BF-08

Business Forecasting ECON2209

Slides 08

Lecturer: Minxian Yang

BF-08 1 my, School of Economics, UNSW

Ch.9 Forecasting Cycles

• Lecture Plan – Big picture:

– Optimal forecast and linear forecast – Forecasting general linear processes – Forecasting ARMA processes with the chain rule – Long run forecasts

BF-08 my, School of Economics, UNSW 2

0)(E ,0 , ,1

===++= ∑=

++ t

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kkttpttttt xsssxsmy


Forecasting Cycles (Ch.9) • Optimal forecast

– As parameters can be estimated, we assume that parameters in ARMA models are known for now. At T, we wish to forecast the SP (or cycle) at T+h.

– Information set ΩT = yT, yT-1, …. When models are well-defined, ΩT = yT, yT-1, …; εT, εT-1, ….

– Forecast for yT+h based on ΩT: yT+h|T . – Expected quadratic loss (MSFE):


].Ω|)[(E)(MSFE 2|| TThThTThT yyy +++ −=


• Optimal forecast eg. ARMA(1,1) forecast (continued)


,112112 ++++ +++= TTTT ycy εθεϕ

,)|()|( |11112|2 TTTTTTTT ycyEcyEy ++++ +=Ω+=Ω= ϕϕ

.|11112|2 TTTTTT ee ++++ ++= ϕεθε Forecast error = yT+2 – yT+2|T

.,

,

|21213|3

|21|3

213213

TTTTTT

TTTT

TTTT

eeycy

ycy

++++

++

++++

++=

+=+++=

ϕεθε

ϕεθεϕ

Forecast error = yT+3 – yT+3|T


• Optimal forecast – Linear forecast: a linear combination of the

elements in ΩT. A linear forecast may be different from E(yT+h|ΩT).

– Best linear forecast (aka. linear projection):

with βs being chosen to minimise MSFE. – If the true model (dgp) is ARMA, the linear

projection coincides with E(yT+h|ΩT). – In general, the linear projection may be used as an

approximation to E(yT+h|ΩT).


+++= −+ 1210| TTThT yyy βββ


• Forecasting a general linear process – General linear process (cf. Wold representation)

– With the information ΩT = εT, εT-1, …, the Linear projection is

yT+h|T = bhεT + bh+1εT-1 + …


).,0(WN~ 1, , 20

0σεε t

iitit bby ==∑

∞

=−

. |

|

:forecast

11

:error forecast

1111

ThT

ThT

y

ThTh

e

ThhThThT

bb

bby

+

+

+++

+++=

−+

+−−+++

εε

εεε

ΩT

yT+h

yT+h|T

eT+h|T

TΩ


• Forecasting a general linear process – Forecast error

– When h → ∞, yT+h|T → E(yT) and σh2 → Var(yT).

– 95% Interval forecast when εt ~ iid N(0, σ2) [ yT+h|T − 1.96σh , yT+h|T + 1.96σh ]

– Conditional density when εt ~ iid N(0, σ2) yT+h | ΩT ~ N(yT+h|T , σh

2)


anceerror vari .)1()Var(

unbiased ,0)()1MA( ,

2221|

2

|

1111|

σσ

εεε

hThTh

ThT

ThhThTThT

bbe

eEhbbe

+++==

=

−+++=

+

+

+−−+++

Chebyshev’s theorem:

P(Y outside kσ) ≤ 1/k2


• Making it feasible with ARMA – Use ARMA to produce a linear projection.

• When dgp is ARMA, the linear projection is optimal; • In general, the linear projection approximates

E(yT+h|ΩT) . – Use estimated ARMA parameters. For large

samples, uncertainty in estimates may be ignored.

– ARMA: future Y = terms in ΩT + terms outside ΩT. • Use a chain rule to separate the terms in ΩT and the

terms outside ΩT.



• Making it feasible with ARMA eg. ARMA(1,1) chain rule


;

,

,

1|1

11|1

1

inside

111

++

+

+

Ω

+

=

++=

+++=

TTT

TTTT

TTTT

e

ycy

ycyT

ε

εθϕ

εεθϕ

.,

,)(

|11112|2

|11|2

|11112

inside

|11

112|1|112

TTTTTT

TTTT

TTTTTT

TTTTTTT

eeycy

eyceycy

T

++++

++

+++

Ω

+

+++++

++=

+=

++++=

++++=

ϕεθε

ϕ

ϕεθεϕ

εθεϕ

yT+h = yT+h|T + eT+h|T


• Making it feasible with ARMA eg. ARMA(1,1) chain rule (continued)


2211|1

11|1

|1|11111

;

,,

σσε

εθφεθεφ

==

++=

+=+++=

++

+

++++

TTT

TTTT

TTTTTTTT

e

ycyeyycy

221

21

21

22112|11|2

|11|2

112|1|112

)1( ;

,,)(

σθσφσεθεφ

φεθεφ

++=++=

+=

++++=

++++

++

+++++

TTTTTT

TTTT

TTTTTTT

ee

ycyeycy

221

22

21

23213|21|3

|21|3

213|2|213

)1( ;

,,)(

σθσφσεθεφ

φεθεφ

++=++=

+=

++++=

++++

++

+++++

TTTTTT

TTTT

TTTTTTT

ee

ycyeycy


• eg. Canadian employment index: 61q1:94q4


80

85

90

95

100

105

110

115

1965 1970 1975 1980 1985 1990

Y

AIC: MA(0-4)

AIC: ARMA(3,1) SIC: ARMA(2,0)

AR(0-4)

SIC: MA(0-4)

AR(0-4)

Search within ARMA(4,4,)




AIC: ARMA(3,1)

0

5

10

15

20

25

-4 -2 0 2 4 6

Series: ResidualsSample 1962Q1 1993Q4Observations 128

Mean 7.38e-05Median 0.041454Maximum 6.754276Minimum -3.629406Std. Dev. 1.412262Skewness 0.547303Kurtosis 6.518596

Jarque-Bera 72.41962Probability 0.000000-4

-2

0

2

4

6

8

80

90

100

110

120

1965 1970 1975 1980 1985 1990

Residual Actual Fitted




80

84

88

92

96

100

104

108

112

116

1990 1992 1994 1996 1998 2000 2002 2004 2006

YYF

YF_UPYF_LO

Forecast 94M1-06M4: ARMA(3,1) Model

80

84

88

92

96

100

104

108

1990 1991 1992 1993 1994

Forecast 94M1-94M4: ARMA(3,1) Model

Forecasts: ARMA(3,1) For a stationary time series y, the long horizon point forecast is approximately the unconditional mean of y. Its forecast error variance is approximately the unconditional variance of y.


• EViews


'caemp.prg wfcreate(wf=null) q 1961:1 2010:4 smpl 1961:1 1994:4 read caemp.dat y 'Plots smpl 1961:1 1993:4 y.line y.correl(16) 'Selecting models within ARMA(4,4) smpl 1962:1 1993:4 matrix(5,5) aic matrix(5,5) sic ls y c sic(1,1)=@schwarz aic(1,1)=@aic ls y c ma(1) sic(1,2)=@schwarz aic(1,2)=@aic ls y c ma(1) ma(2) sic(1,3)=@schwarz aic(1,3)=@aic ls y c ma(1) ma(2) ma(3) sic(1,4)=@schwarz aic(1,4)=@aic ls y c ma(1) ma(2) ma(3) ma(4) sic(1,5)=@schwarz aic(1,5)=@aic

for !k=1 to 4 ls y c y(-1 to -!k) sic(!k+1,1)=@schwarz aic(!k+1,1)=@aic ls y c y(-1 to -!k) ma(1) sic(!k+1,2)=@schwarz aic(!k+1,2)=@aic ls y c y(-1 to -!k) ma(1) ma(2) sic(!k+1,3)=@schwarz aic(!k+1,3)=@aic ls y c y(-1 to -!k) ma(1) ma(2) ma(3) sic(!k+1,4)=@schwarz aic(!k+1,4)=@aic ls y c y(-1 to -!k) ma(1) ma(2) ma(3) ma(4) sic(!k+1,5)=@schwarz aic(!k+1,5)=@aic next aic.bar sic.bar 'Preferred models equation eq1.ls y c ar(1) ar(2) ar(3) ma(1) equation eq2.ls y c ar(1) ar(2)

'Checking residuals eq1.correl(10) eq1.hist 'Forecast exercise smpl 1962:1 1993:4 eq1.makeresids res genr yf=y-res smpl 1994:1 2006:4 eq1.forecast yhat se genr yf=yhat genr yf_up=yhat+1.96*se genr yf_lo=yhat-1.96*se smpl 1990:1 1994:4 group fig1 y yf yf_up yf_lo freeze(Figure1) fig1.line Figure1.draw(shade, bottom) 1994:1 1994:4 show Figure1 smpl 1990:1 2006:4 group fig2 y yf yf_up yf_lo freeze(Figure2) fig2.line Figure2.draw(shade, bottom) 1994:1 2006:4 show Figure2 stop

Use subsample 1962:1 1993:4 for model selection, so that models within ARMA(4,4) use the same sample size.


• Summary – Under MSFE, what is the optimal forecast? – What is a linear forecast? – What is the linear projection? – How do we find the point forecast and forecast

error from a general linear process? – Do you know how to use the chain rule to produce

point forecasts and forecast error variances for ARMA(1,1)?

– How do you make long horizon forecasts for a stationary time series?


BF-08

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