Partial Autocorrelation Function (PACF): Definition and Application

The Partial Autocorrelation Function (PACF) measures the correlation between observations in a time series separated by various lag lengths, ignoring the correlations at shorter lags. It is a crucial tool in identifying the appropriate lag length in time series models.

What is PACF?§

The Partial Autocorrelation Function (PACF) measures the direct relationship between observations in a time series separated by specific lag lengths, after removing the effects of shorter lags. It is a crucial analytical tool in time series analysis, particularly used to identify the number of lags in autoregressive models.

Mathematical Definition§

Mathematically, the PACF at lag $k$, denoted by $\phi_k$, is the correlation between $Y_t$ and $Y_{t-k}$ that is not accounted for by lags $1$ through $k-1$:

$$\phi_k = Cor(Y_t - \hat{Y_t}, Y_{t-k} - \hat{Y_{t-k}})$$

where $\hat{Y_t}$ is the linear regression prediction of $Y_t$ based on $Y_{t-1}, Y_{t-2}, \ldots, Y_{t-(k-1)}$.

Types of Partial Autocorrelation Functions§

Sample PACF§

The sample PACF is an estimate of the true PACF calculated from finite sample data. It is often visualized with a partial autocorrelation plot, showcasing PACF values at different lags.

Theoretical PACF§

The theoretical PACF is derived from the underlying data-generating process of the time series, assuming infinite observations.

Importance and Applications§

Identifying AR Order§

PACF is pivotal in determining the order of autoregressive (AR) models. For an AR(p) process, the PACF plot cuts off after lag $p$, meaning PACF values drop to zero beyond lag $p$.

Model Diagnostics§

PACF helps in diagnosing the adequacy of fitted time series models by examining residuals for remaining autocorrelations.

PACF Calculation§

Box-Jenkins Methodology§

The Box-Jenkins methodology employs PACF alongside the autocorrelation function (ACF) to choose appropriate models by iteratively fitting, diagnosing, and refining.

Yule-Walker Equations§

PACF values can be derived from solving Yule-Walker equations, which involve recursively computing partial autocorrelations.

Examples and Interpretations§

• First-order AR Process (AR(1)):

  $$Y_t = \phi_1 Y_{t-1} + \epsilon_t$$
  PACF: Cuts off sharply after lag 1.

• Second-order AR Process (AR(2)):

  $$Y_t = \phi_1 Y_{t-1} + \phi_2 Y_{t-2} + \epsilon_t$$
  PACF: Cuts off sharply after lag 2.

Interpretation§

If a PACF plot exhibits significant correlations at lags 1 and 2 but not beyond, it suggests that an AR(2) model is appropriate.

Historical Context§

The development of PACF analysis stems from early advancements in statistical techniques for analyzing time series data. The pioneering work by Udny Yule and Gilbert Walker on autoregressive processes laid the groundwork for modern methods.

Related Terms§

• Autocorrelation Function (ACF): ACF measures the linear relationship between time series observations separated by various lags, but unlike PACF, it does not account for other intervening lags.
• Autoregressive Model (AR): An AR model represents a time series as a linear function of its past values.

FAQs§

How is PACF different from ACF?

PACF measures direct relationships after accounting for shorter lags, while ACF measures the overall correlation without distinguishing the direct and indirect effects.

Why is PACF important in ARIMA modeling?

PACF helps in determining the number of AR terms in ARIMA models, guiding the model selection process.

Can PACF be used for non-stationary data?

PACF is typically applied to stationary data. For non-stationary series, differencing or transformation is required before applying PACF.

References§

• Box, G. E. P., Jenkins, G. M., & Reinsel, G. C. (2015). “Time Series Analysis: Forecasting and Control.”
• Hamilton, J. D. (1994). “Time Series Analysis.”

Summary§

The Partial Autocorrelation Function (PACF) is an essential tool in time series analysis, helping identify the direct relationships between observations at specific lags. By understanding and applying PACF, analysts and statisticians can effectively model and forecast time-dependent data.