NAG Library Routine Document
G13ADF calculates preliminary estimates of the parameters of an autoregressive integrated moving average (ARIMA) model from the autocorrelation function of the appropriately differenced times series.
|SUBROUTINE G13ADF (
||MR, R, NK, XV, NPAR, WA, NWA, PAR, RV, ISF, IFAIL)
||MR(7), NK, NPAR, NWA, ISF(4), IFAIL
||R(NK), XV, WA(NWA), PAR(NPAR), RV
Preliminary estimates of the
non-seasonal autoregressive parameters
non-seasonal moving average parameters
may be obtained from the sample autocorrelations relating to lags
, of the differenced
is assumed to follow a (possibly) seasonal ARIMA model (see Section 3
in G13AEF for the specification of an ARIMA model).
are the solutions to the equations
, are obtained from the solutions to the equations
(Cramer Wold-factorization), by setting
are the ‘covariances’ modified in a two stage process by the autoregressive parameters.
seasonal autoregressive parameters
seasonal moving average parameters
are estimated in the same way as the non-seasonal parameters, but each
is replaced in the calculation by
is the seasonal period.
An estimate of the residual variance is obtained by successively reducing the sample variance, first for non-seasonal, and then for seasonal, parameter estimates. If moving average parameters are estimated, the variance is reduced by a multiplying factor of , but otherwise by .
Box G E P and Jenkins G M (1976) Time Series Analysis: Forecasting and Control (Revised Edition) Holden–Day
- 1: MR() – INTEGER arrayInput
On entry: the orders vector of the ARIMA model whose parameters are to be estimated. , , and refer respectively to the number of autoregressive , moving average , seasonal autoregressive and seasonal moving average parameters. , and refer respectively to the order of non-seasonal differencing, the order of seasonal differencing and the seasonal period.
- if , ;
- if , .
- 2: R(NK) – REAL (KIND=nag_wp) arrayInput
On entry: the autocorrelations (starting at lag ), which must have been calculated after the time series has been appropriately differenced.
, for .
- 3: NK – INTEGERInput
: the maximum lag of the autocorrelations in array R
- 4: XV – REAL (KIND=nag_wp)Input
On entry: the series sample variance, calculated after appropriate differencing has been applied to the series.
- 5: NPAR – INTEGERInput
: the exact number of parameters specified in the model by array MR
- 6: WA(NWA) – REAL (KIND=nag_wp) arrayWorkspace
- 7: NWA – INTEGERInput
On entry: the amount of workspace available.
if and and , .
- 8: PAR(NPAR) – REAL (KIND=nag_wp) arrayOutput
: the first NPAR
elements of PAR
contain the preliminary estimates of the ARIMA model parameters, in standard order.
- 9: RV – REAL (KIND=nag_wp)Output
On exit: an estimate of the residual variance of the preliminarily estimated model.
- 10: ISF() – INTEGER arrayOutput
: contains success/failure indicators, one for each of the four types of parameter (autoregressive, moving average, seasonal autoregressive, seasonal moving average).
The indicator has the interpretation:
||No parameter of this type is in the model.
||Parameters of this type appear in the model and satisfactory preliminary estimates of this type were obtained.
||Parameters of this type appear in the model but satisfactory preliminary estimates of this type were not obtainable. The estimates of this type of parameter were set to in array PAR.
- 11: IFAIL – INTEGERInput/Output
must be set to
. If you are unfamiliar with this parameter you should refer to Section 3.3
in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
is recommended. If the output of error messages is undesirable, then the value
is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
. When the value is used it is essential to test the value of IFAIL on exit.
unless the routine detects an error or a warning has been flagged (see Section 6
6 Error Indicators and Warnings
If on entry
, explanatory error messages are output on the current error message unit (as defined by X04AAF
Errors or warnings detected by the routine:
On entry, the orders vector MR
is invalid. One of the constraints in Section 5
has been violated.
On entry, . There are not enough autocorrelations to enable the required model to be estimated.
|On entry,||at least one element of R lies outside the range .|
|On entry,||the workspace array WA is too small. See Section 5 for the minimum size formula.|
Satisfactory parameter estimates could not be obtained for all parameter types in the model. Inspect array ISF
for indicators of the parameter type(s) which could not be estimated.
The performance of the algorithm is conditioned by the roots of the autoregressive and moving average operators. If these are not close to unity in modulus, the errors, , should satisfy where is machine precision.
The time taken by G13ADF is approximately proportional to
This example reads the sample autocorrelations to lag
and the sample variance of the lagged and doubly differenced series of airline passenger totals (Box and Jenkins example series G (see Box and Jenkins (1976)
)). Preliminary estimates of the parameters of the
model are obtained by a call to G13ADF.
9.1 Program Text
Program Text (g13adfe.f90)
9.2 Program Data
Program Data (g13adfe.d)
9.3 Program Results
Program Results (g13adfe.r)