NY Times 23 Sept 2008 - time series of the day

Stat 153 - 23 Sept 2008 D. R. Brillinger

Chapter 4 - Fitting t.s. models in the time domain

sample autocovariance coefficient.

Under stationarity, ...

)/1(),( NOccCov mk

NaskcE k )()(

,...2,1,0 ,/))((1

kNxxxxc kt

kN

ttk

Estimated autocorrelation coefficient

0/ ccr kk

large and .. ,..., If 1 Ndiixx N

NrE k /1)(

NrVar k /1)(

asymptotically normal

interpretation

N/2 :0 around CI 95% eapproximat

Uses of acf

mixing (asymptotically independent)?

MA(q)?

Seasonal component?

ergodic

N

tt NXfENXf

1 asy probabilitin ))((/)(

Estimating the mean

N

tt NXX

1/

unbiased ,)( XE

1

1

2 /)]()1(21[)(N

rNr

Nr

XVar

Can be bigger or less than 2/N

NAR /)11

(approx )1(For 2

Fitting an autoregressive, AR(p)

Easy. Remember regression and least squares

nixY iii ,...,1 ,

2

),( ][min ii xy

normal equations

iiii

ii

yxxx

yx

ˆ ˆ

ˆ ˆ12

... ' ,)(...)( 11 diisZZXXX tptptt

N

ptptptt xxxS

1

2

11 )](...)([

AR(1)

1

1

1

1

2

1 )(/))((ˆN

t

N

tttt xxxxxx

Cp.

011 / ccr

Fitting an MA(q).

Later. There is an R program

Fitting an ARMA(p,q).

Later. There is an R program

Estimating p, q, (p,q).

Later. There is a criterion.

Seasonal ARIMA. seasonal parameter s

SARIMA(p,d,q)(P,D,Q)s

Example

tt ZBXBB )1()1)(1( 1212

1213112 )( tttttt ZZXXXX

t

s

Qqt

s

Pp ZBBWBB )()()()(

Residual analysis.

Paradigm

observation = fitted value plus residual

The parametric models have contained Zt

tz is residual The

1t ˆz

AR(1).

tt xx

Plot residuals vs. t

Acf of residuals

Portmanteau lack-of-fit statistic

K

kkzrNQ

1

2

,

ARMA(p,q) appropriate?

2

qpKQ

Model building

(1) model formulation

(2) model estimation

(3) model checking

All models are wrong but some are useful