Index: The Book of Statistical ProofsStatistical Models ▷ Univariate normal data ▷ Multiple linear regression ▷ Ordinary least squares

Theorem: Given a linear regression model with independent observations

\[\label{eq:MLR} y = X\beta + \varepsilon, \; \varepsilon_i \overset{\mathrm{i.i.d.}}{\sim} \mathcal{N}(0, \sigma^2) \; ,\]

the parameters minimizing the residual sum of squares are given by

\[\label{eq:OLS} \hat{\beta} = (X^\mathrm{T} X)^{-1} X^\mathrm{T} y \; .\]

Proof: The residual sum of squares is defined as

\[\label{eq:RSS} \mathrm{RSS}(\beta) = \sum_{i=1}^n \varepsilon_i = \varepsilon^\mathrm{T} \varepsilon = (y-X\beta)^\mathrm{T} (y-X\beta)\]

which can be developed into

\[\label{eq:RSS-dev} \begin{split} \mathrm{RSS}(\beta) &= y^\mathrm{T} y - y^\mathrm{T} X \beta - \beta^\mathrm{T} X^\mathrm{T} y + \beta^\mathrm{T} X^\mathrm{T} X \beta \\ &= y^\mathrm{T} y - 2 \beta^\mathrm{T} X^\mathrm{T} y + \beta^\mathrm{T} X^\mathrm{T} X \beta \; . \end{split}\]

The derivative of $\mathrm{RSS}(\beta)$ with respect to $\beta$ is

\[\label{eq:RSS-der} \frac{\mathrm{d}\mathrm{RSS}(\beta)}{\mathrm{d}\beta} = - 2 X^\mathrm{T} y + 2 X^\mathrm{T} X \beta\]

and setting this deriative to zero, we obtain:

\[\label{eq:OLS-qed} \begin{split} \frac{\mathrm{d}\mathrm{RSS}(\hat{\beta})}{\mathrm{d}\beta} &= 0 \\ 0 &= - 2 X^\mathrm{T} y + 2 X^\mathrm{T} X \hat{\beta} \\ X^\mathrm{T} X \hat{\beta} &= X^\mathrm{T} y \\ \hat{\beta} &= (X^\mathrm{T} X)^{-1} X^\mathrm{T} y \; . \end{split}\]

Since the quadratic form $y^\mathrm{T} y$ in \eqref{eq:RSS-dev} is positive, $\hat{\beta}$ minimizes $\mathrm{RSS}(\beta)$.

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Metadata: ID: P40 | shortcut: mlr-ols2 | author: JoramSoch | date: 2020-02-03, 18:43.