4.2 Partial autocorrelation function (PACF) and its interpretation
2 min read•july 22, 2024
The (PACF) is a key tool in time series analysis. It measures correlation between observations and their lags, controlling for intermediate lags. This helps identify the order of autoregressive (AR) terms in ARIMA models.
PACF plots display values for different lags. Significant spikes suggest including corresponding lags in the AR model. Used with ACF plots, PACF helps determine appropriate ARIMA models for forecasting and analysis.
Partial Autocorrelation Function (PACF)
Partial autocorrelation function definition
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- Measures the correlation between an observation and its , while controlling for the effects of intermediate lags
- Removes the influence of shorter lags when calculating the correlation for longer lags (lag 1, lag 2)
- Identifies the extent of the lag in an autoregressive (AR) model
- Helps determine the order of the AR term in ARIMA models (AR(1), AR(2))
- Calculated as the correlation that results after removing the effect of any correlation due to terms at shorter lags
- PACF at lag is the correlation after removing the effect of lags 1, 2, ...,
PACF plot interpretation
- Displays the partial autocorrelation values for different lags (lag 1, lag 2, lag 3)
- Significant spikes suggest the inclusion of the corresponding lag in the AR model
- For an AR() model, the PACF will have significant spikes at lags 1, 2, ..., , and cuts off to zero afterwards
- The lag at which the PACF cuts off indicates the order of the AR term (AR(1), AR(2))
- Gradual decay in the PACF suggests a higher-order AR term or the presence of a moving average (MA) term
- Indicates the need for further investigation using the ACF plot
ACF vs PACF comparison
- ACF (Autocorrelation Function) measures the correlation between an observation and its lagged values
- Helps identify the presence of AR and MA terms in a time series model (ARMA, ARIMA)
- PACF focuses on identifying the order of the AR term in ARIMA models
- Controls for the influence of shorter lags when calculating correlations
- ACF and PACF plots are used together to determine the appropriate ARIMA model
- ACF identifies the presence of AR and MA terms (ARMA(1,1), ARIMA(1,1,1))
- PACF determines the order of the AR term (AR(1), AR(2))
PACF for moving average models
- Examine the PACF plot to identify significant spikes at different lags (lag 1, lag 2, lag 3)
- The lag at which the PACF cuts off or the last significant spike occurs indicates the appropriate order of the AR model
- Example: PACF with significant spikes at lags 1 and 2, and cuts off afterwards, suggests an AR(2) model
- Gradual decay in the PACF without a clear cut-off suggests considering higher-order AR models or the presence of MA terms
- Use the ACF plot in conjunction with the PACF to determine the appropriate ARIMA model (ARIMA(1,1,1), ARIMA(2,1,2))
Key Terms to Review (14)
ARIMA Model Identification: ARIMA model identification is the process of determining the appropriate parameters for an Autoregressive Integrated Moving Average (ARIMA) model to accurately represent a given time series. This involves analyzing patterns within the data, such as trends and seasonality, and utilizing tools like autocorrelation function (ACF) and partial autocorrelation function (PACF) to help identify the order of the model components. Proper identification is crucial for building an effective forecasting model that can capture the underlying behavior of the data.
Box-Jenkins methodology: Box-Jenkins methodology is a systematic approach for analyzing and forecasting time series data using ARIMA (AutoRegressive Integrated Moving Average) models. It emphasizes model identification, parameter estimation, and diagnostic checking, allowing for effective predictions and analysis of patterns within time series data. This methodology also incorporates seasonal adjustments, making it especially useful for data with seasonal fluctuations.
Confidence Intervals: A confidence interval is a statistical range, with a specified level of confidence, that is likely to contain the true value of an unknown population parameter. This concept is crucial in statistical analysis as it provides a measure of uncertainty around sample estimates, helping to quantify the reliability of those estimates. The width and position of the interval can provide insights into data variability and the precision of the estimate.
Cut-off property: The cut-off property refers to a characteristic of the partial autocorrelation function (PACF) in time series analysis, where the PACF values become zero beyond a certain lag. This property is crucial for identifying the appropriate order of autoregressive (AR) models, as it helps to determine the maximum number of lags that have a direct relationship with the current value of the series, thus facilitating model selection and interpretation.
Exponential decay: Exponential decay refers to a process where a quantity decreases at a rate proportional to its current value, resulting in a rapid decline that slows over time. This concept is often represented mathematically by the equation $$N(t) = N_0 e^{-kt}$$, where $$N(t)$$ is the amount remaining at time $$t$$, $$N_0$$ is the initial amount, and $$k$$ is the decay constant. Understanding exponential decay is essential for analyzing time series data, particularly when interpreting trends and the effects of autocorrelation.
Lag: In time series analysis, lag refers to the time delay between observations or events, indicating how past values of a series can influence its future values. This concept is crucial for understanding relationships within the data, such as how past values impact future predictions and correlations.
Non-significant lags: Non-significant lags refer to time intervals in a time series analysis where the autocorrelation coefficients do not show a statistically meaningful relationship with the variable being analyzed. In other words, these lags do not contribute valuable information for predicting future values and can often be ignored. Identifying non-significant lags is crucial for simplifying models and enhancing interpretability when working with partial autocorrelation functions.
PACF vs ACF: The Partial Autocorrelation Function (PACF) and the Autocorrelation Function (ACF) are both tools used to measure the correlation of a time series with its own past values. While ACF evaluates the total correlation at different lags, including both direct and indirect correlations, PACF specifically measures the direct correlation between a time series and its lagged values, removing the effects of intervening lags. Understanding the distinction between PACF and ACF is crucial for identifying the appropriate order of autoregressive and moving average components in time series modeling.
Partial autocorrelation: Partial autocorrelation is a statistical measure that quantifies the correlation between a time series and its own lagged values, while controlling for the effects of intermediate lags. This allows for a clearer understanding of the direct relationship between an observation and its previous values, making it useful in identifying the order of autoregressive models. By examining the partial autocorrelation function (PACF), analysts can discern patterns, assess model suitability, and evaluate the presence of seasonality in time series data.
Partial Autocorrelation Function: The partial autocorrelation function (PACF) measures the correlation between observations in a time series at different lags while controlling for the influence of intermediate lags. It helps to identify the direct relationship between an observation and its lagged values, which is essential in determining the appropriate order of autoregressive models. By analyzing PACF, one can gain insights into the underlying structure of a time series and diagnose model adequacy through residual analysis.
Sample pacf: The sample partial autocorrelation function (PACF) measures the correlation between a time series and its own past values, while controlling for the values of the intervening time points. This function helps to identify the direct relationships between a variable and its lags, making it essential for understanding the underlying structure of time series data and is particularly useful in model selection.
Significant lags: Significant lags refer to specific time intervals in a time series where past values have a meaningful correlation with the current value, indicating that these lags can provide valuable information for modeling and predicting future observations. Understanding significant lags is essential for identifying the appropriate structure of time series models, such as ARIMA, and helps in distinguishing between which lags contribute to explaining variability in the data versus those that do not.
Statistical significance: Statistical significance is a measure that helps determine whether the results of a study or experiment are likely to be genuine or if they could have occurred by random chance. It involves the use of p-values to assess the strength of evidence against the null hypothesis, which posits no effect or relationship. In the context of time series analysis, statistical significance plays a crucial role in interpreting the results from models like the partial autocorrelation function (PACF), helping to identify which lags in the series have meaningful correlations.
Theoretical PACF: The theoretical partial autocorrelation function (PACF) measures the correlation between a time series and its lagged values while removing the effects of intermediate lags. This function is critical for understanding the direct relationship between the current value and previous values of the series, thus helping to identify the order of autoregressive models. Theoretical PACF helps distinguish between direct and indirect associations in time series data, which is essential for model selection and parameter estimation.