Linear Time Series

	Attachment	Size
	TUTORIAL-_CORRELOGRAM.xls

Many time series data sets exhibit time interdependency among their values. This is important to detect and eventually factor in to improve the forecast quality of the model.

Autocorrelation Function (ACFⁱ)

The Autocorrelation Function (ACF) computes the correlation factor between a time series and a lagged (back shifted) copy of it.

$\rho_{k}=\frac{\sum_{t=k}^T{(r_{t}-\bar r)(r_{t-k}-\bar r)}}{\sum_{t=k}^T(r_{t}-\bar r)^2}$

Please note that as the lag order increases, the size of the data set used decreases in the same order. This becomes an issue for smaller sized datasets.

Similar to the sample mean, the computed ACF value is an estimate. To represent our uncertainty, we compute upper and lower limits to reflect where values are considered significant rather than noise.

Using the same example as before, we compute the ACF up to lag 28.

ACF Correlogram

TIP: Use the "ACF" function to compute autocorrelation. Use "ACFCI" to compute the upper and lower confidence limits.

In the statistical test section, we saw that daily log returns do exhibit a strong serial correlation. The correlogram above shows a serial correlation for the 1st lag, and weaker ones for the 6th, 11th and 18th lags.

Partial-Autocorrelation Function (PACFⁱ)

The Partial Autocorrelation Function (PACF) gives us a different view of serial correlation by constructing a series of Autoregressive (AR) models. The PACF represents the regression coefficient of the last lagged series of a model.

Interpretation:

Underlying True Model: Autoregressive (AR)
- ACF Correlogram - Autocorrelation is significant and declines in value
- PACF Correlogram - Partial autocorrelation will be significant for a few lags, then drops to zero
Underlying True Model: Moving Average (MA)
- ACF Correlogram - Autocorrelation is significant for a few lags
- PACF Correlogram - Partial autocorrelation is significant, but declining, throughout the entire spectrum

Let's examine the MSFT daily log returns case.

PACF Correlogram

The PACF and ACF Correlograms look similar, so we suspect that an ARMAⁱ type of process is driving the conditional mean.

Using the NumXL Toolbar

Using the NumXL Correlogram toolbar, you can generate the ACF/PACF values and their plots in a few steps.

Using the NumXL toolbar (or menu in Excel 97-2003), select Correlogram.
The Correlogram dialog box pops up. Fill in the location of your data, series time order, output options and location for the table and the graphs be generated in your worksheet.
Once finished, the tool prints out the table (along with the formulas) into the target cells and creates a correlogram plot (if selected).