Autocorrelation in Financial Time Series

Let $\{r_t\}$ be weakly stationary, so the mean $\mu = \E[r_t]$ and every autocovariance depend on lag alone [1]. Two functions summarise its linear dynamics. The autocorrelation function records dependence at each lag, and the partial autocorrelation function strips out the intervening lags. Their sample versions identify autoregressive moving-average order and drive the diagnostic tests that follow.

#Autocorrelation and its estimator

Definition1

The lag- $\ell$ autocovariance and autocorrelation of $\{r_t\}$ are

\gamma_\ell = \Cov(r_t, r_{t-\ell}) = \E[(r_t-\mu)(r_{t-\ell}-\mu)], \qquad \rho_\ell = \frac{\gamma_\ell}{\gamma_0}. \tag{1}

By construction $\gamma_0 = \Var(r_t)$ , $\rho_0 = 1$ , $|\rho_\ell| \le 1$ , and $\gamma_{-\ell}=\gamma_\ell$ , hence $\rho_{-\ell}=\rho_\ell$ . A white noise series, independent and identically distributed with finite variance, has $\rho_\ell = 0$ for every $\ell \neq 0$ . Detecting departures from this benchmark is the object of the analysis, and the natural vehicle is the linear representation that every purely nondeterministic stationary series admits.

Lemma2

Let $r_t = \mu + \sum_{i\ge 0}\psi_i a_{t-i}$ with $\psi_0 = 1$ , $\sum_i \psi_i^2 < \infty$ , and $\{a_t\}$ white noise of variance $\sigma_a^2$ . Then

\gamma_\ell = \sigma_a^2 \sum_{i\ge 0} \psi_i\,\psi_{i+\ell}, \qquad \rho_\ell = \frac{\sum_{i\ge 0}\psi_i\,\psi_{i+\ell}}{\sum_{i\ge 0}\psi_i^2}. \tag{2}

Proof

Center and multiply the representations at times $t$ and $t-\ell$ ,

\gamma_\ell = \E\Bigl[\Bigl(\textstyle\sum_i \psi_i a_{t-i}\Bigr)\Bigl(\sum_j \psi_j a_{t-\ell-j}\Bigr)\Bigr] = \sum_{i}\sum_{j}\psi_i\psi_j\,\E[a_{t-i}a_{t-\ell-j}]. \tag{3}

Because $\E[a_{t-i}a_{t-\ell-j}] = \sigma_a^2$ when $i = \ell+j$ and zero otherwise, only the diagonal $i = \ell + j$ survives, giving $\gamma_\ell = \sigma_a^2\sum_{j\ge 0}\psi_{j+\ell}\psi_j$ . Dividing by $\gamma_0 = \sigma_a^2\sum_i \psi_i^2$ yields $\rho_\ell$ .

The $\psi$ -weights fully determine the autocorrelations, and, as we see next, the asymptotic variance of their estimators.

Definition3

Given $r_1,\dots,r_T$ with sample mean $\bar r$ , the lag- $\ell$ sample autocorrelation is

\hat\rho_\ell = \frac{\sum_{t=\ell+1}^T (r_t-\bar r)(r_{t-\ell}-\bar r)}{\sum_{t=1}^T (r_t-\bar r)^2}. \tag{4}

#The sampling law of the estimator

The vector $\sqrt{T}(\hat\rho_1-\rho_1,\dots,\hat\rho_m-\rho_m)$ is asymptotically normal for any linear process with summable weights and finite fourth moment, and its covariance is fixed by Bartlett's formula [2].

Theorem4

For the linear process of Lemma 2, $\sqrt{T}\,(\hat\rho_\ell - \rho_\ell)$ is asymptotically normal with variance $w_{\ell\ell}$ , where

w_{\ell\ell} = \sum_{k=1}^{\infty}\bigl(\rho_{k+\ell} + \rho_{k-\ell} - 2\rho_\ell\rho_k\bigr)^2, \tag{5}

and $\Cov(\sqrt{T}\hat\rho_i,\sqrt{T}\hat\rho_j) \to w_{ij}$ with the same summand in the two indices.

The formula is the delta method applied to $\hat\rho_\ell = \hat\gamma_\ell/\hat\gamma_0$ . The sample autocovariances $\hat\gamma_\ell$ are jointly asymptotically normal, and for a linear process their limiting fourth-order terms reduce to products of the $\gamma$ 's; propagating that covariance through the ratio produces Equation (5) [2]. Two special cases carry the rest of the article.

Corollary5

If $\{r_t\}$ is MA( $q$ ), so $\rho_j = 0$ for all $j > q$ , then for every $\ell > q$

\Var(\hat\rho_\ell) \approx \frac{1}{T}\Bigl(1 + 2\sum_{i=1}^q \rho_i^2\Bigr). \tag{6}

Proof

Fix $\ell > q$ . In each summand of Equation (5), $\rho_{k+\ell} = 0$ since $k+\ell > q$ , and $\rho_\ell = 0$ since $\ell > q$ , leaving $w_{\ell\ell} = \sum_{k\ge 1}\rho_{k-\ell}^2$ . Reindex by $m = k-\ell$ . As $k$ runs over $1,2,\dots$ , $m$ runs over $1-\ell, 2-\ell, \dots$ , and $\rho_m^2$ is nonzero only for $|m|\le q$ . Because $\ell > q$ forces $1-\ell \le -q$ , the whole window $m = -q,\dots,q$ is reached, at $k = \ell-q,\dots,\ell+q$ , each a positive integer. Hence $w_{\ell\ell} = \sum_{m=-q}^{q}\rho_m^2 = \rho_0^2 + 2\sum_{m=1}^q \rho_m^2 = 1 + 2\sum_{m=1}^q\rho_m^2$ , and $\Var(\hat\rho_\ell) \approx w_{\ell\ell}/T$ .

Corollary6

If $\{r_t\}$ is white noise, then for $\ell \ge 1$ the $\sqrt{T}\,\hat\rho_\ell$ are asymptotically independent standard normals.

Proof

Set $\rho_k = 0$ for all $k \neq 0$ in Equation (5). The summand at index pair $(i,j)$ is $(\rho_{k+i}+\rho_{k-i})(\rho_{k+j}+\rho_{k-j})$ , and with $i,j\ge 1$ the factor $\rho_{k+i}+\rho_{k-i}$ is nonzero only at $k = i$ , where it equals $\rho_0 = 1$ . Thus $w_{ij} = 1$ if $i = j$ and $0$ otherwise, so the limiting covariance is the identity.

The individual test of the null $\rho_\ell = 0$ against dependence up to lag $q$ refers the studentised statistic $\hat\rho_\ell \big/ \sqrt{(1+2\sum_{i=1}^{\ell-1}\hat\rho_i^2)/T}$ to the standard normal, using Equation (6) for the denominator.

#Order identification

The ACF alone cannot separate direct dependence at lag $\ell$ from dependence relayed through the intervening lags. The partial autocorrelation removes the relay.

Definition7

The lag- $\ell$ partial autocorrelation $\phi_{\ell\ell}$ is the coefficient of $r_{t-\ell}$ in the linear projection of $r_t$ on its $\ell$ nearest lags,

r_t = \phi_{\ell 0} + \phi_{\ell 1} r_{t-1} + \cdots + \phi_{\ell\ell} r_{t-\ell} + e_{\ell t}. \tag{7}

The projection coefficients solve the order- $\ell$ Yule-Walker system $\sum_{j=1}^{\ell}\phi_{\ell j}\rho_{|i-j|} = \rho_i$ , $i = 1,\dots,\ell$ , whose last component is $\phi_{\ell\ell}$ . The Durbin-Levinson recursion solves the nested systems in $O(\ell)$ work per order,

\phi_{\ell\ell} = \frac{\rho_\ell - \sum_{j=1}^{\ell-1}\phi_{\ell-1,j}\,\rho_{\ell-j}}{1 - \sum_{j=1}^{\ell-1}\phi_{\ell-1,j}\,\rho_j}, \qquad \phi_{\ell j} = \phi_{\ell-1,j} - \phi_{\ell\ell}\,\phi_{\ell-1,\ell-j}, \tag{8}

started at $\phi_{11} = \rho_1$ , which gives $\phi_{22} = (\rho_2 - \rho_1^2)/(1-\rho_1^2)$ and so on. The two functions are dual order-identification tools. For MA( $q$ ) the ACF cuts off, $\rho_\ell = 0$ for $\ell > q$ , by Lemma 2 with finitely many $\psi_i$ . For AR( $p$ ) the PACF cuts off, $\phi_{\ell\ell} = 0$ for $\ell > p$ , since the order- $p$ projection is already exact, with sample estimate $\hat\phi_{\ell\ell} \sim \mathcal N(0, 1/T)$ beyond $p$ . A sharp truncation in one function names the order in the other.

#The portmanteau test

Theorem8

Under the null that $\{r_t\}$ is white noise,

Q(m) = T(T+2)\sum_{\ell=1}^m \frac{\hat\rho_\ell^2}{T-\ell} \Rightarrow \chi^2_m. \tag{9}

Proof

By Corollary 6 the $\sqrt{T}\,\hat\rho_\ell$ are asymptotically independent standard normals, so $T\hat\rho_\ell^2 \Rightarrow \chi^2_1$ and $T\sum_{\ell=1}^m \hat\rho_\ell^2 \Rightarrow \chi^2_m$ , the Box-Pierce statistic. The weight $(T+2)/(T-\ell)$ replaces the crude scaling $T$ by a factor matching the finite-sample moment $\E[\hat\rho_\ell^2] \approx (T-\ell)/[T(T+2)]$ , which sharpens the small-sample fit while leaving the limit at $\chi^2_m$ [3].

When the $\hat\rho_\ell$ are the residual autocorrelations of a fitted ARMA( $p,q$ ) model, the $p+q$ estimated parameters consume that many degrees of freedom and the reference law becomes $\chi^2_{m-p-q}$ .

#Microstructure-induced autocorrelation

Short-lag autocorrelation in returns need not reflect predictable dynamics. Two microstructure mechanisms manufacture it with opposite sign, and both admit an exact MA(1) computation.

Proposition9

Let the efficient price $P_t^\ast$ follow a random walk with innovation variance $\sigma^2$ , and let the observed transaction price be $P_t = P_t^\ast + \tfrac{S}{2} Q_t$ , where $S$ is the bid-ask spread and $Q_t \in \{-1,+1\}$ is an independent trade-direction indicator with $\Pr(Q_t = \pm 1)=\tfrac12$ , independent of $\{P_t^\ast\}$ . Then observed price changes are MA(1) with

\Var(\Delta P_t) = \sigma^2 + \frac{S^2}{2}, \qquad \gamma_1 = -\frac{S^2}{4}, \qquad \rho_1 = \frac{-S^2/4}{\sigma^2 + S^2/2} \in \Bigl(-\tfrac12,\,0\Bigr). \tag{10}

Proof

Since $\Pr(Q_t = \pm 1) = \tfrac12$ , we have $\E[Q_t] = 0$ and $\E[Q_t^2] = 1$ . Write $\Delta P_t = u_t + \tfrac{S}{2}(Q_t - Q_{t-1})$ with $u_t = \Delta P_t^\ast$ , $\Var(u_t) = \sigma^2$ . Independence of $u$ and $Q$ and $\Var(Q_t - Q_{t-1}) = \E[Q_t^2] + \E[Q_{t-1}^2] = 2$ give $\Var(\Delta P_t) = \sigma^2 + \tfrac{S^2}{4}\cdot 2$ . For the lag-one autocovariance only the shared term $Q_{t-1}$ contributes,

\gamma_1 = \frac{S^2}{4}\,\E\bigl[(Q_t - Q_{t-1})(Q_{t-1}-Q_{t-2})\bigr] = \frac{S^2}{4}\bigl(-\E[Q_{t-1}^2]\bigr) = -\frac{S^2}{4}, \tag{11}

and $\gamma_\ell = 0$ for $\ell \ge 2$ because no innovation is shared. Dividing gives $\rho_1$ , which increases in $\sigma^2$ , approaching $0$ as $\sigma^2\to\infty$ and $-\tfrac12$ as $\sigma^2\to 0$ .

The sign is unambiguously negative and the magnitude identifies the spread, $S = 2\sqrt{-\gamma_1}$ , so a first-lag autocorrelation can measure trading cost rather than signal. Nonsynchronous trading, where a recorded price is a stale last-trade value, smooths the latent return across periods and reverses the sign.

Proposition10

Let latent returns $r_t$ be iid with variance $\sigma^2$ , and let staleness spread each latent return equally over two periods, so the recorded return is $o_t = \tfrac12(r_t + r_{t-1})$ . Then $o_t$ is MA(1) with

\gamma_0 = \frac{\sigma^2}{2}, \qquad \gamma_1 = \frac{\sigma^2}{4}, \qquad \rho_1 = +\tfrac12, \tag{12}

and $\gamma_\ell = 0$ for $\ell \ge 2$ .

Proof

$\gamma_0 = \Var\bigl(\tfrac12(r_t+r_{t-1})\bigr) = \tfrac14(\sigma^2 + \sigma^2) = \sigma^2/2$ . The lag-one term shares only $r_{t-1}$ , so $\gamma_1 = \Cov\bigl(\tfrac12(r_t+r_{t-1}),\tfrac12(r_{t-1}+r_{t-2})\bigr) = \tfrac14\Var(r_{t-1}) = \sigma^2/4$ . No term is shared at lag two or beyond, so $\gamma_\ell = 0$ there, and $\rho_1 = (\sigma^2/4)/(\sigma^2/2) = \tfrac12$ .

The equal-weight case is the boundary. The general nontrading model of [4], where each security trades with a fixed per-period probability, interpolates $\rho_1 \in (0,\tfrac12)$ and induces positive cross-autocorrelation across a portfolio for the same reason.

The ACF and PACF are complementary fingerprints of linear structure, the first truncating at the moving-average order by Lemma 2 and the second at the autoregressive order, and the Ljung-Box statistic turns their sample values into a formal white-noise test through Corollary 6. Because the bid-ask bounce forces $\rho_1 = -\tfrac12$ in the limit and stale prices force $\rho_1 = +\tfrac12$ , short-lag autocorrelation in observed returns must be read against these mechanical sources before it is credited to genuine predictability. The Ornstein-Uhlenbeck process supplies the continuous-time AR(1) benchmark, and the same second-moment machinery underpins the cross-correlation structure exploited in statistical arbitrage.

[1]

R. S. Tsay, Analysis of Financial Time Series, 3rd ed. Wiley, 2010.

[2]

P. J. Brockwell and R. A. Davis, Time Series: Theory and Methods, 2nd ed. Springer, 1991.

[3]

G. M. Ljung and G. E. P. Box, “On a measure of lack of fit in time series models,” Biometrika, vol. 65, no. 2, pp. 297–303, 1978.

[4]

A. W. Lo and A. C. MacKinlay, “An econometric analysis of nonsynchronous trading,” Journal of Econometrics, vol. 45, no. 1–2, pp. 181–211, 1990.

Explore connections

see in the atlas

referenced by (1)

Autoregressive Models

cite

@misc{autocorrelation,
  author = {Zac Kienzle},
  title  = {Autocorrelation in Financial Time Series},
  year   = {2026},
  month  = {07},
  url    = {https://zackienzle.com/blog/autocorrelation}
}

#Autocorrelation and its estimator

Definition1

The lag- $\ell$ autocovariance and autocorrelation of $\{r_t\}$ are

\gamma_\ell = \Cov(r_t, r_{t-\ell}) = \E[(r_t-\mu)(r_{t-\ell}-\mu)], \qquad \rho_\ell = \frac{\gamma_\ell}{\gamma_0}. \tag{1}

Lemma2

Let $r_t = \mu + \sum_{i\ge 0}\psi_i a_{t-i}$ with $\psi_0 = 1$ , $\sum_i \psi_i^2 < \infty$ , and $\{a_t\}$ white noise of variance $\sigma_a^2$ . Then

\gamma_\ell = \sigma_a^2 \sum_{i\ge 0} \psi_i\,\psi_{i+\ell}, \qquad \rho_\ell = \frac{\sum_{i\ge 0}\psi_i\,\psi_{i+\ell}}{\sum_{i\ge 0}\psi_i^2}. \tag{2}

Proof

Center and multiply the representations at times $t$ and $t-\ell$ ,

\gamma_\ell = \E\Bigl[\Bigl(\textstyle\sum_i \psi_i a_{t-i}\Bigr)\Bigl(\sum_j \psi_j a_{t-\ell-j}\Bigr)\Bigr] = \sum_{i}\sum_{j}\psi_i\psi_j\,\E[a_{t-i}a_{t-\ell-j}]. \tag{3}

The $\psi$ -weights fully determine the autocorrelations, and, as we see next, the asymptotic variance of their estimators.

Definition3

Given $r_1,\dots,r_T$ with sample mean $\bar r$ , the lag- $\ell$ sample autocorrelation is

\hat\rho_\ell = \frac{\sum_{t=\ell+1}^T (r_t-\bar r)(r_{t-\ell}-\bar r)}{\sum_{t=1}^T (r_t-\bar r)^2}. \tag{4}

#The sampling law of the estimator

Theorem4

For the linear process of Lemma 2, $\sqrt{T}\,(\hat\rho_\ell - \rho_\ell)$ is asymptotically normal with variance $w_{\ell\ell}$ , where

w_{\ell\ell} = \sum_{k=1}^{\infty}\bigl(\rho_{k+\ell} + \rho_{k-\ell} - 2\rho_\ell\rho_k\bigr)^2, \tag{5}

and $\Cov(\sqrt{T}\hat\rho_i,\sqrt{T}\hat\rho_j) \to w_{ij}$ with the same summand in the two indices.

Corollary5

If $\{r_t\}$ is MA( $q$ ), so $\rho_j = 0$ for all $j > q$ , then for every $\ell > q$

\Var(\hat\rho_\ell) \approx \frac{1}{T}\Bigl(1 + 2\sum_{i=1}^q \rho_i^2\Bigr). \tag{6}

Proof

Corollary6

If $\{r_t\}$ is white noise, then for $\ell \ge 1$ the $\sqrt{T}\,\hat\rho_\ell$ are asymptotically independent standard normals.

Proof

#Order identification

The ACF alone cannot separate direct dependence at lag $\ell$ from dependence relayed through the intervening lags. The partial autocorrelation removes the relay.

Definition7

The lag- $\ell$ partial autocorrelation $\phi_{\ell\ell}$ is the coefficient of $r_{t-\ell}$ in the linear projection of $r_t$ on its $\ell$ nearest lags,

r_t = \phi_{\ell 0} + \phi_{\ell 1} r_{t-1} + \cdots + \phi_{\ell\ell} r_{t-\ell} + e_{\ell t}. \tag{7}

\phi_{\ell\ell} = \frac{\rho_\ell - \sum_{j=1}^{\ell-1}\phi_{\ell-1,j}\,\rho_{\ell-j}}{1 - \sum_{j=1}^{\ell-1}\phi_{\ell-1,j}\,\rho_j}, \qquad \phi_{\ell j} = \phi_{\ell-1,j} - \phi_{\ell\ell}\,\phi_{\ell-1,\ell-j}, \tag{8}

#The portmanteau test

Theorem8

Under the null that $\{r_t\}$ is white noise,

Q(m) = T(T+2)\sum_{\ell=1}^m \frac{\hat\rho_\ell^2}{T-\ell} \Rightarrow \chi^2_m. \tag{9}

Proof

#Microstructure-induced autocorrelation

Short-lag autocorrelation in returns need not reflect predictable dynamics. Two microstructure mechanisms manufacture it with opposite sign, and both admit an exact MA(1) computation.

Proposition9

\Var(\Delta P_t) = \sigma^2 + \frac{S^2}{2}, \qquad \gamma_1 = -\frac{S^2}{4}, \qquad \rho_1 = \frac{-S^2/4}{\sigma^2 + S^2/2} \in \Bigl(-\tfrac12,\,0\Bigr). \tag{10}

Proof

\gamma_1 = \frac{S^2}{4}\,\E\bigl[(Q_t - Q_{t-1})(Q_{t-1}-Q_{t-2})\bigr] = \frac{S^2}{4}\bigl(-\E[Q_{t-1}^2]\bigr) = -\frac{S^2}{4}, \tag{11}

Proposition10

\gamma_0 = \frac{\sigma^2}{2}, \qquad \gamma_1 = \frac{\sigma^2}{4}, \qquad \rho_1 = +\tfrac12, \tag{12}

and $\gamma_\ell = 0$ for $\ell \ge 2$ .

Proof

[1]

R. S. Tsay, Analysis of Financial Time Series, 3rd ed. Wiley, 2010.

[2]

P. J. Brockwell and R. A. Davis, Time Series: Theory and Methods, 2nd ed. Springer, 1991.

[3]

G. M. Ljung and G. E. P. Box, “On a measure of lack of fit in time series models,” Biometrika, vol. 65, no. 2, pp. 297–303, 1978.

[4]

A. W. Lo and A. C. MacKinlay, “An econometric analysis of nonsynchronous trading,” Journal of Econometrics, vol. 45, no. 1–2, pp. 181–211, 1990.

Explore connections

see in the atlas

referenced by (1)

Autoregressive Models

cite

@misc{autocorrelation,
  author = {Zac Kienzle},
  title  = {Autocorrelation in Financial Time Series},
  year   = {2026},
  month  = {07},
  url    = {https://zackienzle.com/blog/autocorrelation}
}