forecasting (prediction) limits example linear deterministic trend estimated by least-squares note!...

Forecasting (prediction) limits

Example Linear deterministic trend estimated by least-squares

t

s

e

t

s

ee

ltltltltlt

ltltlt

tttltlt

ttt

ts

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YVarYVarYYeVar

YYe

ltYYbYYbltbbYYYEY

etY

1

2

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22

1110101

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1 theoryanalysis regression From

ˆtindependen ˆ and ˆ

ˆˆ

,,,,,,ˆ

Note! The average of the numbers 1, 2, … , t is

2

1

2

1111

ttt

ts

t

t

s

Hence, calculated prediction limits for Yt+l become

where c is a quantile of a proper sampling distribution emerging from the use of and the requested coverage of the limits.

t

s

eltt

s

tlt

tcY

1

2

21

21

11ˆˆ

22 of estimator an as ˆ ee

For t large it suffices to use the standard normal distribution and a good approximation is also obtained even if the term

is omitted under the square root

t

s

ts

tlt

t1

2

21

21

1

21,0Pr

ˆˆ

2

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zN

zY elt

ARIMA-models

qp

l

l

j jelt zY

ˆ,,ˆ and ˆ,,ˆ estimates parameter

theof functions are ˆ,,ˆ where

ˆˆˆ

11

10

1

0

22

Using R

ts=arima(x,…) for fitting models

plot.Arima(ts,…) for plotting fitted models with 95% prediction limits

See documentation for plot.Arima . However, the generic command plot can be used.

forecast.Arima Install and load package “forecast”. Givesmore flexibility with respect to prediction limits.

Seasonal ARIMA models

Example “beersales” data

A clear seasonal pattern and also a trend, possibly a quadratic trend

Residuals from detrended data

beerq<-lm(beersales~time(beersales)+I(time(beersales)^2))plot(y=rstudent(beerq),x=as.vector(time(beersales)),type="b",pch=as.vector(season(beersales)),xlab="Time")

Seasonal pattern, but possibly no long-term trend left

SAC and SPAC of the residuals:

SAC

SPAC

Spikes at or close to seasonal lags (or half-seasonal lags)

Modelling the autocorrelation at seasonal lags

Pure seasonal variation:

otherwise0

3624120

circleunit theoutside

01equation sticcharacteri the toRoots 1 if Stationary

model-AR(1) Seasonal

121

1211

12121

,...,,,kρ

eYY

k

k

ttt

otherwise0

121

01

circleunit theoutside

01equation sticcharacteri the toRoots 1 if Invertible

model-MA(1) Seasonal

21

1

1211

12121

k

k

ρ

eeY

k

ttt

Non-seasonal and seasonal variation:

AR(p, P)s or ARMA(p,0)(P,0)s

tsPtPstptptt eYYYYY 111

However, we cannot discard that the non-seasonal and seasonal variation “interact” Better to use multiplicative Seasonal AR Models

ttsP

Psp

p eYBBBB 11 11

Example:

ttttt

tt

tt

eYYYY

eYBBB

eYBB

13121

1312

12

05.02.03.0

2.03.02.03.01

2.013.01

Multiplicative MA(q, Q)s or ARMA(0,q)(0,Q)s

tsQ

Qsq

qt eBBBBY 11 11

Mixed models:

t

sQQ

sqq

tsP

Psp

p

eBBBB

YBBBB

11

11

11

11

Many terms! Condensed expression:

Q

j

jsj

q

j

jj

s

P

i

isi

sp

i

ii

ts

ts

BBBB

BBBB

eBBYBB

11

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1;1

1;1

sQPqp ,,ARMA

Non-stationary Seasonal ARIMA models

Non-stationary at non-seasonal level:

Model dth order regular differences: td

ttd YBYY 1

Non-stationary at seasonal level:

Seasonal non-stationarity is harder to detect from a plotted times-series. The seasonal variation is not stable.

Model Dth order seasonal differences: t

Dstssst

Ds YBYY 1

Example First-order monthly differences:

can follow a stable seasonal pattern

1212 1 ttts

t YYYBY

The general Seasonal ARIMA model

ts

t

Dsds eBBYBBBB 11

It does not matter whether regular or seasonal differences are taken first

sQDPqdp ,,,,ARIMA

Model specification, fitting and diagnostic checking

Example “beersales” data

Clearly non-stationary at non-seasonal level, i.e. there is a long-term trend

Investigate SAC and SPAC of original data

Many substantial spikes both at non-seasonal and at seasonal level-

Calls for differentiation at both levels.

Try first-order seasonal differences first. Here: monthly data

12121 tttt YYYBW

beer_sdiff1 <- diff(beersales,lag=12)

Look at SAC and SPAC again

Better, but now we need to try regular differences

Take first order differences in seasonally differenced data

13112112 111 ttttttttt YYYYWWWBYBBU

beer_sdiff1rdiff1 <- diff(beer_sdiff1,lag=1)

Look at SAC and SPAC again

SAC starts to look “good”, but SPAC not

Take second order differences in seasonally differenced data

Since we suspected a non-linear long-term trend

14213112

21211

1122

2

2

111

tttttt

ttttttt

ttttt

YYYYYY

WWWWWWW

UUUBYBBV

beer_sdiff1rdiff2 <- diff(diff(beer_sdiff1,lag=1),lag=1)

Could be an ARMA(2,0)(0,1)12 or an ARMA(1,1) (0,1)12

Non-seasonal part Seasonal part

These models for original data becomes

ARIMA(2,2,0) (0,1,1)12 and ARIMA(1,2,1) (0,1,1)12

model1 <-arima(beersales,order=c(2,2,0), seasonal=list(order=c(0,1,1),period=12))

Series: beersales ARIMA(2,2,0)(0,1,1)[12]

Coefficients: ar1 ar2 sma1 -1.0257 -0.6200 -0.7092s.e. 0.0596 0.0599 0.0755

sigma^2 estimated as 0.6095: log likelihood=-216.34AIC=438.69 AICc=438.92 BIC=451.42

Diagnostic checking can be used in a condensed way by function tsdiag. The Ljung-Box test can specifically be obtained from function Box.test

tsdiag(model1)standardized residuals

SPAC(standardized residuals)

P-values of Ljung-Box test with K = 24

Box.test(residuals(model1), lag = 12, type = "Ljung-Box", fitdf = 3)

Box-Ljung test

data: residuals(model1) X-squared = 30.1752, df = 9, p-value = 0.0004096

K (how many lags included)

p + q + P + Q (how many degrees of freedom withdrawn from K)

For seasonal data with season length s the L-B test is usually calculated forK = s, 2s, 3s and 4s


Box-Ljung test

data: residuals(model1) X-squared = 57.9673, df = 21, p-value = 2.581e-05


Box-Ljung test



Box-Ljung test


Hence, the residuals from the first model are not satisfactory

model2 <-arima(beersales,order=c(1,2,1), seasonal=list(order=c(0,1,1),period=12))print(model2)

Series: beersales ARIMA(1,2,1)(0,1,1)[12]

Coefficients: ar1 ma1 sma1 -0.4470 -0.9998 -0.6352s.e. 0.0678 0.0176 0.0930

sigma^2 estimated as 0.4575: log likelihood=-192.86AIC=391.72 AICc=391.96 BIC=404.45

Better fit ! But is it good?

tsdiag(model2)

Not good! We should maybe try second-order seasonal differentiation too.

Time series regression models

The classical set-up uses deterministic trend functions and seasonal indices

n variatioseasonal no trend,quatadic

otherwise0

month in is if1 where

datamonthly in endlinear tr

:Examples

2210

12

2,10

ttY

jttx

etxtY

etStmY

t

j

tj

jjst

tt

The classical set-up can be extended by allowing for autocorrelated error terms (instead of white noise). Usually it is sufficient with and AR(1) or AR(2). However, the trend and seasonal terms are still assumed deterministic.

Dynamic time series regression models

To extend the classical set-up with explanatory variables comprising other time series we need another way of modelling.

Note that a stationary ARMA-model

can also be written

tt

tq

tp

p

qtqttptptt

eBYB

eBBYBB

eeeYYY

0

1101

11110

11

tt eB

BY

0

0

0 B

The general dynamic regression model for a response time series Yt with one covariate time series Xt can be written

tt

bt e

B

BXB

B

BCY

0

Special case 1:

Xt relates to some event that has occurred at a certain time points (e.g. 9/11)

It can the either be a step function

or a pulse function

TSTt

TtTX tt

0

1

TPTt

TtTX tt

0

1

Step functions would imply a permanent change in the level of Yt . Such a change can further be constant or gradually increasing (depending on (B) and (B) ). It can also be delayed (depending on b )

Pulse functions would imply a temporary change in the level of Yt . Such a change may be just at the specific time point gradually decreasing (depending on (B) and (B) ).

Strep and pulse functions are used to model the effects of a particular event, as so-called intervention. Intervention models

For Xt being a “regular” times series (i.e. varying with time) the models are called transfer function models

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