Moving average processes

Given a zero mean white noise $e(t) \sim WN(0,\lambda^2)$ we define the moving average process of order $MA(n)$ as: $$y(t)=c_0e(t)+c_1e(t-1)+...+c_ne(t-n)$$

Which is a linear combination of the past values of $e(t)$ from $t$ to $t-n$ with real parameters $c_0$,$c_1$,...,$c_n$.

MA(n) weak characterization and stationarity

Given a moving average process $y(t) \sim MA(n)$ then we can easily demostrate that the mean function is always null:

$$ m_y(t)=\mathbb{E}[y(t)]=\mathbb{E}[c_0e(t)+c_1e(t-1)+...+c_ne(t-n)]=\newline =c_0\mathbb{E}[e(t)]+c_1\mathbb{E}[e(t-1)]+...+c_n\mathbb{E}[e(t-n)]=0 \text{ because } e(t) \sim WN(0,\lambda^2) $$

And the covariance function is (proof in notes 02 23 2022, step by step computation remembering that the white noise at two different time instant is completely uncorrelated): $$ \gamma_y(\tau)= \begin{cases} (c_0^2+c_1^2+...+c_n^2)\lambda^2 &\text{if } \tau=0 \newline (c_0c_1+c_1c_2+...+c_{n-1}c_n)\lambda^2 &\text{if } \tau=\pm1 \newline \text{...} \newline c_0c_n\lambda^2 &\text{if } \tau=\pm n \newline 0 &\text{if } |\tau|>n \end{cases} $$ Which is time invariant so $y(t)$ is stationary.

MA(inf) process

The class of models $MA(\infty)$ is a huge class which covers almost all SSP.

Considering a zero mean white noise $e(t) \sim WN(0,\lambda^2)$ the process $y(t) \sim MA(\infty)$ is: $$y(t)=c_0e(t)+c_1e(t-1)+...+c_ie(t-i)+...=\sum_{i=0}^{+\infty}c_ie(t-i)$$ Which is an infinite sequence of random variables and the convergence to a proper random variance is not always guaranteed, only if: $$\sum_{i=0}^{+\infty}c_i^2 < +\infty$$

A moving average process of order $n$ is equal to a $MA(\infty)$ process with $c_i=0$ for $i>n$.

MA(inf) weak characterization and stationarity

The mean function of a well defined $MA(\infty)$ process is: $$m_y(t)=\mathbb{E}[y(t)]=\mathbb{E}[\sum_{i=0}^{+\infty}c_ie(t-i)]=0$$

While the covariance function: $$ \gamma(t,\tau)=\mathbb{E}[(y(t)-m_y(t))(y(t-\tau)-m_y(t-\tau))]=\newline =\mathbb{E}[y(t)-y(t-\tau)]=\mathbb{E}[\sum_{i=0}^{+\infty}c_ie(t-i)\sum_{j=0}^{+\infty}c_je(t-\tau-j)]=\newline [\text{Using linearity (well defined proces)}]\newline =\sum_{i,j=0}^{+\infty}c_ic_j\mathbb{E}[e(t-i)e(t-\tau-j)] $$ Since $\mathbb{E}[e(t-i)e(t-\tau-j)]=\gamma_e(t-i-(t-\tau-j))=\gamma_e(\tau+j-i)$ wich is always null except $\gamma_e(\tau+j-i)=\lambda^2$ for $i=j+\tau$. So: $$\gamma(t,\tau)=\lambda^2\sum_{j=0}^{+\infty}c_jc_{j+\tau}$$

Which is time invariant so the process is stationary.

Note that for $\tau=0$ we obtain the variance $\gamma_y(0)=\lambda^2\sum_{j=0}^{+\infty}c_j^2$ and the conditions of convergence makes the process well defined because if $\sum_{i=0}^{+\infty}c_i^2 < +\infty$ then $\gamma_{y}(0) < +\infty$ and since the process is stationary then the covariance is bounded by the variance, this implies that $\gamma_y(\tau)$ is well defined for every $\tau$.

Questions from past exams

Give the definition of an MA(inf) process e discuss the conditions under which it would be stationary and well defined. Prove the formula for the computation of the covariance function of an MA(inf) process. Explain why these processes are interesting to study AR and ARMA.

See above.

An $MA(\infty)$ process are interesting to study AR and ARMA processes because the steady state solution of the recursive euqation of those processes is an $MA(\infty)$ process with coefficients that are functions of the parameters of the recursive equation.

This implies also that the steady state solution is well defined when the condition under which a general MA(∞) process is well defined.