1. ARMA models with a mean. The following are candidate ways to write an ARMA(p,q) model with mean \(\mu\). \[\begin{array}{crcl} [A1] & \hspace{15mm} Y_n &=& \mu + \phi_1 Y_{n-1}+\phi_2Y_{n-2}+\dots +\phi_pY_{n-p} \\ & && + \epsilon_n +\psi_1 \epsilon_{n-1} +\dots+\psi_q\epsilon_{n-q} \\ [A2] & Y_n &=& \mu + \phi_1 (Y_{n-1}-\mu)+\phi_2(Y_{n-2}-\mu)+\dots +\phi_p(Y_{n-p}-\mu) \\ & && + \epsilon_n +\psi_1 \epsilon_{n-1} +\dots+\psi_q\epsilon_{n-q} \\ [A3] & \phi(B) Y_n &=& \psi(B)\epsilon_n \\ &\phi(x)&=&1-\phi_1x - \phi_2 x^2 - \dots - \phi_p x^p \\ &\psi(x)&=&1+\psi_1x + \psi_2 x^2 + \dots + \psi_q x^q. \end{array}\] Are both [A1] and [A2] reasonable alternative models? Is there a reason to prefer one to the other? Which one matches [A3]?

  2. Two equivalent MA models. Consider the following moving average models, defined in terms of a Gaussian white noise process \(\epsilon_n \sim \mathrm{iid} \, N(0,\sigma^2)\). \[\begin{array}{crcll} [B1] & \hspace{10mm} Y_n &=& \epsilon_n + 2\epsilon_{n-1}, \hspace{5mm} & \sigma^2=1. \\ [B2] & Y_n &=& \epsilon_n + 0.5\epsilon_{n-1}, & \sigma^2=4. \end{array}\] Why can data never tell us which of [B1] and [B2] is preferable for a particular situation? Is there a practical reason to prefer [B1] or [B2]?

  3. Over-damped AR(2) models. In Chapter 4, we saw that AR(2) models can describe quasi-periodic oscillatory behavior, with an autocorrelation function (ACF) that is an exponentially decaying sinusoid. You could experiment with ARMAacf() to find parameter values \(\phi_1\) and \(\phi_2\) for a causal AR(2) model that does not have a sinusoidal oscillation in its ACF. Instead, the ACF should decay to zero similar to an AR(1) model. Can you say anything about the class of all AR(2) models that do not have any oscillatory behavior? In other words, can you find a mathematical expression for the set of values of \(\phi_1\) and \(\phi_2\)? One way to address this in the context of Chapter 4 is to work out when a deterministic skeleton decays with no oscillations. In physics and engineering, this is called an over-damped system.