Correlogram

of a time series $ x _ {1} \dots x _ {T} $

The set of serial (sample) correlation coefficients

$$ r _ {t} = \ \frac{ { \frac{1}{T - t } } \sum _ {s = 1 } ^ { {T } - t } ( x _ {s} - \overline{x}\; ) ( x _ {s + t } - \overline{x}\; ) }{ { \frac{1}{T} } \sum _ {s = 1 } ^ { T } ( x _ {s} - \overline{x}\; ) ^ {2} } ,\ \ t = 1 \dots T - 1, $$

where $ \overline{x}\; $ is the sample mean of the series, i.e.

$$ \overline{x}\; = \ { \frac{1}{T} } \sum _ {s = 1 } ^ { T } x _ {s} . $$

The term correlogram is sometimes applied to the graph of $ r _ {t} $ as a function of $ t $. It is an empirical measure of the statistical interdependence of the terms of the sequence $ \{ x _ {t} \} $. In time-series analysis, the correlogram is used for statistical inferences concerning a probability model suggested for the description and explanation of an observed sequence of data.

The term theoretical correlogram is sometimes used for the normalized correlation function of a (stationary) random sequence $ \{ X _ {t} \} $:

$$ \rho _ {t} = \ \frac{ \mathop{\rm cov} ( X _ {s} , X _ {s + t } ) }{ {\mathsf D} ( X _ {s} ) } ,\ \ t = 1, 2 \dots $$

where

$$ \mathop{\rm cov} ( X _ {s} , X _ {s + t } ) = \ {\mathsf E} ( X _ {s} - {\mathsf E} X _ {s} ) ( X _ {s + t } - {\mathsf E} X _ {s + t } ) $$

is the covariance of the random variables $ X _ {s} , X _ {s + t } $, and $ {\mathsf D} ( X _ {s} ) $ is the variance of the random variable $ X _ {s} $. If $ \{ x _ {t} \} $ is regarded as a realization of the random sequence $ \{ X _ {t} \} $, then, under fairly general assumptions, the sample correlogram $ \{ r _ {t} \} $ gives consistent and asymptotically normal estimators for the theoretical correlogram $ \{ \rho _ {t} \} $( see [3]).

From a mathematical point of view, the descriptions of a stationary random sequence in correlation and spectral terms are equivalent; in the statistical analysis of time series, however, the correlation and spectral approaches have different fields of application, depending on the initial material and the final aim of the analysis. Whereas spectral analysis gives one an idea of the existence and intensities of periodic components in a time series, correlation methods are more convenient when one is investigating statistical relationships between consecutive values of the observed data. In statistical practice, methods based on the correlogram are usually employed when there are grounds to postulate a fairly simple stochastic model (auto-regression, moving averages or a mixed model with auto-regression and moving averages of relatively low orders) generating the given time series (e.g. in econometrics). In such models, the theoretical correlogram $ \{ \rho _ {t} \} $ possesses special properties (it vanishes for all sufficiently large $ t $- values in the moving-averages model; in auto-regression models it decreases exponentially with possible oscillation). The presence of one such property in the sample correlogram may serve as an indication for a certain hypothetical probability model. In order to check for goodness-of-fit and to estimate the parameters of the selected model, statistical methods have been developed based on the distributions of the serial correlation coefficients.

References

[1]	T.W. Anderson, "The statistical analysis of time series" , Wiley (1971)
[2]	M.G. Kendall, A. Stuart, "The advanced theory of statistics" , 3. Design and analysis , Griffin (1966)
[3]	E.J. Hannan, "Multiple time series" , Wiley (1970)