Extended Autocorrelation Function 📂Statistical Analysis

Extended Autocorrelation Function

Buildup

PACF helps in determining the order of $AR(p)$ , while ACF is helpful in setting the order for $MA(q)$ . However, when applying to $ARMA(p,q)$ model, due to the invertibility of ARMA models, $AR(p)$ may appear as $MA(\infty)$ , and $MA(q)$ may appear as $AR(\infty)$ . Therefore, various methods have been devised to circumvent these issues and find the ARMA model.

Definition

The Extended Autocorrelation Function is one such method, defined as EACF for lags $k$ and $j$ , described as follows: $W_{t,k,j} := Y_{t} - \tilde{\phi}_{1} Y_{t-1} - \cdots - \tilde{\phi}_{k} Y_{t-k}$

Explanation

Understanding EACF just from its definition is impossible, so let’s comprehend it through equations. Given lags $k$ and $j$ , an ARMA model $ARMA(k,j)$ can be expressed as follows: $Y_{t} = \sum_{i = 1}^{k} \phi_{i} Y_{t-i} + e_{t} - \sum_{i = 1}^{j} \theta_{i} e_{t-i}$ Here, performing a multiple regression analysis with $Y_{t}$ as $Y_{t-1} , \cdots , Y_{t-k}$ yields the regression coefficients $\tilde{\phi}_{1} , \cdots , \tilde{\phi}_{k}$ , which are estimates of $\phi_{1} , \cdots , \phi_{k}$ . And the residuals would disastrously appear as the following: $Y_{t} - \sum_{i = 1}^{k} \tilde{\phi}_{i} Y_{t-i} = e_{t} - \sum_{i = 1}^{j} \theta_{i} e_{t-i}$ However, on closer inspection, by setting the left side as $W_{t,k,j}$ , we can achieve the $MA(j)$ model. $W_{t,k,j} = e_{t} - \sum_{i = 1}^{j} \theta_{i} e_{t-i}$ Then, $W_{t,k,j}$ follows a normal distribution $\displaystyle N \left( 0 , {{1} \over {n - k - j}}\right)$ , just as ACF did, and this is used for hypothesis testing.

To simplify, this approach essentially disentangles the complex ARMA model $ARMA(k,j)$ by specifying lag $k$ to eliminate $AR(k)$ , and then by specifying lag $j$ to solely consider $MA(j)$ for targeted analysis. The reason it is aptly named Extended Autocorrelation Function is because it ultimately requires the use of ACF to discern $MA(j)$ .

Exercise

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Since the lags are listed on two axes, unlike ACF or PACF, it’s impossible to draw a correlogram ¹ and instead, a table that merely indicates whether the null hypothesis is rejected or not with O and X is used. However, this table is not based on actual data but is a theoretical schematic that should emerge from analyzing an ideally working $ARMA(1,1)$ model. It’s rarely this clean in reality.

The method to read the table is as follows:

Step 1. Find a vertex where a line continues horizontally to the right from the upper left.
Step 2. From the vertex, find the line that drops diagonally right-downward forming an acute angle with a horizontal line.
Step 3. If both lines are identified, analyze with the corresponding $ARMA(p,q)$ at the vertex.

Following this method, in the given illustration, $ARMA(1,1)$ at the vertex O* becomes a candidate model. As one might guess after studying time series to some extent and reading the explanation thoroughly, the actual analysis often involves a lot of subjective and vague parts. The only way to get accustomed to this is by actively performing many analyses.