Interpreting the AR(2) Stochastic Cycle Length Formula

Summary

The document asks how to interpret a formula for the average length of a stochastic cycle in an autoregressive model of order two. It cites the condition under which the AR(2) characteristic roots form a complex conjugate pair and asks whether the coefficients in the cycle-length expression are the same as coefficients estimated by least squares. It also asks how to find corresponding quantities for higher-order models.

The material presents the question but contains no answer, derivation, or empirical evidence. It therefore identifies a connection between AR(2) coefficients, complex roots, and cyclical behavior without resolving how the formula’s notation maps to an estimated model or how cycle length should be generalized beyond order two. Readers would need additional theory or a source response to establish the interpretation and estimation procedure.

Key ideas

• The document links the AR(2) characteristic roots to the existence of oscillatory behavior.
• It presents a formula for average stochastic cycle length in terms of AR(2) parameters.
• It asks whether those parameters correspond directly to least-squares coefficient estimates.
• It raises the question of extending the cycle-length calculation to autoregressive models of higher order.
• No answer or evidence resolving these questions is included.

Tags

# Defining the Average Length of Business Cycle using AR(p) model


# Defining the Average Length of Business Cycle using AR(p) model












I'm currently reading through Analysis of Financial Time Series by Ruey Tsay. The AR model is introduced in chapter 2 and its properties in 2.4.1. The difference equations are explained and then its stated (for an AR(2)) that if $\phi^2_1 - 4\phi_2 <0$, a complex conjugate pair exists. The average length of the stochastic cycle is defined as:

$k = \frac{2\pi}{cos^{-1}[\phi_1/2\sqrt{-\phi_2}]}$

Are the $\phi_1$ and $\phi_2$ in the average length equation the same as the coefficients found for AR(2) model using least squares? If not, how can we find $\phi_1$ and $\phi_2$ for any p?

It seems im overlooking something obvious. Thanks in advance.

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