Index: The Book of Statistical Proofs ▷ Statistical Models ▷ Univariate normal data ▷ Multiple linear regression ▷ Total, explained and residual sum of squares

Theorem: Assume a linear regression model with independent observations

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

and let $X$ contain a constant regressor $1_n$ modelling the intercept term. Then, it holds that

\[\label{eq:mlr-pss} \mathrm{TSS} = \mathrm{ESS} + \mathrm{RSS}\]

where $\mathrm{TSS}$ is the total sum of squares, $\mathrm{ESS}$ is the explained sum of squares and $\mathrm{RSS}$ is the residual sum of squares.

Proof: The total sum of squares is given by

\[\label{eq:TSS} \mathrm{TSS} = \sum_{i=1}^{n} (y_i - \bar{y})^2\]

where $\bar{y}$ is the mean across all $y_i$. The $\mathrm{TSS}$ can be rewritten as

\[\label{eq:TSS-s1} \begin{split} \mathrm{TSS} &= \sum_{i=1}^{n} (y_i - \bar{y} + \hat{y}_i - \hat{y}_i)^2 \\ &= \sum_{i=1}^{n} \left( (\hat{y}_i - \bar{y}) + (y_i - \hat{y}_i) \right)^2 \\ &= \sum_{i=1}^{n} \left( (\hat{y}_i - \bar{y}) + \hat{\varepsilon}_i \right)^2 \\ &= \sum_{i=1}^{n} \left( (\hat{y}_i - \bar{y})^2 + 2 \, \hat{\varepsilon}_i (\hat{y}_i - \bar{y}) + \hat{\varepsilon}_i^2 \right) \\ &= \sum_{i=1}^{n} (\hat{y}_i - \bar{y})^2 + \sum_{i=1}^{n} \hat{\varepsilon}_i^2 + 2 \sum_{i=1}^{n} \hat{\varepsilon}_i (\hat{y}_i - \bar{y}) \\ &= \sum_{i=1}^{n} (\hat{y}_i - \bar{y})^2 + \sum_{i=1}^{n} \hat{\varepsilon}_i^2 + 2 \sum_{i=1}^{n} \hat{\varepsilon}_i (x_i \hat{\beta} - \bar{y}) \\ &= \sum_{i=1}^{n} (\hat{y}_i - \bar{y})^2 + \sum_{i=1}^{n} \hat{\varepsilon}_i^2 + 2 \sum_{i=1}^{n} \hat{\varepsilon}_i \left( \sum_{j=1}^{p} x_{ij} \hat{\beta}_j \right) - 2 \sum_{i=1}^{n} \hat{\varepsilon}_i \, \bar{y} \\ &= \sum_{i=1}^{n} (\hat{y}_i - \bar{y})^2 + \sum_{i=1}^{n} \hat{\varepsilon}_i^2 + 2 \sum_{j=1}^{p} \hat{\beta}_j \sum_{i=1}^{n} \hat{\varepsilon}_i x_{ij} - 2 \bar{y} \sum_{i=1}^{n} \hat{\varepsilon}_i \\ \end{split}\]

The fact that the design matrix includes a constant regressor ensures that

\[\label{eq:e-est-sum} \sum_{i=1}^{n} \hat{\varepsilon}_i = \hat{\varepsilon}^\mathrm{T} 1_n = 0\]

and because the residuals are orthogonal to the design matrix, we have

\[\label{eq:X-e-orth} \sum_{i=1}^{n} \hat{\varepsilon}_i x_{ij} = \hat{\varepsilon}^\mathrm{T} x_j = 0 \; .\]

Applying \eqref{eq:e-est-sum} and \eqref{eq:X-e-orth} to \eqref{eq:TSS-s1}, this becomes

\[\label{eq:TSS-s2} \mathrm{TSS} = \sum_{i=1}^{n} (\hat{y}_i - \bar{y})^2 + \sum_{i=1}^{n} \hat{\varepsilon}_i^2\]

and, with the definitions of explained and residual sum of squares, it is

\[\label{eq:TSS-s3} \mathrm{TSS} = \mathrm{ESS} + \mathrm{RSS} \; .\]
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Metadata: ID: P76 | shortcut: mlr-pss | author: JoramSoch | date: 2020-03-09, 22:18.