Proof. \(b - \beta = \sum w_tu_t\) with \(w_t = x_t/\sum x_j^2\), so \[ \Var (b) = \sum _t\sum _{t'}w_tw_{t'}\Cov (u_t, u_{t'}) = \frac {\sigma _u^2}{\bigl (\sum x_j^2\bigr )^2}\sum _t\sum _{t'}x_tx_{t'}\rho ^{|t - t'|} . \] The terms with \(t = t'\) give \(\sum x_t^2\); those with \(|t - t'| = s \geq 1\) occur twice and give \(2\rho ^s\sum _{t=1}^{n-s}x_tx_{t+s}\). Dividing out one factor \(\sum x_t^2\) gives the first formula. With \(\sum _t x_tx_{t+s}/\sum x_t^2 \approx \lambda ^s\) and \(n\) large, \[ 1 + 2\sum _{s\geq 1}(\rho \lambda )^s = 1 + \frac {2\rho \lambda }{1 - \rho \lambda } = \frac {1 + \rho \lambda }{1 - \rho \lambda } . \] For \(s^2\), Theorem 5.2.1(iii) gives \(E(e'e) = \tr \bigl (M\sigma _u^2V\bigr )\), where \(V = (1 - \rho ^2)\Omega \) is the correlation matrix of \(U\). Since \(\tr (M V) = \tr V - \tr \bigl ((X'X)^{-1}X'VX\bigr ) = n - (1 + \rho \lambda )/(1 - \rho \lambda )\) approximately, \(E(s^2) \approx \sigma _u^2\bigl [n - (1 + \rho \lambda )/(1 - \rho \lambda )\bigr ]/(n - 1)\), which tends to \(\sigma _u^2\). □

Proof. Multiply \(Y_t - \beta Y_{t-1} = u_t\) by \((1 - \rho L)\), where \(L\) is the lag operator: \((Y_t - \beta Y_{t-1}) - \rho (Y_{t-1} - \beta Y_{t-2}) = \varepsilon _t\), that is \[ Y_t = \phi _1Y_{t-1} + \phi _2Y_{t-2} + \varepsilon _t, \qquad \phi _1 = \beta + \rho , \quad \phi _2 = -\beta \rho . \] Its roots \(\beta \) and \(\rho \) lie inside the unit circle, so \(Y_t\) is a stationary AR(2) process. Multiplying by \(Y_{t-1}\) and taking expectations, with \(\varepsilon _t\) uncorrelated with the past, gives the Yule–Walker equation \(\gamma _1 = \phi _1\gamma _0 + \phi _2\gamma _1\), so the first autocorrelation is \(\rho _1 = \phi _1/(1 - \phi _2) = (\beta + \rho )/(1 + \beta \rho )\). By the law of large numbers for stationary processes, \(\plim b = \plim \bigl (\frac 1n\sum Y_tY_{t-1}\bigr )/\plim \bigl (\frac 1n\sum Y_{t-1}^2\bigr ) = \gamma _1/\gamma _0 = \rho _1\), which is (i).

(ii) Write \(p = \rho _1\) for the probability limit of \(b\), so that the residuals behave like \(e_t = Y_t - pY_{t-1}\). Using \(\gamma _1 = p\gamma _0\), \[ \Var (e_t) = \gamma _0(1 + p^2) - 2p\gamma _1 = \gamma _0(1 - p^2), \qquad \Cov (e_t, e_{t-1}) = \gamma _1(1 + p^2) - p(\gamma _0 + \gamma _2) = \gamma _0\,p\,(p^2 - \rho _2), \] where \(\rho _2 = \phi _1\rho _1 + \phi _2\) is the second Yule–Walker equation. Hence \(\plim \hat r = p(p^2 - \rho _2)/(1 - p^2)\), and substituting for \(p\) and \(\rho _2\) and simplifying gives the stated expression. Finally, for \(\rho \neq 0\) the inequality \(|\plim \hat r| < |\rho |\) is equivalent to \(\beta ^2(\beta + \rho )^2 < (1 + \beta \rho )^2\), and \[ (1 + \beta \rho )^2 - \beta ^2(\beta + \rho )^2 = (1 - \beta ^2)(1 + \beta ^2 + 2\beta \rho ) > 0 , \] since \(1 - \beta ^2 > 0\) and \(1 + \beta ^2 + 2\beta \rho \geq 1 + \beta ^2 - 2|\beta | = (1 - |\beta |)^2 > 0\). □