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L9 ACF and PACF for Some Time Series Processes

For an AR(1) processes, as we did in L8__Autocorrelation and the Partial Autocorrelation Functions. We can infer,

\[ \gamma_{k} = E(Y_t Y_{t-k}) = E[(\phi_{1}Y_{t-1} + e_{t})Y_{t-k}] = \phi_{1}E(Y_{t-1}Y_{t-k}) = \phi_{1}\gamma_{k-1} \]

Doing it recursively we will end up with

\[ \gamma_{k} = \phi_{1}^k \gamma_{0} \]

ACF becomes

\[ \rho_{k} = \dfrac{\gamma_{k}}{\gamma_{0}} = \phi_{1}^k \]
  • For an \(MA(q)\) process
    • \(E(Y_{t}) = c = \mu\)
    • \(V(Y_{t}) = V(e_{t}) + \theta_{1}^{2}V(e_{t-1}) + \dots + \theta^{2}_{q}V(e_{t-q})\)
    • which is \(\sigma_{e}^{2} \sum_{j=0}^q \theta_{j}^{2}\) where \(\theta_{0}=1\)

Simulated Processes

  • For a \(AR(1)\) process with coefficient (\(\phi_{1}\)) = 0.6
    • Plot looks like a normal TS graph
    • ACF is downward sloping (since \(\phi_{1}^k\) keeps getting smaller with \(k\))
  • For an \(MA(1)\) process
    • \(\gamma_{1} = Cov(Y_{t},Y_{t-1}) = \theta_{1}\sigma_{e}^{2}\)
    • \(\gamma_{2}= Cov(Y_{t},Y_{t-2})\) and so, \(\gamma_{k}=0\) for all \(k \gt 1\)
\[ \rho_{k} = \dfrac{\gamma_{k}}{\gamma_{0}} = \begin{cases} \dfrac{\theta_{1}}{(1+\theta_{1}^{2})} & \text{if } k = 1 \\ 0 & \text{if } k \gt 1 \end{cases} \]
  • For an \(MA(2)\) process
    • \(\gamma_{1}=Cov(Y_{t},Y_{t-1}) = \theta_{1}(1+\theta_{2})\sigma_{e}^{2}\)
    • \(\gamma_{2} = \theta_{2}\sigma_{e}^{2}\)
    • for \(k\gt 2\), \(\gamma_{k}= 0\)

  • For an \(MA(q)\) process

    • for \(k \gt q\), \(\gamma_{k} = 0\)
    • Thus, the ACF function \(\gamma_{k}\) can help us identify the possible models to fit into a certain component.

  • For a random walk,

    • \(E(Y_{t}) = t\mu\)
    • \(V(Y_{t}) = \gamma_{0}= t\sigma_{e}^{2}\)
    • \(\gamma_{k}= (t-k)\sigma_{e}^{2}\)
    • \(\rho_{k} = \sqrt{ 1 - \dfrac{k}{t} }\)
    • In general a random walk is non-stationary, mean and variance depend on \(t\).