Chapter 8: Time Series Analysis and Forecasting

SPWF

Computes the Wiener forecast operator for a stationary stochastic process.

Required Arguments

W — Vector of length NOBS containing the stationary time series.   (Input)

WNADJ — White noise adjustment factor.   (Input)
WNADJ must be greater than or equal to zero.

EPS — Bound on the normalized mean square error.   (Input)
EPS must be in the range (0, 1) inclusive.

MLFOP — Maximum length of the forecast operator.   (Input)
MLFOP must be greater than or equal to one and less than NOBS.

LFOP — Length of the estimated forecast operator.   (Output)

FOP — Vector of length LFOP containing the estimated forecast operator coefficients.   (Output)

Optional Arguments

NOBS — Number of observations in the stationary time series W.   (Input)
NOBS must be greater than or equal to two.
Default: NOBS = size (W,1).

IWMEAN — Option for estimation of the mean of W.   (Input)
Default: IWMEAN = 1.

IWMEAN

Action

0

WMEAN is user specified.

1

WMEAN is set equal to the arithmetic mean of W.

WMEAN — Estimate of the mean of the time series W.   (Input, if IWMEAN = 0; output, if IWMEAN = 1)
WMEAN is used to center the time series W prior to estimation of the forecast operator.
Default: WMEAN = 0.0.

FORTRAN 90 Interface

Generic:                              CALL SPWF (W, WNADJ, EPS, MLFOP, LFOP, FOP [,…])

Specific:                             The specific interface names are S_SPWF and D_SPWF.

FORTRAN 77 Interface

Single:                                CALL SPWF (NOBS, W, IWMEAN, WMEAN, WNADJ, EPS, MLFOP, LFOP, FOP)

Double:                              The double precision name is DSPWF.

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, …, n.

Let

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

Consider the autoregressive model of order k defined by

where

and

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

Note that the variance

used in the fitting algorithm is adjusted by the amount δ = WNADJ according to

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 ɸ = (ɸ1, ɸ2, …, ɸk*)T for
k* = LFOP is contained in FOP. See also Craddock (1969).

Comments

1.         Workspace may be explicitly provided, if desired, by use of S2WF/DS2WF. The reference is:

CALL S2WF (NOBS, W, IWMEAN, WMEAN, WNADJ, EPS, MLFOP, LFOP, FOP, CW, WK)

The additional arguments are as follows:

CW — Vector of length NOBS containing the centered time series W.   (Output)

WK — Vector of length 2 * MLFOP + 1.   (Output)

2.         Informational error

Type Code

3         5                  No operator could be found of length less than or equal to MLFOP that produced a normalized mean square error less than EPS.

3          The length of the forecast operator is determined by the arguments EPS and MLFOP. Iteration to a longer forecast operator stops when either the normalized mean square error is less than EPS, or the operator reaches the maximum allowable length, MLFOP.

4.         The white noise adjustment factor, WNADJ, is used to modify the the estimate of the variance of the time series W used in the computation of the autocorrelation function of W. In the absence of white noise, WNADJ should be set to zero.

Example

Consider the Wφlfer 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. Application of routine SPWF to these data produces the following results:

 

      USE GDATA_INT

      USE SPWF_INT

      USE UMACH_INT

 

      IMPLICIT   NONE

      INTEGER    MLFOP, NOBS, NOUT, IMEAN, I

      PARAMETER  (MLFOP=1, NOBS=100)

!     INTEGER    I, IMEAN, LFOP, NCOL, NOUT, NROW

 

      REAL       EPS, FOP(MLFOP), RDATA(176,2), W(NOBS), WMEAN, WNADJ

      REAL       NROW, NCOL, LFOP

!

      EQUIVALENCE (W(1), RDATA(22,2))

!                                 Wolfer Sunspot Data for

!                                 years 1770 through 1869

      CALL GDATA (2, RDATA, NROW, NCOL)

!                                 Center on arithmetic mean

      IMEAN = 0

      WMEAN = 46.976

!                                 White noise adjustment

      WNADJ = 0.0

!                                 Bound on normalized MSE

      EPS = 0.1

!                                 Determine autoregressive model

      CALL SPWF (W, WNADJ, EPS, MLFOP, LFOP, FOP, IWMEAN=IMEAN, &

           WMEAN=WMEAN)

!                                 Print results

      CALL UMACH (2, NOUT)

      WRITE (NOUT,99997) LFOP

99997 FORMAT (/, 1X, 'Forecast operator length, LFOP = ', I2)

      WRITE (NOUT,99998)

99998 FORMAT (/, 1X, ' I           FOP(I)')

      DO 10  I=1, LFOP

         WRITE (NOUT,99999) I, FOP(I)

99999    FORMAT (1X, I2, 2X, F12.4)

   10 CONTINUE

!

      END

Output

 

*** WARNING  ERROR 5 from SPWF.  No operator could be found of length less
***          than or equal to 1 which produced a normalized mean square
***          error less than 1.000000E-01.

Forecast operator length, LFOP =  1

I           FOP(I)
1        0.8063



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