Autocovariance is Autocorrelation by Variance

Theorem

Let $\map S z$ be a stochastic process giving rise to a time series $T$.

Let $S$ be weakly stationary of order $2$ or greater.

Let $\gamma_k$ denote the autocovariance coefficient of $S$ at lag $k$.

Let $\rho_k$ denote the autocorrelation coefficient of $S$ at lag $k$.

Then:

$\gamma_k = \rho_k \sigma_z^2$

where $\sigma_z^2$ denotes the variance of $S$.

Proof

\(\ds \rho_k\)

\(=\)

\(\ds \dfrac {\expect {\paren {z_t - \mu} \paren {z_{t + k} - \mu} } } {\sqrt {\expect {\paren {z_t - \mu}^2} \expect {\paren {z_{t + k} - \mu}^2} } }\)

Definition of Autocorrelation

\(\ds \gamma_k\)

\(=\)

\(\ds \expect {\paren {z_t - \mu} \paren {z_{t + k} - \mu} }\)

Definition of Autocovariance

\(\ds \sigma_z^2\)

\(=\)

\(\ds \expect {\paren {z_t - \mu}^2}\)

Definition of Variance of Stochastic Process

\(\ds \)

\(=\)

\(\ds \expect {\paren {z_{t + k} - \mu}^2}\)

as $S$ is weakly stationary of order $2$ or greater

\(\ds \leadsto \ \ \)

\(\ds \rho_k\)

\(=\)

\(\ds \dfrac {\gamma_k} {\sigma_z^2 \sigma_z^2}\)

Hence the result.

$\blacksquare$

Sources

• 1994: George E.P. Box, Gwilym M. Jenkins and Gregory C. Reinsel: Time Series Analysis: Forecasting and Control (3rd ed.) ... (previous) ... (next):

Part $\text {I}$: Stochastic Models and their Forecasting:

$2$: Autocorrelation Function and Spectrum of Stationary Processes:

$2.1$ Autocorrelation Properties of Stationary Models:

$2.1.4$ Autocovariance and Autocorrelation Functions