g13aa carries out non-seasonal and seasonal differencing on a time series. Information which allows the original series to be reconstituted from the differenced series is also produced. This information is required in time series forecasting.

g13ab computes the sample autocorrelation function of a time series. It also computes the sample mean, the sample variance and a statistic which may be used to test the hypothesis that the true autocorrelation function is zero.

g13ac calculates partial autocorrelation coefficients given a set of autocorrelation coefficients. It also calculates the predictor error variance ratios for increasing order of finite lag autoregressive predictor, and the autoregressive parameters associated with the predictor of maximum order.

g13ad calculates preliminary estimates of the parameters of an autoregressive integrated moving average (ARIMA) model from the autocorrelation function of the appropriately differenced times series.

g13af is an easy-to-use version of (G13AEF not in this release). It fits a seasonal autoregressive integrated moving average (ARIMA) model to an observed time series, using a nonlinear least squares procedure incorporating backforecasting. Parameter estimates are obtained, together with appropriate standard errors. The residual series is returned, and information for use in forecasting the time series is produced for use in g13ag and g13ah.

The estimation procedure is iterative, starting with initial parameter values such as may be obtained using g13ad. It continues until a specified convergence criterion is satisfied or until a specified number of iterations have been carried out. The progress of the iteration can be monitored by means of an optional printing facility.

g13ag accepts a series of new observations of a time series, the model of which is already fully specified, and updates the ‘state set’ information for use in constructing further forecasts. The previous specifications of the time series model should have been obtained by using (G13AEF not in this release) g13af to estimate the relevant parameters. The supplied state set will originally have been produced by (G13AEF not in this release) g13af, but may since have been updated by earlier calls to g13ag.

A set of residuals corresponding to the new observations is returned. These may be of use in checking that the new observations conform to the previously fitted model.

g13ah produces forecasts of a time series, given a time series model which has already been fitted to the time series using (G13AEF not in this release) g13af. The original observations are not required, since g13ah uses as input either the original state set produced by (G13AEF not in this release) g13af or the state set updated by a series of new observations using g13ag. Standard errors of the forecasts are also provided.

g13aj applies a fully specified seasonal ARIMA model to an observed time series, generates the state set for forecasting and (optionally) derives a specified number of forecasts together with their standard deviations.

g13am performs exponential smoothing using either single exponential, double exponential or a Holt–Winters method.

g13as is a diagnostic checking method suitable for use after fitting a Box–Jenkins ARMA model to a univariate time series using (G13AEF not in this release) g13af. The residual autocorrelation function is returned along with an estimate of its asymptotic standard errors and correlations. Also, g13as calculates the Box–Ljung portmanteau statistic and its significance level for testing model adequacy.

g13au calculates the range (or standard deviation) and the mean for groups of successive time series values. It is intended for use in the construction of range-mean plots.

g13ba filters a time series by an ARIMA model.

g13bb filters a time series by a transfer function model.

g13bc calculates cross-correlations between two time series.

g13bd calculates preliminary estimates of the parameters of a transfer function model.

g13be 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, g13be fits a univariate ARIMA model.

g13bg accepts a series of new observations of an output time series and any associated input time series, for which a multi-input model is already fully specified, and updates the ‘state set’ information for use in constructing further forecasts.

The previous specification of the multi-input model will normally have been obtained by using g13be to estimate the relevant transfer function and ARIMA parameters. The supplied state set will originally have been produced by g13be (or possibly g13bj), but may since have been updated by g13bg.

g13bj produces forecasts of a time series (the output series) which depends on one or more other (input) series via a previously estimated multi-input model for which the state set information is not available. The future values of the input series must be supplied. In contrast with (G13BHF not in this release) the original past values of the input and output series are required. Standard errors of the forecasts are produced. If future values of some of the input series have been obtained as forecasts using ARIMA models for those series, this may be allowed for in the calculation of the standard errors.

g13ca calculates the smoothed sample spectrum of a univariate time series using one of four lag windows – rectangular, Bartlett, Tukey or Parzen window.

g13cb calculates the smoothed sample spectrum of a univariate time series using spectral smoothing by the trapezium frequency (Daniell) window.

g13cc calculates the smoothed sample cross spectrum of a bivariate time series using one of four lag windows: rectangular, Bartlett, Tukey or Parzen.

g13cd calculates the smoothed sample cross spectrum of a bivariate time series using spectral smoothing by the trapezium frequency (Daniell) window.

For a bivariate time series, g13ce calculates the cross amplitude spectrum and squared coherency, together with lower and upper bounds from the univariate and bivariate (cross) spectra.

For a bivariate time series, g13cf calculates the gain and phase together with lower and upper bounds from the univariate and bivariate spectra.

For a bivariate time series, g13cg calculates the noise spectrum together with multiplying factors for the bounds and the impulse response function and its standard error, from the univariate and bivariate spectra.

g13dd fits a vector autoregressive moving average (VARMA) model to an observed vector of time series using the method of Maximum Likelihood (ML). Standard errors of parameter estimates are computed along with their appropriate correlation matrix. The method also calculates estimates of the residual series.

g13dj computes forecasts of a multivariate time series. It is assumed that a vector ARMA model has already been fitted to the appropriately differenced/transformed time series using g13dd. The standard deviations of the forecast errors are also returned. A reference vector is set up so that, should future series values become available, the forecasts and their standard errors may be updated by calling g13dk.

g13dk accepts a sequence of new observations in a multivariate time series and updates both the forecasts and the standard deviations of the forecast errors. A call to g13dj must be made prior to calling this method in order to calculate the elements of a reference vector together with a set of forecasts and their standard errors. On a successful exit from g13dk the reference vector is updated so that should future series values become available these forecasts may be updated by recalling g13dk.

g13dl differences and/or transforms a multivariate time series. It is intended to be used prior to g13dd to fit a vector autoregressive moving average (VARMA) model to the differenced/transformed series.

g13dx calculates the zeros of a vector autoregressive (or moving average) operator. This method is likely to be used in conjunction with g05pj g13as g13dd (G13DSF not in this release).

g13fa estimates the parameters of either a standard univariate regression GARCH process, or a univariate regression-type I $\text{AGARCH}\left(p,q\right)$ process (see Engle and Ng (1993)).

g13fb forecasts the conditional variances $h_t$, for $\mathit{t}=T+1,\dots ,T+\xi$, from a type I $\text{AGARCH}\left(p,q\right)$ sequence, where $\xi$ is the forecast horizon and $T$ is the current time (see Engle and Ng (1993)).

g13fc estimates the parameters of a univariate regression-type II $\text{AGARCH}\left(p,q\right)$ process.

g13fd forecasts the conditional variances, $h_t,t=T+1,\dots ,T+\xi$ from a type II $\text{AGARCH}\left(p,q\right)$ sequence, where $\xi$ is the forecast horizon and $T$ is the current time (see Engle and Ng (1993)).

g13fe estimates the parameters of a univariate regression-GJR $\text{GARCH}\left(p,q\right)$ process (see Glosten et al. (1993)).

g13ff forecasts the conditional variances, $h_t$, for $\mathit{t}=T+1,\dots ,T+\xi$ from a GJR $\text{GARCH}\left(p,q\right)$ sequence, where $\xi$ is the forecast horizon and $T$ is the current time (see Glosten et al. (1993)).

g13fg estimates the parameters of a univariate regression-exponential $\text{GARCH}\left(p,q\right)$ process (see Engle and Ng (1993)).

g13fh forecasts the conditional variances, $h_t,t=T+1,\dots ,T+\xi$ from an exponential $\text{GARCH}\left(p,q\right)$ sequence, where $\xi$ is the forecast horizon and $T$ is the current time (see Engle and Ng (1993)).