Correlogram

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 $$\alpha\,$$:


 * $$B=\pm t_{1-\alpha/2} SE(\hat\rho_l)$$ 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 of no autocorrelation is rejected at the confidence level $$\alpha\,$$. Thus, positive (negative) autocorrelation is given.

t is the quantile of the t-distribution; SE is the standard error, which can be computed by Bartlett’s formula for MA(l) processes:


 * $$SE(\hat\rho_1)=\frac {1} {T} $$ bzw. $$ SE(\hat\rho_l)=\sqrt\frac{1+2\sum_{i=1}^{l-1} \hat\rho^2_i}{T}$$ 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.