Class Note for STAT 635 at OSU 13
Class Note for STAT 635 at OSU 13
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Date Created: 02/06/15
STATISTICS 635 SUMMER 2005 STAT635 LECTURE OUTLINE 4 There is nothing training cannot do Nothing is above its reach It can turn bad morals to good it can destroy bad principles and recreate good ones it can lift men to angelship M ark Twain STATIONARY PROCESSES EH Making predictions in time series analysis presupposes that something does not vary with time EH In a deterministic setting a linear function is used for extrapolation if the rst derivative is found to be constant EH As we saw in the previous chapter the time invariance of aspects of a time series was captured by the fundamental concept of stationarity EH If the random component of a time series is stationary then one can develop powerful techniques to forecast its future values Let Xi be a stationary time series with mean function u ES The autocovariance function ACVF of Xi at lag h is 70L covXthXt IE mt We MI ES The autocorrelation function ACF of Xi at lag his ph corrXthXt The ACVF and the ACF play a very important role in stationary time series STATISTICAL ANALYSIS OF TIME SERIES 1 STATISTICS 635 SUMMER 2005 NONNEGATIVE DEFINITE FUNCTIONS A real valued function E de ned on the integers is nonnegative definite if ZZaZMt jaj Z 0 i1 j1 for all positive integers n and vectors a a1 a2 anT BASIC PROPERTIES OF THE ACVF 7 Let Xi be a stationary time series with autocovariance function ACVF The following hold true 53 70 VOW 2 0 EH 7h 3 70 for all h EH 70L 7 h for all h so that is even function EH is nonnegative definite WHAT FUNCTIONS QUALIFY AS ACVF OR ACF EH Theorem 1 A real valued function de ned on the integers is the ACVF ofa stationary time series if and only if it is even and nonnegative definite EH Theorem 2 A real valued function de ned on the integers is the ACF ofa stationary time series if and only if it is even and nonnegative definite and p0 l STATISTICAL ANALYSIS OF TIME SERIES 2 STATISTICS 635 SUMMER 2005 STATIONARY PROCESSES HAVE NONNEGATIVE ACVF EH Let Xi be a stationary time series with ACVF De ne Y En OZtXt tl where n is some integer and 011012 ozn are reals constants EH Using the linearity property of covariances it is easy to show that Vlti0 tXtgt Z n as at covX3 Xi 31 tl ZZasatys t 31 tl ES The above result is a standard property of autocovariance functions EH Since VY is a variance we have VY Z 0 EH Therefore ZZasatys t Z 0 31 tl for all positive integers n and vectors a 011012 oznT ES The ACVF of any stationary time series Xi is nonnegative definite FROM A NONNEGATIVE FUNCTION TO ITS STATIONARY PROCESS ES There exists a stationary time series with ACVF E if E is even realvalued and nonnegative definite This proof is lengthier than the previous one ES If E is even realvalued and nonnegative definite then there exists a sta tionary Gaussian time series Xi with mean 0 and ACVF STATISTICAL ANALYSIS OF TIME SERIES 3 STATISTICS 635 SUMMER 2005 Simple little Exercise Let Y and Z be two uncorrelated random variables both with mean 0 and variance 1 De ne Xi Ycos6t Zsin6t 1 Find the mean of the process Xi 2 Find the autocovariance function of Xi 3 Is Xi a stationary process 4 Is the function KUL cos6h a nonnegative de nite function Remark To verify that a given function is nonnegative de nite it is often easier to to nd a stationary process that has the given function as its ACVF than to verify the conditions of the above theorem STATISTICAL ANALYSIS OF TIME SERIES 4 STATISTICS 635 SUMMER 2005 STATIONARITY REVISITED De nition 1 A time series Xi is said to be weakly stationary if the vector X1 XnT and the time shifted vector X1h 7XnhT have the same mean vectors and the same covariance matrices for all integers h and n 2 1 In other words a time series Xi is weakly stationary if i aXt is independent of t ie aXt aX for all t ii 7Xt h t is independent oft for each h h is called the lag De nition 2 A time series Xi is said to be strictly stationary if the vector X1 XnT and the time shifted vector X1h 7XnhT have the same joint distribution function ie d X1 an X1h Xan for all integers h and n 2 1 Consequences of strict stationarity ES The random variable Xi are identically distributed as XtXthT 2 X1 X1hT for all t and h EH Xi is weakly stationary if ElXt lt 00 for all t EH Weak stationarity does not imply strict stationarity EH An iid sequence is strictly stationary Note Remember that stationarity is crucial to forecasting STATISTICAL ANALYSIS OF TIME SERIES 5 STATISTICS 635 SUMMER 2005 EXCURSION To THE GAUSSIAN WORLD The multivariate normal distribution is one of the most important distribution in the whole eld of statistics and also plays an important role in time series analysis ES Let X X1X2 XnT be a random vector X has a multivariate normal distribution or multivariate Gaussian distribution with mean vector L and variance covariance matrix 2 if m lt2wgt 2ltdetlt2gtgt12exp ax mTz x m EH Notation X N NL 2 EH If X has a multivariate normal distribution then the conditional distribution of any set of components given any order is again multivariate normal EH For instance we can partition as follows X 1 and L lEX XO 2 As a result 2 E E covX 11 12 E312 2322 where IN lEXi and 22 lEXi uiXj INUT EH X and X9 are independent if and only if 212 0 ES The conditional distribution of X given Xe X0 is N 0 21222721062 Ital E311 2122521221 So clearly EX1X2 X2 1 2122231062 0 STATISTICAL ANALYSIS OF TIME SERIES 6 STATISTICS 635 SUMMER 2005 SIMPLE LITTLE EXERCISE Let XT X1X2 be a 2 x 1 random vector Let X have a bivariate normal distribution with mean vector LT 1 ml and covariance matrix 2 011 012 012 022 1 Find the conditional mean IEX2X1 X1 2 Find the conditional variance VX2X1 X1 GAUSSIAN TIME SERIES EH Xi is a Gaussian time series if all of its joint distributions are multivariate normal ie if for any collection of integers t1t239 tn the random vector Xi 1 22 XZn has a multivariate normal distribution EH If Xi is a Gaussian time series then all of its joint distributions are com pletely determined by the mean function Mt ElXt and the autocovariance function 8t covX3 Xi EH If the process is stationary then ut u and Mt ht 70L for all t so that X1 XnT i X1h XnhT for all integers h and n gt 0 EH For a Gaussian time series strict stationarity is equivalent to weak station arity STATISTICAL ANALYSIS OF TIME SERIES 7 STATISTICS 635 SUMMER 2005 ON THE ROLE OF THE ACF IN PREDICTION EH Suppose that Xi is a stationary Gaussian time series with ACF p mean u and variance 02 ES Suppose that we have observed Xn EH Goal Find the function of Xn that gives us the best predictor of Xnh BE A natural and computationally convenient of best 7 is to specify our required predictor to be the function of Xn that minimizes the mean squared error mXn arg min lEXnh mXn2 mEH Fact 1 The function m of Xn that minimizes ElXnh mXn2l is the conditional mean mXn Eanthnl u phXn M With the corresponding mean squared error being new mltXn2l 02a paw PROOF D STATISTICAL ANALYSIS OF TIME SERIES 8 STATISTICS 635 SUMMER 2005 ASPECTS OF PREDICTION WITH GAUSSIAN TIME SERIES EH For Gaussian time series prediction of Xnh in terms of Xn is more accurate as ph gets closer to 1 EH In the limit of p gt ll the mean squared error approaches 0 Remarks ES The above predictor relies on the joint normality of Xnh and Xn PREDICTION WITH NON GAUSSIAN TIME SERIES EH For time series with nonnormal joint distributions the derivation of the best predictor is much more complicated EH Question What do practitioners do EH Popular wisdom Do not look for the best predictor lnstead look for the best linear predictor which is the best predictor of the form Xn aXn b EH ln MSE terms 6Xn arg lEXnh aXn by ES It turns out that the best linear predictor is 13Xn u phXn M EH Xn depends only on the mean and the ACF of the process Xi EH Xn can be derived without detailed knowledge of the joint distributions EH If Xi is Gaussian Xn and mXn are the same In general MSEltmltXngtgt MSElt ltXngtgt STATISTICAL ANALYSIS OF TIME SERIES 9 STATISTICS 635 SUMMER 2005 SOME USEFUL TERMINOLOGY De nition 3 A stationary time series Xi is said to be q dependent if X8 and Xi are independent whenever t 5 gt q EH Example An iid sequence is O dependent De nition 4 A stationary time series Xi is said to be q correlated if7h 0 whenever h gt q For example BE A white noise process is 0 correlated ES The MA1 process is I correlated CONSTRUCTION OF STRICTLY STATIONARY TIME SERIES EH Let Z be an iid sequence which we know to be strictly stationary EH De ne the series Xi as follows Xt gZt Ztih 39 quot 7Zt7q for some real valued function g ES The time series Xi is said to constructed by ltering the iid sequence Zt using filter EH Clearly Xi is strictly stationary since Zth7 39 39 39 7ZthiqlT i Z757 39 39 39 7Ztiq for all integers h EH Xi is clearly q dependent Remark The above method provides one of the simplest ways to construct time series that are strictly stationary STATISTICAL ANALYSIS OF TIME SERIES 10 STATISTICS 635 SUMMER 2005 THE MAq PROCESS AND RELATED RESULTS EH Let Zt be a WN002 process Let 60 1 Let q be any integer and consider the real constants 01 02 64 with 6g y 0 De ne X Z 61Zt71 62Zt72 39 39 39 qutiq OOZI 61Zt71 62Zt72 39 39 39 qutiq q ZOJZH j0 EH Xi is known as the moving average of order q or the MAq process ES The MAq process is q correlated Theorem 3 If Xi is a stationary q correlated time series with a zero mean then it can be represented as a MAq process LINEAR PROCESSES BE A set of real constants IJj j E Z is absolutely summable if 2 M lt 00 j7oo ES The time series Xi is a linear process if it has the representation X Z TJthej 1700 for all t where Z N WN0 02 and IJj is absolutely summable EH We can write Xi as a linear combination of all the noise terms the past current and future EH Absolute summability of the constants guarantees that the in nite sum con verges STATISTICAL ANALYSIS OF TIME SERIES 11 STATISTICS 635 SUMMER 2005 Proposition 1 Let be a stationary time series With mean 0 and covariance function 7y 1162700 lt 00 then the time series Xt Z MYH WBWt F700 is stationary With mean 0 and au tocovariance function 00 00 7X00 Z Z j kwh k J j7ooj7oo In the special case Where Xi is a linear process 7X00 Z j jh02 j7oo B is the backward shift operator BXt XH STATISTICAL ANALYSIS OF TIME SERIES 12
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