CS计算机代考程序代写 AI Covariance Stationarity

Covariance Stationarity

General Theory

Australian National University

(James Taylor) 1 / 5

8 I

Covariance Stationarity

Focusing now on cyclical models without drift

Need to assume the underlying probabilistic structure doesn’t change

If it changes, forecasting would not be possible

Use covariance stationarity (weak second order stationarity)

(James Taylor) 2 / 5

Covariance Stationarity – Definition

Let {yt} = {. . . , y�1, y0, y1, y2, . . .} be a (doubly-infinite) sequence.
{yt} is covariance stationary if

E(yt) = µ, Cov(yt , yt�s) = g(s)

The function g : N ! R is the autocovariance function.
Note g(0) = Var(yt).

(James Taylor) 3 / 5

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Covariance Stationary – Pros and Cons

Covariance stationarity is very strict

Non-e.g. Anything with trend or seasonality

However, we can deal with the trend and seasonality so that what’s

left is covariant stationary

Often Overlooked: These methods allow us to solve the covariant

stationary part only. Still need to deal separately with

trend/seasonality etc.

(James Taylor) 4 / 5

CARMA ARMA

White Noise

White Noise {ut} is a special kind of covariant stationary sequence
where

Eut = 0, Var(ut) = s
2
u < •, Eutus = 0 Show this is indeed covariant stationary This sequence is serially uncorrelated, sometimes serially independent Examples: Sequence of iid N (0, s2), sequence of iid t(n, 0, s2). Examples: Any sequence of iid random variables with mean zero and finite variance (James Taylor) 5 / 5 E IUt m Var Ut 62 i lov ut ht s f s i 3 want tovCut i Ut s 8Ls1 forSFo GvCut i ut s IE I ut Fut Uts TtUt s1 TEl utUts O W L White Noise White Noise {ut} is a special kind of covariant stationary sequence where Eut = 0, Var(ut) = s 2 u < •, Eutus = 0 Show this is indeed covariant stationary This sequence is serially uncorrelated, sometimes serially independent Examples: Sequence of iid N (0, s2), sequence of iid t(n, 0, s2). Examples: Any sequence of iid random variables with mean zero and finite variance (James Taylor) 5 / 5 Covariance Stationarity AR and MA models Australian National University (James Taylor) 1 / 7 8.2 Using White Noise Recall: White Noise {ut} is a covariant stationary sequence where Eut = 0, Var(ut) = s 2 u < •, Eutus = 0 Nice but very limiting, no persistence But using white noise we can build interesting models Example: Moving average of white noise - MA(1) Example: Autoregressive process - AR(1) (James Taylor) 2 / 7 0 Moving Average Process Let {et} be an MA(1) process: et = ut + yut�1 Then {et} is covariant stationary with autocovariance: g(0) = Var(et) = s 2 u + y 2s2u g(1) = Cov(et , et�1) = ys 2 u g(s) = Cov(et , et�s) = 0, s = 2, 3, . . . (James Taylor) 3 / 7 Varlutt Gu IFlutko E UtUs _o for tts GV Ut.us IF utUs yo IELstl ttlutltxftlut.it0 p p e CoV EtEst E EtEs 8 0 GV Et Et Var Et test Stl IF htt xUt1 htt tht 11 lE utttxutut itxtutD 6utdx.at NGG 6utx26u Moving Average Process Let {et} be an MA(1) process: et = ut + yut�1 Then {et} is covariant stationary with autocovariance: g(0) = Var(et) = s 2 u + y 2s2u g(1) = Cov(et , et�1) = ys 2 u g(s) = Cov(et , et�s) = 0, s = 2, 3, . . . (James Taylor) 3 / 7 fLD Cov Et Et l IE Et St I I IE Iut t t Ut l ut i t XUt4 TtI utUt I t t UtUt s t t Ut t t t ut iut if O t f O t46ud tTtO IN6I Moving Average Process Let {et} be an MA(1) process: et = ut + yut�1 Then {et} is covariant stationary with autocovariance: g(0) = Var(et) = s 2 u + y 2s2u g(1) = Cov(et , et�1) = ys 2 u g(s) = Cov(et , et�s) = 0, s = 2, 3, . . . (James Taylor) 3 / 7 0 fly Gv Et Et 2 IELEt Et L IE Cut t xUi i lCUt r t tUt s1 IF utUtL t t UtUt s t t Ut IUt L t X ut IUl s OtO t OtO O Moving Average Process Let {et} be an MA(1) process: et = ut + yut�1 Then {et} is covariant stationary with autocovariance: g(0) = Var(et) = s 2 u + y 2s2u g(1) = Cov(et , et�1) = ys 2 u g(s) = Cov(et , et�s) = 0, s = 2, 3, . . . (James Taylor) 3 / 7 hmm Autoregressive Process Let {et} be an AR(1) process: et = fet�1 + ut with |f| < 1. Then {et} is covariant stationary with autocovariance: g(0) = Var(et) = s2u 1� f2 g(s) = Cov(et , et�s) = f sg(0), s = 1, 2, . . . (James Taylor) 4 / 7 IE Ut IE UtUs ofor tts LetE Et m IE Et E 01St int p IEM_Onto Gvkt.cat GvlEti Etit flo1 ewor M o Ht 8 0 Var Et GV Et Et Tt Et Et Tt 0Et lthtkfst i utl tt 02EE.it201St Ut t Ut 82.810 120 0 64 Tco 028101 64 Ko 6h it to 4 04.8101 Autoregressive Process Let {et} be an AR(1) process: et = fet�1 + ut with |f| < 1. Then {et} is covariant stationary with autocovariance: g(0) = Var(et) = s2u 1� f2 g(s) = Cov(et , et�s) = f sg(0), s = 1, 2, . . . (James Taylor) 4 / 7 811 GV Et Et y IE Et St Tt lolEt I t Ut l Et if IE ofEt I t UtEt i 08 lo t O un Autoregressive Process Let {et} be an AR(1) process: et = fet�1 + ut with |f| < 1. Then {et} is covariant stationary with autocovariance: g(0) = Var(et) = s2u 1� f2 g(s) = Cov(et , et�s) = f sg(0), s = 1, 2, . . . (James Taylor) 4 / 7 0 (James Taylor) 4 / 7 tck Gv Et Et z IE Et Et L It LofEt I tUtt Et if11 013tlol It ol St L t fUt t t Ut Et if E ol Et r t olUt i Et L t UtSt if y r lo t o t o 6 r loI Finding g for MA(1), the xcov function %MA(1) Process N = 1000; u = randn(1,N); psi = 0.7; e = zeros(1,N); e(1) = u(1); for n = 2:N e(n) = u(n) + psi*u(n�1); end [cov e,lags e] = xcov(e,10,'coeff'); stem(lags e,cov e) (James Taylor) 5 / 7 makedata Eh Unt 4 Uh i ai plot Finding g for AR(1), the xcov function %AR(1) Process N = 1000; u = randn(1,N); y = zeros(1,N); ph = 0.7; y(1) = ph*u(1); for n = 2:N y(n) = ph*y(n�1) + u(n); end [cov y,lags y] = xcov(y,10,'coeff'); stem(lags y,cov y) (James Taylor) 6 / 7 makedata yn ofyn it Un f T T T plot Visualising g (a) MA(1) Process (b) AR(1) Process Figure: Autocovariance functions with f = y = 0.7 (James Taylor) 7 / 7 I o7 THI Kil symmetric Wold Representation Theorem Australian National University (James Taylor) 1 / 6 8.3 Wold Representation Theorem Why should we care about moving average models? Because of Wold’s Representation Theorem Every covariant stationary process can be represented as a moving average process (James Taylor) 2 / 6 µA g Et Ut t t Ut I t t LUt L t t XgUt f I The Lag Operator The lag operator L acts on time series Lyt = yt�1 Similarly L2yt = LLyt = Lyt�1 = yt�2 Indeed for any polynomial B(L) where B(L) = b0 + b1L+ b2L 2 + · · ·+ bmLm =) B(L)yt = boyt + b1yt�1 + b2yt�2 + · · ·+ bmyt�m = m  i=0 biyt�i (James Taylor) 3 / 6 Do Lag Polynomials The polynomial B(L) transforms yt into a weighted sum of current and past values The classical di↵erence operator D is a lag polynomial Dyt = yt � yt�1 = (1� L)yt Infinite-order lag polynomials (really formal sums): B(L) = b0 + b1L+ b2L 2 + · · ·+ = •  i=1 biL i (James Taylor) 4 / 6 f exit x f ytLfLyt Yt Yt1 Yt Lyt or Ll 1 Yt Infinite-order Lag Polynomials B(L) = b0 + b1L+ b2L 2 + · · ·+ = •  i=1 biL i Infinite-order lag polynomials seem of little practical interest They have infinity many parameters, so cannot be estimated from finite data But particular ones may not, e.g. B(L) = 1+ bL+ b2L2 + · · · = •  i=0 biLi Also are very important theoretically in Wold’s Theorem (James Taylor) 5 / 6 bo b bz f 1 parameter TARU Wold’s Representation Theorem Wold’s Representation Theorem Theorem: Let {yt} be a zero-mean covariance stationary process. Then there is a representation yt = B(L)ut = •  i=0 biut�i where {ut} is white noise, b0 = 1 and •i=0 b2i < •. (James Taylor) 6 / 6 4 Rational Polynomials Australian National University (James Taylor) 1 / 2 8.3A Rational Polynomials The problem with Wold’s Theorem is too many parameters Sometimes we can reduce the number of parameters B(L) = 1+ bL+ b2L2 + · · · = •  i=0 biLi = 1 1� bL More generally, we have rational polynomials B(L) = Y(L) F(L) = Âqi=1 yiL i Âpj=0 fiL i Has only p + q parameters. If B(L) isn’t really rational, we can approximate with a rational (James Taylor) 2 / 2 0 o.o geometricsequence Not a EIXi Polynomial Rational Polynomials The problem with Wold’s Theorem is too many parameters Sometimes we can reduce the number of parameters B(L) = 1+ bL+ b2L2 + · · · = •  i=0 biLi = 1 1� bL More generally, we have rational polynomials B(L) = Y(L) F(L) = Âqi=1 yiL i Âpj=0 fiL i Has only p + q parameters. If B(L) isn’t really rational, we can approximate with a rational (James Taylor) 2 / 2