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# Introduction to Time Series STAT 153

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This 17 page Class Notes was uploaded by Floy Kub on Thursday October 22, 2015. The Class Notes belongs to STAT 153 at University of California - Berkeley taught by P. Bartlett in Fall. Since its upload, it has received 13 views. For similar materials see /class/226725/stat-153-university-of-california-berkeley in Statistics at University of California - Berkeley.

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Date Created: 10/22/15

Introduction to Time Series Analysis Lecture 12 Peter Bartlett Review Time series modelling and forecasting Parameter estimation Maximum likelihood estimator YuleWalker estimation YuleWalker estimation example Review Lecture 1 Time series modelling and forecasting 1 Plot the time series Look for trends seasonal components step changes outliers 2 Transform data so that residuals are stationary a Remove trend and seasonal components b Differencing c Nonlinear transformations log 7 3 Fit model to residuals 4 Forecast time series by forecasting residuals and inverting any transformations Review Time series modelling and forecasting Stationary time series models ARMApq p 0 MAM o q 02 ARp We have seen that any causal invertible linear process has an MAoo representation from causality and an ARoo representation from invertibility Real data cannot be exactly modelled using a nite number of parameters We choose p q to give a simple but accurate model Review Time series modelling and forecasting How do we use data to decide on p q 1 Use sample ACFPACF to make preliminary choices of model order 2 Estimate parameters for each of these choices 3 Compare predictive accuracycomplexity of each using eg AIC NB We need to compute parameter estimates for several different model orders Thus recursive algorithms for parameter estimation are important We ll see that some of these are identical to the recursive algorithms for forecasting Review Time series modelling and forecasting Model ACF PACF ARp decays zero for h gt p MAq zero for h gt q decays ARMApq decays decays Introduction to Time Series Analysis Lecture 12 Review Time series modelling and forecasting Parameter estimation Maximum likelihood estimator YuleWalker estimation YuleWalker estimation example Parameter estimation I We want to estimate the parameters of an ARMApq model We will assume for now that l The model order p and q is known and 2 The data has zero mean If 2 is not a reasonable assumption we can subtract the sample mean 3 t a zeromean ARMA model BXt 9BWta to the meancorrected time series Xt Yt g and then use Xt g as the model for Yt Parameter estimation Maximum likelihood estimatorI One approach Assume that X is Gaussian that is BXt 6BWt where Wt is iid Gaussian Choose Bj to maximize the likelihood L 6a2 fX1Xn where f is the joint Gaussian density for the given ARMA model of choosing the parameters that maximize the probability of the data Maximum likelihood estimation I Suppose that X1 X2 Xn is drawn from a zero mean Gaussian ARMApq process The likelihood of parameters d E R19 6 E Rq 0121 E R is de ned as the density ofX X1 X2 Xn under the Gaussian model with those parameters 1 1 Lqgt6a exp XT1X mm2 M 2 Where A denotes the determinant of a matrix A and Tn is the variancecovariance matrix of X with the given parameter values The maximum likelihood estimator MLE of o 6 0121 maximizes this quantity Parameter estimation Maximum likelihood estimatorI Advantages of MLE Ef cient low variance estimates Often the Gaussian assumption is reasonable Even if X is not Gaussian the asymptotic distribution of the estimates 62 is the same as the Gaussian case Disadvantages of MLE Di icult optimization problem Need to choose a good starting point often use other estimators for this Preliminary parameter estimates I YuleWalker for ARp Regress Xt onto Xt1 Xtp DurbinLevinson algorithm with 7 replaced by amp YuleWalker for ARMApq Method of moments Not ef cient Innovations algorithm for MAq with 7 replaced by amp HannanRissanen algorithm for ARMApq 1 Estimate highorder AR 2 Use to estimate unobserved noise Wt 3 Regress Xt onto Xt1 Xtp W154 Wtq 4 Regress again with improved estimates of Wt YuleWalker estimation I For a causal ARp model BXt Wt we have 29 E Xt7 Xt Z EXt7Wt fOI39Y 07 p j1 ltgt 70 d7 02 and 7p Fp 07 where d 1 p and we ve used the causal representation Xt Wt l ijWt j 11 to calculate the values EXt Wt YuleWalker estimation I Method of moments We choose parameters for which the moments are equal to the empirical moments In this case we choose d so that 7 W YuleWalker equations for o These are the forecasting equations We can use the DurbinLevinson algorithm Some facts about YuleWalker estimation I o If amp0 gt 0 then fm is nonsingular o In that case d3 l WP de nes the causal model Xt 1th 19pr Wt Wt WN0 62 o If X is an ARp process Qgt w q 2 A 039 1 7 A 1 hh N AN 0 E for h gt p Thus we can use the sample PACF to test for AR order and we can calculate approximate con dence intervals for the parameters p YuleWalker estimation Con dence intervals I If X is an ARp process and n is large 0 wfp op is approximately NO 6213f o with probability 1 04 pm is in the interval q H 6 ml2 19 l aZ p jj 7 Where ltIgt1a2 is the 1 12 quantile of the standard normal YuleWalker estimation Con dence intervals I o with probability 1 oz op is in the ellipsoid A A A 52 a5 6 RP asp q Pp qbp ab Vim where xfap is the 1 oz quantile of the chisquared with p degrees of freedom To see this notice that A A 2 var Pyle2W1 19 P1192 3er pF192 0 wI n Thus U P1192ampP 19 N N06wn1 and so A 221 N X209 0w

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