6.1 The model in matrix form

With \(k - 1\) regressors and an intercept, the model for observation \(i\) is \[ Y_i = \beta _1 + \beta _2X_{2i} + \dots + \beta _kX_{ki} + u_i, \qquad i = 1, \dots , n . \] Stacking the \(n\) observations, \[ Y = X\beta + U, \qquad Y = \begin {pmatrix} Y_1 \\ \vdots \\ Y_n \end {pmatrix}, \quad X = \begin {pmatrix} 1 & X_{21} & \cdots & X_{k1} \\ \vdots & \vdots & & \vdots \\ 1 & X_{2n} & \cdots & X_{kn} \end {pmatrix}, \quad \beta = \begin {pmatrix} \beta _1 \\ \vdots \\ \beta _k \end {pmatrix} . \] Each row of \(X\) is an observation and each column a variable; the first column is \(\iota \), the vector of ones. The assumptions of Definition 2.1.1 become:

(A1)

\(E(U \mid X) = 0\);

(A2–3)

\(\Var (U \mid X) = \sigma ^2I_n\);

(A4)

\(\rank X = k\): no column of \(X\) is an exact linear combination of the others;

(A5)

\(U \mid X \sim N(0, \sigma ^2I_n)\), when exact inference is needed.

(A4) generalises “the \(X_i\) are not all equal”. It fails, for example, if a regression includes income, consumption and saving when saving is defined as income minus consumption, or dummy variables for every category together with an intercept. Its failure is perfect multicollinearity, and least squares then has no unique solution.

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