Time Series Analysis - Autocorrelations

An autocorrelation is the correlation of a series with itself, shifted by a particular lag of k observations. The plot of autocorrelations for various lags is a crucial tool for determining an appropriate model for ARIMA analysis (see also, Time Series Analysis Overview and Time Series Analysis Index). The computations of the autocorrelation coefficients rk follow the standard formulas, as described in most time series references (e.g., Box & Jenkins, 1976).

Standard error of rk. Under the assumption that the true moving average process in the series is of the order k-1, then the approximate standard error of rk is defined as:

StdErr(rk) = Ö{(1/N) * [1+2*Σ(ri2 )]}

(for i = 1 to k-1)

Here N is the number of observations in the series. However, under the assumption that the series is a white noise process, that is, that all autocorrelations are equal to zero, the standard error of rk is defined as:

StdErr(rk) = Ö{(1/N) * [(N-k)/(N+2)]}

Select the White noise standard errors check box to compute the standard errors in this manner.

Box-Ljung Q

At a given lag k the Box-Ljung Q statistic is defined by:

Qk = n*(n+2)*Σ[ri2 /(n-i)]

(for i = 1 to k)

When the number of observations is large, then the Q statistic has a Chi-Square distribution with k-p-q degrees of freedom, where p and q are the number of autoregressive and moving average parameters, respectively.