Formulas and survey Time Series Analisys.pdf

8/2/2019 Formulas and survey Time Series Analisys.pdf

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Avd. Matematisk statistik

Formulas and survey

Time series analysis

Jan Grandell


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INNEHALL 1

Innehall

1 Some notation 2

2 General probabilistic formulas 2

2.1 Some distributions . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Stochastic processes 5

4 Stationarity 6

5 Spectral theory 7

6 Time series models 86.1 ARMA processes . . . . . . . . . . . . . . . . . . . . . . . . . . 9

6.2 ARIMA and FARIMA processes . . . . . . . . . . . . . . . . . . 116.3 Financial time series . . . . . . . . . . . . . . . . . . . . . . . . 11

7 Prediction 127.1 Prediction for stationary time series . . . . . . . . . . . . . . . . 127.2 Prediction of an ARMA Process . . . . . . . . . . . . . . . . . . 14

8 Partial correlation 148.1 Partial autocorrelation . . . . . . . . . . . . . . . . . . . . . . . 14

9 Linear filters 14

10 Estimation in time series 1510.1 Estimation of . . . . . . . . . . . . . . . . . . . . . . . . . . . 1510.2 Estimation of () and () . . . . . . . . . . . . . . . . . . . . . 1610.3 Estimation of the spectral density . . . . . . . . . . . . . . . . . 17

10.3.1 The periodogram . . . . . . . . . . . . . . . . . . . . . . 1710.3.2 Smoothing the periodogram . . . . . . . . . . . . . . . . 17

11 Estimation for ARMA models 1911.1 Yule-Walker estimation . . . . . . . . . . . . . . . . . . . . . . . 1911.2 B urgs algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 19

11.3 The innovations algorithm . . . . . . . . . . . . . . . . . . . . . 2011.4 The HannanRissanen algorithm . . . . . . . . . . . . . . . . . 2111.5 Maximum Likelihood and Least Square estimation . . . . . . . . 2211.6 O rder selection . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

12 Multivariate time series 23

13 Kalman filtering 25

Index 28


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1 SOME NOTATION 2

1 Some notation

R = (, )Z = {0, 1, 2, . . . }C = The complex numbers =

{x + iy; x

R, y

R

}def= means equality by definition.

2 General probabilistic formulas

(, F, P) is a probability space, where: is the sample space, i.e. the set of all possible outcomes of an experiment.

F is a -field (or a -algebra), i.e.(a) F;

(b) ifA1, A2, Fthen1

Ai F;

(c) ifA Fthen Ac F.P is a probability measure, i.e. a function F [0, 1] satisfying(a) P() = 1;

(b) P(A) = 1 P(Ac);

(c) ifA1, A2, Fare disjoint, then P1 Ai = 1 P(Ai).Definition 2.1 A random variable X defined on (, F, P) is a function R such that { : X() x} F for all x R.

Let X be a random variable.

FX (x) = P{X x} is the distribution function (fordelningsfunktionen).fX (), given by FX (x) =

x

fX (y) dy, is the density function(tathetsfunktionen).

pX (k) = P{X = k} is the probability function (sannolikhetsfunktionen).X (u) = E

eiuX

is the characteristic function (karakteristiska funktionen).

Definition 2.2 Let X1, X2, . . . be a sequence of random variables. We say

that Xn converges in probability to the real number a, written XnP a, if for

every > 0,lim

nP(|Xn a| > ) = 0.


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2 GENERAL PROBABILISTIC FORMULAS 3

Definition 2.3 Let X1, X2, . . . be a sequence of random variables with finitesecond moment. We say that Xn converges in mean-square to the randomvariable X, written Xn

m.s. X, if

E[(Xn

X)2]

0 as n

.

An important property of mean-square convergence is that Cauchy-sequencesdo converge. More precisely this means that if X1, X2, . . . have finite secondmoment and if

E[(Xn Xk)2] 0 as n, k ,then there exists a random variable X with finite second moment such thatXn

m.s. X.The space of square integrable random variables is complete under mean-squareconvergence.

2.1 Some distributions

The Binomial Distribution X Bin(n, p) if

pX (k) =

n

k

pk(1 p)nk, k = 0, 1, . . . , n and 0 < p < 1.

E(X) = np, Var(X) = np(1 p).The Poisson Distribution X Po() if

pX (k) = k

k!e, k = 0, 1, . . . and > 0.

E(X) = , Var(X) = , X (u) = e(1eiu).

The Exponential Distribution X Exp() if

fX (x) =

1

ex/ if x 0,0 if x < 0

> 0.

E(X) = , Var(X) = 2.

The Standard Normal Distribution X N(0, 1) if

fX (x) =12

ex2/2, x R.

E(X) = 0, Var(X) = 1, X (u) = eu2/2 .

The density function is often denoted by () and the distribution function by().


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2 GENERAL PROBABILISTIC FORMULAS 4

The Normal Distribution X N(, 2) ifX

N(0, 1), R, > 0.

E(X) = , Var(X) = 2, X (u) = eiueu2

2

/2 .The (multivariate) Normal DistributionY= (Y1, . . . , Y m)

N(, ), if there exists

a vector =

1...m

, a matrix B = b11 . . . b1n...

bm1 . . . bmn

with = BB and a random vector X= (X1, . . . , X n)

with independent and N(0, 1)-distributedcomponents, such that Y= + BX. If

Y1Y2

N

12

,

21 12

12 22

,

then

Y1 conditional on Y2 = y2 N

1 +12

(y2 2), (1 2)21

.

More generally, if

Y1Y2 N12 , 11 1221 22 ,then Y1 conditional on Y2 = y2

N1 + 12122 (y2 2), 11 12122 21 .Asymptotic normality

Definition 2.4 LetY1, Y2, . . . be a sequence of random variables.Yn AN(n, 2n) means that

limnPYn nn x = (x).Definition 2.5 LetY1,Y2, . . . be a sequence of random k-vectors.Yn AN(n, n) means that(a) 1, 2, . . . have no zero diagonal elements;

(b) Yn AN(n,n) for every Rk such thatn > 0 for allsufficiently large n.


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3 STOCHASTIC PROCESSES 5

2.2 Estimation

Let x1, . . . , xn be observations of random variables X1, . . . , X n with a (known)distribution depending on the unknown parameter . A point estimate (punkt-

skattning) of is then the value (x1, . . . , xn). In order to analyze the estimatewe consider the estimator (stickprovsvariabeln) (X1, . . . , X n). Some nice pro-perties of an estimate are the following:

An estimate of is unbiased (vantevardesriktig) ifE((X1, . . . , X n)) = for all .

An estimate of is consistent if P(|(X1, . . . , X n) | > ) 0 forn .

If

and are unbiased estimates of we say that

is more effective

than if Var(

(X1, . . . , X n)) Var((X1, . . . , X n)) for all .

3 Stochastic processes

Definition 3.1 (Stochastic process) A stochastic process is a family ofrandom variables {Xt, t T} defined on a probability space (, F, P).

A stochastic process with T Z is often called a time series.Definition 3.2 (The distribution of a stochastic process) Put

T=

{t

Tn : t1 < t2 1.

and spectral density

f() =2

2(1 + 2 + 2 cos()), .

6.2 ARIMA and FARIMA processes

Definition 6.9 (The ARIMA(p, d, q) process) Letd be a non-negative in-teger. The process {Xt, t Z} is said to be an ARIMA(p, d, q) process if(1 B)dXt is a causal ARMA(p, q) process.

Definition 6.10 (The FARIMA(p, d, q) process) Let 0 < |d| < 0.5. Theprocess {Xt, t Z} is said to be a fractionally integrated ARMA process or aFARIMA(p, d, q) process if {Xt} is stationary and satisfies

(B)(1 B)dXt = (B)Zt, {Zt} WN(0, 2).

6.3 Financial time series

Definition 6.11 (The ARCH(p) process) The process {Xt, t Z} is saidto be an ARCH(p) process if it is stationary and if

Xt = tZt, {Zt} IID N(0, 1),

where2t = 0 + 1X

2t1 + . . . + pX

2tp

and 0 > 0, j 0 for j = 1, . . . , p, and if Zt and Xt1, Xt2, . . . are indepen-dent for all t.

Definition 6.12 (The GARCH(p, q) process) The process {Xt, t Z} issaid to be an GARCH(p, q) process if it is stationary and if

Xt = tZt, {Zt} IID N(0, 1),

where2t = 0 + 1X

2t1 + . . . + pX

2tp + 1

2t1 + . . . + q

2tq

and 0 > 0, j 0 for j = 1, . . . , p, k 0 for k = 1, . . . , q, and if Zt andXt1, Xt2, . . . are independent for all t.


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7 PREDICTION 12

7 Prediction

Let X1, X2, . . . , X n and Y be any random variables with finite means andvariances. Put i = E(Xi), = E(Y),

n =1,1 . . . 1,n...

n,1 . . . n,n

= Cov(X1, X1) . . . Cov(X1, Xn)...Cov(Xn, X1) . . . Cov(Xn, Xn)

and

n =

1...n

=Cov(X1, Y)...

Cov(Xn, Y)

.Definition 7.1 The best linear predictor

Y of Y in terms of X1, X2, . . . , X n

is a random variable of the formY = a0 + a1X1 + . . . + anXnsuch that E

(Y Y)2 is minimized with respect to a0, . . . , an. E(Y Y)2 is

called the mean-squared error.

It is often convenient to use the notation

Psp{1, X1,...,Xn}Ydef= Y .

The predictor is given by

Y = + a1(X1 1) + . . . + an(Xn n)where

an =

a1...an

satisfies n = nan. If n is non-singular we have an =

1n n.

There is no restriction to assume all means to be 0. The predictor

Y of Y is determined by

Cov(Y Y, Xi) = 0, for i = 1, . . . , n .7.1 Prediction for stationary time series

Theorem 7.1 If{Xt} is a zero-mean stationary time series such that(0) > 0and (h) 0 as h , the best linear predictor Xn+1 of Xn+1 in terms ofX1, X2, . . . , X n is

Xn+1 =

ni=1

n,iXn+1i, n = 1, 2, . . . ,


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7 PREDICTION 13

where

n =

n,1...

n,n

= 1n n, n =

(1)...

(n)

and

n =(1 1) . . . (1 n)...

(n 1) . . . (n n)

.The mean-squared error is vn = (0) n1n n.

Theorem 7.2 (The DurbinLevinson Algorithm) If{Xt} is a zero-meanstationary time series such that (0) > 0 and (h) 0 as h , then1,1 = (1)/(0), v0 = (0),

n,n = (n) n1j=1

n1,j(n j)v1n1 n,1...

n,n1

= n1,1...

n1,n1

n,nn1,n1...

n1,1

and

vn = vn1[1 2n,n].

Theorem 7.3 (The Innovations Algorithm) If {Xt} has zero-mean andE(XiXj) = (i, j), where the matrix (1,1) ... (1,n)...

(n,1) ... (n,n)

is non-singular, we haveXn+1 =

0 if n = 0,nj=1

n,j(Xn+1j Xn+1j ) if n 1, (6)and

v0 = (1, 1),

n,nk = v1

k(n + 1, k + 1) k1

j=0

k,kj n,nj vj, k = 0, . . . , n 1,vn = (n + 1, n + 1)

n1j=0

2n,nj vj .


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8 PARTIAL CORRELATION 14

7.2 Prediction of an ARMA Process

Let {Xt} be a causal ARMA(p, q) process defined by (B)Xt = (B)Zt. Then

Xn+1 =

n

j=1 n,j (Xn+1j Xn+1j) if 1 n < m,1Xn + + pXn+1p+

qj=1

n,j (Xn+1j Xn+1j) if n m,where m = max(p, q). The nj:s are obtained by the innovations algorithmapplied to

Wt = 1Xt, if t = 1, . . . , m ,

Wt = 1(B)Xt, if t > m.

8 Partial correlationDefinition 8.1 Let Y1, Y2 and W1, . . . , W k be random variables. The partialcorrelation coefficient of Y1 and Y2 with respect to W1, . . . , W k is defined by

(Y1, Y2)def= (Y1 Y1, Y2 Y2),

where Y1 = Psp{1,W1,...,Wk}Y1 and Y2 = Psp{1,W1,...,Wk}Y2.8.1 Partial autocorrelation

Definition 8.2 Let{

Xt, t

Z

}be a zero-mean stationary time series. The

partial autocorrelation function (PACF) of {Xt} is defined by(0) = 1,

(1) = (1),

(h) = (Xh+1 Psp{X2,...,Xh}Xh+1, X1 Psp{X2,...,Xh}X1), h 2.Theorem 8.1 Under the assumptions of Theorem 7.2 (h) = h,h forh 1.

9 Linear filters

A filter is an operation on a time series {Xt} in order to obtain a new time series{Yt}. {Xt} is called the input and {Yt} the output. The following operation

Yt =

k=

ct,kXk

defines a linear filter. A filter is called time-invariant if ct,k depends only ont k, i.e. if

ct,k = htk.


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10 ESTIMATION IN TIME SERIES 15

A time-invariant linear filter (TLF) is said to by causal if

hj = 0 for j < 0,

A TLF is called stable if

|hk

| 0 we define the innovations estimates

m = m1...mm and vm, m = 1, 2, . . . , n 1,


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11 ESTIMATION FOR ARMA MODELS 21

by the recursion relations

v0 =(0),m,mk =v

1k (m k)

k1

j=0m,mjk,kjvj , k = 0, . . . , m 1,vm =(0) m1j=0

2m,mjvj.This method works also for causal invertible ARMA processes. The followingtheorem gives asymptotic statistical properties of the innovations estimates.

Theorem 11.2 Let {Xt} be the causal invertible ARMA process (B)Xt =(B)Zt, {Zt} IID(0, 2), EZ4t < , and let(z) =

j=0 jz

j = (z)(z) , |z| 1(with0 = 1 and j = 0 for j < 0). Then for any sequence of positive integers{m(n), n = 1, 2, . . . } such that m and m = o(n1/3) as n , we have

for each fixedk, m1...mk AN

1...

k

, n1A ,

where A = (aij)i,j=1,...,k and

aij =

min(i,j)r=1

irjr.

Moreover, vm P 2.11.4 The HannanRissanen algorithm

Let {Xt} be an ARMA(p, q) process:Xt 1Xt1 . . . pXtp = Zt + 1Zt1 + . . . + qZtq, {Zt} IID(0, 2).The HannanRissanen algorithm consists of the following two steps:

Step 1

A high order AR(m) model (with m > max(p, q)) is fitted to the data by Yule-Walker estimation. If m1, . . . ,mm are the estimated coefficients, then Zt isestimated byZt = Xt m1Xt1 . . . mmXtm, t = m + 1, . . . , n .Step 2

The vector = (,) is estimated by least square regression of Xt onto

Xt1, . . . , X tp,

Zt1, . . . ,

Ztq,


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11 ESTIMATION FOR ARMA MODELS 22

i.e. by minimizing

S() =n

t=m+1

(Xt 1Xt1 . . . pXtp 1

Zt1 . . . q

Ztq)

2

with respect to . This gives the HannanRissanen estimator = (ZZ)1ZXn provided ZZ is non-singular,where

Xn =

Xm+1...Xn

and

Z = Xm Xm1 . . . X mp+1 Zm Zm1 . . . Zmq+1... ...Xn1 Xn2 . . . X np Zn1 Zn2 . . . Znq .

The HannanRissanen estimate of the white noise variance 2 is

2HR = S()n m .11.5 Maximum Likelihood and Least Square estimation

It is possible to obtain better estimates by the maximum likelihood method

(under the assumption of Gaussian processes) or by the least square method.In the least square method we minimize

S(,) =n

j=1

(Xj Xj)2rj1

,

where rj1 = vj1/2, with respect to and . The estimates has to be

obtained by recursive methods, and the estimates discussed are natural startingvalues. The least square estimate of 2 is

2LS = S(LS,LS)n p q ,where (LS,LS) is the estimate obtained by minimizing S(,).Let us assume, or at least act as if, the process is Gaussian. Then, for any fixedvalues of, , and 2, the innovations X1 X1, . . . , X n Xn are independentand normally distributed with zero means and variances v0 =

2r0, . . . , vn1 =2rn1. The likelihood function is then

L(, , 2) =n

j=1

fXj bXj(Xj

Xj) =

nj=1

1

22rj1

exp

(Xj

Xj)222rj1

.


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12 MULTIVARIATE TIME SERIES 23

Proceeding in the usual wayae get

ln L(,, 2) = 12

ln((22)nr0 rn1) S(, )22

.

Obviously r0, . . . , rn1 depend on and but they do not depend on

2

. Tomaximize ln L(, , 2) is the same as to minimize

(, ) = ln(n1S(,)) + n1n

j=1

ln rj1,

which has to be done numerically.In the causal and invertible case rn 1 and therefore n1

nj=1 ln rj1 is

asymptotically negligible compared with ln S(,). Thus both methods leastsquare and maximum likelihood give asymptotically the same result in thatcase.

11.6 Order selection

Assume now that we want to fit an ARMA(p, q) process to real data, i.e. wewant to estimate p, q, (, ), and 2. We restrict ourselves to maximum li-kelihood estimation. Then we maximize L(, , 2), or which is the same minimize 2 ln L(, , 2), where L is regarded as a function also of p andq. Most probably we will get very high values of p and q. Such a model willprobably fit the given data very well, but it is more or less useless as a mathe-matical model, since it will probably not be lead to reasonable predictors nordescribe a different data set well. It is therefore natural to introduce a penaltyfactorto discourage the fitting of models with too many parameters. Insteadof maximum likelihood estimation we may apply the AICC Criterion:

Choose p, q, and (p, q), to minimize

AICC = 2 ln L(p, q, S(p,q)/n) + 2(p + q + 1)n/(n p q 2).(The letters AIC stand for Akaikes Information Criterionand the last C forbiased-Corrected.)The AICC Criterion has certain nice properties, but also its drawbacks. Ingeneral one may say the order selection is genuinely difficult.

12 Multivariate time series

Let

Xtdef=

Xt1...Xtm

, t Z,where each component is a time series. In that case we talk about multivariatetime series.


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12 MULTIVARIATE TIME SERIES 24

The second-order properties of{Xt} are specified by the mean vector

tdef= EXt =

t1

...

tm

=

EXt1

...EX

tm

, t Z,

and the covariance matrices

(t+h, t)def= E[(Xt+ht+h)(Xtt)] =

11(t + h, t) . . . 1m(t + h, t)...m1(t + h, t) . . . mm(t + h, t)

where ij(t + h, t)

def= Cov(Xt+h,i, Xt,j).

Definition 12.1 The m-variate time series {Xt, t Z} is said to be (weakly)stationary if

(i) t = for all t Z,(ii) (r, s) = (r + t, s + t) for all r,s,t Z.Item (ii) implies that (r, s) is a function ofrs, and it is convenient to define

(h)def= (h, 0).

Definition 12.2 (Multivariate white noise) An m-variate process

{Zt, t

Z

}is said to be a white noise with mean and covariance matrix | , written

{Zt} WN(, | ),

if EZt = and (h) =

| if h = 0,0 if h = 0.

Definition 12.3 (The ARMA(p, q) process) The process {Xt, t Z} issaid to be an ARMA(p, q) process if it is stationary and if

Xt 1Xt1 . . . pXtp = Zt + 1Zt1 + . . . + qZtq, (12)where {Zt} WN(0, | ). We say that {Xt} is an ARMA(p, q) process withmean if {Xt } is an ARMA(p, q) process.

Equations (12) can be written as

(B)Xt = (B)Zt, t Z,where

(z) = I 1z . . . pzp,


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13 KALMAN FILTERING 25

(z) = I+ 1z + . . . + qzq,

are matrix-valued polynomials.Causality and invertibility are characterized in terms of the generating poly-nomials:

Causality: Xt is causal if det (z) = 0 for all |z| 1;Invertibility: Xt is invertible if det (z) = 0 for all |z| 1.Assume that

h=

|ij(h)| < , i, j = 1, . . . , m . (13)

Definition 12.4 (The cross spectrum) Let {Xt, t Z} be an m-variatestationary time series whose ACVF satisfies (13). The function

fjk () =

1

2

h=

eih

jk (h), , j = k,

is called the cross spectrum or cross spectral density of{Xtj} and{Xtk}. Thematrix

f() =

f11() . . . f 1m()...fm1() . . . f mm()

is called the spectrum or spectral density matrix of {Xt}.The spectral density matrix f() is non-negative definite for all

[

, ].

13 Kalman filtering

We will use the notation

{Zt} WN(0, {| t}),

to indicate that the process {Zt} has mean 0 and that

EZsZt =

| t if s = t,0 otherwise.

Notice this definition is an extension of Definition 12.2 on the page before inorder to allow for non-stationarity.A state-space model is defined by

the state equation

Xt+1 = FtXt + Vt, t = 1, 2, . . . , (14)


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where{Xt} is a v-variate process describing the state of some system,{Vt} WN(0, {Qt}), and{Ft} is a sequence of v v matrices

andthe observation equation

Yt = GtXt + Wt, t = 1, 2, . . . , (15)

where{Yt} is a w-variate process describing the observed state of some system,{Wt} WN(0, {Rt}), and{Gt} is a sequence of w v matrices.Further {Wt} and {Vt} are uncorrelated. To complete the specification it isassumed that the initial state X1 is uncorrelated with {Wt} and {Vt}.Definition 13.1 (State-space representation) A time series {Yt} has astate-space representation if there exists a state-space model for {Yt} as spe-cified by equations (14) and (15).

PutPt(X)

def= P(X | Y0, . . . ,Yt),

i.e. the vector of best linear predictors ofX1, . . . , X v in terms of all componentsofY0, . . . ,Yt.Linear estimation ofXt in terms of

Y0, . . . ,Yt1 defines the prediction problem; Y0, . . . ,Yt defines the filtering problem; Y0, . . . ,Yn, n > t, defines the smoothing problem.

Theorem 13.1 (Kalman Prediction) The predictors Xt def= Pt1(Xt) andthe error covariance matrices

tdef= E[(Xt Xt)(Xt Xt)]

are uniquely determined by the initial conditions

X1 = P(X1 | Y0), 1 def= E[(X1

X1)(X1

X1)

]

and the recursions, for t = 1, . . .,Xt+1 = FtXt + t1t (Yt GtXt) (16)t+1 = FttF

t + Qt t1t t, (17)

where

t = GttGt + Rt,

t = FttGt.

The matrix t1t is called the Kalman gain.


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Theorem 13.2 (Kalman Filtering) The filtered estimates Xt|tdef= Pt(Xt)

and the error covariance matrices

t|tdef= E[(Xt Xt|t)(Xt Xt|t)]

are determined by the relations

Xt|t = Pt1(Xt) + tGt

1t (Yt GtXt)

andt|t+1 = t tGt1t Gtt.

Theorem 13.3 (Kalman Fixed Point Smoothing) The smoothed estimates

Xt|ndef= Pn(Xt) and the error covariance matrices

t|ndef= E[(Xt

Xt|n)(Xt

Xt|n)

]

are determined for fixed t by the recursions, which can be solved successivelyfor n = t, t + 1, . . . :

Pn(Xt) = Pn1(Xt) + t.nGn

1n (Yn GnXn),

t.n+1 = t.n[Fn n1n Gn],t|n = t|n1 t.nGn1n Gnt.n,

with initial conditions Pt1(Xt) =

Xt and t.t = t|t1 = t found from

Kalman prediction.


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Sakregister

ACF, 6ACVF, 6AICC, 23AR(p) process, 10ARCH(p) process, 11ARIMA(p, d, q) process, 11ARMA(p, q) process, 9

causal, 9invertible, 9multivariate, 24

autocorrelation function, 6autocovariance function, 6

autoregressive process, 10

best linear predictor, 12Brownian motion, 5

Cauchy-sequence, 3causality, 9characteristic function, 2convergence

mean-square, 3cross spectrum, 25

density function, 2distribution function, 2DurbinLevinson algorithm, 13

estimationleast square, 22maximum likelihood, 23

FARIMA(p, d, q) process, 11Fourier frequencies, 17

GARCH(p, q) process, 11Gaussian time series, 6generating polynomials, 9

HannanRissanen algorithm, 21

IID noise, 8innovations algorithm, 13invertibility, 9

Kalman filtering, 27

Kalman prediction, 26Kalman smoothing, 27

linear filter, 14causal, 15stable, 15time-invariant, 14

linear process, 8

MA(q) process, 10mean function, 5mean-square convergence, 3mean-squared error, 12

moving average, 10

observation equation, 26

PACF, 14partial autocorrelation, 14partial correlation coefficient, 14periodogram, 17point estimate, 5Poisson process, 6power transfer function, 15

probability function, 2probability measure, 2probability space, 2

random variable, 2

sample space, 2shift operator, 9-field, 2spectral density, 7

matrix, 25

spectral distribution, 7spectral estimator

discrete average, 18state equation, 25state-space model, 25state-space representation, 26stochastic process, 5strict stationarity, 6strictly linear time series, 15

28


31/31

SAKREGISTER 29

time series, 5linear, 8multivariate, 23stationary, 6, 24strictly linear, 15strictly stationary, 6weakly stationary, 6, 24

TLF, 15transfer function, 15

weak stationarity, 6, 24white noise, 8

multivariate, 24Wiener process, 5WN, 8, 24

Yule-Walker equations, 10

Documents

Formulas and survey Time Series Analisys.pdf