Computing ACFs of Causal AR(2) Processes Using Difference Equations#

Stage 3 · 🔗 Linear & ARMA Processes · Lesson 07 of 18 · intermediate

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Important

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The recursion#

For a causal AR(2), \(x_t = \phi_1 x_{t-1} + \phi_2 x_{t-2} + w_t\), the autocorrelations obey the same recursion as the process itself — a homogeneous linear difference equation:

\[\rho(h) = \phi_1\,\rho(h-1) + \phi_2\,\rho(h-2), \qquad h \ge 1.\]

With \(\rho(0) = 1\), the Yule–Walker start gives \(\rho(1) = \phi_1 / (1 - \phi_2)\), and every later lag follows by iterating.

Solving it#

Rather than iterate forever, solve the difference equation in closed form through its characteristic equation — equivalently, the roots \(z_1, z_2\) of the AR polynomial \(1 - \phi_1 z - \phi_2 z^2\). The ACF is then a combination of the terms \(z_i^{-h}\), whose magnitudes are controlled by how far the roots sit outside the unit circle (causality guarantees they do).

Two regimes#

The discriminant \(\phi_1^2 + 4\phi_2\) decides the shape. When it is positive, the roots are real and the ACF is a sum of two damped exponentials (decaying monotonically, with uniform or alternating sign). When it is negative, the roots are complex conjugates and the ACF is a damped sinusoid — a decaying oscillation with system frequency

\[f_0 = \frac{1}{2\pi}\cos^{-1}\!\left( \frac{\phi_1}{2\sqrt{-\phi_2}} \right).\]

What it tells you#

Either way the ACF tails off toward zero but never truly cuts off — the signature of an autoregressive process. (Its partner, the PACF, does cut off, after lag 2.) Reading whether the decay is exponential or oscillatory is a first clue to the underlying dynamics.

Hint