Econ2209 Week 5

7/21/2019 Econ2209 Week 5

1/28

Business Forecasting

ECON2209

Slides 05

Lecturer: Minxian Yang

BF-05 1my, School of Economics, UNSW

7/21/2019 Econ2209 Week 5

2/28

Ch.7 Characterising Cycles

Lecture Plan

Big picture:

Stochastic processes

Strictly stationarity and covariance stationarity

Autocorrelation (ACF) and partial ACF

White noise and its ACF sampling distribution

Wolds theorem

Conditional expectations

BF-05 my, School of Economics, UNSW 2

0)(E,0,, 1 ===++= =++ t

p

kkttpttttt xsssxsmy

7/21/2019 Econ2209 Week 5

3/28


Characterising Cycles (Ch.7)

Notation change:We useytfor the cycle component in this chapter.

Assume thatytis observable.

Cycle, stochastic process, time series

Cycleytis described as a stochastic process (SP).

SP is a random variable dependent on time index.

ytis a random variable for each fixed t.

ytis a sample path (or time series, or realization) foreach fixed outcome.

Observed time series is a realisation (sample-path)of the underlying SP.


7/21/2019 Econ2209 Week 5

4/28




0 10 20 30 40 50

-2

0

2

4

SamplePathsofaStochasticProcessyt

t

y

A random variable

for fixed t(eg. t= 30).

7/21/2019 Econ2209 Week 5

5/28



Discrete-time SP:eg. monthly inflation rate of 2025

Continuous-time SP:

eg. US/AUS exchange rate on Friday

We only cover discrete-time SP.

Main issues

Understand the characteristics of SP via a time

series, which is a realisation of SP.

In particular, find out SPs dependence structure.


,...}.,,,,{...,}{ 32101 yyyyyyt =

}.:{ btayt

7/21/2019 Econ2209 Week 5

6/28


Expectation operations E

Many characteristics of SP are given by the expectedvalues.

Here are some useful rules of mathematical expectations.

LetXand Ybe random variables and abe a constant.


7/21/2019 Econ2209 Week 5

7/28


Strictly stationary process

The distribution of yt generally depends on t.When it does not, yt is said to be strictly

stationary.

Precisely, a SP is strictly stationaryif itsjoint

distributionat any set of points in time is invariantto any time-shift, ie, the joint distribution at t+sis

the same as that at tfor any s.

No matter at which point in time you observe astationary SP, the joint distribution is the same.


7/21/2019 Econ2209 Week 5

8/28



Supposeytis strictly stationary. ThenIts distributions at t= 1, 2, ...are the same;

So are its joint distributions att = (2,6),(3,7), (4,8), ...;

So are its joint distributions at t= (1,3,5), (2,4,6), (3,5,7), ...;

...

Also, its mean and

variance (if exist)are independent of t.


SomeRealisationsofaStationaryProcessyt

Time

Y

t

2 4 6 8 10

-3

-2

-1

0

1

2

7/21/2019 Econ2209 Week 5

9/28



Stationary processes are statistically tractable andrich enough for many applications. But they differ

from random samples.

A random sample from a population consists of many

independent realisations.We learn the population/distribution via a large sample

(LLN and CLT).

A time series sample consists of one sample path, ie,

one realisation.Only when the underlying SP is stationary (and ergodic), we

may learn the properties of SP via a long time series.


7/21/2019 Econ2209 Week 5

10/28



eg. US 10-year treasury bondyield (monthly):

stationary?

eg. Department stores turnover

1982.04 1999.10,

cyclical component:

Stationary?


2

4

6

8

10

12

14

16

65 70 75 80 85 90 95 00 05

10-year T-bond Yield

-.12

-.08

-.04

.00

.04

.08

.12

82 84 86 88 90 92 94 96 98

Cycle

Unlikely, because its behaviour

differs markedly in sub-periods.

Likely.

7/21/2019 Econ2209 Week 5

11/28


Auto-covariance

Covariance ofxandy:

Auto-covariance ofytandyt-:

When = 0, (t,0) is simply the variance ofyt.


).E(),E(

)],()E[(),Cov(

yx

yxyx

yx

yx

==

=

....3,2,1,0,

)],E)(EE[(),Cov(),(

=

==

tttttt yyyyyyt

7/21/2019 Econ2209 Week 5

12/28


Covariance stationary processes

When the mean and autocovariance ofytexist andare independent of t,

{yt} is said to be covariance stationary.

For a covariance stationary process,

Strict stationarity and existence of variance

imply covariance stationarity.


,...3,2,1,0,),(),Cov(,)E( === ttt yyy

....3,2,1,),()(

,)Var()0( 2

==

==

ty

7/21/2019 Econ2209 Week 5

13/28


Sample autocovariance

eg. Department stores turnover, 1982.04 2005.04,

cyclical component


,...3,2,1,0,,))((1)(,111

=== += =

T

t

tt

T

t

t yyT

yT

-.25

-.20

-.15

-.10

-.05

.00

.05

.10

.15

82 84 86 88 90 92 94 96 98 00 02 04

Cyclical Component of Retail Turnover: 1982:04 - 2005:04

-.4

-.3

-.2

-.1

.0

.1

.2

1 2 3 4 5 6 7 8 9 10 11 12

Autocovariance/Var (variance=0.0014)

7/21/2019 Econ2209 Week 5

14/28


Autocorrelation function (ACF)

For a stationary process, the autocovariancedepends only on displacement (not time index t).

The ACF is defined as

free of measurement unit;

-1 () 1;

(-) =().

Sample ACF:


,....3,2,1,0,)0(

)(

)(Var)(Var

),(Cov)( ===

tt

tt

yy

yy

,....3,2,1,0,)0(

)()( ==

7/21/2019 Econ2209 Week 5

15/28


Patterns of ACF Exponential decay with/out cut-off (stationary series)

Close to 1 with very slow decay (non-stationary, unit root)

Seasonal peaks (presence of seasonality)


-1.0

-0.5

0.0

0.5

1.0

1 2 3 4 5 6 7 8 9 10 11 12

ACF: exponential decline with oscillation

0.85

0.90

0.95

1.00

1 2 3 4 5 6 7 8 9 10 11 12

ACF: very slow decline

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

2 4 6 8 10 12 14 16 18 20 22 24

ACF: seasonal effects

-1.2

-0.8

-0.4

0.0

0.4

0.8

1 2 3 4 5 6 7 8 9 10 11 12

ACF: exponential decline with cut-off

7/21/2019 Econ2209 Week 5

16/28


Partial autocorrelation function (PACF)

Chain effect: (2)is non-zero because eitherytis directly related toyt-2;

or ytis related toyt-1that is in turn related toyt-2;

or both.

PACFp(2)measures the direct relation between

ytandyt-2, after controlling for the effects ofyt-1.

In general, PACFp()measures the direct relation

betweenytandyt-, after controlling for the effects

ofyt-1,...,yt-+1.

t-, t-+1,... , t-1, t


7/21/2019 Econ2209 Week 5

17/28


Sample PACF

Computation of

eg.

is the estimated 11in the regression

yt= 10 + 11yt-1+ error.

is the estimated 22in the regressionyt= 20 + 21yt-1 + 22yt-2+ error.

is the estimated 33in the regression

yt= 30 + 31yt-1 + 32yt-2 + 33yt-3+ error.


)( p

.ontcoefficienestimated)(

and],...,,1[onofOLSrun,...,3,2,1eachFor

;1)0(

1

=

=

=

t

ttt

yp

yyy

p

)1(p

)2(p

)3(p

7/21/2019 Econ2209 Week 5

18/28


Theoretical PACF

Theoretical PACF can be described as the limit ofsample PACF as sample size increases to infinity.

Or,p()is defined as a function of autocovariances

[(0), (1), , ()],


.

)(

)2(

)1(

)0()2()1(

)2()0()1(

)1()1()0(

,)(1

2

1

=

=

p

7/21/2019 Econ2209 Week 5

19/28


Example & EViewse.g. Department stores turnover,

1982.04 2005.04,

cyclical component from X12


-.25

-.20

-.15

-.10

-.05

.00

.05

.10

.15

82 84 86 88 90 92 94 96 98 00 02 04


-1

0

1

2

3

4

5

6

7

8

82 84 86 88 90 92 94 96 98 00 02 04

Y_SF

Y_SA

Y_TC

Y_IR

Find the correlogram of y

y.correl(12)

7/21/2019 Econ2209 Week 5

20/28


White noise (WN) process

WN is a building block of time series models.WN, {t}, is a serially-uncorrelatedSP with zero

meanand constant finite variance.

If tis iid (0,2), it is also a WN (called iid WN)

process. But WN is not necessarily iid.

ACF and PACF of a WN process


.0for,0),(Cov,)(Var,0)(E:),0(WN~ 22 === ttttt

.0if,0

0if,1)(;

0if,0

0if,1)(

==

==

p

7/21/2019 Econ2209 Week 5

21/28


Sampling properties of the ACF of WN

Let {1

,2

,,T

} be the sample path of an iid WN process.

The sample ACF

has the sampling distribution

for large T. This is also true for PACF.

Useful to check if residuals from a model are serially

correlated.


,...3,2,1,0,,)0()()(

,))((1

)(,1

11

==

== +=

=

T

t

tt

T

t

tTT

).1(~approx.)(),1,0(~approx.)( 22 TNT

7/21/2019 Econ2209 Week 5

22/28


Test for autocorrelation

Null hypothesis H0: time series {et} is WN.Reject H0if is too large.

Approximate 95% confidence band (Bartlett)

ACF is inside the band with probability 0.95 under H0.

Ljung-Box test

where mshould be less than

Reject H0if QLBis too large.


|)(| T

),(~approx.,)(1

)2( 2

1

2 mT

TTQm

LB

=

+=

.T

]./20,/20[ TT +

7/21/2019 Econ2209 Week 5

23/28


Test for autocorrelation

eg. Department stores turnover, 1982.04 2005.04,

cyclical component, 2/ 0.12

No AC is rejected.


T

-.25

-.20

-.15

-.10

-.05

.00

.05

.10

.15

82 84 86 88 90 92 94 96 98 00 02 04


P(2(1)> 31.946) = 0.000

7/21/2019 Econ2209 Week 5

24/28


Wolds representation theorem

A zero-mean covariance-stationary process {yt}can be represented as the linear process,

where

If E(yt) = 0, the theorem applies to (yt-).

tmay be interpreted as 1-step forecast error of the

best linear predictor based on {yt-1, yt-2,}.

Impulse response:


),,0(WN~, 2

0

ti

itit by

=

=

.1,0

0

2

=

=

Econ2209 Week 5

Documents

Transcript of Econ2209 Week 5