The Partial Autocorrelation Function

The ordinary autocorrelation has a blind spot. Suppose today is correlated with yesterday, and yesterday with the day before. Then today and the day-before-yesterday will look correlated too — even if there is no direct link between them at all. The correlation just leaks through the middle day. So when a correlogram shows a fat bar at lag 2, you cannot tell whether the series genuinely reaches back two steps or whether it is only lag-1 dependence echoing through. The partial autocorrelation function (PACF) is built precisely to remove that leak.

The partial autocorrelation at lag k, written \alpha(k), is the correlation between x_t and x_{t-k} after linearly stripping out the effect of the intervening values x_{t-1}, \dots, x_{t-k+1}. It answers the sharper question: once I already know the days in between, does knowing the value k steps back tell me anything new?

The cleanest definition: the last AR coefficient

There is a strikingly practical way to see the PACF. Fit the best linear predictor of x_t from its previous k values — an AR(k) regression:

x_t = \phi_{k1} x_{t-1} + \phi_{k2} x_{t-2} + \dots + \phi_{kk} x_{t-k} + \text{error}.

The last coefficient, \phi_{kk}, is exactly the partial autocorrelation \alpha(k). It measures the marginal value of the k-th lag given all the shorter lags are already in the model — the intervening lags are the other regressors, so their effect is automatically "partialled out." That is why the double-subscript notation \phi_{kk} appears everywhere the PACF does.

For the first two lags this gives tidy formulas from the ACF:

\alpha(1) = \rho_1, \qquad \alpha(2) = \frac{\rho_2 - \rho_1^{2}}{1 - \rho_1^{2}}.

The lag-2 formula is the intuition made algebra: it takes the raw lag-2 correlation \rho_2 and subtracts off what lag-1 dependence alone (\rho_1^2) would have manufactured.

The signature: PACF of an AR(p) cuts off

Now the payoff, and it is the exact mirror of the MA story. In a true AR(p), once you have conditioned on the first p lags there is nothing left for lag p+1 to explain — the model literally only depends on p past values. So the partial autocorrelations past lag p are all zero.

The PACF is thus the tool that reads an AR order straight off a plot, exactly as the ACF reads an MA order. Used together, the two plots are the heart of identifying an ARMA model.

Seeing the cutoff

Here is the PACF of an AR(2) (\phi_1 = 0.5, \phi_2 = 0.3). There are exactly two non-zero spikes — at lags 1 and 2 — and from lag 3 onward every partial autocorrelation is flat on the axis. Reading the picture: "two spikes, then nothing" says p = 2. (Its ACF, by contrast, would trail off through many lags.)

Worked example: an AR(1) has PACF zero past lag 1

Take an AR(1) with \phi = 0.6. Its ACF is \rho_k = 0.6^{k}, so \rho_1 = 0.6 and \rho_2 = 0.36. Then

\alpha(1) = \rho_1 = 0.6, \qquad \alpha(2) = \frac{\rho_2 - \rho_1^{2}}{1 - \rho_1^{2}} = \frac{0.36 - 0.36}{1 - 0.36} = 0.

The numerator vanishes: the lag-2 correlation 0.36 is entirely manufactured by lag-1 dependence, with nothing extra. So \alpha(2) = 0, and indeed \alpha(k) = 0 for all k \ge 2 — the PACF cuts off at lag 1, the signature of an AR(1). The ACF tails off forever while the PACF stops after one step: the two functions tell you different, complementary things.

The theory gives an exact zero past lag p, but a sample PACF computed from finite data never sits at exactly zero — it wobbles. Software draws significance bands at roughly \pm 1.96/\sqrt{T}; a spike poking just past the band on some far-out lag is very often pure sampling noise, not a real term. If you dutifully add an AR lag for every bar that scrapes the band you will over-fit wildly. Read the plot for the first few clear, decisive spikes and treat distant marginal ones with suspicion. Parsimony wins — a fact the Box–Jenkins loop enforces by checking residuals rather than chasing every wiggle.

Computing a fresh AR(k) regression for every k would be wasteful. The Durbin–Levinson recursion gets all the \phi_{kk} at once: it walks up the orders, using the Yule–Walker structure to turn the order-k solution into the order-k+1 one in a single step, and the "reflection coefficient" it produces at each stage is the partial autocorrelation \alpha(k). So the PACF is not just a diagnostic picture — it falls out for free from the same recursion that solves the AR normal equations, and it reappears under the name "reflection coefficient" in signal processing and speech coding.