g13bef fits a multi-input model relating one output series to the input series with a choice of three different estimation criteria: nonlinear least squares, exact likelihood and marginal likelihood. When no input series are present, g13bef fits a univariate ARIMA model.
Note: the orders appearing in each of the transfer function models and the ARIMA model are not necessarily the same; is the result of applying non-seasonal differencing of order and seasonal differencing of seasonality and order to the series : the differenced series is then of length ; the constant term parameter may optionally be held fixed at its initial value (usually, but not necessarily zero) rather than being estimated.
For the purpose of defining an estimation criterion it is assumed that the series is a sequence of independent Normal variates having mean and variance . An allowance has to be made for the effects of unobserved data prior to the observation period. For the noise component an allowance is always made using a form of backforecasting.
For each transfer function input, you have to decide what values are to be assumed for the pre-period terms and which are in theory necessary to re-create the component series , during the estimation procedure.
The first choice is to assume that all these values are zero. In this case, in order to avoid undesirable transient distortion of the early values , you are advised first to correct the input series by subtracting from all the terms a suitable constant to make the early values , close to zero. The series mean is one possibility, but for a series with strong trend the constant might be simply .
The second choice is to treat the unknown pre-period terms as nuisance parameters and estimate them along with the other parameters. This choice should be used with caution. For example, if and , it is equivalent to fitting to the data a decaying geometric curve of the form , for , along with the other inputs, this being the form of the transient. If the output contains a strong trend of this form, which is not otherwise represented in the model, it will have a tendency to influence the estimate of away from the value appropriate to the transfer function model.
In most applications the first choice should be adequate, with the option possibly being used as a refinement at the end of the modelling process. The number of nuisance parameters is then , with a corresponding loss of degrees of freedom in the residuals. If you align the input with the output by using in its place the shifted series , then setting in the transfer function model, there is some improvement in efficiency. On some occasions when the model contains two or more inputs, each with estimation of pre-period nuisance parameters, these parameters may be co-linear and lead to failure of the routine. The option must then be ‘switched off’ for one or more inputs.
3.2The Estimation Criterion
This is a measure of how well a proposed set of parameters in the transfer function and noise ARIMA models matches the data. The estimation routine searches for parameter values which minimize this criterion. For a proposed set of parameter values it is derived by calculating
(i)the components as the responses to the input series using the equations (a) or (b) above,
(ii)the discrepancy between the output and the sum of these components, as the noise
(iii)the residual series from by reversing the recursive equations (i), (ii) and (iii) above.
This last step again requires treatment of the effect of unknown pre-period values of and other terms in the equations regenerating .
This is identical to the treatment given in Section 3 in g13aef, and leads to a criterion which is a sum of squares function , of the residuals .
It may be shown that the finite algorithm presented there is equivalent to taking the infinite set of past values , as (linear) nuisance parameters.
There is no loss of degrees of freedom however, because the sum of squares function may be expressed as including the corresponding set of past residuals; see page 273 of Box and Jenkins (1976), who prove that
The function is the first of the three possible criteria, and is quite adequate for moderate to long series with no seasonal parameters. The second is the exact likelihood criterion which considers the past set not as simple nuisance parameters, but as unobserved random variables with known distribution. Calculation of the likelihood of the observed set requires theoretical integration over the range of the past set. Fortunately this yields a criterion of the form (whose minimization is equivalent to maximizing the exact likelihood of the data), where is exactly as before, and the multiplier is a function calculated from the ARIMA model parameters. The value of is always , and tends to for any fixed parameter set as the sample size tends to . There is a moderate computational overhead in using this option, but its use avoids appreciable bias in the ARIMA model parameters and yields a better conditioned estimation problem.
The third criterion of marginal likelihood treats the coefficients of the simple inputs in a manner analogous to that given to the past set . These coefficients, together with the constant term used to represent the mean of , are in effect treated as random variables with highly dispersed distributions. This leads to the criterion again, but with a different value of which now depends on the simple input series values . In the presence of a moderate to large number of simple inputs, the marginal likelihood criterion can counteract bias in the ARIMA model parameters which is caused by estimation of the simple inputs. This is particularly important in relatively short series.
g13bef can be used with no input series present, to estimate a univariate ARIMA model for the output alone. The marginal likelihood criterion is then distinct from exact likelihood only if a constant term is being estimated in the model, because this is treated as an implicit simple input.
3.3The Estimation Procedure
This is the minimization of the estimation criterion or objective function (for deviance). The routine uses an extension of the algorithm of Marquardt (1963). The step size in the minimization is inversely related to a parameter , which is increased or decreased by a factor at successive iterations, depending on the progress of the minimization. Convergence is deemed to have occurred if the fractional reduction of in successive iterations is less than a value , while .
Certain model parameters (in fact all excluding the s) are subject to stability constraints which are checked throughout to within a specified tolerance multiple of machine accuracy. Using the least squares criterion, the minimization may halt prematurely when some parameters ‘stick’ at a constraint boundary. This can happen particularly with short seasonal series (with a small number of whole seasons). It will not happen using the exact likelihood criterion, although convergence to a point on the boundary may sometimes be rather slow, because the criterion function may be very flat in such a region. There is also a smaller risk of a premature halt at a constraint boundary when marginal likelihood is used.
A positive, or zero number of iterations can be specified. In either case, the value of the objective function at iteration zero is presented at the initial parameter values, except for estimation of any pre-period terms for the input series, backforecasts for the noise series, and the coefficients of any simple inputs, and the constant term (unless this is held fixed).
At any later iteration, the value of is computed after re-estimation of the backforecasts to their optimal values, corresponding to the model parameters presented at that iteration. This is not true for any pre-period terms for the input series which, although they are updated from the previous iteration, may not be precisely optimal for the parameter values presented, unless convergence of those parameters has occurred. However, in the case of marginal likelihood being specified, the coefficients of the simple inputs and the constant term are also re-estimated together with the backforecasts at each iteration, to values which are optimal for the other parameter values presented.
The residual variance is taken as , where (total number of parameters estimated), is the residual degrees of freedom.
The pre-period nuisance parameters for the input series are included in the reduction of , as is the constant if it is estimated.
The covariance matrix of the vector of model parameter estimates is given by
where is the linearized least squares matrix taken from the final iteration of the algorithm of Marquardt. From this expression are derived the vector of standard deviations, and the correlation matrix of parameter estimates. These are approximations which are only valid asymptotically, and must be treated with great caution when the parameter estimates are close to their constraint boundaries.
The residual series is available upon completion of the iterations over the range corresponding to the differenced noise series .
Because of the algorithm used for backforecasting, these are only true residuals for , provided this is positive. Estimation of pre-period terms for the inputs will also tend to reduce the magnitude of the early residuals, sometimes severely.
The model component series and may optionally be returned in place of the supplied series values, in order to assess the effects of the various inputs on the output.
For the purpose of constructing forecasts of the output series at future time points using g13bhf, it is not necessary to use the whole set of observations and , for . It is sufficient to retain a limited set of quantities constituting the ‘state set’ as follows: for each series which appears with lagged subscripts in equations (a), (b), (i), (ii) and (iii) above, include the values at times for up to the maximum lag associated with that series in the equations. Note that (i) implicitly includes past values of and intermediate differences of such as .
If later observations of the series become available, it is possible to update the state set (without re-estimating the model) using g13bgf. If time series data is supplied with a previously estimated model, it is possible to construct the state set (and forecasts) using g13bjf.
Box G E P and Jenkins G M (1976) Time Series Analysis: Forecasting and Control (Revised Edition) Holden–Day
Marquardt D W (1963) An algorithm for least squares estimation of nonlinear parameters J. Soc. Indust. Appl. Math.11 431
1: – Integer arrayInput
On entry: the orders vector of the ARIMA model for the output noise component.
, , 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: – IntegerInput
On entry: the total number of input and output series. There may be any number of input series (including none), but always one output series.
if there are no parameters in the model (that is, and ), .
3: – Integer arrayInput
On entry: the transfer function model orders , and of each of the input series. The order parameters for input series are held in column . Row holds the value , row 2 holds the value and row 3 holds the value . For a simple input, .
Row 4 holds the value , where for a simple input, for a transfer function input for which no allowance is to be made for pre-observation period effects, and for a transfer function input for which pre-observation period effects will be treated by estimation of appropriate nuisance parameters.
When , any nonzero contents of rows 1, 2, and 3 of column are ignored.
, for .
4: – Real (Kind=nag_wp) arrayInput/Output
On entry: initial values of the multi-input model parameters. These are in order, firstly the ARIMA model parameters: values of parameters, values of parameters, values of parameters and values of parameters. These are followed by initial values of the transfer function model parameters , for the first of any input series and similarly for each subsequent input series. The final component of para is the initial value of the constant , whether it is fixed or is to be estimated.
On exit: the latest values of the estimates of these parameters.
5: – IntegerInput
On entry: the exact number of , and parameters.
, the summation being over all the supplied in mt. must be included, whether fixed or estimated.
6: – IntegerInput
On entry: must be set to if the constant is to remain fixed at its initial value, and if it is to be estimated.
7: – IntegerInput
On entry: the (common) length of the original, undifferenced input and output time series.
8: – Real (Kind=nag_wp) arrayInput/Output
On entry: the columns of xxy must contain the nxxy original, undifferenced values of each of the input series and the output series in that order.
If , the columns of xxy hold the corresponding values of the input component series in place of and the output noise component in place of , in that order.
9: – IntegerInput
On entry: the first dimension of the array xxy as declared in the (sub)program from which g13bef is called.
10: – IntegerInput
On entry: indicates the likelihood option.
Gives least squares.
Gives exact likelihood.
Gives marginal likelihood.
, or .
11: – IntegerInput
On entry: the maximum required number of iterations.
No change is made to any of the model parameters in array para except that the constant (if ) and any relating to simple input series are estimated. (Apart from these, estimates are always derived for the nuisance parameters relating to any backforecasts and any pre-observation period effects for transfer function inputs.)
12: – IntegerInput
On entry: must be set to if the routine is to use the input values of zsp in the minimization procedure, and to any other value if the default values of zsp are to be used.
13: – Real (Kind=nag_wp) arrayInput/Output
On entry: if , then zsp must contain the four values used to control the strategy of the search procedure.
Contains , the value used to constrain the magnitude of the search procedure steps.
Contains , the multiplier which regulates the value of .
Contains , the value of the stationarity and invertibility test tolerance factor.
Contains , the value of the convergence criterion.
If before entry, default values of zsp are supplied by the routine. These are , , and , respectively.
On exit: contains the values, default or otherwise, used by the routine.
14: – IntegerOutput
On exit: the number of iterations carried out.
Indicates that the only estimates obtained up to this point have been for the nuisance parameters relating to backforecasts, unless the marginal likelihood option is used, in which case estimates have also been obtained for simple input coefficients and for the constant (if ). This value of itc usually indicates a failure in a consequent step of estimating transfer function input pre-observation period nuisance parameters.
Indicates that estimates have been obtained up to this point for the constant (if ), for simple input coefficients and for the nuisance parameters relating to the backforecasts and to transfer function input pre-observation period effects.
15: – Real (Kind=nag_wp) arrayOutput
On exit: the npara values of the standard deviations corresponding to each of the parameters in para. When the constant is fixed its standard deviation is returned as zero. When the values of para are valid, the values of sd are usually also valid. However, if an exit value of , or , then the contents of sd will be indeterminate.
16: – Real (Kind=nag_wp) arrayOutput
On exit: the first npara rows and columns of cm contain the correlation coefficients relating to each pair of parameters in para. All coefficients relating to the constant will be zero if the constant is fixed. The contents of cm will be indeterminate under the same conditions as sd.
17: – IntegerInput
On entry: the first dimension of the array cm as declared in the (sub)program from which g13bef is called.
18: – Real (Kind=nag_wp)Output
On exit: the residual sum of squares, , at the latest set of valid parameter estimates.
19: – Real (Kind=nag_wp)Output
On exit: the objective function, , at the latest set of valid parameter estimates.
20: – IntegerOutput
On exit: the number of degrees of freedom associated with .
21: – IntegerInput
On entry: must not be set to , if the values of the input component series and the values of the output noise component are to overwrite the contents of xxy on exit, and must be set to if xxy is to remain unchanged.
22: – Real (Kind=nag_wp) arrayOutput
On exit: the values of the residuals relating to the differenced values of the output series. The remainder of the first nxxy terms in the array will be zero.
On entry: the dimension of the array sttf as declared in the (sub)program from which g13bef is called.
, where over all input series for which .
25: – IntegerOutput
On exit: the number of values in the state set array sttf.
26: – Real (Kind=nag_wp) arrayOutput
27: – IntegerInput
28: – Integer arrayOutput
29: – IntegerInput
These arguments are no longer accessed by g13bef. Workspace is provided internally by dynamic allocation instead.
30: – IntegerInput
On entry: must not be set to , if it is required to monitor the course of the optimization. The course of the optimization is monitored by printing out at each iteration the iteration count (itc), the residual sum of squares (s), the objective function (d) and a description and value for each of the parameters in the para array. The descriptions are PHI for , THETA for , SPHI for , STHETA for , OMEGA/SI for in a simple input, OMEGA for in a transfer function input, DELTA for and CONSTANT for . In addition SERIES 1, SERIES 2, etc. indicate the input series relevant to the OMEGA and DELTA parameters.
kpriv must be set to if the print-out of the above information is not required.
31: – IntegerInput/Output
On entry: ifail must be set to , or to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of means that an error message is printed while a value of means that it is not.
If halting is not appropriate, the value or is recommended. If message printing is undesirable, then the value is recommended. Otherwise, the value is recommended since useful values can be provided in some output arguments even when on exit. When the value or is used it is essential to test the value of ifail on exit.
On exit: unless the routine detects an error or a warning has been flagged (see Section 6).
6Error Indicators and Warnings
If on entry or , explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
Note: in some cases g13bef may return useful information.
On entry, .
Constraint: , or .
On entry, .
Constraint: or .
On entry, and .
On entry, and .
On entry, .
On entry, .
On entry, and there are no parameters in the model.
On entry, and .
Constraint: , or .
On entry, .
On entry, and .
Constraint: npara, kfc, mr and mt must be consistent.
The routine has failed to converge after iterations. If steady decreases in the objective function, , were monitored up to the point where this exit occurred, then the exit probably occurred because nit was set too small, so the calculations should be restarted from the final point held in para.
On entry, . isttf is too small and the process failed to converge after nit iterations.
An unexpected error has been triggered by this routine. Please
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
The computation used is believed to be stable.
8Parallelism and Performance
g13bef is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g13bef makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
The time taken by g13bef is approximately proportional to .
After the full iterations, the following are computed and printed out: the final values of the para parameters and their standard errors, the correlation matrix, the residuals for the differenced values, the values of and , the values of the state set and the number of degrees of freedom.