Introduction toTime Series and Forecasting,Second EditionPeter J. BrockwellRichard A....

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Introduction toTime Series and Forecasting,Second EditionPeter J. BrockwellRichard A. DavisSpringer
The Bartlett Press, Inc. brockwel 8·i·2002 1:59 p.m. Page iSpringer Texts in StatisticsAdvisors:George Casella Stephen Fienberg Ingram OlkinSpringerNew YorkBerlinHeidelbergBarcelonaHong KongLondonMilanParisSingaporeTokyo
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The Bartlett Press, Inc. brockwel 8·i·2002 1:59 p.m. Page iiiPeter J. Brockwell Richard A. DavisIntroductionto Time Seriesand ForecastingSecond EditionWith 126 IllustrationsIncludes CD-ROM13
The Bartlett Press, Inc. brockwel 8·i·2002 1:59 p.m. Page ivPeter J. BrockwellRichard A. DavisDepartment of StatisticsDepartment of StatisticsColorado State UniversityColorado State UniversityFort Collins, CO 80523Fort Collins, CO [email protected]@stat.colostate.eduEditorial BoardGeorge CasellaStephen FienbergIngram OlkinDepartment of StatisticsDepartment of StatisticsDepartment of StatisticsGriffin-Floyd HallCarnegie Mellon UniversityStanford UniversityUniversity of FloridaPittsburgh, PA 15213-3890Stanford, CA 94305P.O. Box 118545USAUSAGainesville, FL 32611-8545USALibrary of Congress Cataloging-in-Publication DataBrockwell, Peter J.Introduction to time series and forecasting / Peter J. Brockwell and Richard A. Davis.—2nd ed.p.cm. — (Springer texts in statistics)Includes bibliographical references and index.ISBN 0-387-95351-5 (alk. paper)1. Time-series analysis. I. Davis, Richard A. II. Title. III. Series.QA280.B757 2002519.5'5—dc212001049262Printed on acid-free paper.© 2002, 1996 Springer-Verlag New York, Inc.All rights reserved. This work may not be translated or copied in whole or in part without the written permission ofthe publishers (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for briefexcerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storageand retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known orhereafter developed is forbidden.The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former arenot especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks andMerchandise Marks Act, may accordingly be used freely by anyone.Production managed by MaryAnn Brickner; manufacturing supervised by Joe Quatela.Typeset by The Bartlett Press, Inc., Marietta, GA.Printed and bound by R.R. Donnelley and Sons, Harrisonburg, VA.Printed in the United States of America.987654321ISBN 0-387-95351-5SPIN 10850334Springer-Verlag New York Berlin HeidelbergA member of BertelsmannSpringer Science+Business Media GmbHDisclaimer: This eBook does not include the ancillary media that waspackaged with the original printed version of the book.
The Bartlett Press, Inc. brockwel 8·i·2002 1:59 p.m. Page vTo Pam and Patti

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Robert · Accepted Answer

1. We are given that        (  )      (  ) where A and B are independent 
random variables with mean 0 and variance 1. We note that  ( )   ( )   (  )    
and  (  )   (  )   . ( follows from given mean, variance and 0 correlation) 
Now,   (  )   ( )   (  )   ( )   (  )   . Hence, the mean of the series is not 
dependent on time. 
Also, the autocorrelation function for lag   is given by 
  ( )     (       )   (      ) 
  (  )   (  )   (     )    (  )   (  )   (     )
  (  )   (  )   (     )   (  )   (  )   (     )
    (        )      (  ) 
Using theorem 2.1.1 and the fact that    ( ) is an auto-covariance function and 
  ( )     (  )     (   )    (  ), we get the fact that   ( ) is nonnegative 
definite.
2. a) We are given that                     where       (   ). We note that 
 (  )   (      )    and  (   
 )   . ( Follows from given mean, variance and 0 
correlation) 
Now, the autocorrelation function for lag   is given by 
  ( )     (       )   (      ) 
  (  
 )      (    
 )      (    
 )       [ Using the fact that  (      )   . Hence 
all cross product terms become 0.] 
b) We are given that                       where       (     ). We note that 
 (  )   (      )    and  (   
 )     . ( Follows from given mean, variance and 0 
correlation) 
Now, the autocorrelation function for lag   is given by 
  ( )     (       )   (      ) 
  (  
 )       (    
 )       (    
 )             [ Using the fact that 
 (      )    and  (   
 )     . Hence all cross product terms become 0.]
3.

Introduction toTime Series and Forecasting,Second EditionPeter J. BrockwellRichard A. DavisSpringer The Bartlett Press, Inc. brockwel 8·i·2002 1:59 p.m. Page iSpringer Texts in...

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