The time series modeling commands are used to fit theoretical models to the series and use the fitted model to predict (forecast) future values of the series. These commands also produce statistics and residuals that allow you to ascertain the adequacy of the model you have elected to use. You can select the modeling commands repeatedly. Each time you select a model, a report of the results of the fit and a forecast is added to the platform results.
The fit of each model begins with a dialog that lets you specify the details of the model being fit as well as how it will be fit. Each general class of models has its own dialog, as discussed in their respective sections. The models are fit by maximizing the likelihood function, using a Kalman filter to compute the likelihood function. The ARIMA, seasonal ARIMA, and smoothing models begin with the following report tables.
Model Comparison shows the Model Comparison Report.
The Model Comparison table summarizes the fit statistics for each model. You can use it to compare several models fitted to the same time series. Each row corresponds to a different model. The models are sorted by the AIC statistic. The Model Comparison table shown above summarizes the ARIMA models (1, 0, 0), (0, 0, 1), (1, 0, 1), and (1, 1, 1) respectively. Use the Report checkbox to show or hide the Model Report for a model.
opens a window giving the settings of the model. You can change the settings to fit a different model.
provides one simulation of the model out k time periods. The simulation is shown on the Model Comparison time series plot. To change k, use the Number of Forecast Periods option on the platform red-triangle menu.
provides the specified number of simulations of the model out k time periods. The simulations are shown on the Model Comparison time series plot. To change k, use the Number of Forecast Periods option on the platform red-triangle menu.
generates simulations for the given model, and stores the results in a data table. You specify the random seed, number of simulations, and the number of forecast periods.
The Model Comparison report provides plots for a model when the Graph checkbox is selected. Model Plots shows the plots for the ARIMA(1,1,1) model.
The top plot is a time series plot of the data, forecasts, and confidence limits. Below that are plots of the autocorrelation and partial autocorrelation functions.
Each model fit generates a Model Summary table, which summarizes the statistics of the fit. In the formulae below, n is the length of the series and k is the number of fitted parameters in the model.
the unconditional sum of squares (SSE) divided by the number of degrees of freedom, SSE / (n – k). This is the sample estimate of the variance of the random shocks at, described in the section ARIMA Model.
is the square root of the variance estimate. This is a sample estimate of the standard deviation of at, the random shocks.
If the model fits the series badly, the model error sum of squares, SSE might be larger than the total sum of squares, SST and R2 will be negative.
The adjusted R2 is
is minus two times the natural log of the likelihood function evaluated at the best-fit parameter estimates. Smaller values are better fits.
indicates whether the autoregressive operator is stable. That is, whether all the roots of lie outside the unit circle.
indicates whether the moving average operator is invertible. That is, whether all the roots of lie outside the unit circle.
There is a Parameter Estimates table for each selected fit, which gives the estimates for the time series model parameters. Each type of model has its own set of parameters. They are described in the sections on specific time series models. The Parameter Estimates table has these terms:
lists the name of the parameter. These are described below for each model type. Some models contain an intercept or mean term. In those models, the related constant estimate is also shown. The definition of the constant estimate is given under the description of ARIMA models.
lists the factor of the model that contains the parameter. This is only shown for multiplicative models. In the multiplicative seasonal models, Factor 1 is nonseasonal and Factor 2 is seasonal.
lists the degree of the lag or backshift operator that is applied to the term to which the parameter is multiplied.
lists the estimates of the standard errors of the parameter estimates. They are used in constructing tests and confidence intervals.
lists the test statistics for the hypotheses that each parameter is zero. It is the ratio of the parameter estimate to its standard error. If the hypothesis is true, then this statistic has an approximate Student’s t-distribution. Looking for a t-ratio greater than 2 in absolute value is a common rule of thumb for judging significance because it approximates the 0.05 significance level.
lists the observed significance probability calculated from each t-ratio. It is the probability of getting, by chance alone, a t-ratio greater (in absolute value) than the computed value, given a true hypothesis. Often, a value below 0.05 (or sometimes 0.01) is interpreted as evidence that the parameter is significantly different from zero.
The Parameter Estimates table also gives the Constant Estimate, for models that contain an intercept or mean term. The definition of the constant estimate is given under ARIMA Model.
Each model has its own Forecast plot. The Forecast plot shows the values that the model predicts for the time series. It is divided by a vertical line into two regions. To the left of the separating line the one-step-ahead forecasts are shown overlaid with the input data points. To the right of the line are the future values forecast by the model and the confidence intervals for the forecasts.
You can control the number of forecast values by changing the setting of the Forecast Periods box in the platform launch dialog or by selecting Number of Forecast Periods from the Time Series drop-down menu. The data and confidence intervals can be toggled on and off using the Show Points and Show Confidence Interval commands on the model’s popup menu.
The graphs under the residuals section of the output show the values of the residuals based on the fitted model. These are the actual values minus the one-step-ahead predicted values. In addition, the autocorrelation and partial autocorrelation of these residuals are shown. These can be used to determine whether the fitted model is adequate to describe the data. If it is, the points in the residual plot should be normally distributed about the zero line and the autocorrelation and partial autocorrelation of the residuals should not have any significant components for lags greater than zero.
The model parameter estimation is an iterative procedure by which the log-likelihood is maximized by adjusting the estimates of the parameters. The iteration history for each model you request shows the value of the objective function for each iteration. This can be useful for diagnosing problems with the fitting procedure. Attempting to fit a model which is poorly suited to the data can result in a large number of iterations that fail to converge on an optimum value for the likelihood. The Iteration History table shows the following quantities:
The title bar for each model you request has a popup menu, with the following options for that model:
submits code to SAS that duplicates the model analysis. If you are not connected to a SAS server, prompts guide you through the connection process.
controls which displays of residual statistics are shown for the model. These displays are described in the section Time Series Commands; however, they are applied to the residual series.