Correlogram

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Example for a correlogramm
Example for a correlogramm

A correlogram is graph of the autocorrelations \rho_l\, versus l\, (the time lags). If cross-correlation is used, it is called a cross-correlogram.

In the same graph one can draw upper and lower bounds for autocorrelation with significance level \alpha\,:


B=\pm z_{1-\alpha/2} SE(\hat\rho_l) with \hat\rho_l\, as the estimated autocorrelation in period l\,.


If the autocorrelation is higher (lower) than this upper (lower) bound, the null hypothesis that there is no autocorrelation at and beyond a given lag is rejected at a significance level of \alpha\,. This test is an approximate one and assumes that the time-series is Gaussian.

In the above, z1-α/2 is the quantile of the Normal distribution; SE is the standard error, which can be computed by Bartlett’s formula for MA(l) processes:


SE(\hat\rho_1)=\frac {1} {T}
SE(\hat\rho_l)=\sqrt\frac{1+2\sum_{i=1}^{l-1} \hat\rho^2_i}{T} for l>1\,


In the picture above we can reject the null that there is no autocorrelation between periods that are close to each other. For the other periods one cannot reject the null of no autocorrelation.

[edit] Further reading

Hanke, John E./Reitsch, Arthur G./Wichern, Dean W. (2001): Business forecasting, 7th edition, Upper Saddle River, NJ: Prentice Hall, 2001.

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