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By Christopher F. Baum

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9 A time series with a periodic component can be constructed from xt = U1 sin(2πω0 t) + U2 cos(2πω0 t), where U1 and U2 are independent random variables with zero means and E(U12 ) = E(U22 ) = σ 2 . The constant ω0 determines the period or time it takes the process to make one complete cycle. Show that this series is weakly stationary with autocovariance function γ(h) = σ 2 cos(2πω0 h). 10 Suppose we would like to predict a single stationary series xt with zero mean and autocorrelation function γ(h) at some time in the future, say, t + , for > 0.

7 in Appendix A. 1: Large Sample Distribution of the ACF Under general conditions, if xt is white noise, then for n large, the sample ACF, ρx (h), for h = 1, 2, . . , H, where H is fixed but arbitrary, is approximately normally distsributed with zero mean and standard deviation given by 1 σρˆx (h) = √ . 6: Estimation of Correlation 31 √ outside the interval ±2/ n (or plus/minus two standard errors); for a white noise sequence, approximately 95% of the sample ACFs should be within these limits. The applications of this property develop because many statistical modeling procedures depend on reducing a time series to a white noise series by various kinds of transformations.

6 Estimation of Correlation Although the theoretical autocorrelation and cross-correlation functions are useful for describing the properties of certain hypothesized models, most of the analyses must be performed using sampled data. This limitation means the sampled points x1 , x2 , . . , xn only are available for estimating the mean, autocovariance, and autocorrelation functions. From the point of view of classical statistics, this poses a problem because we will typically not have iid copies of xt that are available for estimating the covariance and correlation functions.

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A review of Stata 8.1 and its time series by Christopher F. Baum

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