Autocorrelation Functions and Persistence

Why Autocorrelation Describes Persistence

A time series can vary around a stable mean while still carrying information from one period to the next. If a series is unusually high today, for example, it may tend to remain high in nearby periods. The autocorrelation function (ACF) summarizes the strength and direction of this linear relationship at different time separations, or lags. Its pattern helps describe persistence: how long associations remain evident, and whether they tend to continue in the same direction or alternate.

The definition here is for a weakly stationary process. Weak stationarity means that the process has a finite, time-invariant mean, finite variance, and an autocovariance that depends on the time separation rather than on the calendar date. This assumption makes it meaningful to describe the relationship at lag $h$ with one quantity, rather than needing a different quantity for every pair of dates.

Definition: Let $\{X_t\}$ be a weakly stationary process with mean $\mu$ and finite, positive variance. Its autocovariance at integer lag $h$ is $$ \gamma(h)=\operatorname{Cov}(X_t,X_{t-h}) =E[(X_t-\mu)(X_{t-h}-\mu)]. $$ Its autocorrelation at lag $h$ is $$ \rho(h)=\frac{\gamma(h)}{\gamma(0)}, $$ where $\gamma(0)=\operatorname{Var}(X_t)>0$. The ACF is the collection of these values over integer lags $h$.

With this convention, $h=1$ compares the series in one period with the series one period earlier; $h=2$ compares values two periods apart. Negative lags are included in the mathematical definition because they help reveal a symmetry property. Since $\gamma(0)$ is the variance, the denominator puts autocovariances onto a unit-free scale. The value $\rho(h)$ is a correlation, not a causal effect or a measure of every possible form of dependence.

Properties of the ACF

Three facts follow directly from the definition: the ACF equals one at lag zero, it is symmetric around zero, and its values lie between $-1$ and $1$. Together, these make the ACF interpretable as a sequence of correlations across time.

Theorem: For a weakly stationary process with positive variance, $$ \rho(0)=1,\qquad \rho(-h)=\rho(h),\qquad -1\leq \rho(h)\leq 1. $$

At lag zero, the two variables in the covariance are the same: $\gamma(0)=\operatorname{Var}(X_t)$. Dividing this variance by itself gives $\rho(0)=1$. For symmetry, covariance is unchanged when its two arguments are exchanged. Thus $$ \gamma(-h)=\operatorname{Cov}(X_t,X_{t+h}) =\operatorname{Cov}(X_{t+h},X_t)=\gamma(h), $$ where the last equality uses weak stationarity: both covariances concern the same separation $h$. Dividing by the same positive $\gamma(0)$ gives $\rho(-h)=\rho(h)$.

For the bound, the Cauchy–Schwarz inequality gives $$ |\operatorname{Cov}(X_t,X_{t-h})| \leq \sqrt{\operatorname{Var}(X_t)\operatorname{Var}(X_{t-h})} =\gamma(0). $$ The final equality uses the constant variance required by weak stationarity. Dividing both sides by $\gamma(0)>0$ proves $|\rho(h)|\leq 1$. The bound is not a restriction unique to time series; it is the usual bound for a correlation.

Location and Scale Do Not Change the ACF

An important feature of the ACF is that it does not depend on the units in which the series is recorded, or on a constant added to every observation. Suppose $Y_t=a+bX_t$, where $a$ and $b$ are constants and $b\neq 0$. If $X_t$ is weakly stationary, then $Y_t$ is too: its mean is $a+b\mu$, and its variance is $b^2\gamma(0)$. Centering $Y_t$ removes the added constant, so $$ \gamma_Y(h) =E\{b(X_t-\mu)b(X_{t-h}-\mu)\} =b^2\gamma_X(h). $$ Therefore $$ \rho_Y(h)=\frac{b^2\gamma_X(h)}{b^2\gamma_X(0)} =\rho_X(h). $$ This holds for negative as well as positive $b$. A negative rescaling reverses the signs of both variables in each correlation pair, leaving the correlation unchanged. If $b=0$, $Y_t$ has zero variance and its ACF is undefined.

Worked Example: Rescaling a Series

Suppose a hypothetical weekly series $X_t$, measured in thousands of units, is weakly stationary with $\gamma_X(0)=4$ and $\gamma_X(1)=2$, in squared thousands of units. A reporting system records $Y_t=10+3X_t$. Find the lag-one ACF for both series.

Check the assumptions. The example stipulates weak stationarity and positive variance for $X_t$. An affine transformation with nonzero multiplier preserves weak stationarity, and $b=3$ is nonzero, so the ACF of $Y_t$ is defined.

For $X_t$, the lag-one ACF is $$ \rho_X(1)=\frac{\gamma_X(1)}{\gamma_X(0)} =\frac{2}{4}=0.5. $$ For $Y_t$, the transformed autocovariance and variance are $$ \gamma_Y(1)=3^2(2)=18,\qquad \gamma_Y(0)=3^2(4)=36. $$ Thus $$ \rho_Y(1)=\frac{18}{36}=0.5. $$ The added 10 shifts the series’ level, and multiplication by 3 changes its units and spread, but neither changes the lag-one correlation.

Reading Persistence Patterns

Persistence is a description of how the ACF behaves as the lag grows. A positive correlation indicates that values separated by that lag tend to move together: values above the mean are associated with other values above the mean, and values below the mean with other values below it. A negative correlation indicates an opposite-direction linear association. The size and pattern of the correlations across many lags matter; no single universal cutoff divides persistent from nonpersistent behavior.

A geometrically decaying ACF has correlations that decline by a roughly constant proportion from one lag to the next. When the correlations are positive, this pattern represents associations that weaken steadily with time separation. An oscillating ACF changes sign or varies periodically across lags, indicating alternating or cyclical linear association. A slowly decaying ACF remains appreciable across more distant lags and declines gradually rather than quickly. These are descriptions of covariance patterns, not by themselves explanations of what caused the pattern.

Worked Example: Geometric Decay

Suppose a hypothetical weakly stationary process, measured in points, has autocovariance $\gamma(h)=9(0.7)^{|h|}$ at integer lags. Calculate and interpret the ACF through lag 3.

Check the assumptions. The example stipulates a weakly stationary process with this covariance pattern. Its variance is $\gamma(0)=9>0$, so the ACF is defined. The pattern is a valid geometric covariance pattern; the calculations below use the stipulated covariance and do not require estimating it from observations.

At lag zero, $\gamma(0)=9(0.7)^0=9$. For $h=1,2,3$, divide each autocovariance by 9: $$ \rho(h)=\frac{9(0.7)^h}{9}=(0.7)^h,\qquad h\geq 0. $$ In particular, $$ \rho(1)=0.7,\qquad \rho(2)=0.7^2=0.49,\qquad \rho(3)=0.7^3=0.343. $$ The positive correlations weaken with the lag, but are still visible at lag 3. For example, 0.343 describes a positive linear association between values three periods apart. It does not say that one value causes the other. The same ACF values hold at the corresponding negative lags by symmetry.

Worked Example: An Oscillating ACF

Consider the hypothetical process $X_t=A\cos(\pi t/3)+B\sin(\pi t/3)$, where $A$ and $B$ are independent, have mean zero, and each have variance 2. Find its mean, variance, and ACF at lags 1 through 3.

Check the assumptions. The process has mean zero because $A$ and $B$ have mean zero. Using their independence and equal variances, its covariance at lag $h$ is $2\cos(\pi h/3)$, which depends only on the lag; its variance is the constant value 2. It is therefore weakly stationary, and its positive variance makes the ACF well-defined.

To see the covariance, write $\theta=\pi/3$. Products involving one $A$ and one $B$ have expectation zero by independence and zero means. The remaining terms give $$ \gamma(h)=2\{\cos(\theta t)\cos(\theta(t-h))+ \sin(\theta t)\sin(\theta(t-h))\} =2\cos(\theta h). $$ Consequently, $\gamma(0)=2$ and $\rho(h)=\cos(\pi h/3)$. The values are $$ \rho(1)=\cos(\pi/3)=0.5,\quad \rho(2)=\cos(2\pi/3)=-0.5,\quad \rho(3)=\cos(\pi)=-1. $$ The signs alternate, so the pattern describes oscillating association. At lag 3, the correlation is $-1$: values three periods apart are perfectly negatively linearly related in this particular process. It is not a decaying pattern; the correlations repeat as the lag increases.

Slow decay is distinct from oscillation: correlations can stay positive and decline gradually, or their signs can alternate while their magnitudes decline. Nor does “slowly decaying” mean “never decays.” The comparison is about how quickly the magnitude of the ACF diminishes across lags. In empirical work, an estimated ACF is calculated from a finite sample and will vary from sample to sample; it should not be read as the exact population ACF.

Worked Example: Slow Decay Across Lags

Suppose a hypothetical weakly stationary process has $\rho(h)=1/(1+|h|)$. Compare its correlations at lags 1, 5, and 10.

Check the assumptions. The example stipulates a covariance-stationary process with this ACF. The specified sequence is a valid ACF: for nonnegative lags, $1/(1+h)=\int_0^1 r^h\,dr$, a mixture of geometric correlation sequences $r^{|h|}$. Its lag-zero value is 1, and the process is assumed to have positive finite variance.

Substitution gives $$ \rho(1)=\frac{1}{1+1}=0.5,\qquad \rho(5)=\frac{1}{1+5}\approx 0.167,\qquad \rho(10)=\frac{1}{1+10}\approx 0.0909. $$ All three correlations are positive, and the value at lag 10 is smaller than at lag 5, which is smaller than at lag 1. But the association remains positive even at lag 10 and declines more gradually than a geometric pattern such as $0.7^h$. This is an example of slowly decaying persistence, not evidence that correlations must remain positive forever in every process.

Common Mistakes and Practical Interpretation

The ACF is useful because it condenses a time series’ second-moment dependence into a lag-by-lag summary. It is also important to keep its scope clear. Each value is a standardized autocovariance and therefore describes linear association between two dates. It does not summarize every feature of the joint distribution, establish a mechanism, or show that a change in one date causes a later change.

Confusing autocovariance with autocorrelation. Autocovariance has squared measurement units; autocorrelation is dimensionless. Divide by the variance $\gamma(0)$ to obtain the ACF.
Forgetting lag-zero normalization. If the variance is positive, $\rho(0)=1$, not an unknown quantity. A process with zero variance has no defined ACF.
Reading negative lags as a separate pattern. For a weakly stationary process, $\rho(-h)=\rho(h)$. The ACF is symmetric, so its negative-lag values repeat the positive-lag values.
Treating a large positive correlation as causation. The ACF describes an association between observations at different times; it does not identify why they are associated.
Calling every decline “geometric” or equating slow decay with no decay. Geometric decay has a roughly constant proportional decline; slow decay can remain noticeable over longer lags, while still tending downward.

When reading an ACF, first check the lag-zero normalization and symmetry, then look across a range of positive lags. Ask whether the pattern is mostly positive, alternates in sign, or declines gradually. Those observations provide a disciplined description of persistence without claiming more than correlation can establish.

Check Your Understanding

Answer each question using the definition and properties of the ACF.

A weakly stationary process has $\gamma(0)=16$ and $\gamma(2)=4$. What is $\rho(2)$, and what is $\rho(-2)$?
Explain why adding a constant to every value of a series does not change its ACF.
If $\rho(1)=0.8$, must $\rho(2)$ equal 0.64? Explain what additional pattern would be needed for that conclusion.
What kind of persistence is suggested by correlations that alternate in sign while their magnitudes decline?
In context, what does a positive ACF at lag 4 describe, and what does it not establish?