By Daniel Peña; George C Tiao; Ruey S Tsay

ISBN-10: 047136164X

ISBN-13: 9780471361640

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**Extra info for A course in time series analysis**

**Sample text**

2 ζ 3. Cov(z,_*, Zt) = y ,k depends only the separation lag k and not on t. z The sequence y _k is the autocovariance function of the series and, dropping the suffix ζ for simplicity, ρ* = γ*/·Υο is the autocorrelation function. 4 between the first 100 values of the series of steel production and the values at, respectively, lags of 1,10 and 16 quarters. The first plot for example has the values Z 2 . - - . z i o o plotted vertically against ζ ι Z99 horizontally. Over the earlier period this series appears to be approximately stationary, although in the longer term there is certainly a change of structure.

5 the peaks in the sample acf at lags 4, 8, 12, . . draw attention to the quarterly seasonality in the series which requires modeling by means of some such transformation. 5. C Y C L E S A N D T H E P E R I O D O G R A M Periodic or cyclic behaviour is a feature of many time series. In this section we define first the periodogram, which is a general tool for revealing unknown periodicities as well as confirming known periodicities in an observed series. The periodogram provides another statistical summary of an observed time series, just like the sample acf and pacf.

The sample spectrum reflects the white-noise component in the fairly uniform values at frequencies away from zero, and the random-walk component in the peak close to frequency zero. 8. STATE-SPACE M O D E L S A N D T H E K A L M A N F I L T E R In the last section of this chapter we use the simple signal-noise model of the previous section as an introduction to linear state-space models. Such models are also called structural models if they are believed to explain the statistical behaviour of the series in terms of more simple components, and a further example, of a seasonal structural model, is presented to illustrate this.

### A course in time series analysis by Daniel Peña; George C Tiao; Ruey S Tsay

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