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ACF - Autocorrelation Function

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Mohamad
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Calculates the sample autocorrelation function (ACF) of a stationary time series.

Syntax

ACF(X, Order, K, Method)
X
is the univariate time series data (a one-dimensional array of cells (e.g., rows or columns)).
Order
is the time order in the data series (i.e., the first data point's corresponding date (earliest date=1 (default), latest date=0)).
Order Description
1 ascending (the first data point corresponds to the earliest date) (default)
0 descending (the first data point corresponds to the latest date)
K
is the lag order (e.g. 0=no lag, 1=1st lag, etc.). If missing, the default lag order of zero (i.e., Lag=0) is assumed.
Method
is the calculation method for estimating the autocorrelation function (0= Sample Autocorrelation (Default), 2=Periodogram-based estimate , 2=Cross correlation).
Value Method
0 Sample autocorrelation method.(default)
1 Periodogram-based estimate. method
2 Cross-correlation method

Remarks

  1. The time series is homogeneous or equally spaced.
  2. The time series may include missing values (e.g., #N/A) at either end.
  3. The lag order (k) must be less than the time series size, or an error value (#VALUE!) is returned.
  4. The ACF values are bound between -1 and 1, inclusive.
  5. Method 1: The sample autocorrelation is computed as:$$\hat{\rho}(h)=\frac{\sum_{k=h}^T{(y_{k}-\bar y)(y_{k-h}-\bar y)}}{\sum_{k=1}^T(y_{k}-\bar y)^2}$$Where:
    • $y_{t}$ is the value of the time series at time t.
    • $h$ is the lag order.
    • $T$ is the number of non-missing values in the time series data.
    • $\bar y$ is the sample average/mean of the time series.
      $\bar y=\frac{\sum_{i=1}^{N} y_i}{N}$
  6. Note that we subtract the full sample mean $\bar y$.
  7. Method 2: Periodogram estimate.
    In this method, we compute the spectral density (periodogram is an estimator) of the sample data set, and use it to calculate the sample auto-correlation.
  8. Although the periodogram-based ACF estimate is usually biased, it generally exhibits a smaller standard error.
  9. Method 3: cross-correlation method:$$\rho(h)=\frac{\sum_{i=1}^{N-h}(y_i-\bar y)\times (y_{i+h}-\bar y_h)}{\sqrt{\sum_{i=1}^{N-h}(y_i-\bar y)^2 \times\sum_{j=h}^N (y_j-\bar y_h)^2}}$$Where:
    • $y_{t}$ is the value of the time series at time t.
    • $h$ is the lag order.
    • $\bar y=\frac{\sum_{i=1}^{N-h} y_i}{N-h}$
    • $\bar y_h=\frac{\sum_{i=h}^N y_i}{N-h}$
    • $T$ is the number of non-missing values in the time series data.
    • $\bar y$ is the sample average/mean of the time series.
  10. Special Cases:
    • By definition, $\hat{\rho}(0) \equiv 1.0 $

 

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