Proof. (i) Expanding the square, \[ \sum _{t=2}^n(e_t - e_{t-1})^2 = \sum _{t=2}^ne_t^2 + \sum _{t=1}^{n-1}e_t^2 - 2\sum _{t=2}^ne_te_{t-1} = 2\sum _{t=1}^ne_t^2 - e_1^2 - e_n^2 - 2\sum _{t=2}^ne_te_{t-1} ; \] divide by \(\sum e_t^2\). The bounds follow from \(0 \leq (e_t - e_{t-1})^2 \leq 2e_t^2 + 2e_{t-1}^2\), summed over \(t\), since each \(e_t^2\) appears at most twice.

(ii) \(M\) is symmetric idempotent of rank \(n - k\), so \(M = CC'\) with \(C'C = I_{n-k}\). Then \(e = MU = C\xi \), where \(\xi = C'U \sim N(0, \sigma ^2I_{n-k})\), and \(e'e = \xi '\xi \), \(e'Ae = \xi 'C'AC\xi \). Diagonalise \(C'AC = GNG'\) with \(G\) orthogonal and \(N = \operatorname {diag}(\nu _i)\), and put \(\eta = G'\xi /\sigma \sim N(0, I_{n-k})\). Then \(d = \eta 'N\eta /\eta '\eta \). The matrix \(C\), and with it the \(\nu _i\), is determined by \(X\).

(iii) Write \(d = \eta 'N\eta /\eta '\eta \). For a standard normal vector, the direction \(\eta /\|\eta \|\) is independent of the length \(\|\eta \|\), by the spherical symmetry of its density, and \(d\) depends only on the direction. Hence \(d\) and \(\eta '\eta \) are independent, and \[ \tr N = E(\eta 'N\eta ) = E(d\cdot \eta '\eta ) = E(d)\,E(\eta '\eta ) = (n - k)E(d) . \] Finally \(\tr N = \tr (C'AC) = \tr (ACC') = \tr (AM) = \tr (MA)\). □

Proof. Conditionally on the initial observations, the log-likelihood is \(\ell = \text {const} - \frac n2\log \sigma ^2 - \frac {1}{2\sigma ^2}\sum _t\varepsilon _t^2\), where \[ \varepsilon _t(\beta , \rho ) = (Y_t - x_t'\beta ) - \sum _{j=1}^p\rho _j(Y_{t-j} - x_{t-j}'\beta ) . \] Let \(\theta = (\beta ', \rho _1, \dots , \rho _p)'\) and \(z_t = -\partial \varepsilon _t/\partial \theta \). At \(\rho = 0\), \(\partial \varepsilon _t/\partial \beta = -x_t\) and \(\partial \varepsilon _t/\partial \rho _j = -u_{t-j}\), so \(z_t = (x_t', u_{t-1}, \dots , u_{t-p})'\). The score is \(\partial \ell /\partial \theta = \sigma ^{-2}\sum _t\varepsilon _tz_t\), and the information matrix is asymptotically equivalent to \(\sigma ^{-2}\sum _tz_tz_t'\): the other term of the second derivative, \(\sigma ^{-2}\sum \varepsilon _t\,\partial ^2\varepsilon _t/\partial \theta \partial \theta '\), has expectation zero, because \(\varepsilon _t\) is independent of the past values that make up the second derivatives. The restricted estimates under \(H_0\) are least squares: \(\varepsilon _t = e_t\), \(\tilde \sigma ^2 = e'e/n\), and \(z_t\) becomes the row \((x_t', e_{t-1}, \dots , e_{t-p})\) of a matrix \(Z\). Hence \[ \text {LM} = s'I^{-1}s = \frac {e'Z}{\tilde \sigma ^2}\Bigl (\frac {Z'Z}{\tilde \sigma ^2}\Bigr )^{-1}\frac {Z'e}{\tilde \sigma ^2} = n\,\frac {e'Z(Z'Z)^{-1}Z'e}{e'e} , \] which is \(n\) times the \(R^2\) of the regression of \(e\) on \(Z\). (\(R^2\) is uncentred here, but \(e\) has mean zero when \(x_t\) contains an intercept, so it is the usual \(R^2\).) The \(\chi ^2_p\) limit is the general property of LM statistics stated in Section 9.3. □

Sketch of proof. For each \(k\), the products \(w_{kt} = u_tu_{t-k}\) have mean zero and variance \(\sigma ^4\), and \(E(w_{kt} \mid \text {past}) = u_{t-k}E(u_t) = 0\). The same calculation shows \(E(w_{kt}w_{js}) = 0\) unless \(k = j\) and \(t = s\), so the vectors \((w_{1t}, \dots , w_{mt})\) are uncorrelated with covariance \(\sigma ^4I_m\). The central limit theorem for such martingale differences gives \(n^{-1/2}\sum _t(w_{1t}, \dots , w_{mt}) \xrightarrow {d} N(0, \sigma ^4I_m)\), and dividing by \(\frac 1n\sum u_t^2 \xrightarrow {p} \sigma ^2\) gives the result by Slutsky’s theorem. \(Q\) is then a sum of \(m\) asymptotically independent squared standard normals. □