MAMME - IMSL Fortran 905.0 Beta Documentation

IMSL STAT/LIBRARY Chapter 8: Time Series Analysis and Forecasting · 723 Wt= q0 + f1Wt-1 + ¼ + fpWt-p + At - q1At-1 - ¼ - qqAt-q, t Î{0, ± 1, ±2, K}

where p = NPAR and q = NPMA. Since Wt depends only on the innovations At that have occurred up through time t, the p autoregressive parameters are related to the autocovariances of lags k = q + 1, …, q + p by the set of equations

s(q + 1) = f1s(q) + f2s(q - 1) + ¼ + fps(q - p + 1) s(q + 2) = f1s(q + 1) + f2s(q) + ¼ + fps(q - p + 2) .

. .

s(q + p) = f1s(q + p - 1) + f2s(q + p - 2) + ¼ + fps(q)

This general system of linear equations is called the extended Yule-Walker equations. For q = 0, the system is referred to as the Yule-Walker equations. The equivalent matrix version is given by

Sf = s where

( ,1 , )

( ) , 1, ,

( ) 1, ,

T p ij

q i j i j p

q i i p

f f f

s s s

å = + - =

= + =

K K The overall constant q0 is defined by

( )

0

1

_i^p _i

0

p p q m

m = f

ìï =

= íïî - å >

where

is the mean of W

.

In practice, the autocovariance function is estimated by the sample autocovariances

( )

ˆ

k s

for k = 1, …, K. The solution of the extended Yule-Walker equations using these sample moments yields the method of moments estimates of the autoregressive parameters. The overall constant may then be estimated given an estimate of

. Note that the extended Yule-Walker equations may be analogously defined in terms of autocorrelations instead of autocovariances.

See Box and Jenkins (1976, pages 189–191) for some comments concerning the initial

estimation of autoregressive parameters using the Yule-Walker equations.

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

Required Arguments

MAXLAG

— Maximum lag of the sample autocovariances of the time series W. (Input)

MAXLAG

must be greater than or equal to

NPAR

+

NPMA

.

ACV

— Vector of length

MAXLAG

+ 1 containing the sample autocovariances of W. (Input) The k-th sample autocovariance of W is denoted by

ACV

(k), k = 0, 1, …,

MAXLAG

.

PAR

— Vector of length

NPAR

containing the estimates of the autoregressive parameters.

(Input)

PMA

— Vector of length

NPMA

containing the estimates of the moving average parameters.

(Output) Optional Arguments

IPRINT

— Printing option. (Input) Default

: IPRINT

= 0.

IPRINT Action

0 No printing is performed.

1 Prints the estimates of the moving average parameters.

NPAR

— Number of autoregressive parameters. (Input)

NPAR

must be greater than or equal to zero.

Default:

NPAR

= size (

PAR

,1).

RELERR

— Stopping criterion for use in the nonlinear equation solver. (Input)

If

RELERR

= 0.0, then the default value

RELERR

= 100.0

*AMACH

(4) is used. See the documentation for routine

AMACH

(in the Reference Material section of this manual).

Default:

RELERR

= 0.0.

MAXIT

— The maximum number of iterations allowed in the nonlinear equation solver.

(Input)

If

MAXIT

= 0, then the default value

MAXIT

= 200 is used.

Default:

MAXIT

= 0.

NPMA

— Number of moving average parameters. (Input)

NPMA

must be greater than or equal to one.

Default:

NPMA

= size (

PMA

,1).

FORTRAN 90 Interface

Generic:

CALL MAMME(MAXLAG, ACV, PAR, PMA [,…])

Specific: The specific interface names are

S_MAMME

and

D_MAMME

.

IMSL STAT/LIBRARY Chapter 8: Time Series Analysis and Forecasting · 725

FORTRAN 77 Interface

Single:

CALL MAMME (MAXLAG, ACV, IPRINT, NPAR, PAR, RELERR, MAXIT, NPMA, PMA)

Double: The double precision name is

DMAMME

. Example

Consider the Wölfer Sunspot Data (Box and Jenkins 1976, page 530) consisting of the number of sunspots observed each year from 1770 through 1869. Routine

MAMME

is invoked to compute the method of moments estimates for the moving average parameter of an ARMA(2,1) model given the sample autocovariances computed from routine

ACF (page 697) and given the

estimated autoregressive parameters computed from routine

ARMME (page 720).

USE GDATA_INT USE ACF_INT USE ARMME_INT USE MAMME_INT

INTEGER IMEAN, IPRINT, ISEOPT, LDX, MAXLAG, NDX, NOBS, &

NOPRIN, NPAR, NPMA

PARAMETER (IMEAN=1, IPRINT=1, ISEOPT=0, LDX=176, MAXLAG=4, &

NDX=2, NOBS=100, NOPRIN=0, NPAR=2, NPMA=1)

INTEGER MAXIT, NCOL, NROW

REAL AC(0:MAXLAG), ACV(0:MAXLAG), PAR(2), PMA(1), &

RDATA(LDX,NDX), RELERR, SEAC(1), W(100), WMEAN

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

! Wolfer Sunspot Data for

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

! Compute sample ACV CALL ACF (W, MAXLAG, AC, ACV=ACV)

! Compute estimates of autoregressive

! parameters for ARMA(2,1) model CALL ARMME (MAXLAG, ACV, NPMA, NPAR, PAR)

! Convergence parameters

! Compute estimate of moving average

! parameter for ARMA(2,1) model CALL MAMME (MAXLAG, ACV, PAR, PMA, IPRINT=IPRINT)

END

Output

Output PMA from MAMME/M2MME -0.1241

Comments

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

M2MME/DM2MME

. The

reference is:

726 · Chapter 8: Time Series Analysis and Forecasting IMSL STAT/LIBRARY CALL M2MME (MAXLAG, ACV, IPRINT, NPAR, PAR, RELERR, MAXIT, NPMA, PMA, PARWK, ACVMOD, TAUINI, TAU, FVEC, FJAC, R, QTF, WKNLN)

The additional arguments are as follows:

PARWK

— Work vector of length equal to

NPAR

+ 1.

ACVMOD

— Work vector of length equal to

NPMA

+ 1.

TAUINI

— Work vector of length equal to

NPMA

+ 1.

TAU

— Work vector of length equal to

NPMA

+ 1.

FVEC

— Work vector of length equal to

NPMA

+ 1.

FJAC

— Work vector of length equal to (

NPMA

+ 1)

.

— Work vector of length equal to (

NPMA

+ 1) * (

NPMA

+ 2)/2.

QTF

— Work vector of length equal to

NPMA

+ 1.

WKNLN

— Work vector of length equal to 5 * (

NPMA

+ 1).

2. Informational error Type Code

4 1 The nonlinear equation solver did not converge to

RELERR

within

MAXIT

iterations.

3. The sample autocovariance function may be computed using the routine

ACF (page 697).

4. The autoregressive parameter estimates may be computed using the routine

ARMME (page 720).

Description

Routine

MAMME

estimates the moving average parameters of an ARMA process based on a system of nonlinear equations given K =

MAXLAG

autocovariances

(k) for k = 1, …, K and

p = NPAR

autoregressive parameters

for i = 1, …, p.

Suppose the time series {W

} is generated by an ARMA(p,q) model

(B)W

=

+

(B)A

,

t Î

2 {0,

1,

2,

} where p =

NPAR

and q =

NPMA

Let

' ( )

t t

W =f B W

then the autocovariances of the derived moving average process Wt = q(B)At are given by

IMSL STAT/LIBRARY Chapter 8: Time Series Analysis and Forecasting · 727

( )

0 0 0

( ) 0

( ) _i^p ^p_j _i _j 1, 1

k p

k k i j p

s s ₌ ₌ f f s f

ìï =

¢ = íå åïî + - ³ º -

where s(k) denotes the autocovariance function of the original Wt process. The iterative procedure for determining the moving average parameters is based on the relation

2 2 2

1 1

(1 ) 0

( ) ( ) 1

q A

k k q k q A

k k

q q s

s q q q + q q s-

ì + + + =

¢ = íïïî - + + + ³

L L Let t = (t0, t1, …, tq)T and f = (f0, f1, …, fq)T where

/ 1, ,

A j

j q

t sq t ìï =

= í-ïî = K

and

( ) 0,1, ,

q j

j i i j

f ^- t t₊ s j j q

å

- ¢ = ^K

Then, the value of t at the (i + 1)-th iteration is determined by ti⁺¹ = ti -(Ti)^-1fi The estimation procedure begins with the initial value

0 ( (0), 0, , 0)^T

t = s¢ K

and terminates at iteration i when either ||f i|| is less than RELERR or i equals MAXIT. The moving average parameters are determined from the final estimate of t by setting qj = -tj/t0 for j = 1,

…, q. The random shock variance is determined according to

2 1

2 0

(0) ( ) 0

i i

i q q

s f s

s t

ì - å= =

= íïïî ³

In practice, both the autocovariances and the autoregressive parameters are estimated. The solution of the system of nonlinear equations using these sample moments yields the method of moments estimates of the moving average parameters and the random shock variance. Note that autocorrelations r(k) may be used instead of autocovariances s(k) to compute s¢(k) for k = 1,

…, K. See Box and Jenkins (1976, pages 203–204) for additional motivation concerning the initial estimation of moving average parameters using a Newton-Raphson algorithm.

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