Why use autocorrelation instead of autocovariance when examining stationary time series?

Autocorrelation and autocovariance are one of the most critical metrics in financial time series econometrics. Both functions are based on covariance and correlation metrics.

What is Autocorrelation? Autocorrelation measures the degree of similarity between a given time series and the lagged version of that time series over successive time periods. It is similar to calculating the correlation between two different variables except in Autocorrelation we calculate the correlation between two different versions Xt and Xt-k of the same time series.

What is Autocovariance? Autocovariance is defined as the covariance between the present value (xt) with the previous value (xt-1) and the present value (xt) with (xt-2). And it is denoted as γ. In a stationary time series, data are challenging and are likely to change considering the assumptions of linear programming. In such a situation, the residual errors can be correlated in order to eliminate inconsistency. When the autocovariance is utilized, its variables can change over time which can lead to invalid results.

In stationary time series, the statistical properties of the data do not change considerably over time. Hence autocorrelation will be much applicable when compared to autocovariance.

Autocorrelation is used because it is the autocovariance divided by the variance: it has no units, always lies between −1 and +1, and so can be read and compared easily across lags and across different series. Autocovariance values depend on the scale of the data.

For a stationary series with mean \(\mu\):

\[\gamma_k = \text{Cov}(X_t, X_{t-k})\]

\[\rho_k = \frac{\gamma_k}{\gamma_0},\quad \gamma_0 = \text{Var}(X_t)\]

Reasons

• Scale-free: if sales are measured in rupees instead of thousands of rupees, \(\gamma_k\) changes by a factor of \(10^6\), but \(\rho_k\) stays the same.
• Bounded: \(-1 \le \rho_k \le 1\) and \(\rho_0 = 1\), so the strength of dependence is easy to judge (e.g. 0.8 is strong); a raw autocovariance of 250 tells nothing by itself.
• Comparable: ACF plots of different series can be compared directly.
• Significance testing: standard confidence bands (about \(\pm 1.96/\sqrt{n}\)) and tests such as Ljung–Box are based on autocorrelations.
• Model identification: the Box-Jenkins method reads the ACF and PACF patterns to choose AR and MA orders.

Stationarity matters here because only then do \(\gamma_k\) and \(\rho_k\) depend on the lag \(k\) alone and not on time \(t\), so a single ACF describes the whole series.