Partial Autocorrelation Function

Partial Autocorrelation Function (PACF)

The partial autocorrelation function at lag $h$, written ${\varphi }_{hh}$, measures the correlation between $X_t$ and $X_{t-h}$ that remains after the linear effect of the intermediate observations $X_{t-1},X_{t-2},\dots ,X_{t-h+1}$ has been accounted for. The construction is to project $X_t$ linearly onto the intermediate observations, producing a fitted value ${\hat{X}}_t$, and to do the same for $X_{t-h}$, producing ${\hat{X}}_{t-h}$. The two residuals $X_t-{\hat{X}}_t$ and $X_{t-h}-{\hat{X}}_{t-h}$ are the parts of $X_t$ and $X_{t-h}$ that the intermediate observations cannot linearly explain, and the partial autocorrelation is the ordinary correlation between those residuals,
${\varphi }_{hh}=\text{corr}(X_t-{\hat{X}}_t,X_{t-h}-{\hat{X}}_{t-h}).$
At lag $1$ there are no intermediate observations to project away, so ${\varphi }_{11}$ reduces to the ordinary autocorrelation $\rho (1)$. For a stationary process, ${\varphi }_{hh}$ depends only on the lag $h$, not on $t$.

The PACF is the diagnostic counterpart to the ordinary autocorrelation. The autocorrelation $\rho (h)$ counts both the direct dependence between $X_t$ and $X_{t-h}$ and the indirect dependence routed through intermediate values; the PACF subtracts the indirect part and reports only what is left. Concretely: in an AR(1) process $X_t=\varphi X_{t-1}+W_t$, the values $X_t$ and $X_{t-2}$ are correlated because both are correlated with $X_{t-1}$, but $X_{t-2}$ adds no direct predictive information once $X_{t-1}$ is given. The autocorrelation $\rho (2)={\varphi }^2$ is non-zero, while the partial autocorrelation ${\varphi }_{22}$ is exactly zero. This distinction is what makes the PACF, paired with the ACF, the cleanest diagnostic for identifying ARMA model orders from data.

The data signature is straightforward to read. For an AR($p$) process, the PACF is exactly zero past lag $p$, while the ACF tails off geometrically, so the order $p$ can be read directly off the last significant lag of the sample PACF. For an MA(q) process, the roles swap: the ACF cuts off past lag $q$ and the PACF tails off geometrically. For ARMA(p, q), both functions tail off without exact zero, and order selection moves from visual inspection to information criteria like AIC and BIC.

PACF is a Linear Diagnostic linear partial dependence in exactly the same sense that the ACF measures linear total dependence. A non-linear AR-style process (for example a self-exciting threshold AR with two regimes) will produce a PACF whose pattern looks like an ARMA fingerprint but reflects only the linear shadow of the underlying nonlinear dependence. The ACF/PACF identification rules are therefore necessary but not sufficient: residual diagnostics and out-of-sample checks remain essential after fitting any model whose structure these tools picked.

The PACF measures

Intuition

The autocorrelation at lag $h$ confounds two sources of dependence: the direct path from $X_{t-h}$ to $X_t$, and the indirect paths routed through whichever intermediate values lie on the chain. In an AR(1) process this confounding is dramatic: the only direct dependence is at lag 1, but the autocorrelation continues to be non-zero at every lag because the lag-1 dependence chains forward through the recursion. The partial autocorrelation strips out the chained, indirect contribution by asking: once the intermediate observations have been used up to predict $X_t$, does $X_{t-h}$ have any new linear information to add?

For an AR($p$) process this question has a sharp answer at every lag larger than $p$: no. The defining equation $X_t={\varphi }_1{X}_{t-1}+\cdots +{\varphi }_p{X}_{t-p}+W_t$ says that the most recent $p$ values, plus the current shock, are everything that linearly determines $X_t$. So once $X_{t-1},\dots ,X_{t-h+1}$ have been used as predictors, no value at lag $h>p$ can contribute linearly above and beyond what those intermediate values already provide, and the corresponding partial correlation is exactly zero. The PACF therefore cuts off cleanly at lag $p$, which is the diagnostic that identifies the AR order from a sample plot.

The reverse situation holds for MA processes. An MA(q) is a finite combination of shocks, not of past observations, so writing it as a regression on past observations requires the AR($\infty$) representation $X_t=-{\sum }_{j\ge 1}{\pi }_j{X}_{t-j}+W_t$ with infinitely many non-zero coefficients. Every past observation contributes something to the prediction of $X_t$, the coefficients ${\pi }_j$ decay geometrically rather than vanishing, and so the partial autocorrelations decay rather than cutting off. This is the structural reason why ACF and PACF flip roles between the AR and MA families.

Properties

(AR(p) PACF Cutoff) For a causal AR(p), ${\varphi }_{hh}$ is exactly zero at every lag past $p$.

The AR(p) defining equation makes the optimal one-step-ahead linear prediction ${\hat{X}}_t$ from any set of past observations equal to ${\varphi }_1{X}_{t-1}+\cdots +{\varphi }_p{X}_{t-p}$, regardless of how many lags are included. Once the lag count reaches $p$, additional past observations carry no incremental linear information, and the partial correlation between $X_t$ and any further-back value $X_{t-h}$ vanishes after projecting away the intermediates. The last non-zero PACF spike sits at exactly lag $p$, and in a sample plot, “exactly zero” is read as “inside the $\pm 1.96/\sqrt{T}$ confidence band under the white-noise null.” This makes the PACF the cleanest single diagnostic for choosing the AR order.

(MA(q) PACF Tails Off) For an invertible MA(q), the PACF is non-zero at every lag and decays geometrically.

The invertible MA(q) admits an AR($\infty$) representation $W_t=X_t-{\sum }_{j\ge 1}{\pi }_j{X}_{t-j}$ in which the weights ${\pi }_j$ decay geometrically (or as a damped oscillation when MA roots are complex), but never reach exact zero. So when the question “what’s the partial correlation at lag $h$ after removing intermediates?” is asked, every lag contributes through these infinite AR coefficients, and none cleanly vanishes. The PACF tails off as a sum of geometric or damped-oscillatory terms: the structural mirror image of the autocorrelation’s behavior in the AR family.

(ARMA(p,q) Both Tail Off) For a causal, invertible ARMA(p, q), the PACF tails off in the same way as the ACF.

Because an ARMA(p, q) admits both an MA($\infty$) and an AR($\infty$) representation with infinitely many non-zero weights, neither the autocorrelation nor the partial autocorrelation cuts off. The lag-domain diagnostics no longer identify $(p,q)$ uniquely from cutoff lags alone, and order selection moves to information criteria: try a few small candidate orders, compute AIC and BIC for each fit, and pick the smallest model whose residuals look like white noise on a Ljung-Box test. The diagnostic table from this regime is “tails off both ways,” in contrast to “ACF tails, PACF cuts” for AR and “ACF cuts, PACF tails” for MA.

(Yule-Walker Computation) The PACF is computed by solving a Yule-Walker system at each lag.

For each lag $h$, the Yule-Walker equations applied to $\{\rho (0),\rho (1),\dots ,\rho (h)\}$ produce a length-$h$ vector of AR coefficients $(a_1^{(h)},\dots ,a_h^{(h)})$ representing the best linear predictor of $X_t$ from the previous $h$ values. The last entry $a_h^{(h)}$ is the PACF at lag $h$:
${\varphi }_{hh}=a_h^{(h)}.$
Equivalently, the Durbin-Levinson recursion updates the coefficient vector from order $h-1$ to $h$ with $O(h)$ work per step, producing the full PACF up to lag $H$ in $O(H^2)$ total. This is stable, efficient, and built into every standard time-series package (statsmodels.tsa.stattools.pacf, R’s pacf).

(Variance Reduction Interpretation) The squared PACF measures the marginal variance reduction from adding lag $h$ to the predictor set.

If ${\sigma }_h^2$ denotes the variance of the residual after projecting $X_t$ onto the previous $h$ observations, then ${\sigma }_h^2={\sigma }_{h-1}^2(1-{\varphi }_{hh}^2)$. So ${\varphi }_{hh}^2$ is the proportion of the remaining variance that adding the lag-$h$ observation explains beyond what the closer lags already cover. The PACF is therefore the time-series analogue of partial $R^2$ in regression, answering “how much variance does adding lag $h$ explain that the closer lags don’t?” The total variance shrinkage from including all of $X_{t-1},\dots ,X_{t-h}$ is the cumulative product, ${\sigma }_h^2={\sigma }_0^2{\prod }_{j=1}^h(1-{\varphi }_{jj}^2)$.

(All PACF Values Bounded by 1) Every ${\varphi }_{hh}$ is a correlation coefficient and therefore satisfies $|{\varphi }_{hh}|\le 1$.

This is immediate from the definition: ${\varphi }_{hh}$ is the correlation between two specific residuals, and any correlation lies in $[-1,1]$. The bound has a useful consequence in the variance-reduction formula above: the cumulative product ${\prod }_j(1-{\varphi }_{jj}^2)$ is always between $0$ and $1$, confirming that adding more predictors can only shrink (never inflate) the optimal prediction error variance.

Examples

PACF of AR(1): predicted vs sample

For the process $X_t=0.7X_{t-1}+W_t$, what is the population PACF, and what should a sample PACF from $T=200$ observations look like?

Solution ${\varphi }_{11}=\rho (1)=0.7$. At every higher lag, the AR(1) defining equation makes $X_t$ depend linearly only on $X_{t-1}$, so the partial correlation ${\varphi }_{hh}$ between $X_t$ and $X_{t-h}$ after projecting away $X_{t-1},\dots ,X_{t-h+1}$ is exactly zero for $h\ge 2$. The corresponding sample PACF should show a single significant spike at lag 1 near value $0.7$, with subsequent lags falling inside the confidence band $\pm 1.96/\sqrt{200}\approx \pm 0.139$. About $5\%$ of higher lags will randomly poke outside the band by chance, which is not a problem unless they form a coherent pattern.

At lag 1, the PACF coincides with the autocorrelation:

PACF of MA(1) at $\theta =0.6$

What does the PACF of an MA(1) at $\theta =0.6$ look like at the first three lags?

Solution $\rho (1)=0.6/(1+0.36)\approx 0.441$ and $\rho (h)=0$ for $h\ge 2$. The lag-1 PACF coincides with $\rho (1)$, so ${\varphi }_{11}\approx 0.441$. For lag 2, the Yule-Walker system on $\{\rho (0),\rho (1),\rho (2)\}$ becomes $\left(\begin{array}{cc}1& 0.441\\ 0.441& 1\end{array}\right)\left(\begin{array}{c}{a}_1\\ a_2\end{array}\right)=\left(\begin{array}{c}0.441\\ 0\end{array}\right),$ and solving gives $a_2=-0.441\cdot 0.441/(1-0.441^2)\approx -0.244$, so ${\varphi }_{22}\approx -0.244$. Iterating the Durbin-Levinson recursion gives ${\varphi }_{33}\approx 0.142$ and ${\varphi }_{44}\approx -0.084$. The PACF tails off geometrically with alternating sign: the visual signature of an MA(1) with positive $\theta$. The PACF does not cut off; it never reaches exact zero.

The MA(1) autocorrelation is

Order selection from sample plots

The sample ACF of a series is $\hat{\rho }=(1.00,0.62,0.39,0.25,0.16,0.10,\dots )$, decaying smoothly toward zero. The sample PACF is $(1.00,0.62,0.01,-0.03,0.05,0.02,\dots )$, with only the lag-1 entry significant. What model fits?

Solution $X_{t-1}$ is in the predictor set, no further past observation adds linear information. The ACF tails off geometrically with ratio approximately $0.62$, matching the AR(1) prediction $\rho (h)=0.62^h$ ($\rho (2)\approx 0.38,\rho (3)\approx 0.24,\rho (4)\approx 0.15,\rho (5)\approx 0.09$, all matching the observed values to within the sampling noise). Fit AR(1) with $\hat{\varphi }\approx 0.62$, then confirm by checking the residuals: the residual ACF should be flat (Ljung-Box test not significant) and the residual PACF likewise. If the residuals retain structure, the model order needs to be raised, but the diagnostic plot is clean enough here that AR(1) is the obvious choice.

The PACF cuts off cleanly after lag 1, which is the AR(1) signature: once

See Also

• 2.2.3 Autocorrelation Function: the total-correlation diagnostic that the PACF complements; the two together form the Box-Jenkins identification machinery.
• 2.2.2 Autocovariance Function: the unnormalized version of autocorrelation, the underlying second-moment object.
• 2.5 ARIMA Models: the family the ACF/PACF identification table is designed to discriminate among.
• 2.5.2 AR(p) Process: the family whose order is read off the PACF cutoff.
• 2.5.3 MA(q) Process: the family whose order is read off the ACF cutoff.
• 2.5.4 ARMA(p,q) Process: the family where both tail off and information criteria take over.