asymptoticsprimerslides2

asymptoticsprimerslides2 - Time Series Concepts A...

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Time Series Concepts A stochastic process { Y t } t =1 is a sequence of random variables indexed by time t : { ...,Y 1 ,Y 2 ,...,Y t ,Y t +1 ,... } A realization of a stochastic process is the sequence of observed data { y t } t =1 : { ...,Y 1 = y 1 ,Y 2 = y 2 ,...,Y t = y t ,Y t +1 = y t +1 ,... }
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We are interested in the conditions under which we can treat the stochastic process like a random sample, as the sample size goes to in f nity. Under such conditions, at any point in time t 0 , the ensemble average 1 N N X k =1 Y ( k ) t 0 will converge to the sample time average 1 T T X t =1 Y t as N and T go to in f nity. If this result occurs then the stochastic process is called ergodic .
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Stationary Stochastic Processes De f nition 1 Strict stationarity A stochastic process { Y t } t =1 is strictly stationary if, for any given f nite integer r and for any set of subscripts t 1 ,t 2 ,...,t r the joint distribution of ( Y t ,Y t 1 ,Y t 2 ,...,Y t r ) depends only on t 1 t, t 2 t,. ..,t r t but not on t.
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Remarks 1. For example, the distribution of ( Y 1 ,Y 5 ) is the same as the distribution of ( Y 12 ,Y 16 ) . 2. For a strictly stationary process, Y t has the same mean, variance (mo- ments) for all t. 3. Any function/transformation g ( · ) of a strictly stationary process, { g ( Y t ) } is also strictly stationary.
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Example 1 iid sequence If { Y t } is an iid sequence, then it is strictly stationary. Let { Y t } be an iid sequence and let X N (0 , 1) independent of { Y t } . Let Z t = Y t + X. Then the sequence { Z t } is strictly stationary. Since { Z t } is strictly stationary, { Z 2 t } is also strictly stationary.
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De f nition 2 Covariance (Weak) stationarity A stochastic process { Y t } t =1 is covariance stationary (weakly stationary) if 1. E [ Y t ]= μ does not depend on t 2. cov ( Y t .Y t j )= γ j exists, is f nite, and depends only on j but not on t for j =0 , 1 , 2 ,... Remark: A strictly stationary process is covariance stationary if the mean and variance exist and the covariances are f nite.
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For a weakly stationary process { Y t } t =1 de f ne the following moments: γ j =c o v ( Y t ,Y t j )= j th order autocovariance γ 0 =v a r ( Y t )= variance ρ j = γ j 0 = j th order autocorrelation Remark A weakly stationary process is uniquely determined by its mean, variance and autocovariances.
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De f nition 3 Ergodicity Loosely speaking, a stochastic process { Y t } t =1 is ergodic if any two collections of random variables partitioned far apart in the sequence are almost indepen- dently distributed. The formal de f nition of ergodicity is highly technical (see Hayashi 2000, p. 101 and note typo from errata).
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asymptoticsprimerslides2 - Time Series Concepts A...

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