7.3 Testing linear restrictions
A set of \(q\) linear restrictions is written \[ H_0\colon R\beta = r , \] where \(R\) is a known \(q \times k\) matrix of rank \(q\) and \(r\) a known \(q\)-vector. With \(k = 3\) and \(\beta = (\beta _1, \beta _2, \beta _3)'\):
| Hypothesis | \(R\) | \(r\) |
| \(\beta _2 = 0\) | \((0\ \ 1\ \ 0)\) | \(0\) |
| \(\beta _2 = 1\) (unit elasticity) | \((0\ \ 1\ \ 0)\) | \(1\) |
| \(\beta _2 + \beta _3 = 1\) (constant returns) | \((0\ \ 1\ \ 1)\) | \(1\) |
| \(\beta _2 = \beta _3\) | \((0\ \ 1\ \ {-1})\) | \(0\) |
| \(\beta _2 = \beta _3 = 0\) | \(\begin {pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \end {pmatrix}\) | \(\begin {pmatrix} 0 \\ 0 \end {pmatrix}\) |
Theorem 7.3.1. Under (A1)–(A5):
- (i)
- \(E(Rb - r) = R\beta - r\) and \(\Var (Rb - r) = \sigma ^2R(X'X)^{-1}R'\), so \(Rb - r \sim N\bigl (R\beta - r,\ \sigma ^2R(X'X)^{-1}R'\bigr )\);
- (ii)
- under \(H_0\colon R\beta = r\), \[ W = (Rb - r)'\bigl [\sigma ^2R(X'X)^{-1}R'\bigr ]^{-1}(Rb - r) \sim \chi ^2_q ; \]
- (iii)
- under \(H_0\), with \(\sigma ^2\) replaced by \(s^2\), \[ F = \frac {(Rb - r)'\bigl [R(X'X)^{-1}R'\bigr ]^{-1}(Rb - r)/q}{s^2} \sim F_{q,\ n-k} . \]
Proof. (i) \(Rb - r\) is a linear function of \(b\); apply Theorem 5.2.1(i) and Theorem 7.1.1(iv). \(R(X'X)^{-1}R'\) is pd because \((X'X)^{-1}\) is pd and \(R\) has full row rank \(q\).
(ii) Under \(H_0\), \(Rb - r \sim N\bigl (0, \sigma ^2R(X'X)^{-1}R'\bigr )\), and Theorem 5.4.2(ii) gives the \(\chi ^2_q\) distribution.
(iii) \(F = (W/q)\big /\bigl ((n - k)s^2/\sigma ^2\bigr /(n - k)\bigr )\). The numerator is a \(\chi ^2_q\) over \(q\). The denominator is a \(\chi ^2_{n-k}\) over \(n - k\) by Theorem 5.5.1(v). They are independent because \(W\) is a function of \(b\), and \(b\) is independent of \(s^2\) by Theorem 5.5.1(vi). □
\(W\) is not usable directly because \(\sigma ^2\) is unknown, but it matters in its own right. In large samples \(s^2 \xrightarrow {p} \sigma ^2\), so \(qF \xrightarrow {d} \chi ^2_q\) without normality, and the \(\chi ^2\) form is the basis of the Wald tests of the section on maximum likelihood. Reject \(H_0\) at level \(\alpha \) if \(F\) exceeds the upper \(\alpha \) point of \(F_{q, n-k}\).
Corollary 7.3.2. For a single restriction \(c'\beta = r\), with \(q = 1\), \(F\) is the square of \[ t = \frac {c'b - r}{s\sqrt {c'(X'X)^{-1}c}} \sim t_{n-k} . \] In particular the \(t\) ratio for \(\beta _j = 0\) is \(b_j/\operatorname {se}(b_j)\), and the test of \(\beta _2 + \beta _3 = 1\) uses \(\operatorname {se}(b_2 + b_3) = \sqrt {\widehat \Var (b_2) + \widehat \Var (b_3) + 2\widehat \Cov (b_2, b_3)}\).
Proof. With \(R = c'\), \(R(X'X)^{-1}R' = c'(X'X)^{-1}c\) is a scalar, and \[ F = \frac {(c'b - r)^2}{s^2c'(X'X)^{-1}c} = t^2 . \] That \(t \sim t_{n-k}\) follows as in Theorem 2.4.1: a \(N(0, 1)\) over the square root of an independent \(\chi ^2_{n-k}/(n - k)\). □
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