Abstract
We characterize the stationary process {X t } by its mean and its (matrix) covariance function. In the Gaussian case, this already characterizes the whole distribution. The estimation of these entities becomes crucial in the empirical analysis. As it turns out, the results from the univariate process carry over analogously to the multivariate case. If the process is observed over the periods t = 1, 2, …, T, then a natural estimator for the mean μ is the arithmetic mean or sample average:
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Notes
- 1.
The theorem may also be used to conduct a causality test between two times series (see Sect. 15.1).
- 2.
The order of the AR processes are set arbitrarily equal to 10 which is more than enough to obtain white noise residuals.
- 3.
The quarterly data are taken from Berndt (1991). They cover the period from the first quarter 1956 to the fourth quarter 1975. In order to achieve stationarity, we work with first differences.
- 4.
The order of the AR processes are set arbitrarily equal to 10 which is more than enough to obtain white noise residuals.
- 5.
We use data for Switzerland as published by the State Secretariat for Economic Affairs SECO.
- 6.
With quarterly data it is wise to set to order as a multiple of four to account for possible seasonal movements. As it turns out p = 8 is more than enough to obtain white noise residuals.
- 7.
During the interpretation of the cross-correlations be aware of the ordering of the variables because \(\rho _{12}(1) =\rho _{21}(-1)\neq \rho _{21}(1)\).
References
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Haugh LD (1976) Checking the independence of two covariance stationary time series: A univariate residual cross-correlation approach. J Am Stat Assoc 71:378–385
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Neusser, K. (2016). Estimation of Mean and Covariance Function. In: Time Series Econometrics. Springer Texts in Business and Economics. Springer, Cham. https://doi.org/10.1007/978-3-319-32862-1_11
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DOI: https://doi.org/10.1007/978-3-319-32862-1_11
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