10.1: Ordinary Least Squares
- Page ID
- 57750
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)First, let us review ordinary least squares (OLS). When formulating OLS estimation of the classical linear model (CLM), we made the assumption that the residuals are independent and identically distributed Normal with constant zero expected value and variance.
In symbols, this is written as either
\begin{equation}
\varepsilon_i \stackrel{\text{iid}}{\sim} N\left(0;\, \sigma^2\right)
\end{equation}
or as
\begin{equation}
\mathbf{E} \sim N_n\left(\mathbf{0};\, \sigma^2\mathbf{I}\right)
\end{equation}
The two statements are different ways of saying the exact same thing.
Note that the covariance matrix of \(\mathbf{E}\) is \(\sigma^2\mathbf{I}\):
\begin{equation}
V[\mathbf{E}]\ =\ \sigma^2\ \left[
\begin{matrix}
1 & 0 & 0 & \cdots & 0 \\
0 & 1 & 0 & \cdots & 0 \\
0 & 0 & 1 & & 0 \\
\vdots & \vdots & & \ddots & \vdots \\
0 & 0 & 0 & \cdots & 1 \\
\end{matrix}
\right]\ =\ \left[
\begin{matrix}
\sigma^2 & 0 & 0 & \cdots & 0 \\
0 & \sigma^2 & 0 & \cdots & 0 \\
0 & 0 & \sigma^2 & & 0 \\
\vdots & \vdots & & \ddots & \vdots \\
0 & 0 & 0 & \cdots & \sigma^2 \\
\end{matrix}
\right]
\end{equation}
The values along the diagonal represent the variances of each residual in the population. That they are the same value, \(\sigma^2\), indicates that the variance of the residuals is constant.
The values off the diagonal represent the covariance between the residuals. For instance, the value at position 1,2 is the covariance between \(\varepsilon_1\) and \(\varepsilon_2\), which we symbolized as \(\sigma_{1,2}\) in The Appendix of Statistics. Since that value is \(0\), we are specifying that the two are linearly uncorrelated (a.k.a. independent). Thus, the covariance matrix above specifies that the variances of the residuals are constant and that the residuals are independent of each other. If this requirement is met, then we should use ordinary least squares regression. However, not always is this requirement met.