Displays characteristics of time series models.

### Options

`PRINT` = string tokens |
What to print (`autocorrelations, expansion, impulse, piweight, psiweight` ); default `*` |
---|---|

`GRAPH` = string tokens |
What to display with graphs (`autocorrelations, impulse, piweight, psiweight` ); default `*` |

`MAXLAG` = scalar |
Maximum lag for results; default 30 |

### Parameters

`TSM` = TSMs |
Models to be displayed |
---|---|

`AUTOCORRELATIONS` = variates |
To save theoretical autocorrelations |

`IMPULSERESPONSE` = variates |
To save impulse-response function |

`STEPFUNCTION` = variates |
To save step function from impulse |

`PIWEIGHTS` = variates |
To save pi-weights |

`PSIWEIGHTS` = variates |
To save psi-weights |

`EXPANSION` = TSMs |
To save expanded models |

`VARIANCE` = scalars |
To save variance of each TSM |

### Description

The `TSUMMARIZE`

directive helps you investigate time-series models by displaying or saving various characteristics. These are the theoretical autocorrelation function of an ARIMA model, and the pi-weights and psi-weights; also the impulse-response function of a transfer-function model. `TSUMMARIZE`

can derive the expanded form of a model, in which all seasonal terms are combined with the non-seasonal term.

For an ARIMA model in the `TSM`

parameter, you can set only the `AUTOCORRELATIONS`

, `PSIWEIGHTS`

and `PIWEIGHTS`

parameters. Also, you can set the `IMPULSERESPONSE`

parameter only for a transfer-function model. You can set the `EXPAND`

parameter for either type of model. The TSMs in any `TSUMMARIZE`

statement must be completely defined; that is, you must have set the orders and parameters, and the lags if you are using them. The only exceptions are that Genstat takes the transformation parameter to be 1.0 if it is missing, and that the innovation variance of an ARIMA model need not be set.

The `MAXLAG`

option specifies the maximum lag to which Genstat is to do calculations: this applies to autocorrelations, psi-weights, pi-weights and impulse responses. If `MAXLAG`

is unset, the maximum lag is defined implicitly as the length of the first variate in the parameters. However, if the length of this variate is also undefined, the maximum lag cannot be defined and Genstat reports a fault.

You can set the `PRINT`

and `GRAPH`

options independently of the parameters: these store results, and display the various characteristics of models.

The `AUTOCORRELATIONS`

parameter allows you to store the theoretical autocorrelation function of an ARIMA model. Such a model uniquely defines an autocorrelation function whose values *r*_{0} … *r _{m}* are assigned by Genstat to the variate

`R`

, where *m*is the maximum lag. If the model has differencing parameters

*d*=

*D*=0, then the autocorrelation function is that of a series

*y*that follows this model.

_{t}If either *d*>0 or *D*>0, then the theoretical autocorrelations are calculated as if *d*=*D*=0, and so they correspond to those of the differenced *y _{t}* series. This is because the autocorrelations of

*y*are undefined for non-stationary models.

_{t}The `PSIWEIGHTS`

parameter allows you to store the theoretical psi-weights ψ_{0} … ψ_{m} of an ARIMA model. These are used internally by Genstat when error limits are calculated for forecasts obtained using the model. You will need them for example if you want to calculate the variance of the total of the forecast values up to some specified maximum lead time. They are defined for a non-seasonal model by

1 + ψ_{1}*B* + ψ_{2}*B*^{2} + … = θ(*B*) / { φ(*B*)∇^{d} }

The `PIWEIGHTS`

parameter allows you to store the theoretical pi-weights π_{0} … π_{m} of an ARIMA model: these show explicitly how past values contribute to a forecast. The weights are defined by:

1 – π_{1}*B* – π_{2}*B*^{2} – … = { φ(*B*)∇^{d} } / θ(*B*)

The `IMPULSERESPONSE`

parameter allows you to store the theoretical impulse-response function, *v*_{0} … *v _{m}*, of a transfer-function model. This function can help you interpret the model. The sequence is defined for a non-seasonal transfer-function model by:

ν_{0} + ν_{1}*B* + ν_{2}*B*^{2} + … = ω(*B*)*B ^{b}* / { δ(

*B*)∇

^{d}}

For an ARIMA model you can combine into one generalized autoregressive operator all the differencing operators, the non-seasonal autoregressive operators, and the seasonal autoregressive operators. The non-seasonal and seasonal moving-average operators may similarly be combined. This expanded model can be printed using the `expansion`

setting of `PRINT`

and saved using the `EXPANSION`

parameter. It can be used to help you understand a series. But you might also want to re-estimate the parameters in the expanded model, to test whether the differencing operators or seasonal factors unnecessarily constrain the structure of the original model. If you have not previously defined one of the identifiers supplied by the `EXPANSION`

parameter, Genstat will automatically define it to be a TSM, and its component variates will be set up to have the length defined by the corresponding model in the `TSM`

parameter. The expansion does not change the transformation parameter of the model, nor the constant term, nor the innovation variance. If the model that you have supplied contains non-zero differencing orders, then the generalized model does not satisfy the stationarity constraint on the parameters; neither does the constant term have the same interpretation as it had in the supplied model. The expansion of transfer-function models exactly parallels that of ARIMA models.

Options: `PRINT`

, `GRAPH`

, `MAXLAG`

.

Parameters: `TSM`

, `AUTOCORRELATIONS`

, `IMPULSERESPONSE`

, `STEPFUNCTION`

, `PIWEIGHTS`

, `PSIWEIGHTS`

, `EXPANSION`

, `VARIANCE`

.

### See also

Directives: `TSM`

, `FTSM`

, `TDISPLAY`

, `TFILTER`

, `TFIT`

, `TFORECAST`

, `TKEEP`

, `TRANSFERFUNCTION`

, `CORRELATE`

.

Procedures: `BJESTIMATE`

, `BJFORECAST`

, `BJIDENTIFY`

.

Commands for: Time series.

### Example

" Examples 2:7.7.3-4 " " Display the autocorrelations of an AR[2] model." TSM AR[2]; ORDERS=!(2,0,0); PARAMETERS=!(1,15,2.5,0.5,-0.5) TSUMMARIZE [MAXLAG=12; PRINT=autocorrelations] AR[2] " Expand the seasonal ARIMA model used for modelling the number of airline passengers in Section 7.3.7." VARIATE [VALUES=0,1,1, 0,1,1,12] Ord & [VALUES=0,0,0.00143, 0.34, 0.54] Par TSM Airpass; ORDERS=Ord; PARAMETERS=Par PRINT Airpass TSUMMARIZE [PRINT=expansion] Airpass