TSM directive

Declares one or more TSM data structures.

Option

MODELTYPE = string token	Type of model (arima, transfer); default arim

Parameters

IDENTIFIER = identifiers	Identifiers of the TSMs
ORDERS = variates	Orders of the autoregressive, integrated and moving-average parts of each TSM
PARAMETERS = variates	Parameters of each TSM
LAGS = variates	Lags, if not default

Description

The TSM structure stores a time-series model which you can use with directives such as TFIT for Box-Jenkins modelling of time series. The information that you give to specify the model is stored in two variates, called the orders and the parameters; an optional third variate contains lags. The elements of a TSM are thus

[1] or ['Orders'];

[2] or ['Parameters'];

[3] or ['Lags'].

The labels of the TSM can be specified in either upper or lower case, or any mixture.

To declare a TSM you use the TSM directive. You set the type of model by the MODELTYPE option. The default setting defines an ARIMA model. This is an equation relating the present value y_t of an observed time series to past values. The equation includes lagged values not only of the series itself, but also of an unobserved series of innovations, a_t ; you can interpret the innovations as the error in predicting y_t from past values y_t–₁, y_t–₂ …. The usual statistical model assumes that the innovations are a series of independent Normal deviates with mean zero and constant variance. The residuals obtained from fitting the model can be used to estimate the innovations.

Using the notation of Box & Jenkins (1970), the simple non-seasonal ARIMA model for the time series y_t is

φ(B) {∇^dy_t^(λ) – c} = θ(B)a_t

where B is the backward shift operator B^py_t =y_t-p ,

∇ is the differencing operator ∇y_t =y_t–y_t–₁ , ∇^dy_t =∇^d-1 (y_t–y_t–₁ ), and

φ(B) = 1 – φ₁B – … – φ_pB^p

θ(B) = 1 – θ₁B – … – θ_qB^q

The parameter λ specifies a Box-Cox power transformation defined by

y_t^(λ) = (y_t^λ – 1) / λ, λ ≠ 0

y_t⁽⁰⁾ = log(y_t)

However, in the default case when λ is fixed and not estimated, the value λ=1 implies no transformation and then y_t⁽¹⁾=y_t rather than y_t-1. If λ≠1 or if λ is to be estimated, then Genstat will not let you have values of y_t ≤0. The usual case however is that λ=1 and is not to be estimated, so that y_t may take any values.

The ORDERS parameter is a list of variates, one for each of the models. For each simple ARIMA model, the variate contains the three values p, d and q.

The PARAMETERS parameter is a list of variates, one for each of the models. For each simple ARIMA model, the variate contains (3+p+q) values: λ, c, σ_a², φ₁…φ_p, θ₁…θ_q. You must always include the first three parameters. The parameter σ_a² is the innovation variance.

Whenever a TSM is used, Genstat checks its values. The orders must all be non-negative. The parameters λ and c can take any values, but σ_a² must be non-negative. The next p+q values specify the autoregressive and moving-average parameters: they must satisfy the stationarity and invertibility conditions for ARIMA models (see Box & Jenkins 1970). An exception is that before estimation the model parameters may be unset, in which case Genstat sets them to default values. You can omit the PARAMETERS parameter, in which case an unnamed structure is defined to contain the default values. However, you should usually specify the variate of parameters, and if possible assign good preliminary values before estimation (see FTSM) as this will speed up the model fitting process.

The LAGS parameter is a list of variates, one for each of the models. For each simple ARIMA model, this variate contains p+q values, one corresponding to each of the autoregressive and moving-average parameters. Genstat then modifies the ARIMA model by defining

φ(B) = 1 – φ₁ B**l₁ – … – φ_p B**l_p

θ(B) = 1 – θ₁ B**m₁ – … – θ_q B**m_q

The LAGS parameter for this model contains l₁…l_p, m₁…m_q. The sequences of lags l₁…l_p must be positive integers that are strictly increasing; the default values are 1…p if LAGS is not set. The same rule applies to m₁…m_q.

The seasonal ARIMA model for the time series y_t is an extension of the simple model, to the form

φ(B) Φ(B^s) { ∇^d∇_s^Dy_t^(λ) – c } = θ(B) Θ(B^s) a_t

where the extra, seasonal, operators associated with seasonal period s are of three types:

Φ(B^s) = 1 – Φ₁ B^s – … – Φ_P B**P_s

which is seasonal autoregression of order P;

∇_s^D

which is seasonal differencing of order D; and

Θ(B^s) = 1 – Θ₁ B^s – … – Θ_Q B**Q_s

which is seasonal moving average of order Q.

When seasonal terms are to be included, you must extend the ORDERS parameter so that it contains p, d, q, P, D, Q and s. Even if the non-seasonal part of the model has p=d=q=0, these parameters must still be included at the beginning of the list. The seasonal orders must satisfy P≥0, D≥0, Q≥0 and s≥1.

You must also extend the PARAMETERS parameter to contain:

λ, c, σ_a², φ₁…φ_p, θ₁…θ_q, Φ₁…Φ_P, Θ₁…Θ_Q

You can modify the seasonal model to allow other lags:

Φ(B^s) = 1 – Φ₁ B**L₁ – … – Φ_P B**L_p

Θ(B^s) = 1 – Θ₁ B**M₁ – … – Θ_Q B**M_Q

The sequence of lags L₁…L_P must be strictly increasing and must be positive-integer multiples of the period s; the default values are s, 2s … Ps. The same rules apply to M₁…M_Q. For any seasonal model, you must extend the LAGS parameter, if supplied, so that it contains

l₁ … l_p, m₁ … m_q, L₁ … L_P, M₁ … M_Q.

You can use multiple seasonal periods, by extending the variate of ORDERS with further seasonal orders P′, D′, Q′ and s′. You must correspondingly extend the variates of PARAMETERS and LAGS. It is also possible to set the seasonal periods to 1, which means you can estimate non-seasonal models with factored operators.

You can declare an ORDERS variate to have more values than is necessary, provided that the extra values are filled with zeroes, and that the number of values is 3+4k, k being the number of seasonal periods. The same applies to PARAMETERS and LAGS variates, except that Genstat ignores the extra values whatever they may be. Thus you can extend a simple model to a seasonal model, simply by resetting the extra values.

Setting MODELTYPE=transferfunction defines a transfer-function model. The simple non-seasonal transfer-function model relates a component z_t of the output series to the corresponding input series x_t, by the equation

δ(B) ∇^d z_t = ω(B) B^b {x_t^(λ) – c}

where

δ(B) = 1 – δ₁ B – … – δ_p B^P

ω(B) = ω₀ – ω₁ B – … – ω_q B^q .

The integer b>0 defines a pure delay, and the integer d>0 defines the order of differencing in the transfer function.

The parameter λ specifies a Box-Cox power transformation for the input series, and the parameter c specifies a reference level for the transformed input. There is no mean correction of the input series when transfer-function models are estimated, and you should use a value of c close to the series mean so as to improve the numerical conditioning of the estimation procedure. However, if the input series x_t is trend-like rather than stationary, you could alternatively use a value for c close to the early series values, because this reduces the transient errors that arise when the transfer function is applied. The PRIORMETHOD parameter of TRANSFERFUNCTION, described below, provides further means of handling these transients.

The parameters λ and c are not estimated unless you specify otherwise by the BOXCOXMETHOD parameter of TRANSFERFUNCTION or the FIX option of TFIT. Often c in the transfer-function model is aliased with the constant term in the ARIMA errors, and so they should not both be estimated. In some circumstances, however, they both could be estimated, for example in a differenced transfer-function model with stationary noise.

The ORDERS parameter for the simple transfer-function model described above specifies a variate containing the four values b, p, d and q.

The PARAMETERS parameter specifies a variate containing 3+p+q values: λ, c, δ₁, … δ_p, ω₀, ω₁ … ω_q. You must always include the parameters λ, c and ω₀. When you use a transfer-function model, Genstat will check its parameter values. In particular the operator δ(B) must satisfy the stability or stationarity condition.

The LAGS parameter is optional, and may be used to change the lags associated with the parameters, from the default values of 1 … p, 1 … q. The variate of lags contains values corresponding to the parameters δ₁ … δ_p, ω₁ … ω_q. They have the same interpretation as the lags in ARIMA models, and must satisfy the same conditions. Note that there is no lag associated with ω₀, because the delay b provides the necessary flexibility for this.

You can also have seasonal extensions of transfer-function models:

δ(B)Δ(B^s)∇^d∇_s^Dz_t = ω(B)Ω(B^s)B^b{x_t^(λ) – c}

Δ(B^s) = 1 – Δ₁ B^s – … – Δ_P B^Ps

Ω(B^s) = 1 – Ω₁ B^s – … – Ω_Q B^Qs

Note that there is no Ω₀ coefficient, because ω₀ is always present in the model and provides sufficient flexibility.

The ORDERS parameter here contains b, p, d, q, P, D, Q and s, and the PARAMETERS parameter contains λ, c, δ₁ … δ_p, ω₀ … ω_q, Δ₁ … Δ_P, Ω₁ … Ω_Q. You can analogously extend the LAGS parameter. You can also have extensions to multiple seasonal periods, as for ARIMA models.

Option: MODELTYPE.

Parameters: IDENTIFIER, ORDERS, PARAMETERS, LAGS.

Reference

Box, G.E.P. & Jenkins, G.M. (1970). Time Series Analysis, Forecasting and Control. Holden-Day, San Francisco.

See also

Directives: FTSM, TDISPLAY, TFILTER, TFIT, TFORECAST, TKEEP, TRANSFERFUNCTION, TSUMMARIZE, CORRELATE, FOURIER, POINTER.

Procedures: BJESTIMATE, BJFORECAST, BJIDENTIFY, MOVINGAVERAGE, PERIODTEST, PREWHITEN, REPPERIODOGRAM, SMOOTHSPECTRUM.

Commands for: Data structures, Time series.

Example

" Example TFIT-1: Fitting a seasonal ARIMA model"

VARIATE time; VALUES=!(1...120)
FILEREAD [NAME='%gendir%/examples/TFIT-1.DAT'] apt

" Display the correlation structure of the logged data"
CALCULATE lapt = LOG(apt)
BJIDENTIFY [GRAPHICS=high; WINDOWS=!(5,6,7,8)] lapt

" Calculate the autocorrelations of the differences and seasonally
  differenced series"
CALCULATE ddslapt = DIFFERENCE(DIFFERENCE(lapt; 12); 1)
CORRELATE [PRINT=auto; MAXLAG=48] ddslapt; AUTO=ddsr

" Define a model for the series: 
  IMA(1) (that is, a model with a single moving-average parameter
          applied to the differences of the series)
  plus a seasonal IMA(1) component"
TSM [MODELTYPE=arima] airpass; ORDERS=!((0,1,1)2,12)
" Form preliminary estimates of the parameters, using a log transformation
  (BOXCOX=0 is equivalent to log)"
FTSM [PRINT=model] airpass; ddsr; BOXCOX=0
" Get the best estimates, fixing the constant"
TFIT [CONSTANT=fix] SERIES=apt; TSM=airpass

" Graph the residuals against time"
TKEEP RESID=resids
DGRAPH [WINDOW=3; KEYWINDOW=0; TITLE='Residuals vs Time'] resids; time

" Test the independence of the residuals"
CORRELATE [GRAPH=auto; MAXLAG=48] resids; TEST=S
PRINT 'Test statistic for independence of the residuals',S