Partial autocorrelation function

Partial correlation of a time series with its lagged values

Partial autocorrelation function of Lake Huron's depth with confidence interval

In time series analysis, the partial autocorrelation function (PACF) gives the partial correlation of a stationary time series with its own lagged values, regressed the values of the time series at all shorter lags. It contrasts with the autocorrelation function, which does not control for other lags.

This function plays an important role in data analysis aimed at identifying the extent of the lag in an autoregressive (AR) model. The use of this function was introduced as part of the Box–Jenkins approach to time series modelling, whereby plotting the partial autocorrelative functions one could determine the appropriate lags p in an AR (p) model or in an extended ARIMA (p,d,q) model.

## Definition[edit]

Given a time series  z t {\displaystyle z_{t}} , the partial autocorrelation of lag  k {\displaystyle k} , denoted  ϕ k , k {\displaystyle \phi _{k,k}} , is the autocorrelation between  z t {\displaystyle z_{t}} and  z t + k {\displaystyle z_{t+k}} with the linear dependence of  z t {\displaystyle z_{t}} on  z t + 1 {\displaystyle z_{t+1}} through  z t + k − 1 {\displaystyle z_{t+k-1}} removed. Equivalently, it is the autocorrelation between  z t {\displaystyle z_{t}} and  z t + k {\displaystyle z_{t+k}} that is not accounted for by lags  1 {\displaystyle 1} through  k − 1 {\displaystyle k-1} , inclusive.[1]

ϕ 1 , 1 = corr ⁡ ( z t + 1 , z t ) , for k = 1 , {\displaystyle \phi _{1,1}=\operatorname {corr} (z_{t+1},z_{t}),{\text{ for }}k=1,}

ϕ k , k = corr ⁡ ( z t + k − z ^ t + k , z t − z ^ t ) , for k ≥ 2 , {\displaystyle \phi _{k,k}=\operatorname {corr} (z_{t+k}-{\hat {z}}_{t+k},\,z_{t}-{\hat {z}}_{t}),{\text{ for }}k\geq 2,}

where  z ^ t + k {\displaystyle {\hat {z}}_{t+k}} and  z ^ t {\displaystyle {\hat {z}}_{t}} are linear combinations of  { z t + 1 , z t + 2 , . . . , z t + k − 1 } {\displaystyle \\{z_{t+1},z_{t+2},...,z_{t+k-1}\\}} that minimize the mean squared error of  z t + k {\displaystyle z_{t+k}} and  z t {\displaystyle z_{t}} respectively. For stationary processes, the coefficients in  z ^ t + k {\displaystyle {\hat {z}}_{t+k}} and  z ^ t {\displaystyle {\hat {z}}_{t}} are the same, but reversed:[2]

z ^ t + k = β 1 z t + k − 1 + ⋯ + β k − 1 z t + 1 and z ^ t = β 1 z t + 1 + ⋯ + β k − 1 z t + k − 1 . {\displaystyle {\hat {z}}_{t+k}=\beta _{1}z_{t+k-1}+\cdots +\beta _{k-1}z_{t+1}\qquad {\text{and}}\qquad {\hat {z}}_{t}=\beta _{1}z_{t+1}+\cdots +\beta _{k-1}z_{t+k-1}.}

## Calculation[edit]

The theoretical partial autocorrelation function of a stationary time series can be calculated by using the Durbin–Levinson Algorithm:

ϕ n , n = ρ ( n ) − ∑ k = 1 n − 1 ϕ n − 1 , k ρ ( n − k ) 1 − ∑ k = 1 n − 1 ϕ n − 1 , k ρ ( k ) {\displaystyle \phi _{n,n}={\frac {\rho (n)-\sum _{k=1}^{n-1}\phi _{n-1,k}\rho (n-k)}{1-\sum _{k=1}^{n-1}\phi _{n-1,k}\rho (k)}}}

where  ϕ n , k = ϕ n − 1 , k − ϕ n , n ϕ n − 1 , n − k {\displaystyle \phi _{n,k}=\phi _{n-1,k}-\phi _{n,n}\phi _{n-1,n-k}} for  1 ≤ k ≤ n − 1 {\displaystyle 1\leq k\leq n-1} and  ρ ( n ) {\displaystyle \rho (n)} is the autocorrelation function.[3][4][5]

The formula above can be used with sample autocorrelations to find the sample partial autocorrelation function of any given time series.[6][7]

## Examples[edit]

The following table summarizes the partial autocorrelation function of different models:[5][8]

Model  | PACF

|

White noise | The partial autocorrelation is 0 for all lags.
Autoregressive model | The partial autocorrelation for an AR(p) model is nonzero for lags less than or equal to p and 0 for lags greater than p.
Moving-average model | If  ϕ 1 , 1 > 0 {\displaystyle \phi _{1,1}>0} , the partial autocorrelation oscillates to 0.
If  ϕ 1 , 1 < 0 {\displaystyle \phi _{1,1}<0} , the partial autocorrelation geometrically decays to 0.
Autoregressive–moving-average model | An ARMA(p, q) model's partial autocorrelation geometrically decays to 0 but only after lags greater than p.

The behavior of the partial autocorrelation function mirrors that of the autocorrelation function for autoregressive and moving-average models. For example, the partial autocorrelation function of an AR(p) series cuts off after lag p similar to the autocorrelation function of an MA(q) series with lag q. In addition, the autocorrelation function of an AR(p) process tails off just like the partial autocorrelation function of an MA(q) process.[2]

## Autoregressive model identification[edit]

Sample partial autocorrelation function with confidence interval of a simulated AR(3) time series

Partial autocorrelation is a commonly used tool for identifying the order of an autoregressive model.[6] As previously mentioned, the partial autocorrelation of an AR(p) process is zero at lags greater than p.[5][8] If an AR model is determined to be appropriate, then the sample partial autocorrelation plot is examined to help identify the order.

The partial autocorrelation of lags greater than p for an AR(p) time series are approximately independent and normal with a mean of 0.[9] Therefore, a confidence interval can be constructed by dividing a selected z-score by  n {\displaystyle {\sqrt {n}}} . Lags with partial autocorrelations outside of the confidence interval indicate that the AR model's order is likely greater than or equal to the lag. Plotting the partial autocorrelation function and drawing the lines of the confidence interval is a common way to analyze the order of an AR model. To evaluate the order, one examines the plot to find the lag after which the partial autocorrelations are all within the confidence interval. This lag is determined to likely be the AR model's order.[1]

## References[edit]

1. ^ a b "6.4.4.6.3. Partial Autocorrelation Plot". www.itl.nist.gov. Retrieved 2022-07-14.
2. ^ a b Shumway, Robert H.; Stoffer, David S. (2017). Time Series Analysis and Its Applications: With R Examples. Springer Texts in Statistics. Cham: Springer International Publishing. pp. 97–99. doi:10.1007/978-3-319-52452-8. ISBN 978-3-319-52451-1.
3. ^ Durbin, J. (1960). "The Fitting of Time-Series Models". Revue de l'Institut International de Statistique / Review of the International Statistical Institute. 28 (3): 233–244. doi:10.2307/1401322. ISSN 0373-1138. JSTOR 1401322.
4. ^ Shumway, Robert H.; Stoffer, David S. (2017). Time Series Analysis and Its Applications: With R Examples. Springer Texts in Statistics. Cham: Springer International Publishing. pp. 103–104. doi:10.1007/978-3-319-52452-8. ISBN 978-3-319-52451-1.
5. ^ a b c Enders, Walter (2004). Applied econometric time series (2nd ed.). Hoboken, NJ: J. Wiley. pp. 65–67. ISBN 0-471-23065-0. OCLC 52387978.
6. ^ a b Box, George E. P.; Reinsel, Gregory C.; Jenkins, Gwilym M. (2008). Time Series Analysis: Forecasting and Control (4th ed.). Hoboken, New Jersey: John Wiley. ISBN 9780470272848.
7. ^ Brockwell, Peter J.; Davis, Richard A. (1991). Time Series: Theory and Methods (2nd ed.). New York, NY: Springer. pp. 102, 243–245. ISBN 9781441903198.
8. ^ a b Das, Panchanan (2019). Econometrics in Theory and Practice : Analysis of Cross Section, Time Series and Panel Data with Stata 15. 1. Singapore: Springer. pp. 294–299. ISBN 978-981-329-019-8. OCLC 1119630068.
9. ^ Quenouille, M. H. (1949). "Approximate Tests of Correlation in Time-Series". Journal of the Royal Statistical Society, Series B (Methodological). 11 (1): 68–84. doi:10.1111/j.2517-6161.1949.tb00023.x.

* v
* t
* e

Statistics

* Outline
* Index

Descriptive statistics

| Continuous data|

| Center|

* Mean
* Arithmetic
* Arithmetic-Geometric
* Cubic
* Generalized/power
* Geometric
* Harmonic
* Heronian
* Heinz
* Lehmer
* Median
* Mode

|

Dispersion|

* Average absolute deviation
* Coefficient of variation
* Interquartile range
* Percentile
* Range
* Standard deviation
* Variance

Shape|

* Central limit theorem
* Moments
* Kurtosis
* L-moments
* Skewness

Count data|

* Index of dispersion

Summary tables|

* Contingency table
* Frequency distribution
* Grouped data

Dependence|

* Partial correlation
* Pearson product-moment correlation
* Rank correlation
* Kendall's τ
* Spearman's ρ
* Scatter plot

Graphics|

* Bar chart
* Biplot
* Box plot
* Control chart
* Correlogram
* Fan chart
* Forest plot
* Histogram
* Pie chart
* Q–Q plot
* Radar chart
* Run chart
* Scatter plot
* Stem-and-leaf display
* Violin plot

Data collection

| Study design|

* Effect size
* Missing data
* Optimal design
* Population
* Replication
* Sample size determination
* Statistic
* Statistical power

|

Survey methodology|

* Sampling
* Cluster
* Stratified
* Opinion poll
* Questionnaire
* Standard error

Controlled experiments|

* Blocking
* Factorial experiment
* Interaction
* Random assignment
* Randomized controlled trial
* Randomized experiment
* Scientific control

Adaptive designs|

* Adaptive clinical trial
* Stochastic approximation
* Up-and-down designs

Observational studies|

* Cohort study
* Cross-sectional study
* Natural experiment
* Quasi-experiment

Statistical inference

| Statistical theory|

* Population
* Statistic
* Probability distribution
* Sampling distribution
* Order statistic
* Empirical distribution
* Density estimation
* Statistical model
* Model specification
* Lp space
* Parameter
* location
* scale
* shape
* Parametric family
* Likelihood (monotone)
* Location–scale family
* Exponential family
* Completeness
* Sufficiency
* Statistical functional
* Bootstrap
* U
* V
* Optimal decision
* loss function
* Efficiency
* Statistical distance
* divergence
* Asymptotics
* Robustness

|

Frequentist inference|

| Point estimation|

* Estimating equations
* Maximum likelihood
* Method of moments
* M-estimator
* Minimum distance
* Unbiased estimators
* Mean-unbiased minimum-variance
* Rao–Blackwellization
* Lehmann–Scheffé theorem
* Median unbiased
* Plug-in

|

Interval estimation|

* Confidence interval
* Pivot
* Likelihood interval
* Prediction interval
* Tolerance interval
* Resampling
* Bootstrap
* Jackknife

Testing hypotheses|

* 1- & 2-tails
* Power
* Uniformly most powerful test
* Permutation test
* Randomization test
* Multiple comparisons

Parametric tests|

* Likelihood-ratio
* Score/Lagrange multiplier
* Wald

Specific tests|

* Z-test (normal)
* Student's t-test
* F-test

Goodness of fit|

* Chi-squared
* G-test
* Kolmogorov–Smirnov
* Anderson–Darling
* Lilliefors
* Jarque–Bera
* Normality (Shapiro–Wilk)
* Likelihood-ratio test
* Model selection
* Cross validation
* AIC
* BIC

Rank statistics|

* Sign
* Sample median
* Signed rank (Wilcoxon)
* Hodges–Lehmann estimator
* Rank sum (Mann–Whitney)
* Nonparametric anova
* 1-way (Kruskal–Wallis)
* 2-way (Friedman)
* Ordered alternative (Jonckheere–Terpstra)
* Van der Waerden test

Bayesian inference|

* Bayesian probability
* prior
* posterior
* Credible interval
* Bayes factor
* Bayesian estimator
* Maximum posterior estimator

* Correlation
* Regression analysis

| Correlation|

* Pearson product-moment
* Partial correlation
* Confounding variable
* Coefficient of determination

|

Regression analysis|

* Errors and residuals
* Regression validation
* Mixed effects models
* Simultaneous equations models
* Multivariate adaptive regression splines (MARS)

Linear regression|

* Simple linear regression
* Ordinary least squares
* General linear model
* Bayesian regression

Non-standard predictors|

* Nonlinear regression
* Nonparametric
* Semiparametric
* Isotonic
* Robust
* Heteroscedasticity
* Homoscedasticity

Generalized linear model|

* Exponential families
* Logistic (Bernoulli) / Binomial / Poisson regressions

Partition of variance|

* Analysis of variance (ANOVA, anova)
* Analysis of covariance
* Multivariate ANOVA
* Degrees of freedom

Categorical / Multivariate / Time-series / Survival analysis

| Categorical|

* Cohen's kappa
* Contingency table
* Graphical model
* Log-linear model
* McNemar's test
* Cochran–Mantel–Haenszel statistics

|

Multivariate|

* Regression
* Manova
* Principal components
* Canonical correlation
* Discriminant analysis
* Cluster analysis
* Classification
* Structural equation model
* Factor analysis
* Multivariate distributions
* Elliptical distributions
* Normal

Time-series|

| General|

* Decomposition
* Trend
* Stationarity
* Seasonal adjustment
* Exponential smoothing
* Cointegration
* Structural break
* Granger causality

|

Specific tests|

* Dickey–Fuller
* Johansen
* Q-statistic (Ljung–Box)
* Durbin–Watson
* Breusch–Godfrey

Time domain|

* Autocorrelation (ACF)
* partial (PACF)
* Cross-correlation (XCF)
* ARMA model
* ARIMA model (Box–Jenkins)
* Autoregressive conditional heteroskedasticity (ARCH)
* Vector autoregression (VAR)

Frequency domain|

* Spectral density estimation
* Fourier analysis
* Least-squares spectral analysis
* Wavelet
* Whittle likelihood

Survival|

| Survival function|

* Kaplan–Meier estimator (product limit)
* Proportional hazards models
* Accelerated failure time (AFT) model
* First hitting time

|

Hazard function|

* Nelson–Aalen estimator

Test|

* Log-rank test

Applications

| Biostatistics|

* Bioinformatics
* Clinical trials / studies
* Epidemiology
* Medical statistics

|

Engineering statistics|

* Chemometrics
* Methods engineering
* Probabilistic design
* Process / quality control
* Reliability
* System identification

Social statistics|

* Actuarial science
* Census
* Crime statistics
* Demography
* Econometrics
* Jurimetrics
* National accounts
* Official statistics
* Population statistics
* Psychometrics

Spatial statistics|

* Cartography
* Environmental statistics
* Geographic information system
* Geostatistics
* Kriging

* Category
* Mathematics portal
* Commons
* WikiProject

*[v]: View this template
*[t]: Discuss this template
*[e]: Edit this template