Index: The Book of Statistical Proofs ▷ Statistical Models ▷ Univariate normal data ▷ Multiple linear regression ▷ Ordinary least squares for two regressors

Theorem: Consider a linear regression model in which the design matrix has two columns:

\[\label{eq:mlr-tr} y = X\beta + \varepsilon \quad \text{where} \quad y \in \mathbb{R}^{n \times 1} \quad \text{and} \quad X = \left[ \begin{matrix} x_1 & x_2 \end{matrix} \right] \in \mathbb{R}^{n \times 2} \; .\]

Then,

1) the ordinary least squares estimates for $\beta_1$ and $\beta_2$ are given by

\[\label{eq:mlr-ols-tr} \hat{\beta}_1 = \frac{x_2^\mathrm{T} x_2 x_1^\mathrm{T} y - x_1^\mathrm{T} x_2 x_2^\mathrm{T} y}{x_1^\mathrm{T} x_1 x_2^\mathrm{T} x_2 - x_1^\mathrm{T} x_2 x_2^\mathrm{T} x_1} \quad \text{and} \quad \hat{\beta}_2 = \frac{x_1^\mathrm{T} x_1 x_2^\mathrm{T} y - x_2^\mathrm{T} x_1 x_1^\mathrm{T} y}{x_1^\mathrm{T} x_1 x_2^\mathrm{T} x_2 - x_1^\mathrm{T} x_2 x_2^\mathrm{T} x_1}\]

2) and, if the two regressors are orthogonal to each other, they simplify to

\[\label{eq:mlr-ols-tr-orth} \hat{\beta}_1 = \frac{x_1^\mathrm{T} y}{x_1^\mathrm{T} x_1} \quad \text{and} \quad \hat{\beta}_2 = \frac{x_2^\mathrm{T} y}{x_2^\mathrm{T} x_2}, \quad \text{if} \quad x_1 \perp x_2 \; .\]

Proof: The model in \eqref{eq:mlr-tr} is a special case of multiple linear regression and the ordinary least squares solution for multiple linear regression is:

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

1) Plugging $X = \left[ \begin{matrix} x_1 & x_2 \end{matrix} \right]$ into this equation, we obtain:

\[\label{eq:mlr-ols-tr-s1} \begin{split} \hat{\beta} &= \left( \left[ \begin{matrix} x_1^\mathrm{T} \\ x_2^\mathrm{T} \end{matrix} \right] \left[ \begin{matrix} x_1 & x_2 \end{matrix} \right] \right)^{-1} \left[ \begin{matrix} x_1^\mathrm{T} \\ x_2^\mathrm{T} \end{matrix} \right] y \\ & = \left( \begin{matrix} x_1^\mathrm{T} x_1 & x_1^\mathrm{T} x_2 \\ x_2^\mathrm{T} x_1 & x_2^\mathrm{T} x_2 \end{matrix} \right)^{-1} \left( \begin{matrix} x_1^\mathrm{T} y \\ x_2^\mathrm{T} y \end{matrix} \right) \; . \end{split}\]

Using the inverse of of a $2 \times 2$ matrix

\[\label{eq:inv-2x2} \left[ \begin{matrix} a & b \\ c & d \end{matrix} \right]^{-1} = \frac{1}{a d - b c} \left[ \begin{matrix} d & -b \\ -c & a \end{matrix} \right] \; ,\]

this can be further developped into

\[\label{eq:mlr-ols-tr-s2} \begin{split} \hat{\beta} &= \frac{1}{x_1^\mathrm{T} x_1 x_2^\mathrm{T} x_2 - x_1^\mathrm{T} x_2 x_2^\mathrm{T} x_1} \left( \begin{matrix} x_2^\mathrm{T} x_2 & -x_1^\mathrm{T} x_2 \\ -x_2^\mathrm{T} x_1 & x_1^\mathrm{T} x_1 \end{matrix} \right) \left( \begin{matrix} x_1^\mathrm{T} y \\ x_2^\mathrm{T} y \end{matrix} \right) \\ &= \frac{1}{x_1^\mathrm{T} x_1 x_2^\mathrm{T} x_2 - x_1^\mathrm{T} x_2 x_2^\mathrm{T} x_1} \left( \begin{matrix} x_2^\mathrm{T} x_2 x_1^\mathrm{T} y - x_1^\mathrm{T} x_2 x_2^\mathrm{T} y \\ x_1^\mathrm{T} x_1 x_2^\mathrm{T} y - x_2^\mathrm{T} x_1 x_1^\mathrm{T} y \end{matrix} \right) \end{split}\]

which can also be written as

\[\label{eq:mlr-ols-tr-qed} \begin{split} \hat{\beta}_1 &= \frac{x_2^\mathrm{T} x_2 x_1^\mathrm{T} y - x_1^\mathrm{T} x_2 x_2^\mathrm{T} y}{x_1^\mathrm{T} x_1 x_2^\mathrm{T} x_2 - x_1^\mathrm{T} x_2 x_2^\mathrm{T} x_1} \\ \hat{\beta}_2 &= \frac{x_1^\mathrm{T} x_1 x_2^\mathrm{T} y - x_2^\mathrm{T} x_1 x_1^\mathrm{T} y}{x_1^\mathrm{T} x_1 x_2^\mathrm{T} x_2 - x_1^\mathrm{T} x_2 x_2^\mathrm{T} x_1} \; . \end{split}\]

2) If two regressors are orthogonal to each other, this means that the inner product of the corresponding vectors is zero:

\[\label{eq:reg-orth} x_1 \perp x_2 \quad \Leftrightarrow \quad x_1^\mathrm{T} x_2 = x_2^\mathrm{T} x_1 = 0 \; .\]

Applying this to equation \eqref{eq:mlr-ols-tr-qed}, we obtain:

\[\label{eq:mlr-ols-tr-orth-qed} \begin{split} \hat{\beta}_1 &= \frac{x_2^\mathrm{T} x_2 x_1^\mathrm{T} y}{x_1^\mathrm{T} x_1 x_2^\mathrm{T} x_2} = \frac{x_1^\mathrm{T} y}{x_1^\mathrm{T} x_1} \\ \hat{\beta}_2 &= \frac{x_1^\mathrm{T} x_1 x_2^\mathrm{T} y}{x_1^\mathrm{T} x_1 x_2^\mathrm{T} x_2} = \frac{x_2^\mathrm{T} y}{x_2^\mathrm{T} x_2} \; . \end{split}\]
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Metadata: ID: P418 | shortcut: mlr-olstr | author: JoramSoch | date: 2023-10-06, 12:48.