TS_Fall_2011_Lecture_2

TS_Fall_2011_Lecture_2 - FIN 271 FINANCIAL MODELING AND...

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FIN 271 FINANCIAL MODELING AND ECONOMETRICS TIME SERIES MODELING LECTURE SET 2 REFIK SOYER THE GEORGE WASHINGTON UNIVERSITY
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2 STOCHASTIC TIME SERIES MODELS In the general model ] œ 0Ð Ñ  t t t % when the effect of time is embodied in we have a "stochastic process" (SP) with % t Ê a probabilistic structure. What is a SP ? It is a sequence of random variables (RV's) indexed by time. In analyzing stochastic TS, we have a long sequence of random variables: á ] ] ] ] ] . , , , , , , . . . . 2 1 0 1 2 Ê an infinite series going indefinitely into the future and past. The sequence of n observations , , , is a sample from this long series. ] ] á ] 1 2 n Goal longer : to make inference about the probability structure of the series (on the basis of sample). 1
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3 LINEAR STOCHASTIC PROCESS Consider an , that is, autoregressive process of order one ]œ ] tt 1 t 9% Ö] × t - observed values of a random variable. Ö× % t - white-noise error terms; unobserved random variables. 9 - unknown constant. We can write from the above that ] ] œ œ t-1 t-2 t-1 t t-1 t t-2 t-1 t 9 % , ( ) and if we continue this way we obtain ] œ  t t t-1 t-2 t-3 23 % 9 %9 % . . . . . Ê] the effect of time is embodied in , that is, 's are correlated as a result of sharing % common error terms. A stochastic process generated by taking a linear combination of (moving average of) the white noise terms is known as a . linear process 2
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4 STATIONARY PROCESSES Wold's theorem: any stationary process can be represented as a decomposition of a linear process and a deterministic part. We will consider some special cases of linear processes. We need some simplifying assumptions about the probability structure of the longer series (about the underlying process): Stationarity Assumption. Ê • What is a Stationary TS ? The behavior of a set of random variables at one point in time is the probabilistically same as the behavior of a set at another point in time the probabilistic structure of the TS is not time-dependent. Ê • Why do we need this ? You can take your from any portion of the long series and the probability finite sample structure will be the same. Note that what we have (the sample) is one realization from the long-series. 3
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5 IMPLICATIONS OF STATIONARITY • What are the (practical) implications of stationarity ? (1) The mean (expected value) of the TS is constant over time. I ] œ ( ) for all t t . (2) The variance of the TS will be constant over time. Z +< ] œ œ ( ) for all t. t 0 2 5 # (3) The autocovariance of the TS at any lag does not depend ontime only a function of the lag (not of time). Ê G9@ ] ] œ G9@ ] ] œ ( , ) ( , ) ( , ) for all t. 1 1 k 10 10 k t t k k # Note that the autocovariance function is symmetric, that is, . # # k k œ If a TS satisfy (1), (2) and (3) then the TS is weakly stationary (covariance Ê stationary or second-order stationary). If a TS is weakly stationary and if 's are ] ] t t normally distributed, then the TS is stationary (strong stationarity).
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This note was uploaded on 02/29/2012 for the course FINA 6271 taught by Professor Phillipwirtz,refiksoyer during the Fall '11 term at GWU.

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TS_Fall_2011_Lecture_2 - FIN 271 FINANCIAL MODELING AND...

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