L8 Autocorrelation and the Partial Autocorrelation Functions

Different notations

\[ \gamma_{k}(t) = Cov(Y_{t},Y_{t-k}) = E(Y_{t}Y_{t+k}) - \mu_{t}\mu_{t+k} \]

gives the covariance between value at \(t\) and \(t+k\).

Similarly,

\[ \rho_{k} = Corr(Y_{t},Y_{t+k}) \]

\(\rho_{k} = \rho_{-k}\)
\(\lvert \rho_{k} \rvert \leq 1\)
Non-uniqueness of ACF: A stationary normal process is completely determined by its mean, variance and ACF.
- It is possible to find several non-normal processes with the same ACF.

\[ \rho_{k} = \begin{cases} 1 & \text{if } k=0 \\ 0 & \text{otherwise} \end{cases} \]

Sample version of the correlation function

Sample ACF at lag \(1\) is given by

\[ r_{1} = \dfrac{\sum_{t=1}^{n-1}(y_{t}-\bar{y})(y_{t+1}-\bar{y})}{\sum_{t=1}^n(y_{t}-\bar{y})^{2}} \]

Sample ACF at lag \(k\) is given by

\[ r_{1} = \dfrac{\sum_{t=1}^{n-k}(y_{t}-\bar{y})(y_{t+k}-\bar{y})}{\sum_{t=1}^n(y_{t}-\bar{y})^{2}} \]

Graph of \(r_{k}\) against \(k\) is called as a Correlogram.

PACF of any order \(k\), \(\alpha_{k}\) is the partial correlation coefficient between \(Y_{t}, Y_{t-k}\) conditional on the intermediate values.

\[ \alpha_{k} = Corr(Y_{t},kY_{t-k}\vert Y_{t-1},Y_{t-2},\dots,Y_{t-k+1}) \]

For an \(AR(p)\) process, we assume \(E(Y_{t})= E(Y_{t-k})\) for all \(k\). Applying expectation to the \(AR(p)\) form, we will get

\[ \mu = \dfrac{c}{1-\phi_{1}-\phi_{2}\dots-\phi_{p}} \]

Thus, given \(Y_{t-1}\), \(Y_{t}\) becomes independent of past lags.

This is called the Markovian property, when the mean of the time variable is dependent on only on one of its past lags.

The process is variance stationary.

\[ \gamma_{0} = V(Y_{t}) = V(Y_{t-1}) = \dots = \sigma_{y}^{2} \]

And,

\[ \sigma_{y}^{2} = E[\phi_{1}Y_{t-1} + e_{t}]^2 = \phi_{1}^{2}E(Y_{t-1}^{2}) + E(e_{t}^{2}) + 2\phi_{1}E(Y_{t-1}e_{t}) \]

\[ = \phi_{1}^{2}\sigma_{y}^{2} + \sigma_{e}^{2} \]

So,

\[ \sigma_{y}^{2} = \dfrac{\sigma_{e}^{2}}{(1-\phi_{1}^{2})} \]