NSBJF - IMSL Fortran 905.0 Beta Documentation

752 · Chapter 8: Time Series Analysis and Forecasting IMSL STAT/LIBRARY

Description

Routine SPWF performs least-squares estimation of parameters for successive autoregressive models of a stationary stochastic process given a sample of n = NOBS observations {Wt} for t = 1, ¼, n.

Let

ˆ WMEAN

m =

be the estimate of the mean m of the stochastic process {Wt} where

known

ˆ 1

unknown

n t t

n W

m m

ìï

= íïî

å

Consider the autoregressive model of order k defined by

( ) 0

k B Wt At k

f % = ³

where

t t

ˆ

W% =W -m

and

1 2

( ) 1 ^k 1

k B kB kB kkB k

f = -f -f - -L f ³

Successive AR(k) models are fit to the centered data using Durbin’s algorithm (1960) based on the sample autocovariances

ˆ ( ) 1

^{n k}

(

ˆ )(

_{t k}

ˆ ) 0

k W W k

s n ^- m + m

å

- - ³

Note that the variance

ˆ (0)

s^*

used in the fitting algorithm is adjusted by the amount

=

WNADJ

according to ˆ (0) (1 ) (0) ˆ

s^* = +d s

See Robinson (1967, page 96).

Iteration to the next higher order model terminates when either the expected mean square error

of the model is less than

EPS

or when k =

MLFOP

. The forecast operator

= (

,

, …,

fk*

)

for

k^*

=

LFOP

is contained in

FOP

. See also Craddock (1969).

IMSL STAT/LIBRARY Chapter 8: Time Series Analysis and Forecasting · 753

Required Arguments

— Vector of length

NOBS

containing the time series. (Input)

PAR

— Vector of length

NPAR

containing the autoregressive parameters. (Input)

LAGAR

— Vector of length

NPAR

containing the order of the autoregressive parameters.

(Input)

The elements of

LAGAR

must be greater than zero.

PMA

— Vector of length

NPMA

containing the moving average parameters. (Input)

LAGMA

— Vector of length

NPMA

containing the order of the moving average parameters.

(Input)

The elements of

LAGMA

must be greater than zero.

ICNST

— Option for including the overall constant in the model. (Input)

ICNST Action

0 No overall constant is included.

1 The overall constant is included.

CNST

— Estimate of the overall constant. (Input)

AVAR

— Estimate of the random shock variance. (Input)

AVAR

must be greater than 0.

ALPHA

— Value in the exclusive interval (0, 1) used to specify the 100(1

_-ALPHA

)%

probability limits of the forecasts. (Input) Typical choices for

ALPHA

are 0.10, 0.05, and 0.01.

MXBKOR

— Maximum backward origin. (Input)

MXBKOR

must be greater than or equal to zero and less than or equal to

NOBS-

max(

MAXAR

,

MAXMA

) where

MAXAR

= max(

LAGAR

(i)) and

MAXMA

= max(

LAGMA

(j)). Forecasts at origins

NOBS-MXBKOR

through

NOBS

are generated.

MXLEAD

— Maximum lead time for forecasts. (Input)

MXLEAD

must be greater than zero.

FCST

—

MXLEAD

by (

MXBKOR

+ 3) matrix defined below. (Output)

754 · Chapter 8: Time Series Analysis and Forecasting IMSL STAT/LIBRARY Column

Content

Forecasts for lead times l = 1, …,

MXLEAD

at origins

NOBS-MXBKOR-

1 + j, j = 1, …,

MXBKOR

+ 1.

MXBKOR

+ 2 Deviations from each forecast that give the 100(1

-ALPHA

)%

probability limits.

MXBKOR

+ 3 Psi weights of the infinite order moving average form of the model.

Optional Arguments

NOBS

— Number of observations in the time series

. (Input)

NOBS

must be greater than

ICONST

+ max(

LAGAR

(i)) + max(

LAGMA

(j)).

Default:

^NOBS

= size (

,1).

IPRINT

— Printing option. (Input) Default:

^IPRINT

= 0.

IPRINT Action

0 No printing is performed.

1 Prints the forecasts for lead times l = 1, …,

MXLEAD

at each origin

t = (NOBS-MXBKOR

), …,

NOBS

, the 100(1

-ALPHA

)% probability limit deviations, and the psi weights.

NPAR

— Number of autoregressive parameters. (Input)

NPAR

must be greater than or equal to zero.

Default:

NPAR

= size (

PAR

,1).

NPMA

— Number of moving average parameters. (Input)

NPMA

must be greater than or equal to zero.

Default:

^NPMA

= size (

^PMA

,1).

LDFCST

— Leading dimension of

FCST

exactly as specified in the dimension statement in the calling program. (Input)

LDFCST

must be greater than or equal to

MXLEAD

. Default:

LDFCST

= size (

FCST

,1).

FORTRAN 90 Interface

Generic:

CALL NSBJF(W, PAR, LAGAR, PMA, LAGMA, ICNST, CNST, AVAR, ALPHA, MXBKOR, MXLEAD, FCST [,…])

Specific: The specific interface names are

S_NSBJF

and

D_NSBJF

.

IMSL STAT/LIBRARY Chapter 8: Time Series Analysis and Forecasting · 755

FORTRAN 77 Interface

Single:

CALL NSBJF (NOBS, W, IPRINT, NPAR, PAR, LAGAR, NPMA, PMA, LAGMA, ICNST, CNST, AVAR, ALPHA, MXBKOR, MXLEAD, FCST, LDFCST)

Double: The double precision name is

DNSBJF

. Example

Consider the Wölfer Sunspot Data (Anderson 1971, page 660) consisting of the number of sunspots observed each year from 1749 through 1924. The data set for this example consists of the number of sunspots observed from 1770 through 1869. Routine

NSBJF

is used to computed forecasts and 95% probability limits for the forecasts for an

ARMA

(2, 1) model fit using routine

NSPE (page 727). With MXBKOR

= 3, columns one through four of

FCST

give forecasts given the data through 1866, 1867, 1868, and 1869, respectively. Column 5 gives the deviations from the forecast for computing probability limits, and column six gives the psi weights, which can be used to update forecasts once more data is available. For example, the forecast for the 102-nd observation (year 1871) given the data through the 100-th observation (year 1869) is 77.21, and 95% probability limits are given by 77.21

56.30. After observation 101 (W

101

for year 1870) is available, the forecast can be updated by using equation 7 with the psi weight (

= 1.37) and the one-step-ahead forecast error for observation 101 (W

101-

83.72) to give

77.21 + 1.37 (W

₁₀₁_-

83.72)

Since this updated forecast is one step ahead, the 95% probability limits are now given by the forecast

33.22.

USE GDATA_INT USE NSPE_INT USE NSBJF_INT USE WRRRL_INT

INTEGER LDFCST, MXBKOR, MXLEAD, NOBS, NPAR, NPMA

PARAMETER (MXBKOR=3, MXLEAD=12, NOBS=100, NPAR=2, NPMA=1, &

LDFCST=MXLEAD)

INTEGER ICNST, LAGAR(NPAR), LAGMA(NPMA), NCOL, NROW

REAL ALPHA, AVAR, CNST, FCST(LDFCST,MXBKOR+3), PAR(NPAR), &

PMA(NPMA), RDATA(176,2), W(NOBS), WMEAN CHARACTER CLABEL(MXBKOR+4)*40, RLABEL(1)*6

EQUIVALENCE (W(1), RDATA(22,2))

DATA LAGAR(1), LAGAR(2)/1, 2/

DATA LAGMA(1)/1/

DATA RLABEL/’NUMBER’/, CLABEL/’%/Lead%/Time’, &

’%/Forecast%/From 1866’, ’%/Forecast%/From 1867’, &

’%/Forecast%/From 1868’, ’%/Forecast%/From 1869’, &

’ Deviation %/ for 95% %/Prob. Limits’, ’%/%/Psi’/

! Wolfer Sunspot Data for

! years 1770 through 1869 CALL GDATA (2, RDATA, NROW, NCOL)

756 · Chapter 8: Time Series Analysis and Forecasting IMSL STAT/LIBRARY

! Compute preliminary parameter

! estimates for ARMA(2,1) model CALL NSPE (W, CNST, PAR, PMA, AVAR)

! Include constant in forecast model ICONST = 1

! Specify 95 percent probability

! limits for forecasts ALPHA = 0.05

! Compute forecasts CALL NSBJF (W, PAR, LAGAR, PMA, LAGMA, &

ICNST, CNST, AVAR, ALPHA, MXBKOR, MXLEAD, FCST)

! Print results

CALL WRRRL (’FCST’, FCST, RLABEL, CLABEL, FMT=’(5F9.2, F6.3)’)

END

Output

FCST

Deviation Lead Forecast Forecast Forecast Forecast for 95%

Time From 1866 From 1867 From 1868 From 1869 Prob. Limits Psi 1 18.28 16.62 55.19 83.72 33.22 1.368 2 28.92 32.02 62.76 77.21 56.30 1.127 3 41.01 45.83 61.89 63.46 67.62 0.616 4 49.94 54.15 56.46 50.10 70.64 0.118 5 54.09 56.56 50.19 41.38 70.75 -0.208 6 54.13 54.78 45.53 38.22 71.09 -0.326 7 51.78 51.17 43.32 39.30 71.91 -0.286 8 48.84 47.71 43.26 42.46 72.53 -0.169 9 46.53 45.47 44.46 45.77 72.75 -0.045 10 45.35 44.69 45.98 48.08 72.77 0.041 11 45.21 44.99 47.18 49.04 72.78 0.077 12 45.71 45.82 47.81 48.91 72.82 0.072

Comments

1. Workspace may be explicitly provided, if desired, by use of

N2BJF/DN2BJF

. The reference is:

CALL N2BJF (NOBS, W, IPRINT, NPAR, PAR, LAGAR, NPMA, PMA, LAGMA, ICNST, CNST, AVAR, ALPHA, MXBKOR, MXLEAD, FCST, LDFCST, PARH, PMAH, PSIH, PSI, LAGPSI)

The additional arguments are as follows:

PARH

— Work vector of length equal to

IARDEG

+ 1.

PMAH

— Work vector of length equal to

IMADEG

+ 1.

PSIH

— Work vector of length equal to

MXLEAD

+ 1.

PSI

— Work vector of length equal to

MXLEAD

+ 1.

IMSL STAT/LIBRARY Chapter 8: Time Series Analysis and Forecasting · 757 LAGPSI

— Work vector of length equal to

MXLEAD

+ 1.

2. If the

series has been transformed using a Box-Cox transformation with parameters

POWER

and

SHIFT

, the forecasts and probability limits for the original series may be obtained by application of routine

BCTR (page 689) with the same parameters and

argument

IDIR

set equal to one.

Description

Routine

NSBJF

computes Box-Jenkins forecasts and their associated probability limits for a nonseasonal ARMA model given a sample of n =

NOBS

observations {W

} for t = 1, 2, …, n.

Suppose the time series {W

} is generated by a nonseasonal ARMA model of the form ( )

B W_t 0

( )

B A_t t

{0, 1, 2, }

f = +q q Î ± ± K

where B is the backward shift operator,

=

CONST

,

(1) (2) ( )

1 2

(1) (2) ( )

1 2

( ) 1 ( ) 1

l l l p

l l l q

B B B B

f f f

q q q

f f f f

q q f q

= - - - -

L L

p = NPAR and q = NPMA. Without loss of generality, we assume

1 (1) (2) ( )

l l l p

l l l q

f f f

q q q

£ £ £ £

L L

so that the nonseasonal ARMA model is of order (p¢, q¢) where p¢ = lf(p) and q¢ = lq(q). Note that the usual hierarchal model assumes

( ) 1

l i i i p

l j j j q

f q

= £ £

The Box-Jenkins forecast at origin t for lead time l of Wt+l is defined in terms of the difference equation

0 1 (1) ( )

1 (1) ( )

ˆ ( ) [ ] [ ]

[ ] [ ] [ ]

t t l l p t l l p

t l t l l q t l l q

W l W W

A A A

f f

q q

q f f

q q

+ - + -

+ + - + -

= + + +

+ - - -

L L where

0, 1, 2,

[ ]

ˆ ( ) 1, 2,

ˆ (1) 0, 1, 2,

[ ]

0 1, 2,

t k t k

t k

W k

W W k k

W W k

A k

+ +

+ + -

= - -

= íìïïî =

ì - = - -

= íïïî =

K K

K K The 100(1 - a)% probability limits for Wt+l are given by

758 · Chapter 8: Time Series Analysis and Forecasting IMSL STAT/LIBRARY

1 1/ 2 2 / 2

ˆ ( )

1

^l _j _A

W l z_a ^- y s

ì ü

± í + ý

å

where z

(1-a/2)

is the 100(1

-a

/2) percentile of the standard normal distribution,

AVAR

sA =

and {

} are the parameters of the random shock form of the difference equation. Note that the forecasts are computed for lead times l = 1, 2, …, L at origins t = (n