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Unformatted text preview: Chapter 3: Probability Models for Time Series Li Chen Department of Mathematics University of Bristol 1 / 16 Outline I Introduction I Stationary processes I The autocorrelation function I Some useful stochastic processes I Mixed models I Integrated models 2 / 16 3.1 Introduction I To fully specify a time series it is necessary to write down the joint probability distribution of X t 1 , . . . , X tn for any set of times t 1 , . . . , t n and any value of n . But this can be complicated. I A more useful way of describing a series is through the moments of a process [particularly the first and second moments: mean, variance and autocovariance]. I Mean (the mean function): μ x ( t ) = E( X t ) I Variance (the variance function): σ 2 x ( t ) = Var( x t ) I Covariance is defined by γ ( t 1 , t 2 ) = Cov( X t 1 , X t 2 ) = E[ { X t 1 E( X t 1 ) }{ X t 2 E( X t 2 ) } ] 3 / 16 3.2 Stationary Processes: Strict Stationarity A strictly stationary time series is one where the joint distribution of X ( t 1 ) , . . . , X ( t n ) is the same as the joint distribution of X ( t 1 + τ ) , . . . , X ( t n + τ ) for all t 1 , . . . , t n , τ and n . Properties For a strictly stationary time series, I the mean and variance do not depend on t . Therefore μ ( t ) = μ and σ 2 ( t ) = σ 2 . I the autocovariance function only depends on the lag and can be written as γ ( τ ) = E[ { X ( t ) μ }{ X ( t + τ ) μ } ] . This is called the autocovariance coefficient at lag τ . 4 / 16 3.2.1 Stationary Processes: SecondOrder Stationarity I This is less restrictive than strict stationarity....
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This note was uploaded on 10/19/2009 for the course MATH 611 taught by Professor Jsdkasj during the Spring '09 term at Kansas.
 Spring '09
 jsdkasj
 Correlation, Probability

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