moment matrix

A multiple linear regression model can be written as

:$y\; =\; \backslash beta\_\{0\}\; +\; \backslash beta\_\{1\}\; x\_\{1\}\; +\; \backslash beta\_\{2\}\; x\_\{2\}\; +\; \backslash dots\; \backslash beta\_\{k\}\; x\_\{k\}\; +\; u$

where $y$ is the dependent variable, $x\_\{1\},\; x\_\{2\}\; \backslash dots,\; x\_\{k\}$ are the independent variables, $u$ is the error, and $\backslash beta\_\{0\},\; \backslash beta\_\{1\}\; \backslash dots,\; \backslash beta\_\{k\}$ are unknown coefficients to be estimated. Given observations $\backslash left\backslash \{\; y\_\{i\},\; x\_\{i1\},\; x\_\{i2\},\; \backslash dots,\; x\_\{ik\}\; \backslash right\backslash \}\_\{i=1\}^\{n\}$, we have a system of $n$ linear equations that can be expressed in matrix notation.{{cite book |first=David S. |last=Huang |title=Regression and Econometric Methods |location=New York |publisher=John Wiley & Sons |year=1970 |isbn=0-471-41754-8 |pages=52–65 |url=https://books.google.com/books?id=5IxRAAAAMAAJ&pg=PA52 }}

:

or

:$\backslash mathbf\{y\}\; =\; \backslash mathbf\{X\}\; \backslash boldsymbol\{\backslash beta\}\; +\; \backslash mathbf\{u\}$

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

:

and

:$\backslash mathbf\{X\}^\{\backslash mathsf\{T\}\}\; \backslash mathbf\{y\}\; =\; \backslash begin\{bmatrix\}\; \backslash sum\; y\_\{i\}\; \backslash \backslash \; \backslash sum\; x\_\{i1\}\; y\_\{i\}\; \backslash \backslash \; \backslash vdots\; \backslash \backslash \; \backslash sum\; x\_\{ik\}\; y\_\{i\}\; \backslash end\{bmatrix\}$

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

• Design matrix
• Gramian matrix
• Projection matrix

{{Reflist}}

• {{springer|title=Moment matrix|id=p/m130190}}

{{Matrix classes}}

Category:Matrices (mathematics)

Category:Least squares

{{Linear-algebra-stub}}