The AR(2) is stationary iff \(|\lambda_1| < 1\) and \(|\lambda_2| < 1\), where \(\lambda_1, \lambda_2\) are roots of \(p(\lambda) = \lambda^2 - \alpha_1\lambda - \alpha_2\). Note that \(p(\lambda)\) is an upward-opening parabola in \(\lambda\), and we can write: \[
p(\lambda) = (\lambda - \lambda_1)(\lambda - \lambda_2)
\]
The condition \(|\lambda_1| < 1\) and \(|\lambda_2| < 1\) is equivalent to requiring \(p(\lambda) > 0\) at \(\lambda = 1\) and \(\lambda = -1\) (since the parabola is positive outside both roots when both roots are inside \((-1,1)\)), plus the product of the roots satisfying \(|\lambda_1 \lambda_2| < 1\).
Evaluating at \(\lambda = 1\): \[
p(1) = 1 - \alpha_1 - \alpha_2 > 0 \iff \alpha_1 + \alpha_2 < 1 \tag{14.35}
\]
Evaluating at \(\lambda = -1\): \[
p(-1) = 1 + \alpha_1 - \alpha_2 > 0 \iff \alpha_2 - \alpha_1 < 1 \tag{14.36}
\]
From the properties of quadratic equations, \(\lambda_1 \lambda_2 = -\alpha_2\). Requiring \(|\lambda_1\lambda_2| < 1\): \[
|\!-\alpha_2| < 1 \iff |\alpha_2| < 1 \iff -1 < \alpha_2 < 1
\]
The upper bound \(\alpha_2 < 1\) is already implied by (14.35) (since \(\alpha_1 + \alpha_2 < 1\) and \(\alpha_1 \geq 0\) in many cases, but more generally \(\alpha_2 < 1 - \alpha_1\), and the parabola argument ensures this is not binding independently). The binding constraint is therefore: \[
\alpha_2 > -1 \tag{14.37}
\]
The three conditions \(p(1) > 0\), \(p(-1) > 0\), and \(\lambda_1\lambda_2 > -1\) are necessary and sufficient for both roots to lie inside the unit circle, giving (14.35)–(14.37).
