{T}}\mathbf{X}\right)}^{-1}{\mathbf{X}}^{\mathsf{T}}\mathbf{y}}$, involving the two moment matrices${\displaystyle {\mathbf{X}}^{\mathsf{T}}\mathbf{X}}$ and${\displaystyle {\mathbf{X}}^{\mathsf{T}}\mathbf{y}}$ defined as

and

${\displaystyle {\mathbf{X}}^{\mathsf{T}}\mathbf{y}=\left[\begin{array}{c}\sum y_i\\ \sum x_{i1}{y}_i\\ ⋮\\ \sum x_{ik}{y}_i\end{array}\right]}$

where${\displaystyle {\mathbf{X}}^{\mathsf{T}}\mathbf{X}}$ is a squarenormal matrix of dimension${\displaystyle (k+1)\times (k+1)}$, and${\displaystyle {\mathbf{X}}^{\mathsf{T}}\mathbf{y}}$ is a vector of dimension${\displaystyle (k+1)\times 1}$.

• Design matrix
• Gramian matrix
• Projection matrix

• "Moment matrix",Encyclopedia of Mathematics,EMS Press, 2001 [1994]

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