Correlogram

Example correlogram

A correlogram is the graphical representation of the autocorrelation of a time series . For this purpose, the correlation coefficients are plotted against the duration of the time shift . ${\displaystyle \rho _{l}}$ ${\displaystyle l}$

If it exceeds (falls below) the upper limit (lower limit) , the null hypothesis that there is no autocorrelation is rejected with an error probability of : ${\displaystyle \rho _{l}}$ ${\displaystyle B}$ ${\displaystyle \alpha }$

${\displaystyle B=\pm t_{1-\alpha /2}\,SE({\hat {\rho }}_{l})}$,

With

• the corresponding quantile  t of the t-distribution
• the standard error  SE; In this context, this is usually calculated using Bartlett's formula for MA (l) processes ( moving average , see ARMA model ):

for : ${\displaystyle l=1}$ ${\displaystyle SE({\hat {\rho }}_{1})={\frac {1}{T}}}$

for : ${\displaystyle l>1}$ ${\displaystyle SE({\hat {\rho }}_{l})={\sqrt {\frac {1+2\sum _{i=1}^{l-1}{\hat {\rho }}_{i}^{2}}{T}}}}$

• the estimated autocorrelation between observations that are periods apart. ${\displaystyle {\hat {\rho }}_{l}}$ ${\displaystyle l}$

In the picture above, the null hypothesis that there is no autocorrelation between neighboring periods is rejected; Observations of successive periods thus correlate significantly . For the remaining delays, however, the null hypothesis of missing autocorrelation cannot be rejected.

literature

• John E. Hanke, Arthur G. Reitsch, Dean W. Wichern: Business forecasting. 7th ed. Prentice Hall, Upper Saddle River, NJ 2001, ISBN 0-13-087810-3 .